3. Virial Equation of State

class scithermo.eos.virial.Virial(pow: callable = <ufunc 'power'>, exp: callable = <ufunc 'exp'>)

Parameters

• R (float, hard-coded) – gas constant, set to SI units

• pow (callable, optional) – function for computing power, defaults to numpy.power

• exp (callable, optional) – function for computing logarithm, defaults to numpy.exp

d_B0_d_Tr_expr(T_r)

\[\frac{\mathrm{d} B^0}{\mathrm{d}T_\mathrm{r}}\]

d_B1_d_Tr_expr(T_r)

\[\frac{\mathrm{d} B^1}{\mathrm{d}T_\mathrm{r}}\]

hat_phi_i_expr(*args)

expression for fugacity coefficient

class scithermo.eos.virial.SecondVirial(dippr_no: str = None, compound_name: str = None, cas_number: str = None, pow: callable = <ufunc 'power'>, **kwargs)

Virial equation of state for one component. See [GP07][SVanNessA05]

G_R_RT_expr(P, T)

Dimensionless residual gibbs

\[\frac{G^\mathrm{R}}{RT} = (B^0 + \omega B^1)\frac{P_\mathrm{r}}{T_\mathrm{r}}\]

Returns

Expression for residual gibbs free (divided by RT) – dimensionless

H_R_RT_expr(P, T)

Dimensionless residual enthalpy

\[\frac{H^\mathrm{R}}{RT} = P_\mathrm{r}\left[ \frac{B^0}{T_\mathrm{r}} - \frac{\mathrm{d} B^0}{\mathrm{d}T_\mathrm{r}} + \omega\left(\frac{B^1}{T_\mathrm{r}} - \frac{\mathrm{d}B^1}{\mathrm{d}T_\mathrm{r}}\right) \right]\]

Returns

Expression for residual enthalpy (divided by RT) – dimensionless

S_R_R_expr(P, T)

Dimensionless residual entropy

\[\frac{S^\mathrm{R}}{R} = -P_\mathrm{r}\left( \frac{\mathrm{d} B^0}{\mathrm{d}T_\mathrm{r}} + \omega\frac{\mathrm{d}B^1}{\mathrm{d}T_\mathrm{r}} \right)\]

Returns

Expression for residual entropy (divided by R) – dimensionless

ln_hat_phi_i_expr(P, T)

logarithm of fugacity coefficient

Note

single-component version

\[\ln\hat{\phi}_i = \frac{PB}{RT}\]

Parameters

• P (float) – pressure in Pa

• T (float) – temperature in K

plot_Z_vs_P(T, P_min, P_max, symbol='o', ax=None, **kwargs)

Plot compressibility as a function of pressure

Parameters

• T (float) – temperature [K]

• P_min (float) – minimum pressure for plotting [Pa]

• P_max (float) – maximum pressure for plotting [Pa]

• phase (str) – phase type (liquid or vapor), defaults to vapor

• symbol (str) – marker symbol, defaults to ‘o’

• ax (plt.axis) – matplotlib axes for plotting, defaults to None

• kwargs – keyword arguments for plotting

class scithermo.eos.virial.BinarySecondVirial(i_kwargs=None, j_kwargs=None, k_ij=0.0, pow: callable = <ufunc 'power'>, exp: callable = <ufunc 'exp'>)

Second virial with combining rules from [PLdeAzevedo86]

\[\begin{split}\begin{align} w_{ij} &= \frac{w_i + w_j}{2} \\ T_{\mathrm{c},ij} &= \sqrt{T_{\mathrm{c},i}T_{\mathrm{c},j}}(1-k_{ij}) :label:eq_Tcij\\ P_{\mathrm{c},ij} &= \frac{Z_{\mathrm{c},ij}RT_{\mathrm{c},ij}}{V_{\mathrm{c},ij}} \\ \end{align}\end{split}\]

where

\[\begin{split}\begin{align} Z_{\mathrm{c},ij} &= \frac{Z_{\mathrm{c},i} + Z_{\mathrm{c},j}}{2} \\\ V_{\mathrm{c},ij} &= \left(\frac{V_{\mathrm{c},i}^{1/3} + V_{\mathrm{c},j}^{1/3}}{2}\right)^{3} \end{align}\end{split}\]

Parameters

• k_ij (float) – equation of state mixing rule in calculation of critical temperautre, see Equation eq_Tcij. When \(i=j\) and for chemical similar species, \(k_{ij}=0\). Otherwise, it is a small (usually) positive number evaluated from minimal \(PVT\) data or, in the absence of data, set equal to zero.

• cas_pairs (list(tuple(str))) – pairs of cas registry numbers, derived from cas_numbers calculated

• other_cas (dict) – map from one cas number to other

B_ij_expr(cas_1, cas_2, T)

Returns \(B_{ij}\) considering that i=j or i!=j

If i=j, find the component \(k\) for which k=i=j (all cas no’s equal). Then return \(B_{kk}\) where

\[B_{kk}=\frac{RT_{\mathrm{c},k}}{P_{\mathrm{c},k}}\left[ B^0\left(\frac{T}{T_{\mathrm{c},k}}\right) + \omega_k\left(\frac{T}{T_{\mathrm{c},k}}\right) \right]\]

Otherwise, if i!=j, return \(B_{ij}\), where

\[B_{ij}=\frac{RT_{\mathrm{c},ij}}{P_{\mathrm{c},ij}}\left[ B^0\left(\frac{T}{T_{\mathrm{c},ij}}\right) + \omega_{ij}\left(\frac{T}{T_{\mathrm{c},ij}}\right) \right]\]

This is implemented in a simplified fashion usintg BinarySecondVirial.get_w_Tc_Pc() and then calling the generic expression for \(B\)

Parameters

• cas_1 (str) – cas number of first component

• cas_2 (str) – cas number of second component

• T – temperature [K]

B_mix_expr(y_k, T)

Parameters

• y_k (dict[cas_numbers, float]) – mole fractions of each component \(k\)

• T – temperature in K

Returns

\(B\) [m^3/mol], where \(Z = 1 + BP/R/T\)

G_R_RT(*args)

Residual free energy of mixture \(G^\mathrm{R}\)

H_R_RT(*args)

Residual enthalpy of mixture \(H^\mathrm{R}\)

S_R_R(*args)

Residual entropy of mixture \(S^\mathrm{R}\)

Tstar_d_lnphi_dTstar(cas_i, y_i, P, T)

Returns

\[\begin{split}\begin{align} T^\star \frac{\partial \ln{\hat{\phi}_i}}{\partial T^\star} &= \frac{T T_\text{ref}}{T_\text{ref}}\frac{\partial \ln{\hat{\phi}_i}}{\partial T} = \frac{T}{T_\text{c}}\frac{\partial \ln{\hat{\phi}_i}}{\partial T_\text{r}} \\ &= \frac{T}{T_\text{c}}\left[ \frac{P}{RT}\left( \frac{\partial B_{ii}}{\partial T_r} + (1-y_i)^2\frac{\partial \delta_{ij}}{\partial T_r} \right) - \frac{T_c\ln\hat{\phi}_i}{T} \right] \\ &= \frac{P}{R T_\text{c}}\left( \frac{\partial B_{ii}}{\partial T_r} + (1-y_i)^2\frac{\partial \delta_{ij}}{\partial T_r} \right) - \ln\hat{\phi}_i \\ &= -\frac{\bar{H}_i^\text{R}}{RT} \end{align}\end{split}\]

where \(\frac{\partial \delta_{ij}}{\partial T_r}\) is given by (3)

Parameters

• cas_i (str) – cas number for component of interest

• y_i (float) – mole fraction of component of interest

• P (float) – pressure in Pa

• T (float) – temperature in K

X_R_dimensionless(method: callable, cas_i: str, y_i: float, P: float, T: float)

Residual property of \(X\) for mixture.

Parameters

method (callable) – function to compute partial molar property of compound

bar_GiR_RT(*args)

Dimensionless residual partial molar free energy of component \(i\)

(1)\[\frac{\bar{G}^\mathrm{R}_i}{RT} = \ln\hat{\phi}_i\]

bar_HiR_RT(cas_i, y_i, P, T)

Dimensionless residual partial molar enthalpy of component \(i\)

\[\begin{split}\begin{align} \frac{\bar{H}^\mathrm{R}_i}{RT} &= -T \left(\frac{\partial \ln\hat{\phi}_i}{\partial T}\right)_{P,y} \\ &= - \frac{T}{T_c} \left(\frac{\partial \ln\hat{\phi}_i}{\partial T_r}\right)_{P,y} \\ &= - \frac{T}{T_c}\left( \frac{P}{RT}\left[ \frac{\partial B_{ii}}{\partial T_r} + (1-y_i)^2\frac{\partial \delta_{ij}}{\partial T_r} \right] - \frac{T_c\ln\hat{\phi}_i}{T} \right) \end{align}\end{split}\]

where \(\frac{\partial \delta_{ij}}{\partial T_r}\) is given by (3) so that we obtain

(2)\[\frac{\bar{H}^\mathrm{R}_i}{RT} = -\frac{P}{RT_c}\left[ \frac{\partial B_{ii}}{\partial T_r} + (1-y_i)^2\frac{\partial \delta_{ij}}{\partial T_r} \right] + \ln\hat{\phi}_i\]

Parameters

• cas_i (str) – cas number for component of interest

• y_i (float) – mole fraction of component of interest

• P (float) – pressure in Pa

• T (float) – temperature in K

bar_SiR_R(cas_i, y_i, P, T)

Dimensionless residual partial molar entropy of component \(i\)

Since

\[G^\mathrm{R} = H^\mathrm{R} - T S^\mathrm{R}\]

In terms of partial molar properties, then

\[\begin{split}\begin{align} \bar{S}_i^\mathrm{R} &= \frac{\bar{H}_i^\mathrm{R} - \bar{G}_i^\mathrm{R}}{T} \\ \frac{\bar{S}_i^\mathrm{R}}{R} &= \frac{\bar{H}_i^\mathrm{R}}{RT} - \frac{\bar{G}_i^\mathrm{R}}{RT} \\ \end{align}\end{split}\]

By comparing Equation (1) and (2) it is observed that

\[\frac{\bar{S}_i^\mathrm{R}}{R} = -\frac{P}{RT_c}\left[ \frac{\partial B_{ii}}{\partial T_r} + (1-y_i)^2\frac{\partial \delta_{ij}}{\partial T_r} \right]\]

where \(\frac{\partial \delta_{ij}}{\partial T_r}\) is given by (3)

Parameters

• cas_i (str) – cas number for component of interest

• y_i (float) – mole fraction of component of interest

• P (float) – pressure in Pa

• T (float) – temperature in K

bar_ViR_RT(cas_i, y_i, P, T)

residual Partial molar volume for component i

\[\begin{split}\begin{align} \frac{\bar{V}_i^\mathrm{R}}{RT} &= \left(\frac{\partial \ln\hat{\phi}_i}{\partial P}\right)_{T,y}\\ &= \frac{B_{ii} + (1-y_i)^2\delta_{ij}}{RT} \end{align}\end{split}\]

Note

This expression does not depend on \(P\)

Parameters

• cas_i (str) – cas number for component of interest

• y_i (float) – mole fraction of component of interest

• P (float) – pressure in Pa

• T (float) – temperature in K

Returns

\(\bar{V}_i^\mathrm{R}/R/T\)

calc_Z(y_k, P, T)

Parameters

• y_k (dict) – mole fractions of each component \(k\)

• P – pressure in Pa

• T – temperature in K

Returns

\(Z\) [mol/m^3], where \(Z = 1 + BP/R/T\)

d_dij_d_Tr(T)

(3)\[\frac{\partial \delta_{ij}}{\partial T_r} = 2\frac{\partial B_{ij}}{\partial T_r}-\frac{\partial B_{ii}}{\partial T_r}-\frac{\partial B_{jj}}{\partial T_r}\]

Parameters

T – temperature [K]

d_ij_expr(T)

(4)\[\delta_{ij} = 2B_{ij} - B_{ii} - B_{jj}\]

Parameters

T – temperature [K]

Returns

\(\delta_{ij}\) [m**3/mol]

fugacity_i_expr(cas_i, y_i, P, T)

Fugacity of component i in mixture \(f_i=\hat{\phi}_i y_i P\)

Parameters

• cas_i (str) – cas number for component of interest

• y_i (float) – mole fraction of component of interest

• P (float) – pressure in Pa

• T (float) – temperature in K

get_w_Tc_Pc(cas_i, cas_j=None)

Returns critical constants for calculation based off of whetner i = j or not

Returns

(\(w\), \(T_c\), \(P_c\))

Return type

tuple

ln_hat_phi_i_expr(cas_i, y_i, P, T)

logarithm of fugacity coefficient

\[\ln\hat{\phi}_i = \frac{P}{RT}\left[B_{ii} + (1-y_i)^2\delta_{ij}\right]\]

Parameters

• cas_i (str) – cas number for component of interest

• y_i (float) – mole fraction of component of interest

• P (float) – pressure in Pa

• T (float) – temperature in K

where \(\delta_{ij}\) is given by Equation (4)