Unit Operations

Mixer

Overview

The Mixer is a unit operation that combines two or more material streams into a single outlet stream. It performs rigorous mass and energy balances across all connected feeds, computing the outlet composition, temperature, pressure, and enthalpy from first principles. The Mixer is a passive unit. It adds no shaft work, heat, or pressure, and it models an ideal adiabatic mixing point.

Connections

The Mixer accepts up to six material stream inlets and produces exactly one material stream outlet. At least one inlet and the outlet must be connected for the calculation to proceed.

Port	Direction	Type	Description
Inlet 1–6	Inlet	Material	Up to six feed streams
Outlet	Outlet	Material	Combined mixed stream

Mixer Connections

Calculation

The Mixer solves the following balances across all \(N\) connected inlet streams, indexed by \(k\) .

Total mass flow:

\[ \dot{m}_{\text{out}}=\sum_{k=1}^{N}\dot{m}_{k} \]

Component mass fractions (mixing rule by mass):

\[ w_{i,\text{out}}=\frac{{\displaystyle \sum_{k=1}^{N}\dot{m}_{k}\,w_{i,k}}}{\dot{m}_{\text{out}}} \]

where \(w_{i,k}\) is the mass fraction of component \(i\) in stream \(k\) .

Specific enthalpy (energy balance, adiabatic):

\[ h_{\text{out}}=\frac{{\displaystyle \sum_{k=1}^{N}\dot{m}_{k}\,h_{k}}}{\dot{m}_{\text{out}}} \]

Outlet pressure (user-configurable)

The outlet temperature and phase state are not set directly. The outlet stream is sent to a flash calculation at the computed \((P_{\text{out}},h_{\text{out}})\) , and the thermodynamic property package resolves the temperature and phase fractions.

When only a single inlet stream carries a non-zero flow, the outlet is assigned directly from that stream without a mixing flash, improving computational efficiency.

Pressure Calculation Mode

Because inlet streams may arrive at different pressures, the user must choose how the outlet pressure is determined:

Mode	Description
Minimum (default)	The outlet pressure equals the lowest pressure among all connected inlet streams. This is the physically conservative choice, as it represents the pressure level at which all streams can coexist without back-flow.
Maximum	The outlet pressure equals the highest pressure among all inlet streams. Useful when modeling a mixing point downstream of a check valve where the highest-pressure stream sets the system pressure.
Average	The outlet pressure is the arithmetic mean of all inlet pressures. Suitable as an approximation when inlet pressures are close to each other.

Mixer Pressure Calculation Modes

Dynamic Mode

The Mixer supports dynamic simulation. In dynamic mode, the outlet pressure is propagated back to all inlet streams (rather than being derived from them), so the upstream equipment drives the pressure. The mass and energy balances remain identical to steady-state.

Splitter

Overview

The Splitter divides a single material stream into two or three outlet streams. All outlets share the same temperature, pressure, composition, and specific enthalpy as the inlet. The Splitter performs no phase separation, heat exchange, or composition change. It is a purely mechanical flow distribution device, analogous to a pipe tee or manifold.

Connections

The Splitter has one material stream inlet and up to three material stream outlets. Outlet ports must be connected sequentially: port 2 cannot be used unless port 1 is already connected, and port 3 cannot be used unless port 2 is already connected.

Port	Direction	Type	Description
Inlet	Inlet	Material	Single feed stream
Outlet 1	Outlet	Material	First product stream
Outlet 2	Outlet	Material	Second product stream (optional)
Outlet 3	Outlet	Material	Third product stream (optional)

Splitter Connections

Operating Modes

The Splitter supports three operating modes that control how the outlet flows are specified:

Mode	Description
Split Ratios (default)	The user specifies dimensionless split fractions \(\alpha_1, \alpha_2, \ldots\) for each outlet. The fractions must sum to 1. The last fraction is always calculated automatically as the complement of the others.
Stream Mass Flow Specification	The user specifies the mass flow rate (kg/s) for outlets 1 and (optionally) 2. The remaining outlet receives the balance of the inlet mass flow. The specified flows must not exceed the total inlet mass flow.
Stream Mole Flow Specification	The user specifies the molar flow rate (kmol/s) for outlets 1 and (optionally) 2. The remaining outlet receives the balance of the inlet molar flow. The specified flows must not exceed the total inlet molar flow.

Splitter Operating Modes

Calculation

Split Ratios mode: outlet mass flows

\[ \dot{m}_{i}=\alpha_{i}\,\dot{m}_{\text{in}},\qquad\sum_{i=1}^{n}\alpha_{i}=1 \]

The last split fraction is always computed as the complement:

\[ \alpha_{n}=1-\sum_{i=1}^{n-1}\alpha_{i} \]

Stream Mass Flow Specification mode:

\[ \begin{align} \dot{m}_{1} & =\dot{m}_{\text{spec},1}\\ \dot{m}_{2} & =\dot{m}_{\text{spec},2}\quad\text{(if three outlets)}\\ \dot{m}_{n} & =\dot{m}_{\text{in}}-\sum_{i=1}^{n-1}\dot{m}_{\text{spec},i} \end{align} \]

with the constraint \(\dot{m}{\text{spec},1}+\dot{m}{\text{spec},2}\leq\dot{m}_{\text{in}}\) .

Stream Mole Flow Specification mode

Identical in structure to mass flow specification, applied to molar flows \(\dot{n}_{i}\) .

Property propagation

For every outlet \(i\) , regardless of mode:

\[ T_{i}=T_{\text{in}},\quad P_{i}=P_{\text{in}},\quad h_{i}=h_{\text{in}},\quad x_{c,i}=x_{c,\text{in}}\;\forall c \]

No flash calculation is performed on the outlet streams: they inherit the inlet phase state and composition identically.

Split Fraction Specifications

Property	Access	Description
\(\alpha_1\) (SR1)	Read/Write	Split fraction for Outlet 1
\(\alpha_2\) (SR2)	Read/Write	Split fraction for Outlet 2 (three-outlet mode only)
\(\alpha_3\) (SR3)	Read only	Auto-calculated complement: \(1 - \alpha_1 - \alpha_2\)

Split Fraction Properties

Dynamic Mode

The Splitter supports dynamic simulation. In dynamic mode, the inlet flow is back-calculated as the sum of all outlet flows, allowing downstream pressure specifications to drive the split. Composition and thermal properties continue to propagate from inlet to all outlets unchanged.

Constraints and Validation

Outlet ports must be connected sequentially.

In Split Ratios mode, all fractions must lie in \([0,1]\) and sum to 1.

In mass or mole flow specification modes, the sum of specified outlet flows must not exceed the total inlet flow; the simulation raises an error if this constraint is violated.

Separator Vessel

Overview

The Separator Vessel, also known as a Flash Drum, is a unit operation that separates a mixed feed stream into its vapor and liquid phases based on thermodynamic equilibrium. It is one of the most commonly used pieces of equipment in chemical process simulation, representing vessels where pressure reduction or heat addition/removal causes a feed to partially vaporize or condense, allowing the resulting phases to be collected separately.

In DWSIM, the Separator Vessel accepts up to six material inlet streams and one optional energy (heat) stream. It produces up to four material outlet streams, one for the vapor phase, one or two for liquid phases, and one auxiliary outlet, plus an optional energy outlet stream.

Operating Modes

The Separator Vessel supports two phase-separation modes:

• Two-Phase Separation: The vessel separates the feed into a vapor phase and a single liquid phase. This is the standard mode for most flash drum applications.

• Three-Phase Separation: The vessel separates the feed into a vapor phase, a light liquid phase, and a heavy liquid phase. This mode is used when two immiscible liquid phases form (e.g., an aqueous phase and an organic phase). The lighter liquid (lower density) is directed to the first liquid outlet, and the heavier liquid (higher density) is directed to the second liquid outlet. If solid-phase material is present, it is distributed proportionally between the liquid outlets based on their respective mass flow ratios.

Calculation Modes

Four calculation modes are available, each defining how the flash equilibrium conditions are determined:

• Adiabatic: The vessel operates with no heat exchange with the surroundings. When multiple inlet streams are present, they are mixed and a pressure-enthalpy (PH) flash is performed at the mixed conditions. For a single inlet stream, the outlet conditions match the inlet without additional equilibrium calculation.

• Legacy: The separation is performed at a specified temperature and pressure. The user may override the inlet temperature, the inlet pressure, or both. If an energy stream is connected, its heat duty is added to the system before the flash calculation.

• Isothermic (Heating/Cooling): A specified amount of heat is added to or removed from the system while maintaining constant temperature. The vessel adjusts its operating pressure to satisfy the energy balance at the given temperature.

• Isobaric (Heating/Cooling): A specified amount of heat is added to or removed from the system while maintaining constant pressure. The vessel adjusts its operating temperature to satisfy the energy balance at the given pressure.

Inlet Pressure Handling

When multiple inlet streams are connected, the vessel must determine a single operating pressure for the flash calculation. Three options are available:

• Minimum (default): Uses the lowest pressure among all connected inlet streams.

• Maximum: Uses the highest pressure among all connected inlet streams.

• Average: Uses the arithmetic average of all inlet stream pressures.

Heat Transfer and Energy Streams

An optional energy inlet stream can be connected to supply or remove heat from the vessel. Additionally, a rigorous heat balance option is available, which accounts for heat transfer through the vessel walls, including wall material thermal conductivity, insulation layers, ambient temperature, and optionally solar radiation. When enabled, the rigorous heat balance calculates internal and external heat transfer coefficients and adjusts the vessel’s energy balance accordingly.

Vessel Sizing

The Separator Vessel includes built-in sizing calculations for both vertical and horizontal orientations. The sizing routine estimates the vessel diameter and length (or height) based on the vapor disengaging velocity, which is a function of the liquid-to-vapor density difference. Key sizing parameters include:

• Dimension Ratio: The length-to-diameter ratio (default: 3).

• Surge Factor: An oversizing multiplier applied to the calculated volume (default: 1.2).

• Residence Time: The required liquid residence time in seconds (default: 5 s).

• Wall Thickness: The vessel shell thickness in meters (default: 0.01 m).

• Wall Material: Options include Steel, Carbon Steel, Cast Iron, Stainless Steel, and Commercial Copper.

• Head Type: The vessel end-cap geometry, such as Ellipsoidal (2:1), Hemispherical, or Torispherical (ASME F&D).

Nozzle diameters for the inlet, vapor outlet, and liquid outlet are also estimated based on velocity constraints.

Dynamic Mode

The Separator Vessel supports dynamic (time-dependent) simulation. In dynamic mode, the vessel tracks the holdup of liquid and vapor inside the vessel over time, and the liquid level changes as inlet and outlet flow rates vary. Key dynamic parameters include vessel volume, liquid level, operating pressure, and a minimum pressure constraint. The vessel contents can be initialized from an inlet stream or reset to empty at the start of a dynamic run.

Safety Valve (Pressure Safety Valve / Relief Valve)

Overview

The Safety Valve, also referred to as a Pressure Safety Valve (PSV) or Relief Valve, is a unit operation designed for dynamic simulations. It models a spring-loaded pressure relief device intended to protect pressurized equipment (such as separator vessels, pipes, tanks, and reactors) from exceeding safe operating pressure limits. When the inlet pressure rises above a user-defined set point, the valve opens progressively, allowing fluid to discharge until the pressure is relieved. When the pressure falls back below the set point, the valve closes.

In DWSIM, the Safety Valve is classified as a pressure changer and uses the naming prefix "PSV-". It connects to the pressurized equipment on its inlet side and to a downstream relief header, blowdown drum, or atmosphere on its outlet side. Because it is purpose-built for dynamic simulation, its steady-state Calculate method performs no calculation: the valve only activates during dynamic integrator runs.

Connections

The Safety Valve has two connection ports:

Port	Direction	Description
Inlet	In	Connected to the pressurized equipment (vessel, pipe, etc.)
Outlet	Out	Connected to the relief destination (blowdown system, flare, atmosphere)

Safety Valve Connection Ports

Both the inlet and outlet streams must have their dynamic specification type set to Pressure for the valve to operate correctly during dynamic simulation.

Pressure Set Points

Two pressure thresholds govern the valve’s behavior:

• Set Point Pressure: The pressure at which the valve begins to open. Below this pressure, the valve remains fully closed and no flow passes through it.

• Fully Opened Pressure: The pressure at which the valve reaches its maximum opening. Between the set point and this value, the valve opening varies continuously.

The valve opening percentage is calculated as:

\[ \text{Opening} (\%) = \frac{P_{\text{inlet}} - P_{\text{set point}}}{P_{\text{fully opened}} - P_{\text{set point}}} \]

This value is clamped between 0% (fully closed) and 100% (fully open), providing a modulating action rather than a simple on/off behavior.

Letter	Area (in2)	Area (cm2)
D	0.11	0.71
E	0.20	1.26
F	0.31	1.98
G	0.50	3.24
H	0.79	5.06
J	1.29	8.30
K	1.84	11.85
L	2.85	18.40
M	3.60	23.23
N	4.34	28.00
P	6.38	41.16
Q	11.05	71.29
R	16.00	103.22
T	26.00	167.74

API 526 Standard Orifice Sizes

Correction Coefficients

Three dimensionless coefficients allow the user to account for real-world deviations from ideal nozzle flow:

• Discharge Coefficient ( \(K_{d}\) ): Accounts for losses due to the actual valve geometry compared to an ideal nozzle. Default: 1.0.

• Back Pressure Coefficient ( \(K_{b}\) ): Corrects for the effect of downstream (back) pressure on the valve’s relieving capacity, particularly important for balanced-bellows or pilot-operated valves. Default: 1.0.

• Viscosity Coefficient ( \(K_{v}\) ): Corrects for the effect of fluid viscosity on flow through the orifice, relevant for highly viscous liquids. Default: 1.0.

Valve Characteristic Curves

The relationship between the valve’s opening percentage and its effective flow coefficient ( \(K_{v}/K_{v,\text{max}}\) ) can be configured using one of five characteristic types:

• Linear: The flow coefficient is directly proportional to the opening percentage. This is the default.

• Equal Percentage: The flow coefficient follows an exponential curve, producing small flow changes at low openings and large changes near full opening. A characteristic parameter controls the curve shape.

• Quick Opening: The flow coefficient follows a square-root curve, producing a large initial flow increase at low openings that tapers off as the valve opens further.

• User-Defined Expression: A mathematical expression is evaluated at runtime. The variable \(OP\) represents the opening percentage (0–100), and standard math functions are available.

• Data Table: The user provides a table of opening percentage versus \(K_{v}/K_{v,\text{max}}\) percentage pairs, and the valve interpolates between them during simulation.

The corresponding equations are:

\[ K_{vc} = \frac{OP}{100} \quad \text{(Linear)} \]

\[ K_{vc} = R^{\left(\frac{OP}{100} - 1\right)} \quad \text{(Equal Percentage, where } R \text{ is the characteristic parameter)} \]

\[ K_{vc} = \sqrt{\frac{OP}{100}} \quad \text{(Quick Opening)} \]

Flow Calculation

During each dynamic time step, the valve determines the mass flow rate through the orifice based on the phase of the inlet fluid.

• Vapor (gas) flow: The valve first checks whether the flow is choked (sonic) by comparing the outlet-to-inlet pressure ratio against the critical pressure ratio:

\[ r_c = \left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma}{\gamma - 1}} \]

where \(\gamma=C_{p}/C_{v}\) is the heat capacity ratio of the gas.

• Choked flow (when \(P_{2}/P_{1}\geq r_{c}\) ):

\[ \dot{m} = A \cdot K_{vc} \cdot K_d \cdot K_b \cdot \sqrt{\frac{P_1 \cdot \gamma}{v_1} \cdot \left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma - 1}{\gamma + 1}}} \]

• Non-choked flow (when \(P_{2}/P_{1}<r_{c}\) ):

\[ \dot{m} = A \cdot K_{vc} \cdot K_d \cdot \sqrt{\frac{P_1}{v_1} \cdot \frac{2\gamma}{\gamma + 1} \cdot \left[\left(\frac{P_2}{P_1}\right)^{\frac{2}{\gamma}} - \left(\frac{P_2}{P_1}\right)^{\frac{\gamma + 1}{\gamma}}\right]} \]

• Liquid flow:

\[ \dot{m} = A \cdot K_{vc} \cdot K_d \cdot K_v \cdot \sqrt{2 \cdot (P_1 - P_2) \cdot \rho} \]

where \(A\) is the orifice area, \(P_{1}\) and \(P_{2}\) are the inlet and outlet pressures, \(v_{1}\) is the inlet specific volume, and $\rho$ is the inlet liquid density.

The expansion across the valve is treated as isenthalpic: the outlet stream enthalpy equals the inlet stream enthalpy, and the fluid composition is preserved unchanged.

Note: Two-phase (mixed vapor-liquid) flow through the valve is not currently supported. If the inlet vapor fraction falls between 0.01 and 0.99, the calculation will raise an error.

Key Specifications Summary

Property	Description	Default
Set Point Pressure	Pressure at which the valve begins to open	0 Pa
Fully Opened Pressure	Pressure at which the valve is 100% open	0 Pa
Orifice Area	Effective flow area of the valve nozzle	0.71 cm2 (letter D)
Discharge Coefficient (\(K_d\))	Geometric flow loss correction	1.0
Back Pressure Coefficient (\(K_b\))	Downstream pressure correction	1.0
Viscosity Coefficient (\(K_v\))	Fluid viscosity correction	1.0
Characteristic Type	Opening-to-\(K_v\) relationship curve	Linear

Safety Valve Key Specifications

Usage Notes

• The Safety Valve is only active during dynamic simulation runs. In steady-state mode, it performs no calculation.

• Connect the inlet port to a pressurized unit operation (separator vessel, pipe segment, tank) and the outlet to the relief destination.

• Both connected streams must use the Pressure dynamic specification type.

• Set the set point pressure to the equipment’s maximum allowable working pressure (MAWP) or the desired relief threshold, and the fully opened pressure to the accumulation pressure (typically 10–21% above set point, per applicable codes).

• Select the appropriate API 526 orifice size or use the built-in PSV sizing utility to determine the correct orifice area for your scenario.

• Adjust the correction coefficients based on the valve manufacturer’s data or applicable standard (API 520/521).

Orifice Area

The orifice area defines the effective flow area of the valve when fully open. The user can enter a custom value or select from a set of standard API 526 orifice letter designations:

Tank

In DWSIM, the Tank model applies a user-specified pressure drop to the process fluid at constant temperature (adiabatic operation).

Input Parameters

• Pressure drop: pressure difference between the outlet and inlet streams.

Calculation Method

The outlet pressure is computed as the inlet pressure minus the specified pressure drop. The temperature is assumed constant (no heat exchange). A TP flash is then performed at the outlet conditions to determine the phase distribution and thermodynamic properties of the outlet stream.

Output Parameters

There are no output parameters for this object.

Pipe Segment

The Pipe Segment unit operation models single- and two-phase fluid flow through piping systems, computing pressure drop and heat transfer along the pipe length. Several widely used pressure-drop correlations are available (see below). The thermal profile can be calculated rigorously by specifying heat-transfer coefficients and ambient conditions. By combining multiple Pipe Segments with the Recycle logical operation, complex piping networks (e.g., water distribution systems) can be modeled.

The pipe is subdivided into sections, each of which can represent a straight tube, a fitting (valve, elbow, tee, etc.), or a change in elevation. Each section is further discretized into a user-specified number of increments for the numerical integration of the pressure and temperature profiles.

Input Parameters

• Hydraulic profile: the pipe hydraulic profile editor allows the user to define each section’s type (straight tube, fitting, etc.), number of computational increments, pipe material, length, elevation change, and internal and external diameters.

• Pressure drop correlation: select the model to be used for the pressure drop calculation in the pipe segment.

• Thermal profile: In the thermal profile editor it is possible to define how the temperature profile in the pipe should be calculated. The configurations in this window are valid for the entire pipe segment. Changes are saved automatically.

Calculation Method

The pipe segment is solved by marching incrementally along the pipe length, performing coupled mass and energy balances at each increment. The algorithm uses three nested iteration loops: the outer loop advances through successive increments, the middle loop converges the temperature, and the inner loop converges the pressure. The procedure at each increment is as follows:

1. The inlet temperature and pressure are used to estimate the increment outlet pressure and temperature.

2. Fluid properties are evaluated at the arithmetic mean of the inlet and estimated outlet conditions.

3. The calculated properties and the inlet pressure are used to calculate the pressure drop. With it, the outlet pressure is calculated.

4. The calculated outlet pressure is compared with the initial estimate; if the difference exceeds the convergence tolerance, the estimate is updated and steps 2–3 are repeated.

5. Once the internal loop has converged, the outlet temperature is calculated. If the global heat transfer coefficient (U) was given, the outlet temperature is calculated from the following equation:\

\[ Q=UA\Delta T_{ml} \]

\
where: $Q$ = heat transferred, $A$ = heat transfer area (external surface) and $\Delta T_{ml}$ = logarithmic mean temperature difference.

1. The calculated temperature is compared to the estimated one, and if their difference exceeds the specified tolerance, a new temperature is estimated and new properties are calculated (return to step 2).

2. Once both the pressure and temperature have converged within their respective tolerances, the computed outlet conditions become the inlet conditions for the next increment, and the procedure repeats.

Output Parameters

• Delta-T: temperature variation in the pipe segment.

• Delta-P: pressure variation in the pipe segment.

• Heat exchanged: amount of heat exchanged with the environment, or lost by friction in the pipe walls.

• Results (table): results are show section by section in a table.

• Results (graph): a graph shows the temperature, pressure, liquid holdup, velocity and heat exchanged profiles.

Description of calculation methods

Mass and Heat balance

For each section on each segment, do:

Step 1 Read properties from the fluid entering the section

Step 2 For emulsion viscosity calculation:

\[ \phi=\frac{Q_{l1}}{Q_{l1}+Q_{l2}} \]

\[ \eta_{lh}=\eta_{l1}\exp\left[3.6\left(1-\phi\right)\right] \]

\[ \eta_{ll}=\eta_{l2}\left(1+2.5\phi\frac{\left(\eta_{l1}+0.4\eta_{l2}\right)}{\left(\eta_{l1}+\eta_{l2}\right)}\right) \]

\[ \begin{align} \phi & >0.5:\eta_{liq}=\eta_{lh}\\ \phi & <0.33:\eta_{liq}=\eta_{ll}\\ 0.33<\phi & <0.5:\eta_{liq}=\frac{\phi-0.33}{0.17}\eta_{lh}+\left(1-\frac{\phi-0.33}{0.17}\right)\eta_{ll} \end{align} \]

Step 3 For slurry viscosity calculation:

\[ \phi_{s}=\frac{Q_{s}}{Q_{l}+Q_{s}} \]

\[ \eta_{r}=1+3\frac{\phi_{s}}{1-\frac{\phi_{s}}{0.52}} \]

\[ \eta_{slurry}=\eta_{liq}\times\eta_{r} \]

Step 4 Start heat balance for the section, with initial estimate for \(T_{2}\) :

\[ T_{ext}>T_{1}:T_{2}=1.005T_{1} \]

\[ T_{ext}<T_{1}:T_{2}=\nicefrac{T_{1}}{1.005} \]

Step 5 Temperature convergence loop begins

Step 6 Calculate pressure drop for section using Beggs-Brill (), Lockhart-Martinelli () or Petalas-Aziz () models. Calculation defaults to Darcy-Weisbach correlation for single phase fluids:

\[ \Delta P_{f}=f\frac{Lv^{2}}{2gD} \]

\[ \begin{align} Re & <2100:f=\frac{64}{Re}\\ Re & >4000:f=\log\left[\frac{\left(\frac{k}{D}\right)}{2.8257}^{1.1096}+\left(\frac{5.8506}{Re}\right)^{0.8961}\right]\\ 2100\leq Re & \leq4000:f=8\left(\frac{\left(\frac{8}{Re}\right)^{12}}{\left(\left(2.457\log\left(\frac{1}{\left(\frac{7}{Re}\right)^{0.9}+0.27\frac{\varepsilon}{D}}\right)^{16}\right)+\left(\frac{37530}{Re}\right)^{16}\right)^{1.5}}\right)^{\frac{1}{12}} \end{align} \]

\[ \Delta P_{h}=9.8\rho\sin\left(\arcsin\left(\frac{\Delta h}{L}\right)\right)L \]

\[ \Delta P_{t}=\Delta P_{h}+\Delta P_{f} \]

Step 7 Calculate outlet pressure \(P_{2}\) and compare with previous calculation. After 3 pressure iterations, accelerate convergence with a secant procedure:

\[ P_{2}=P_{1}-\Delta P_{t} \]

\[ n_{iterations,P}>3:P_{2}=P_{2}-f_{P}\frac{P_{2}-P_{2,i-1}}{f_{P}-f_{P,i-1}} \]

where

\[ f_{P}=P_{2}-P_{2,i-1} \]

\[ f_{P,i-1}=P_{2,i-1}-P_{2,i-2} \]

Step 8 Calculate Overall Heat Transfer Coefficient ( \(U\) )

If \(U\) was defined by the user, read value directly from input. Otherwise, calculate it with

\[ \frac{1}{U}=\frac{1}{h_{i}}+\frac{D_{i}}{\log\left(\frac{D_{e}}{D_{i}}\right)k_{wall}}+\frac{D_{e}}{\ln\left(\frac{D_{e}+2L_{ins}}{D_{e}}\right)k_{ins}}+\frac{D_{i}}{\left(D_{e}+2L_{ins}\right)h_{e}} \]

\[ h_{i}=\frac{k_{fluid}}{D_{i}}\frac{f}{8}\left(Re-1000\right)\frac{Pr}{1+12.7\sqrt{\frac{f}{8}}\left(Pr^{2/3}-1\right)} \]

\[ h_{e}=0.25\frac{k_{fluid,e}}{D_{e}}Re^{0.6}Pr^{0.38} \]

Step 9 Calculate Heat Exchanged ( \(Q\) )

\[ \Delta Q=\frac{\left(T_{2}-T_{1}\right)}{\ln\left(\frac{T_{ext}-T_{1}}{T_{ext}-T_{2}}\right)}UA \]

\[ A=\pi D_{e}L \]

If Solar Irradiation is included, then

\[ \Delta Q=\Delta Q+Q_{solar} \]

\[ Q_{solar}=\frac{S_{r}}{t_{flux}}A \]

\[ t_{flux}=\pi\frac{D_{e}^{2}}{4}L\left(Q_{v}+Q_{l}+Q_{s}\right) \]

\(Q_{v}\) , \(Q_{l}\) and \(Q_{s}\) are the volumetric flow rates of vapor, liquid and solid, respectively.

Step 8 Calculate outlet temperature \(T_{2}\) and compare with previous calculation:

\[ f_{T}=T_{2}-T_{2,i-1} \]

If \(f_{T}\) is less than the convergence tolerance, the temperature has converged and the algorithm proceeds to the next section. Otherwise, calculate the new average fluid properties with \(P_{avg}=\frac{\left(P_{1}-P_{2}\right)}{2}\) and \(T_{avg}=\frac{\left(T_{1}-T_{2}\right)}{2}\) and return to Step 2.

Valve

The Valve models an isenthalpic throttling process: the fluid undergoes a pressure reduction at constant enthalpy, and the outlet temperature and phase state are determined by a Pressure–Enthalpy (PH) flash at the reduced pressure.

Input Parameters

• Pressure drop: pressure difference between the outlet and inlet streams.

Calculation Method

The outlet pressure is computed as the inlet pressure minus the specified pressure drop. A Pressure–Enthalpy (PH) flash at the outlet pressure and the inlet enthalpy then determines the outlet temperature and phase state. Because the process is isenthalpic, the outlet temperature is typically equal to or lower than the inlet temperature (Joule–Thomson cooling), except for fluids whose Joule–Thomson coefficient is negative at the operating conditions.

Output Parameters

• Delta-T: temperature drop observed in the valve expansion process.

Opening (OP)/Flow Coefficient (Kv[Cv]) Relationship Types

• Linear

\[ K_{V}=K_{Vmax}\times OP \]

• Quick Opening

\[ K_{V}=K_{Vmax}\times\sqrt{OP} \]

• Equal Percentage

\[ K_{V}=K_{Vmax}\times R^{OP-1} \]

where \(R\) is the characteristic parameter (range: 20 - 50)

• User-Defined Expression

\[ K_{V}=K_{Vmax}\times f(OP) \]

• Data Table

Actual flow coefficient value is determined by interpolating data from a user-defined table.

Pump

The Pump increases the pressure of a liquid stream by converting shaft work into hydraulic energy. The ideal (isentropic) work is corrected by a user-specified adiabatic efficiency to account for irreversibilities (friction, recirculation losses, etc.).

Calculation Modes

The calculation mode decides which values the user specifies and which ones the pump calculates.

• Pressure Increase: the pressure rise is specified; the outlet pressure, the power and the outlet temperature are calculated.

• Outlet Pressure: the discharge pressure is specified; the rest follows as in the previous mode.

• Power: the shaft power is specified; the pressure rise is calculated.

• Energy Stream: as in the Power mode, with the power read from the energy stream connected to the pump.

• Performance Curves: the operating point is read from the head, efficiency, power and NPSHr curves (see below).

• Positive Displacement: the flow rate comes from the displacement, the speed and the volumetric efficiency (see below).

Input Parameters

• Efficiency (%): pump adiabatic efficiency. It is used in every mode except Performance Curves with an efficiency curve.

• Pressure Increase, Outlet Pressure or Power: the specification of the selected mode.

• Performance Curves: the curves and the operating speed.

• Positive Displacement: the displacement per revolution, the volumetric efficiency, the relief valve setting, the operating speed and the outlet pressure.

Calculation Method

In the Pressure Increase and Outlet Pressure modes the discharge pressure is known:

\[ P_{2}=P_{1}+\Delta P \]

The shaft power and the outlet enthalpy follow from the pressure rise, the liquid density at the inlet and the efficiency:

\[ Pot=\frac{W\,\Delta P}{\rho\,\eta} \]

\[ H_{2}=H_{1}+\frac{Pot}{W} \]

The work lost to the efficiency stays in the liquid as heat, so the outlet is slightly warmer than the inlet. The outlet temperature comes from a PH flash at \(P_{2}\) and \(H_{2}\) .

In the Power and Energy Stream modes the power is known and the pressure rise is calculated. The efficient part of the power raises the pressure, and all of it reaches the liquid as enthalpy, the rest as heat, as in the other two modes:

\[ P_{2}=P_{1}+\frac{\rho\,\eta\,Pot}{W} \]

\[ H_{2}=H_{1}+\frac{Pot}{W} \]

The outlet temperature again comes from a PH flash. The available NPSH is calculated from the bubble pressure of the feed at the inlet temperature:

\[ NPSH_{a}=\frac{P_{1}-P_{bub}\left(T_{1}\right)}{\rho\,g} \]

where:

\(Pot\) shaft power

\(W\) mass flow

\(\rho\) liquid density at the inlet

\(\eta\) pump efficiency

\(H_{1},H_{2}\) inlet and outlet specific enthalpy

\(P_{1},P_{2}\) inlet and outlet pressure

\(P_{bub}\) bubble pressure of the feed

\(g\) gravitational acceleration

Outlet Parameters

• Pressure Increase and Outlet Pressure, when they are not the specification.

• Power Required: shaft power.

• Temperature Difference: temperature rise of the liquid.

• Head: pressure rise expressed as a height of liquid.

• Available NPSH.

Performance Curves

In the Performance Curves calculation mode the pump is described by its measured curves instead of a fixed efficiency: head, power, efficiency and NPSHr against flow rate, each a table of points with its own units, edited in the curve editor of either interface. The head curve is required; with the efficiency curve disabled the pump falls back to the efficiency typed in the editor, and with the power curve disabled the power follows from the head and the efficiency. The pump refuses to extrapolate: a flow rate outside the range of a curve is reported as an error rather than guessed.

A pump driven by a variable-frequency drive runs at a speed other than the one its curves were measured at. The Operating Speed field sets that speed; left at zero the pump runs at the speed of its curves. When the pump carries a single set of curves, the other speeds are covered by the affinity laws: the operating point is read at the flow rate divided by the speed ratio, head and NPSHr are scaled by the square of that ratio, power by its cube, and efficiency is taken as invariant along the affinity parabola. This is the right approximation for a pump with one published curve.

A manufacturer usually publishes a dataset for a variable-frequency pump: one set of curves per measured speed. The curve editor takes any number of them, each keyed by the speed it was measured at and labelled with the supply frequency that would drive the pump there, which follows from the frequency given for the reference speed. Between two measured speeds the pump reads the operating point off both sets at the actual flow rate and interpolates linearly between the two readings, so the measured efficiency and NPSHr are followed instead of assumed. Above the highest measured speed or below the lowest, the nearest set is scaled with the affinity laws and the flowsheet log records that the curves were extrapolated. The sets are saved with the simulation, and the classic curve editor exports and imports the whole dataset as a JSON file.

Positive Displacement

A metering, dosing or plunger pump is the opposite of a centrifugal one: it delivers the volume it sweeps per revolution, times its speed and its volumetric efficiency, so the flow rate is linear in the rotation and does not depend on the discharge pressure. The Positive Displacement calculation mode models the machine that way. The displacement, the volumetric efficiency and the relief setting are entered with the other pump data; the pressure at the discharge is the one the line imposes, limited by the relief setting, which stands for the relief valve every such machine needs. The shaft power follows from the delivered flow rate, the pressure rise and the pump efficiency.

The delivered volumetric and mass flow rates are reported as properties of the pump, and the flowsheet log says when the feed stream brings a flow rate different from the one the pump displaces, which in a real installation is what the suction accumulator or the relief valve absorbs. In dynamic mode the machine acts as the flow source of its line: the delivered flow rate follows the drive speed through the motor inertia and torque, so a flow controller writing the speed is the loop that sets the dose. The affinity laws do not apply to this machine, and there is no head curve to scale.

Compressor/Expander

The Compressor/Expander models the compression or expansion of a vapor-phase stream. The ideal reference process is isentropic (constant entropy); irreversibilities are accounted for through a user-specified efficiency. The user selects between an adiabatic (isentropic) or polytropic thermodynamic path, depending on the available performance data.

Input Parameters

• Delta-P: pressure change in the equipment.

• Efficiency: adiabatic/polytropic efficiency;

• Ignore liquid in the inlet stream: defines if the calculator should ignore any liquid in the inlet stream;

• Thermodynamic path: select the thermodynamic path according to the experimental/field data available.

Calculation Method

Isentropic (Adiabatic) or Polytropic power is calculated from:

\[ P=\frac{H_{2s}-H_{1}}{\eta}W \]

for compressor, and

\[ P=\left(H_{2s}-H_{1}\right)\times W\times\eta \]

for expander, where:

\(H_{2s}\) Outlet Enthalpy for Isentropic Process

\(H_{1}\) Inlet Enthalpy

\(W\) Mass Flow

\(\eta\) Adiabatic or Polytropic Efficiency

Isentropic (Adiabatic) and Polytropic Coefficients are calculated from:

\[ n_{i}=\frac{\ln\left(P_{2}/P_{1}\right)}{\ln\left(\rho_{2i}/\rho_{1}\right)} \]

\[ n_{p}=\frac{\ln\left(P_{2}/P_{1}\right)}{\ln\left(\rho_{2}/\rho_{1}\right)} \]

where:

\(\rho_{2i}\) Outlet Gas Density calculated with Inlet Gas Entropy

Adiabatic and Polytropic Heads are calculated from:

\[ H=\frac{P}{W\times g} \]

where:

\(H\) Adiabatic or Polytropic Head

\(P\) Adiabatic or Polytropic Power

\(W\) Mass Flow

\(\eta\) Adiabatic or Polytropic Efficiency

\(g\) Gravitational Constant (9.8 m/s2)

Heater/Cooler

The Heater and Cooler are single-sided heat exchange models: they add or remove thermal energy from a process stream without explicitly modeling the utility-side fluid. They are used to represent furnaces, electric heaters, cooling-water exchangers, or any other device whose duty or outlet condition is known but whose utility stream need not be simulated.

Calculation Modes

• Energy Stream: the energy flow from a connected stream is used to heat or cool the inlet stream.

• Define Outlet Temperature: the outlet temperature is defined and the amount of heat added or removed is calculated and written to the energy stream.

• Define Outlet Vapor Fraction: the quality of the outlet fluid is defined and the required amount of heat to add or remove is written to the energy stream.

• Temperature Change: the temperature difference is defined and the required amount of heat to add or remove is written to the energy stream.

• Heat Added/Removed: the heat added or removed is defined directly from the unit operation.

Input/Output Parameters

• Pressure Drop (input only): defines the pressure drop in the heater/cooler.

• Heating/Cooling: used as input in Heat Added/Removed calculation mode, otherwise it is a calculated value.

• Efficiency (input only): heating/cooling efficiency.

• Outlet Vapor Fraction: used as input in Define Outlet Vapor Fraction mode, otherwise it is a calculated value.

• Outlet Temperature: used as input in Define Outlet Temperature mode, otherwise it is a calculated value.

• Temperature Change: used as input in Temperature Change mode, otherwise it is a calculated value.

Shortcut Column

Overview

The Shortcut Column is a unit operation that performs approximate distillation design calculations using classical shortcut methods. Rather than solving the full stage-by-stage MESH equations of a rigorous column, it applies the Fenske–Underwood–Gilliland (FUG) method to rapidly estimate the minimum number of stages, minimum reflux ratio, actual number of ideal stages, optimal feed stage location, internal flow rates, and condenser and reboiler duties. It is well suited for early-stage process design, equipment sizing surveys, and generating initial estimates for rigorous column simulations.

The Shortcut Column accepts a single multicomponent feed stream and produces a distillate and a bottoms product. The separation is characterized by specifying a light key and a heavy key component, whose recoveries in the top and bottom products define the desired split. The column uses the connected thermodynamic property package to compute K-values, relative volatilities, and product enthalpies at the operating conditions.

Connections

Port	Direction	Type	Description
Feed	Inlet	Material	Single feed stream
Distillate	Outlet	Material	Overhead product
Bottoms	Outlet	Material	Bottom product
Condenser Duty	Outlet	Energy	Heat removed by condenser (optional)
Reboiler Duty	Inlet	Energy	Heat added at reboiler (optional)

Shortcut Column Connections

Condenser Types

Type	Description
Total Condenser	All overhead vapor is condensed. The distillate is withdrawn as a saturated or subcooled liquid. The reflux is returned as liquid. The distillate temperature is calculated at the bubble point.
Partial Condenser	The overhead vapor is only partially condensed. The distillate is withdrawn as a saturated vapor in equilibrium with the reflux liquid. The distillate temperature is calculated at the dew point.

Condenser Types

Key Component Specification and Recovery

The user designates one component as the light key (LK), the most volatile component that should primarily report to the bottoms, and one as the heavy key (HK), the least volatile component that should primarily report to the distillate. All components lighter than the LK are assumed to report entirely to the distillate (light non-keys); all components heavier than the HK are assumed to report entirely to the bottoms (heavy non-keys).

Recovery is controlled indirectly through two purity specifications:

• Heavy key mole fraction in distillate ( \(x_{D,\text{HK}}\) ): controls how much HK is allowed to appear in the overhead product (default: 0.01).

• Light key mole fraction in bottoms ( \(x_{B,\text{LK}}\) ): controls how much LK is allowed to appear in the bottoms product (default: 0.01).

The distribution of all remaining (distributed) non-key components between the two products is then solved iteratively.

Calculation Sequence

Step 1: Feed Characterization

The feed thermal condition parameter \(q\) is computed from the feed enthalpy relative to its bubble-point and dew-point enthalpies:

\[ q=1+\frac{h_{\text{bub}}-h_{F}}{h_{\text{dew}}-h_{\text{bub}}} \]

where \(h_{F}\) is the molar enthalpy of the feed, \(h_{\text{bub}}\) is the enthalpy at the bubble point, and \(h_{\text{dew}}\) is the enthalpy at the dew point at the feed pressure.

Step 2: Relative Volatilities

K-values for all components are evaluated at the feed temperature and pressure using the selected property package. The relative volatility of each component is defined with respect to the heavy key:

\[ \alpha_{i}=\frac{K_{i}}{K_{\text{HK}}} \]

Components are then classified as light non-keys ( \(\alpha_{i}>\alpha_{\text{LK}}\) ), distributed non-keys ( \(\alpha_{\text{HK}}<\alpha_{i}<\alpha_{\text{LK}}\) ), or heavy non-keys ( \(\alpha_{i}<\alpha_{\text{HK}}\) ).

Step 3: Minimum Number of Stages (Fenske Equation)

The minimum number of ideal stages at total reflux is:

\[ N_{\min}=\frac{\ln\!\left(\dfrac{x_{D,\text{LK}}}{x_{D,\text{HK}}}\cdot\dfrac{x_{B,\text{HK}}}{x_{B,\text{LK}}}\right)}{\ln\!\left(\dfrac{\alpha_{\text{LK}}}{\alpha_{\text{HK}}}\right)} \]

The Fenske equation is also used iteratively to distribute each non-key component between the two products, consistent with the specified key recoveries.

Step 4: Minimum Reflux Ratio (Underwood’s Method)

The Underwood equation finds the root $\theta$ (in the interval \(\alpha_{\text{HK}}<\theta<\alpha_{\text{LK}}\) ) of:

\[ \sum_{i=1}^{C}\frac{\alpha_{i}\,z_{i}}{\alpha_{i}-\theta}=1-q \]

The minimum reflux ratio is then:

\[ R_{\min}=\sum_{i=1}^{C}\frac{\alpha_{i}\,x_{D,i}}{\alpha_{i}-\theta}-1 \]

When distributed non-keys are present, multiple \(\theta\) roots exist (one between each pair of adjacent component \(\alpha\) values) and the resulting system of equations is solved by matrix inversion. If the specified reflux ratio is lower than \(R_{\min}\) , the calculation raises an error.

Step 5: Actual Number of Stages (Gilliland Correlation)

The actual number of ideal stages is obtained from the Gilliland correlation:

\[ X=\frac{R-R_{\min}}{R+1} \]

\[ Y=0.75\left(1-X^{0.5668}\right) \]

\[ N=\frac{Y+N_{\min}}{1-Y} \]

where \(R\) is the specified operating reflux ratio.

Step 6: Internal Flow Rates

The molar flow rates in each column section are:

\[ \begin{align} L &= R \cdot D & &\text{(rectifying liquid)} \\ V &= D + L & &\text{(rectifying vapor)} \\ L' &= L + q \cdot F & &\text{(stripping liquid)} \\ V' &= L' - B & &\text{(stripping vapor)} \end{align} \]

where \(D\) is the distillate molar flow, \(B\) is the bottoms molar flow, and \(F\) is the feed molar flow.

Step 7: Optimal Feed Stage (Kirkbride/Fenske)

The optimal feed stage location from the top is estimated as:

\[ N_{F}=\frac{N_{\min,S}}{N_{\min}}\cdot N \]

where \(N_{\min,S}\) is the Fenske minimum stages for the stripping section alone, calculated using the stripping-section relative volatility and the bottoms key component compositions.

Step 8: Product Temperatures and Enthalpies

The distillate temperature is computed by flashing the distillate composition at the condenser pressure to a vapor fraction of 0 (total condenser) or 1 (partial condenser).

The bottoms temperature is computed by flashing the bottoms composition at the reboiler pressure to a vapor fraction of 0 (bubble point).

Step 9: Heat Duties

Condenser duty (negative, heat removal):

For a total condenser:

\[ Q_{C}=-\left(h_{L}-h_{V}^{\text{sat}}\right)\left(L+D\right) \]

For a partial condenser:

\[ Q_{C}=-\left(h_{L}-h_{D}\right)L \]

where \(h_{L}\) is the saturated liquid enthalpy at the condenser pressure and \(h_{V}^{\text{sat}}\) (or \(h_{D}\) ) is the saturated vapor enthalpy.

Reboiler duty (positive, heat input), from the overall column energy balance:

\[ Q_{R}=D\,h_{D}+B\,h_{B}-F\,h_{F}-Q_{C} \]

All enthalpies are on a molar basis.

Rigorous Distillation Column

Overview

The Rigorous Distillation Column is a unit operation that models the separation of a multicomponent mixture into two or more product streams by exploiting differences in component volatilities across a series of equilibrium stages. Unlike the Shortcut Column, which relies on approximate correlations (Fenske, Underwood, Gilliland), the rigorous model solves the full set of MESH equations (Material balances, Equilibrium relations, Summation constraints, and Heat (enthalpy) balances) simultaneously for every stage.

In DWSIM, the Distillation Column supports multiple feed streams at arbitrary stages, liquid and vapor side draws, inter-stage heat exchangers, and pump-arounds. The column is bounded by a condenser at the top and a reboiler at the bottom. It can also be configured as a Reboiled Absorber (no condenser) or a Refluxed Absorber (no reboiler) to model stripping and enriching sections independently.

Column Structure

A rigorous distillation column consists of \(N\) equilibrium stages numbered from top to bottom. Stage 1 is the condenser and stage \(N\) is the reboiler. Each stage \(j\) receives liquid from the stage above ( \(L_{j-1}\) ), vapor from the stage below ( \(V_{j+1}\) ), and optionally a feed stream ( \(F_{j}\) ). Liquid and vapor side draws ( \(U_{j}\) and \(W_{j}\) , respectively) and a stage heat duty ( \(Q_{j}\) ) can also be specified.

MESH Equations

For each stage \(j\) and each component \(i\) , the rigorous solver satisfies the following system of equations:

• Material balance (M):

\[ M_{i,j}=L_{j-1}\,x_{i,j-1}+V_{j+1}\,y_{i,j+1}+F_{j}\,z_{i,j}-(L_{j}+U_{j})\,x_{i,j}-(V_{j}+W_{j})\,y_{i,j}=0 \]

where \(x_{i,j}\) and \(y_{i,j}\) are the liquid and vapor mole fractions of component \(i\) on stage $j$, \(z_{i,j}\) is the feed composition, \(U_{j}\) is the liquid side-draw flow, and \(W_{j}\) is the vapor side-draw flow.

• Equilibrium relation (E):

\[ E_{i,j}=y_{i,j}-\eta_{j}\,K_{i,j}\,x_{i,j}-(1-\eta_{j})\,y_{i,j}^{*}=0 \]

where \(K_{i,j}\) is the vapor-liquid equilibrium ratio evaluated by the selected thermodynamic property package and \(\eta_{j}\) is the Murphree stage efficiency (default: 1.0 for an ideal stage).

• Summation constraints (S):

\[ S_{j}^{L}=\sum_{i=1}^{C}x_{i,j}-1=0\qquad\qquad S_{j}^{V}=\sum_{i=1}^{C}y_{i,j}-1=0 \]

• Energy balance (H):

\[ H_{j}=L_{j-1}\,h_{j-1}^{L}+V_{j+1}\,h_{j+1}^{V}+F_{j}\,h_{j}^{F}-(L_{j}+U_{j})\,h_{j}^{L}-(V_{j}+W_{j})\,h_{j}^{V}-Q_{j}=0 \]

where \(h^{L}\) , \(h^{V}\) , and \(h^{F}\) are the liquid, vapor, and feed molar enthalpies.

Condenser Types

The column condenser can be configured in one of three modes:

Type	Description
Total Condenser	All overhead vapor is condensed to liquid. The distillate
	is withdrawn as a subcooled or saturated liquid. An optional
	subcooling \(\Delta T\) can be specified.
Partial Condenser	Only part of the overhead vapor is condensed. The distillate
	is withdrawn as a vapor in equilibrium with the reflux liquid.
Full Reflux	All condensed liquid is returned to the column as reflux.
	No distillate product is withdrawn.

Condenser Types

Two independent specifications are required to fully define the column: one associated with the condenser and one with the reboiler. The available specification types are:

Specification Type	Description
Heat Duty	Condenser or reboiler duty (\(Q\), in W)
Product Molar Flow Rate	Total distillate or bottoms molar flow (mol/s)
Product Mass Flow Rate	Total distillate or bottoms mass flow (kg/s)
Component Molar Flow Rate	Molar flow of a specific component in the product (mol/s)
Component Mass Flow Rate	Mass flow of a specific component in the product (kg/s)
Component Fraction	Mole or mass fraction of a component in the product
Component Recovery	Fraction of a feed component recovered in the product (%)
Stream Ratio	Reflux ratio (\(L/D\)) or boilup ratio (\(V/B\))
Temperature	Stage temperature (K)
Feed Recovery	Overall recovery of feed in the product (%)

Column Specification Types

Solving Methods

DWSIM provides four built-in solving methods for the rigorous column, each with different strengths:

Method	Description
Wang-Henke (Bubble Point)	Classic tri-diagonal matrix method. Uses stage temperatures \(T_j\) and vapor flows \(V_j\) as tear variables. Solves the linearized material balance with the Thomas algorithm and updates temperatures via bubble-point calculations. Best suited for narrow-boiling systems where temperature is a strong function of composition.
Modified Wang-Henke	Enhanced variant of the bubble-point method with improved convergence behavior for difficult or wide-boiling systems.
Naphtali-Sandholm (Simultaneous Correction)	Solves all MESH equations simultaneously using a Newton-Raphson method with a full Jacobian matrix. More robust for highly non-ideal systems, columns with multiple feeds and side draws, and problems where the bubble-point method fails to converge.
Burningham-Otto (Sum-Rates)	Updates liquid flows from the summation of component material balances rather than from energy balances. Designed for absorbers and strippers where the temperature profile is relatively flat and the bubble-point method is ill-conditioned.

Available Solving Methods

External (third-party) solvers can also be registered through the plug-in interface.

Initialization Strategies

Good initial estimates are critical for convergence of rigorous column calculations. DWSIM offers four initialization schemes:

• Direct: Uses the raw problem specification without modification.

• Ideal K-values: Initializes equilibrium ratios from Raoult’s law ( \(K_{i}=P_{i}^{\text{sat}}/P\) ), providing a simple starting point for the composition profile.

• Ideal Enthalpies: Initializes stage enthalpies from ideal mixing rules.

• Ideal K-values and Enthalpies: Combines both idealizations for the broadest initial simplification.

The user can also supply manual initial estimates for stage temperatures, vapor and liquid flows, and compositions. If the option Auto-Update Initial Estimates is enabled, the converged solution from the previous run is stored and reused as the starting point for subsequent calculations.

Convergence Parameters

Parameter	Description	Default
Maximum Iterations	Upper limit on solver iterations	100
Internal Loop Tolerance	Convergence criterion for inner loop	\(1 \times 10^{-5}\)
External Loop Tolerance	Convergence criterion for outer loop	\(1 \times 10^{-5}\)
Broyden Acceleration	Use Broyden’s method to accelerate	Enabled
	successive substitution updates

Convergence Settings

A detailed convergence report can optionally be generated, recording the error norm at each iteration.

Stage Properties

Each equilibrium stage has the following configurable properties:

Property	Description	Default
Pressure (\(P_j\))	Stage operating pressure (Pa)	—
Efficiency (\(\eta_j\))	Murphree tray efficiency (0–1)	1.0
Heat Duty (\(Q_j\))	External heat added or removed (W)	0
Liquid Side Draw (\(U_j\))	Liquid withdrawal rate (mol/s)	0
Vapor Side Draw (\(W_j\))	Vapor withdrawal rate (mol/s)	0

Stage Properties

Additional tray hydraulic parameters are available for detailed design: dry tray pressure drop coefficient, total hole area, downcomer length, downcomer height, and liquid flow equation coefficients.

Connections

Port	Direction	Type	Description
Feed(s)	Inlet	Material	One or more feeds at any stage
Distillate	Outlet	Material	Top liquid product (total condenser)
			or top vapor product (partial condenser)
Bottoms	Outlet	Material	Bottom liquid product
Overhead Vapor	Outlet	Material	Vapor from partial condenser (if applicable)
Liquid Side Draw(s)	Outlet	Material	Liquid withdrawn from intermediate stages
Vapor Side Draw(s)	Outlet	Material	Vapor withdrawn from intermediate stages
Condenser Duty	Outlet	Energy	Heat removed by the condenser
Reboiler Duty	Inlet	Energy	Heat supplied to the reboiler
Inter-Exchangers	In/Out	Energy	Stage-level heating or cooling duties

Distillation Column Connections

Physical Dimensions

The column estimates its physical dimensions from the converged solution:

• Diameter: Estimated from the maximum vapor and liquid traffic using flooding correlations.

• Height: Calculated as \(H=N_{s}\cdot\Delta h+h_{\text{top}}+h_{\text{bottom}}\) , where \(\Delta h\) is the tray spacing (default 0.5 m), \(h_{\text{top}}\) is the top clearance (default 0.1 m), and \(h_{\text{bottom}}\) is the bottom sump height (default 0.5 m).

Dynamic Mode

The Distillation Column supports dynamic (time-dependent) simulation. In dynamic mode, the solver tracks the liquid and vapor holdup on each stage over time using per-stage accumulation streams. Three parameters control the maximum allowable change per time step:

Parameter	Description	Default
Max. P change (%)	Maximum pressure change per step	10%
Max. L change (%)	Maximum liquid flow change per step	10%
Max. V change (%)	Maximum vapor flow change per step	10%

Dynamic Mode Parameters

The column can be initialized from its steady-state solution before starting a dynamic run.

Rigorous Absorption Column

Overview

The Absorption Column models gas-liquid absorption and liquid-liquid extraction operations using a rigorous stage-by-stage approach. It shares the same mathematical framework and base class as the Distillation Column but operates without a condenser or reboiler by default. The overhead product exits as a vapor (in absorption mode) and the bottoms product exits as a liquid, with no reflux or boilup generated internally. Separation is driven entirely by the contact between the feed gas and the solvent.

Operating Modes

The Absorption Column supports two operating modes:

Mode	Description
Absorber	Gas-liquid absorption. A gas feed enters near the bottom and a lean solvent enters near the top. Soluble components transfer from the gas phase into the liquid phase as the two streams flow countercurrently through the stages. The overhead product is the scrubbed gas (vapor) and the bottoms product is the rich solvent (liquid).
Extractor	Liquid-liquid extraction. Two immiscible liquid feeds contact each other countercurrently. Components transfer between the two liquid phases based on their partition coefficients. The solver uses liquid-liquid equilibrium (LLE) calculations and requires trial compositions for both liquid phases to initialize.

Absorption Column Operating Modes

Differences from the Distillation Column

Feature	Distillation Column	Absorption Column
Condenser	Yes (Total, Partial, or Full Reflux)	None
Reboiler	Yes	None
Default solver	Wang-Henke (Bubble Point)	Burningham-Otto (Sum-Rates)
Top product	Distillate (liquid or vapor)	Overhead vapor (or liquid
		in extractor mode)
LLE extraction	Not supported	Supported
Side draws	Supported	Supported
Dynamic mode	Supported	Supported

Comparison: Distillation Column vs. Absorption Column

Two variant configurations bridge the gap between the two column types:

• Reboiled Absorber: An absorption column with a reboiler at the bottom stage, used for stripping operations where additional vapor generation is needed to strip dissolved components from the rich solvent.

• Refluxed Absorber: An absorption column with a condenser at the top stage, used when partial condensation of the overhead vapor is needed to improve separation.

Solving Methods

The Absorption Column uses the following solving methods:

• Burningham-Otto (Sum-Rates): the default method. It updates liquid flows from the summation of component material balances, which is well-suited for absorbers and strippers where the temperature profile is nearly flat and the energy balance has a weak influence on the solution. If convergence difficulties arise, the solver automatically retries with relaxation of temperature and composition updates.

• Naphtali-Sandholm (Simultaneous Correction): available as an alternative for more strongly coupled systems.

For the Extractor mode, the solver requires multiple sets of trial compositions for the two liquid phases. It iterates through each set of trial estimates until one leads to convergence; if all trials fail, the calculation raises an error.

Connections

Port	Direction	Type	Description
Feed(s)	Inlet	Material	Gas and solvent feeds at any stage
Overhead Product	Outlet	Material	Scrubbed gas (vapor) or extract (liquid)
Bottoms Product	Outlet	Material	Rich solvent (liquid) or raffinate
Liquid Side Draw(s)	Outlet	Material	Liquid from intermediate stages
Vapor Side Draw(s)	Outlet	Material	Vapor from intermediate stages

Absorption Column Connections

Convergence Parameters

The Absorption Column uses the same convergence settings as the Distillation Column:

Parameter	Description	Default
Maximum Iterations	Upper limit on solver iterations	100
Convergence Tolerance	Error norm threshold for convergence	\(1 \times 10^{-5}\)
Broyden Acceleration	Accelerate successive substitution	Enabled
Generate Report	Output detailed convergence report	Disabled

Absorption Column Convergence Settings

Stage Properties

Each stage in the Absorption Column has the same configurable properties as in the Distillation Column: pressure, Murphree efficiency, heat duty, and liquid/vapor side-draw rates. Tray hydraulic parameters (hole area, downcomer geometry, dry tray pressure drop coefficient) are also available for detailed design.

Dynamic Mode

The Absorption Column supports dynamic simulation with the same per-stage accumulation tracking and maximum change constraints (pressure, liquid flow, vapor flow) as the Distillation Column. A bottoms accumulation stream tracks the liquid holdup at the column sump. The column can be initialized from its steady-state solution before starting a transient run.

Usage Notes

For gas absorption, connect the gas feed to a stage near the bottom and the lean solvent to a stage near the top so that the two phases flow countercurrently.

• For liquid-liquid extraction, provide trial compositions for both liquid phases in the initial estimates to help the LLE solver converge.

• If the Burningham-Otto solver fails to converge, try the Naphtali-Sandholm method or adjust the stage pressures and initial temperature profile.

• For stripping applications requiring a heat source at the bottom, use the Reboiled Absorber variant. For applications requiring partial condensation at the top, use the Refluxed Absorber variant.

• The stage pressure profile should be specified consistently: DWSIM does not compute hydraulic pressure drops between stages automatically; the user must set the pressure on each stage or specify an overall column pressure drop.

Heat Exchanger

The Heat Exchanger models a two-stream, countercurrent heat exchange device. It supports phase change (boiling, condensation) and multiphase flow on either side, evaluating enthalpy changes from the selected property package to correctly account for latent heat effects.

Input Parameters

The heat exchanger in DWSIM has seven calculation modes:

1. Calculate hot fluid outlet temperature: you must provide the cold fluid outlet temperature and the exchange area to calculate the hot fluid temperature.

2. Calculate cold fluid outlet temperature: in this mode, DWSIM needs the hot fluid outlet temperature and the exchange area to calculate the cold fluid temperature.

3. Calculate both temperatures: in this mode, DWSIM needs the exchange area and the heat exchanged to calculate both temperatures.

4. Calculate area: in this mode you must provide the HTC and both temperatures to calculate the exchange area.

5. Rate a Shell and Tube exchanger: in this mode you must provide the exchanger geometry and DWSIM will calculate output temperatures, pressure drop on the shell and tubes, overall HTC, LMTD, and exchange area.

6. Pinch-Point (minimum temperature difference between outlet streams)

7. Specify Outlet Vapor Fraction (Stream 1 or Stream 2)

Calculation Mode

All calculation modes are based on the fundamental heat-transfer rate equation:

\[ Q=UA\Delta T_{ml}, \]

\ where: \(Q\) = heat transferred, \(A\) = heat transfer area (external surface) and \(\Delta T_{ml}\) = Logarithmic Mean Temperature Difference (LMTD). The energy balance for each stream is:

\[ Q=\dot{m}\,\Delta H, \]

\ where \(\dot{m}\) is the mass flow rate and \(\Delta H\) is the specific enthalpy change between outlet and inlet.

Depending on the selected mode, the unknowns are resolved as follows:

1. Calculate hot fluid outlet temperature: HTC (Heat Transfer Coefficient), hot fluid outlet temperature, heat load and LMTD.

2. Calculate cold fluid outlet temperature: HTC, cold fluid outlet temperature, heat load and LMTD.

3. Calculate both temperatures: HTC, cold and hot fluid outlet temperatures and LMTD.

4. Calculate area: exchange area and LMTD.

5. Rate Shell and Tube exchanger: exchanger geometry information.

Results

The output quantities computed by the heat exchanger depend on the selected mode:

1. Calculate hot fluid outlet temperature: overall HTC, hot fluid outlet temperature, heat load and LMTD.

2. Calculate cold fluid outlet temperature: overall HTC, cold fluid outlet temperature, heat load and LMTD.

3. Calculate both temperatures: overall HTC, cold and hot fluid outlet temperatures and LMTD.

4. Calculate area: exchange area and LMTD.

5. Rate Shell and Tube exchanger: area, LMTD, LMTD correction factor (F), overall HTC, hot fluid outlet temperature, cold fluid outlet temperature, hot fluid pressure drop (shell/tubes only), cold fluid pressure drop (shell/tubes only).

Description of calculation methods

Shell and Tube

The calculation method for Shell and Tube Design and Rating is based on the method of Tinker ().

# Fundamental Equations

The heat transfer \(Q\) between the hot and cold fluids in a shell and tube heat exchanger can be written:

\[ Q=m_{t}C_{pt}(T_{t1}-T_{t2}) \]

\[ Q=m_{s}C_{ps}(T_{s1}-T_{s2}) \]

\[ Q=h_{i}A_{ti}(T_{t}-T_{ti}) \]

\[ Q=\frac{2k_{t}\pi nL}{\ln\frac{d_{e}}{d_{i}}}(T_{ti}-T_{te}) \]

\[ Q=h_{e}A_{te}(T_{te}-T_{s}) \]

\[ Q=UA_{te}\Delta T_{m} \]

Heat losses for the environment are not considered.

Symbols:

\(m_{t}\) tube-side flow rate

\(C_{pt}\) tube-side fluid average heat capacity

\(T_{t1}\) tube-side inlet fluid temperature

\(T_{t2}\) tube-side outlet fluid temperature

\(m_{s}\) shell-side flow rate

\(C_{ps}\) shell-side fluid average heat capacity

\(T_{s1}\) shell-side inlet fluid temperature

\(T_{s2}\) shell-side outlet fluid temperature

\(h_{i}\) average film coefficient at the tube inner wall

\(A_{ti}\) internal tube surface heat exchange area

\(T_{t}\) average tube fluid temperature

\(T_{ti}\) average internal tube surface temperature

\(k_{t}\) tube thermal conductivity

\(L\) tube total length

\(n\) total number of tubes in the exchanger

\(d_{i}\) tube internal diameter

\(d_{e}\) tube external diameter

\(T_{te}\) average external tube surface temperature

\(h_{e}\) average film coefficient at the tube outer wall

\(A_{te}\) external tube surface heat exchange area

\(T_{s}\) shell-side fluid average temperature

\(U\) overall heat exchange coefficient

\(\triangle T_{m}\) mean temperature difference

The mass flux \(G_{t}\) for the tube-side flow can be written:

\[ G_{t}=\rho_{t}V_{t}=\frac{m_{t}}{\frac{n}{N_{t}}S_{ti}} \]

where:

\(\rho_{t}\) tube-side fluid density

\(V_{t}\) tube-side flow velocity

\(N_{t}\) number of tube passes

\(S_{ti}\) tube internal cross-flow section area:

\[ S_{ti}=\frac{\pi d_{i}^{2}}{4} \]

The mass flux \(G_{s}\) for the shell-side flow can be written:

\[ G_{s}=\frac{m_{b}}{S_{s}} \]

where:

\(m_{b}\) fraction of total flow that crosses the tube bundle

\(S_{s}\) cross-flow section area through the tube bundle:

\[ S_{s}=C_{a}lD_{f} \]

where:

\(l\) distance between two adjacent baffles

\(D_{f}\) bundle diameter

\[ C_{a}=C_{b}\frac{s-d_{e}}{s} \]

where \(s\) is the tube pass and

\(C_{b}=0.97\) for ▷ and □

\(C_{b}=1.37\) for ◇

To determine the film coefficient for the shell-side flow, we have:

\[ G_{sh}=\frac{m_{s}}{S_{sh}} \]

\[ S_{sh}=\frac{S_{s}M}{F_{h}} \]

\[ F_{h}=\frac{1}{1+N_{h}\sqrt{\frac{D_{i}}{s}}} \]

\(S_{sh}\) effective cross-section area for heat exchange

\(F_{h}\) fraction of total flow that crosses \(S_{s}\)

\(M\) and \(N_{h}\) are correction factors obtained from the tables in Figures 46, 47, 48, 49, 50 and 51.

Shell-side heat transfer characteristic for triangle layout.

Shell-side pressure drop characteristic for triangle layout.

Shell-side heat transfer characteristic for square layout.

Shell-side pressure drop characteristic for square layout.

Shell-side heat transfer characteristic for rotated square layout.

Shell-side pressure drop characteristic for rotated square.

# Overall Heat Transfer Coefficient

The Overall HTC \(U\) is given by the expression

\[ U=\frac{1}{\frac{d_{e}}{h_{i}d_{i}}+\frac{R_{di}d_{e}}{d_{i}}+\frac{d_{e}}{2k_{t}}\ln\frac{d_{e}}{d_{i}}+R_{de}+\frac{1}{h_{e}}} \]

where

\(R_{di}\) resistance due to deposits in the internal tube surface

\(R_{de}\) resistance due to deposits in the external tube surface

# Mean Temperature Difference

The basic equation for heat transfer to be used in heat exchanger design is

\[ U=\intop U\Delta TdA \]

Fluid temperatures are not constant, changing at each point as heat is transferred from the hot fluid to the cold fluid, resulting in a variation in the temperature differences between the fluids through the exchanger.

Associated to the temperature variations are the fluid and materials’ thermal properties, which implies in variations in the thermal resistances, and thus in the overall coefficient \(U\) .

When designing an exchanger, though, we usually calculate an average value for \(U\) , and the properties of each fluid are evaluated at the arithmetic mean of the end temperatures, and the result value is assumed to be constant. Thus, we can write:

\[ Q=UA\Delta T_{m} \]

where

\[ \Delta T_{m}=\frac{1}{A}\intop_{0}^{A}\Delta TdA \]

By knowing how \(\Delta T\) changes inside the exchanger, the above expression can be integrated, resulting in something like

\[ \Delta T_{m}=F\times LMTD \]

where \(LMTD\) is the log mean temperature difference for the exchanger conditions, calculated as if the exchanger is countercurrent, with only one shell pass and one tube pass.

\(F\) is a correction factor which is given in formulas and charts such as in Figure 52, where P and R are given by

\[ P=\frac{T_{t2}-T_{t1}}{T_{s1}-T_{t1}} \]

\[ R=\frac{T_{s1}-T_{s2}}{T_{t2}-T_{t1}} \]

Correction factor for LMTD.

# Film Coefficient

The film coefficient \(h_{e}\) is obtained from the charts in Figures 46, 48 and 50 as a function of the Reynolds number \(Re_{h}\) and the \(\nicefrac{s}{d_{e}}\) :

\[ Re_{h}=\frac{G_{sh}d_{e}}{\mu_{s}} \]

# Tube-side Pressure Drop

\[ P_{1}-P_{2}=f_{D}\frac{L}{d_{i}}\frac{\rho_{t}V_{t}^{2}}{2} \]

\[ \frac{1}{\sqrt{f_{d}}}=-2\log\left(\frac{\varepsilon/d_{i}}{3.7}+\frac{2.51}{Re\sqrt{f_{D}}}\right) \]

Initial value for \(f_{D}\) :

\[ f_{D0}=0.25\left[\log\left(\frac{\varepsilon/d_{i}}{3.7}+\frac{5.74}{\sqrt{Re}}\right)\right]^{-2} \]

Viscosity correction:

\[ f_{Dc}=f_{D}\left(\frac{\mu_{ti}}{\mu_{t}}\right)^{0.14} \]

# Shell-side Pressure Drop

\[ \Delta P_{s}=4f_{s}\frac{G_{sf}^{2}}{2\rho_{c}}C_{x}\left(1-\frac{H}{D_{i}}\right)\frac{D_{i}}{s}N'_{B}\left(1+\frac{Ys}{D_{i}}\right)\left(\frac{\mu_{te}}{\mu_{s}}\right)^{0.14} \]

\(C_{x}=1.154\) for ▷ arrangement

\(C_{x}=1.0\) for □ arrangement

\(C_{x}=1.414\) for ◇ arrangement

\(\left(1+\frac{Ys}{D_{i}}\right)\) is obtained from the charts in Figures 46, 48 and 50. \(N'_{B}\) is the number of spaces between baffles and is given by \(N'_{B}=N_{B}+1\) where \(N_{B}\) is the number of baffles.

Air Cooler

Air Cooler model

The Air Cooler models a fin-fan heat exchanger in which the process fluid flows through tube bundles and ambient air is the cooling medium on the shell (external) side. The calculation is based on a simplified variant of the shell-and-tube heat-transfer model.

Input Parameters

The Air Cooler model in DWSIM has three calculation modes:

• Specify Outlet Temperature: you must provide the fluid outlet temperature and DWSIM will calculate Overall UA and Heat Exchanged.

• Specify Tube Geometry: in this mode you must provide the tube geometry and DWSIM will calculate output temperatures, pressure drop at the tubes, overall HTC (U), LMTD, and exchange area. This calculation mode uses a simplified version of Tinker’s method for Shell and Tube exchanger calculations, with a modification for the outside heat transfer coefficient (convection).

• Specify Overall UA: in this mode you must provide the Overall UA and DWSIM will calculate the Heat Exchanged and Outlet Temperature.

You can provide the pressure drop for the hot fluid in the exchanger for modes 1 and 3 only.

Calculation Mode

The Air Cooler is calculated using the fundamental heat-transfer rate equation:

\[ Q=UA\Delta T_{ml}, \]

\ where: \(Q\) = heat transferred, \(A\) = heat transfer area (external surface) and \(\Delta T_{ml}\) = Logarithmic Mean Temperature Difference (LMTD). The energy balance for each stream is:

\[ Q=\dot{m}\,\Delta H, \]

\ where \(\dot{m}\) is the mass flow rate and \(\Delta H\) is the specific enthalpy change between outlet and inlet.

Component Separator

The Component Separator is an idealized mass-balance unit operation that partitions the feed components between two product streams according to user-specified split fractions or absolute flow rates. No thermodynamic equilibrium is solved for the separation itself; after the component split is applied, the energy balance and phase states of the outlet streams are computed via flash calculations.

Input Parameters

• Specified stream: sets the stream to which the separation specifications will be applied. ”0” corresponds to the Outlet stream 1 (overhead) and ”1” corresponds to the Outlet stream 2 (bottoms).

Results

• Energy imbalance: Difference between enthalpy of outlet and inlet streams. in some cases it can be interpreted as the energy necessary to do the separation.

Orifice Plate

This model implements the ISO 5167 standard for thin-plate orifice flow meters. It computes the differential pressure across the orifice as well as the permanent pressure loss. When used in conjunction with the Adjust logical operation, it can be employed to back-calculate stream flow rates from a measured differential pressure.

Input Parameters

• Pressure tappings: select the option which corresponds to the arrangement of the tappings for pressure reading.

• Orifice diameter: inner diameter of the plate.

• Beta (d/D): ratio between plate’s inner and outer diameters.

• Correction factor: multiplier for the mass flow rate used in the calculation of the pressure drop across the orifice. Default is 1.

Results

• Orifice pressure drop: Pressure drop across the orifice. This is the value that is read through the tappings.

• Overall pressure drop: permanent pressure loss after downstream pressure recovery. This value is always less than or equal to the orifice differential pressure.

• Delta T: temperature drop across the orifice, considering that the process is an adiabatic expansion.

Custom Unit Operation

The Custom Unit Operation allows the user to define the calculation logic through a script that is executed inside the Calculate() method called by the flowsheet solver. Up to six material streams are available: three inlets and three outlets.

Supported Languages are IronPython, IronRuby, VBScript and JScript. You can use some predefined reference variables inside your script, defined as shortcuts to the most common objects:

ims1 Input Material Stream in slot 1 (MaterialStream class instance)

ims2 Input Material Stream in slot 2 (MaterialStream class instance)

ims3 Input Material Stream in slot 3 (MaterialStream class instance)

oms1 Output Material Stream in slot 1 (MaterialStream class instance)

oms2 Output Material Stream in slot 2 (MaterialStream class instance)

oms3 Output Material Stream in slot 3 (MaterialStream class instance)

Me Reference variable to the Custom UO object instance (CustomUO UnitOperation class)

Flowsheet Reference variable to the active flowsheet object (FormChild class)

Solver Flowsheet solver class instance, used to send commands to the calculator (COMSolver class)

Spreadsheet Reference variable to the Spreadsheet object.

Solids Separator

The Solids Separator splits a multiphase feed stream into a liquid product and a solids product, based on user-specified separation efficiencies for each phase.

Input Parameters

• Solids Separation Efficiency: defines the amount of solids in the liquid stream. 100% efficiency means no solids in the liquid stream.

• Liquids Separation Efficiency: defines the amount of liquid in the solids stream. 100% efficiency means no liquid in the solids stream.

Calculation Method

The solids separator performs a component mass balance, distributing the solid and liquid phases of the inlet stream into two distinct product streams according to the specified efficiencies.

Continuous Cake Filter

In a continuous filter, the feed, filtrate and cake move at steady constant rates. It is evident that the process consists of several steps in series - cake formation, washing, drying and discharging - and that each step involves progressive and continual change in conditions. The pressure drop across the filter during cake formation is, however, held constant.

Calculation Method

For a continuous cake filter, the equation that relates the filter characteristics with the rate of solids production is

\[ \frac{\dot{m_{c}}}{A_{T}}=\frac{\left[2c\alpha\triangle Pfn/\mu+\left(nR_{m}\right)^{2}\right]^{0,5}-nR_{m}}{\alpha}, \]

where:

\(\dot{m_{c}}\) rate of solids production, \(kg/s\)

\(A_{T}\) total filter area, \(m^{2}\)

\(\Delta P\) total pressure drop, \(Pa\)

\(f\) fraction of filter area available for cake formation

\(c\) solids concentration in the solids stream

\(\alpha\) specific cake resistance, \(m/kg\)

\(R_{m}\) filter medium resistance, \(m^{-1}\)

\(n\) drum speed, \(s^{-1}\)

Input and Output Parameters

• Filter Medium Resistance: filter medium flow resistance;

• Specific Cake Resistance: specific cake flow resistance;

• Cycle Time: filter cycle time.

• Cake Relative Humidity: filter cake moisture in % wet basis;

• Filter Calculation Mode: Design or Simulation. If Design is selected, DWSIM will calculate the filter area given the total pressure drop. If Simulation is selected, it will do the opposite;

• Total Filter Area: filter area measured perpendicularly to the direction of flow;

• Pressure Drop: total pressure drop across the filter (cake + medium).

Excel Unit Operation

The Excel (Spreadsheet) Unit Operation allows user-defined calculation models to be implemented in an external Microsoft Excel workbook.

Each instance of this unit operation is associated with a separate Excel file that follows a predefined template. At each solver iteration, DWSIM writes the inlet stream data and user-defined parameters to the workbook, triggers the spreadsheet calculation, and reads the results back to populate the outlet streams. User-defined parameters can be transferred in both directions. Up to four material streams may be connected to both the inlet and outlet sides. An energy stream receives the calculated energy from the overall enthalpy balance.

Calculation Parameters

Calculation parameters are defined inside the Excel definition file as input parameters.

Editor: displays the Excel file associated with this unit operation. Click the file name to open the unit editor. After the spreadsheet model is evaluated, DWSIM computes the overall enthalpy balance between all inlet and outlet streams to determine the net heat added or removed. Additional output parameters defined in the workbook are also read back into DWSIM.

Search Button: to search Excel file to be associated with Unit Operation

Create New Button: creates a new definition file from the built-in template. The file can then be opened and edited directly in Excel; changes are saved from within Excel.

Excel definition file: the Excel definition file has a fixed structure that must not be altered. Modifying the layout will prevent DWSIM from correctly reading and writing data.

Input Tab

DWSIM writes the properties of all connected inlet streams into the blue area. From line 12 downward all components and their molar flows are listed. The red area contains the parameters which are required for calculation. This parameters are displayed in DWSIM as "Calculation Parameters" inside the property tab of ExcelUO. You may list as many properties as you want here. DWSIM starts to read the list below the heading downwards until it finds an empty cell. Each parameter features a name, a value, a unit and an annotation.

Output Tab

The output tab has the same structure as the input tab. Molare flows of each component of every output stream are to be written into their fields by the user defined calculation procedures. You also have to calculate temperature and pressure of all streams leaving the unit. The enthalpy of each stream is calculated by DWSIM automatically after finishing Excel calculations.

Results

DWSIM is calculating the enthalpy balance of the unit from enthalpy of outlet streams minus enthalpy of inlet streams. The result of this calculation is written to the energy stream. After finishing the calculations in Excel, DWSIM checks the mass balance of that unit. If mass balance is not ok DWSIM will issue an error message.

Flowsheet Unit Operation

The Flowsheet Unit Operation allows you to run a XML simulation file as a block inside another flowsheet. This can be useful if you have a large simulation and want to split it in several, smaller blocks which can be run as independent simulations, making it easier to mantain, make modifications and fix errors in the smaller blocks.

Mass transfer between flowsheets is done in a per-compound basis, that is, any compound that doesn’t match in both simulations will have its flow data erased in the inner flowsheet and will be ignored in the outer one. The mass is transferred to and from the inner flowsheet by matching inlet and outlet

Material Streams connected to the Flowsheet UO with streams in the inner flowsheet. Besides mass/mole flow information, temperature, pressure and enthalpy are also written and read to and from the inner flowsheet.

DWSIM uses the Property Package models defined in the inner flowsheet to do the mass and energy balances. Only settings like Parallel CPU and Parallel GPU calculations affect the way that DWSIM does its calculations inside the block, since it will use the parameters defined when the inner flowsheet was last saved.

You can select Parameters/Properties from objects in the inner flowsheet to expose them to the outer flowsheet, allowing usage of these parameters in Optimization and Sensitivity Analysis cases, Script blocks and for displaying in the outer flowsheet as well.

Connections

Ten inlet and ten outlet Material Stream ports are available for connecting with Material Streams from the inner flowsheet.

After connecting streams to the ports, you must open the Control Panel to map the connected streams to streams in the inner flowsheet.

Calculation Parameters

Simulation file: selects the XML simulation file to use as the inner flowsheet.

Control Panel: opens the Control Panel to initialize the flowsheet, define stream mapping, expose input and output parameters from the inner flowsheet and define the mass transfer mode.

Initialize on load: if true, initializes the inner flowsheet during the opening of the main flowsheet.

Update process data on saving: when saving the main simulation file, DWSIM will update the process data from the inner flowsheet in the selected XML file. Only object process data calculated by the solver will be updated. Other settings will remain unchanged.

Redirect calculator messages: show calculation details from the inner flowsheet in the main flowsheet’s log window.

View flowsheet: shows the inner flowsheet PFD.

Linked Input Parameters

This section will display the parameters that you’ve selected in the control panel to use as input parameters in the outer flowsheet. They can be changed anytime and will trigger the calculation of the flowsheet block.

Linked Output Parameters

This section will display the parameters that you’ve selected in the control panel to use as output parameters in the outer flowsheet. They are read-only and will be updated only after the flowsheet block is calculated successfully.

Results

Mass balance error: shows the mass balance error in %. It can be useful to detect orphan streams in the inner flowsheet, that is, streams that work as inlet or outlet streams in the inner flowsheet but aren’t connected to any stream in the main flowsheet, as this may lead to large mass balance errors.

Control Panel: Use the Control Panel to initialize the flowsheet. Only after initialization you’ll be able to make the connection and expose parameters from the inner flowsheet. You can also use the "Initialize/Reload" button to reload the simulation file after you’ve done changes in the simulation by opening it in another DWSIM window.

Viewing the inner flowsheet

By clicking on the "View Flowsheet" button in the property grid you’ll be able to view the inner flowsheet, check object properties and the overall layout. You can’t change anything here, so any attempt to do so will result in an error.

PEM Fuel Cell

PEM Fuel Cell model

Proton-exchange membrane fuel cells (PEMFC), also known as polymer electrolyte membrane (PEM) fuel cells, are a type of fuel cell being developed mainly for transport applications, as well as for stationary fuel-cell applications and portable fuel-cell applications. Their distinguishing features include lower temperature/pressure ranges (50 to 100 °C) and a special proton-conducting polymer electrolyte membrane. PEMFCs generate electricity and operate on the opposite principle to PEM electrolysis, which consumes electricity.

The PEM Fuel Cell model in DWSIM is an interface for the Amphlett Static Model from the OPEM Python Library (https://www.ecsim.ir/opem/).

The Amphlett static model has been used to predict the performance of proton exchange membrane fuel cell. Key concepts in Amphlett static model are Nernst voltage, activation polarization loss, ohmic polarization loss and concentration polarization loss. Amphlett static model has a mechanistic and empirical approach to describe the performance of proton exchange membrane fuel cell. The ideal standard potential of an H2/O2 fuel cell is 1.229 V with liquid water product. The actual cell potential is decreased from its reference potential because of irreversible losses.

For more information about the model inputs and outputs, please visit https://www.ecsim.ir/opem/doc/Static/Amphlett.html.

Water Electrolyzer

Water Electrolyzer model

Electrolysis of water, also known as electrochemical water splitting, is the process of using electricity to decompose water into oxygen and hydrogen gas by a process called electrolysis. Hydrogen gas released in this way can be used as hydrogen fuel, or remixed with the oxygen to create oxyhydrogen gas, which is used in welding and other applications.

Electrolysis of water requires a minimum potential difference of 1.23 volts, though at that voltage external heat is required from the environment.

# Setup and Calculation Guide

• The Water Electrolyzer model requires Water, Hydrogen and Oxygen added to the simulation with Liquid Water present in the inlet stream.

• Input Parameters: Total Voltage and Number of Cells.

• Output Parameters: Cell Voltage, Current, Electron Transfer and Waste Heat.

• After the calculation, the generated power is directed to the energy stream, while the waste heat is added to the outlet material stream, increasing its temperature.

Hydroelectric Turbine

Hydroelectric Turbine model

A Hydroelectric Turbine is a rotary machine that converts kinetic energy and potential energy of water into mechanical work.

Water turbines were developed in the 19th century and were widely used for industrial power prior to electrical grids. Now, they are mostly used for electric power generation. Water turbines are mostly found in dams to generate electric power from water potential energy.

Setup and Calculation Guide

• The Hydroelectric Turbine model converts head and velocity from the inlet stream into usable energy for the process.

• Input parameters: Static Head, Inlet Velocity, Outlet Velocity and Efficiency.

• Output parameters: Velocity Head, Total Head, Generated Power.

• The generated power is calculated from

\[ P=\eta\,\rho\,g\,h\,q \]

where:

\(P\) generated power (W)

\(\eta\) efficiency (0.00-1.00)

\(\rho\) fluid density (kg/m3)

\(g\) acceleration of gravity (9.81 m/s2)

\(h\) total head (m)

• Total head is calculated from

\[ h=h_{s}+h_{v} \]

where

\(h_{s}\) static head (m)

\(h_{v}\) velocity head (m), calculated from

\[ h_{v}=\frac{v_{in}^{2}-v_{out}^{2}}{2g} \]

Wind Turbine

Wind Turbine model

A wind turbine is a device that converts the kinetic energy of wind into electrical energy. Hundreds of thousands of large turbines, in installations known as wind farms. They are an increasingly important source of intermittent renewable energy, and are used in many countries to lower energy costs and reduce reliance on fossil fuels.

Setup and Calculation Guide

• The Wind Turbine model converts energy from air (wind) into usable energy for the process.

• Input parameters: Wind Speed, Atmospheric Temperature and Pressure, Relative Humidity, Rotor Diameter, Efficiency and Number of Units.

• Output parameters: Generated Power, Maximum Theoretical Power and Calculated Air Density.

• Atmospheric conditions are used to calculate the density of air.

• Conservation of mass requires that the amount of air entering and exiting a turbine must be equal. Accordingly, Betz’s law gives the maximal achievable extraction of wind power by a wind turbine as 16⁄27 (59.3%) of the rate at which the kinetic energy of the air arrives at the turbine (ref. T̈he Physics of Wind Turbines Kira Grogg Carleton College, 2005, p. 8̈).

• The maximum theoretical power output of a wind machine is thus 16/27 times the rate at which kinetic energy of the air arrives at the effective disk area of the machine. If the effective area of the disk is \(A\) , and the wind velocity \(v\) , the maximum theoretical power output \(P_{max}\) is:

\[ {\displaystyle P_{max}=\frac{16}{27}\frac{1}{2}\rho v^{3}A=\frac{8}{27}\rho v^{3}A}, \]

where \(ρ\) is the air density. The actual generated power is given by

\[ {\displaystyle P=}n\,\eta\,P_{max} \]

Solar Panel

Solar Panel model

A solar cell panel, solar electric panel, photo-voltaic (PV) module or solar panel is an assembly of photo-voltaic cells mounted in a framework for installation. Solar panels use sunlight as a source of energy to generate direct current electricity. A collection of PV modules is called a PV panel, and a system of PV panels is called an array. Arrays of a photovoltaic system supply solar electricity to electrical equipment.

Setup and Calculation Guide

• The Solar Panel converts energy from solar irradiation into usable energy for the process.

• Input parameters: Solar Irradiation (kW/m2), Panel Area, Panel Efficiency and Number of Panels.

• Output parameters: Generated Power.

• Generated Power is calculated from

\[ {\displaystyle P=}\eta\,n\,S\,A\, \]

where

\(P\) generated power (kW)

\(\eta\) panel efficiency

\(n\) number of panels

\(S\) solar irradiation (kW/m2)

\(A\) panel area (m2)

Fired Heater

Overview

The Fired Heater (also referred to as a process furnace or direct-fired heater) is a unit operation used extensively in petroleum refining and petrochemical processes. It transfers heat released by the combustion of a fuel, typically refinery fuel gas or natural gas, to a process fluid flowing through tubes inside the furnace.

This DWSIM implementation models a two-zone fired heater consisting of a radiant section, an optional shield (shock) section, and a convection section, topped by a stack (chimney). The model accepts two material streams as input: the process fluid to be heated and the fuel gas stream. It produces two output streams: the heated process fluid and the flue gas exhaust.

Supported furnace configurations

The model supports three furnace geometries: vertical cylindrical upfired (cylindrical shell with a vertical tube coil and floor-fired burners), box horizontal (rectangular cabin with horizontal tube passes), and box vertical (rectangular cabin with vertical tube passes along the side walls and floor-fired burners, the most common configuration in petroleum refining).

Operating modes

The furnace can be specified in three modes. In outlet temperature mode, the user specifies the desired process outlet temperature and the model calculates the required fuel consumption. In duty mode, the user specifies the total absorbed duty and the model calculates the fuel consumption and process outlet temperature. In fuel flow rate mode, the user specifies the fuel stream flow rate as a material stream and the model calculates the absorbed duty and process outlet temperature.

Connections

The unit operation has four stream connections. The process inlet is a material stream carrying the cold process fluid entering the convection section, requiring temperature, pressure, flow rate, and composition. The fuel inlet is a material stream carrying the fuel gas or liquid fuel entering the burners, whose composition is used for stoichiometric combustion calculations. The process outlet is a material stream carrying the hot process fluid exiting the radiant section. The flue gas outlet is a material stream carrying the combustion products exiting the convection section, with composition including \(CO_{2}\) , \(H_{2}O\) , \(N_{2}\) , \(O_{2}\) , and \(SO_{2}\) .

Combustion Model

The combustion model calculates the stoichiometric combustion of the fuel stream based on its molar composition. Each combustible component undergoes complete oxidation.

Combustion reactions

For a general hydrocarbon \(\text{C}_{n}\text{H}_{m}\) :

\[ \text{C}_{n}\text{H}_{m}+\left(n+\frac{m}{4}\right)\text{O}_{2}\rightarrow n\,\text{CO}_{2}+\frac{m}{2}\,\text{H}_{2}\text{O} \]

For hydrogen:

\[ \text{H}_{2}+\frac{1}{2}\,\text{O}_{2}\rightarrow\text{H}_{2}\text{O} \]

For carbon monoxide:

\[ \text{CO}+\frac{1}{2}\,\text{O}_{2}\rightarrow\text{CO}_{2} \]

For hydrogen sulfide:

\[ \text{H}_{2}\text{S}+\frac{3}{2}\,\text{O}_{2}\rightarrow\text{SO}_{2}+\text{H}_{2}\text{O} \]

Supported fuel components

Component	Formula	\(\nu_{\text{O}_{2}}\)	\(\nu_{\text{CO}_{2}}\)	\(\nu_{\text{H}_{2}\text{O}}\)	LHV [kJ/mol]
Methane	CH\(_{4}\)	2.0	1	2	802.3
Ethane	C\(_{2}\)H\(_{6}\)	3.5	2	3	1427.8
Propane	C\(_{3}\)H\(_{8}\)	5.0	3	4	2043.9
n-Butane	C\(_{4}\)H\(_{10}\)	6.5	4	5	2657.3
n-Pentane	C\(_{5}\)H\(_{12}\)	8.0	5	6	3272.1
Hydrogen	H\(_{2}\)	0.5	0	1	241.8
Carbon monoxide	CO	0.5	1	0	283.0
Hydrogen sulfide	H\(_{2}\)S	1.5	0	1	518.0
Ethylene	C\(_{2}\)H\(_{4}\)	3.0	2	2	1323.1
Propylene	C\(_{3}\)H\(_{6}\)	4.5	3	3	1926.4

Stoichiometric coefficients and lower heating values for supported fuel components.

Air requirement

The total stoichiometric oxygen requirement is:

\[ \dot{n}_{\text{O}_{2},\text{stoich}}=\sum_{i}\dot{n}_{f,i}\;\nu_{\text{O}_{2},i} \]

where \(\dot{n}_{f,i}\) is the molar flow of fuel component \(i\) , \(\nu_{\text{O}_{2},i}\) is the stoichiometric coefficient of \(\text{O}_{2}\) for that component, and the summation runs over all combustible species in the fuel stream.

The actual air molar flow, accounting for fractional excess air \(e_{a}\) , is:

\[ \dot{n}_{\text{air}}=\frac{\dot{n}_{\text{O}_{2},\text{stoich}}}{y_{\text{O}_{2},\text{air}}}\left(1+e_{a}\right) \]

where \(y_{\text{O}_{2},\text{air}}=0.2095\) is the mole fraction of \(\text{O}_{2}\) in dry air and \(e_{a}\) is the fractional excess air (e.g. \(e_{a}=0.15\) for \(15\%\) excess). The air composition is assumed as \(20.95\%\) \(\text{O}_{2}\) , \(78.08\%\) \(\text{N}_{2}\) , and \(0.93\%\) \(\text{Ar}\) on a molar basis, with approximately \(1\%\) moisture.

Heat released

The total heat released by combustion on a lower heating value basis is:

\[ Q_{\text{rel}}=\sum_{i}\dot{n}_{f,i}\cdot\text{LHV}_{i} \]

where \(\text{LHV}_{i}\) is the lower heating value of fuel component \(i\) in \(\text{J}/\text{mol}\) .

Adiabatic flame temperature

The adiabatic flame temperature \(T_{\text{flame}}\) is obtained by solving the enthalpy balance iteratively:

\[ Q_{\text{rel}}+\dot{n}_{\text{air}}\int_{T_{\text{ref}}}^{T_{\text{air}}}C_{p,\text{air}}(T)\,dT=\sum_{j}\dot{n}_{fg,j}\int_{T_{\text{ref}}}^{T_{\text{flame}}}C_{p,j}(T)\,dT \]

where \(T_{\text{ref}}=298.15\;\text{K}\) is the reference temperature, \(T_{\text{air}}\) is the combustion air inlet temperature, \(C_{p,j}(T)\) are polynomial heat capacity functions for each flue gas species \(j\) , and \(\dot{n}_{fg,j}\) is the molar flow of flue gas species \(j\) . The model uses Newton–Raphson iteration to solve for \(T_{\text{flame}}\) .

Heat capacity correlations

The molar heat capacities \(C_{p}\) [ \(\text{J}/(\text{mol}\cdot\text{K})\) ] are expressed as third-order polynomials in temperature \(T\) [K]:

\[ C_{p}(T)=a+bT+cT^{2}+dT^{3} \]

where \(a\) , \(b\) , \(c\) , and \(d\) are species-specific coefficients listed in Table \(~§\) .

Species	\(a\)	\(b\)	\(c\)	\(d\)
\(\text{CO}_{2}\)	22.26	\(5.981\times10^{-2}\)	\(-3.501\times10^{-5}\)	\(7.469\times10^{-9}\)
\(\text{H}_{2}\text{O}\)	32.24	\(1.924\times10^{-3}\)	\(1.055\times10^{-5}\)	\(-3.596\times10^{-9}\)
\(\text{N}_{2}\)	28.90	\(-1.571\times10^{-3}\)	\(8.081\times10^{-6}\)	\(-2.873\times10^{-9}\)
\(\text{O}_{2}\)	25.48	\(1.520\times10^{-2}\)	\(-7.156\times10^{-6}\)	\(1.312\times10^{-9}\)
\(\text{SO}_{2}\)	25.78	\(5.795\times10^{-2}\)	\(-3.812\times10^{-5}\)	\(8.612\times10^{-9}\)
\(\text{Ar}\)	20.786	0	0	0

Polynomial coefficients for molar heat capacity \(C_{p}\) [\(\text{J}/(\text{mol}\cdot\text{K})\)].

Radiant Section Model

The radiant section is the primary heat transfer zone where the process fluid absorbs heat predominantly by thermal radiation from the hot combustion gases and refractory walls. The model uses the Lobo–Evans method (Lobo and Evans, 1939).

Mean beam length

The mean beam length \(L_{b}\) for the combustion gas volume is calculated as:

\[ L_{b}=0.9\times\frac{3.6\,V_{\text{chamber}}}{A_{\text{chamber}}} \]

where \(V_{\text{chamber}}\) is the chamber volume [ \(\text{m}^{3}\) ] and \(A_{\text{chamber}}\) is the total internal surface area [ \(\text{m}^{2}\) ]. The factor \(0.9\) is the correction from the optically thin limit to the actual mean beam length.

Gas emissivity (Hottel method)

The total emissivity of the combustion gases \(\varepsilon_{g}\) is calculated as the sum of contributions from \(\text{CO}_{2}\) and \(\text{H}_{2}\text{O}\) with an overlap correction \(\Delta\varepsilon\) :

\[ \varepsilon_{g}=\varepsilon_{\text{CO}_{2}}+\varepsilon_{\text{H}_{2}\text{O}}-\Delta\varepsilon \]

The individual emissivities \(\varepsilon_{\text{CO}_{2}}\) and \(\varepsilon_{\text{H}_{2}\text{O}}\) are functions of the partial-pressure–path-length product \(pL\) [ \(\text{atm}\cdot\text{m}\) ] and the gas temperature, evaluated from simplified Hottel correlations. The overlap correction \(\Delta\varepsilon\) accounts for spectral band interference between \(\text{CO}_{2}\) and \(\text{H}_{2}\text{O}\) :

\[ \Delta\varepsilon=0.02\;\frac{p_{\text{H}_{2}\text{O}}L}{(p_{\text{CO}_{2}}+p_{\text{H}_{2}\text{O}})L}\;\left(1-\frac{p_{\text{H}_{2}\text{O}}L}{(p_{\text{CO}_{2}}+p_{\text{H}_{2}\text{O}})L}\right)\;(p_{\text{CO}_{2}}+p_{\text{H}_{2}\text{O}})L \]

where \(p_{\text{CO}_{2}}\) and \(p_{\text{H}_{2}\text{O}}\) are the partial pressures of \(\text{CO}_{2}\) and \(\text{H}_{2}\text{O}\) in the flue gas [atm] and \(L\) is the mean beam length \(L_{b}\) [m].

Exchange factor

The total exchange factor \(\mathcal{F}\) accounts for the geometric arrangement of tubes and refractory walls using the Lobo–Evans formulation:

\[ \mathcal{F}=\frac{1}{\dfrac{1}{\varepsilon_{g}}+\dfrac{A_{cp}}{A_{cp}+A_{r}}\left(\dfrac{1}{\varepsilon_{t}}-1\right)} \]

where \(\varepsilon_{g}\) is the gas emissivity, \(\varepsilon_{t}\) is the tube surface emissivity (typically \(0.9\) for oxidized steel), \(A_{cp}\) is the cold-plane area (the projected area of the tube bank as seen from the flame) [ \(\text{m}^{2}\) ], and \(A_{r}\) is the exposed refractory area [ \(\text{m}^{2}\) ].

Cold-plane area factor (alpha)

The fraction of the tube plane that intercepts radiation is characterized by the alpha factor \(\alpha\) , which depends on the tube outer diameter \(D_{o}\) and the centre-to-centre tube pitch \(S\) :

\[ \alpha=1-\left(1-\frac{D_{o}}{S}\right)\sqrt{1-\left(\frac{D_{o}}{S}\right)^{2}}-\frac{1}{\pi}\arcsin\left(\frac{D_{o}}{S}\right) \]

The effective cold-plane area is then \(A_{cp,\text{eff}}=\alpha\;A_{cp}\) , where \(A_{cp}=N_{t}\;D_{o}\;L_{t}\) is the total projected plane area, \(N_{t}\) is the number of tubes, and \(L_{t}\) is the effective tube length.

Radiant heat transfer

The heat absorbed in the radiant section \(Q_{\text{rad}}\) comprises a dominant radiative component and a minor convective component:

\[ Q_{\text{rad}}=\mathcal{F}\;\sigma\;A_{cp,\text{eff}}\left(T_{g,\text{mean}}^{4}-T_{t}^{4}\right)+h_{c,\text{rad}}\;A_{t}\left(T_{g,\text{mean}}-T_{t}\right) \]

where \(\sigma=5.67\times10^{-8}\) \(\text{W}/(\text{m}^{2}\cdot\text{K}^{4})\) is the Stefan–Boltzmann constant, \(T_{g,\text{mean}}=(T_{\text{flame}}+T_{bw})/2\) is the mean gas temperature in the radiant section, \(T_{t}\) is the average tube surface temperature [K], \(A_{t}\) is the total tube external surface area \(A_{t}=N_{t}\;\pi\;D_{o}\;L_{t}\) [ \(\text{m}^{2}\) ], and \(h_{c,\text{rad}}\approx11.4\) \(\text{W}/(\text{m}^{2}\cdot\text{K})\) is the natural convection coefficient inside the radiant chamber.

Bridgewall temperature

The bridgewall temperature \(T_{bw}\) (gas temperature at the exit of the radiant section, entering the convection section) is found by iterative solution of the energy balance:

\[ Q_{\text{rad}}+\dot{n}_{fg}\;C_{p,fg}(T_{bw})\;\left(T_{bw}-T_{\text{ref}}\right)=Q_{\text{rel}}\left(1-f_{\text{wall}}\right) \]

where \(\dot{n}_{fg}\) is the total flue gas molar flow [ \(\text{mol}/\text{s}\) ], \(C_{p,fg}(T_{bw})\) is the mean molar heat capacity of the flue gas mixture evaluated at \(T_{bw}\) [ \(\text{J}/(\text{mol}\cdot\text{K})\) ], \(T_{\text{ref}}=298.15\;\text{K}\) is the reference temperature, \(Q_{\text{rel}}\) is the total heat released by combustion [W], and \(f_{\text{wall}}\) is the fractional heat loss through the furnace walls (typically \(0.02\) ). The iteration is performed using the Newton–Raphson method.

Tube skin temperature

The average tube metal temperature (skin temperature) \(T_{\text{skin}}\) is critical for metallurgical design and is calculated from the inside-out thermal resistance model:

\[ T_{\text{skin}}=T_{\text{process}}+\dot{q}_{\text{avg}}\left(\frac{D_{o}}{D_{i}\;h_{\text{int}}}+\frac{D_{o}}{2k_{w}}\ln\frac{D_{o}}{D_{i}}\right) \]

where \(T_{\text{process}}\) is the bulk process fluid temperature [K], \(\dot{q}_{\text{avg}}=Q_{\text{rad}}/A_{t}\) is the average heat flux [ \(\text{W}/\text{m}^{2}\) ], \(D_{o}\) is the tube outer diameter [m], \(D_{i}\) is the tube inner diameter [m], \(h_{\text{int}}\) is the internal film coefficient of the process fluid [ \(\text{W}/(\text{m}^{2}\cdot\text{K})\) ], and \(k_{w}\) is the thermal conductivity of the tube wall material [ \(\text{W}/(\text{m}\cdot\text{K})\) ].

The maximum skin temperature \(T_{\text{skin,max}}\) accounts for the circumferential heat flux variation with a factor \(f_{\text{circ}}\approx1.5\) for single-row tubes backed by refractory:

\[ T_{\text{skin,max}}=T_{\text{process}}+f_{\text{circ}}\;\dot{q}_{\text{avg}}\left(\frac{D_{o}}{D_{i}\;h_{\text{int}}}+\frac{D_{o}}{2k_{w}}\ln\frac{D_{o}}{D_{i}}\right) \]

Shield Section Model

The shield section consists of one or two rows of bare (unfinned) tubes located between the radiant and convection sections. These tubes receive both direct radiation from the radiant chamber and convective heat transfer from the flue gases crossing them. The total duty absorbed by the shield \(Q_{\text{shield}}\) is:

\[ Q_{\text{shield}}=f_{r}\;\varepsilon_{g}\;\sigma\;A_{\text{shield}}\left(T_{bw}^{4}-T_{t,s}^{4}\right)+h_{c,s}\;A_{\text{shield}}\left(T_{bw}-T_{t,s}\right) \]

where \(f_{r}\) is the fraction of radiant heat reaching the shield (typically \(0.08\) - \(0.12\) ), \(A_{\text{shield}}\) is the total tube surface area in the shield section [ \(\text{m}^{2}\) ], \(T_{t,s}\) is the tube surface temperature in the shield [K], and \(h_{c,s}\) is the convective coefficient for flue gas crossing the shield tubes [ \(\text{W}/(\text{m}^{2}\cdot\text{K})\) ].

The gas temperature exiting the shield \(T_{g,\text{out,shield}}\) is obtained from the energy balance on the gas side:

\[ T_{g,\text{out,shield}}=T_{bw}-\frac{Q_{\text{shield}}}{\dot{n}_{fg}\;C_{p,fg}(T_{bw})} \]

Convection Section Model

The convection section recovers additional heat from the flue gases as they flow across a bank of tubes (often finned) before exiting through the stack. The model uses the Zukauskas correlation for crossflow over tube banks.

External heat transfer coefficient

The Nusselt number \(\text{Nu}\) for crossflow over a bank of tubes is:

\[ \text{Nu}=C_{1}\;C_{2}\;\text{Re}^{m}\;\text{Pr}^{0.36} \]

where \(\text{Re}=\rho_{g}\;V_{\max}\;D_{o}/\mu_{g}\) is the Reynolds number based on the maximum velocity \(V_{\max}\) in the minimum flow area, \(\text{Pr}=\mu_{g}\;C_{p,g}/(k_{g}\;M_{fg})\) is the Prandtl number of the flue gas, \(\rho_{g}\) is the gas density [ \(\text{kg}/\text{m}^{3}\) ], \(\mu_{g}\) is the dynamic viscosity [ \(\text{Pa}\cdot\text{s}\) ], \(k_{g}\) is the thermal conductivity [ \(\text{W}/(\text{m}\cdot\text{K})\) ], and the constants \(C_{1}\) and \(m\) depend on the Reynolds range and tube arrangement as shown in Table \(~§\) .

Zukauskas correlation constants for crossflow over tube banks.

Re range	Inline			Staggered
2-3	\(C_{1}\)	\(m\)		\(C_{1}\)	\(m\)
\(\text{Re}<500\)	0.9	0.4		1.04	0.4
\(500\leq\text{Re}<10^{3}\)	0.52	0.5		0.71	0.5
\(10^{3}\leq\text{Re}<2\times10^{5}\)	0.27	0.63		0.35	0.63
\(\text{Re}\geq2\times10^{5}\)	0.033	0.8		0.031	0.8

The row correction factor \(C_{2}\) accounts for the number of tube rows \(N_{r}\) :

\[ C_{2}\approx0.70+0.30\;\frac{N_{r}}{20}\quad\text{for }N_{r}<20;\quad C_{2}=1\quad\text{for }N_{r}\geq20 \]

The external heat transfer coefficient \(h_{o}\) [ \(\text{W}/(\text{m}^{2}\cdot\text{K})\) ] is then:

\[ h_{o}=\frac{\text{Nu}\;k_{g}}{D_{o}} \]

where \(k_{g}\) is the thermal conductivity of the flue gas [ \(\text{W}/(\text{m}\cdot\text{K})\) ] and \(D_{o}\) is the tube outer diameter [m].

Fin efficiency

For finned tubes, the annular fin efficiency \(\eta_{f}\) is calculated using the Harper-Brown solution:

\[ \eta_{f}=\frac{\tanh(m\,r_{1}\,\phi)}{m\,r_{1}\,\phi} \]

where the parameter \(m\) and the geometric correction \(\phi\) are defined as:

\[ \begin{align} m & =\sqrt{\frac{2\,h_{o}}{k_{f}\,t_{f}}}\\ \phi & =\left(\frac{r_{2}}{r_{1}}-1\right)\left(1+0.35\ln\frac{r_{2}}{r_{1}}\right) \end{align} \]

with \(r_{1}=D_{o}/2\) being the tube outer radius [m], \(r_{2}=r_{1}+H_{f}\) the fin tip radius [m], \(H_{f}\) the fin height [m], \(k_{f}\) the fin material thermal conductivity [ \(\text{W}/(\text{m}\cdot\text{K})\) ], and \(t_{f}\) the fin thickness [m].

Overall heat transfer coefficient

The overall heat transfer coefficient \(U\) [ \(\text{W}/(\text{m}^{2}\cdot\text{K})\) ] based on the external surface area is:

\[ U=\frac{1}{\dfrac{1}{h_{o}\,\eta_{f}}+R_{\text{foul}}+\dfrac{D_{o}}{2k_{w}}\ln\dfrac{D_{o}}{D_{i}}+\dfrac{D_{o}}{D_{i}\,h_{\text{int}}}} \]

where \(h_{o}\) is the external (gas-side) heat transfer coefficient, \(\eta_{f}\) is the fin efficiency (equal to \(1.0\) for bare tubes), \(R_{\text{foul}}\) is the fouling resistance [ \(\text{m}^{2}\cdot\text{K}/\text{W}\) ] (typically \(0.0002\) for clean flue gas service), \(k_{w}\) is the tube wall thermal conductivity [ \(\text{W}/(\text{m}\cdot\text{K})\) ], \(D_{o}\) and \(D_{i}\) are the tube outer and inner diameters [m], and \(h_{\text{int}}\) is the internal (process-side) film coefficient [ \(\text{W}/(\text{m}^{2}\cdot\text{K})\) ].

Duty calculation

The heat transferred in the convection section \(Q_{\text{conv}}\) [W] is:

\[ Q_{\text{conv}}=U\;A_{\text{eff}}\;\Delta T_{\text{lm}} \]

where \(A_{\text{eff}}\) is the effective (fin-weighted) surface area [ \(\text{m}^{2}\) ] and \(\Delta T_{\text{lm}}\) is the log-mean temperature difference for counter-current flow [K]:

\[ \Delta T_{\text{lm}}=\frac{\left(T_{g,\text{in}}-T_{p,\text{out}}\right)-\left(T_{g,\text{out}}-T_{p,\text{in}}\right)}{\ln\dfrac{T_{g,\text{in}}-T_{p,\text{out}}}{T_{g,\text{out}}-T_{p,\text{in}}}} \]

Here \(T_{g,\text{in}}\) is the gas inlet temperature to the convection bank (equal to \(T_{g,\text{out,shield}}\) , the outlet of the shield section) [K], \(T_{g,\text{out}}\) is the gas outlet temperature [K], \(T_{p,\text{in}}\) is the process fluid inlet temperature to the convection section [K], and \(T_{p,\text{out}}\) is the process fluid temperature exiting the convection section and entering the radiant section [K].

Draft Model

Natural draft

The available draft \(\Delta P_{\text{draft}}\) [Pa] from the stack is generated by the density difference between the ambient air and the hot flue gases:

\[ \Delta P_{\text{draft}}=g\;H_{\text{stack}}\left(\rho_{\text{amb}}-\rho_{fg}\right) \]

where \(g=9.81\) \(\text{m}/\text{s}^{2}\) is the gravitational acceleration, \(H_{\text{stack}}\) is the stack height [m], \(\rho_{\text{amb}}\) is the ambient air density [ \(\text{kg}/\text{m}^{3}\) ], and \(\rho_{fg}\) is the flue gas density at stack temperature [ \(\text{kg}/\text{m}^{3}\) ], calculated from the ideal gas law:

\[ \rho_{fg}=\frac{P\;M_{fg}}{R\;T_{\text{stack}}} \]

where \(P\) is the absolute pressure [Pa], \(M_{fg}\) is the mean molar mass of the flue gas [ \(\text{kg}/\text{mol}\) ], \(R=8.314\) \(\text{J}/(\text{mol}\cdot\text{K})\) is the universal gas constant, and \(T_{\text{stack}}\) is the flue gas temperature at the stack [K]. The ideal gas law is appropriate for the flue gas and ambient air density calculations because these gases are at near-atmospheric pressure and elevated temperature, where the compressibility factor \(Z\approx1.000\) . The process-side density, in contrast, is obtained from the property package (see Process-Side Pressure Drop below).

Pressure drops

The convection section pressure drop \(\Delta P_{\text{conv}}\) [Pa] is estimated using an Euler number approach:

\[ \Delta P_{\text{conv}}=\text{Eu}\;N_{r}\;\frac{\rho_{g}\;V_{\max}^{2}}{2} \]

where \(\text{Eu}\) is the Euler number (approximately \(1.0\) for staggered and \(0.7\) for inline arrangements), \(N_{r}\) is the number of tube rows, \(\rho_{g}\) is the gas density at the mean convection temperature [ \(\text{kg}/\text{m}^{3}\) ], and \(V_{\max}\) is the maximum gas velocity in the minimum flow cross-section [ \(\text{m}/\text{s}\) ].

The stack friction loss \(\Delta P_{\text{stack}}\) [Pa] is estimated by the Darcy–Weisbach equation:

\[ \Delta P_{\text{stack}}=f\;\frac{H_{\text{stack}}}{D_{\text{stack}}}\;\frac{\rho_{fg}\;V_{\text{stack}}^{2}}{2} \]

where \(f\) is the Darcy friction factor (typically \(0.02\) ), \(D_{\text{stack}}\) is the stack inner diameter [m], \(\rho_{fg}\) is the flue gas density at stack conditions [ \(\text{kg}/\text{m}^{3}\) ], and \(V_{\text{stack}}\) is the flue gas velocity inside the stack [ \(\text{m}/\text{s}\) ].

The net draft \(\Delta P_{\text{net}}\) [Pa] is:

\[ \Delta P_{\text{net}}=\Delta P_{\text{draft}}-\Delta P_{\text{conv}}-\Delta P_{\text{rad}}-\Delta P_{\text{stack}} \]

where \(\Delta P_{\text{rad}}\) is the pressure drop across the radiant chamber (typically small, around \(5\;\text{Pa}\) for an open firebox). A positive value of \(\Delta P_{\text{net}}\) indicates the system is self-drafting; a negative value means forced or induced draft is required.

Thermal Efficiency

The overall thermal efficiency \(\eta\) is:

\[ \eta=1-\frac{Q_{\text{stack}}}{Q_{\text{rel}}}-f_{\text{wall}} \]

where \(Q_{\text{rel}}\) is the total heat released by combustion [W], \(f_{\text{wall}}\) is the fractional wall loss, and the stack loss \(Q_{\text{stack}}\) [W] is:

\[ Q_{\text{stack}}=\dot{n}_{fg}\;C_{p,fg}(T_{\text{stack}})\;\left(T_{\text{stack}}-T_{\text{amb}}\right) \]

where \(\dot{n}_{fg}\) is the total flue gas molar flow [ \(\text{mol}/\text{s}\) ], \(C_{p,fg}(T_{\text{stack}})\) is the mean molar heat capacity of the flue gas evaluated at the stack temperature [ \(\text{J}/(\text{mol}\cdot\text{K})\) ], \(T_{\text{stack}}\) is the flue gas exit temperature [K], and \(T_{\text{amb}}\) is the ambient temperature [K]. Typical thermal efficiencies for well-designed process furnaces range from \(80\%\) to \(92\%\) , depending on stack temperature and excess air.

Emissions

\(CO_{2}\) and \(SO_{2}\)

These are calculated directly from the combustion stoichiometry:

\[ \begin{align} \dot{m}_{\text{CO}_{2}} & =\dot{n}_{\text{CO}_{2}}\;M_{\text{CO}_{2}}\\ \dot{m}_{\text{SO}_{2}} & =\dot{n}_{\text{SO}_{2}}\;M_{\text{SO}_{2}} \end{align} \]

where \(\dot{n}_{\text{CO}_{2}}\) and \(\dot{n}_{\text{SO}_{2}}\) are the molar flows from the combustion model [ \(\text{mol}/\text{s}\) ], \(M_{\text{CO}_{2}}=0.04401\) \(\text{kg}/\text{mol}\) is the molar mass of carbon dioxide, and \(M_{\text{SO}_{2}}=0.06406\) \(\text{kg}/\text{mol}\) is the molar mass of sulfur dioxide.

\(NO_{x}\)

\(\text{NO}_{x}\) formation is estimated using an empirical correlation that accounts for the peak flame temperature \(T_{\text{flame}}\) and excess oxygen concentration \(y_{\text{O}_{2}}\) :

\[ C_{\text{NO}_{x}}\;\text{[mg/Nm}^{3}\text{]}=4.0\times10^{-8}\;\exp\left(0.01\;T_{\text{flame}}\right)\;\sqrt{y_{\text{O}_{2}}\times100} \]

where \(T_{\text{flame}}\) is the adiabatic flame temperature [K] and \(y_{\text{O}_{2}}\) is the mole fraction of excess \(\text{O}_{2}\) in the flue gas. This is a simplified estimate suitable for conventional gas-fired burners. For low- \(\text{NO}_{x}\) burner performance, the user should consult manufacturer data.

Process-Side Pressure Drop

The pressure drop through the process tubes \(\Delta P_{\text{process}}\) [Pa] is calculated using the Darcy–Weisbach equation with corrections for return bends:

\[ \Delta P_{\text{process}}=f\;\frac{L_{\text{total}}+L_{\text{eq,bends}}}{D_{i}}\;\frac{\rho_{p}\;V_{p}^{2}}{2} \]

where \(f\) is the Darcy friction factor, \(L_{\text{total}}=L_{\text{tube}}\times N_{\text{passes}}\) is the total tube length [m], \(L_{\text{eq,bends}}=(N_{\text{passes}}-1)\times30\;D_{i}\) is the equivalent length for return bends [m], \(D_{i}\) is the tube inner diameter [m], \(\rho_{p}\) is the process fluid density [ \(\text{kg}/\text{m}^{3}\) ] obtained from the property package (overall phase density of the process inlet stream), \(V_{p}\) is the process fluid velocity in the tubes [ \(\text{m}/\text{s}\) ], \(L_{\text{tube}}\) is the effective tube length per pass [m], and \(N_{\text{passes}}\) is the number of process fluid passes through the furnace. The process fluid velocity is calculated as:

\[ V_{p}=\frac{\dot{m}_{p}}{\rho_{p}\;A_{\text{flow}}} \]

where \(\dot{m}_{p}\) is the total process mass flow rate [ \(\text{kg}/\text{s}\) ] and \(A_{\text{flow}}=N_{\text{tubes/pass}}\;\pi\;D_{i}^{2}/4\) is the total flow cross-sectional area per pass [ \(\text{m}^{2}\) ]. Using the property-package density ensures correct pressure-drop estimation for both liquid and gas-phase process fluids, including high-pressure or supercritical services where the ideal gas law would be inaccurate.

Solution Algorithm

The overall calculation procedure follows these steps:

1. Read process inlet and fuel inlet stream properties (temperature, pressure, flow, composition).

2. Based on the operating mode, estimate the fuel flow rate or outlet temperature.

3. Calculate the combustion stoichiometry, heat released \(Q_{\text{rel}}\) , and adiabatic flame temperature \(T_{\text{flame}}\) .

4. Assume an initial radiant/convective duty split ( \(70/30\) ).

5. Iterate the radiant section model to find the bridgewall temperature \(T_{bw}\) .

6. Calculate the shield section duty \(Q_{\text{shield}}\) and gas outlet temperature \(T_{g,\text{out,shield}}\) .

7. Calculate the convection section duty \(Q_{\text{conv}}\) , gas outlet temperature \(T_{g,\text{out}}\) , and overall coefficient \(U\) .

8. Update the duty split and intermediate process temperature; repeat from step 5 until convergence (temperature change \(<0.5\;\text{K}\) between iterations).

9. Calculate draft \(\Delta P_{\text{net}}\) , emissions ( \(\dot{m}_{\text{CO}_{2}}\) , \(\dot{m}_{\text{SO}_{2}}\) , \(C_{\text{NO}_{x}}\) ), and thermal efficiency \(\eta\) .

10. Update all output streams.

Convergence is typically achieved in \(10\) – \(30\) outer iterations with a sub-relaxation factor of \(0.4\) for stability.

Input Parameters

Parameter	Description	Unit
Length	Internal chamber length	m
Width	Internal chamber width	m
Height	Internal chamber height	m
TubeOuterDiameter	Tube outer diameter (\(D_{o}\))	m
TubeWallThickness	Tube wall thickness	m
TubeEffectiveLength	Heated tube length (\(L_{t}\))	m
NumberOfTubes	Total tubes in radiant section (\(N_{t}\))	–
NumberOfPasses	Process fluid passes (\(N_{\text{passes}}\))	–
TubePitch	Centre-to-centre spacing (\(S\))	m
TubeToWallDistance	Centre of tube to refractory wall	m
TubeThermalConductivity	Tube material conductivity (\(k_{w}\))	\(\text{W}/(\text{m}\cdot\text{K})\)
TubeEmissivity	Oxidised tube surface emissivity (\(\varepsilon_{t}\))	–
RefractoryEmissivity	Refractory wall emissivity	–
RefractoryThickness	Refractory lining thickness	m

Radiant section geometry parameters.

Parameter	Description	Unit
TubeOuterDiameter	Tube outer diameter (\(D_{o}\))	m
TubeWallThickness	Tube wall thickness	m
TubeEffectiveLength	Tube length	m
NumberOfTubes	Total tubes	–
NumberOfRows	Tube rows perpendicular to gas flow (\(N_{r}\))	–
NumberOfPasses	Process fluid passes	–
TransversePitch	Pitch perpendicular to gas flow (\(S_{T}\))	m
LongitudinalPitch	Pitch along gas flow (\(S_{L}\))	m
Arrangement	Inline or staggered	–
FinType	Bare, solid fin, or serrated fin	–
FinHeight	Fin height from tube surface (\(H_{f}\))	m
FinThickness	Fin thickness (\(t_{f}\))	m
FinDensity	Fins per unit length	\(1/\text{m}\)
FinThermalConductivity	Fin material conductivity (\(k_{f}\))	\(\text{W}/(\text{m}\cdot\text{K})\)

Convection section geometry parameters.

Parameter	Description	Unit
StackHeight	Chimney height (\(H_{\text{stack}}\))	m
StackInnerDiameter	Chimney inner diameter (\(D_{\text{stack}}\))	m
AmbientTemperature	Surrounding air temperature (\(T_{\text{amb}}\))	K
AmbientPressure	Atmospheric pressure (\(P\))	Pa
ExcessAir	Fractional excess air (\(e_{a}\), e.g. \(0.15=15\%\))	–
WallHeatLossFraction	Heat loss as fraction of \(Q_{\text{rel}}\) (\(f_{\text{wall}}\))	–
DraftType	Natural, forced, or induced	–

Stack and general operating parameters.

Output Variables

The model reports the following results: total duty absorbed \(Q_{\text{total}}\) [W], thermal efficiency \(\eta\) [ \(\%\) ], fuel consumption [ \(\text{kg}/\text{s}\) ], bridgewall temperature \(T_{bw}\) [K], average and maximum tube skin temperatures \(T_{\text{skin}}\) and \(T_{\text{skin,max}}\) [K], average and maximum heat flux \(\dot{q}_{\text{avg}}\) and \(\dot{q}_{\max}\) [ \(\text{W}/\text{m}^{2}\) ], flue gas outlet temperature \(T_{g,\text{out}}\) [K], available and net draft \(\Delta P_{\text{draft}}\) and \(\Delta P_{\text{net}}\) [Pa], stack gas velocity \(V_{\text{stack}}\) [ \(\text{m}/\text{s}\) ], \(\text{CO}_{2}\) emission rate \(\dot{m}_{\text{CO}_{2}}\) [ \(\text{kg}/\text{h}\) ], \(\text{SO}_{2}\) emission rate \(\dot{m}_{\text{SO}_{2}}\) [ \(\text{g}/\text{h}\) ], \(\text{NO}_{x}\) concentration \(C_{\text{NO}_{x}}\) [ \(\text{mg}/\text{Nm}^{3}\) ], and the complete temperature profile across all sections.

Zeolite Adsorber

Overview

The Zeolite Adsorber is a general-purpose gas-phase adsorption unit operation that models the separation of multicomponent gas mixtures on zeolite (or other microporous) adsorbents. The model supports two operating modes:

• Equilibrium mode: a steady-state shortcut calculation based on working capacity at specified adsorption and desorption conditions.

• PSA Cycle mode: a simplified four-step Skarstrom pressure-swing (or temperature-swing) adsorption cycle that yields cycle-averaged raffinate and desorbate flows.

Three isotherm families are available: single-site Langmuir, dual-site Langmuir (DSL), and Freundlich. Multicomponent competition is handled through the extended (competitive) Langmuir and DSL mixing rules.

Stream Topology

The unit operation has the following connection ports:

Port	Direction	Description
Feed Gas In	Inlet (material)	Mixed-gas feed stream
Raffinate Out	Outlet (material)	Less- or non-adsorbed product
Desorbate Out	Outlet (material)	Adsorbed product (regeneration gas)

Isotherm Models

All isotherm calculations use partial pressure \(p_i = y_i P\) as the independent variable, where \(y_i\) is the mole fraction of component \(i\) and \(P\) is the total pressure. Loadings are expressed in mol kg\(^{-1}\) (mol of adsorbate per kg of dry adsorbent).

Temperature Dependence of Affinity Constants

For the Langmuir and DSL models, the affinity constant follows a van’t Hoff relationship:

\[ b_i(T) = b_{0,i} \exp\!\left(\frac{\Delta H_i}{R T}\right) \]

where \(b_{0,i}\) is the pre-exponential factor (Pa\(^{-1}\)), \(\Delta H_i > 0\) is the isosteric heat of adsorption (J mol\(^{-1}\), sign convention: positive for exothermic adsorption), \(R = 8.314\) J mol\(^{-1}\) K\(^{-1}\) is the universal gas constant, and \(T\) is the absolute temperature (K). This convention ensures that \(b_i\) increases as temperature decreases, which is physically correct for physisorption.

Single-Site Langmuir (SSL)

The pure-component single-site Langmuir isotherm is

\[ q_i = \frac{q_{\mathrm{sat},i}\, b_i(T)\, p_i} {1 + b_i(T)\, p_i} \]

where \(q_{\mathrm{sat},i}\) (mol kg\(^{-1}\)) is the saturation capacity.

For a multicomponent mixture the extended Langmuir mixing rule is applied:

\[ q_i = \frac{q_{\mathrm{sat},i}\, b_i(T)\, p_i} {1 + \displaystyle\sum_{j=1}^{N} b_j(T)\, p_j} \]

This expression is thermodynamically consistent (satisfies Gibbs–Duhem) only when all saturation capacities are equal; it is used as a practical approximation for unequal \(q_{\mathrm{sat}}\) values.

Dual-Site Langmuir (DSL)

Zeolites often present two structurally distinct adsorption sites (e.g. cation sites and window sites in zeolite 5A, or cage and window sites in zeolite 13X). The pure-component DSL isotherm is

\[ q_i = \frac{q_{\mathrm{sat}1,i}\, b_{1,i}(T)\, p_i} {1 + b_{1,i}(T)\, p_i} + \frac{q_{\mathrm{sat}2,i}\, b_{2,i}(T)\, p_i} {1 + b_{2,i}(T)\, p_i} \]

For a mixture, each site type is treated as independent and the competitive Langmuir mixing rule is applied site-by-site:

\[ q_i = \frac{q_{\mathrm{sat}1,i}\, b_{1,i}(T)\, p_i} {1 + \displaystyle\sum_j b_{1,j}(T)\, p_j} + \frac{q_{\mathrm{sat}2,i}\, b_{2,i}(T)\, p_i} {1 + \displaystyle\sum_j b_{2,j}(T)\, p_j} \]

Freundlich

The Freundlich isotherm is an empirical power-law expression:

\[ q_i = K_i\, p_i^{1/n_i} \]

where \(K_i\) (mol kg\(^{-1}\) Pa\(^{-1/n}\)) is the Freundlich pre-factor and \(n_i > 0\) is the heterogeneity index. For \(n_i > 1\) the isotherm is sub-linear (concave), which is typical for heterogeneous surfaces. Because no rigorous IAST extension is available for the Freundlich isotherm, an additive (independent) approximation is used for mixtures; this is valid only when inter-component competition is weak.

Note on temperature dependence

The current implementation treats \(K_i\) as temperature-independent. Users who need to account for temperature effects should supply \(K_i\) values measured at the desired operating temperature.

Working Capacity

The working (or delta) capacity is the difference in equilibrium loading between adsorption and desorption conditions:

\[ \Delta q_i = q_i(T_{\mathrm{ads}}, p_i^{\mathrm{ads}}) - q_i(T_{\mathrm{des}}, p_i^{\mathrm{des}}) \]

where the desorption partial pressures are approximated as

\[ p_i^{\mathrm{des}} = y_i^{\mathrm{feed}}\, P_{\mathrm{des}} \]

This assumes the feed mole fractions remain unchanged during regeneration, which is an approximation that is most accurate for dilute systems and for PSA processes where blowdown is fast.

Equilibrium (Shortcut) Mode

In Equilibrium mode the model converts the cycle working capacity into an equivalent steady-state molar flow of adsorbed species using the cycle time \(t_{\mathrm{cyc}}\) (s) as a time basis:

\[ \dot{n}_{i}^{\mathrm{ads}} = \frac{\Delta q_i\, M_{\mathrm{ads}}} {t_{\mathrm{cyc}}} \]

where \(M_{\mathrm{ads}}\) (kg) is the total adsorbent mass.

The raffinate flow before purge correction is

\[ \dot{n}_{i}^{\mathrm{raff,0}} = \dot{n}_{i}^{\mathrm{feed}} - \dot{n}_{i}^{\mathrm{ads}} \]

A fraction \(\phi_{\mathrm{purge}}\) of the net raffinate is recycled as purge gas to assist regeneration. The final raffinate and desorbate flows are

\[ \begin{align} \dot{n}_{i}^{\mathrm{raff}} &= \dot{n}_{i}^{\mathrm{raff,0}} \left(1 - \phi_{\mathrm{purge}}\right) \\[4pt] \dot{n}_{i}^{\mathrm{des}} &= \dot{n}_{i}^{\mathrm{ads}} + \phi_{\mathrm{purge}}\, \dot{n}_{i}^{\mathrm{raff,0}} \end{align} \]

A physical constraint is applied so that the adsorbed flow cannot exceed the feed flow (\(\dot{n}_{i}^{\mathrm{ads}} \le \dot{n}_{i}^{\mathrm{feed}}\)).

PSA Cycle Mode

The PSA Cycle mode simulates a simplified four-step Skarstrom cycle operating at local equilibrium (sharp-front approximation). The four steps and their mass balances are described below. All mole quantities refer to a single bed; cycle-averaged flows are obtained by dividing by \(t_{\mathrm{cyc}}\).

The cycle time is partitioned as follows. Let \(f_{\mathrm{press}}\) be the fraction of the cycle allocated to pressurisation; the remaining time is split equally between adsorption/feed and blowdown+purge:

\[ \begin{align*} t_{\mathrm{press}} &= f_{\mathrm{press}}\, t_{\mathrm{cyc}} \\ t_{\mathrm{feed}} &= \tfrac{1}{2}(1 - f_{\mathrm{press}})\, t_{\mathrm{cyc}} \\ t_{\mathrm{blow}} &= t_{\mathrm{purge}} = \tfrac{1}{4}(1 - f_{\mathrm{press}})\, t_{\mathrm{cyc}} \end{align*} \]

Step 1 – Pressurisation

The bed is pressurised from \(P_{\mathrm{des}}\) to \(P_{\mathrm{ads}}\) using raffinate (product-end) gas. The total moles consumed are

\[ \Delta n_{\mathrm{press}} = \underbrace{\frac{V_{\mathrm{void}}(P_{\mathrm{ads}}-P_{\mathrm{des}})}{Z\,R\,T_{\mathrm{ads}}}}_{\text{void filling}} + \underbrace{f_{\mathrm{press}} \sum_i \Delta q_i\, M_{\mathrm{ads}}}_{\text{adsorbent loading}} \]

where \(Z\) is the gas compressibility factor at adsorption conditions, derived from the property-package molar density: \(Z = P_{\mathrm{ads}} / (\rho_{\mathrm{mol}}\, R\, T_{\mathrm{ads}})\). where \(V_{\mathrm{void}} = V_{\mathrm{bed}}\,\varepsilon\) is the void volume of the bed (m\(^3\)) and \(\varepsilon\) is the void fraction. The bed volume is

\[ V_{\mathrm{bed}} = \frac{M_{\mathrm{ads}}}{\rho_{\mathrm{bulk}}(1-\varepsilon)} \]

No product is withdrawn during this step.

Step 2 – Feed / Adsorption

Feed gas flows into the bed at \(P_{\mathrm{ads}}\). The heavy (more-adsorbed) component loads the adsorbent and the light product exits as raffinate. Moles adsorbed per component during this step:

\[ \Delta n_{i}^{\mathrm{feed}} = \frac{1-f_{\mathrm{press}}}{2} \Delta q_i\, M_{\mathrm{ads}} \]

The per-component raffinate from this step is

\[ \Delta n_{i}^{\mathrm{raff}} = \max\!\left(0,\; \dot{n}_{i}^{\mathrm{feed}}\, t_{\mathrm{feed}} - \Delta n_{i}^{\mathrm{feed}}\right) \]

Step 3 – Blowdown

Pressure drops from \(P_{\mathrm{ads}}\) to \(P_{\mathrm{des}}\) co-currently. Gas released comprises void-space gas and desorbed adsorbate. Half of the working capacity is attributed to this step:

\[ \begin{align} \Delta n_{\mathrm{void}}^{\mathrm{blow}} &= \frac{V_{\mathrm{void}}(P_{\mathrm{ads}}-P_{\mathrm{des}})}{Z\,R\,T_{\mathrm{des}}} \\[4pt] \Delta n_{i}^{\mathrm{des,blow}} &= \tfrac{1}{2}\,\Delta q_i\, M_{\mathrm{ads}} \end{align} \]

The blowdown effluent is added to the desorbate stream.

Step 4 – Purge

A fraction \(\phi_{\mathrm{purge}}\) of the net raffinate flow is routed counter-currently through the bed at \(P_{\mathrm{des}}\) to strip residual adsorbate. The remaining half of the working capacity is desorbed during this step. The purge exhaust (purge gas in \(+\) desorbed gas) forms part of the desorbate stream.

Cycle-Averaged Mass Balance

The cycle-averaged molar flows (mol s\(^{-1}\)) returned to the DWSIM streams are:

\[ \begin{align} \dot{n}_{i}^{\mathrm{des}} &= \frac{\Delta n_{i}^{\mathrm{blow}} + \Delta n_{i}^{\mathrm{purge,out}}} {t_{\mathrm{cyc}}} \\[4pt] \dot{n}_{i}^{\mathrm{raff}} &= \max\!\left(0,\;\frac{\Delta n_{i}^{\mathrm{raff}}}{t_{\mathrm{cyc}}} - \frac{\Delta n_{\mathrm{press}}}{t_{\mathrm{cyc}}}\,y_i - \phi_{\mathrm{purge}}\,\dot{n}_{i}^{\mathrm{raff}}\right) \end{align} \]

where the pressurisation penalty is distributed to the raffinate stream in proportion to feed mole fractions.

Separation Performance Indicators

The model reports the following key performance indicators for the most-adsorbed (key) component \(k\):

\[ \begin{align} \text{Recovery} &= \frac{\dot{n}_k^{\mathrm{des}}} {\dot{n}_k^{\mathrm{feed}}} \\[6pt] \text{Purity (desorbate)} &= \frac{\dot{n}_k^{\mathrm{des}}} {\displaystyle\sum_i \dot{n}_i^{\mathrm{des}}} \\[6pt] \text{Purity (raffinate)} &= \frac{\dot{n}_k^{\mathrm{raff}}} {\displaystyle\sum_i \dot{n}_i^{\mathrm{raff}}} \end{align} \]

Pressure Drop (Ergun Equation)

When the bed geometry parameters (vessel diameter, bed length, particle diameter, particle sphericity) are specified, the model computes the pressure drop across the packed bed using the Ergun equation :

\[ \frac{\Delta P}{L} = \frac{150\,\mu\,u\,(1-\varepsilon)^2} {\varphi^2\,d_p^2\,\varepsilon^3} + \frac{1.75\,\rho\,u^2\,(1-\varepsilon)} {\varphi\,d_p\,\varepsilon^3} \]

where \(L\) (m) is the bed length, \(\mu\) (Pa s) is the gas dynamic viscosity, \(u\) (m s\(^{-1}\)) is the superficial gas velocity, \(\varepsilon\) is the bed void fraction, \(\varphi\) is the particle sphericity, \(d_p\) (m) is the mean particle diameter, and \(\rho\) (kg m\(^{-3}\)) is the gas density.

The first term represents viscous (Blake–Kozeny) losses and the second term represents inertial (Burke–Plummer) losses. The total pressure drop is

\[ \Delta P = \frac{\Delta P}{L}\, L \]

The superficial velocity is computed per bed from the total feed flow divided by the number of parallel beds \(N_{\mathrm{beds}}\):

\[ u = \frac{Q_{\mathrm{vol}}}{A_{\mathrm{cross}}} \;,\qquad Q_{\mathrm{vol}} = \frac{\dot{m}_{\mathrm{total}}} {\rho\, N_{\mathrm{beds}}} \;,\qquad A_{\mathrm{cross}} = \frac{\pi}{4}\,D^2 \]

where \(D\) (m) is the vessel diameter, \(\dot{m}_{\mathrm{total}}\) (kg s\(^{-1}\)) is the total mass flow rate, and \(\rho\) (kg m\(^{-3}\)) is the gas density obtained from the property package (vapor phase).

The outlet (raffinate) pressure is set to \(P_{\mathrm{raff}} = P_{\mathrm{ads}} - \Delta P\), subject to a minimum floor of 1 kPa. If any geometry parameter is zero or unset, the pressure drop is taken as zero and the outlet pressure equals the inlet pressure.

Bed Geometry Parameters

Parameter	Symbol	SI Unit	Default
Vessel diameter	\(D\)	m	1.0
Bed length	\(L\)	m	3.0
Particle diameter	\(d_p\)	m	0.002
Particle sphericity	\(\varphi\)	–	1.0

Heat of Adsorption Estimate

An approximate heat duty associated with adsorption is computed from the isosteric heats and the cycle-averaged adsorbed flows:

\[ \dot{Q}_{\mathrm{ads}} = \sum_i \dot{n}_i^{\mathrm{ads}}\,\Delta H_i \]

This estimate does not account for the sensible heat required to heat the bed during TSA regeneration. For rigorous energy balances, a detailed dynamic model is recommended.

Model Parameters

Bed Parameters

Parameter	Symbol	SI Unit	Default
Adsorbent mass	\(M_{\mathrm{ads}}\)	kg	1000
Void fraction	\(\varepsilon\)	–	0.40
Bulk density	\(\rho_{\mathrm{bulk}}\)	kg m\(^{-3}\)	700
Number of beds	\(N_{\mathrm{beds}}\)	–	2

Cycle Parameters

Parameter	Symbol	Unit	Default
Cycle time	\(t_{\mathrm{cyc}}\)	s	600
Purge fraction	\(\phi_{\mathrm{purge}}\)	–	0.10
Pressurisation time fraction	\(f_{\mathrm{press}}\)	–	0.10
Adsorption pressure	\(P_{\mathrm{ads}}\)	Pa	(from feed stream)
Desorption pressure	\(P_{\mathrm{des}}\)	Pa	20 000
Adsorption temperature	\(T_{\mathrm{ads}}\)	K	(from feed stream)
Desorption temperature	\(T_{\mathrm{des}}\)	K	423.15

Isotherm Parameters (per component)

Parameter	Symbol	Applicable Model	Unit
Saturation capacity (site 1)	\(q_{\mathrm{sat}1}\)	SSL, DSL	mol kg\(^{-1}\)
Pre-exp. affinity (site 1)	\(b_{0,1}\)	SSL, DSL	Pa\(^{-1}\)
Isosteric heat (site 1)	\(\Delta H_1\)	SSL, DSL	J mol\(^{-1}\)
Saturation capacity (site 2)	\(q_{\mathrm{sat}2}\)	DSL only	mol kg\(^{-1}\)
Pre-exp. affinity (site 2)	\(b_{0,2}\)	DSL only	Pa\(^{-1}\)
Isosteric heat (site 2)	\(\Delta H_2\)	DSL only	J mol\(^{-1}\)
Freundlich pre-factor	\(K\)	Freundlich	mol kg\(^{-1}\) Pa\(^{-1/n}\)
Freundlich exponent	\(n\)	Freundlich	–

Built-In Zeolite Presets

Indicative Langmuir and DSL parameters for common zeolite–gas systems are provided as starting-point presets. These values are drawn from published literature and should be replaced by experimental data before engineering calculations are performed.

Preset	Pore size	Target separation
Zeolite 3A	\(\approx 3\) Å	Water removal (unsaturated hydrocarbons)
Zeolite 4A	\(\approx 4\) Å	General drying, CO\(_2\) removal
Zeolite 5A	\(\approx 5\) Å	O\(_2\)/N\(_2\) air separation
Zeolite 13X	\(\approx 10\) Å	CO\(_2\)/CH\(_4\) biogas upgrading

Assumptions and Limitations

1. Local equilibrium: the model assumes instantaneous equilibrium between the gas phase and the adsorbed phase (infinite mass-transfer rate). Real columns exhibit dispersive mass-transfer zones; the shortcut result represents the best achievable performance for a given set of equilibrium data.

2. Steady-state cycle average: the PSA Cycle mode converts a cyclic process to steady-state equivalent flows. Instantaneous concentration profiles within a cycle are not resolved.

3. Real gas in void space: the void-space inventory uses the compressibility factor \(Z\) derived from the property-package molar density at adsorption conditions. The same \(Z\) is applied at desorption conditions as an approximation.

4. Simplified desorption composition: the desorption partial pressures are approximated using feed mole fractions ([eq:des_partial]). In reality the desorbate is enriched in the heavy component; a more rigorous treatment requires solving the column material balance iteratively.

5. Isothermal operation: for PSA calculations the bed temperature is held constant. Heat effects due to adsorption and desorption are not fed back into the energy balance; they are reported separately as \(\dot{Q}_{\mathrm{ads}}\) ([eq:heat_ads]).

6. Ergun pressure drop: the pressure drop is estimated from the Ergun equation ([eq:ergun]) using the gas density and viscosity from the property package. The calculation assumes uniform, isothermal, single-phase gas flow through a homogeneous packed bed.

Numerical Solution Procedure

1. Resolve feed stream conditions: \(T\), \(P\), \(\{y_i\}\), \(\dot{n}_{\mathrm{total}}\).

2. Synchronise the component isotherm data list with the compound list from the feed stream.

3. Compute partial pressures \(p_i = y_i P_{\mathrm{ads}}\).

4. Evaluate equilibrium loadings at adsorption conditions \(q_i^{\mathrm{ads}}\) using the selected isotherm ([eq:langmuir_multi], [eq:DSL_multi], or [eq:freundlich]).

5. Evaluate equilibrium loadings at desorption conditions \(q_i^{\mathrm{des}}\).

6. Compute working capacities \(\Delta q_i\) ([eq:working_capacity]).

7. Obtain the gas molar density \(\rho_{\mathrm{mol}}\) from the property-package vapor-phase density and the feed mass and molar flows. Derive \(Z = P/({\rho_{\mathrm{mol}}\,R\,T})\) for void-space calculations.

8. Equilibrium mode: apply [eq:ads_flow]–[eq:des_flow].\ PSA Cycle mode: execute the four Skarstrom steps and compute cycle-averaged flows ([eq:pressurisation]–[eq:raff_avg]).

9. Compute bed pressure drop via the Ergun equation ([eq:ergun]) if geometry parameters are specified.

10. Set outlet stream temperatures, pressures and molar flows. The raffinate pressure is reduced by \(\Delta P\).

11. Compute separation performance indicators ([eq:recovery]–[eq:purity_raff]) and heat of adsorption estimate ([eq:heat_ads]).

No iteration is required for either mode; the calculation is explicit given the isotherm parameters and operating conditions.

Typical Usage Workflow

1. Add the Zeolite Adsorber block to the flowsheet and connect the feed, raffinate, and desorbate material streams.

2. Specify the operating mode (Equilibrium or PSA Cycle) and isotherm model on the Parameters tab.

3. Select a zeolite preset or enter custom isotherm parameters on the Isotherm Parameters tab. If a feed stream is connected, the component table is populated automatically.

4. Set the adsorption pressure (taken from the feed stream) and the desorption conditions (\(P_{\mathrm{des}}\), \(T_{\mathrm{des}}\)).

5. For PSA mode, adjust the cycle time, purge fraction, and pressurisation time fraction.

6. Run the simulation and review the Results tab for stream summaries, per-component loadings, and separation performance indicators.

Copper Bed Mercury Adsorber

Overview

The Copper Bed Mercury Adsorber models a fixed-bed guard vessel used to remove elemental mercury (\(\ce{Hg^0}\)) from natural gas, NGL, and LNG process streams. The unit represents a once-through, non-regenerable sorbent bed based on copper sulphide (CuS/Al\(_2\)O\(_3\)), metallic copper on activated carbon (Cu/C), or sulphur-impregnated activated carbon (SIAC).

Mercury occurs in natural gas at trace concentrations (typically 0.001–10,000 μg/Nm\(^3\)) and must be removed to protect aluminium heat exchangers, catalyst beds, and downstream equipment, as well as to comply with product-quality specifications . The primary removal mechanism is irreversible chemisorption:

\[ \begin{align} \ce{Hg^0 + CuS &-> HgS + Cu} \\ \ce{Hg^0 + Cu &-> Cu\text{--}Hg} \quad \text{(amalgam)} \end{align} \]

Because the reaction is essentially irreversible, regeneration is not practised; the bed is replaced when the mercury capacity is exhausted.

Two calculation modes are available:

• Capacity-Based mode: a simplified sizing model that uses the vendor-rated maximum mercury capacity \(q_{\max}\) to compute the bed lifetime at a given inlet concentration and gas flow rate. Full removal (outlet concentration equal to the breakthrough specification) is assumed until the capacity is exhausted.

• Wheeler-Jonas mode: a rigorous breakthrough model based on the Wheeler-Jonas equation , combined with a Langmuir or Freundlich adsorption isotherm. The model predicts the time-varying outlet Hg concentration as a function of bed age, inlet conditions, and mass-transfer kinetics.

Stream Topology

The unit operation has the following connection ports:

Port	Direction	Description
Feed Gas In	Inlet (material)	Raw gas containing trace mercury
Treated Gas Out	Outlet (material)	Cleaned gas at reduced Hg content

No second outlet is provided because the sorbent is a consumable; the spent bed is removed from service rather than regenerated.

Mercury is identified in the feed stream by the configurable compound name (default: Mercury). If that compound is absent from the component list, the user may instead specify the inlet Hg concentration directly in μg/Nm\(^3\).

Mercury Concentrations and Unit Conversions

Natural gas mercury concentrations are most commonly reported at normal conditions (0 °C, 101 325 Pa) in units of μg/Nm\(^3\). The model uses actual conditions (operating \(T\), \(P\)) internally and converts for display.

Mole fraction to actual concentration

The mass concentration of mercury at operating conditions is

\[ C_{\mathrm{act}} \; [\mu\text{g/m}^3] = y_{\mathrm{Hg}}\, \rho_{\mathrm{mol}}\, M_{\mathrm{Hg}} \times 10^6 \]

where \(y_{\mathrm{Hg}}\) is the mercury mole fraction, \(\rho_{\mathrm{mol}}\) (mol m\(^{-3}\)) is the gas molar density from the property package, and \(M_{\mathrm{Hg}} = 200.59\) g mol\(^{-1}\). The molar density is computed as \(\rho_{\mathrm{mol}} = \dot{n}_{\mathrm{total}} / Q\), where \(Q\) is the actual volumetric flow obtained from the mass flow and the property-package density.

Actual to normal conditions

\[ C_{\mathrm{Nm}^3} \; [\mu\text{g/Nm}^3] = C_{\mathrm{act}} \cdot \frac{\rho_{\mathrm{mol}}^{\mathrm{NTP}}} {\rho_{\mathrm{mol}}} \]

where \(\rho_{\mathrm{mol}}^{\mathrm{NTP}} = P_{\mathrm{NTP}}/(R\,T_{\mathrm{NTP}}) \approx 44.6\) mol m\(^{-3}\) is the ideal-gas molar density at normal conditions (\(T_{\mathrm{NTP}} = 273.15\) K, \(P_{\mathrm{NTP}} = 101{,}325\) Pa). The inverse conversion is used to bring user-specified inlet concentrations (given at normal conditions) to actual conditions for the mass balance.

Isotherm Models

Both isotherm models use the Hg partial pressure \(p_{\mathrm{Hg}} = y_{\mathrm{Hg}}\,P\) as the independent variable. Loadings are expressed in mol kg\(^{-1}\) and converted to mg Hg g\(^{-1}\) for display via

\[ q \; [\text{mg Hg/g}] = q \; [\text{mol/kg}] \times M_{\mathrm{Hg}} \; [\text{g/mol}] \]

Langmuir Isotherm

The single-site Langmuir model with temperature-dependent affinity constant is

\[ q = \frac{q_{\mathrm{sat}}\, b(T)\, p_{\mathrm{Hg}}} {1 + b(T)\, p_{\mathrm{Hg}}} \]

where \(q_{\mathrm{sat}}\) (mol kg\(^{-1}\)) is the saturation capacity and the temperature-dependent affinity constant follows a van’t Hoff relationship:

\[ b(T) = b_0 \exp\!\left(\frac{\Delta H}{R T}\right) \]

Here \(b_0\) (Pa\(^{-1}\)) is the pre-exponential factor and \(\Delta H > 0\) (J mol\(^{-1}\)) is the isosteric heat of chemisorption (sign convention: positive for exothermic). For copper-sulphide chemisorption, \(\Delta H\) is typically in the range 50,000–80,000 J mol\(^{-1}\), reflecting a very strong Hg–sulphur bond.

At the trace concentrations found in natural gas (\(p_{\mathrm{Hg}} \ll 1/b(T)\)), the Langmuir isotherm approaches Henry’s-law behaviour:

\[ q \approx q_{\mathrm{sat}}\, b(T)\, p_{\mathrm{Hg}} \qquad (b\,p_{\mathrm{Hg}} \ll 1) \]

Conversely, at elevated concentrations or with strongly chemisorptive sorbents (\(b\,p_{\mathrm{Hg}} \gg 1\)), the loading approaches \(q_{\mathrm{sat}}\), indicating the sorbent is operating at or near saturation.

Freundlich Isotherm

The empirical Freundlich model is

\[ q = K_F\, p_{\mathrm{Hg}}^{1/n_F} \]

where \(K_F\) (mol kg\(^{-1}\) Pa\(^{-1/n_F}\)) is the pre-factor and \(n_F > 0\) is the heterogeneity index. For chemisorption at trace concentrations, \(n_F > 1\) gives a favourable (concave) isotherm. The current implementation treats \(K_F\) and \(n_F\) as temperature-independent; users should supply values measured at the design operating temperature.

Capacity-Based Mode

In this simplified mode the sorbent is characterised solely by its maximum working capacity \(q_{\max}\) (mg Hg g\(^{-1}\)), which is typically obtained from the sorbent vendor’s datasheet or from accelerated laboratory tests.

The total mercury storage capacity of the bed is

\[ \hat{N} = q_{\max} \; [\text{mg/g}] \times W_s \; [\text{kg}] \times 10^3 \; [\text{g/kg}] \quad [\text{mg Hg}] \]

The actual volumetric gas flow at operating conditions is

\[ Q = \frac{\dot{m}_{\mathrm{total}}}{\rho} \]

where \(\dot{m}_{\mathrm{total}}\) (kg s\(^{-1}\)) is the total mass feed flow and \(\rho\) (kg m\(^{-3}\)) is the gas density from the property package.

The bed lifetime (time from start-up to breakthrough) is

\[ t_b = \frac{\hat{N} \; [\text{mg Hg}] \times 10^3 \; [\mu\text{g/mg}]} {C_{\mathrm{in}} \; [\mu\text{g/m}^3] \times Q \; [\text{m}^3/\text{s}]} = \frac{q_{\max}\, W_s \times 10^6} {C_{\mathrm{in}}\, Q} \]

The outlet concentration is set equal to the breakthrough specification \(C_b\) (corresponding to a fresh-to-breakthrough average), which represents the design limit for a guard bed in service.

Wheeler-Jonas Breakthrough Model

The Wheeler-Jonas equation provides a closed-form estimate of the breakthrough time for a fixed-bed adsorber with first-order mass-transfer kinetics and a favourable (concave) isotherm. It is widely used for sizing gas-phase sorbent systems including mercury guard beds in natural gas service .

Stoichiometric breakthrough time

The stoichiometric breakthrough time corresponds to the ideal (infinitely sharp) mass-transfer front and equals the ratio of total bed capacity to the Hg mass feed rate:

\[ t_s = \frac{q_e \; [\text{mg/g}] \times W_s \; [\text{kg}] \times 10^6} {C_{\mathrm{in}} \; [\mu\text{g/m}^3] \times Q \; [\text{m}^3/\text{s}]} \]

where \(q_e\) is the equilibrium loading at the inlet concentration, obtained from Eq. [eq:langmuir_hg] or Eq. [eq:freundlich_hg] and converted by Eq. [eq:loading_conv]. The factor \(10^6\) converts kg mg/g \(\to\) μg consistently with \(C_{\mathrm{in}}\) expressed in μg/m\(^3\).

Wheeler-Jonas breakthrough time

The mass-transfer zone (MTZ) shifts the breakthrough curve relative to the stoichiometric front. The Wheeler-Jonas corrected breakthrough time is

\[ t_b = t_s - \frac{1}{k_v} \ln\!\left(\frac{C_{\mathrm{in}}}{C_b} - 1\right) \]

where \(k_v\) (s\(^{-1}\)) is the overall first-order volumetric mass-transfer coefficient and \(C_b\) (μg/m\(^3\)) is the breakthrough concentration (outlet specification at actual conditions).

The second term in Eq. [eq:wj_time] is the mass-transfer correction \(\delta t = k_v^{-1} \ln(C_{\mathrm{in}}/C_b - 1)\), which is negative when \(C_b < C_{\mathrm{in}}/2\) (typical of stringent specifications), meaning that real beds break through earlier than the stoichiometric prediction. Equations [eq:t_stoich] and [eq:wj_time] are valid provided \(C_b < C_{\mathrm{in}}\).

Breakthrough concentration profile

At any elapsed service time \(t\) (s) the outlet concentration is given by the logistic (S-shaped) breakthrough curve

\[ C_{\mathrm{out}}(t) = \frac{C_{\mathrm{in}}} {1 + \exp\!\left[k_v \left(t_b - t\right)\right]} \]

This expression reproduces the expected behaviour: \(C_{\mathrm{out}} \to 0\) for \(t \ll t_b\) (fresh bed) and \(C_{\mathrm{out}} \to C_{\mathrm{in}}\) for \(t \gg t_b\) (exhausted bed). The steepness of the S-curve is controlled by \(k_v\); as \(k_v \to \infty\) the profile approaches the sharp-front limit.

Bed saturation fraction

The fraction of the bed capacity consumed at age \(t\) is approximated as

\[ f_{\mathrm{sat}} = \min\!\left(1,\; \frac{t}{t_b}\right) \]

which is reported for monitoring purposes and is used to set the Bed Saturation output property.

Mercury Mass Balance

The molar flow of mercury removed is computed from the actual concentration difference and the volumetric gas flow:

\[ \begin{align} \dot{m}_{\mathrm{Hg}}^{\mathrm{removed}} \; [\mu\text{g/s}] &= \left(C_{\mathrm{in}} - C_{\mathrm{out}}\right) \times Q \\[6pt] \dot{n}_{\mathrm{Hg}}^{\mathrm{removed}} \; [\text{mol/s}] &= \frac{\dot{m}_{\mathrm{Hg}}^{\mathrm{removed}}}{M_{\mathrm{Hg}} \times 10^6} \end{align} \]

where the factor \(10^6\) converts μg mol\(^{-1}\) to g mol\(^{-1}\). The outlet Mercury molar flow is

\[ \dot{n}_{\mathrm{Hg}}^{\mathrm{out}} = \max\!\left(0,\; \dot{n}_{\mathrm{Hg}}^{\mathrm{feed}} - \dot{n}_{\mathrm{Hg}}^{\mathrm{removed}}\right) \]

All other components pass through the bed unchanged.

Removal efficiency

\[ \eta = \frac{C_{\mathrm{in}} - C_{\mathrm{out}}}{C_{\mathrm{in}}} \]

Pressure Drop (Ergun Equation)

When bed geometry parameters (vessel diameter, bed length, particle diameter, particle sphericity) are specified, the pressure drop across the packed bed is computed using the Ergun equation :

\[ \frac{\Delta P}{L} = \frac{150\,\mu\,u\,(1-\varepsilon)^2} {\varphi^2\,d_p^2\,\varepsilon^3} + \frac{1.75\,\rho\,u^2\,(1-\varepsilon)} {\varphi\,d_p\,\varepsilon^3} \]

where \(L\) (m) is the bed length, \(\mu\) (Pa s) is the gas dynamic viscosity, \(u\) (m s\(^{-1}\)) is the superficial gas velocity, \(\varepsilon\) is the bed void fraction, \(\varphi\) is the particle sphericity, \(d_p\) (m) is the mean particle diameter, and \(\rho\) (kg m\(^{-3}\)) is the gas density.

The first term accounts for viscous (Blake–Kozeny) losses and the second term for inertial (Burke–Plummer) losses. The total pressure drop is \(\Delta P = (\Delta P / L)\, L\).

The superficial velocity is computed from the mass flow and the gas density obtained from the property package:

\[ u = \frac{Q}{A} \;,\qquad Q = \frac{\dot{m}_{\mathrm{total}}}{\rho} \;,\qquad A = \frac{\pi}{4}\,D^2 \]

where \(D\) (m) is the vessel diameter, \(\dot{m}_{\mathrm{total}}\) (kg s\(^{-1}\)) is the total mass flow, and \(\rho\) (kg m\(^{-3}\)) is the gas density from the property package (vapor phase).

The outlet pressure is set to \(P_{\mathrm{out}} = P_{\mathrm{in}} - \Delta P\), subject to a minimum floor of 1 kPa. If any geometry parameter is zero or unset, the pressure drop defaults to zero.

Bed Geometry Parameters

Parameter	Symbol	SI Unit	Default
Vessel diameter	\(D\)	m	1.0
Bed length	\(L\)	m	3.0
Particle diameter	\(d_p\)	m	0.002
Particle sphericity	\(\varphi\)	–	1.0

Sorbent Presets

Representative parameters for three common sorbent types are provided as starting-point presets. Values are drawn from published literature and from commercially available guard-bed datasheets. These parameters should always be replaced with site-specific experimental data before engineering calculations are performed.

Preset	Mechanism	\(q_{\max}\) [mg/g]	\(k_v\) [s\(^{-1}\)]
CuS / Al\(_2\)O\(_3\)	Hg + CuS \(\to\) HgS + Cu	100	0.002
Cu / Activated Carbon	Amalgam + sulphide	200	0.003
Sulphur-Impregnated Carbon	Hg + S \(\to\) HgS	150	0.003

The Langmuir isotherm parameters for each preset correspond to near-saturation behaviour at typical natural gas conditions, reflecting the essentially irreversible chemisorption mechanism: \(\Delta H \approx 55\text{--}60\) kJ/mol with very large \(b(T)\) values (bed operates in the plateau region of the isotherm).

Model Parameters

Bed Parameters

Parameter	Symbol	SI Unit	Default
Sorbent mass	\(W_s\)	kg	1000
Bulk density	\(\rho_b\)	kg m\(^{-3}\)	700
Void fraction	\(\varepsilon\)	–	0.40

Capacity-Based Parameters

Parameter	Symbol	Unit	Default
Max. Hg capacity	\(q_{\max}\)	mg Hg/g	100

Isotherm Parameters (Wheeler-Jonas mode)

Parameter	Symbol	Model	Unit	Default
Saturation capacity	\(q_{\mathrm{sat}}\)	Langmuir	mol kg\(^{-1}\)	0.5
Pre-exp. affinity constant	\(b_0\)	Langmuir	Pa\(^{-1}\)	\(2\times 10^{-6}\)
Isosteric heat	\(\Delta H\)	Langmuir	J mol\(^{-1}\)	60,000
Freundlich pre-factor	\(K_F\)	Freundlich	mol kg\(^{-1}\) Pa\(^{-1/n_F}\)	10
Freundlich exponent	\(n_F\)	Freundlich	–	3

Wheeler-Jonas and Operating Parameters

Parameter	Symbol	Unit	Default
Mass-transfer coefficient	\(k_v\)	s\(^{-1}\)	0.002
Bed age	\(t\)	h	0
Breakthrough spec.	\(C_b\)	μg/Nm\(^3\)	1.0
Inlet Hg concentration\(^\dagger\)	\(C_{\mathrm{in}}\)	μg/Nm\(^3\)	100
Operating temperature	\(T\)	K	(from feed stream)

\(^\dagger\)Used only when the Mercury compound is absent from the feed stream; otherwise the concentration is derived from the mole fraction via Eq. [eq:c_act].

Assumptions and Limitations

1. Irreversible chemisorption: the sorbent is modelled as non-regenerable. The capacity-based mode assumes that the sorbent is fully effective (100 % removal) until breakthrough; deactivation kinetics or competing reactions are not modelled.

2. Elemental mercury only: only elemental Hg\(^0\) is considered. Organomercury compounds (e.g. dimethylmercury) and ionic species (Hg\(^{2+}\)) have different adsorption behaviour and require separate treatment.

3. Local equilibrium in Wheeler-Jonas mode: the model assumes that the axial dispersion and external film resistance are lumped into the single parameter \(k_v\). Rigorous mass-transfer analysis (e.g. linear driving force or pore-diffusion models) is beyond the scope of this unit operation.

4. Ergun pressure drop: the pressure drop is estimated from the Ergun equation ([eq:ergun_hg]) using the gas density and viscosity from the property package. The calculation assumes uniform, isothermal, single-phase gas flow through a homogeneous packed bed.

5. Isothermal operation: the outlet gas temperature is set equal to the inlet value. The heat released by chemisorption (\(\Delta H \approx 50\text{--}80\) kJ/mol) is not fed back into the energy balance; it is reported implicitly through the removed molar flow and the isosteric heat parameter.

6. Real-gas properties: all volumetric flows, gas densities, and mercury concentration conversions use the gas molar density from the property package, which accounts for real-gas compressibility effects. Normal-condition quantities (Nm\(^3\)) use the ideal-gas molar density at NTP as per the standard definition.

7. No competitive adsorption: the isotherm parameters describe the Hg–sorbent interaction only. Competitive adsorption by H\(_2\)S, COS, or heavy hydrocarbons (which can reduce the effective Hg capacity) is not modelled.

8. Uniform concentration profile: the Wheeler-Jonas equation assumes an axially uniform initial Hg loading. It does not resolve the spatial concentration profile within the bed.

Numerical Solution Procedure

1. Resolve feed stream conditions: \(T\), \(P\), \(\{y_i\}\), \(\dot{n}_{\mathrm{total}}\).

2. Determine the inlet Hg concentration \(C_{\mathrm{in,act}}\) (μg/m\(^3\)) from the stream mole fraction (Eq. [eq:c_act]) or from the user-specified value converted by the inverse of Eq. [eq:c_norm].

3. Compute the actual volumetric flow \(Q\) from the mass flow and property-package density (Eq. [eq:vol_flow]), and derive the gas molar density \(\rho_{\mathrm{mol}} = \dot{n}_{\mathrm{total}} / Q\).

4. Convert the breakthrough specification \(C_b\) from μg/Nm\(^3\) to actual conditions using Eq. [eq:c_norm].

5. Capacity-Based mode:

   1. Compute total bed capacity \(\hat{N}\) (Eq. [eq:total_cap]).

   2. Compute bed lifetime \(t_b\) (Eq. [eq:lifetime_cap]).

   3. Set \(C_{\mathrm{out}} = C_b\) (design assumption).

6. Wheeler-Jonas mode:

   1. Compute \(p_{\mathrm{Hg}} = y_{\mathrm{Hg}} P\) and evaluate the equilibrium loading \(q_e\) from the selected isotherm (Eqs. [eq:langmuir_hg] or [eq:freundlich_hg]).

   2. Convert \(q_e\) to mg/g (Eq. [eq:loading_conv]).

   3. Compute the stoichiometric breakthrough time \(t_s\) (Eq. [eq:t_stoich]) and the Wheeler-Jonas breakthrough time \(t_b\) (Eq. [eq:wj_time]).

   4. Compute the outlet concentration at the specified bed age (Eq. [eq:wj_profile]).

   5. Compute the bed saturation fraction (Eq. [eq:saturation]).

7. Compute Hg removal efficiency \(\eta\) (Eq. [eq:removal_eff]) and the removed molar flow (Eqs. [eq:hg_removed_mass] and [eq:hg_removed_mol]).

8. Compute bed pressure drop via the Ergun equation ([eq:ergun_hg]) if geometry parameters are specified.

9. Set the outlet stream: all non-Hg components unchanged; Mercury molar flow set to \(\dot{n}_{\mathrm{Hg}}^{\mathrm{out}}\) (Eq. [eq:hg_out]); \(T\) equal to inlet value; \(P_{\mathrm{out}} = P_{\mathrm{in}} - \Delta P\).

The calculation is explicit (no iteration required) in both modes.

Typical Usage Workflow

1. Add the Copper Bed Hg Adsorber block to the flowsheet and connect the feed and treated-gas material streams.

2. On the Parameters tab, select the operating mode (CapacityBased or WheelerJonas).

3. Choose a sorbent preset or enter custom parameters. If the gas stream includes a Mercury compound, ensure the Hg Compound Name field matches the DWSIM compound name and set Hg Concentration Source to From Stream. Otherwise select Specified and enter the inlet concentration in μg/Nm\(^3\).

4. Enter the bed geometry (mass, bulk density, void fraction) and the breakthrough specification \(C_b\).

5. For WheelerJonas mode, enter the isotherm parameters (use a preset as a starting point) and the mass-transfer coefficient \(k_v\). To obtain the outlet concentration at a given bed age, set the Bed Age field to the elapsed service hours.

6. Run the simulation and review the Results tab for inlet and outlet Hg concentrations, removal efficiency, equilibrium capacity, total bed capacity, and bed lifetime.

7. For design studies, use the DWSIM Sensitivity or Optimizer tool to assess how bed lifetime varies with sorbent mass, inlet concentration, or breakthrough specification.

Worked Example

A natural gas stream at 50 bar and 40 °C with an inlet mercury concentration of 100 μg/Nm\(^3\) flows at 500 Nm\(^3\)/h through a CuS guard bed loaded with 500 kg of sorbent (\(q_{\max} = 100\) mg/g).

The actual volumetric flow (assuming \(Z \approx 0.9\) from the property package at 50 bar, 313 K) is

\[ Q = \frac{500\,\text{Nm}^3/\text{h} \times \tfrac{1}{3600}\,\text{h/s} \times 313.15\,\text{K}} {273.15\,\text{K}} \times \frac{101\,325\,\text{Pa}}{50 \times 10^5\,\text{Pa}} \times Z \approx 2.91 \times 10^{-3}\,\text{m}^3/\text{s} \]

(In practice, the model obtains \(Q\) directly from the stream mass flow and property-package density; the calculation above is for illustration.)

The actual inlet concentration uses the property-package molar density \(\rho_{\mathrm{mol}} = P/(ZRT) \approx 2186\) mol/m\(^3\):

\[ C_{\mathrm{in,act}} = 100 \times \frac{\rho_{\mathrm{mol}}}{\rho_{\mathrm{mol}}^{\mathrm{NTP}}} \approx 100 \times \frac{2186}{44.6} \approx 4899\,\mu\text{g/m}^3 \]

From Eq. [eq:lifetime_cap] the bed lifetime in Capacity-Based mode is

\[ t_b = \frac{100 \times 500 \times 10^6}{4899 \times 2.91 \times 10^{-3}} \approx 3.51 \times 10^9\,\text{s} \approx 40{,}600\,\text{days} \]

This unrealistically long lifetime reveals that for this low flow rate the bottleneck is not volumetric capacity but rather the mass of mercury accumulated. Reducing the bed mass to 10 kg gives a more typical result of \(\approx 828\) days.

Pipe Network

Overview

The Pipe Network unit operation performs a rigorous steady-state simulation of fluid flow, pressure distribution, and heat transfer in arbitrarily connected piping systems. The model resolves the simultaneous mass, momentum, and energy balances for all segments and junction nodes in the network, supporting single-phase and two-phase (gas–liquid) flows with full thermodynamic property integration.

The network is built by placing and connecting a set of network objects (pipes, nodes, pumps, compressors, valves, separators, sources and sinks) on a graphical canvas. A nonlinear equation solver then determines the mass flow rates, pressures, and temperatures throughout the network that satisfy all governing balances simultaneously.

Network Objects

Object	Inlets / Outlets	Role
Source	0 in / 1 out	Boundary-condition inlet
Sink	1 in / 0 out	Boundary-condition outlet
Node	\(\le10\) in / \(\le10\) out	Flow-splitting/mixing junction
Pipe	1 in / 1 out	Pressure-drop and heat-transfer segment
Pump	1 in / 1 out	Liquid pressure booster
Compressor	1 in / 1 out	Gas pressure booster
Valve	1 in / 1 out	Throttling or control element
Separator	1 in / 2 out	Gas–liquid flash separator
Bridge	1 in / 1 out	Non-mixing bypass connection

Source and Sink

Sources and sinks impose boundary conditions on the network. Each can be specified independently in seven modes:

Specification mode	Fixed quantities
Pressure only	\(P\)
Mass flow only	\(\dot{m}\)
Molar flow only	\(\dot{n}\)
Volumetric flow only	\(\dot{V}\)
Pressure & mass flow	\(P\), \(\dot{m}\)
Pressure & molar flow	\(P\), \(\dot{n}\)
Pressure & vol. flow	\(P\), \(\dot{V}\)
None (fully free)	—

Node

Junction nodes mix or split streams. The model enforces per-node overall mass, pressure, and energy balances (see 2.31.3). An optional rigorous heat balance mode uses the full stream enthalpy from the thermodynamic property package rather than an ideal mixing approximation.

Pipe

The pipe segment calculates the pressure drop and thermal profile for a single pipe section given its geometry, orientation, and selected two-phase flow correlation (see 2.31.4). A per-segment equilibrium flash can be performed at configurable intervals to update stream thermodynamic properties along the pipe length.

Pump, Compressor, and Valve

These objects wrap the corresponding DWSIM base unit operations in \(\Delta P\) calculation mode, allowing them to be embedded directly in the network without separate flowsheet connections.

A valve is solved on its flow coefficient, and how the opening reaches that coefficient is the valve’s own business: with the opening versus \(K_v\) relationship enabled, the coefficient follows the opening through the characteristic chosen there (linear, equal percentage, quick opening, a table or an expression), which is what lets a controller, or the actuator of 2.31.7, move the valve during a run. With the relationship disabled, which is the default, the coefficient entered in the valve stands whatever the opening says. A coefficient of zero is a shut branch.

Separator

Performs an adiabatic flash split. The vapour outlet supplies the gas phase stream and the liquid outlet supplies the liquid phase stream to the downstream network.

Additional Blocks (Nodal Solver)

With the nodal Newton solver (2.31.6) the palette adds blocks for water distribution and petroleum production: a Water Pipe (a lightweight single-phase pipe using the Hazen–Williams  or Darcy–Weisbach correlation with static head, for water grids), a Reservoir/Tank fixed-head boundary, a Pressure Control Valve (a reducing PRV holding the downstream pressure, or a sustaining PSV holding the upstream pressure), an Inflow Performance (IPR) well block, and a Choke bean restriction (see 2.31.8).

Node Balance Equations

For a node \(k\) with \(n_{\mathrm{in}}\) inlets and \(n_{\mathrm{out}}\) outlets, three dimensionless residuals are formed and minimised by the network solver.

Mass Balance

\[ r_{m,k} = \frac{\displaystyle\sum_{i \in \mathrm{in}} \dot{m}_i - \displaystyle\sum_{j \in \mathrm{out}} \dot{m}_j} {\dot{m}_{\mathrm{total}}} \]

Pressure Balance

All streams leaving a node share the same node pressure \(P_k\). Streams arriving at the node are assumed to match \(P_k\) after pressure-drop elements upstream. The residual is defined as

\[ r_{P,k} = \frac{1}{n_{\mathrm{in}}} \sum_{i \in \mathrm{in}} \left(\frac{P_i}{P_k}\right)^{\!2} - \frac{1}{n_{\mathrm{out}}} \sum_{j \in \mathrm{out}} \left(\frac{P_j}{P_k}\right)^{\!2} \]

Energy Balance

\[ r_{E,k} = \frac{\displaystyle\sum_{i \in \mathrm{in}} \dot{m}_i h_i - \displaystyle\sum_{j \in \mathrm{out}} \dot{m}_j h_j} {\dot{H}_{\mathrm{total}}} \]

where \(h_i\) is the specific enthalpy of stream \(i\) and \(\dot{H}_{\mathrm{total}}\) is a reference enthalpy scale for normalisation.

Pipe Pressure-Drop Models

The user selects a pressure-drop correlation independently for each pipe segment. Three two-phase correlations are available, plus the underlying single-phase Darcy–Weisbach equation.

Single-Phase Flow (Darcy–Weisbach)

For a pipe of length \(L\), internal diameter \(D\), and friction factor \(f\) carrying a fluid of density \(\rho\) at mean velocity \(u\):

\[ \Delta P = f \frac{L}{D} \frac{\rho u^2}{2} + \rho g L \sin\theta \]

where \(\theta\) is the pipe inclination angle from the horizontal and \(g\) is the gravitational acceleration. The Fanning friction factor \(f\) is evaluated from the Colebrook–White implicit equation :

\[ \frac{1}{\sqrt{f}} = -2\log_{10}\!\left( \frac{\varepsilon}{3.7\,D} + \frac{2.51}{Re\sqrt{f}} \right) \]

where \(\varepsilon\) is the pipe roughness and \(Re = \rho u D / \mu\) is the Reynolds number. Equation [eq:colebrook] is solved iteratively (or via the explicit Swamee–Jain approximation for initialisation).

Beggs and Brill (1973)

The Beggs–Brill correlation is the default method for two-phase gas–liquid flow. It predicts the in-situ liquid holdup \(H_L\) and a two-phase friction multiplier \(\phi_{tp}\) from the mixture Froude number \(Fr_m\), input liquid volume fraction \(\lambda_L\), and velocity numbers \(N_{vL}\), \(N_{vG}\).

The total pressure gradient is decomposed as

\[ \left.\frac{dP}{dz}\right|_{\mathrm{total}} = \left.\frac{dP}{dz}\right|_{\mathrm{fric}} + \left.\frac{dP}{dz}\right|_{\mathrm{el}} + \left.\frac{dP}{dz}\right|_{\mathrm{acc}} \]

The friction term uses the mixture density \(\rho_m\) and the two-phase friction factor \(f_{tp}\):

\[ \left.\frac{dP}{dz}\right|_{\mathrm{fric}} = f_{tp}\,\frac{\rho_m u_m^2}{2D} \]

The elevation (hydrostatic) term uses the in-situ average density:

\[ \left.\frac{dP}{dz}\right|_{\mathrm{el}} = \bar{\rho}\,g\sin\theta, \qquad \bar{\rho} = \rho_L H_L + \rho_G (1 - H_L) \]

The flow-pattern map identifies four regimes (segregated, intermittent, distributed and transition) and the holdup correlation is applied per regime with an inclination correction factor \(\psi(\theta, H_L)\). The friction factor is corrected by an empirical multiplier \(e^S\) that depends on \(\lambda_L / H_L^2\):

\[ f_{tp} = f_{ns}\,e^S \]

where \(f_{ns}\) is the no-slip friction factor evaluated at the mixture Reynolds number.

Lockhart and Martinelli (1949)

The Lockhart–Martinelli correlation relates the two-phase pressure gradient to the single-phase liquid gradient via the two-phase multiplier \(\phi_L^2\):

\[ \left.\frac{dP}{dz}\right|_{tp} = \phi_L^2 \left.\frac{dP}{dz}\right|_{L} \]

The multiplier is correlated against the Martinelli parameter \(X\):

\[ X = \sqrt{\frac{(dP/dz)_L}{(dP/dz)_G}} \]

The Chisholm  parameterisation is used for \(\phi_L^2\):

\[ \phi_L^2 = 1 + \frac{C}{X} + \frac{1}{X^2} \]

where the constant \(C\) depends on whether each phase is in laminar (\(C\) = 5 or 10) or turbulent (\(C\) = 12 or 20) flow.

Petalas and Aziz (2000)

The Petalas–Aziz model is a mechanistic unified approach that uses a comprehensive flow-pattern classification and separate closure relationships for each regime, including stratified, annular-mist, slug, and dispersed bubble flow. It is recommended for high-pressure and high-GOR applications where empirical correlations may be less reliable.

Thermal Model

The temperature profile along each pipe segment is determined from the steady-state energy balance:

\[ \dot{m}\,\frac{dh}{dz} = q(z) - \dot{m}\,g\sin\theta \]

where \(q(z) = U_o \pi D_o [T_{\mathrm{amb}}(z) - T(z)]\) is the heat flux per unit length, \(U_o\) is the overall heat-transfer coefficient based on the outer diameter \(D_o\), and \(T_{\mathrm{amb}}(z)\) is the ambient temperature.

An optional Joule-Thomson correction accounts for the isenthalpic temperature change that accompanies pressure drop in compressible fluids. The Joule-Thomson coefficient is evaluated from the equation of state via

\[ \mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_h = -\frac{1}{\dot{m}\,c_P}\left(\frac{\partial H}{\partial P}\right)_T \]

For emulsified oil-water flows, an optional emulsion viscosity correction (Yoshida et al.) adjusts the mixture viscosity to account for the droplet-induced viscosity enhancement, which can significantly affect the friction pressure drop in oil-continuous emulsions.

Network Solver

Problem Formulation

The network is cast as a nonlinear optimisation problem. The decision variables \(\mathbf{x}\) are the unknown source pressures and/or mass flow rates. To ensure physically meaningful (strictly positive) values, variables are log-transformed:

\[ x_i = \ln\!\left(\frac{v_i}{s_i}\right), \qquad v_i = s_i\,e^{x_i} \]

where \(v_i\) is the physical quantity (Pa or kg s\(^{-1}\)) and \(s_i\) is a scale factor (\(P_{\max}\) or \(\dot{m}_{\max}\)).

The objective function to be minimised is the sum of squared node residuals:

\[ F(\mathbf{x}) = \sum_k \left(r_{m,k}^2 + r_{P,k}^2 + r_{E,k}^2\right) \]

For each evaluation of \(F\), all network blocks are calculated sequentially (sources first, then pipes, pumps, compressors, valves and separators) and the node residuals are assembled from the resulting stream conditions.

Solver Options

Two numerical methods are available:

• Simplex (default): derivative-free Nelder–Mead simplex method . Robust for moderate-size networks and does not require gradient information.

• IPOPT: interior-point optimisation  using numerical gradients computed by finite differences. May converge faster for large or stiff networks.

The two methods above pose the network as a least-squares problem over the boundary pressures and flows and are retained for backward compatibility; they do not represent looped networks or flow reversal and are slower than the nodal Newton solver described next.

Nodal Newton Solver (Todini–Pilati)

The recommended default for new networks is a sparse nodal Newton method based on the Global Gradient Algorithm of Todini and Pilati . The unknowns are the nodal pressures, obtained by solving a reduced symmetric positive-definite linear system at each iteration; the branch flows follow from the nodal pressures. Because each branch flow carries a sign, looped (meshed) networks and flow reversal are handled automatically, whatever direction the pipe was drawn in, and the method converges in a few iterations to machine-precision continuity.

Flow Models

The nodal Newton solver offers two flow models. In the incompressible (single-phase) model each pipe is a closed-form pressure-drop law, Hazen–Williams  or Darcy–Weisbach with the Churchill friction factor , plus the static head; this is the model for water distribution grids. In the compositional (multiphase) model each pipe wraps the full two-phase pipe segment (2.31.4) as a black box, and an outer loop refreshes the pressure and temperature of every branch while the inner Newton step resolves the hydraulics; this is the model for petroleum gathering and production networks.

Spatial Discretisation and Richardson Extrapolation

In the compositional flow model each pipe is walked increment by increment, and that walk is first order in the increment size: halving the increment removes about half of the remaining error. For a single-phase liquid line the point is moot, because the pressure gradient is essentially constant along the pipe and any discretisation returns the same answer. For multiphase flow it is not: holdup, in-situ density and phase velocities all vary along the pipe, and on a gathering or production network the computed rate can move by a few percent between the discretisation a network was drawn with and a grid fine enough to have converged. Refining the grid pays for that badly, since first-order convergence means eight times the work buys roughly six times less error.

Two grids and an extrapolation buy considerably more. Writing \(f(h)\) for the outlet state computed with increment size \(h\), a first-order error gives

\[ f_{\mathrm{exact}} \;\approx\; 2\,f(h/2) - f(h), \]

which cancels the leading term. With Richardson extrapolation enabled, every pipe is evaluated on its own grid and on one twice as fine, and [eq:richardson] is applied to the outlet pressure and temperature. It costs three pipe calculations where there was one, and in exchange it typically reaches a converged answer from the discretisation the network already carries.

The extrapolation is applied only where it is meaningful. When the two grids disagree by more than a quarter of the pressure drop, the pair is not in the asymptotic range, there is no single leading term to cancel, and the finer grid is delivered unextrapolated; the same fallback applies if the extrapolated state would be unphysical. This also sets the honest expectation for the feature: it corrects a discretisation that is already close, and it is not a substitute for one that is too coarse to be in the asymptotic range at all. A network whose rate is still moving substantially between successive grid refinements needs more increments, not extrapolation.

The option is off by default, because it triples the cost of every pipe evaluation and it changes the result of an existing network.

Degrees of Freedom

Before solving, the model checks that the network is properly specified. For each source the number of fixed quantities (pressure, mass flow, or both) determines the degrees of freedom contributed to the system. A network with unconnected sources or insufficient boundary conditions will not converge.

Dynamic Mode

The Pipe Network takes part in a dynamic (time-domain) simulation. It is a quasi-steady participant: at every pressure-flow step of the integration the network is re-solved in steady state against the boundary conditions of that instant, using the solver and flow model configured for it. The network itself holds no inventory, so nothing accumulates in it between steps and the only memory it carries from one step to the next is its converged solution.

Validity

The quasi-steady treatment is valid whenever the dynamics of interest are slower than the transit time of the line, which covers everything driven by control: level and pressure loops, valve and choke movements, well shut-ins, ramping demand. It is not a transient hydraulic model. Line pack, surge and water hammer are not represented, and a case whose answer depends on them must be posed differently. A natural arrangement is therefore one where the inventory sits elsewhere on the flowsheet, in a vessel or a tank, and the network supplies the hydraulics: the line settles in seconds while the vessel takes tens of minutes, so the network is the algebraic part of the problem and the vessel holds the state.

Step Structure

DWSIM’s integrator separates each step into a pressure-flow part and an equilibrium part, and can run them at different rates. The network moves only on the pressure-flow part, because with no inventory there is nothing for the equilibrium part to relax. On a step where equilibrium is not scheduled the network still solves its hydraulics but skips the flash of its boundary streams.

Warm Start

Each step seeds the solver with the previous step’s node pressures, branch flows, temperatures and mixed compositions. This is what makes dynamic operation practical on compositional networks: a field case that needs minutes from a cold start settles a step in seconds once it is following its own solution.

State Between Attempts

An implicit or step-doubling integration solves the same interval more than once from the same starting point, restoring every object’s state in between. The network hands over the warm start, the commanded actuator targets, the queue of commands still inside their dead time, and the current actuator positions. Restoring all four is what makes the repeated attempts start from the same guess and the same geometry; without it the error estimate would be measuring the solver’s initialisation rather than the integration.

Because the network carries no inventory, it reports no contents to the adaptive integrator and therefore does not vote on the step-size error estimate. Step size is governed by the objects that do accumulate.

Actuator Dynamics

A controller writing a step change to a network block is a discontinuity: the solver is handed a network that jumped, the warm start no longer describes it, and the branch model may be pushed into a region where it offers no gradient to follow. Real final control elements cannot step either. The blocks whose manipulated quantity is normally driven by a controller therefore move towards a commanded value through a first-order lag with dead time.

Block	Actuated quantity
Valve	Opening
Choke	Bean diameter
Pump (ESP)	Operating frequency
Pressure Control Valve	Setting pressure
Gas Lift	Surface casing pressure

A command issued at time \(t\) becomes the target \(x_{\mathrm{sp}}\) after the dead time \(t_d\) has elapsed, and the actuator position \(x\) then advances over an integration step \(\Delta t\) by the exact solution of \(\tau\,\dot{x} = x_{\mathrm{sp}} - x\):

\[ x(t + \Delta t) = x(t) + \left[x_{\mathrm{sp}} - x(t)\right] \left(1 - e^{-\Delta t / \tau}\right) \]

Using the closed form rather than an explicit increment means the result does not depend on how the integrator chose to slice the interval, and a large step cannot overshoot the target. The time constant \(\tau\) and the dead time \(t_d\) are properties of each block and are saved with the flowsheet. Both default to zero, which reproduces instantaneous movement exactly, so a network built before actuators existed behaves as it always did. Neither has any effect on a steady-state solve.

Controlling the Network

Every block inside the network publishes its own properties to the flowsheet under a composite name of the form <block>: <property>. A PID controller placed on the flowsheet can therefore read a node pressure or a branch flow as its process variable and write a valve opening, a choke bean or a pressure control valve setting as its manipulated variable, without the controller needing to know that the two live inside a single unit operation.

When the controlled and manipulated quantities are in different units and of different magnitude, which is the usual case here (a flow in kg s\(^{-1}\) held by a pressure in Pa), the controller’s manipulated variable span should be set so that its output is scaled on the manipulated variable’s own scale about a bias, rather than about the setpoint of the controlled variable.

Specification Rules

Dynamic mode is stricter than steady state about boundary conditions, and the rules are checked before the first step rather than in the middle of a run. At least one boundary must fix a pressure: with every boundary on flow the network has no pressure level and the nodal system is singular, something a steady-state solve only ever closed by accident. The network writes the mass flow of its boundary streams, so those streams should carry a flow dynamics specification; a boundary stream left on a pressure specification competes with the network for the same variable and the answer becomes dependent on calculation order.

Settings

Setting	Unit	Default
Pressure-flow calculation rate	steps	1
Fail mode	—	Hold the last solution
Maximum solve time	s	0 (uncapped)
Diagram refresh rate	steps	1
Actuator time constant \(\tau\)	s	0
Actuator dead time \(t_d\)	s	0

The first four are properties of the network and are set on the Dynamics tab of its editor; the last two are properties of each actuated block. Raising the pressure-flow calculation rate re-solves the network only every \(N\) steps, which is worth doing when a large compositional network is too slow to solve at every step and its hydraulics are much faster than the loop being studied. The fail mode decides what happens when a step does not solve: holding the last solution and warning lets an integration survive a single bad step, whereas aborting stops the run at it. The maximum solve time is a wall-clock ceiling on one step, and the diagram refresh rate limits how often the open network editor is rebuilt during a run.

Running a pipe network in dynamic mode requires the higher subscription tier. Unlike the choice of solver, which degrades to a slower method, there is nothing to degrade to here.

Producing Wells and Nodal Analysis

On the nodal Newton solver with the compositional flow model, the Pipe Network doubles as a steady-state production-network and nodal-analysis tool. A producing well is assembled from an Inflow Performance Relationship (IPR) at the sandface, a tubing string of one or more multiphase pipe segments, and a wellhead Choke; manifolds and export flowlines then gather several wells into a common system.

Inflow Performance (IPR)

The IPR block sets the rate the reservoir delivers to the bottomhole as a function of the drawdown \(P_r - P_{wf}\), where \(P_r\) is the reservoir pressure and \(P_{wf}\) the flowing bottomhole pressure . Two forms are available: a linear productivity index,

\[ q = J\,(P_r - P_{wf}), \]

and Vogel’s solution-gas-drive relationship for saturated reservoirs,

\[ \frac{q}{q_{\max}} = 1 - 0.2\,\frac{P_{wf}}{P_r} - 0.8\left(\frac{P_{wf}}{P_r}\right)^{\!2}. \]

Placed between the reservoir boundary and the bottomhole node, the IPR closes the inflow–outflow (IPR \(\times\) VLP) analysis together with the tubing that lifts the fluid to surface.

Wellhead Choke

The choke is a subcritical restriction defined by a bean diameter and a discharge coefficient. For critical (sonic) flow it offers a mechanistic orifice model, which caps the rate at the critical pressure drop through the bean, and a Gilbert-type wellhead-choke correlation ,

\[ P_1 = \frac{A\,q_L\,\mathrm{GLR}^{B}}{d^{C}}, \]

where \(P_1\) is the upstream pressure, \(q_L\) the gross liquid rate, GLR the producing gas–liquid ratio and \(d\) the bean size, with selectable coefficient sets (Gilbert, Ros, Baxendell, Achong, or a tunable custom set). The GLR and liquid rate come from a standard-conditions flash of the produced fluid, so the Gilbert model applies on a compositional network; a dry-gas or dead-oil stream falls back to the mechanistic orifice.

Artificial Lift

A Pump block on the tubing represents an electrical submersible pump (ESP). Its optional free-gas derating models the head loss from free gas at the intake: the intake gas volume fraction (GVF) is read from a flash at the live intake pressure, and the developed head is scaled by a factor that falls linearly from unity (below a tunable onset GVF) to zero at a tunable gas-lock GVF . Because the intake pressure is taken live from the solve, the derating tracks the operating point. Gas-lift injection is modelled at a node, where the injected gas mixes and flashes with the produced fluid; production then follows the classic gas-lift performance curve, rising with injection to an optimum and falling as the added gas raises the friction gradient .

Flow Assurance Screens

Selecting a solved pipe segment and choosing Flow Assurance from the designer’s Tools menu screens the pipe against a set of integrity limits along its length. Every screen reads the segment’s converged hydraulic profile, so the results are consistent with the solved network; a fluid lacking the phase or data a screen needs simply reports no risk for that screen.

• Erosion: the API RP 14E erosional-velocity limit , \(V_e = C/\sqrt{\rho_{ns}}\), evaluated on the no-slip mixture density \(\rho_{ns} = \lambda_L\rho_L + (1-\lambda_L)\rho_g\); the plot draws the mixture velocity against the limit and flags any increment where the ratio reaches unity.

• Hydrate: the hydrate formation temperature at the pipe pressures, from the natural-gas-hydrate models (van der Waals–Platteeuw as implemented by Parrish and Prausnitz, Klauda–Sandler, and Chen–Guo)  , interpolated along the traverse; the flowing temperature is overlaid and the stretch that cools into the hydrate region is shaded. Requires water in the fluid.

• Wax: the wax appearance temperature (cloud point), taken as the highest temperature at which a solid wax phase first precipitates at the pipe pressure in the engine’s solid–liquid equilibrium flash ; increments where the flowing temperature drops below it are flagged. Requires compounds carrying fusion data (heavy paraffins).

• Asphaltene: an indicative de Boer stability screening  that classifies the oil (no problem, slight-to-moderate, or severe) from its stock-tank density and the supersaturation \(P_r - P_b\) at the inlet, and draws the rigorously computed bubble-point profile against the flowing pressure, flagging where the pipe crosses the bubble point. DWSIM has no first-principles asphaltene model, so the de Boer boundaries are an indicative screen to be tuned to field experience. Requires an oil with a bubble point.

• Liquid loading: for gas wells, the Turner droplet criterion , with the critical gas velocity

\[ V_c = \frac{C\,\sigma^{1/4}(\rho_L - \rho_g)^{1/4}}{\rho_g^{1/2}} \]

(SI constant \(C = 6.558\), the 20%-adjusted value; Coleman’s unadjusted value is 5.464) drawn against the superficial gas velocity; increments where the gas velocity falls below \(V_c\) are flagged as loading. Requires two-phase gas–liquid flow.

Scaling and Corrosion Analysis

Selecting a solved pipe segment and choosing Scaling & Corrosion from the Tools menu couples the electrolyte water chemistry to the flowline, a capability standard nodal-network tools lack. Increment by increment, from the in-situ pH and ionic speciation, it computes the \(\mathrm{CO_{2}}\)/\(\mathrm{H_{2}S}\)/\(\mathrm{O_{2}}\) corrosion rate (de Waard–Milliams with the NORSOK M-506 film factor, and NACE MR0175 sulfide-stress-cracking screening)  and the mineral scaling tendency, expressed as saturation indices for calcite, barite, gypsum, anhydrite, celestite, siderite and others,

\[ \mathrm{SI} = \log_{10}\!\left(\frac{\mathrm{IAP}}{K_{sp}}\right) > 0 \]

indicating supersaturation . The view plots the corrosion rate and the saturation indices against distance and shows the extension’s full report, including inhibitor dosing and API 570/579 remaining-life estimates . It requires the produced water to carry the brine ion compounds (the (ion) species of DWSIM’s electrolyte database) together with \(\mathrm{CO_{2}}\)/\(\mathrm{H_{2}S}\), and the Corrosion & Scaling extension to be installed.

Black Oil Fluids and PVT Calibration

For petroleum networks a fluid can be modelled with the Black Oil property package, which represents the produced fluid by its solution gas–oil ratio \(R_s\), oil formation volume factor \(B_o\), bubble point \(P_b\), and phase viscosities rather than a full compositional description. The correlations are Standing’s for \(R_s\), \(P_b\) and \(B_o\) , Beggs–Robinson for oil viscosity , and Dranchuk–Abou-Kassem for the gas compressibility factor .

Because those correlations are generic, the Black Oil compound creator offers a lab-PVT calibration step. Enter the measured PVT points (pressure, temperature, and any of \(R_s\), \(B_o\) and oil viscosity) together with a measured bubble point and the reservoir temperature, and Calibrate fits a correction multiplier for each quantity as the mean of the measured-to-correlated ratios. The four multipliers (default unity, meaning uncalibrated) are stored with the compound (BO_RsMult, BO_BoMult, BO_PbMult, BO_OilViscMult) and applied to \(R_s\), \(B_o\), \(P_b\) and oil viscosity at every evaluation and in the flash split, so a black-oil fluid can be matched to a lab report without leaving the black-oil model.

Analysis Tools and Plots

The designer’s Tools menu collects the analysis views. Besides Flow Assurance and Scaling & Corrosion it provides:

• Plot Profiles of Selected Pipes: orders the selected pipe segments head-to-tail by their shared nodes and concatenates their per-increment profiles by cumulative distance, giving a single traverse along a well string or flow path (pressure, temperature, holdup, phase velocities and more).

• Nodal Analysis Plot (IPR \(\times\) VLP): for a producing well, plots the inflow (IPR) and outflow (VLP) curves in flowing bottomhole pressure versus rate and marks the operating point at their intersection. The curves are built from the same branch models the solver uses , so the operating point matches the solved network.

• Field Report, Gas Lift Allocation and Field Target complete the production-analysis tools.

Flare and Relief Networks

A flare network is a pipe network whose sources are pressure relief devices and whose only pressure boundary is the atmosphere behind the flare tip. The nodal solver then returns the built-up backpressure at every device implicitly. Four blocks and one flow model serve this service.

Relief device

A specialised source (PSV, balanced-bellows or pilot-operated valve, rupture disk) that injects its relieving rate, composition (its associated stream) and temperature into the node at its outlet flange. From the solved node pressure it reports, per API 520 Part I , the total and built-up backpressure as a percentage of the gauge set pressure against the allowable of its type: the built-up backpressure of a conventional valve may not exceed its allowable overpressure (10 % at 10 % overpressure), the total backpressure of a balanced-bellows valve is held to 30 % (with the Figure 30 capacity correction \(K_b\) for the allowable overpressure up to 50 %) and that of a pilot-operated valve to 50 %. It also reports the flow regime (critical when the backpressure is below the critical-flow pressure), the effective area its rate needs (gas, liquid with the viscosity correction and the Figure 31 factor \(K_w\), or two-phase by the omega method), the API 526 orifice  that holds it, and the rated capacity of the installed orifice at that backpressure. In the Rated Capacity mode the injected rate is the installed orifice’s capacity; in the Sizing mode the rate is given and the orifice is reported.

Compressible-gas flow model

For gas headers the network runs the Compressible Gas flow model: every pipe is marched in increments with the momentum balance

\[ -\mathrm{d}p = \frac{f}{D}\,\frac{\rho v^2}{2}\,\mathrm{d}x + K_f\,\frac{\rho v^2}{2}\,\frac{\mathrm{d}x}{L} + G^2\,\mathrm{d}\!\left(\frac{1}{\rho}\right) + \rho g\,\mathrm{d}z \]

where \(G\) is the mass flux, \(f\) the Churchill friction factor and \(K_f\) the fittings resistance spread along the length; the gas state follows the isothermal path by default, which API 521 recommends for discharge piping as the slightly more conservative one, or the adiabatic (Fanno) path. Each pipe reports its maximum and outlet Mach number against a per-pipe limit (0.5 in headers, 0.7 in laterals for short peak loads ), the sonic exit pressure when it chokes, the minimum temperature along it and the friction, acceleration and static parts of its pressure drop. A choked pipe cannot pass more than its sonic rate, which the solver enforces through a smooth barrier so the network still converges; two-phase relief keeps the compositional model.

Knockout drum, seal drum and flare tip

The knockout drum is the inline separator with the API 521 sizing check: the allowable vapour velocity from the settling of the design droplet (300 to 600 \(\mu\)m) against the actual velocity, and the holdup volume the scenario’s liquid needs against the one available. The seal drum is a one-way branch whose pressure drop is the seal head \(\rho_s g h\) plus a velocity-head loss. The flare tip drops the exit velocity head at the exit density, caps the rate at the sonic value of the tip and checks the exit Mach number; it also computes the heat release from the mixture’s lower heating value, the flame length \(L = 0.00326\,Q^{0.478}\), the flame bent by the wind per the API 521 distortion chart (its horizontal and vertical extents against the ratio of the wind to the exit velocity, the flame centre at half of each), the radiant heat at a target

\[ K = \frac{\tau F Q}{4 \pi D^2} \]

with the API 521 transmissivity \(\tau\) and fraction radiated \(F\), the stack height that brings \(K\) down to a limit (4.73 kW m\(^{-2}\) where emergency actions of a few minutes are needed, 1.58 kW m\(^{-2}\) for continuous exposure), and the sound pressure level at 30 m, \(L_{30} = L(\mathrm{PR}) + 10\log_{10}(\tfrac{1}{2} m c^2)\), attenuated by \(20\log_{10}(r/30)\) to the target .

Relief scenarios and load cases

A scenario is one contingency: the devices that relieve together and their rates; a device the scenario does not name stays shut. Relief Scenarios on the Tools menu keeps the device-by-scenario table, runs every scenario in turn (each solve warm-starting from the last) and names the governing scenario per device (the largest margin over the allowable backpressure), per pipe (the highest Mach number) and overall; the network is left solved under the governing one, and the Active Relief Scenario property selects the scenario every later solve imposes. Relief Load Cases computes the rates from the flowsheet by the API 521 methods: fire on a wetted vessel (\(Q = C F A^{0.82}\) on the wetted area within the fire zone, \(W = Q/\lambda\) at the relieving pressure), fire on a gas-filled vessel (wall-temperature form), blocked outlet behind a pump (head curve) or a wide-open control valve (ISA gas, liquid and two-phase forms), control valve fail-open and gas blow-by, loss of cooling and abnormal heat input (\(W = Q/\lambda\)), thermal expansion of a blocked-in liquid (\(W = \alpha_v Q/c_p\)), exchanger tube rupture (twice the tube area: choked or subsonic gas, orifice liquid, omega two-phase) and composite cases that sum others. The protected flowsheet object supplies the geometry, duty, curve and pressures; a case written into a scenario is recomputed on every batch run. The Flare Network Report collects the device, pipe, equipment and radiation tables of all scenarios.

Design

Size Flare Network grows the pipes marked for sizing through the ASME B36.10 nominal sizes  (standard or extra-strong wall), running every scenario on each pass, plus one scenario per device alone at the rated capacity of its installed orifice, since API 520 Part II (Section 5.3)  sizes the laterals and tailpipes of a device on its rated capacity and the common headers on the required capacities, until no pipe exceeds its Mach limit or chokes and every device is within its allowable backpressure: a Mach violation jumps the bore by \(\sqrt{\mathrm{Ma}/\mathrm{Ma}_{\max}}\), a backpressure violation grows the largest-drop pipe on the device’s path to the flare. Devices in the Sizing mode receive the API 526 orifice their governing scenario needs. The drum, seal and tip findings stay in the report for the engineer.

Flare stack sizing tool

Size Flare Stack on the Tools menu takes the selected flare tip (its gas from the last solve) or typed inputs and works through Section 4.4.2 and 5.4.3 of API RP 521 : the tip diameter for a design Mach number (5.4.3.1.1, equation 24; 0.2 for normal flows, 0.5 for a short peak), the tip pressure drop, the heat release and flame length (Figure 9), the flame bent by the wind (Figure 10), the stack height that holds a radiation limit at a target (equation 20 with the C-2 transmissivity, solar radiation added), and the radial profiles downwind and upwind of the stack of the radiant heat, of the equilibrium temperature of an exposed surface (the heat balance of 4.4.2.3.2, absorbed radiation against convection and re-radiation), of the exposure time to the pain threshold (Table 7) and of the noise (5.4.4.3). It also marches the gas up the riser, isothermal or adiabatic with friction, fittings and gravity, and reports the pressure, temperature, velocity and Mach profile and the base pressure the stack puts on the header. The sized tip and stack can be written to the block.

Knockout drum sizing tool

Size Knockout Drum follows 5.4.2.1: the dropout velocity of the design droplet from the drag coefficient (Figure 20, equations 30 and 32), a vertical drum by the vapour area and the liquid holdup (equations 46 to 49), and the horizontal drum by the trial table of the standard (Tables 13 and 14): for each diameter the slop and holdup liquid segments, the vapour space left, the dropout time across it, the vapour velocity of one or two passes and the minimum cylindrical length; the smallest shell is picked and any feasible trial can be written to the block.

Validation against the API worked examples

The relief-network calculations were checked against the worked examples of API RP 520 Part I  and API RP 521 , run through the same code the blocks use. Table 7 lists the cases, the section of the standard each one comes from, the value the standard gives and the value DWSIM returns.

Case	Reference	Standard	DWSIM
Gas relief area, critical flow (24 260 kg/h, \(M\) = 65, 348 K, 670 kPa, \(Z\) = 0.84, \(k\) = 1.09)	API 520 Pt I, 3.6.2.2	3179 mm\(^2\)	3185 mm\(^2\)
Gas relief area, subcritical flow (back pressure 532 kPa)	API 520 Pt I, 3.6.3.2	3610 mm\(^2\)	3644 mm\(^2\)
Critical flow pressure of the same case	API 520 Pt I, Table 7	395 kPa	393 kPa
Liquid relief area (6814 L/min, \(G\) = 0.90, \(K_d\) = 0.65, \(K_w\) = 0.97)	API 520 Pt I, 3.8.2	3066 mm\(^2\)	3067 mm\(^2\)
Same, with the viscosity correction \(K_v\) = 0.964	API 520 Pt I, 3.8.2.5	3180 mm\(^2\)	3188 mm\(^2\)
Balanced-bellows \(K_b\), 10 % overpressure, 40 % of set	API 520 Pt I, Fig. 30	0.85	0.85
Balanced-bellows \(K_w\) (liquid), 50 % of set	API 520 Pt I, Fig. 31	0.67	0.67
Flame length at \(Q\) = 6.3 \(\times\) 10\(^8\) W	API 521, Fig. 9	52 m	52.4 m
Flame distortion, \(U_\infty/U_j\) = 0.156 (\(\Sigma\Delta x\), \(\Sigma\Delta y\))	API 521, C.2.4, Fig. 10	44.2 m, 18.2 m	44.3 m, 18.1 m
Flame distortion, \(U_\infty/U_j\) = 0.062	API 521, C.2.4, Fig. 10	37.4 m, 27.6 m	37.5 m, 27.5 m
Stack height for 6.3 kW/m\(^2\) at 45.7 m, tip Mach 0.2	API 521, C.2.5	33.7 m	33.7 m
Stack height for 6.3 kW/m\(^2\) at 45.7 m, tip Mach 0.5	API 521, C.2.5	27 m	27.0 m
Distance from the flame centre for 9.5 kW/m\(^2\) (\(Q\) = 6.3 \(\times\) 10\(^6\) kW, \(F\) = 0.3)	API 521, C.3.4	126 m	125.8 m
Transmissivity, 50 % humidity, 30.5 m	API 521, eq. C-2	0.825	0.825
Tip diameter for Mach 0.2 and 0.5 (45 455 kg/h, \(M\) = 46.1, 422 K)	API 521, C.2.2	0.46 m, 0.29 m	0.457 m, 0.289 m
Knockout drum dropout velocity (300 \(\mu\)m, 496.6 / 2.9 kg/m\(^3\), 0.01 cP)	API 521, 5.4.2.1	0.71 m/s	0.69 m/s
Horizontal drum trial, \(D\) = 2.44 m: vapour area, vapour velocity, minimum length	API 521, Table 14	1.89 m\(^2\), 3.9 m/s, 5.6 m	1.89 m\(^2\), 3.88 m/s, 5.83 m
Vertical drum diameter for the same case	API 521, eq. 49	3.6 m	3.68 m
Flare noise at 30 m (14.6 kg/s, \(c\) = 353 m/s, PR = 3)	API 521, 5.4.4.3	114 dB	113.6 dB

Flare network calculations against the API RP 520 / 521 worked examples

The gas subcritical area and the dropout velocity differ from the standard by the reading of a chart: the standard reads \(F_2\) = 0.86 from Figure 34 and the drag coefficient \(C\) = 1.3 from Figure 20, where DWSIM evaluates the \(F_2\) expression and the standard sphere drag curve. The other differences come from rounding in the standard’s arithmetic. The relief loads follow API 521 Section 3: fire on a wetted vessel by equations 3 and 4 with the Table 5 environment factors (3.15.2.1), fire on a gas-filled vessel by equation 8 (3.15.2.1.2), the 50 Btu/lb minimum latent heat near the critical point (3.15.3.1), tube rupture as two orifices (3.18.3), and power failure as the sum of the affected loads (3.17). The discharge piping follows Section 5.4.1.3: isothermal flow by default (5.4.1.3.2), the critical pressure at a pipe outlet (equations 25 and 26), the Table 11 fitting resistances entered as \(K_f\), the built-up backpressure limit of about 10 % of set for conventional valves and 30 to 50 % for balanced valves (5.4.1.3.1), and the tip velocity of up to Mach 0.5 for peak flows (5.4.3.1.1). The seal depth follows equation 51 (5.4.2.2) and the drum check the dropout-velocity procedure of 5.4.2.1.

Model Parameters

Solver Settings

Parameter	Symbol	Unit	Default
Solver method	—	—	Nodal Newton
Maximum iterations	\(N_{\max}\)	—	1000
Convergence tolerance	\(\epsilon\)	—	\(10^{-4}\)
Richardson extrapolation	—	—	Off

Pipe Segment Parameters

Parameter	Symbol	Unit	Description
Internal diameter	\(D\)	m	Pipe bore
Length	\(L\)	m	Segment length
Wall roughness	\(\varepsilon\)	m	Absolute roughness
Inclination angle	\(\theta\)	\(^\circ\)	Angle from horizontal
Ambient temperature	\(T_{\mathrm{amb}}\)	K	Surrounding temperature
Overall HTC	\(U_o\)	W m\(^{-2}\) K\(^{-1}\)	Based on outer diameter
Increments per section	—	—	Cells the segment is walked in
Pressure-drop model	—	—	BB / LM / PA
Joule-Thomson correction	—	—	On / Off
Emulsion correction	—	—	On / Off
Max. segment iterations	—	—	Per-pipe convergence limit

Abbreviations:

BB = Beggs & Brill; LM = Lockhart & Martinelli; PA = Petalas & Aziz; HTC = heat-transfer coefficient.

Results Reported

For each network object the solver reports:

Quantity	Symbol	Unit
Mass flow rate	\(\dot{m}\)	kg s\(^{-1}\)
Molar flow rate	\(\dot{n}\)	mol s\(^{-1}\)
Volumetric flow rate	\(\dot{V}\)	m\(^3\) s\(^{-1}\)
Pressure (in/out/avg)	\(P\)	Pa
Temperature (in/out/avg)	\(T\)	K
Pressure drop	\(\Delta P\)	Pa
Temperature change	\(\Delta T\)	K

For each node, the solver also reports the dimensionless mass, pressure, and energy balance residuals (\(r_m\), \(r_P\), \(r_E\)) as convergence indicators. A converged network solution has all three residuals below the specified tolerance \(\epsilon\).

Assumptions and Limitations

1. Steady state, or quasi-steady: the model does not resolve transient behaviour such as surge, water hammer, or slug initiation. All flows and pressures represent time-averaged steady-state conditions. In a dynamic simulation the network still solves in steady state at each step, against that instant’s boundary conditions (2.31.7); what changes with time are the boundaries and the actuator positions, not the state of the fluid in the line.

2. One-dimensional flow: each pipe segment is treated as a 1-D plug-flow element. Radial temperature and concentration gradients within the pipe cross-section are neglected.

3. Homogeneous mixture in pipes: unless a rigorous two-phase correlation is selected, the two phases are treated as a homogeneous mixture for property evaluation. Slip between phases is captured by the holdup correlations in the Beggs–Brill and Lockhart–Martinelli methods.

4. No condensation or vaporisation along pipes by default: phase change within a pipe is accounted for only when the per-segment equilibrium flash option is enabled. Without it, the overall stream composition entering each segment is assumed constant.

5. Instantaneous mixing at nodes: streams mixing at a junction node are assumed to reach thermodynamic equilibrium instantaneously. Phase separation at nodes is not modelled; use a Separator object for this purpose.

6. Adiabatic pump/compressor/valve by default: thermal effects in pump, compressor, and valve elements follow the standard DWSIM base unit operation assumptions.

7. Single composition throughout: the network does not currently support reactions. Composition changes arise only from phase equilibrium at separator or equilibrium-flash-enabled pipe objects.

8. Spatial discretisation: the increment walk along a pipe is first order, so the number of increments a segment is divided into is an accuracy setting and not only a reporting resolution. On multiphase flow the computed rate can move by a few percent between a coarse discretisation and a converged one, which on such a network is a larger error than any of the numerical tolerances. Refine a segment until the answer stops moving, or enable the Richardson extrapolation described in 2.31.6. A single-phase liquid line is unaffected, its gradient being essentially constant along the pipe.

9. Pressure-drop correlation range: the empirical correlations (Beggs–Brill, Lockhart–Martinelli) were developed from data sets at specific pressure, velocity, and fluid-property ranges. Extrapolation beyond these ranges may reduce accuracy. The Petalas–Aziz mechanistic model generally has wider applicability.

Numerical Solution Procedure

1. Parse the network topology: identify all objects, connections, sources, sinks, and nodes.

2. Check degrees of freedom: count independent boundary conditions against unknown pressures and flow rates.

3. Initialise all internal stream conditions by propagating boundary values from sources through the network.

4. Formulate the decision variable vector \(\mathbf{x}\) using log-transformed source pressures and/or flow rates ([eq:log_transform]).

5. At each solver iteration:

   1. Recover physical values \(v_i = s_i\,e^{x_i}\).

   2. Assign recovered values to the corresponding source streams.

   3. Calculate all network blocks sequentially: pipes, pumps, compressors, valves, separators.

   4. Assemble node residuals \(r_{m,k}\), \(r_{P,k}\), \(r_{E,k}\) ([eq:mass_balance]–[eq:energy_balance]).

   5. Evaluate the objective function \(F(\mathbf{x})\) ([eq:objective]).

6. Continue until \(F < \epsilon^2\) or the maximum iteration count \(N_{\max}\) is reached.

7. Update all network object results and report convergence status.

Typical Usage Workflow

1. Open a new or existing DWSIM flowsheet and add the Pipe Network block from the unit operations palette.

2. Double-click the block to open the network canvas editor.

3. Drag Source and Sink objects onto the canvas and specify their boundary conditions (pressure, flow rate, and composition). Stream composition is inherited from connected DWSIM material streams.

4. Add Pipe, Node, Pump, and other objects as needed to build the network topology. Connect objects by drawing links between inlet and outlet ports.

5. For each pipe segment, specify the geometry (diameter, length, roughness, angle), thermal conditions (ambient temperature, overall HTC), and preferred pressure-drop correlation.

6. Select the solver method and convergence settings on the Solver tab.

7. Return to the main flowsheet and run the simulation. The solver iterates until the objective function falls below the convergence tolerance or the iteration limit is reached.

8. Inspect results by opening the network editor: each object displays its pressure, temperature, and flow results, and nodes show their balance residuals as convergence indicators.

9. To carry the converged network into a dynamic run, set the boundary streams to a flow specification, give the actuated blocks a time constant on their editors, review the Dynamics tab of the network editor, and drive it from the flowsheet’s Dynamics Manager (2.31.7).

Restriction Orifice

Overview

The Restriction Orifice unit operation models a concentric sharp-edged orifice plate installed in a gas pipeline. The block serves two purposes simultaneously: it acts as a flow-restriction element by imposing a user-specified permanent pressure drop on the portion of the stream that passes through the orifice, and as a flow splitter by routing the remaining fraction of the inlet flow to a bypass outlet at the original inlet pressure.

The discharge coefficient follows the ISO 5167-2 standard for concentric orifice plates with corner taps . A compressibility expansion factor \(Y\) accounts for gas density changes through the orifice. Two operating modes are available: sizing mode (calculate the orifice diameter for a target flow rate) and operation mode (calculate the actual flow rate through a known orifice).

Stream Topology

Port	Direction	Description
Inlet Port 1	Inlet (material)	Upstream gas stream at \(P_1\), \(T_1\)
Outlet Port 1	Outlet (material)	Through-orifice stream at \(P_2 = P_1 - \Delta P_{\mathrm{perm}}\)
Outlet Port 2	Outlet (material)	Bypass stream at \(P_1\)

The inlet stream must be 100 % vapour phase. Both outlet streams inherit the inlet composition and temperature; only the mass flow and pressure of Outlet 1 are modified.

Calculation Modes

1. Sizing mode (SizingMode = True): the user specifies the pipe internal diameter \(D\), the permanent pressure drop \(\Delta P_{\mathrm{perm}}\), and the target mass flow rate \(\dot{m}_{\mathrm{spec}}\); the model iterates to find the required orifice diameter \(d_o\).

2. Operation mode (SizingMode = False): the user specifies \(D\), \(\Delta P_{\mathrm{perm}}\), and the orifice diameter \(d_o\); the model iterates to find the actual volumetric flow rate \(Q\) (and hence the mass flow \(\dot{m}\)) through the orifice.

Orifice Geometry

The diameter ratio (beta ratio) is

\[ \beta = \frac{d_o}{D} \]

and the orifice cross-sectional area is

\[ A_o = \frac{\pi d_o^2}{4} \]

Pressure Relations

The user specifies the permanent (irrecoverable) pressure drop \(\Delta P_{\mathrm{perm}}\). The differential pressure across the orifice taps, \(\Delta P_o\), is related to the permanent loss by

\[ \Delta P_o = \frac{\Delta P_{\mathrm{perm}}}{1 - \beta^2} \]

The downstream pressure at Outlet 1 is

\[ P_2 = P_1 - \Delta P_{\mathrm{perm}} \]

and the pressure ratio used in the expansion factor calculation is

\[ \Pi = \frac{P_2}{P_1} \]

Orifice Reynolds Number

The Reynolds number is based on the orifice diameter and the local flow conditions:

\[ Re = \frac{\dot{m}\,d_o}{A_o\,\mu} \]

where \(\mu\) (Pa s) is the dynamic viscosity of the gas phase evaluated at the inlet conditions.

Discharge Coefficient (ISO 5167-2, Corner Taps)

The discharge coefficient \(C\) is computed from the ISO 5167-2 correlation for corner taps (\(L_1 = L_2 = 0\)) . The auxiliary roughness parameter is first evaluated:

\[ A_1 = \left(\frac{19000\,\beta}{Re}\right)^{0.8} \]

The three additive components of \(C\) are:

Base term \(c_1\)

\[ c_1 = 0.5961 + 0.0261\beta^2 - 0.216\beta^8 + 0.000521\!\left(\frac{10^6\beta}{Re}\right)^{\!0.7} + \left(0.0188 + 0.0063 A_1\right)\beta^{3.5} \left(\frac{10^6}{Re}\right)^{\!0.3} \]

Corner-tap pressure-recovery term \(c_2\)

For corner taps the tap-position coefficients in ISO 5167-2 evaluate to zero, giving

\[ c_2 = (0.043 + 0.080 - 0.123)(1 - 0.11 A_1) \frac{\beta^4}{1-\beta^4} = 0 \]

Small-orifice correction \(c_3\)

\[ c_3 = \begin{cases} 0.011(0.75 - \beta) \!\left(2.8 - \dfrac{d_o\,[\text{mm}]}{25.4}\right) & d_o < 71.12\ \text{mm} \\[8pt] 0 & d_o \geq 71.12\ \text{mm} \end{cases} \]

Combined discharge coefficient

\[ C = c_1 + c_2 + c_3 = c_1 + c_3 \]

Typical values lie in the range \(0.60 < C < 0.75\) for \(0.10 < \beta < 0.75\) and \(Re > 5\times10^3\).

Expansion Factor (ISO 5167-2)

The expansion factor \(Y\) corrects for the reduction in gas density as it accelerates through the orifice . Two sub-correlations are used depending on the pressure ratio \(\Pi\):

Subsonic flow (\(\Pi \geq 0.63\))

\[ Y = 1 - \frac{1-\Pi}{\kappa}\left(0.41 + 0.35\beta^4\right) \]

High pressure drop (\(\Pi < 0.63\))

\[ Y = 1 - \frac{0.4604}{\kappa} - \frac{0.413}{\kappa}\beta^4 + \left(0.49 + 0.45\beta^4\right)\frac{\Pi}{\kappa} \]

In both equations \(\kappa = C_p/C_v\) is the isentropic exponent of the gas evaluated at the inlet conditions. For an incompressible fluid \(Y \to 1\).

Fundamental Flow Equation

Combining all factors, the volumetric flow rate through the orifice is

\[ Q = A_o\,C\,Y\,\sqrt{\frac{2\,\Delta P_o}{\rho_1}} \]

and the corresponding mass flow rate is

\[ \dot{m} = \rho_1\,Q \]

where \(\rho_1\) (kg m\(^{-3}\)) is the gas-phase density at the inlet.

Iterative Solution Procedure

Because \(C\) depends on \(Re\) and \(Re\) depends on \(\dot{m}\) (or \(d_o\)), the system is solved by fixed-point iteration.

Sizing Mode

Rearranging [eq:ro_Q] for \(d_o\) yields the iteration update:

\[ d_o^{(k+1)} = \left[ \frac{4Q_{\mathrm{spec}}}{\pi\,C^{(k)}\,Y^{(k)}} \sqrt{\frac{\rho_1}{2\,\Delta P_o^{(k)}}} \right]^{1/2} \]

where \(Q_{\mathrm{spec}} = \dot{m}_{\mathrm{spec}}/\rho_1\).

Operation Mode

Substituting the fixed \(d_o\) directly into [eq:ro_Q] gives the update:

\[ Q^{(k+1)} = A_o\,C^{(k)}\,Y^{(k)} \sqrt{\frac{2\,\Delta P_o}{\rho_1}} \]

Convergence

Both modes iterate until the absolute change in the primary variable falls below \(10^{-8}\) (m for \(d_o\); m\(^3\) s\(^{-1}\) for \(Q\)):

\[ \left|x^{(k+1)} - x^{(k)}\right| < 10^{-8} \]

The iteration is capped at 100 steps; convergence is typically achieved in 5–15 iterations.

Outlet Stream Assignment

After convergence, the two outlet streams are set as follows:

Property	Outlet 1 (through orifice)	Outlet 2 (bypass)
Mass flow	\(\dot{m}\)	\(\dot{m}_{\mathrm{in}} - \dot{m}\)
Pressure	\(P_2 = P_1 - \Delta P_{\mathrm{perm}}\)	\(P_1\)
Temperature	\(T_1\) (adiabatic)	\(T_1\)
Composition	Same as inlet	Same as inlet
Flash spec	Pressure & enthalpy	—

A physical constraint enforces \(\dot{m} \le \dot{m}_{\mathrm{in}}\); an exception is raised if the calculated or specified flow rate exceeds the inlet.

Model Parameters

Parameter	Symbol	Unit	Default
Pipe internal diameter	\(D\)	m	0.0254
Permanent pressure drop	\(\Delta P_{\mathrm{perm}}\)	Pa	0
Orifice diameter	\(d_o\)	m	0.005
Target mass flow (sizing)	\(\dot{m}_{\mathrm{spec}}\)	kg s\(^{-1}\)	0
Sizing mode	—	—	True

Results Reported

Quantity	Symbol	Unit
Beta ratio	\(\beta\)	—
Orifice area	\(A_o\)	m\(^2\)
Reynolds number	\(Re\)	—
Discharge coefficient	\(C\)	—
Expansion factor	\(Y\)	—
Orifice pressure drop	\(\Delta P_o\)	Pa
Pressure ratio	\(\Pi = P_2/P_1\)	—
Orifice diameter (sizing)	\(d_o\)	m
Mass flow rate (operation)	\(\dot{m}\)	kg s\(^{-1}\)

Assumptions and Limitations

1. Gas phase only: the inlet stream must be 100 % vapour. Liquid or two-phase streams are rejected with an error message.

2. Corner taps: the discharge coefficient is evaluated for corner-tap geometry (\(L_1 = L_2 = 0\)). The pressure-recovery term \(c_2\) vanishes identically for this configuration ([eq:ro_c2]).

3. ISO 5167 applicability range: the correlation is validated for \(0.10 < \beta < 0.75\) and \(Re > 4000\) (turbulent flow). Results outside these ranges are unreliable.

4. Subsonic flow: no choked-flow (sonic) limit is modelled. The expansion factor correlations assume \(\Pi > 0\); for very high pressure drops (\(\Pi \to 0\)) the model may not converge or may produce non-physical results.

5. Adiabatic, no Joule-Thomson correction: the outlet temperature equals the inlet temperature. The isentropic temperature drop through the orifice is not computed.

6. Constant inlet properties: density \(\rho_1\), viscosity \(\mu\), and the isentropic exponent \(\kappa\) are evaluated once at the inlet conditions and held fixed during the iteration.

7. No fin or discharge length effects: the model is strictly applicable to a thin, sharp-edged plate. Nozzles, venturi elements, or long-bore orifices require different coefficients.

8. Horizontal pipe assumed: no gravitational head correction is applied.

9. Premium requirement: the calculation routine requires an active DWSIM Premium Supporter subscription.

Typical Usage Workflow

1. Connect a gas material stream to Inlet Port 1 and two material streams to Outlet Port 1 (restricted flow) and Outlet Port 2 (bypass).

2. On the Parameters tab, enter the pipe internal diameter and the permanent pressure drop.

3. Sizing mode: enable Sizing Mode, enter the desired mass flow rate, and run the simulation. Read the calculated orifice diameter from the results.

4. Operation mode: disable Sizing Mode, enter the orifice diameter from the design step, and run the simulation. Read the actual flow through the orifice and the bypass flow from the outlet streams.

5. Verify that \(\beta\) lies within 0.10–0.75 and \(Re > 4000\) in the results panel; adjust geometry if necessary.

Advanced Heat Exchanger

Overview

The Advanced Heat Exchanger is a rigorous shell-and-tube heat exchanger model that performs incremental (zone-by-zone) integration of heat transfer and pressure drop along the exchanger length. Unlike the standard DWSIM heat exchanger, this unit operation evaluates local thermophysical properties at each integration increment, handles phase changes (condensation and vaporisation) on both the shell and tube sides, and uses the full Bell–Delaware method  for shell-side thermal–hydraulic calculations.

Four calculation modes are available:

• Rating: Compute the heat duty, outlet temperatures, and pressure drops from a fully specified geometry.

• Design: Size the exchanger (number of tubes and shell diameter) to meet a specified outlet temperature.

• Simulation: Use a user-supplied overall coefficient \(U\) to compute the duty and outlet temperatures.

• Fouling Factor: Back-calculate the overall fouling resistance from known inlet and outlet temperatures.

Stream Topology

Port	Direction	Description
Hot Side In	Inlet (material)	Hot-side inlet stream
Cold Side In	Inlet (material)	Cold-side inlet stream
Hot Side Out	Outlet (material)	Hot-side outlet stream
Cold Side Out	Outlet (material)	Cold-side outlet stream

Either fluid may be assigned to the shell side or the tube side via the Shell_Fluid and Tube_Fluid settings.

Geometry

The exchanger geometry is defined by the STHXPropertiesAdvanced data class. The principal user inputs are:

Shell parameters

Shell inside diameter \(D_s\) (mm), number of shell passes, number of shells in series, TEMA shell type designation, baffle type (single-segmental, double-segmental, or no-tubes-in-window), baffle cut \(B_c\) (%), central baffle spacing \(L_b\) (m), inlet and outlet baffle spacings \(L_{bi}\) and \(L_{bo}\) (m), shell-to-baffle diametral clearance \(D_{sb}\) (mm), number of sealing strip pairs \(N_{ss}\), and shell-side fouling factor \(R_{f,s}\) (m\(^2\) K/W).

Tube parameters

Tube outside diameter \(d_o\) (mm), inside diameter \(d_i\) (mm), tube length \(L_t\) (m), number of tubes \(N_t\), tube passes \(N_{tp}\), tube pitch \(P_t\) (mm), tube layout pattern (30° triangular, 90° square, 60° rotated triangular, or 45° rotated square), tube wall thermal conductivity \(k_w\) (W/m K), inside roughness \(\varepsilon\) (mm), tube-to-baffle clearance \(d_{tb}\) (mm), and tube-side fouling factor \(R_{f,t}\) (m\(^2\) K/W).

Nozzle parameters

Inside diameters of the shell-side inlet/outlet and tube-side inlet/outlet nozzles (mm).

Derived quantities

The following intermediate quantities are computed once from the user inputs :

\[ \begin{align} D_\mathit{otl} &= D_s - D_{sb}, \\ D_\mathit{ctl} &= D_\mathit{otl} - d_o, \\ N_b &= \left\lfloor\frac{L_t - L_{bi} - L_{bo}}{L_b}\right\rfloor + 1, \\ S_m &= L_b\left(D_s - D_\mathit{otl} + D_\mathit{ctl}\,\frac{P_t' - d_o}{P_t'}\right), \\ A_\mathit{sb} &= \pi\,D_s\,\frac{D_{sb}}{2}\,(1 - B_c), \\ A_\mathit{tb} &= \frac{\pi}{4}\bigl[(d_o + d_{tb})^2 - d_o^2\bigr]\,N_t\,(1-F_w), \end{align} \]

where \(P_t'\) is the effective pitch (adjusted for rotated layouts), \(F_w\) is the fraction of tubes in the window zone, and \(S_m\) is the cross-flow area at the bundle centreline.

The total outside heat-transfer area is:

\[ A = N_t\,\pi\,d_o\,L_t. \]

Incremental Integration (Rating Mode)

The exchanger is divided into \(N\) equal-area increments. At each increment \(k\), local thermophysical properties (density, viscosity, thermal conductivity, heat capacity, vapor fraction) are evaluated by performing a pressure–temperature flash at the local conditions. The local heat duty is:

\[ \delta Q_k = U_k\,\delta A\,\Delta T_k, \]

where \(U_k\) is the local overall coefficient, \(\delta A = A/N\) is the incremental area, and \(\Delta T_k = T_{h,k} - T_{c,k}\) is the local temperature difference.

After each increment, the new enthalpies are:

\[ \begin{align} H_{h,k+1} &= H_{h,k} - \frac{\delta Q_k}{\dot{m}_h}, \\ H_{c,k+1} &= H_{c,k} + \frac{\delta Q_k}{\dot{m}_c}, \end{align} \]

and the local temperatures are recovered by a pressure–enthalpy flash. The iteration is repeated until the total duty \(Q = \sum_k \delta Q_k\) converges.

Shell-Side Heat Transfer (Bell–Delaware Method)

Ideal cross-flow coefficient

The ideal (un-corrected) shell-side heat-transfer coefficient is computed from the Taborek \(j\)-factor correlation :

\[ h_\mathit{id} = j_i\,\frac{c_p\,G_s}{\Pr^{2/3}} \left(\frac{\mu}{\mu_w}\right)^{0.14}, \]

where \(G_s = \dot{m}_s / S_m\) is the shell-side mass velocity, \(\Pr\) is the Prandtl number, \(\mu/\mu_w\) is the viscosity correction, and the \(j\)-factor is:

\[ j_i = a_1\left(\frac{1.33}{P_t/d_o}\right)^{a}\,\Re_s^{a_2}, \qquad a = \frac{a_3}{1 + 0.14\,\Re_s^{a_4}}, \]

with constants \(a_1\)–\(a_4\) tabulated by Taborek for each tube layout and Reynolds number range (\(\Re_s = d_o\,G_s/\mu\)).

Correction factors

The actual shell-side coefficient is:

\[ h_s = h_\mathit{id}\,J_c\,J_l\,J_b\,J_s\,J_r, \]

where the five Bell–Delaware correction factors account for the following effects:

Factor	Description
\(J_c = 0.55 + 0.72\,F_c\)	Baffle-cut correction; \(F_c\) is the fraction of tubes in the cross-flow zone.
\(J_l\)	Baffle-leakage correction for shell–baffle (\(A_{sb}\)) and tube–baffle (\(A_{tb}\)) clearance streams.
\(J_b = \exp\!\bigl[-C_{bh}\,F_{sbp}\,(1 - (2N_{ss}/N_{cl})^{1/3})\bigr]\)	Bundle-bypass correction; \(C_{bh} = 1.35\) (laminar) or \(1.25\) (turbulent); \(F_{sbp}\) is the bypass fraction; \(N_{ss}\) is the number of sealing strip pairs.
\(J_s\)	Unequal baffle-spacing correction for inlet/outlet vs. central spacings.
\(J_r\)	Laminar-flow correction (\(\Re_s < 100\)) for adverse temperature gradients.

Tube-Side Heat Transfer

The single-phase tube-side coefficient is computed from the Gnielinski correlation  in the turbulent regime (\(\Re \geq 10\,000\)), the Sieder–Tate correlation  in the laminar regime (\(\Re \leq 2300\)), and a linear interpolation in the transition zone:

Gnielinski (turbulent)

\[ \Nu = \frac{(f/8)\,(\Re - 1000)\,\Pr} {1 + 12.7\,\sqrt{f/8}\,(\Pr^{2/3} - 1)} \left(\frac{\mu}{\mu_w}\right)^{0.14}, \]

where \(f = (0.79\ln\Re - 1.64)^{-2}\) is the Petukhov friction factor.

Sieder–Tate (laminar)

\[ \Nu = \max\!\left(3.66,\; 1.86\,\Gz^{1/3}\left(\frac{\mu}{\mu_w}\right)^{0.14}\right), \]

where \(\Gz = \Re\,\Pr\,(d_i/L_t)\) is the Graetz number.

Two-Phase Correlations

Flow regime detection

At each integration increment the local vapor fractions on both sides are compared with the previous increment to classify the flow regime as single-phase liquid, single-phase vapor, condensing, vaporising, or two-phase (quality unchanged). The appropriate two-phase correlation is then selected.

Lockhart–Martinelli parameter

The turbulent–turbulent Martinelli parameter is used by several correlations:

\[ X_\mathit{tt} = \left(\frac{1-x}{x}\right)^{0.9} \left(\frac{\rho_V}{\rho_L}\right)^{0.5} \left(\frac{\mu_L}{\mu_V}\right)^{0.1}, \]

where \(x\) is the local vapor quality.

Shell-side condensation

The shell-side condensation coefficient is taken as the maximum of a gravity-controlled (Nusselt film) contribution and a shear-controlled (McNaught/Boyko–Kruzhilin) contribution :

\[ \begin{align} h_\mathrm{grav} &= 0.725\left[\frac{\rho_L(\rho_L - \rho_V)\,g\,h_{fg}\,k_L^3} {N_r\,\mu_L\,d_o\,\Delta T_\mathrm{film}}\right]^{0.25}, \\ h_\mathrm{shear} &= h_{L}\sqrt{\frac{\rho_L}{\rho_\mathit{TP}}}, \end{align} \]

where \(N_r\) is the number of tube rows, \(h_{fg}\) is the latent heat, and \(\rho_\mathit{TP}\) is the two-phase mixture density.

Tube-side condensation (Shah)

The Shah  condensation correlation modifies the liquid-only coefficient:

\[ h_\mathrm{cond} = h_L\left(0.55 + \frac{2.09}{X_\mathit{tt}^{0.38}}\right). \]

Shell-side vaporisation (Chen)

Shell-side boiling uses the Chen  superposition of nucleate pool boiling (Mostinski correlation ) and forced convective enhancement:

\[ h_\mathrm{boil} = S\,h_\mathit{nb} + F\,h_L, \]

where \(h_\mathit{nb}\) is the Mostinski nucleate boiling coefficient:

\[ h_\mathit{nb} = 0.00417\,P_c^{0.69}\,q^{0.7}\,F_P, \qquad F_P = 1.8\,P_r^{0.17} + 4\,P_r^{1.2} + 10\,P_r^{10}, \]

and the Chen enhancement factor \(F\) and suppression factor \(S\) are:

\[ \begin{align} F &= \max\!\left(1,\; 2.35\,(1/X_\mathit{tt} + 0.213)^{0.736}\right), \\ S &= \frac{1}{1 + 2.53 \times 10^{-6}\,\Re_\mathit{TP}^{1.17}}, \quad \Re_\mathit{TP} = \Re_L\,F^{1.25}. \end{align} \]

Tube-side vaporisation (Chen/Forster–Zuber)

Tube-side flow boiling uses the full Chen  correlation with the Forster–Zuber nucleate boiling coefficient:

\[ h_\mathit{FZ} = 0.00122\, \frac{k_L^{0.79}\,c_{p,L}^{0.45}\,\rho_L^{0.49}} {\sigma^{0.5}\,\mu_L^{0.29}\,h_{fg}^{0.24}\,\rho_V^{0.24}} \,\Delta T_\mathrm{sat}^{0.24}\,\Delta P_\mathrm{sat}^{0.75}, \]

and \(h_\mathrm{boil} = S\,h_\mathit{FZ} + F\,h_L\) with the same \(F\) and \(S\) factors as above.

Critical heat flux

The maximum (critical) heat flux is checked using the Zuber  correlation:

\[ q_\mathrm{max} = 0.149\,h_{fg}\,\rho_V \left[\frac{\sigma\,g\,(\rho_L - \rho_V)}{\rho_V^2}\right]^{0.25} \!\cdot f_b, \]

where \(f_b \approx 0.5\) is a bundle correction factor.

Overall Heat-Transfer Coefficient

The local overall coefficient based on the outside tube area is:

\[ \frac{1}{U} = \frac{1}{h_s} + R_{f,s} + \frac{d_o\,\ln(d_o/d_i)}{2\,k_w} + \frac{d_o}{d_i}\,R_{f,t} + \frac{d_o}{d_i}\,\frac{1}{h_t}, \]

where \(h_s\) and \(h_t\) are the shell-side and tube-side coefficients, \(k_w\) is the tube wall thermal conductivity, and \(R_{f,s}\), \(R_{f,t}\) are the shell-side and tube-side fouling resistances.

LMTD and F-Correction Factor

The log-mean temperature difference for counter-current flow is:

\[ \Delta T_\mathrm{lm} = \frac{(T_{h,\mathrm{in}} - T_{c,\mathrm{out}}) - (T_{h,\mathrm{out}} - T_{c,\mathrm{in}})} {\ln\!\dfrac{T_{h,\mathrm{in}} - T_{c,\mathrm{out}}} {T_{h,\mathrm{out}} - T_{c,\mathrm{in}}}}. \]

For multi-pass configurations the LMTD correction factor \(F\) is computed using the \(P\)–\(R\) method for a 1–\(N\) shell-and-tube exchanger:

\[ P = \frac{T_{c,\mathrm{out}} - T_{c,\mathrm{in}}} {T_{h,\mathrm{in}} - T_{c,\mathrm{in}}}, \qquad R = \frac{T_{h,\mathrm{in}} - T_{h,\mathrm{out}}} {T_{c,\mathrm{out}} - T_{c,\mathrm{in}}}, \]

with the analytical formula for \(F\) as a function of \(P\), \(R\), and the number of shell passes.

Pressure Drop

Shell-side pressure drop (Bell–Delaware)

The total shell-side pressure drop comprises four contributions:

\[ \Delta P_s = \Delta P_\mathrm{cross} + \Delta P_\mathrm{window} + \Delta P_\mathrm{end} + \Delta P_\mathrm{nozzle}. \]

The ideal cross-flow drop per baffle space is:

\[ \Delta P_\mathrm{id} = 2\,f_i\,N_{cl}\,\frac{G_s^2}{\rho_s}, \]

where \(f_i\) is the ideal friction factor (tabulated by layout and \(\Re_s\)) and \(N_{cl}\) is the number of tube rows crossed. The cross-flow, window, and end-zone contributions are then :

\[ \begin{align} \Delta P_\mathrm{cross} &= (N_b - 1)\,\Delta P_\mathrm{id}\,R_l\,R_b, \\ \Delta P_\mathrm{window} &= N_b\,(2 + 0.6\,N_{tw})\,\frac{G_w^2}{2\,\rho_s}\,R_l, \\ \Delta P_\mathrm{end} &= 2\,\Delta P_\mathrm{id} \left(1 + \frac{N_{cw}}{N_{cl}}\right) R_b, \end{align} \]

where \(G_w = \dot{m}_s / \sqrt{S_m\,A_w}\) is the geometric mean mass velocity in the window zone, \(N_{tw}\) is the number of tube rows in the window, \(N_{cw}\) is the number of cross-flow rows in the end zone, and \(R_l\), \(R_b\) are the leakage and bypass correction factors (numerically equal to \(J_l\) and \(J_b\)). Nozzle losses are computed as \(\frac{1}{2}\rho v_n^2\) for each nozzle.

Tube-side pressure drop

The tube-side pressure drop is computed using the Darcy–Weisbach equation with the Churchill  friction factor (valid for all flow regimes):

\[ \Delta P_t = N_{tp}\,f\,\frac{L_t}{d_i}\,\frac{\rho\,v^2}{2} + N_{tp}\,4\,\frac{\rho\,v^2}{2} + \Delta P_\mathrm{nozzle}, \]

where the first term is the friction loss, the second the return-bend loss (\(4\times\) velocity head per pass), and nozzle losses are again \(\frac{1}{2}\rho v_n^2\) for each nozzle.

Two-phase pressure drop

When a two-phase flow is detected, single-phase pressure drops are multiplied by the Lockhart–Martinelli  two-phase multiplier:

\[ \phi_L^2 = 1 + \frac{C}{X_\mathit{tt}} + \frac{1}{X_\mathit{tt}^2}, \]

with \(C = 20\) (turbulent–turbulent). On the shell side, the Grant correlation provides an alternative multiplier:

\[ \phi_\mathrm{Grant}^2 = 1 + x\left(\frac{\rho_L}{\rho_V} - 1\right). \]

Vibration Analysis

A tube vibration check is performed at the end of every rating calculation. Three mechanisms are evaluated:

1. Natural frequency. The first-mode natural frequency of a fixed–fixed tube span between baffles is:

\[ f_n = \frac{22.4}{2\pi}\sqrt{\frac{EI}{m_\mathrm{total}\,L_b^4}}, \]

where $E$ is the tube Young’s modulus (default 200 GPa, carbon steel), $I$ is the second moment of area, and $m_\mathrm{total}$ is the mass per unit length (tube wall + internal fluid + external added mass).

1. Vortex shedding. The vortex shedding frequency \(f_{vs} = 0.2\,v_\mathrm{cross}/d_o\) is compared to \(f_n\). A warning is issued when \(0.8 < f_{vs}/f_n < 1.2\).

2. Fluid-elastic instability (Connors criterion). The critical cross-flow velocity is:

\[ v_\mathrm{crit} = 3.3\,f_n\,d_o \sqrt{\frac{2\pi\,\zeta\,m_\mathrm{total}}{\rho_s\,d_o^2}}, \]

where $\zeta = 0.01$ is the assumed damping ratio. A warning is issued when $v_\mathrm{cross}/v_\mathrm{crit} > 0.7$.

Design Mode

In Design mode the user specifies one outlet temperature. The model first estimates the required area from a shortcut LMTD calculation with an assumed \(U = 500\) W/m\(^2\) K, computes the initial tube count as \(N_t = A_\mathrm{req}/(\pi\,d_o\,L_t)\), and derives the shell diameter from the tube-count rule \(D_\mathrm{otl}^2 = (4/\pi)(C_L/C_\mathrm{TP})\,N_t\,p_t^2\) (\(C_L = 1\) for square and \(0.866\) for triangular layouts, \(C_\mathrm{TP}\) the pass-partition factor), the same rule the geometry check applies, with a 2% margin. It then iterates (up to 20 times) by running a full Rating calculation and adjusting \(N_t\) by the ratio \(Q_\mathrm{req}/Q_\mathrm{calc}\) until convergence within 2%. If a maximum shell-side pressure drop constraint is active, the baffle spacing is increased by 20% in each iteration where the constraint is violated.

Kettle Reboiler

The exchanger can be run as a kettle reboiler (TEMA K shell), the usual choice for the reboiler of a distillation column. The option is on the Shell Side page. In this mode the boiling process fluid is the cold fluid and has to be on the shell side; the heating medium (steam, thermal oil, hot process stream) goes through the tubes.

Thermal model

The shell side of a kettle is a pool of liquid at one temperature, with no baffles and no flow across the bundle. The incremental integration of Section 2.33.4 is kept for the tube side, and the shell side is taken as a well-mixed pool at the outlet state of the process fluid: the pool temperature and vapor fraction are the outlet temperature and vapor fraction, and are settled with the outer iterations of the rating. The coefficient on the bundle is the nucleate boiling coefficient of Mostinski  at the pseudo-critical pressure of the mixture (the mole-fraction average of the critical pressures), solved together with the local heat flux, once the pool is at its bubble point; while the pool is still subcooled the coefficient is that of natural convection over a horizontal tube (Churchill and Chu). The Bell–Delaware corrections, the shell-side pressure drop and the vibration checks do not apply to a pool and are not computed.

Kettle shell

The shell inside diameter of the geometry page is the port that holds the tube bundle. The kettle shell around it is sized from the service:

• the liquid level is the bundle diameter plus the bottom clearance plus the height the weir keeps above the top row of tubes (75 mm by default), which is the weir height;

• the vapor velocity across the free liquid surface (the chord of the shell at the liquid level times the tube length) must stay below the allowable value \(u_\mathrm{v} = K \sqrt{(\rho_L - \rho_V)/\rho_V}\), with \(K = 0.2\) m/s by default ;

• the vapor space above the liquid must not be smaller than the minimum given (300 mm by default).

When the kettle shell diameter is left at zero, the largest of 1.6 times the bundle diameter, the liquid level plus the minimum vapor space, and the diameter that satisfies the velocity limit is taken, rounded up to 10 mm. When a diameter is given, the checks are reported against it. The results give the kettle shell diameter and its ratio to the bundle, the weir height, the vapor space, the free surface area, the vapor velocity and its allowable value, the vapor generated and the fraction of the feed vaporized, the pool temperature, and the liquid inventory up to the weir (the circular segment below the liquid level less the tubes). The shell-side volume and the weights follow the kettle shell.

Heat flux

The design heat flux \(q = Q/A\) is compared with the critical heat flux of the bundle,

\[ q_{c,b} = \phi_b \, q_{c,1}, \qquad q_{c,1} = 3.67 \times 10^{4}\, P_c\, P_r^{0.35}\,(1 - P_r)^{0.9}, \qquad \phi_b = \min\!\left(1,\; 3.1\,\frac{D_b}{N_t\, d_o}\right), \]

where \(q_{c,1}\) is the single-tube critical flux of Mostinski (\(P_c\) in bar, result in W/m\(^2\)) and \(\phi_b\) the bundle factor of Palen and Small , the ratio of the bundle envelope to the tube surface: the inner tubes of a big bundle are blanketed by the vapor of the tubes below them and the bundle reaches its maximum flux well before a single tube would. A warning is issued when \(q\) exceeds the chosen fraction of \(q_{c,b}\) (70% by default), and an absolute cap on the flux can be set for the company practices that limit it (Kern’s 40 kW/m\(^2\) for organics, for instance).

Design of a kettle

In Design mode the duty can be given as the molar fraction of the bottoms liquid that has to be vaporized, the number the column gives (the vapor returned to the column over the liquid to the reboiler); the duty follows from a pressure and vapor fraction flash of the shell fluid. The hot outlet temperature specification of Section 2.33.12 remains available. After the tube count is settled for the duty, the heat flux is checked; when it is above the limit the tube count is raised until the flux is within it (\(q \propto 1/N_t\) while \(q_{c,b} \propto 1/\sqrt{N_t}\)), and a warning tells that the area is set by the flux and that the heating medium has to be throttled to hold the specified vaporization.

Methodology

For the preliminary sizing of the reboiler of a column: take the reboiler duty, the liquid to the reboiler and the vapor generated from the solved column; put the bottoms liquid on the shell side and the heating medium on the tubes of an Advanced Heat Exchanger with the kettle option; run Design mode with the vapor fraction of the column; read the number and length of tubes, the bundle diameter, the kettle shell diameter, the weir height and the liquid inventory for the equipment list and the datasheet. The kettle shell is a conceptual size; the mechanical design belongs to the fabricator.

Simulation Mode

In Simulation mode the user provides a constant overall coefficient \(U\). A full Rating calculation is performed to obtain the geometry-dependent LMTD and \(F\)-factor, then the duty is recomputed as \(Q = U\,A\,F\,\Delta T_\mathrm{lm}\), and the outlet temperatures are recovered by pressure–enthalpy flashes.

Fouling Factor Mode

In Fouling Factor mode both outlet temperatures are specified. The model computes \(Q\) from the hot-side energy balance, then the “dirty” overall coefficient \(U_\mathrm{dirty} = Q/(A\,F\,\Delta T_\mathrm{lm})\). A Rating calculation is repeated with zero fouling to obtain \(U_\mathrm{clean}\). The overall fouling resistance is:

\[ R_f = \frac{1}{U_\mathrm{dirty}} - \frac{1}{U_\mathrm{clean}}. \]

Parameters Summary

Table 8 lists the principal user-configurable parameters.

Principal parameters of the Advanced Heat Exchanger.

Parameter	Units	Default	Description
Parameter	Units	Default	Description
Continued on next page
Calculation Mode	—	Rating	Rating, Design, Simulation, or FoulingFactor
Flow Direction	—	Counter-current	Counter-current or Co-current
Number of Increments	—	20	Integration resolution
Max Iterations	—	100	Outer convergence loop limit
Tolerance	—	\(10^{-6}\)	Relative duty convergence tolerance
Overall Coefficient	W/m\(^2\)K	—	User-specified \(U\) (Simulation mode)
Shell
Shell \(D_s\)	mm	500	Shell inside diameter
Shell passes	—	1	Number of shell passes
Shells in series	—	1	Number of shells in series
TEMA type	—	AEL	TEMA designation
Baffle type	—	Single-seg.	Single-segmental, double-segmental, NTIW
Baffle cut \(B_c\)	%	25	Baffle cut percentage
Baffle spacing \(L_b\)	m	0.25	Central baffle spacing
Shell fouling	m\(^2\)K/W	0.0002	Shell-side fouling resistance
Sealing strips	—	1	Pairs of sealing strips
Tubes
Tube \(d_o\)	mm	25.4	Tube outside diameter
Tube \(d_i\)	mm	21.2	Tube inside diameter
Tube length \(L_t\)	m	4.88	Tube length
Number of tubes \(N_t\)	—	100	Tubes per shell
Tube passes \(N_{tp}\)	—	2	Tube passes per shell
Tube pitch \(P_t\)	mm	31.75	Centre-to-centre pitch
Tube layout	—	30° tri.	Layout pattern
Tube \(k_w\)	W/m K	50	Tube wall thermal conductivity
Tube fouling	m\(^2\)K/W	0.0002	Tube-side fouling resistance

Results

After a successful calculation the following results are available:

• Heat duty \(Q\), hot-side and cold-side outlet temperatures.

• Shell-side and tube-side heat-transfer coefficients (local and average).

• Overall coefficient \(U\) (calculated or specified).

• LMTD and \(F\)-correction factor.

• Thermal efficiency \(\varepsilon\).

• Shell-side and tube-side pressure drops.

• Overall fouling factor (Fouling Factor mode).

• Temperature, duty, and vapor-fraction profiles along the exchanger.

• Vibration warnings (if any).

• Kettle reboiler: shell diameter, weir height, vapor space, free surface, vapor velocity, vapor generated, heat flux against the bundle critical flux, liquid inventory and the kettle checks (Section 2.33.13).

Vapor Compression Chiller

Overview

The Vapor Compression Chiller is a custom unit operation that simulates a complete multi-stage mechanical refrigeration cycle. It is designed for refinery and petrochemical applications where process streams must be cooled below the temperature achievable by cooling water or air alone, as in LPG recovery, gas dewpoint control, alkylation feed chilling, and amine-unit intercooling.

The model covers the full thermodynamic cycle (evaporation, compression, condensation and expansion) together with preliminary sizing of the major equipment items: compressors and heat exchangers. All thermodynamic calculations are performed through the flowsheet property package, so any equation of state or activity-coefficient model available in DWSIM can be used for the refrigerant side.

Stream Topology

The unit operation has the following connection ports:

Port	Direction	Type	Description
Process In	Inlet	Material	Process fluid to be cooled (evaporator shell side)
Cooling Fluid In	Inlet	Material	Condenser cooling medium (water, glycol, air, etc.)
Shaft Power In	Inlet	Energy	Electrical/mechanical power driving the compressors (optional)
Process Out	Outlet	Material	Cooled process fluid leaving the evaporator
Cooling Fluid Out	Outlet	Material	Heated cooling medium leaving the condenser

Configuration

Compression stages

Between one and three compression stages may be specified. Interstage pressures are distributed geometrically (equal pressure ratio per stage):

\[ r = \left(\frac{P_{\mathrm{cond}}}{P_{\mathrm{evap}}}\right)^{1/N}, \qquad P_i = P_{\mathrm{evap}}\,r^{i}, \quad i = 0, 1, \ldots, N, \]

where \(N\) is the number of stages, \(P_{\mathrm{evap}}\) is the evaporator pressure, and \(P_{\mathrm{cond}}\) is the condenser pressure.

Flash economizers

An open-cycle flash economizer may optionally be installed between any two adjacent stages. The liquid refrigerant leaving the upstream condenser or interstage liquid header is flashed to the interstage pressure. The resulting vapor is injected into the suction of the next stage, reducing the refrigerant circulation rate at the evaporator level and improving the cycle coefficient of performance. The vapor fraction at the flash condition is obtained from an isenthalpic flash:

\[ H_{\mathrm{in}} = H_{\mathrm{liq}}\,(1-\psi) + H_{\mathrm{vap}}\,\psi, \]

where \(\psi\) is the molar vapor fraction and \(H_{\mathrm{in}}\) is the enthalpy of the liquid entering the economizer at constant enthalpy (isenthalpic valve upstream).

Refrigerant

The refrigerant composition is defined by selecting compounds already present in the flowsheet and specifying their mole fractions. Any property package registered in the flowsheet can be assigned to the refrigerant side independently. Peng–Robinson EOS is recommended for hydrocarbon and common HFC refrigerants.

Thermodynamic Model

State points

The refrigerant cycle is described by four canonical state points per stage:

• Point 1: Evaporator exit: saturated (or slightly superheated) vapor at \(P_{\mathrm{evap}}\), obtained from a dew-point flash.

• Point 2s: Isentropic compressor discharge: entropy equals Point 1, pressure equals stage discharge pressure. Obtained from a \(P\)–\(S\) flash.

• Point 2: Actual compressor discharge: enthalpy corrected for isentropic efficiency.

• Point 3: Condenser exit: saturated liquid at \(P_{\mathrm{cond}}\), obtained from a bubble-point flash.

• Point 4: Expansion valve exit: isenthalpic flash to \(P_{\mathrm{evap}}\).

Compressor

The actual specific work and discharge state for each stage are:

\[ \begin{align} w_{\mathrm{is}} &= H_{2\mathrm{s}} - H_1, \\ w_{\mathrm{act}} &= \frac{w_{\mathrm{is}}}{\eta_{\mathrm{is}}}, \\ H_2 &= H_1 + w_{\mathrm{act}}, \\ \dot{W}_{\mathrm{shaft},i} &= \dot{n}_{\mathrm{ref},i}\,\frac{w_{\mathrm{act}}}{\eta_{\mathrm{mech}}}, \end{align} \]

where \(\eta_{\mathrm{is}}\) and \(\eta_{\mathrm{mech}}\) are the isentropic and mechanical efficiencies, respectively, and \(\dot{n}_{\mathrm{ref},i}\) is the molar flow rate of refrigerant through stage \(i\).

Refrigerant circulation rate

The molar flow rate at the evaporator level is determined from the process-stream heat balance:

\[ \dot{n}_{\mathrm{ref}} = \frac{\dot{Q}_{\mathrm{evap}}}{H_1 - H_4}, \]

where \(\dot{Q}_{\mathrm{evap}} = \dot{n}_{\mathrm{proc}}\,(H_{\mathrm{proc,in}} - H_{\mathrm{proc,out}})\) is the process-side heat load. When economizers are present, the flow through each subsequent stage is augmented by the injected flash vapor.

Energy balance

The overall energy balance is:

\[ \dot{Q}_{\mathrm{cond}} = \dot{Q}_{\mathrm{evap}} + \sum_{i=1}^{N} \dot{W}_{\mathrm{shaft},i}. \]

Coefficient of performance

\[ \mathrm{COP} = \frac{\dot{Q}_{\mathrm{evap}}}{\displaystyle\sum_{i=1}^{N} \dot{W}_{\mathrm{shaft},i}}. \]

Evaporator and Condenser Specifications

Evaporator

The user may specify either the evaporator temperature \(T_{\mathrm{evap}}\) (from which \(P_{\mathrm{evap}}\) is obtained by a dew-point calculation) or the evaporator pressure \(P_{\mathrm{evap}}\) directly.

Condenser

Three specification modes are available:

1. Fixed condenser temperature \(T_{\mathrm{cond}}\).

2. Fixed condenser pressure \(P_{\mathrm{cond}}\).

3. Approach temperature difference: \(T_{\mathrm{cond}} = T_{\mathrm{cool,in}} + \Delta T_{\mathrm{app}}\), where \(T_{\mathrm{cool,in}}\) is the inlet temperature of the cooling stream.

Equipment Sizing

Compressor sizing

For each stage the actual volumetric flow at suction conditions is:

\[ \dot{V}_{\mathrm{suc}} = \dot{n}_{\mathrm{ref},i}\,\hat{V}_{m,\mathrm{suc}}, \]

where \(\hat{V}_{m,\mathrm{suc}}\) is the molar volume at suction conditions from the property package.

The polytropic index \(n\) is estimated from the isentropic temperature ratio and the actual temperature ratio:

\[ \gamma \approx 1 + \frac{\ln r}{\ln\!\left(T_{2\mathrm{s}}/T_1\right)}, \qquad \eta_p = \frac{\gamma - 1}{\gamma}\,\frac{\ln r}{\ln\!\left(T_2/T_1\right)}, \qquad \frac{n}{n-1} = \eta_p\,\frac{\gamma}{\gamma - 1}, \]

where \(r = P_{\mathrm{dis}}/P_{\mathrm{suc}}\) is the stage pressure ratio.

The polytropic head (Schultz method, simplified) is:

\[ H_p = \frac{R\,T_{\mathrm{suc}}}{M_w}\,\frac{n}{n-1} \left[\left(\frac{P_{\mathrm{dis}}}{P_{\mathrm{suc}}}\right)^{(n-1)/n} - 1\right], \]

in J/kg, where \(M_w\) is the refrigerant molar mass.

For centrifugal compressors, the dimensionless specific speed is:

\[ N_s = \frac{n_{\mathrm{rev}}\,\sqrt{\dot{V}_{\mathrm{suc}}}}{H_p^{3/4}}, \]

with \(n_{\mathrm{rev}}\) in rev/s, \(\dot{V}\) in m\(^3\)/s, and \(H_p\) in m.

For reciprocating compressors, the volumetric efficiency is:

\[ \eta_{\mathrm{vol}} = 1 + c - c\,r^{1/n}, \]

where \(c\) is the clearance ratio (user input). The required piston displacement rate is:

\[ \dot{V}_{\mathrm{disp}} = \frac{\dot{V}_{\mathrm{suc}}}{\eta_{\mathrm{vol}}}. \]

Heat exchanger sizing

Both the evaporator and condenser are sized by the LMTD method. For counter-current flow:

\[ \Delta T_{\mathrm{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}, \]

where \(\Delta T_1\) and \(\Delta T_2\) are the terminal temperature differences at each end of the exchanger.

The required heat transfer area is:

\[ A = \frac{\dot{Q}}{U\,\Delta T_{\mathrm{lm}}}, \]

with the overall heat transfer coefficient:

\[ \frac{1}{U} = \frac{1}{h_{\mathrm{ref}}} + \frac{1}{h_{\mathrm{fluid}}} + R_f, \]

where \(R_f\) is the combined fouling resistance (m\(^2\) K/W).

# Evaporator: refrigerant-side boiling.

The boiling heat transfer coefficient is estimated by the Cooper reduced-pressure pool boiling correlation :

\[ h_{\mathrm{boil}} = 55\,P_r^{0.12}\,\bigl(-\log_{10} P_r\bigr)^{-0.55} M_w^{-0.5}\,q^{0.67}, \]

in W/m\(^2\) K, where \(P_r = P/P_c\) is the reduced pressure, \(M_w\) is the molar mass in g/mol, and \(q\) is the heat flux in W/m\(^2\). Since \(h = q/\Delta T_{\mathrm{lm}}\), substituting \(q = h\,\Delta T_{\mathrm{lm}}\) and solving for \(h\) yields:

\[ h_{\mathrm{boil}} = \left[55\,P_r^{0.12}\,\bigl(-\log_{10} P_r\bigr)^{-0.55} M_w^{-0.5} \,\Delta T_{\mathrm{lm}}^{0.67}\right]^{1/0.33}. \]

The critical pressure of a refrigerant mixture is estimated by Kay’s mixing rule: \(P_{c,\mathrm{mix}} = \sum_i z_i\,P_{c,i}\).

# Condenser: refrigerant-side condensation.

The condensing heat transfer coefficient is estimated by the Shah correlation  evaluated at a mean vapor quality of \(x = 0.5\):

\[ h_{\mathrm{cond}} = h_L\left(0.55 + \frac{2.09}{P_r^{0.38}}\right), \]

where \(h_L\) is the liquid-phase single-flow heat transfer coefficient, approximated as 800 W/m\(^2\) K for typical refrigerants at condensing conditions. This correlation is adequate for preliminary area estimation; rigorous sizing requires fluid-specific transport properties.

# Process and cooling fluid sides.

The heat transfer coefficient on the process (evaporator) or cooling (condenser) fluid side may be supplied directly by the user or left at a conservative default value. When default values are used, the model adopts 2000 W/m\(^2\) K for the process side (liquid) and selects between 80 W/m\(^2\) K (gas/air stream, identified by low density) and 4000 W/m\(^2\) K (liquid stream) for the cooling side.

Parameters Summary

Table 9 lists all user-configurable parameters.

Parameters of the Vapor Compression Chiller.

Parameter	Units	Default	Description
Parameter	Units	Default	Description
Continued on next page
Number of stages	—	1	Integer 1–3
Economizer mask	—	0	Bitmask: bit 0 = after stage 1, bit 1 = after stage 2
Refrigerant PP	—	—	Property package name for refrigerant
Refrigerant comp.	mol/mol	—	Mole fractions of flowsheet compounds
Evaporator
Evaporator spec.	—	Temperature	Temperature or Pressure
Evaporator \(T\)	K	258.15	Evaporator temperature (if spec = T)
Evaporator \(P\)	Pa	2  10\(^5\)	Evaporator pressure (if spec = P)
HX type (evap.)	—	Shell&Tube	Shell-and-tube or Plate
\(U\) override (evap.)	W/m\(^2\)K	0	0 = use correlation
Fouling (evap.)	m\(^2\)K/W	2  10\(^{-4}\)	Combined fouling resistance
Process-side \(h\)	W/m\(^2\)K	0	0 = use default (2000)
Condenser
Condenser spec.	—	Fixed \(T\)	Fixed T, fixed P, or approach \(\Delta T\)
Condenser \(T\)	K	318.15	(if spec = Fixed T)
Condenser \(P\)	Pa	12  10\(^5\)	(if spec = Fixed P)
Approach \(\Delta T\)	K	5.0	(if spec = Approach \(\Delta T\))
HX type (cond.)	—	Shell&Tube	Shell-and-tube or Plate
\(U\) override (cond.)	W/m\(^2\)K	0	0 = use correlation
Fouling (cond.)	m\(^2\)K/W	2  10\(^{-4}\)	Combined fouling resistance
Cooling-side \(h\)	W/m\(^2\)K	0	0 = auto-detect
Compressors (per stage, \(i\) = 1, 2, 3)
Type\(_i\)	—	Centrifugal	Centrifugal or Reciprocating
\(\eta_{\mathrm{is},i}\)	—	0.75	Isentropic efficiency
\(\eta_{\mathrm{mech},i}\)	—	0.95	Mechanical efficiency
Speed\(_i\)	rpm	3000	Rotational speed (centrifugal)
Clearance\(_i\)	—	0.05	Clearance ratio (reciprocating)

Results

After a successful calculation the following results are available in the unit operation report and in the Results tab of the editor:

• Cycle: COP, evaporator duty, condenser duty, total shaft power, refrigerant molar flow rate, interstage pressures.

• Per stage: suction and discharge temperature, pressure, enthalpy, entropy; pressure ratio; shaft power; polytropic index and head; polytropic efficiency; volumetric suction flow; compressor-type-specific sizing result (specific speed or volumetric efficiency and displacement).

• Per economizer: flash vapor fraction, vapor injected, liquid carryover.

• Evaporator and condenser: LMTD, refrigerant-side \(h\), fluid-side \(h\), overall \(U\), fouling resistance, required heat transfer area, terminal temperatures.

Air Cooler 2

Overview.

The Air Cooler 2 models a forced-draft air-cooled heat exchanger in which a process fluid (hot side) flows through horizontal tubes and ambient air (cold side) is forced across the outside of the tube bundle by a fan. The model is a simplification of the TEMA shell-and-tube calculation method adapted for air cooling.

Stream Topology.

Port	Direction	Description
Fluid Inlet	Inlet (material)	Hot process fluid
Power Inlet	Inlet (energy)	Fan shaft power (optional)
Fluid Outlet	Outlet (material)	Cooled process fluid

The air-side streams are generated internally by the model from the ambient conditions specified by the user and are not connected directly to the flowsheet.

Fan Air-Flow Model.

The actual volumetric air flow is proportional to the fan speed relative to a reference condition:

\[ \dot{V}_{\mathrm{air}} = \dot{V}_{\mathrm{ref}} \frac{N_{\mathrm{actual}}}{N_{\mathrm{ref}}} \]

where \(\dot{V}_{\mathrm{ref}}\) (m\(^3\) s\(^{-1}\)) and \(N_{\mathrm{ref}}\) (rpm) are user-specified reference values.

Calculation Modes.

Three calculation modes are available:

1. Specify Outlet Temperature: the user fixes the hot-fluid outlet temperature \(T_{h,\mathrm{out}}\); the model computes the heat load \(Q\), the air outlet temperature \(T_{c,\mathrm{out}}\), and the product \(UA\).

2. Specify Geometry: the user provides the tube-bundle geometry; the model iterates to find \(T_{h,\mathrm{out}}\) and \(T_{c,\mathrm{out}}\) using the simplified Tinker method  for the shell-and-tube calculation.

3. Specify Overall UA: the user provides the product \(UA\) (W K\(^{-1}\)); the model applies the \(\varepsilon\)-NTU method to find the outlet temperatures and \(Q\).

Overall Heat Balance.

All modes use the fundamental heat-exchanger equations:

\[ Q = U A F \,\Delta T_{\mathrm{lm}} \]

\[ Q = -\dot{m}_h \bigl(h_{h,\mathrm{out}} - h_{h,\mathrm{in}}\bigr) = \dot{m}_c \bigl(h_{c,\mathrm{out}} - h_{c,\mathrm{in}}\bigr) \]

where \(U\) (W m\(^{-2}\) K\(^{-1}\)) is the overall heat-transfer coefficient, \(A\) (m\(^2\)) is the external tube surface area, and \(F\) is the log-mean temperature-difference correction factor.

LMTD and Correction Factor.

The log-mean temperature difference for counter-current flow is

\[ \Delta T_{\mathrm{lm}} = \frac{(T_{h,\mathrm{in}}-T_{c,\mathrm{out}}) - (T_{h,\mathrm{out}}-T_{c,\mathrm{in}})} {\ln\!\dfrac{T_{h,\mathrm{in}}-T_{c,\mathrm{out}}} {T_{h,\mathrm{out}}-T_{c,\mathrm{in}}}} \]

The correction factor \(F\) accounts for the multi-pass (1 shell–2 tube pass) geometry . Defining

\[ R = \frac{T_{h,\mathrm{in}}-T_{h,\mathrm{out}}} {T_{c,\mathrm{out}}-T_{c,\mathrm{in}}}, \qquad P = \frac{T_{c,\mathrm{out}}-T_{c,\mathrm{in}}} {T_{h,\mathrm{in}}-T_{c,\mathrm{in}}} \]

and the auxiliary variable \(S = \dfrac{(1-RP)/(1-P)-1}{(1-RP)/(1-P)-R}\), the correction factor for \(R \neq 1\) is

\[ F = \frac{\sqrt{R^2+1}\ln\!\dfrac{1-S}{1-RS}} {(R-1)\ln\!\dfrac{2-S\!\left(R+1-\sqrt{R^2+1}\right)} {2-S\!\left(R+1+\sqrt{R^2+1}\right)}} \]

When \(R = 1\), the limiting form \(F = \dfrac{S\sqrt{2}}{(1-S)\ln\! \dfrac{2(1-S)+S\sqrt{2}}{2(1-S)-S\sqrt{2}}}\) is applied.

Tube-Side Heat Transfer (Gnielinski / Petukhov).

The internal heat-transfer coefficient \(h_i\) (W m\(^{-2}\) K\(^{-1}\)) is evaluated from the Gnielinski–Petukhov correlation :

\[ h_i = \frac{k_h}{D_i}\, \frac{(f/8)(Re_t - 1000)\,Pr_t} {1 + 12.7\sqrt{f/8}\,(Pr_t^{2/3}-1)} \]

The tube-side Darcy friction factor \(f\) is computed using an explicit approximation ; for laminar flow (\(Re \le 3250\)), \(f = 64/Re\).

Tube-Side Pressure Drop.

\[ \Delta P_{\mathrm{tube}} = f\,\frac{L\,n_{\mathrm{pass}}}{D_i} \,\frac{\rho_h u_t^2}{2} \]

where \(u_t = \dot{m}_h / (\rho_h\,n_t\,\pi D_i^2/4)\) is the tube velocity and \(n_t = N_{\mathrm{tubes}} / n_{\mathrm{pass}}\) is the number of tubes per pass.

Air-Side Heat Transfer (Holman).

The external (air-side) heat-transfer coefficient \(h_e\) (W m\(^{-2}\) K\(^{-1}\)) over the tube bundle is evaluated from the Holman correlation :

\[ \frac{h_e D_e}{k_c} = 0.287\,Re_e^{0.61}\,Pr_c^{1/3} \]

where \(Re_e = G_s D_e / \mu_c\) is the shell-side Reynolds number and \(G_s = \dot{m}_c / S_{\mathrm{flow}}\) is the mass flux through the minimum free-flow area \(S_{\mathrm{flow}}\).

Overall Heat-Transfer Coefficient.

The overall heat-transfer coefficient based on the external area is

\[ \frac{1}{U} = \frac{D_e}{h_i D_i} + r_f \frac{D_e}{D_i} + \frac{D_e}{2k_t}\ln\frac{D_e}{D_i} + \frac{1}{h_e} \]

where \(r_f\) (m\(^2\) K W\(^{-1}\)) is the tube-side fouling resistance and \(k_t\) (W m\(^{-1}\) K\(^{-1}\)) is the tube-wall thermal conductivity.

External Surface Area.

\[ A = N_{\mathrm{tubes}}\,\pi D_e (L - 2D_e) \]

\(\varepsilon\)-NTU Method (Mode 3).

When \(UA\) is specified, the outlet temperatures are found from the number of transfer units \(NTU = UA / (W \cdot c_p)\) and the dimensionless temperature-change parameter \(P\):

\[ P = \frac{1 - \exp[(R-1)\,NTU]}{1 - R\exp[(R-1)\,NTU]} \]

where \(R = W_c c_{p,c}/(W_h c_{p,h})\) is the heat-capacity-rate ratio. The outlet temperatures are updated iteratively until the heat load \(Q\) converges.

Model Parameters.

Parameter	Symbol	Unit	Default
Tube internal diameter	\(D_i\)	mm	50
Tube external diameter	\(D_e\)	mm	60
Tube length	\(L\)	m	5
Tube pitch	\(p_t\)	mm	80
Number of tubes per shell	\(N_{\mathrm{tubes}}\)	—	160
Passes per shell	\(n_{\mathrm{pass}}\)	—	1
Tube fouling resistance	\(r_f\)	m\(^2\) K W\(^{-1}\)	0
Tube wall roughness	\(\varepsilon\)	mm	0.045
Tube thermal conductivity	\(k_t\)	W m\(^{-1}\) K\(^{-1}\)	70
Air inlet temperature	\(T_{c,\mathrm{in}}\)	K	298.15
Air pressure	\(P_c\)	Pa	101 325
Reference air flow	\(\dot{V}_{\mathrm{ref}}\)	m\(^3\) s\(^{-1}\)	1
Reference fan speed	\(N_{\mathrm{ref}}\)	rpm	100
Actual fan speed	\(N_{\mathrm{actual}}\)	rpm	100
Hot-side pressure drop	\(\Delta P_h\)	Pa	0

Results Reported.

Quantity	Symbol	Unit
Heat load	\(Q\)	kW
Overall UA	\(UA\)	W K\(^{-1}\)
Exchange area	\(A\)	m\(^2\)
LMTD	\(\Delta T_{\mathrm{lm}}\)	K
LMTD correction factor	\(F\)	—
Air outlet temperature	\(T_{c,\mathrm{out}}\)	K
Maximum heat exchange	\(Q_{\max}\)	kW
Exchanger efficiency	\(Q/Q_{\max}\)	—

Falling Film Evaporator

Overview.

The Falling Film Evaporator (FFE) models a vertical shell-and-tube evaporator in which the feed liquid is distributed at the top of the heating tubes and descends as a thin film under gravity and co-current vapour flow. Heat applied to the tube exterior partially evaporates the film; the resulting vapour–liquid mixture is separated at the bottom of the calandria. A stepwise enthalpy-integration procedure is used, making the model independent of a specific heat-transfer correlation and relying instead on the thermodynamic property package for phase equilibrium and enthalpy calculations.

Stream Topology.

Port	Direction	Description
Material Inlet	Inlet (material)	Liquid (or partial vapour) feed
Energy Inlet	Inlet (energy)	Heating duty \(Q\) (kW)
Vapour Outlet	Outlet (material)	Evaporated vapour product
Concentrated Liquid	Outlet (material)	Residual liquid concentrate

Calculation Modes.

1. Outlet Temperature: the user specifies the exit temperature \(T_{\mathrm{out}}\); the model integrates the enthalpy in \(N_{\mathrm{steps}}\) equal temperature increments and reports the total heat duty.

2. Outlet Vapour Fraction: the user specifies the exit vapour mole fraction \(\psi_{\mathrm{out}}\); the model integrates in \(N_{\mathrm{steps}}\) equal vapour-fraction increments.

3. Energy Stream: the heat duty is read directly from the connected inlet energy stream; the model advances in enthalpy increments until the specified duty is consumed.

Stepwise Integration.

The evaporation path is divided into \(N_{\mathrm{steps}}\) intervals. At each step \(i\), the stream is flashed at the local conditions \((T_i, P_i)\) or \((P_i, H_i)\) using the flowsheet property package. The heat added at step \(i\) is

\[ \delta Q_i = \dot{m}\,(h_i - h_{i-1}) \]

and the total heat duty is

\[ Q = \sum_{i=1}^{N_{\mathrm{steps}}} \delta Q_i = \dot{m}\,(h_{\mathrm{out}} - h_{\mathrm{in}}) \]

The pressure decreases linearly along the tube length:

\[ P_i = P_{\mathrm{in}} - \frac{\Delta P_{\mathrm{tube}}}{N_{\mathrm{steps}}}\,i \]

Phase Separation.

After integration, the outlet stream is split at the exit vapour mass fraction \(\psi_v\):

\[ \begin{align} \dot{m}_{\mathrm{vapour}} &= \dot{m}\,\psi_v \\ \dot{m}_{\mathrm{liquid}} &= \dot{m}\,(1 - \psi_v) \end{align} \]

Evaporation Profile.

At each integration step the model records a profile item containing: temperature, pressure, cumulative heat added, heat of vaporisation (\(\Delta h_{\mathrm{vap}} = h_{\mathrm{vapour}} - h_{\mathrm{liquid}}\)), and vapour/liquid phase fractions, densities, enthalpies, heat capacities, and thermal conductivities. This profile can be exported or used for detailed exchanger sizing calculations.

Model Parameters.

Parameter	Symbol	Unit	Default
Calculation mode	—	—	Outlet Vapour Fraction
Outlet temperature	\(T_{\mathrm{out}}\)	K	300
Outlet vapour fraction	\(\psi_{\mathrm{out}}\)	—	0.3
Number of integration steps	\(N_{\mathrm{steps}}\)	—	10
Tube pressure drop	\(\Delta P_{\mathrm{tube}}\)	Pa	0

Energy Mixer

Overview.

The Energy Mixer sums up to six inlet energy streams into a single outlet energy stream. The calculation is a simple energy balance with no user-adjustable parameters.

Stream Topology.

Up to 6 energy inlets; 1 energy outlet.

Governing Equation.

\[ \dot{E}_{\mathrm{out}} = \sum_{i=1}^{N} \dot{E}_i \]

where \(N \le 6\) is the number of connected inlet streams and \(\dot{E}_i\) (kW) is the energy flow of each stream.

Energy Splitter

Overview.

The Energy Splitter divides one inlet energy stream into up to three outlet energy streams.

Stream Topology.

1 energy inlet; 1–3 energy outlets.

Calculation Modes.

1. Split Ratios: the user specifies the fractions \(r_1, r_2, r_3\) with the constraint \(\sum r_i = 1\). Each outlet receives

\[ \dot{E}_i = r_i\,\dot{E}_{\mathrm{in}} \]

1. Energy Flow Specification: the user specifies the energy flows of the first (and optionally the second) outlet stream; the remaining stream is determined by the energy balance:

\[ \begin{align} \dot{E}_1 &= \dot{E}_{1,\mathrm{spec}} \\ \dot{E}_2 &= \dot{E}_{2,\mathrm{spec}} \\ \dot{E}_3 &= \dot{E}_{\mathrm{in}} - \dot{E}_{1,\mathrm{spec}} - \dot{E}_{2,\mathrm{spec}} \end{align} \]

The specified values must not exceed the inlet energy flow.

Energy Stream Switch

Overview.

The Energy Stream Switch routes an inlet energy stream to one of two outlets depending on the result of a user-defined Boolean expression.

Stream Topology.

1 energy inlet; 2 energy outlets.

Routing Logic.

The user supplies an expression string that is evaluated at run time. The available variable is:

Variable	Meaning	Unit
HF	Inlet energy flow	kW

The routing rule is:

\[ \dot{E}_{\mathrm{out\,1}} = \begin{cases} \dot{E}_{\mathrm{in}} & \text{if expression is TRUE} \\ 0 & \text{otherwise} \end{cases} \quad \dot{E}_{\mathrm{out\,2}} = \dot{E}_{\mathrm{in}} - \dot{E}_{\mathrm{out\,1}} \]

Standard arithmetic and comparison operators (+, -, *, /, >, <, =, AND, OR, NOT) and all functions in System.Math are supported in the expression.

Material Stream Switch

Overview.

The Material Stream Switch routes an inlet material stream to one of two outlet material streams based on a user-defined Boolean expression evaluated against the inlet stream properties.

Stream Topology.

1 material inlet; 2 material outlets.

Available Variables.

Variable	Meaning	SI Unit
T	Temperature	K
P	Pressure	Pa
W	Mass flow rate	kg s\(^{-1}\)
M	Molar flow rate	mol s\(^{-1}\)
Q	Volumetric flow rate	m\(^3\) s\(^{-1}\)
VF	Vapour phase mole fraction	—
LF	Liquid phase mole fraction	—
SF	Solid phase mole fraction	—

Routing Logic.

\[ \text{Outlet 1} \leftarrow \begin{cases} \text{inlet stream} & \text{if expression is TRUE} \\ \text{zero flow} & \text{otherwise} \end{cases} \quad \text{Outlet 2} \leftarrow \begin{cases} \text{zero flow} & \text{if expression is TRUE} \\ \text{inlet stream} & \text{otherwise} \end{cases} \]

All stream properties (composition, temperature, pressure, enthalpy) are copied to the active outlet. The inactive outlet is assigned zero mass flow with the same composition and conditions as the inlet.

Example Expressions.

• T > 373.15: route to Outlet 1 if temperature exceeds 100 \(^\circ\)C.

• VF > 0.5 AND P < 500000: route to Outlet 1 if the stream is predominantly vapour at sub-5 bar pressure.

• W > 1.0: route to Outlet 1 if mass flow exceeds 1 kg s\(^{-1}\).

Material Stream Mapper

Overview.

The Material Stream Mapper copies an inlet material stream to an outlet stream and optionally overrides selected properties and per-compound flow rates. It is particularly useful for:

• connecting streams that use different compound lists (compound mapping);

• forcing a specific temperature, pressure, flow rate, or flash specification downstream without changing the upstream stream;

• scaling individual compound amounts by a fixed fraction or absolute value.

Stream Topology.

1 material inlet (Source Stream); 1 material outlet (Target Stream).

Calculation Procedure.

1. Copy all properties from the inlet to the outlet: \(\text{outlet} \leftarrow \text{inlet}\).

2. Apply optional property overrides in the following order:

   1. Flash specification (T-P, P-H, etc.)

   2. Temperature override: \(T_{\mathrm{out}} = T_{\mathrm{spec}}\)

   3. Pressure override: \(P_{\mathrm{out}} = P_{\mathrm{spec}}\)

   4. Total flow override (mass, molar, or volumetric)

3. Apply per-compound amount overrides for each compound \(c\):

\[ \dot{m}_c^{\mathrm{out}} = \begin{cases} \dot{m}_{c,\mathrm{spec}} & \text{(Mass Flow mode)} \\ \dot{n}_{c,\mathrm{spec}} M_c & \text{(Molar Flow mode)} \\ \dot{m}_c^{\mathrm{in}} \times v_c/100 & \text{(Percentage of Source mode)} \end{cases} \]

where $v_c$ is the user-specified percentage and $M_c$ is the molar mass of compound $c$.

1. Apply compound mappings: transfer the mass flow of compound \(c_1\) to compound \(c_2\) and zero out \(c_1\):

\[ \dot{m}_{c_2}^{\mathrm{out}} \mathrel{+}= \dot{m}_{c_1}^{\mathrm{in}}, \qquad \dot{m}_{c_1}^{\mathrm{out}} = 0 \]

After all overrides are applied, the outlet stream is flagged for recalculation so that the thermodynamic property package updates the equilibrium state consistently.

Override Options.

Override	Description
Flash specification	Change the flash type (e.g. T-P, P-H, P-VF)
Temperature	Fix outlet temperature to a specified value
Pressure	Fix outlet pressure to a specified value
Flow rate	Fix total mass, molar, or volumetric flow rate
Compound amount	Fix per-compound mass flow, molar flow, or percentage of inlet flow
Compound map	Rename/reassign a compound by mass-flow transfer

Thermo Property Editor

Overview.

The Thermo Property Editor is a pass-through block that provides a graphical interface to view and modify the thermodynamic property package parameters (binary interaction coefficients, equation-of-state parameters, activity-model parameters, etc.) associated with a material stream in the flowsheet.

Stream Topology.

1 material inlet; 1 material outlet.

Calculation.

The outlet stream is a direct copy of the inlet stream (\(\text{outlet} \leftarrow \text{inlet}\)); no material transformation is performed. The block’s sole purpose is to expose the property package settings for interactive editing without requiring the user to navigate to the global property package editor.

Typical Use Cases.

• Adjusting binary interaction parameters (\(k_{ij}\)) for a specific part of the flowsheet without altering the global property package.

• Inspecting pure-component parameters (critical properties, acentric factor, etc.) for components in a specific stream.

• Comparing property-package predictions with experimental data at a convenient location in the flowsheet.

Assumptions and Limitations (Additional Unit Operations)

1. Air Cooler 2, pure-air assumption: the cold-side fluid is treated as 100 % air using the Raoult property package. Humid air or alternative cooling media are not supported.

2. Air Cooler 2, no fin model: the current model computes the bare-tube external surface area (Eq. [eq:ac_area]) and does not account for extended surfaces (fins). Users with finned tubes should apply an equivalent fin-efficiency correction to \(A\) externally.

3. Air Cooler 2, cross-flow correction: the correction factor \(F\) is derived for a one-shell, two-tube-pass arrangement. Other configurations require manual adjustment of \(F\).

4. Falling Film Evaporator, no wall-temperature model: the heat flux is distributed uniformly over the tube length (linear pressure drop); local dry-out or nucleation effects are not captured.

5. Falling Film Evaporator, equilibrium flash at each step: the model assumes thermodynamic equilibrium at every integration step, which is equivalent to assuming an infinitely long residence time. Mass-transfer limitations are not modelled.

6. Stream switches, instantaneous evaluation: the Boolean expression is evaluated once per solver iteration using the current stream properties. No hysteresis or deadband logic is built in; users requiring hysteresis must implement it via a custom Python script.

7. Material Stream Mapper, no energy balance: property overrides (temperature, pressure, flow) are applied directly without checking an overall energy or mass balance around the block. It is the user’s responsibility to ensure that overridden values are physically consistent.

8. Premium requirement: all additional unit operations require an active DWSIM Premium Supporter subscription.

Free-Radical Polymerization Reactor

Overview

The Polymerization Reactor models a homogeneous, isothermal free-radical polymerization of one or two monomers. It solves the steady-state or transient population balances by the method of moments and reports the monomer conversion, the number- and weight-average molar masses (\(M_n\), \(M_w\)), the polydispersity index (PDI), and, for two monomers, the copolymer composition. A single vessel can be operated in four ways:

• Continuous stirred tank (CSTR): a perfectly mixed reactor solved at steady state; the composition is fixed at the outlet condition.

• Plug flow / batch (PFR): the balances are integrated along the residence time; the composition drifts as the more reactive monomer depletes.

• Semibatch: an initial charge plus a metered feed; feeding the reactive monomer holds the copolymer composition constant.

• Dynamic mode: the reactor is driven by the DWSIM dynamic integrator as a well-mixed holdup, producing the transient conversion and molar-mass trajectories.

The kinetic model follows the standard free-radical scheme : initiator decomposition, propagation, termination by combination and by disproportionation, and chain transfer to monomer and to a solvent or chain-transfer agent. For two monomers the terminal model is used, with the molar-mass averages obtained through the pseudo-kinetic rate-constant method . The auto-acceleration (gel) effect is available as an optional conversion-dependent reduction of the rate constants.

Stream Topology

Port	Direction	Description
Feed	Inlet (material)	Monomer(s), initiator, optional solvent / CTA
Product	Outlet (material)	Unreacted feed plus the polymer
Energy	Inlet (energy)	Heat duty (isothermal / outlet-temperature modes)

The monomer, initiator, optional solvent, and polymer product are identified by configurable compound names. The polymer product is a non-volatile pseudo-compound already present in the flowsheet (for example a PC-SAFT polymer); its molar mass is set to the computed \(M_n\). The heat-capacity correlations of a pseudo-compound are stored per kmol at its molar mass, so the reactor scales them by the same factor as the molar mass: the specific heat capacity of the polymer, and with it its specific enthalpy, stays that of its chemistry whatever the chain length. Setting a second monomer switches the reactor to the binary copolymerization model.

Kinetic Scheme

Every rate constant is of the Arrhenius form \(k = A\,\exp(-E/RT)\), with the pre-exponential \(A\) in s\(^{-1}\) (initiator) or L mol\(^{-1}\) s\(^{-1}\) (bimolecular) and the activation energy \(E\) in J mol\(^{-1}\). For a single monomer the elementary steps are

\[ \begin{align} \text{Initiation:} &\quad \ce{I ->[k_d] 2R^{.}}, \qquad \ce{R^{.} + M ->[k_i] P_1^{.}} \\ \text{Propagation:} &\quad \ce{P_n^{.} + M ->[k_p] P_{n+1}^{.}} \\ \text{Termination:} &\quad \ce{P_n^{.} + P_m^{.} ->[k_{tc}] D_{n+m}}, \quad \ce{P_n^{.} + P_m^{.} ->[k_{td}] D_n + D_m} \\ \text{Transfer:} &\quad \ce{P_n^{.} + M ->[k_{trM}] D_n + P_1^{.}}, \quad \ce{P_n^{.} + S ->[k_{trS}] D_n + P_1^{.}} \end{align} \]

where \(I\) is the initiator, \(M\) the monomer, \(S\) the solvent or chain-transfer agent, \(P_n^{.}\) a live radical of length \(n\), and \(D_n\) a dead chain. The total termination constant is \(k_t = k_{tc} + k_{td}\).

Radical population

The live-radical concentration is obtained from the quasi-steady-state assumption (the radical lifetime is far shorter than the reactor time), giving the classical result

\[ \mu_0 = \sqrt{\frac{f\,k_d\,[I]}{k_t}} \]

where \(f\) is the initiator efficiency and \(\mu_0\) is the total live-radical concentration (the zeroth live moment).

Steady-State CSTR

In a perfectly mixed reactor of volume \(V\) fed at volumetric rate \(Q\), the residence time is \(\theta = V/Q\). The initiator decomposes by first order, so its outlet is closed form,

\[ [I] = \frac{[I]_{\mathrm{in}}}{1 + k_d\,\theta} \]

and the monomer, consumed by propagation and transfer to monomer, follows

\[ [M] = \frac{[M]_{\mathrm{in}}}{1 + \theta\,(k_p + k_{trM})\,\mu_0}, \qquad X = 1 - \frac{[M]}{[M]_{\mathrm{in}}}, \qquad R_p = k_p\,\mu_0\,[M] \]

where \(X\) is the conversion and \(R_p\) the rate of polymerization.

Method of moments

With the most-probable closure (exact in the long-chain limit), the live moments are set by the propagation probability

\[ \alpha = \frac{k_p\,[M]}{k_p\,[M] + \Psi}, \qquad \Psi = k_t\,\mu_0 + k_{trM}\,[M] + k_{trS}\,[S] \]

where \(\Psi\) is the total chain-stopping rate. The first two live moments are

\[ \mu_1 = \frac{\mu_0}{1-\alpha}, \qquad \mu_2 = \frac{\mu_0\,(1+\alpha)}{(1-\alpha)^2} \]

The dead chains are generated by termination and transfer. Writing the transfer rate as \(\tau = k_{trM}\,[M] + k_{trS}\,[S]\), the moment source terms are

\[ \begin{align} G_0 &= \tau\,\mu_0 + \left(k_{td} + \tfrac{1}{2}k_{tc}\right)\mu_0^2 \\ G_1 &= \tau\,\mu_1 + k_t\,\mu_0\,\mu_1 \\ G_2 &= \tau\,\mu_2 + k_t\,\mu_0\,\mu_2 + k_{tc}\,\mu_1^2 \end{align} \]

The \(\tfrac{1}{2}k_{tc}\) factor in \(G_0\) counts one dead chain per combination event, while the \(k_{tc}\,\mu_1^2\) term in \(G_2\) is the convolution of two combining chains. In a CSTR the dead chains are only generated and swept out, so each dead moment is explicit,

\[ \lambda_k = \theta\,G_k \]

and the molar-mass averages follow directly:

\[ M_n = M_0\,\frac{\lambda_1}{\lambda_0}, \qquad M_w = M_0\,\frac{\lambda_2}{\lambda_1}, \qquad \mathrm{PDI} = \frac{\lambda_0\,\lambda_2}{\lambda_1^2} \]

where \(M_0\) is the monomer molar mass. In the long-chain limit the polydispersity reaches the theoretical values of \(3/2\) for termination purely by combination and \(2\) for termination purely by disproportionation ; chain transfer broadens it towards \(2\).

Binary Copolymerization (Terminal Model)

With a second monomer the reactor uses the terminal model, in which the reactivity of a growing chain depends only on the terminal unit. Propagation is described by the two homo-propagation constants \(k_{p,11}\), \(k_{p,22}\) and the two reactivity ratios

\[ r_1 = \frac{k_{p,11}}{k_{p,12}}, \qquad r_2 = \frac{k_{p,22}}{k_{p,21}} \]

The fraction of radicals ending in monomer 1 follows from the steady state on the radical types,

\[ \phi_1 = \frac{k_{p,21}\,[M_1]}{k_{p,21}\,[M_1] + k_{p,12}\,[M_2]} \]

and the instantaneous copolymer composition is the Mayo-Lewis equation

\[ F_1 = \frac{r_1 f_1^2 + f_1 f_2} {r_1 f_1^2 + 2 f_1 f_2 + r_2 f_2^2} \]

where \(f_i = [M_i]/([M_1]+[M_2])\) is the monomer mole fraction and \(F_1\) is the mole fraction of monomer 1 in the copolymer formed. The reactor recovers \(F_1\) directly from the monomer consumption rates, so it is identical to Eq. [eq:poly_mayolewis] by construction.

Molar mass by the pseudo-kinetic method

The molar-mass averages are obtained by treating the copolymerization as a homopolymerization with pseudo-kinetic (radical-fraction-averaged) rate constants . The pseudo-propagation constant is \(\bar{k}_p = R_p/(\mu_0\,[M])\) with \([M] = [M_1]+[M_2]\), the average chain-transfer-to-monomer constant is \(\bar{k}_{trM} = \phi_1 k_{trM,1} + \phi_2 k_{trM,2}\), and the average repeat-unit mass is

\[ \bar{M} = F_1\,M_{0,1} + (1 - F_1)\,M_{0,2} \]

Equations [eq:poly_alpha]–[eq:poly_mnmw] then apply with \(k_p \to \bar{k}_p\), \(M_0 \to \bar{M}\), and the transfer terms built from the averaged constants. The reactivity ratios are unchanged by chain transfer and by the gel effect, so the composition and the molar mass are decoupled: the composition always follows Eq. [eq:poly_mayolewis], while transfer and the gel effect act only on the chain length.

Gel (Trommsdorff) Effect

At high conversion the medium thickens and chain termination becomes diffusion-controlled, causing auto-acceleration . This is represented by a multiplicative factor \(g(X) \in (0,1]\) on the termination constant, and optionally on the propagation constant near vitrification (the glass effect), of the empirical exponential-polynomial form

\[ g(X) = \exp\!\left[-\left(c_1 X + c_2 X^2 + c_3 X^3\right)\right] \]

so that \(k_t(X) = k_{t,0}\,g_t(X)\) and \(k_p(X) = k_{p,0}\,g_p(X)\). Because the factors depend on the conversion, which depends on them, they are resolved by a fixed-point iteration around the reactor balances; with the default model disabled (\(g \equiv 1\)) the kinetics are unchanged. A lower \(k_t\) raises both the conversion and the molar mass, reproducing the observed auto-acceleration.

Plug-Flow and Batch Operation

A plug-flow reactor (equivalently a batch reactor, with the residence time read as the reaction time) is not perfectly mixed in the direction of flow, so the composition drifts as conversion builds. The monomer, initiator, dead-chain moment, and incorporated-monomer balances are integrated along the residence time by a fourth-order Runge-Kutta method, with the instantaneous kinetics of Sections 2.44.4 and 2.44.5 evaluated pointwise. The reactor reports both the instantaneous composition (the Mayo-Lewis value at the local monomer ratio, which moves along the reactor) and the cumulative composition (the average over all polymer formed),

\[ F_1^{\mathrm{cum}} = \frac{\Psi_1}{\Psi_1 + \Psi_2} \]

where \(\Psi_i\) is the total moles of monomer \(i\) incorporated into chains. For a non-azeotropic feed the two diverge as the more reactive monomer depletes; at the azeotropic composition (\(F_1 = f_1\)) there is no drift. The cumulative polydispersity broadens beyond the instantaneous combination limit of \(3/2\) as the batch accumulates chains formed under changing conditions.

Semibatch Operation and Composition Control

In a semibatch reactor the holdup grows as feed is added, so the balances are written on total amounts (moles, volume) rather than concentrations and are integrated in time. The feed policy is an initial charge plus constant molar and volumetric feed rates over a feed window. Metering the more reactive monomer in during the run holds the reactor monomer ratio, and hence the instantaneous copolymer composition, roughly constant. That is the industrial route to a uniform copolymer. In the monomer-starved limit (a high radical flux and a slow feed) the monomers react as fast as they are fed, so the copolymer composition equals the feed composition rather than the Mayo-Lewis value of that ratio, and the drift is suppressed.

Dynamic Mode

When the flowsheet is solved in dynamic mode, the DWSIM integrator advances the reactor as a well-mixed holdup: at each integration step the inlet feed is added to the holdup, the reaction is integrated over the step, and the conversion, molar-mass averages, and composition are updated from the accumulated state. Charging the vessel and then cutting the feed reproduces the batch trajectory; metering the feed in reproduces the semibatch trajectory. The step is sub-integrated internally, so a coarse integration step remains accurate. Operation is isothermal.

Energy Balance

The heat released by polymerization is proportional to the monomer converted,

\[ \dot{Q}_{\mathrm{gen}} = \dot{n}_{\mathrm{conv}}\,(-\Delta H_p) \]

where \(\dot{n}_{\mathrm{conv}}\) is the molar rate of monomer added to chains (both monomers in copolymer mode) and \(\Delta H_p < 0\) is the heat of polymerization per mole of monomer. In isothermal or outlet-temperature operation the duty holds the reactor at the set temperature, \(\dot{Q} = \dot{m}\,c_p\,(T_r - T_{\mathrm{in}}) - \dot{Q}_{\mathrm{gen}}\) (negative when heat is removed, the usual case for an exothermic polymerization). In adiabatic operation no heat is removed and the temperature rise and the conversion are coupled through Eq. [eq:poly_qgen] and the Arrhenius constants; they are solved together to a fixed point.

Mass balance

The polymer formed carries the mass of the reacted monomer, taken at the molar mass of the monomer compound, plus the mass of the consumed initiator, whose fragments become chain end groups. Polymer already present in the feed keeps its mass and leaves as moles of the same compound at the new \(M_n\). The product mass flow therefore equals the feed mass flow, and the unreacted monomer in the product matches the reported conversion.

Molar-Mass Distribution Emission

By default the polymer leaves the reactor as a single lumped compound whose molar mass is set to \(M_n\). Optionally the reactor emits a real molar-mass distribution: the Schulz-Zimm or log-normal distribution reproducing the computed \(M_n\) and \(M_w\) is discretized into a set of pseudo-component cuts that share the base polymer’s parameters, and the reacted mass is distributed over the cuts so that both the total mass and the number-average molar mass are preserved. A non-volatile cut set lets the downstream property package resolve devolatilization (stripping residual monomer while the polymer stays in the liquid). Each cut is a clone of the base polymer at its own molar mass, with its heat-capacity correlations scaled in the same way, so all cuts share the base polymer’s specific heat capacity.

Model Parameters

Configuration

Parameter	Symbol	SI Unit
Reactor volume	\(V\)	m\(^3\)
Operating temperature	\(T_r\)	K
Heat of polymerization	\(\Delta H_p\)	J mol\(^{-1}\)
Initiator efficiency	\(f\)	–

Arrhenius rate constants

Each constant is entered as a pre-exponential \(A\) and an activation energy \(E\): initiator decomposition (\(k_d\)), propagation (\(k_p\)), termination by combination (\(k_{tc}\)) and by disproportionation (\(k_{td}\)), transfer to monomer (\(k_{trM}\)), and transfer to solvent (\(k_{trS}\)). In copolymer mode the second monomer adds its propagation constant \(k_{p,22}\), its transfer-to-monomer constant, its molar mass, and the two reactivity ratios \(r_1\), \(r_2\). A styrene/AIBN preset (homopolymer) and a styrene/methyl methacrylate preset (copolymer) are provided as starting points and should be replaced with data for the system of interest.

Gel effect and distribution

The gel model is off by default. When enabled, the termination coefficients \(c_1\), \(c_2\), \(c_3\) (and optionally the propagation coefficients) of Eq. [eq:poly_gel] are supplied by the user. The distribution emission is controlled by the number of cuts and the distribution shape.

Assumptions and Limitations

1. Homogeneous, isothermal medium: the reactor contents are a single well-mixed phase (CSTR, semibatch, dynamic) or a plug-flow stream (PFR), at a uniform temperature. Emulsion, suspension, and precipitation polymerizations are not represented.

2. Quasi-steady-state radicals: the live-radical population follows Eq. [eq:poly_mu0]; the radical lifetime is assumed far shorter than the reactor or step time.

3. Long-chain / most-probable closure: the live-radical moments are closed with the most-probable distribution, exact in the long-chain limit.

4. Terminal copolymerization model: reactivity depends only on the terminal unit; penultimate-unit effects are not included. The molar mass uses the pseudo-kinetic averaging, and transfer to monomer is taken independent of which monomer is abstracted.

5. Constant density: the volumetric flow (and, in semibatch, the holdup volume) are additive in the feed; volume change on reaction is neglected.

6. Empirical gel effect: the gel and glass factors are an empirical correlation in conversion (Eq. [eq:poly_gel]); the coefficients are system-specific and must be fitted to data.

Numerical Solution Procedure

1. Read the feed conditions and the monomer, initiator, and optional solvent molar flows; form the inlet concentrations and the residence time \(\theta = V/Q\).

2. Determine the reaction temperature \(T_r\) from the operating mode (isothermal, outlet-temperature, or adiabatic); for adiabatic operation iterate \(T_r\) and the conversion to a fixed point through the energy balance.

3. Solve the kinetics at \(T_r\):

   1. CSTR: the closed-form balances (Eqs. [eq:poly_initiator]–[eq:poly_deadmoments]), with an outer fixed point for the gel factors and, in copolymer mode, an inner iteration for the coupled monomer balances.

   2. PFR / batch: integrate the balances along the residence time by Runge-Kutta.

   3. Semibatch / dynamic: integrate the total-amount balances in time, adding the feed each step.

4. Form \(M_n\), \(M_w\), PDI (Eq. [eq:poly_mnmw]) and, in copolymer mode, the composition (Eq. [eq:poly_mayolewis] or [eq:poly_cumcomp]).

5. Apply the energy balance (Eq. [eq:poly_qgen]) and set the heat duty.

6. Write the outlet: unreacted monomer(s) and initiator, plus the polymer (the reacted monomer and the consumed initiator, and any feed polymer at its own mass) as a lumped compound at \(M_n\) or as the emitted distribution.

Typical Usage Workflow

1. Add the polymer product as a non-volatile compound to the flowsheet (for example a PC-SAFT polymer), and add the monomer(s) and initiator.

2. Drop the Polymerization Reactor block and connect the feed, product, and (for isothermal or outlet-temperature operation) energy streams.

3. On the editor, select the flow model (CSTR, PFR/batch) and the operation mode (isothermal, adiabatic, outlet temperature). Assign the monomer, initiator, optional solvent, and polymer product compounds; for a copolymer, also assign the second monomer and the reactivity ratios.

4. Load a preset or enter the Arrhenius constants, the initiator efficiency, the heat of polymerization, and (optionally) the gel coefficients.

5. To emit a molar-mass distribution, set the number of cuts and the shape, generate the cuts, and enable distribution emission.

6. Run the simulation and read the conversion, \(M_n\), \(M_w\), PDI, and copolymer composition on the Results tab. For a transient study, configure a dynamic schedule and monitor the conversion and molar mass over time.

Worked Example

A bulk styrene polymerization initiated by AIBN at 60 °C in a CSTR with a one-hour residence time and 0.02 mol L\(^{-1}\) initiator gives a few percent conversion, a number-average molar mass of order \(10^5\) g mol\(^{-1}\), and a polydispersity near the combination limit of \(1.5\), broadened slightly by transfer to monomer. Replacing styrene with a styrene/methyl methacrylate feed and assigning the reactivity ratios \(r_1 = 0.52\), \(r_2 = 0.46\) produces a near-alternating copolymer (both ratios below one): for an equimolar feed the instantaneous composition is close to \(0.5\), and the same feed run as a batch to high conversion shows the styrene fraction drifting downward as the faster monomer depletes, while a starved semibatch feed at the target ratio holds the composition constant.

Vessel Depressurization (Blowdown)

Overview

The Vessel Depressurization tool (Dynamics menu, both the classic and the cross-platform interfaces) computes the pressure, temperature, released flow and wall temperature history of a vessel that is blown down through a restriction orifice or a blowdown valve, or that is exposed to a pool fire while it relieves. It answers the questions a depressurization study asks in the sense of API Standard 521 : how long the vessel takes to reach a target pressure, what the lowest fluid and metal temperatures are (the minimum design metal temperature check), what relief rate a fire imposes and how hot the unwetted metal gets.

The tool takes a material stream of the flowsheet as its source, which supplies the composition and the property package (on the classic interface it can also be attached to a stream through Utilities, Add Utility). Everything else is entered in the utility itself: initial pressure and temperature, initial liquid level, vessel orientation and dimensions, head type, wall thickness and material, the outlet nozzle height, the orifice bore and discharge coefficient (or a valve \(C_v\)), the back pressure, the valve opening time, the heat case (adiabatic, fire or isothermal), the ambient temperature, the API 521 fire parameters, the time step and the duration. The results are a summary (time to the stop pressure, peak flow, released mass, minimum fluid and wall temperatures, maximum dry wall temperature), three plots (pressure, temperatures, released flow) and the full time series, which can be exported.

The utility runs the same dynamic vessel and valve models that a dynamic flowsheet uses; it builds a private flowsheet with a vessel, a blowdown valve and a sink, and integrates it with the standard integrator. The dynamic properties described below are therefore also available on a Vessel block in a dynamic simulation.

Vessel content

The vessel is a rigid volume \(V\) holding a mass \(m\) of the fluid at a uniform temperature \(T\) and pressure \(P\). Over a time step \(\Delta t\) the content changes by the inlet and outlet flows,

\[ m_{1} = m_{0} + \sum_{\text{in}} \dot{m}_{i}\,\Delta t - \sum_{\text{out}} \dot{m}_{o}\,\Delta t , \]

and, with the Rigorous Energy Balance (UV) property on, the internal energy changes by the enthalpy carried in and out and by the heat received through the wall,

\[ U_{1} = m_{0}\,h_{0} - P_{0}\,V + \sum_{\text{in}} \dot{m}_{i} h_{i}\,\Delta t - \sum_{\text{out}} \dot{m}_{o} h_{o}\,\Delta t + Q\,\Delta t . \]

The new state is the solution of a volume-internal energy flash: the temperature and pressure at which the mixture of mass \(m_{1}\) occupies exactly \(V\) with the internal energy \(U_{1}\). The flash is solved as an outer temperature search closed by a volume-temperature flash for the pressure, so the content is always on the property package’s own pressure-volume-temperature-energy surface. A gas expanding through the orifice therefore cools along the isentrope corrected for the heat it receives, a boiling liquid cools along its bubble curve, and a fire heats the content and raises its pressure. With the energy balance off the legacy behaviour is kept: the content is flashed at constant temperature, which is the isothermal case.

Two details of the volume flashes matter for a blowdown. A pure compound (or a mixture with more than 99.9 % of one compound, which a stripped CO\(_2\) or steam inventory becomes) has no two-phase pressure window: at a given temperature it is liquid above its vapour pressure, vapour below it and any split exactly at it. The volume-temperature flash handles that case with the package’s bubble pressure and the lever rule between the saturated liquid and vapour volumes. A compressed liquid takes its saturated volume from the liquid density correlation of the package and its compression from the equation of state, so that pressure, volume and internal energy come from one consistent surface; the pressure correction of the density correlations is far too stiff near the critical point, and used on its own it makes a liquid-full vessel lose most of its pressure in the first step. Above the mixture’s critical locus, where no bubble point exists, the dense phase takes its volume from the equation of state.

The Minimum Pressure property is a floor: a vessel vented to the atmosphere, or fitted with a vacuum breaker, stays at that pressure and keeps its liquid level once the blowdown is over.

Outlets

The liquid level follows from the liquid volume of the flash and the vessel geometry (vertical or horizontal cylinder with the selected heads). The gas outlet carries vapour while the level is below its nozzle and liquid once the level has reached it, with a mass-weighted blend across a short transition band so that the integration never sees a step. With the nozzle at the top (the default) a liquid-full vessel blows down as liquid until a gas space forms; with the nozzle at mid-height, a pipe blown down through a hole at its end passes liquid until the level falls below the hole; a nozzle below the top is taken as a hole spanning its own diameter, so the outlet passes a blend while the level is within it. With the Gas Outlet Homogeneous option the outlet takes the content at its bulk quality while both phases exist, which is the homogeneous two-phase flow of a pipe blown down through a hole at its end, where the flow sweeps the liquid to the hole. The liquid outlet is the mirror image: it carries liquid while the level is above its nozzle and gas once the level has dropped below it, which is the gas blow-by case a downstream low-pressure system has to be checked for. The gas fraction of the liquid outlet and the liquid fraction of the gas outlet are reported as read-only properties.

Blowdown orifice

The blowdown valve is a Valve block in \(K_v\) mode with the Use Orifice Flow option, given by its bore \(d\) and discharge coefficient \(C_d\). The mass flow is that of a homogeneous-equilibrium nozzle on the property package’s own isentrope. The fluid expands from the vessel state \((h_{0}, s_{0})\) to a throat pressure \(P_{t}\) along the isentrope, and the mass flux is

\[ G(P_{t}) = \frac{\sqrt{2\,\left[h_{0} - h(P_{t}, s_{0})\right]}} {v(P_{t}, s_{0})} , \qquad \dot{m} = C_d\,\frac{\pi d^{2}}{4}\,\max_{P_{2} \le P_{t} \le P_{1}} G(P_{t}) , \]

where \(v\) is the specific volume of the (possibly two-phase) mixture at the throat. The flux rises as the throat pressure falls until the mixture reaches its own speed of sound; the flow is choked there and no longer responds to the back pressure. For an ideal gas this reduces to the textbook nozzle formula with the critical pressure ratio \((2/(k+1))^{k/(k-1)}\). For a dense supercritical fluid or a flashing liquid the choke sits far above the ideal-gas ratio, which is the case a blowdown valve or a leak on a high-pressure line has to be sized for . The throat pressure is searched by a golden-section method over the logarithm of the pressure, or by a short hill climb from the previous time step’s throat, and is reported by the valve. When the isentropic flash cannot be followed the closed forms are used as a fallback: the ideal-gas nozzle for a gas and \(C_d A \sqrt{2 \rho \Delta P}\) for a liquid. The valve opening can be ramped linearly over a given time, which scales the open area.

A valve given by its flow coefficient uses the ISA/IEC 60534 sizing forms of the Valve block instead, with choking for gas and the two-phase form for a mixed inlet.

Wall and heat transfer

With the Split Wall (Wetted/Dry) property on, the wall is two metal segments, the one wetted by the liquid and the one in contact with the vapour, with areas that follow the liquid level. Each segment is a one-dimensional conduction slab across the wall thickness, discretized in three to twelve nodes and integrated implicitly (backward Euler with a tridiagonal solve), so that thin walls need no smaller time step than the vessel itself. The inner node exchanges heat with the fluid through the film coefficient of the phase it touches, the outer node with the ambient through the external coefficient and receives any solar or fire flux. The inner-surface temperatures are the ones reported and tracked as the minimum wetted and dry wall temperatures, which is what a thermocouple or a minimum design metal temperature check looks at; the hottest node of the dry segment is the maximum dry wall temperature of the fire case.

A still vessel has no forced convection, so the film coefficients are those of turbulent natural convection on a wall,

\[ \mathrm{Nu} = 0.13\,(\mathrm{Gr}\,\mathrm{Pr})^{1/3} \quad \Rightarrow \quad h = 0.13\,k\left(\frac{g\,\beta\,|T_{w} - T|\,\rho^{2}}{\mu^{2}}\,\mathrm{Pr}\right)^{1/3} , \]

in which the length scale cancels . The properties are those of the liquid for the wetted segment and of the vapour for the dry one, evaluated at the vessel pressure, so the coefficient on the gas side falls with the density as the vessel empties. A floor of 50 and 5 W/(m\(^2\)K) keeps the exchange from switching off. The Internal Heat Transfer Factor multiplies both coefficients; it is the knob for a sensitivity study, and the validation below shows what a factor of 0.5 to 0.7 does to a cold blowdown. When the thermal profile of the vessel is set to a user-defined overall coefficient, that value is used for both segments instead.

In the fire case the heat absorbed by the liquid follows API 521 ,

\[ Q_{\text{fire}} = C\,F\,A_{w}^{0.82} , \]

with \(C = 43\,200\) W/m\(^{1.64}\) when there is adequate drainage and prompt firefighting and \(70\,900\) otherwise, \(F\) the environment factor and \(A_{w}\) the wetted area within 7.6 m of grade, counted from the Vessel Bottom Elevation. The wetted metal stays at the liquid temperature, as API 521 assumes for a wall backed by boiling liquid. The dry wall receives the user-given Fire Dry Wall Heat Flux on its outer face and passes heat to the vapour through the conduction slab, so the vapour superheats and the unwetted metal temperature is tracked.

Integration

The vessel is integrated with an explicit first-order step of the chosen length; the flashes and the wall are solved implicitly inside the step. A step of 0.5 to 1 s is adequate for a gas blowdown of minutes; a liquid-full vessel, whose pressure falls by megapascals per kilogram released, needs 0.1 to 0.2 s until a gas space forms. The run stops at the given duration or when the stop pressure is reached, and a warning is issued if the integration stops early.

Validation

The model was checked against a closed-form solution, against its own property package and against the published cases below. All runs use the Peng-Robinson package. The tests are part of the engine test suite (DepressurizationValidationTests), so the figures are reproduced on every build.

Ideal-gas blowdown

Adiabatic blowdown of an ideal gas through a choked orifice has a closed form: with \(a = (C_d A/V)\,c_0\,\psi\), \(\psi = (2/(k+1))^{(k+1)/(2(k-1))}\) and \(c_0 = \sqrt{k R T_0/M}\),

\[ \frac{P}{P_0} = \left[1 + \tfrac{k-1}{2}\,a\,t\right]^{-2k/(k-1)}, \qquad \frac{T}{T_0} = \left[1 + \tfrac{k-1}{2}\,a\,t\right]^{-2} . \]

Methane at 10 bar and 300 K is within 2 % of ideal. Over a 280 s blowdown to 2 bar the model follows the pressure curve within 0.8 % and the temperature within 5 K (the closed form takes a constant \(k\); the package’s \(c_p\) varies with temperature).

Real-gas isentrope

With the wall switched off, the content of a blowdown must stay on the isentrope of the property package. A 60 to 5 bar blowdown of a real gas stays within 1.4 K of the temperature that a pressure-entropy flash gives at the same pressure, which is the drift of the explicit step.

Nitrogen blowdown of Haque et al.

Haque, Richardson, Saville, Chamberlain and Shirvill blew down nitrogen from 150 bar and about 20 \(^{\circ}\)C in a 0.273 m ID \(\times\) 1.524 m vertical vessel with a 25 mm carbon steel wall through a 6.35 mm top orifice; the record is redrawn in the review of Shafiq et al. , Fig. 5b, as changes from the initial temperature. Table 10 compares the run with the figure. The wall is reproduced, the pressure is reproduced, and the gas minimum comes out 15 to 20 K warmer than measured with the natural-convection coefficient as estimated; a factor of 0.6 to 0.7 on the coefficient reproduces the measured gas curve, which is why the factor is exposed. The adiabatic bound is given for reference.

Quantity	Measured	Model, factor 1.0	Model, factor 0.5	Adiabatic
Gas temperature drop at its minimum	108 K at \(\approx\)40 s	90 K at 32 s	115 K at 42 s	230 K
Gas temperature drop at 100 s	60 K	43 K	76 K	214 K
Inner wall temperature drop	5.5 K	4 K	2.6 K	0
Time to atmospheric pressure	\(\approx\)100 s	2 bar at 82 s, 1 bar at 100 s	2 bar at 84 s	2 bar at 78 s

Nitrogen blowdown of Haque et al. : 150 bar, 6.35 mm orifice, 0.0892 m\(^3\), 25 mm wall.

CO\(_2\) blowdown of Fredenhagen and Eggers

Fredenhagen and Eggers blew down CO\(_2\) with a nitrogen impurity from a top-vented, liquid-full 0.05 m\(^3\) vessel (0.242 m ID) at 14 MPa and 298 K through a 17 mm\(^2\) orifice; pressure and temperature transients are redrawn in , Figs. 10 and 11. The nitrogen content is not given in the review and was inferred from the kink of the pressure record at 8 MPa, the end of the liquid-full stage: 6 mol% gives a bubble pressure of 7.3 MPa at 286 K, and the Peng-Robinson bubble curve ends between 7 and 8 % at that temperature. The vessel was run with a 25 mm carbon steel wall and \(C_d = 0.8\). Table 11 gives the comparison for pure CO\(_2\) and for the inferred mixture. The two-phase stage, which is the one of interest, is reproduced within 0.5 MPa and 5 K to 100 s. The duration of the liquid-full stage is right for pure CO\(_2\) and too short for the mixture, whose dense phase Peng-Robinson makes too compressible near the critical locus; at the end of the run the mixture cools too fast as it approaches the triple point of CO\(_2\), which the model does not represent.

Quantity	Measured	Pure CO\(_2\)	CO\(_2\) + 6 % N\(_2\)
End of the liquid-full stage	\(\approx\)3 s, at 8 MPa	3.5 s, at 4.8 MPa	\(<\)1 s, at 6.7 MPa
Pressure at 20 / 40 / 80 s, MPa	5.6 / 3.9 / 2.05	4.0 / 3.3 / 2.2	4.8 / 3.4 / 1.9
Temperature at 20 / 40 / 80 s, K	277 / 270 / 255	279 / 271 / 257	275 / 266 / 251
Temperature at 150 s, K	236	241	225

CO\(_2\) blowdown of Fredenhagen and Eggers : 0.05 m\(^3\), 14 MPa, 298 K, 17 mm\(^2\) orifice, liquid-full.

Ethylene hole flows of Saville, Richardson and Barker

Saville, Richardson and Barker computed with the BLOWDOWN program the flow of ethylene at 10 \(^{\circ}\)C from a pipeline through 10 and 50 mm holes to the atmosphere, with the fluid supercritical (90, 79, 69 and 59.6 bar) or gaseous (44 and 28 bar), and found the hole choked even for the all-liquid dense phase, at a throat pressure of 41 to 45 bar, far above the ideal-gas critical ratio. These are predictions of a homogeneous-equilibrium model, not measurements, so the comparison in Table 12 is between two implementations of the same physics. The orifice model of equation ([eq:blowdown_hem]) with \(C_d = 1\) reproduces the flows within 8 % for the dense phase and 3 % for the gas, and the dense-phase throat within 4 bar. The model runs 3 to 8 % high in the dense phase, consistent with the Peng-Robinson density at 90 bar (367 kg/m\(^3\), some 6 % below the real value). In the gas cases the flux curve is bimodal, with a gas-side maximum near 27 bar and a two-phase one near 15 bar of the same height; the paper reports the two-phase one.

\(P_0\) (bar)	Hole (mm)	State	\(\dot{m}\) BLOWDOWN (kg/s)	\(\dot{m}\) DWSIM (kg/s)	Deviation	\(P_t\) BLOWDOWN (bar)	\(P_t\) DWSIM (bar)
90	10	dense	4.40	4.75	+7.9 %	41.4	37.9
79	10	dense	3.80	4.05	+6.5 %	43.1	39.4
69	10	dense	3.20	3.31	+3.4 %	45.4	41.1
59.6	10	dense	2.30	2.45	+6.4 %	n/a	43.6
44	10	gas	0.95	0.98	+2.7 %	15.5	27.3
28	10	gas	0.55	0.56	+2.6 %	n/a	15.7
90	50	dense	114	118	+3.6 %	45.4	37.8
44	50	gas	24	24.4	+1.7 %	17.7	27.4

Ethylene at 283 K through a hole to the atmosphere, \(C_d = 1\): BLOWDOWN predictions against the orifice model. \(\dot{m}\) is the mass flow and \(P_t\) the throat pressure.

LPG pipeline blowdown of Richardson and Saville

Richardson and Saville compared BLOWDOWN with the Isle of Grain tests, in which 100 m lines full of LPG (95 mol% propane, 5 % butane) were blown down through orifices at their end. Test P47 used the 154 mm ID line (7.3 mm wall, 1.86 m\(^3\)) at 21.3 bar and 14.6 \(^{\circ}\)C, ambient 15.4 \(^{\circ}\)C, with a 50 mm nominal orifice for which the paper adopts an equivalent diameter of 70.4 mm and \(C_d = 0.80\). The orifice is small against the bore, and the paper notes that the open- and closed-end pressures nearly coincide in this test, so the line can be treated as a vessel. BLOWDOWN takes the flow along the line as homogeneous two-phase, and Table 13 compares the measured record (Fig. 4 of the paper, closed end) with the model run the same way, the outlet taking the bulk quality, and with the stratified run in which the hole passes the phase at its level. The homogeneous run follows the measurement to about 60 s, within 0.6 bar, 3 K and 0.07 t; it then empties the line, as the BLOWDOWN prediction also did, whereas the measured inventory keeps a residual of about 0.1 t because the orifice acts as a dam, and the measured pressure and temperature tail off more slowly. The stratified run holds the liquid back below the hole and lets the pressure fall too fast, which is why the homogeneous option exists.

Isle of Grain test P47 : LPG, 21.3 bar, 14.6 \(^{\circ}\)C, 70.4 mm equivalent orifice, \(C_d = 0.80\). Closed-end pressure and temperature; inventory in tonnes. The measured pressure at 0 s is the value after the sub-second expansion of the compressed liquid.

Time (s)	Measured			Model, homogeneous outlet
	\(P\) (bar)	\(T\) (\(^{\circ}\)C)	Inventory (t)	\(P\) (bar)	\(T\) (\(^{\circ}\)C)	Inventory (t)
0	7.4	14.6	0.95	21.3	14.6	0.97
20	7.0	11	0.60	6.4	11.7	0.64
40	6.3	5	0.37	5.8	8.1	0.36
60	5.0	–3	0.22	4.5	–0.5	0.15
80	3.3	–13	0.15	2.1	–23	0.03
100	2.0	–24	0.11	1.0	–40	0.01
Minimum		–33			–40
Stratified outlet, for comparison: 4.3 bar, –1.4 \(^{\circ}\)C and 0.50 t at 40 s; 1.8 bar, –26 \(^{\circ}\)C and 0.37 t at 80 s.

Limitations

The content is one well-mixed gas zone and one well-mixed liquid zone at a common temperature; the model has no thermal stratification of the gas, no non-equilibrium (metastable) flashing at the orifice, no solid formation (dry ice, hydrates), no free-water zone and no pressure drop along a pipeline. A pipe is represented as a vessel only when the hole is small against the bore. The wall is a slab: the heads take the shell thickness, and there is no axial conduction. The natural-convection correlation runs warm on the gas minimum of a cold blowdown, as shown above; a conservative minimum fluid temperature is obtained with the heat transfer factor at 0.5.

Column Internals (Trays and Packings)

Overview

The Column Internals tool (Utilities menu, both the classic and the cross-platform interfaces) rates the internals of a rigorous column stage by stage after the column has been solved. For a trayed section it reports the fraction of flood, the pressure drop, the weeping margin, the fractional entrainment, the downcomer backup and the downcomer residence time of every tray, plus the valve state of a valve tray and the slot opening of a bubble-cap tray; for a packed section it reports the fraction of flood, the pressure drop per metre, the liquid holdup, the wetting ratio, the HETP and the bed height the stages of the section need. A section whose diameter is left at zero is sized so that its worst stage sits at a target fraction of flood; a section with a diameter is rated as it is.

The tool also exists as a utility attached to a column (Add Utility on the classic interface, the Utilities tab of the column editor on the cross-platform one). The attached utility keeps the case in the simulation file, publishes the highest fraction of flood, the column pressure drop, the required diameter and the internals height among the column’s properties, and rates the column again whenever the flowsheet is solved with the update option on.

The tool reads everything it needs from the column’s last solution: the vapour rising into each stage, the liquid leaving it, and the densities, viscosities and surface tension of both phases at the stage temperature, pressure and compositions, computed by the column’s property package. The column is divided into sections, ranges of stages that carry one kind of internal (sieve, valve or bubble-cap tray, random or structured packing) with one geometry. Stage 1 is the top stage; on a distillation column the condenser and the reboiler are stages 1 and \(N\) and are normally left out of the sections.

The inputs are the design targets (fraction of flood for sizing trays and packings, minimum downcomer residence time, turndown ratio checked for weeping), the settings of the iteration with the solver, and, per section, the geometry: for every tray the tray spacing, downcomer area fraction, weir height, downcomer clearance, plate thickness, the flooding correlation and a system (foaming) factor; for sieve trays the hole diameter and hole area fraction; for valve trays the valves per unit of active area, the deck hole diameter, the valve thickness, material density, legs and orifice shape; for bubble-cap trays the cap and riser diameters, the pitch, the slots (number, width, height), the static seal, the skirt clearance and, optionally, the liquid gradient; for packings the packing from the built-in catalogue or a user-defined one, the bed height, the capacity and pressure drop model, the HETP model and the diffusivities of the transferring component. Cases are saved to .dwint files and a case file that carries the flowsheet’s own name next to it is loaded when the tool opens.

Sieve trays

The tray rating follows the design procedure of Towler and Sinnott (also Coulson and Richardson volume 6), with the alternatives Kister recommends . All heads are in millimetres of clear liquid, as the books tabulate them.

Areas

With the column diameter \(D_c\), the total area \(A_c\), the downcomer area \(A_d = f_d A_c\) (a segment of the circle; the weir length \(l_w\) follows from the chord geometry, \(A_d/A_c = (\theta - \sin\theta)/2\pi\) with \(l_w/D_c = \sin(\theta/2)\)), the net area \(A_n = A_c - A_d\) available to the vapour above the tray, the active area \(A_a = A_c - 2A_d\) and the hole area \(A_h = f_h A_a\).

Entrainment flooding

The flow parameter is \(F_{LV} = (L_w/V_w)\sqrt{\rho_V/\rho_L}\) on the mass flows. With the default Fair (1961) correlation the flooding velocity on the net area is

\[ u_f = K_1 \left(\frac{\sigma}{0.02}\right)^{0.2} \sqrt{\frac{\rho_L - \rho_V}{\rho_V}} , \]

where \(K_1(F_{LV}, l_t)\) is the Fair chart (Towler Figure 11.29) in the analytical form of Lygeros and Magoulas ,

\[ K_1 = 0.0105 + 8.127\times10^{-4}\, l_t^{\,0.755} \exp\left(-1.463\,F_{LV}^{\,0.842}\right) , \]

with \(l_t\) the tray spacing in millimetres and \(K_1\) in m/s. The fit reproduces the chart within 3 % over \(0.01 \le F_{LV} \le 1\). The alternative is the Kister and Haas correlation , which Kister recommends for sieve and valve trays,

\[ C_{SB} = 0.144 \left(\frac{d_h^2 \sigma}{\rho_L}\right)^{0.125} \left(\frac{\rho_V}{\rho_L}\right)^{0.1} \left(\frac{S}{h_{ct}}\right)^{0.5} , \]

in ft/s with \(d_h\), \(S\) and \(h_{ct}\) in inches, \(\sigma\) in dyn/cm (capped at 25) and the densities in lb/ft\(^3\); \(h_{ct}\) is the clear liquid height at the froth-to-spray transition from the Jeronimo and Sawistowski correlation with the Kister and Haas property correction (Kister equations 6.68 to 6.70). For valve trays the open valve area replaces the hole area. The fraction of flood is \(u_n/u_f\) with \(u_n\) the vapour velocity on the net area, after the system factor.

Entrainment

The fractional entrainment \(\psi\) (kg entrained per kg of gross liquid flow) is read from Fair’s chart (Towler Figure 11.31), digitised as a table in \(F_{LV}\) and percent of flood. Its effect on the tray efficiency is Colburn’s \(E_a = E_{mv}/[1 + E_{mv}\psi/(1-\psi)]\).

Weeping

The weir crest is the Francis formula for a segmental weir, \(h_{ow} = 750\,[L_w/(\rho_L l_w)]^{2/3}\), and the weep-point hole velocity is

\[ u_{h,\min} = \frac{K_2 - 0.90\,(25.4 - d_h)}{\sqrt{\rho_V}} , \]

with \(d_h\) in mm and \(K_2\) the Eduljee constant read from Towler Figure 11.32 at the clear liquid depth \(h_w + h_{ow}\) of the turndown rate. The tool reports the ratio of the actual hole velocity to the weep point at the design rate and at turndown.

Pressure drop

The dry plate drop is the orifice form \(h_d = 51\,(u_h/C_0)^2\,\rho_V/\rho_L\) with the Liebson coefficient \(C_0\) (Towler Figure 11.36) as a function of the plate thickness to hole diameter ratio and of the hole area fraction; the residual head is \(h_r = 12500/\rho_L\); the total head is \(h_t = h_d + h_w + h_{ow} + h_r\) and the pressure drop \(9.81\times10^{-3}\,h_t\rho_L\) Pa.

Downcomer

The head loss under the apron is \(h_{dc} = 166\,[L_w/(\rho_L A_m)]^2\), with \(A_m\) the smaller of the downcomer area and the area under the apron; the backup is \(h_b = h_w + h_{ow} + h_t + h_{dc}\) and must stay below half the tray spacing plus the weir height; the residence time is \(A_d h_b \rho_L/L_w\) and should exceed 3 s.

Efficiency

Each tray also reports the O’Connell overall efficiency in the Eduljee form used by Towler and Sinnott (equation 11.67), \(E_o = 0.51 - 0.325\log_{10}(\mu_L \alpha)\), with \(\mu_L\) the liquid viscosity in mPa\(\cdot\)s and \(\alpha\) the relative volatility of the two key components on the stage (the component transferring most to the vapour against the one transferring most to the liquid).

Writing the rating back into the column

Two buttons send the rating to the column. Pressures to column keeps the top stage pressure and adds the rated pressure drop of each tray (or of each packed stage’s share of the bed) to the stage below, and switches the column’s linear pressure drop off so the solver uses the stage pressures. Efficiencies to column writes the O’Connell value of every rated tray into the stage efficiency. The column is left to be solved again; rating it once more closes the loop, and a second pass usually changes the pressure drop by a few percent only.

Packed beds and the number of stages

A packed section whose bed height is given carries its own number of theoretical stages: bed height divided by the average HETP of the section. Stages to column sets it on the column, inserting theoretical stages evenly into the section or removing stages that carry no feed, draw or duty, so every stream keeps the stage it is connected to; the stage ranges of the other sections move to follow. The rating also writes the sized diameter, the internals height and the height of every rated stage (the tray spacing, or the HETP of a packed stage) into the column, where the costing and the dynamic holdups read them, and sets the efficiency of a packed stage to 1, since the HETP already carries the efficiency of the bed. The case itself (sections, geometry, models) is kept in the column and saved with the simulation; the tool loads it back when the column is picked, before looking for a case file next to the flowsheet.

Packed stages in the dynamic column

The rating also marks every stage of a packed section as packed and records its packing (the catalogue name, or the constants of a user-defined packing) and its capacity model on the stage, and these are saved with the column. In dynamic mode such a stage is a slice of bed of height equal to its HETP: the vapour flowing up through it comes from the bed pressure drop correlation of the section (Robbins, Billet and Schultes or Rocha, Bravo and Fair, at the liquid rate of the moment) inverted for the pressure difference between the stage below and the stage, and the liquid draining from it comes from the bed holdup correlation (Rocha, Bravo and Fair for a structured packing with its corrugation side, Billet and Schultes otherwise) inverted for the liquid the slice holds, in place of the orifice and weir equations of a tray. When the dynamics start, the content of a packed stage is scaled so that its liquid volume equals the holdup of the bed at the steady-state liquid rate. The Flooding alarm of the column trips when a packed stage passes the flood point of its packing correlation and the Weeping alarm when it falls below the minimum wetting rate of the packing; tray stages keep the Souders-Brown check.

Iteration with the solver

Rate and iterate with the solver closes that loop on its own: it rates the column, re-stages the packed beds that have a bed height (this can be switched off), writes the pressures and the efficiencies (each can be switched off), solves the flowsheet, rates again and repeats until the column pressure drop changes by less than the tolerance (2 % by default), no stage efficiency moves by more than 0.01 and the number of stages stops changing, or until the passes run out (six by default). The summary lists every pass. On the extractive distillation sample the profile settles after a single pass.

Valve trays

Valve trays share the areas, the weir crest, the downcomer and the entrainment of the sieve tray procedure, and either flooding correlation: Fair, or Kister and Haas on the open valve area. The dry pressure drop follows Klein , the procedure Kister recommends (section 6.3.2 and Table 6.9 of ), which fine-tunes the Bolles (1976) model. The hole velocity \(u_h\) is taken on the deck holes, \(A_h = n_v\,\pi d_v^2/4\) with \(n_v\) the number of valves and \(d_v\) the deck hole diameter (standardised at 1.5 in). Two balance points come from the weight of the valve: at the closed balance point the vapour starts to lift the valves,

\[ u_{h,CBP} = \sqrt{t_v R_{vw}\,\frac{C_{vw}}{K_c}\,\frac{\rho_{vm}}{\rho_V}} \quad\text{(ft/s, $t_v$ in inches)}, \]

with \(t_v\) the valve thickness, \(R_{vw}\) the ratio of the weight with legs to the weight without (1.23, 1.34 and 1.00 for flat valves with three legs, four legs and caged; 1.29, 1.45 and 1.00 for venturi valves), \(C_{vw} = 1.3\) the eddy loss coefficient, \(K_c\) the closed loss coefficient (6.154 flat, 3.077 venturi) and \(\rho_{vm}\) the valve metal density; at the open balance point all valves are open, \(u_{h,OBP} = u_{h,CBP}\sqrt{K_c/K_o}\) with the open loss coefficient \(K_o\) (0.448 for venturi valves; 0.821, 0.931 and 1.104 for flat valves on 0.134, 0.104 and 0.074 in decks, scaled with \(1/\sqrt{t}\) at other deck thicknesses). The dry drop is then \(h_d = K\,(\rho_V/\rho_L)\,u_h^2\) in inches with \(K = K_c\) below the closed balance point, \(K = K_o\) above the open balance point and, between them, the constant value of the closed balance point. The aerated liquid is \(h_L = \beta\,(h_w + h_{ow})\) with the aeration factor \(\beta = 0.5 + 0.48\exp(-1.4 F_{va})\), a fit of the Fair and Bolles curve (Kister Figure 6.22) through the valve tray point of Klein’s example, \(F_{va} = u_a\sqrt{\rho_V}\) in ft/s(lb/ft\(^3\))\(^{0.5}\) on the active area. Weeping is read against the closed balance point: below it the valves sit on the deck and the liquid finds the crevices, so the tray warns when \(u_h\) at turndown falls under \(u_{h,CBP}\), and when the unit reference \(u_h/u_{h,OBP}\) at turndown falls under 40 % (Kister asks 40, 60 and 80 % for one-, two- and four-pass trays to avoid vapour channelling). The table reports the valve state and the unit reference of every tray.

Glitsch procedure

The alternative for valve trays is the procedure of the Glitsch Ballast Tray Design Manual , in the units of the manual (vapour load \(V_{load} = CFS\sqrt{D_V/(D_L - D_V)}\) in ft\(^3\)/s, liquid in gpm, areas in ft\(^2\), lengths in inches, heads in inches of liquid). The vapour capacity factor at zero liquid load, \(CAF_0\), is read from Figure 5 of the manual for the tray spacing and the vapour density (the chart’s curves for 12 to 48 in, tabulated and interpolated; the low-density equation \(CAF_0 = TS^{0.63} D_V^{1/6}/12\) below 0.17 lb/ft\(^3\); the limit line above 4 lb/ft\(^3\); the smallest of the three), \(CAF = CAF_0\) times the system factor, and the per cent of flood at constant V/L is

\[ \frac{\%\,\text{flood}}{100} = \frac{V_{load} + GPM\,FPL/13000}{AA\,CAF} , \]

with \(AA\) the active area and \(FPL\) the flow path length of a single-pass tray, \(12 D_T\) less twice the downcomer width. A weir taller than 15 % of the spacing shortens the spacing used for \(CAF_0\) by the excess. The downcomer is checked against its design velocity, the smallest of 250, \(41\sqrt{D_L - D_V}\) and \(7.5\sqrt{TS}\sqrt{D_L - D_V}\) gpm/ft\(^2\) times the system factor. The dry pressure drop is the larger of \(1.35\,t_m D_m/D_L + K_1 V_H^2 D_V/D_L\) (units part open) and \(K_2 V_H^2 D_V/D_L\) (units fully open), with the hole velocity \(V_H\) on \(NU/78.5\) ft\(^2\) of holes for \(NU\) Ballast units, \(K_1 = 0.20\) and \(K_2 = 1.18\), 0.95, 0.86, 0.67 and 0.61 on decks of 0.074 to 0.250 in for V-1 (flat orifice) units, \(K_1 = 0.10\) and \(K_2 = 0.68\) for V-4 (venturi) units; the total is \(\Delta P = \Delta P_{dry} + 0.4\,(GPM/L_{wi})^{2/3} + 0.4 H_w\) and the backup \(H_{dc} = H_w + 0.4\,(GPM/L_{wi})^{2/3} + (\Delta P + H_{ud})\,D_L/(D_L - D_V)\) with \(H_{ud} = 0.65 V_{ud}^2\) under the downcomer, limited to 40, 50 or 60 % of the spacing as the vapour density is above 3, between 1 and 3 or below 1 lb/ft\(^3\). The leakage point is the manual’s table of \(V_H\sqrt{D_V/D_L}\) against the liquid level for V-1 and V-4 units, and a dry drop above 0.2 times the spacing is the manual’s capacity limit by pressure drop.

Bubble-cap trays

Bubble-cap trays follow the Bolles (1956) method as Ludwig presents it (chapter 8, equations 8-225 to 8-245, Figures 8-108, 8-113 and 8-114), in the US units of the original: heads in inches of liquid, \(V\) the vapour load in ft\(^3\)/s, areas in ft\(^2\). The caps are laid on a triangular pitch over the active area, \(N_c\) caps of inside diameter \(d_c\) with risers of inside diameter \(d_r\) (cap walls 2 mm, riser walls 1 mm). The riser, reversal and annulus drop is

\[ h_{pc} = K_c\,\frac{\rho_V}{\rho_L - \rho_V}\left(\frac{V}{A_r}\right)^2 , \]

with \(A_r\) the total riser area and \(K_c\) read from Bolles’ chart against the annular to riser area ratio (0.65 at 1.0, 0.515 at 1.2, 0.42 at 1.5); the slot opening of rectangular slots, \(N_s\) per cap of width \(w_s\) (in inches) and height \(H_s\),

\[ h_s = 32\left(\frac{\rho_V}{\rho_L - \rho_V}\right)^{1/3} \left(\frac{V}{N_c N_s w_s}\right)^{2/3} , \]

is compared with the slot height (Bolles designs the slots 50 to 60 % open; a fully open slot blows vapour under the cap skirt, an opening under 0.5 in at turndown makes the tray pulse) and the vapour load with the maximum slot capacity \(V_m = 0.79 A_s [H_s(\rho_L - \rho_V)/\rho_V]^{1/2}\). Trapezoidal slots, with the top width \(R_s\) times the base width, take Bolles’ equation 8-227 for the capacity, \(V_m = 2.36 A_s\,[\tfrac{2}{3}R_s/(1+R_s) + \tfrac{4}{15}(1-R_s)/(1+R_s)] [H_s(\rho_L - \rho_V)/\rho_V]^{1/2}\) (0.63, 0.74 and 0.79 times \(A_s[\ldots]^{1/2}\) for \(R_s\) = 0, 0.5 and 1), and the opening from the generalised correlation of his Figure 8-107, which follows from integrating the orifice flow over the open strip of the slot with the liquid depth below the top of the slot as the head,

\[ \frac{V}{V_m} = \frac{\tfrac{2}{3}R_s\,x^{3/2} + \tfrac{4}{15}(1-R_s)\,x^{5/2}} {\tfrac{2}{3}R_s + \tfrac{4}{15}(1-R_s)} , \qquad x = \frac{h_s}{H_s} ; \]

for \(R_s = 1\) this is \(x = (V/V_m)^{2/3}\), the rectangular slot equation above, and for a triangular slot \(x = (V/V_m)^{0.4}\). The total tray drop is Bolles’ equation 8-239, \(h_t = h_{pc} + h_s + h_{ss} + h_{ow} + \Delta/2\), with \(h_{ss}\) the static seal (top of the weir above the top of the slots) and \(\Delta\) the liquid gradient across the tray; \(h_{ss} + h_{ow} + \Delta/2\) is the dynamic seal Bolles tabulates by operating pressure.

The modified Dauphine relations (Bolles after Dauphine, Ludwig equations 8-232 to 8-237) are the alternative cap pressure drop: the riser drop \(h_r = 0.111\,(d_r/\rho_L)\,[\rho_V^{1/2} V/A_r]^{2.09}\) when the reversal area exceeds the riser area (with \((a_r/a_x)^{1/2}\) and the exponent 2.1 otherwise), the reversal and annulus drop \(h_{ra} = 0.68/\rho_L\,[(2a_r^2/a_x a_c)\,\rho_V^{1/2} V/A_r]^{1.71}\) for risers taller than 2.5 in, and the dry slot drop \(h'_s = 0.163/\rho_L\,[(d_c\rho_V)^{1/2} V/A_s]^{1.73}\), with \(d_r\) and \(d_c\) the riser and cap inside diameters in inches and \(a_r\), \(a_x\) and \(a_c\) the riser, reversal (the cylinder between the top of the riser and the cap ceiling, which needs the riser height and the cap inside height) and cap inside areas per cap. The dry cap drop \(h'_c = h_r + h_{ra} + h'_s\) over the wet cap correction \(C_w\) of Dauphine’s chart (Ludwig Figure 8-115, against \((V/A_s)\,[(\rho_V/\rho_L)(a_s/a_{an})]^{1/2}\)) is the wet cap drop \(h_c\), and the tray drop is \(h_c + h_{ss} + h_{ow} + \Delta/2\). The cap must not exceed \(h_r + h_{ra}\) plus the slot height (and the shroud ring), or the vapour blows under the shroud ring; the tool warns. The gradient is entered by the user or estimated by the Davies equation as Bolles charts it: the gradient per row of caps \(\Delta'_r\) solves

\[ \frac{Q/L_w}{C_d} = 25.8\,\frac{\gamma}{1+\gamma}\,\Delta_r'^{\,1/2} \left[1.6\Delta'_r + 3\left(h_l + \frac{0.3 s}{\gamma}\right)\right] , \]

the left-hand side read from Bolles’ Figure 8-108 against the liquid load per foot of mean tray width, \(\gamma\) the gap between caps over the cap diameter, \(h_l\) the clear liquid depth and \(s\) the skirt clearance in inches; the gradient is the per-row value times the rows of caps along the flow path, times the Davies vapour load correction (Figure 8-113). Bolles keeps the vapour distribution ratio \(\Delta/h_c\), with \(h_c = h_{pc} + h_s\) the cap drop, below 0.5; above it the inlet caps stop bubbling, and the tool warns. The downcomer backup is Ludwig’s equation 8-245, \(H_d = h_w + h_{ow} + \Delta + h_{dc} + h_t\). Flooding is by Fair’s chart, which was drawn for bubble-cap and sieve trays; the positive slot seal removes the weeping check.

Packings

Robbins and Kister-Gill

The default capacity route needs only the packing factor. The pressure drop is the Robbins correlation in the general forms of Perry’s Handbook :

\[ \begin{align} \Delta P &= C_3 G_f^2 10^{C_4 L_f} + 0.4 \left(\frac{L_f}{20000}\right)^{0.1} \left[C_3 G_f^2 10^{C_4 L_f}\right]^4 , \\ G_f &= 986\,F_s \left(\frac{F_{pd}}{20}\right)^{0.5} 10^{0.3\rho_G} , \end{align} \]

with \(C_3 = 7.4\times10^{-8}\), \(C_4 = 2.7\times10^{-5}\), the liquid loading factor \(L_f = L\,(62.4/\rho_L)(F_{pd}/20)^{0.5}\mu_L^{0.2}\) for \(F_{pd} > 200\) or \(L\,(62.4/\rho_L)(20/F_{pd})^{0.5}\mu_L^{0.1}\) below, and the US units of the original (\(\Delta P\) in inches of water per foot, \(F_s\) in ft/s(lb/ft\(^3\))\(^{0.5}\), \(L\) in lb/h ft\(^2\), \(\mu_L\) in cP, \(F_{pd}\) in 1/ft). The flood point is the vapour velocity at which this pressure drop reaches the Kister and Gill value , \(\Delta P_{flood} = 0.115\,F_p^{0.7}\) inches of water per foot with \(F_p\) in 1/ft.

Billet and Schultes

When the packing has the Billet and Schultes constants , their model gives the holdup below the loading point,

\[ h_L = \left(12\,\frac{Fr_L}{Re_L}\right)^{1/3} \left(\frac{a_h}{a}\right)^{2/3} , \qquad \frac{a_h}{a} = C_h Re_L^{0.15} Fr_L^{0.1}\ (Re_L < 5),\quad 0.85\,C_h Re_L^{0.25} Fr_L^{0.1}\ (Re_L \ge 5), \]

the loading velocity (Seader and Henley , equations 6-105 to 6-108), the flooding velocity \(u_{V,f} = u_{V,l}/0.7\), the dry bed pressure drop \(\Delta P_0/l = \Psi_0\,(a/\varepsilon^3)(u_V^2\rho_V/2)/K_W\) with \(\Psi_0 = C_p\,(64/Re_V + 1.8/Re_V^{0.08})\) and the wall factor \(K_W\), and the irrigated pressure drop \(\Delta P/\Delta P_0 = [\varepsilon/(\varepsilon - h_L)]^{1.5} \exp(13300\,Fr_L^{0.5}/a^{1.5})\). The heights of a transfer unit are their equations 6-132 and 6-133 with the constants \(C_L\) and \(C_V\) and the interface area ratio of equations 6-136 to 6-140.

Rocha, Bravo and Fair

Structured (corrugated sheet) packings are rated with the model of Rocha, Bravo and Fair , in the form Kooijman and Taylor list it . The geometry is the corrugation side \(S\) (catalogue values of Fair and Bravo for Flexipac 2, Gempak 2A, Intalox 2T, Montz B1-200, Mellapak 250Y and Sulzer BX; \(4.5\varepsilon/a\) for the others, which reproduces those within about 15 %), the corrugation angle \(\theta\) (45\(^\circ\), 60\(^\circ\) for the X types), the void fraction and the specific area. The holdup correction factor

\[ F_t = \frac{29.12\,(We_L Fr_L)^{0.15} S^{0.359}} {Re_L^{0.2}\,\varepsilon^{0.6}\,(\sin\theta)^{0.3}\,(1 - 0.93\cos\gamma)} , \]

with \(\cos\gamma = 0.9\) below 0.0453 N/m and \(5.211\times10^{-16.835\sigma}\) above, gives the holdup \(h_t = (4F_t/S)^{2/3}[3\mu_L u_L/(\rho_L\sin\theta\,\varepsilon\,g_{eff})]^{1/3}\) at the effective gravity \(g_{eff} = g\,[(\rho_L - \rho_V)/\rho_L]\,[1 - (\Delta P/\Delta z)/(\Delta P/\Delta z)_{flood}]\), and the pressure drop

\[ \frac{\Delta P}{\Delta z} = \left[\frac{0.177\rho_V}{S\varepsilon^2\sin^2\theta}\,u_V^2 + \frac{88.774\mu_V}{S^2\varepsilon\sin\theta}\,u_V\right] \left(\frac{1}{1 - K_2 h_t}\right)^5 , \qquad K_2 = 0.614 + 71.35 S . \]

Holdup and pressure drop are solved together; the flood pressure drop is the Kister and Gill value of the packing factor (1025 Pa/m, the authors’ figure, when the packing has no factor), and the flooding velocity is the vapour rate above which the pair has no solution below it. For the HETP, the effective velocities \(u_{Le} = u_L/(\varepsilon h_t\sin\theta)\) and \(u_{Ge} = u_V/[\varepsilon(1 - h_t)\sin\theta]\) give \(k_G = 0.054\,(D_G/S)\,Re_G^{0.8}Sc_G^{0.33}\) with \(Re_G\) on \(u_{Ge} + u_{Le}\), \(k_L = 2\sqrt{D_L C_E u_{Le}/(\pi S)}\) with \(C_E = 0.9\), and the effective area \(a_e = F_{SE}F_t a_p\) with the surface enhancement factor \(F_{SE} = 0.35\) of embossed sheet metal. A structured section takes this model by default; a random packing asked for it falls back to Robbins and Onda with a note.

Onda

For random packings without mass transfer constants the HETP comes from the Onda, Takeuchi and Okumoto correlations for the wetted area, \(k_L\) and \(k_G\), with \(H_L = u_L/(k_L a_w)\), \(H_G = u_V/(k_G a_w)\), \(H_{OG} = H_G + \lambda H_L\) and \(\text{HETP} = H_{OG}\ln\lambda/(\lambda - 1)\), where \(\lambda = K V/L\) is the stripping factor of the component transferring to the vapour on the stage.

Rules of thumb and wetting

The rule-of-thumb HETP is always reported beside the model value, as Kister advises : \(1.5\,d_p\) for Pall rings and similar random packings, \(100/a + 4/12\) ft (with \(a\) in ft\(^2\)/ft\(^3\)) for structured packings, and at least the column diameter below 2 ft (Seader equations 6-116 to 6-118). The liquid velocity is checked against the minimum wetting rates of the packing material (ceramic 0.15, oxidized metal 0.3, bright metal 0.9, plastic 1.2 mm/s). The bed height of a section is the sum of the HETPs of its stages, or the height given by the user.

Catalogue

The packing catalogue carries the random and structured packings of Seader and Henley Table 6.8 (specific area, void fraction, \(F_p\) and the Billet and Schultes constants) and of Perry’s Handbook Tables 14-7a and 14-7b (\(F_p\) of Kister and Gill and \(F_{pd}\) of Robbins), each row with its source, plus the corrugation sides of Fair and Bravo for the structured packings they measured. A user-defined packing takes the same fields, with the corrugation side, angle and surface enhancement factor for a structured one.

Rate-based column

With Rate-based switched on in the column editor, the rigorous column keeps its equilibrium-stage solvers (Wang-Henke, Naphtali-Sandholm) but takes the Murphree vapour efficiency of every component on every stage from mass transfer, and lets it follow the solution: the column solves, rates its stages from the flows, compositions and properties it just found, solves again with the new efficiencies and repeats until no efficiency moves by more than the tolerance (0.01 by default, eight passes at most). The solvers carry the component-wise efficiencies in their equilibrium equations, \(y_{ij} = E_{ij} K_{ij} x_{ij} + (1 - E_{ij})\,y_{i+1,j}\), in the Naphtali-Sandholm Jacobian included. The tray geometry and the packings come from the column internals case saved in the column; a stage outside the case is a standard sieve tray (50 mm weir, 12 % downcomer) on the column’s estimated diameter.

Trays

The point efficiency is \(E_{OG,j} = 1 - \exp(-N_{OG,j})\) with \(1/N_{OG,j} = 1/N_{G,j} + \lambda_j/N_{L,j}\) and \(\lambda_j = K_j V/L\) the stripping factor of the component on the stage (Perry’s Handbook , equations 14-132 to 14-142). The transfer units are those of the AIChE Bubble-Tray Design Manual, \(N_{G} = (0.776 + 4.57 h_w - 0.238 F_{va} + 104.8 Q_L/W_l)/\sqrt{Sc_G}\) and \(N_L = 19700\sqrt{D_L}\,(0.4 F_{va} + 0.17)\,t_L\) (SI, with \(F_{va} = u_a\sqrt{\rho_V}\) on the active area and \(t_L = h_L A_a/Q_L\)), or, for the gas phase of sieve trays, Chan and Fair, \(k_G a = 316\sqrt{D_G}\,(1030 f - 867 f^2)/\sqrt{h_L}\) with \(h_L\) in mm and \(f\) the fraction of flood from the tray rating, \(N_G = k_G a\,t_G\) with \(t_G = (1 - \phi_e) h_L A_a/(\phi_e Q_G)\). The clear liquid height is Bennett’s, \(h_L = \phi_e [h_w + C (Q_L/W_l\phi_e)^{0.67}]\) with the froth density \(\phi_e = \exp(-12.55 K_s^{0.91})\) and \(C = 0.5 + 0.438 e^{-137.8 h_w}\). The Murphree tray efficiency follows from the point efficiency with the partial-mixing relation of Gerster et al. (Seader and Henley , equations 6-34 to 6-36), with the Peclet number \(Pe = Z_L^2/(D_E t_L)\) and the AIChE eddy diffusivity \(D_E = (3.93\times10^{-3} + 0.0171 u_a + 3.67 Q_L/W_l + 0.18 h_w)^2\). A long flow path with little back-mixing gives a Murphree efficiency above 1 (Seader’s example reaches 1.25) and the column applies it as computed, kept between 0.02 and 3. On a component the stage strips (its equilibrium vapour value below the vapour arriving from the stage below) an efficiency above 1 overshoots that value, and past \(y_{n+1}/(y_{n+1} - y^*)\) the outlet fraction would turn negative; the solvers stop the overshoot at half the equilibrium value, so trace components stay positive. The clear liquid height, the transfer units, the point efficiency, the Peclet number and the Murphree efficiency of every tray are listed in the stage notes of the properties report.

Packed stages

A packed stage is a slice of bed of height \(H\) (its stage height, set by the rating to the HETP). For each component the packing model of the section (Onda, Billet and Schultes or Rocha, Bravo and Fair) gives the HETP with that component’s diffusivities and stripping factor, the slice holds \(n_j = H/\text{HETP}_j\) theoretical stages of it, and the equivalent Murphree efficiency is the Lewis relation inverted, \(E_j = (\lambda_j^{n_j} - 1)/(\lambda_j - 1)\) (equal to \(n_j\) at \(\lambda_j = 1\)), kept between 0.02 and 1.

Diffusivities

Unless the internals section gives them, the gas diffusivities come from Fuller, Schettler and Giddings, \(D_{AB} = 1.013\times10^{-2}\,T^{1.75}(1/M_A + 1/M_B)^{1/2}/[P (v_A^{1/3} + v_B^{1/3})^2]\), combined into the diffusivity of each component through the mixture by the Wilke and Fairbanks rule, and the liquid diffusivities from Wilke and Chang with the mixture as the solvent (association factor 2.6 when water carries more than half the moles). The molar volumes at the boiling point are Tyn and Calus estimates from the critical volume, \(V_b = 0.285 V_c^{1.048}\), and the Fuller diffusion volumes are taken as 0.8 of them, so these are order-of-magnitude values; entering measured diffusivities in the section is the way to do better. The efficiencies of every pass and the final component values are listed in the column’s properties report.

Validation

The tray hydraulics reproduce Example 11.11 of Towler and Sinnott , the bottom plate of an acetone-water column (Table 14). The differences come from the chart readings of the book and from its rounding of the areas (the book rounds the net area to 0.44 m\(^2\) and the hole area to 0.038 m\(^2\)).

Quantity	Book	DWSIM
Flow parameter \(F_{LV}\)	0.14	0.137
\(K_1\) at 0.5 m spacing (chart, fit)	0.075	0.078
Flooding velocity, m/s	3.38	3.49
Weir crest \(h_{ow}\), mm	27	27.6
\(K_2\) at 72 mm, weep point velocity, m/s	30.6, 14	30.6, 14.5
Hole velocity, m/s	29.7	30.3
\(C_0\), dry drop \(h_d\), mm	0.84, 48	0.839, 50
Total head \(h_t\), mm	138	141
Downcomer backup \(h_b\), mm	221	223
Downcomer residence time, s	3.1	3.09
Percent of flood	76	75
Entrainment \(\psi\)	0.018	0.016
Diameter for 85 % flood, m	0.77	0.74

Sieve plate of Towler and Sinnott Example 11.11: 0.79 m diameter, 0.5 m spacing, 12 % downcomer, 50 mm weir, 5 mm holes at 10 % of the active area; vapour 0.81 kg/s at 0.72 kg/m\(^3\), liquid 4.06 kg/s at 954 kg/m\(^3\), \(\sigma\) = 57 mN/m.

The Kister and Haas transition height matches the first trial of Kister’s sizing example (section 6.5 of ): with 0.5 in holes, a 10 % hole area and 6.45 gpm per inch of weir the water value is 0.937 in in the book and 0.950 in here.

The Billet and Schultes implementation reproduces the worked examples of Seader and Henley (Table 15). In Example 6.15 the book prints a holdup of 0.0128 for the 1.5 in Pall-like rings, while its own equations 6-97 and 6-101 with the stated numbers give 0.0182; the transfer units in the table were checked with the book’s holdup so that equations 6-132 and 6-133 are compared on their own.

Quantity	Book	DWSIM
Ex. 6.12, holdup of 50 mm metal Hiflow rings, m\(^3\)/m\(^3\)	0.0637	0.0637
Ex. 6.12, holdup of Montz B1-200, m\(^3\)/m\(^3\)	0.0722	0.0721
Ex. 6.14, 25 mm metal Bialecki rings: loading velocity, m/s	1.46	1.463
Ex. 6.14, flooding velocity, m/s	2.09	2.09
Ex. 6.14, holdup at the loading point	0.0440	0.0440
Ex. 6.14, wall factor \(K_W\)	0.944	0.945
Ex. 6.14, dry and irrigated pressure drop, Pa/m	281, 331	282, 332
Ex. 6.15, interface area ratio \(a_{Ph}/a\)	0.242	0.242
Ex. 6.15, \(H_L\), \(H_G\), m	0.26, 1.03	0.260, 1.031
Ex. 6.15, \(H_{OG}\), HETP, ft	3.96, 4.73	3.97, 4.75

Billet and Schultes examples of Seader and Henley.

For the Robbins route, Example 6.13 of the same book rates 1 in metal IMTP at 70 % of flood and \(F_{LV}\) = 0.092: the GPDC chart gives 0.88 inches of water per foot and the manufacturer’s data 0.63; Robbins gives 0.56, and the Kister and Gill flood point puts the flooding velocity at 2.58 m/s against the 8.5 ft/s (2.59 m/s) the book reads from the chart.

The stage properties are taken from the column exactly as the column’s own properties profile computes them; a test on the extractive distillation sample checks that every tray between the condenser and the reboiler carries both phases with physical densities, viscosities and surface tension, and that a sieve tray section sized for 80 % flood lands its worst stage on that value.

The valve tray balance points reproduce Klein’s worked example (Ludwig Example 8-41): venturi valves of 16 gauge with four legs in carbon steel, \(\rho_V = 1.91\) and \(\rho_L = 31.0\) lb/ft\(^3\), give 3.06 ft/s at the closed and 8.01 ft/s at the open balance point and a dry drop of 1.77 in of liquid while the valves open, the aeration factor 0.61 at \(F_{va} = 1.04\). The Glitsch procedure reproduces the manual’s own design example (a C\(_3\) splitter on 20 in spacing, vapour at 2.75 and liquid at 29.33 lb/ft\(^3\)): \(CAF_0\) 0.395 from the chart, a downcomer design velocity of 170 gpm/ft\(^2\), 68.6 % of flood, a dry drop of 1.75 in through 534 V-1 units, 3.88 in in total and a backup of 7.88 in. The bubble-cap tray reproduces the top tray of Ludwig’s Example 8-36 (the 6 ft vacuum finishing tower with 129 caps of 3 7/8 in on 5.5 in centres, 50 slots of 1/8 by 1.5 in, 0.5 in static seal): with the cap count laid out by the tool (132), \(h_{pc}\) 0.11 against 0.118 in, \(h_s\) 0.61 against 0.626 in, \(h_{ow}\) 0.10 against 0.099 in, a gradient of 0.09 against 0.12 in and a total of 1.37 against 1.50 in; the tool, like the book, finds the slot opening on the low side. By the modified Dauphine relations the same tray gives \(h_r\) 0.063, \(h_{ra}\) 0.045 and \(h'_s\) 0.031 in as the book, \(C_w\) 0.18 against 0.16, a wet cap drop within 0.1 in of the book’s 0.87 in and a total within 0.2 in of its 1.53 in. Rocha, Bravo and Fair on Mellapak 250Y at total reflux with cyclohexane / n-heptane-like properties at 1 atm gives a flood F-factor of 2.5 Pa\(^{0.5}\), 212 Pa/m and a holdup of 0.08 at \(F = 2\), and an HETP of 0.41 m against the 0.50 m rule of thumb. The tray mass transfer reproduces Example 12 of Perry’s Handbook (ethylbenzene-styrene sieve tray at 74 % of flood): \(N_G\) 1.51, \(N_L\) 18.6 and a point efficiency of 0.75 by Chan and Fair. On the extractive distillation sample the rate-based column settles its efficiencies in a few passes at values of the same order as O’Connell’s.

Dynamic model of the column

The tray and packing geometry the rating writes on the stages is what the dynamic model of the column runs on. Every stage holds a content (a material stream flashed at its own pressure and enthalpy), the condenser stage doubles as the reflux drum (its volume is the column area times the Bottom Spacing, its liquid leaves through the weir relation, so a short Downcomer Length with a Downcomer Height equal to the dead height of the drum makes a reflux line whose flow answers the drum level gently) and, in the quasi-steady formulation, the reboiler stage doubles as the sump (it keeps its liquid, its volume is the stage height plus the Top Spacing, it takes the reboiler duty and gives the bottoms product, as a kettle or thermosiphon reboiler does with the column bottoms). The stage levels, the sump level, the stage temperatures and the vapour and liquid rates every stage sends up and down are properties of the column (Stage_LiquidLevel_i, Sump_LiquidLevel, Stage_Temperature_i, Stage_VaporFlow_i, Stage_LiquidFlow_i) that an integrator can monitor and a controller can read.

Quasi-steady vapour

With Quasi-Steady Vapor on, the liquid holdups are the states and the vapour is not: a stage keeps the vapour its free volume holds at its pressure and sends the rest up within the sub-step, swept from the sump to the top, so a change of boilup reaches the condenser at once, as it does in a real column where the vapour transit time is far below any liquid time constant. The condenser drum is the one pressure state, at the bubble pressure of its liquid (a hair above it), and every stage below sits at the drum pressure plus the dry drop of the vapour through the tray above it and the liquid head on that tray, top-down, from the vapour rates of the previous sub-step. The explicit formulation (the option off) integrates the vapour holdup of every tray against the pressure-driven flow law, which has a time constant of milliseconds on a real column and cannot be run at any step a user would choose. The integration step is divided in sub-steps (Time step discretization); one-second sub-steps are enough once the vapour is quasi-steady, the liquid time constant of a tray being of the order of ten seconds.

Initial state

The first dynamic step seeds the holdups from the steady state: every stage at its steady-state composition and temperature, the trays at the level that passes the steady-state liquid rate over the weir, the vapour space full at the stage pressure, the sump half full. With Calibrate Tray Coefficients the dry tray pressure drop coefficient of every tray is set so the steady-state vapour rate crosses its holes at the steady-state stage pressure drop, and the column then stays where it is when nothing changes, which is the first thing to check of any dynamic column.

Startup from an empty column

With Start Empty the first step seeds an empty column instead: a film of liquid of the feed composition on every stage and in the sump (or the Initial Sump Level), at the Initial Pressure (the inert blanket) and the Initial Temperature, with no vapour and no flows. The run is then a startup driven by the schedule the way an operator would do it: feed on, the liquid falls through the holes of the trays the vapour does not yet hold up (Tray Weeping: the orifice flow of the clear liquid head through the hole area, scaled by how far the dry pressure drop of the vapour falls short of 0.4 of that head, Fair’s weep point), the reboiler duty ramps once its stage is covered, the vapour climbs and fills the drum, the reflux starts on the drum level, the column pressurises from the blanket to the setpoint of the pressure controller, and the level controllers are switched from manual to automatic by events (the ManualOverride property of the PID controller; in manual the controller writes its manual output, zero by default, to the valve). The Minimum Pressure is the pressure the drum cannot fall below, an inert blanket or a vent to atmosphere; the Coolant Temperature keeps the condenser from removing heat from a holdup already colder than its coolant, and a duty stage that holds only a film of liquid warms or cools by at most 25 K per sub-step. Both the startup and the weeping need the quasi-steady vapour. The benzene-toluene column of the case library is started this way from an empty, cold column to its design steady state in two hours of simulated time.

Scope and limits

Valve trays take Klein’s dry pressure drop and the Bolles extension of the weep point through the closed balance point, with the flooding velocity from Fair or from Kister and Haas on the open valve area, or the Glitsch procedure for single-pass V-1 and V-4 Ballast trays; the Glitsch multipass allocation of downcomer areas and its tray efficiency chart are not included. Bubble-cap trays take the Bolles method with rectangular or trapezoidal slots and, as an alternative, the modified Dauphine relations with the wet cap correction; Bolles’ gradient charts by cap spacing are not included, the gradient coming from the Davies equation they were drawn from (or from the user). The Rocha, Bravo and Fair model needs the corrugation side, which the catalogue carries for six packings and estimates for the rest; gauze packings take the same equations with the sheet metal surface enhancement factor. The rate-based column is a nonequilibrium model through component efficiencies computed from the mass transfer on each stage, with the equilibrium at the interface implied by the K-values; it does not solve the interface compositions and the film equations of a Maxwell-Stefan model, so the coupling among the fluxes of a multicomponent mixture is not represented. The iteration with the solver writes the O’Connell efficiencies of the trays and the pressure profile; it does not resize the sections between passes.