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112 lines
3.0 KiB
ReStructuredText
112 lines
3.0 KiB
ReStructuredText
Forming the Phase Boundary
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==========================
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Overview
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--------
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The analysis in this section follows the methodologies proposed in the GERG 2004 monograph published in 2007
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System of Equations
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-------------------
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Our residual vector :math:`\mathbf{F}` is equal to
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.. math::
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F_i = \ln\phi(T,p,\mathbf{y})-\ln \phi(T,p,\mathbf{x})+\ln K_i=0, i=1,2,3... N
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.. math::
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F_{N+1} = \sum_{i=1}^{N}(y_i-x_i)=0
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.. math::
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x_i = \frac{z_i}{1-\beta+\beta K_i}
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and
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.. math::
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y_i = \frac{K_iz_i}{1-\beta+\beta K_i}
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.. math::
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F_{N+2} = X_s - S = 0
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and the system to be solved is equal to
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.. math::
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\mathbf{J}\mathbf{\Delta X}= -\mathbf{F}
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Building the Jacobian matrix
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----------------------------
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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.
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.. math::
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\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]
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.. math::
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\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]
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.. math::
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\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
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where :math:`\zeta = 0` for i:math:`\neq`j , and :math:`\zeta = 0` for i=j. Also
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.. math::
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\phi_{ij} = n\left( \frac{\partial \ln \phi_i}{\partial n_j}\right)_{T,p}
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For the :math:`F_{N+1}` term,
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.. math::
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\frac{\partial F_{N+1}}{\partial \ln K_j}=\frac{K_jz_j}{(1-\beta+\beta K_j)^2}
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and all other partials of :math:`F_{N+1}` in the Jacobian are zero. For the specified term
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.. math::
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\frac{\partial F_{N+2}}{X_s}=1
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and all other partials of :math:`F_{N+2}` in the Jacobian are zero.
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..
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Onwards...
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Gerg 2004 Monograph, Eqn 7.27:
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.. math::
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\ln \phi_i = \left( \frac{\partial n\alpha^r}{\partial n_i}\right)_{T,V,n_j}-\ln Z
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and (Kunz, 2012, Table B4)
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.. math::
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\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}
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so
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.. math::
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\ln \phi_i = \alpha^r + n\left( \frac{\partial \alpha^r}{\partial n_i}\right)_{T,V,n_j}-\ln Z
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and its derivative w.r.t T can be obtained analytically. What about pressure?
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The fugacity coefficient can be obtained from (Kunz, 2012, equation 29)
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From GERG Monograph p. 60:
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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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Pandoc
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------
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pandoc --mathjax -s -f rst -t html5 -o phase_boundary.html phase_boundary.rst |