phase_boundary.rst

Forming the Phase Boundary

Overview

The analysis in this section follows the methodologies proposed in the GERG 2004 monograph published in 2007

System of Equations

Our residual vector \mathbf{F} is equal to

F_i = \ln\phi(T,p,\mathbf{y})-\ln \phi(T,p,\mathbf{x})+\ln K_i=0,  i=1,2,3... N

F_{N+1} = \sum_{i=1}^{N}(y_i-x_i)=0

x_i = \frac{z_i}{1-\beta+\beta K_i}

and

y_i = \frac{K_iz_i}{1-\beta+\beta K_i}

F_{N+2} = X_s - S = 0

and the system to be solved is equal to

\mathbf{J}\mathbf{\Delta X}= -\mathbf{F}

Building the Jacobian matrix

This is the trickiest part of this method. There are a lot of derivatives to implement, and we want to implement all of them analytically.

\frac{\partial F_i}{\partial \ln T} = T\left[ \left(\frac{\partial \ln \phi_i}{\partial T}\right)''_{p,\mathbf{n}} -\left(\frac{\partial \ln \phi_i}{\partial T}\right)'_{p,\mathbf{n}}\right]

\frac{\partial F_i}{\partial \ln p} = p\left[ \left(\frac{\partial \ln \phi_i}{\partial p}\right)''_{T,\mathbf{n}} -\left(\frac{\partial \ln \phi_i}{\partial p}\right)'_{T,\mathbf{n}}\right]

\frac{\partial F_i}{\partial \ln K_j} = \frac{K_jz_j}{(1-\beta+\beta K_j)^2}[(1-\beta)\phi_{ij}''+\beta\phi_{ij}']+\zeta

where \zeta = 0 for i:math:neq`j , and :math:zeta = 0` for i=j. Also

\phi_{ij} = n\left( \frac{\partial \ln \phi_i}{\partial n_j}\right)_{T,p}

For the F_{N+1} term,

\frac{\partial F_{N+1}}{\partial \ln K_j}=\frac{K_jz_j}{(1-\beta+\beta K_j)^2}

and all other partials of F_{N+1} in the Jacobian are zero. For the specified term

\frac{\partial F_{N+2}}{X_s}=1

and all other partials of F_{N+2} in the Jacobian are zero.

Onwards...

Gerg 2004 Monograph, Eqn 7.27:

\ln \phi_i  = \left( \frac{\partial n\alpha^r}{\partial n_i}\right)_{T,V,n_j}-\ln Z

and (Kunz, 2012, Table B4)

\left( \frac{\partial n\alpha^r}{\partial n_i}\right)_{T,V,n_j} = \alpha^r + n\left( \frac{\partial \alpha^r}{\partial n_i}\right)_{T,V,n_j}

so

\ln \phi_i  = \alpha^r + n\left( \frac{\partial \alpha^r}{\partial n_i}\right)_{T,V,n_j}-\ln Z

and its derivative w.r.t T can be obtained analytically. What about pressure?

The fugacity coefficient can be obtained from (Kunz, 2012, equation 29)

From GERG Monograph p. 60:

Since the two phases of a non-critical mixture are characterised by different compositions resulting in different values for the reducing functions and the corresponding reduced variables, a simple integral criterion which connects all phase equilibrium properties in a single relation such as Eq. (4.11) does not exist for mixtures

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