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Forming the Phase Envelope¶

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

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

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

where

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

and

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

DO NOT NORMALIZE $x$ and $y$ !!!!

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

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.

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

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

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

$\zeta$ is the Kronecker delta or $\zeta = 0$ for $i\neq j$ , and $\zeta = 0$ 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.

From 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)

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

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

Density marching phase envelope construction(T,P)¶

Another two alternatives have been proposed in A DENSITY MARCHING METHOD FOR CALCULATING PHASE ENVELOPES Gadhiraju Venkatarathnam, I&ECR, 2014

In this paper, density marching methods are proposed rather than methods that march in temperature, pressure, or K-factor.

The system of equations to be solved is similar to that of the GERG 2004 formulation, where the unknowns are $\ln(T)$ , $\ln(p)$ , and $\ln(K_i)$

(A1) - OK

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

(A2) - OK

$F_{N+1} = \sum_{i=1}^{N}\frac{z_i(K_i-1)}{1-\beta+\beta K_i}=0$

(A3) - TYPO, should be $\ln(\rho)$ rather than $\rho$ , and should be all on left-hand-side

$F_{N+2} = \ln \rho ''-\ln\rho''_{S} = 0$

(A6) - TYPO, missing an n to multiply the terms $\left( \frac{\partial \ln \phi_i}{\partial n_j}\right)_{T,p}$

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

$\zeta$ is the Kronecker delta or $\zeta = 0$ for $i\neq j$ , and $\zeta = 0$ for $i=j$ .

(A7) - OK

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

(A8) - OK

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

(A9) - OK

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

(A11) - OK

$\frac{\partial F_{N+2}}{\partial \ln K_j} = \frac{K_jz_j(1-\beta)\beta}{(1-\beta+\beta K_j)^2}\left(n\left(\frac{\partial \rho}{\partial n_j}\right)''_{T,p}\right)\left(\frac{1}{\rho''_{S}}\right)$

(A12) - OK

$\frac{\partial F_{N+2}}{\partial \ln T}=\left(\frac{\partial \rho}{\partial T}\right)''_{p,n}\frac{T}{\rho''_{S}}$

(A13) - OK

$\frac{\partial F_{N+2}}{\partial \ln p}=\left(\frac{\partial \rho}{\partial p}\right)''_{T,n}\frac{p}{\rho''_{S}}$

Density marching phase envelope construction (density)¶

From Gadhiraju Venkatarathnam, 2014, Density Marching Method for Calculating Phase Envelopes I&ECR (http://pubs.acs.org/doi/abs/10.1021/ie403633d)

The system of equations to be solved is similar to that of the GERG 2004 formulation, here the unknowns are $\ln(K_i)$ , $\ln(T)$ , and $\ln(\rho')$ . The vapor density $\rho''$ is imposed

(A14) - OK

$F_i = \ln K_i+\ln\phi(T,\rho'',\mathbf{y})-\ln \phi(T,\rho',\mathbf{x})=0, i=0,1,2,3... N-1$

(A15) - OK

$F_{N} = \sum_{i=1}^{N}\frac{z_i(K_i-1)}{1-\beta+\beta K_i}=0$

(A16) - OK

$F_{N+1} = p(T,\rho',\mathbf{x})-p(T,\rho'',\mathbf{y}) = 0$

(A17) - TYPO, missing an n to multiply the terms $\left( \frac{\partial \ln \phi_i}{\partial n_j}\right)_{T,p}$

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

$\zeta$ is the Kronecker delta or $\zeta = 0$ for $i\neq j$ , and $\zeta = 0$ for $i=j$ .

(A18) - OK

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

(A19) - OK

$\frac{\partial F_i}{\partial \ln \rho'} = -\rho'\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)'_{T,n}$

(A20) - OK

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

(A22) - TYPO Second derivative of ln(phi) with respect to rho' needs constraints, first needs to have the constraints in the right place

$\frac{\partial F_{N+1}}{\partial \ln K_j} = \frac{RTK_jz_j}{(1-\beta+\beta K_j)^2}\left[(1-\beta)\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)''_{T,n}+\beta\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)'_{T,n}\right]$

(A23) - TYPO Should be A23

$\frac{\partial F_{N+1}}{\partial \ln T}=T\left[\left(\frac{\partial p}{\partial T}\right)'_{\rho',n}-\left(\frac{\partial p}{\partial T}\right)''_{\rho'',n} \right]$

(A24) - TYPO Should be A24

$\frac{\partial F_{N+1}}{\partial \ln p'}=\rho'\left(\frac{\partial p}{\partial \rho}\right)'_{T,n}$

Other analytic derivatives¶

Three analyic derivatives are not provided in GERG and need to be rederived. The last is for T,p marching, the first two are for density marching:

$\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)_{T,n}$

$\left(\frac{\partial \ln \phi_i}{\partial T}\right)_{\rho,n}$

$n\left(\frac{\partial \rho}{\partial n_j}\right)_{T,p}$

Derivations¶

$\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)_{T,n} = \left(\frac{\partial \ln \phi_i}{\partial \delta}\right)_{\tau,\bar z}\frac{d\delta}{d\rho} = \left(\frac{\partial \ln \phi_i}{\partial \delta}\right)_{\tau,\bar z}\frac{1}{\rho_r}$

$\left(\frac{\partial \ln \phi_i}{\partial T}\right)_{\rho,n} = \left(\frac{\partial \ln \phi_i}{\partial \tau}\right)_{\delta,\bar z}\frac{d\tau}{dT} = \left(\frac{\partial \ln \phi_i}{\partial \tau}\right)_{\delta,\bar z}\frac{-T_r}{T^2}$

The fact that derivatives are taken with $n$ constant implies that the mole fractions of the components are constant, and therefore the reducing temperatures and densities are also constant.

For $n\left( \frac{\partial \rho}{\partial n_i}\right)_{T,p,n_j}$ ¶

GERG 2007 Monograph Equation 7.32 gives

$\hat v_i = \left( \frac{\partial V}{\partial n_i}\right)_{T,p,n_j} = \dfrac{-\left(\dfrac{\partial p}{\partial n_i}\right)_{T,V,n_j}}{\left(\dfrac{\partial p}{\partial V}\right)_{T,n}}$

expand the left hand side with

$V = vn = \dfrac{n}{\rho}$

n held constant in derivative, so get

$\left( \frac{\partial V}{\partial n_i}\right)_{T,p,n_j} = n\left( \frac{\partial (1/\rho)}{\partial n_i}\right)_{T,p,n_j} = -\frac{n}{\rho^2}\left( \frac{\partial \rho}{\partial n_i}\right)_{T,p,n_j}$

$n\left( \frac{\partial \rho}{\partial n_i}\right)_{T,p,n_j} = -\rho^2\left( \frac{\partial V}{\partial n_i}\right)_{T,p,n_j}$

From Michelsen and Mollerup

p. 30 Eq. III

$RT \left( \frac{\partial \ln \phi_i}{\partial p}\right)_{T,\mathbf n} = \hat v_i -\frac{RT}{p} = F_{P}^{(i)}$

p. 30 Eq. IV

$RT \left(\frac{\partial \ln \phi_i}{\partial n_i}\right)_{T,P} = \left(\frac{\partial^2 n\alpha^r}{\partial n_i \partial n_j}\right)_{T,V} + \hat v_i\hat v_j\left(\frac{\partial p}{\partial V}\right)_{T,\mathbf{n}} + \frac{RT}{n} = F_j^{(i)}$

p. 30 Eq. VII

$RT \left( \frac{\partial \ln \phi_i}{\partial n_i}\right)_{T,V} = F_{j}^{(i)}+F_{P}^{(i)}\left(\frac{\partial p}{\partial n_j}\right)_{T,V}$

Divide through by RT and multiply through by n

$n \left( \frac{\partial \ln \phi_i}{\partial n_i}\right)_{T,V} = n\frac{F_{j}^{(i)}}{RT} + n\frac{F_{P}^{(i)}}{RT}\left(\frac{\partial p}{\partial n_j}\right)_{T,V}$

Derivation of derivatives of $F_i$ ¶

$F_i = \ln\phi(T,p,\mathbf{y}) -\ln\phi(T,p,\mathbf{x}) + \ln K_i$

$x_j = \frac{z_j}{1-\beta+\beta K_j},\,\,\, y_j = \frac{K_jz_j}{1-\beta+\beta K_j}$

Derivatives needed for product rule:

$\frac{\partial x_j}{\partial K_j} = \frac{-z_j\beta}{(1-\beta+\beta K_j)^2}$

$\frac{\partial y_j}{\partial K_j} = \frac{z_j(1-\beta+\beta K_j)-K_jz_j\beta}{(1-\beta+\beta K_j)^2} = \frac{z_j(1-\beta)}{(1-\beta+\beta K_j)^2}$

$\frac{d \ln K_j}{d K_j} = \frac{1}{K_j}$

Derivative of the residual function

$\left. \frac{\partial F_i}{\partial K_j}\right|_{T,p} = \frac{\partial \ln\phi_i''}{\partial y_j}\frac{d y_j}{d K_j}\frac{d K_j}{d \ln K_j} - \frac{\partial \ln\phi_i'}{\partial x_j}\frac{d x_j}{d K_j}\frac{d K_j}{d \ln K_j} + D_{ij}$

$\left. \frac{\partial F_i}{\partial K_j}\right|_{T,p} = \left.\frac{\partial \ln\phi_i''}{\partial y_j}\right|_{T,p}\frac{z_j(1-\beta)}{(1-\beta+\beta K_j)^2}K_j - \left.\frac{\partial \ln\phi_i'}{\partial x_j}\right|_{T,p}\frac{-z_j\beta}{(1-\beta+\beta K_j)^2}K_j + D_{ij}$

Grouping terms like GERG

$\left. \frac{\partial F_i}{\partial K_j}\right|_{T,p} = \frac{K_jz_j}{(1-\beta+\beta K_j)^2}\left[(1-\beta)\left.\frac{\partial \ln\phi_i''}{\partial y_j}\right|_{T,p} + \beta\left.\frac{\partial \ln\phi_i'}{\partial x_j}\right|_{T,p}\right] + D_{ij}$

Magic key:

$x_j = \frac{n_j}{n}$

therefore

$\frac{dx_j}{dn_j} = \frac{1}{n}$ and thus multiply

$\left.\frac{\partial \ln\phi_i}{\partial x_j}\right|_{T,p}$ by unity (

$n\frac{dx_j}{dn_j}$ ) to get

$\left.\frac{\partial \ln\phi_i}{\partial x_j}\right|_{T,p} = n\left.\frac{\partial \ln\phi_i}{\partial n_j}\right|_{T,p}$

Appendix¶

GERG 2004 monograph Eqn. 7.36

$\left( \frac{\partial \ln\left(f_i/n_i\right)}{\partial T}\right)_{V,\bar n} = \frac{1}{T}+\left(\frac{\partial^2n\alpha_r}{\partial T \partial n_i}\right)_{V,n_j}$

Since derivatives are taken with $\rho$ and $n$ held constant, that also implies that $V$ is held constant in the derivative.

$\frac{\partial \left[\ln\left(\frac{p}{n}\right)\right]}{\partial T} = \frac{1}{pn}\left(\frac{\partial p}{\partial T}\right)_{\rho,n}$

Equivalent fugacities for the $i$ -th component

$F_k^A = \ln f_i(T,p,\mathbf{x})-\ln f_i(T,p,\mathbf{y}) = 0\mbox{ for } k = i = 1...N$

Material balance

$F_k^B = \frac{z_i-x_i}{y_i-x_i}-\frac{z_{N-1}-x_{N-1}}{y_{N-1}-x_{N-1}}\mbox{ for }i=1..N-2; k = i+N; k = N+1..2N-2$

The independent variables to be obtained are

For $\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)_{T,n}$ and $\left(\frac{\partial \ln \phi_i}{\partial T}\right)_{\rho,n}$ ¶

Lemmon 2009 (propane) FOR PURE FLUID ONLY (I think)

$\ln \phi = Z-1-\ln Z +\alpha^r$

GERG 2007 Monograph 7.27

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

GERG 2007 Monograph 7.34

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

thus equating factors of $\left(\frac{\partial n \alpha^r}{\partial n_i} \right)_{T,V,n_j}$ yields

$\ln \phi_i = \ln \left(\frac{f_i}{n_i}\right) - \ln\left(\frac{RT}{V}\right) - \ln \left(Z\right)$

and $Z = (pV)/(nRT)$ ,

$\ln \phi_i = \ln \left(\frac{f_i}{n_i}\right) - \ln\left(\frac{RT}{V}\right) - \ln \left(\frac{pV}{nRT}\right)$

$\ln \phi_i = \ln \left(\frac{f_i}{n_i}\right) -\ln\left(\frac{p}{n}\right)$

$K_i = \frac{\ln\phi'}{\ln\phi''}$

$\beta = \frac{n''}{n} = \frac{n''}{n'+n''}$

$K_i = y_i/x_i$

$K_i = \frac{n_i''/n''}{n_i'/n'}$

$n_i' = \frac{n'}{K_i}\frac{n_i''}{n''} = \frac{n(1-\beta)}{K_i}\frac{n_i''}{n\beta}$

In [ ]:

Conversion of derivatives¶

To convert partial with $T$ , $V$ , $x_k$ held constant to one with $\tau$ , $\delta$ , $x_k$ held constant, use Gernert 3.118, or

$\frac{\partial}{\partial x_j} [Y]_{T,V,x_k} = \frac{\partial}{\partial x_j} [Y]_{\tau,\delta,x_k}+\left(\frac{\partial\delta}{\partial x_j}\right)_{T,V,x_k}\left.\frac{\partial Y}{\partial\delta}\right|_{\tau,\bar x}+\left(\frac{\partial\tau}{\partial x_j}\right)_{T,V,x_k}\left.\frac{\partial Y}{\partial\tau}\right|_{\delta,\bar x}$

To convert pressure,

$p=\rho R T(1+\delta \alpha_\delta)$

convert $\rho$ and $T$ to reduced variables

$p=\rho_r(\bar x)\delta R \frac{T_r(\bar x)}{\tau}(1+\delta \alpha_\delta) = \rho_r(\bar x)R \frac{T_r(\bar x)}{\tau}\delta (1+\delta \alpha_\delta)$

All the derivatives

$\frac{dp}{d\tau}\times\frac{d\tau}{dx_j} = -\rho_r(\bar x)\delta R \frac{T_r(\bar x)}{\tau^2}(1+\delta \alpha_\delta) \times \frac{1}{T}\frac{\partial T_r}{\partial x_j}|_{T,V,x_k}$

$\frac{dp}{d\delta}\times\frac{d\delta}{dx_j} = \rho_r(\bar x) R \frac{T_r(\bar x)}{\tau}[ (1+\delta \alpha_\delta)+ \delta(\alpha_\delta+\delta \alpha_{\delta\delta})] \times \frac{-\delta}{\rho_r}\frac{\partial \rho_r}{\partial x_j}|_{T,V,x_k}$

$\frac{\partial p}{\partial x_j}|_{\tau,\delta,x_k} = \frac{\delta R}{\tau}\left[\rho_r(\bar x)T_r(\bar x)(\delta \frac{\partial}{\partial x_j}[\alpha_\delta]_{\tau, \delta, x_k})+(1+\delta \alpha_\delta)\left(\rho_r \frac{\partial T_r}{\partial x_j}+T_r \frac{\partial \rho_r}{\partial x_j}\right)\right]$

with $\delta$ factored out

Turn back into normal variables

$\frac{dp}{d\tau}\times\frac{d\tau}{dx_j} = -\frac{\rho R}{\tau}(1+\delta \alpha_\delta)\frac{\partial T_r}{\partial x_j}|_{T,V,x_k}$

$\frac{dp}{d\delta}\times\frac{d\delta}{dx_j} = - R T[ (1+\delta \alpha_\delta)+ \delta(\alpha_\delta+\delta \alpha_{\delta\delta})]\delta\frac{\partial \rho_r}{\partial x_j}|_{T,V,x_k}$

$\frac{\partial p}{\partial x_j}|_{\tau,\delta,x_k} = R\left[\rho T (\delta \frac{\partial}{\partial x_j}[\alpha_\delta]_{\tau, \delta, x_k})+(1+\delta \alpha_\delta)\left(\frac{\rho}{\tau} \frac{\partial T_r}{\partial x_j}+\delta T \frac{\partial \rho_r}{\partial x_j}\right)\right]$

First term cancels with a term in the third one, yielding

$\frac{dp}{d\delta}\times\frac{d\delta}{dx_j} = -\delta R T[ (1+\delta \alpha_\delta)+ \delta(\alpha_\delta+\delta \alpha_{\delta\delta})]\frac{\partial \rho_r}{\partial x_j}|_{T,V,x_k}$

$\frac{\partial p}{\partial x_j}|_{\tau,\delta,x_k} = \rho R T (\delta \frac{\partial}{\partial x_j}[\alpha_\delta]_{\tau, \delta, x_k})+(1+\delta \alpha_\delta)\delta R T \frac{\partial \rho_r}{\partial x_j}$

First term in first line cancels with term at end of second line, yielding

$\frac{dp}{d\delta}\times\frac{d\delta}{dx_j} = -\delta R T[ \delta(\alpha_\delta+\delta \alpha_{\delta\delta})]\frac{\partial \rho_r}{\partial x_j}|_{T,V,x_k}$

$\frac{\partial p}{\partial x_j}|_{\tau,\delta,x_k} = \rho R T (\delta \frac{\partial}{\partial x_j}[\alpha_\delta]_{\tau, \delta, x_k})$

Total equation is

$\frac{\partial p}{\partial x_j}|_{T,V,x_k} = \rho R T \left(\delta \frac{\partial}{\partial x_j}[\alpha_\delta]_{\tau, \delta, x_k}-\frac{\delta}{\rho_r}(\alpha_\delta+\delta \alpha_{\delta\delta})\frac{\partial \rho_r}{\partial x_j}|_{T,V,x_k}\right)$

$p=\rho R T(1+\delta \alpha_\delta)$

$p=\rho R T(1+\frac{\rho}{\rho_r} \alpha_\delta)$

In [1]:

import numpy as np
import matplotlib.pyplot as plt
import CoolProp.CoolProp as CP
%matplotlib inline

$\frac{\partial p}{\partial x_j}|_{T,V,x_k} = \rho R T \left(\rho\frac{-1}{\rho_r^2}\left(\frac{\partial \rho_r}{\partial x_j}\right)_{x_{k\neq j}}\alpha_\delta + \frac{\rho}{\rho_r}\left(\frac{\partial}{\partial x_j}\left(\frac{\partial \alpha_r}{\partial \delta}_{\tau,\bar x}\right)\right)_{T,V,x_{k\neq j}}\right)$

$\frac{\partial p}{\partial x_j}|_{T,V,x_k} = \delta\rho R T \left(\frac{-1}{\rho_r}\left(\frac{\partial \rho_r}{\partial x_j}\right)_{x_{k\neq j}}\alpha_\delta + \left(\frac{\partial}{\partial x_j}\left(\frac{\partial \alpha_r}{\partial \delta}_{\tau,\bar x}\right)\right)_{T,V,x_{k\neq j}}\right)$

In [2]:

p = np.linspace(1000, 12000, 100)
mix = 'REFPROP-MIX:Water[0.7]&Ethanol[0.3]'
rhoL = CP.Props('D','P',p,'Q',0,mix)
rhoV = CP.Props('D','P',p,'Q',1,mix)

In [3]:

plt.plot(rhoL,p,rhoV,p)

Out[3]:

[<matplotlib.lines.Line2D at 0x5d5e350>,
 <matplotlib.lines.Line2D at 0x5d5e530>]

In [3]:

K_i=\frac{y_i}{x_i}

Forming the Phase Envelope¶

Overview¶

Building the Jacobian matrix¶

Density marching phase envelope construction(T,P)¶

Density marching phase envelope construction (density)¶

Other analytic derivatives¶

Derivations¶

For n(∂ρ∂ni)T,p,njn\left( \frac{\partial \rho}{\partial n_i}\right)_{T,p,n_j}¶

Derivation of derivatives of FiF_i¶

Appendix¶

For (∂lnϕi∂ρ)T,n\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)_{T,n} and (∂lnϕi∂T)ρ,n\left(\frac{\partial \ln \phi_i}{\partial T}\right)_{\rho,n}¶

Conversion of derivatives¶

For $n\left( \frac{\partial \rho}{\partial n_i}\right)_{T,p,n_j}$ ¶

Derivation of derivatives of $F_i$ ¶

For $\left(\frac{\partial \ln \phi_i}{\partial \rho}\right)_{T,n}$ and $\left(\frac{\partial \ln \phi_i}{\partial T}\right)_{\rho,n}$ ¶