Transport Properties

Density

Liquid Phase

For non-EOS property packages, the liquid-phase molar volume is calculated using the modified Rackett equation when experimental data are not available ,

\[ V_{s}=\frac{RT_{C}}{P_{C}}Z_{RA}^{\left[1+\left(1-T_{r}\right)^{2/7}\right]}, \]

where:

\(V_{s}\) Saturated molar volume (m³/mol)

\(T_{c}\) Critical temperature (K)

\(P_{c}\) Critical pressure (Pa)

\(T_{r}\) Reduced temperature

\(Z_{RA}\) Rackett constant of the component (or the mixture)

\(R\) Ideal Gas constant (8,314 J/[mol.K])

image	If \(T\geqslant T_{cm},\) the Rackett method does not provide a value for \(V_{s}\) and, in this case, DWSIM uses the EOS-generated compressibility factor to calculate the density of the liquid phase.

For mixtures, the equation [eq:liqdens] becomes

\[ V_{s}=R\left(\sum\frac{x_{i}T_{c_{i}}}{P_{c_{i}}}\right)Z_{RA}^{\left[1+\left(1-T_{r}\right)^{2/7}\right]}, \]

with \(\:\) \(T_{r}=T/T_{cm}\) , and

\[ T_{c_{m}}=\sum\sum\phi_{i}\phi_{j}T_{c_{ij}}, \]

\[ \phi_{i}=\frac{x_{i}V_{c_{i}}}{\sum x_{i}V_{c_{i}}}, \]

\[ T_{c_{ij}}=\left[\frac{8\left(V_{c_{i}}V_{c_{j}}\right)^{1/2}}{\left(V_{c_{i}}^{1/3}+V_{c_{j}}^{1/3}\right)^{3}}\right]\left(T_{c_{i}}T_{c_{j}}\right)^{1/2}, \]

where:

\(x_{i}\) Molar fraction

\(V_{c_{i}}\) Critical volume (m³/mol)

If \(Z_{RA}\) isn’t available, it is calculated from the component acentric factor,

\[ Z_{RA}=0.2956-0.08775\omega, \]

If the component (or mixture) isn’t saturated, a correction is applied in order to account for the effect of pressure in the volume,

\[ V=V_{s}\left[1-\left(0.0861488+0.0344483\omega\right)\ln\frac{\beta+P}{\beta+P_{vp}}\right], \]

with

\[ \begin{eqnarray} \frac{\beta}{P} & = & -1-9.070217\left(1-T_{r}\right)^{1/3}+62.45326\left(1-T_{r}\right)^{2/3}-135.1102\left(1-T_{r}\right)+ \\ & & +\exp\left(4.79594+0.250047\omega+1.14188\omega^{2}\right)\left(1-T_{r}\right)^{4/3}, \end{eqnarray} \]

where:

\(V\) Compressed liquid volume (m³/mol)

\(P\) Pressure (Pa)

\(P_{vp}\) Vapor pressure / Bubble point pressure (Pa)

Finally, density is calculated from the molar volume by the following relation:

\[ \rho=\frac{MM}{1000V}, \]

where:

\(\rho\) Density (kg/m³)

\(V\) Specific volume of the fluid (m³/mol)

\(MM\) Liquid phase molecular volume (kg/kmol)

Vapor Phase

image	For the Ideal Gas Property Package, the compressibility factor is considered to be equal to 1.

Vapor phase density is calculated from the compressiblity factor generated by the EOS model, according with the following equation:

\[ \rho=\frac{MM\,P}{1000ZRT}, \]

where:

\(\rho\) Density (kg/m³)

\(MM\) Molecular weight of the vapor phase (kg/kmol)

\(P\) Pressure (Pa)

\(Z\) Vapor phase compressibility factor

\(R\) Ideal Gas constant (8,314 J/[mol.K])

\(T\) Temperature (K)

For ideal gases, the same equation is used, with Z = 1.

Mixture

If there are two phases at system temperature and pressure, the density of the mixture is calculated by the following expression:

\[ \rho_{m}=f_{l}\rho_{l}+f_{v}\rho_{v}, \]

where:

\(\rho_{m,l,v}\) Density of the mixture / liquid phase / vapor phase (kg/m³)

\(f_{l,v}\) Volume fraction of the liquid / vapor phase

Viscosity

Liquid Phase

When experimental mixture data are not available, the liquid-phase mixture viscosity is estimated using a logarithmic mixing rule:

\[ \eta_{L}=\exp\left(\sum_{i}x_{i}\ln\eta_{i}\right), \]

where \(\eta_{i}\) is the pure-component liquid viscosity at the system temperature, evaluated from the correlation

\[ \eta=\exp\left(A+B/T+C\ln T+DT^{E}\right), \]

where A, B, C, D and E are experimental coefficients (or generated by DWSIM in the case of pseudocomponents or hypotheticals).

Vapor Phase

Vapor phase viscosity is calculated in two steps. First, when experimental data is not available, the temperature dependence is given by the Lucas equation ,

\[ \eta\xi=\left[0,.807T_{r}^{0,618}-0.357\exp(-0.449T_{r})+0.34\exp(-4.058T_{r})+0.018\right] \]

\[ \xi=0,176\left(\frac{T_{c}}{MM^{3}P_{c}^{4}}\right)^{1/6}, \]

where

\(\eta\) Viscosity ( \(\mu P\) )

\(T_{c},P_{c}\) Component (or mixture) critical properties

\(T_{r}\) Reduced temperature, \(T/T_{c}\)

\(MM\) Molecular weight (kg/kmol)

In the second step, the experimental or calculated viscosity with the Lucas method is corrected to take into account the effect of pressure, by the Jossi-Stiel-Thodos method ,

\[ \begin{eqnarray} \left[\left(\eta-\eta_{0}\right)\left(\frac{T_{c}}{MM^{3}P_{c}^{4}}\right)^{1/6}+1\right]^{1/4} & = & 1.023+0.23364\rho_{r}+ \\ & + & 0.58533\rho_{r}^{2}-0.40758\rho_{r}^{3}+0.093324\rho_{r}^{4}, \end{eqnarray} \]

where

\(\eta,\eta_{0}\) Corrected viscosity / Lucas method calculated viscosity ( \(\mu P\) )

\(T_{c},P_{c}\) Component critical properties

\(\rho_{r}\) Reduced density, \(\rho/\rho_{c}=V/V_{c}\)

\(MM\) Molecular weight (kg/kmol)

If the vapor phase contains more than a component, the viscosity is calculated by the same procedure, but with the required properties calculated by a molar average.

Surface Tension

When experimental data is not available, the liquid phase surface tension is calculated by doing a molar average of the individual component tensions, which are calculated with the Brock-Bird equation ,

\[ \frac{\sigma}{P_{c}^{2/3}T_{c}^{1/3}}=\left(0.132\alpha_{c}-0.279\right)\left(1-T_{r}\right)^{11/9} \]

\[ \alpha_{c}=0.9076\left[1+\frac{T_{br}\ln(P_{c}/1.01325)}{1-T_{br}}\right], \]

where

\(\sigma\) Surface tension (N/m)

\(T_{c}\) Critical temperature (K)

\(P_{c}\) Critical pressure (Pa)

\(T_{br}\) Reduced normal boiling point, \(T_{b}/T_{c}\)

Isothermal Compressibility

Isothermal compressiblity of a given phase is calculated following the thermodynamic definition:

\[ \beta=-\frac{1}{V}\frac{\partial V}{\partial P} \]

The above expression is calculated rigorously by the PR and SRK equations of state. For the other models, a numerical derivative approximation is used.

Bulk Modulus

The Bulk Modulus of a phase is defined as the inverse of the isothermal compressibility:

\[ K=\frac{1}{\beta} \]