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Cleaned up mixtures docs a bit

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Signed-off-by: Ian Bell <ian.h.bell@gmail.com>
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Ian Bell
2014-12-01 21:30:02 -05:00
parent 0c44768973
commit eb315f1cba
3 changed files with 104 additions and 220 deletions
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104
Web/fluid_properties/Mixtures.rst
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@@ -57,6 +57,110 @@ Phase Envelope
plt.ylabel('Pressure [Pa]')
plt.tight_layout()
Reducing Parameters
-------------------
From Lemmon :cite:`Lemmon-JPCRD-2000` for the properties of Dry Air, and also from Lemmon :cite:`Lemmon-JPCRD-2004` for the properties of R404A, R410A, etc.
.. math::
\rho_r(\bar x) = \left[ \sum_{i=1}^m\frac{x_i}{\rho_{c_i}}+\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}x_ix_j\zeta_{ij}\right]^{-1}
.. math::
T_r(\bar x) = \sum_{i=1}^mx_iT_{c_i}+\sum_{i=1}^{m-1}\sum_{j=i+1}^mx_ix_j\xi_{ij}
From the GERG 2008 formulation :cite:`Kunz-JCED-2012`
.. math::
T_r(\bar x) = \sum_{i=1}^{N}x_i^2T_{c,i} + \sum_{i=1}^{N-1}\sum_{j=i+1}^{N}2x_ix_j\beta_{T,ij}\gamma_{T,ij}\frac{x_i+x_j}{\beta_{T,ij}^2x_i+x_j}(T_{c,i}T_{c,j})^{0.5}
.. math::
\frac{1}{\rho_r(\bar x)}=v_r(\bar x) = \sum_{i=1}^{N}x_i^2\frac{1}{\rho_{c,i}} + \sum_{i=1}^{N-1}\sum_{j=i+1}^N2x_ix_j\beta_{v,ij}\gamma_{v,ij}\frac{x_i+x_j}{\beta^2_{v,ij}x_i+x_j}\frac{1}{8}\left(\frac{1}{\rho_{c,i}^{1/3}}+\frac{1}{\rho_{c,j}^{1/3}}\right)^{3}
Excess Helmholtz Energy Terms
-----------------------------
From Lemmon :cite:`Lemmon-JPCRD-2004` for the properties of R404A, R410A, etc.
.. math::
\alpha^E(\delta,\tau,\mathbf{x}) = \sum_{i=1}^{m-1} \sum_{j=i+1}^{m} \left [ x_ix_jF_{ij} \sum_{k}N_k\delta_{d_k}\tau^{t_k}\exp(-\delta^{l_k})\right]
where the terms :math:`N_k,d_k,t_k,l_k` correspond to the pair given by the indices :math:`i,j`
From Lemmon :cite:`Lemmon-JPCRD-2000` for the properties of Dry Air
.. math::
\alpha^E(\delta,\tau,\mathbf{x}) = \left \lbrace \sum_{i=1}^{2} \sum_{j=i+1}^{3} x_ix_jF_{ij}\right\rbrace \left[-0.00195245\delta^2\tau^{-1.4}+0.00871334\delta^2\tau^{1.5} \right]
From Kunz and Wagner :cite:`Kunz-JCED-2012` for GERG 2008 formulation
.. math::
\alpha^E(\delta,\tau,\mathbf{x}) = \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} x_ix_jF_{ij}\alpha_{ij}^r(\delta,\tau)
where
.. math::
\alpha_{ij}^r(\delta,\tau) = \sum_{k=1}^{K_{pol,ij}}\eta_{ij,k}\delta^{d_{ij,k}}\tau^{t_{ij,k}}+\sum_{k=K_{pol,ij}+1}^{K_{pol,ij}+K_{Exp,ij}}\eta_{ij,k}\delta^{d_{ij,k}}\tau^{t_{ij,k}}\exp[-\eta_{ij,k}(\delta-\varepsilon_{ij,k})^2-\beta_{ij,k}(\delta-\gamma_{ij,k})]
and is for the particular binary pair given by the indices :math:`i,j`. This term is similar in form to other Helmholtz energy terms for pure fluids though the derivatives are slightly special.
Appendix
--------
To convert from the form from Lemmon for HFC and Air to that of GERG 2008, the following steps are required:
.. math::
x_0T_{c0}+(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = x_0^2T_{c0}+(1-x_0)^2T_{c1} + 2x_0(1-x_0)\beta\gamma_T\frac{x_0+(1-x_0)}{\beta x_0 + (1-x_0)}\sqrt{T_{c0}T_{c1}}
set :math:`\beta=1`, solve for :math:`\gamma`. Equate the terms
.. math::
x_0T_{c0}+(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = x_0^2T_{c0}+(1-x_0)^2T_{c1} + 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Move to LHS
.. math::
[x_0-x_0^2]T_{c0}+[(1-x_0)-(1-x_0)^2]T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Factor
.. math::
x_0(1-x_0)T_{c0}+(1-x_0)[1-(1-x_0)]T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Expand
.. math::
x_0(1-x_0)T_{c0}+x_0(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Cancel factors of :math:`x_0(1-x_0)`
.. math::
T_{c0}+T_{c1}+\xi_{01} = 2\gamma_T\sqrt{T_{c0}T_{c1}}
Answer:
.. math::
\boxed{\gamma_T = \dfrac{T_{c0}+T_{c1}+\xi_{01}}{2\sqrt{T_{c0}T_{c1}}}}
Same idea for the volume
.. math::
\boxed{\gamma_v = \dfrac{v_{c0}+v_{c1}+\zeta_{01}}{\frac{1}{4}\left(\frac{1}{\rho_{c,i}^{1/3}}+\frac{1}{\rho_{c,j}^{1/3}}\right)^{3}}}
References
----------

112
Web/mixtures/phase_boundary.rst
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@@ -1,112 +0,0 @@
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 :math:`\mathbf{F}` is equal to
.. math::
F_i = \ln\phi(T,p,\mathbf{y})-\ln \phi(T,p,\mathbf{x})+\ln K_i=0, i=1,2,3... N
.. math::
F_{N+1} = \sum_{i=1}^{N}(y_i-x_i)=0
.. math::
x_i = \frac{z_i}{1-\beta+\beta K_i}
and
.. math::
y_i = \frac{K_iz_i}{1-\beta+\beta K_i}
.. math::
F_{N+2} = X_s - S = 0
and the system to be solved is equal to
.. math::
\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.
.. math::
\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]
.. math::
\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]
.. math::
\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 :math:`\zeta = 0` for i:math:`\neq`j , and :math:`\zeta = 0` for i=j. Also
.. math::
\phi_{ij} = n\left( \frac{\partial \ln \phi_i}{\partial n_j}\right)_{T,p}
For the :math:`F_{N+1}` term,
.. math::
\frac{\partial F_{N+1}}{\partial \ln K_j}=\frac{K_jz_j}{(1-\beta+\beta K_j)^2}
and all other partials of :math:`F_{N+1}` in the Jacobian are zero. For the specified term
.. math::
\frac{\partial F_{N+2}}{X_s}=1
and all other partials of :math:`F_{N+2}` in the Jacobian are zero.
..
Onwards...
Gerg 2004 Monograph, Eqn 7.27:
.. math::
\ln \phi_i = \left( \frac{\partial n\alpha^r}{\partial n_i}\right)_{T,V,n_j}-\ln Z
and (Kunz, 2012, Table B4)
.. math::
\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
.. math::
\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
Pandoc
------
pandoc --mathjax -s -f rst -t html5 -o phase_boundary.html phase_boundary.rst

108
Web/mixtures/reducing_parameters.rst
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@@ -1,108 +0,0 @@
Mixtures Information
====================
Reducing Parameters
-------------------
From Lemmon, JPCRD, 2000 for the properties of Dry Air, and also from Lemmon, JPCRD, 2004 for the properties of R404A, R410A, etc.
.. math::
\rho_r(\bar x) = \left[ \sum_{i=1}^m\frac{x_i}{\rho_{c_i}}+\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}x_ix_j\zeta_{ij}\right]^{-1}
.. math::
T_r(\bar x) = \sum_{i=1}^mx_iT_{c_i}+\sum_{i=1}^{m-1}\sum_{j=i+1}^mx_ix_j\xi_{ij}
From the GERG 2008 formulation (Kunz and Wagner, JCED, 2012)
.. math::
T_r(\bar x) = \sum_{i=1}^{N}x_i^2T_{c,i} + \sum_{i=1}^{N-1}\sum_{j=i+1}^{N}2x_ix_j\beta_{T,ij}\gamma_{T,ij}\frac{x_i+x_j}{\beta_{T,ij}^2x_i+x_j}(T_{c,i}T_{c,j})^{0.5}
.. math::
\frac{1}{\rho_r(\bar x)}=v_r(\bar x) = \sum_{i=1}^{N}x_i^2\frac{1}{\rho_{c,i}} + \sum_{i=1}^{N-1}\sum_{j=i+1}^N2x_ix_j\beta_{v,ij}\gamma_{v,ij}\frac{x_i+x_j}{\beta^2_{v,ij}x_i+x_j}\frac{1}{8}\left(\frac{1}{\rho_{c,i}^{1/3}}+\frac{1}{\rho_{c,j}^{1/3}}\right)^{3}
Excess Helmholtz Energy Terms
-----------------------------
From Lemmon, JPCRD, 2004 for the properties of R404A, R410A, etc.
.. math::
\alpha^E(\delta,\tau,\mathbf{x}) = \sum_{i=1}^{m-1} \sum_{j=i+1}^{m} \left [ x_ix_jF_{ij} \sum_{k}N_k\delta_{d_k}\tau^{t_k}\exp(-\delta^{l_k})\right]
where the terms :math:`N_k,d_k,t_k,l_k` correspond to the pair given by the indices :math:`i,j`
From Lemmon, JPCRD, 2000 for the properties of Dry Air
.. math::
\alpha^E(\delta,\tau,\mathbf{x}) = \left \lbrace \sum_{i=1}^{2} \sum_{j=i+1}^{3} x_ix_jF_{ij}\right\rbrace \left[-0.00195245\delta^2\tau^{-1.4}+0.00871334\delta^2\tau^{1.5} \right]
From Kunz and Wagner, JCED, 2012 for GERG 2008
.. math::
\alpha^E(\delta,\tau,\mathbf{x}) = \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} x_ix_jF_{ij}\alpha_{ij}^r(\delta,\tau)
where
.. math::
\alpha_{ij}^r(\delta,\tau) = \sum_{k=1}^{K_{pol,ij}}\eta_{ij,k}\delta^{d_{ij,k}}\tau^{t_{ij,k}}+\sum_{k=K_{pol,ij}+1}^{K_{pol,ij}+K_{Exp,ij}}\eta_{ij,k}\delta^{d_{ij,k}}\tau^{t_{ij,k}}\exp[-\eta_{ij,k}(\delta-\varepsilon_{ij,k})^2-\beta_{ij,k}(\delta-\gamma_{ij,k})]
and is for the particular binary pair given by the indices :math:`i,j`. This term is similar in form to other Helmholtz energy terms for pure fluids though the derivatives are slightly special.
Appendix
--------
To convert from the form from Lemmon for HFC and Air to that of GERG 2008, the following steps are required:
.. math::
x_0T_{c0}+(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = x_0^2T_{c0}+(1-x_0)^2T_{c1} + 2x_0(1-x_0)\beta\gamma_T\frac{x_0+(1-x_0)}{\beta x_0 + (1-x_0)}\sqrt{T_{c0}T_{c1}}
set :math:`\beta=1`, solve for :math:`\gamma`. Equate the terms
.. math::
x_0T_{c0}+(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = x_0^2T_{c0}+(1-x_0)^2T_{c1} + 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Move to LHS
.. math::
[x_0-x_0^2]T_{c0}+[(1-x_0)-(1-x_0)^2]T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Factor
.. math::
x_0(1-x_0)T_{c0}+(1-x_0)[1-(1-x_0)]T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Expand
.. math::
x_0(1-x_0)T_{c0}+x_0(1-x_0)T_{c1}+x_0(1-x_0)\xi_{01} = 2x_0(1-x_0)\gamma_T\sqrt{T_{c0}T_{c1}}
Cancel factors of :math:`x_0(1-x_0)`
.. math::
T_{c0}+T_{c1}+\xi_{01} = 2\gamma_T\sqrt{T_{c0}T_{c1}}
Answer:
.. math::
\boxed{\gamma_T = \dfrac{T_{c0}+T_{c1}+\xi_{01}}{2\sqrt{T_{c0}T_{c1}}}}
Same idea for the volume
.. math::
\boxed{\gamma_v = \dfrac{v_{c0}+v_{c1}+\zeta_{01}}{\frac{1}{4}\left(\frac{1}{\rho_{c,i}^{1/3}}+\frac{1}{\rho_{c,j}^{1/3}}\right)^{3}}}