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Improved docs for mixture derivatives

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Signed-off-by: Ian Bell <ian.h.bell@gmail.com>
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Ian Bell
2014-08-25 11:34:33 +02:00
parent 4299a27490
commit d9ef19a3a8
1 changed files with 43 additions and 20 deletions
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63
src/Backends/Helmholtz/HelmholtzEOSMixtureBackend.h
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@@ -256,34 +256,40 @@ public:
long double mixderiv_d2alphar_dxi_dDelta(int i);
long double mixderiv_d2alphardxidxj(int i, int j);
/*! The derivative term
/*! \brief GERG 2004 Monograph equation 7.61
*
* The derivative term
\f[
\left(\frac{\partial p}{\partial T} \right)_{V,\bar n} = \rho R(1+\delta \alpha_{\delta}^r-\delta \tau \alpha^r_{\delta\tau})
\f]
GERG 2004 Monograph equation 7.61
*/
long double mixderiv_dpdT__constV_n();
long double mixderiv_dpdrho__constT_n();
/*! The derivative term
/** \brief GERG 2004 Monograph equation 7.62
*
* The derivative term
\f[
n\left(\frac{\partial p}{\partial V} \right)_{T,\bar n} = -\rho^2 RT(1+2\delta \alpha_{\delta}^r+\delta^2\alpha^r_{\delta\delta})
\f]
GERG 2004 Monograph equation 7.62
*/
long double mixderiv_ndpdV__constT_n();
/*! The derivative term
/** \brief GERG 2004 Monograph equation 7.63
*
* The derivative term
\f[
n\left(\frac{\partial p}{\partial n_i} \right)_{T,V,n_j} = \rho RT\left[1+\delta\alpha_{\delta}^r\left[2- \frac{1}{\rho_r}\cdot n\left( \frac{\partial \rho_r}{\partial n_i}\right)_{n_j}\right] +\delta\cdot n\left(\frac{\partial\alpha_{\delta}^r}{\partial n_i}\right)_{T,V,n_j}\right]
\f]
GERG 2004 Monograph equation 7.63
*/
long double mixderiv_ndpdni__constT_V_nj(int i);
/// GERG 2004 monograph Eqn. 7.32
/*! The partial molar volume
/** GERG 2004 monograph Eqn. 7.32
* The partial molar volume
\f[
\hat v_i = \left( \frac{\partial V}{\partial n_i}\right)_{T,p,n_j} = \frac{-\left(\dfrac{\partial p}{\partial n_i}\right)_{T,V,n_j}}{\left(\dfrac{\partial p}{\partial V}\right)_{T,\bar n}}
\f]
@@ -352,15 +358,28 @@ public:
long double mixderiv_ndln_fugacity_coefficient_dnj__constT_p(int i, int j);
/// Gernert Equation 3.115
/// Catch test provided
/** The derivative term
\f[
\left(\frac{\partial \ln \phi_i}{\partial x_j}\right)_{T,p,x_{k\neq j}} = \left(\frac{\partial^2n\alpha^r}{\partial x_j \partial n_i} \right)_{T,V}+\frac{1}{RT}\frac{\left(\frac{\partial p}{\partial n_i}\right)_{T,V,n_{k\neq i}}\left(\frac{\partial p}{\partial x_j}\right)_{T,V,x_{k\neq j}}}{\left(\frac{\partial p}{\partial V}\right)_{T,\bar n}}
\f]
*/
long double mixderiv_dln_fugacity_coefficient_dxj__constT_p_xi(int i, int j);
/// Gernert Equation 3.130
/// Catch test provided
/** The derivative term
\f[
\left(\frac{\partial p}{\partial x_j} \right)_{T,V,x_{k\neq j}} = \rho RT\left(-\frac{1}{\rho_r}\left(\frac{\partial \rho_r}{\partial x_j}\right)_{x_{k\neq j}} \delta\alpha_{\delta}^r + \delta\left(\frac{\partial}{\partial x_j}\left(\left( \frac{\partial \alpha^r}{\partial \delta}\right)_{\tau,\bar x}\right)\right)_{T,V,x_{k\neq j}}\right)
\f]
*/
long double mixderiv_dpdxj__constT_V_xi(int j);
/// Gernert Equation 3.117
long double mixderiv_d2nalphar_dni_dxj__constT_V(int i, int j){ return mixderiv_d_ndalphardni_dxj__constT_V_xi(i, j) + mixderiv_dalphar_dxj__constT_V_xi(j);};
/** The derivative term
\f[
\left(\frac{\partial^2n\alpha^r}{\partial x_i\partial n_j} \right)_{T,V} = \left(\frac{\partial}{\partial x_j}\left(n\left(\frac{\partial \alpha^r}{\partial n_i}\right)_{T,V,n_{j\neq i}}\right)\right)_{T,V,x_{k\neq j}} +\left(\frac{\partial \alpha^r}{\partial x_j}\right)_{T,V,x_{k\neq j}}
\f]
*/
long double mixderiv_d2nalphar_dxi_dxj__constT_V(int i, int j){ return mixderiv_d_ndalphardni_dxj__constT_V_xi(i, j) + mixderiv_dalphar_dxj__constT_V_xi(j);};
/// Gernert Equation 3.119
/// Catch test provided
@@ -403,8 +422,9 @@ public:
*/
long double mixderiv_d_ndalphardni_dTau(int i);
/// GERG 2004 Monograph Equation 7.50 and Table B4, Kunz, JCED, 2012
/** The derivative term
/** \brief GERG 2004 Monograph Equation 7.50 and Table B4, Kunz, JCED, 2012
*
The derivative term
\f{eqnarray*}{
\left(\frac{\partial }{\partial \delta} \left( n\left(\frac{\partial \phi^r}{\partial n_i} \right)_{T,V,n_j} \right)\right)_{\tau,\bar x} &=& (\alpha_{\delta}^r+\delta\alpha_{\delta\delta}^r)\left[1-\frac{1}{\rho_r}\cdot n\left(\frac{\partial \rho_r}{\partial n_i}\right)_{n_j} \right] \\
&+&\tau\alpha^r_{\delta\tau}\frac{1}{T_r}\cdot n\left(\frac{\partial T_r}{\partial n_i}\right)_{n_j}\\
@@ -413,7 +433,8 @@ public:
*/
long double mixderiv_d_ndalphardni_dDelta(int i);
/** GERG 2004 Monograph equation 7.41:
/** \brief GERG 2004 Monograph equation 7.41
* The derivative term
\f[
n\left(\frac{\partial^2n\alpha^r}{\partial n_i \partial n_j} \right)_{T,V} = n\left( \frac{\partial}{\partial n_j}\left(\frac{\partial n\alpha^r}{\partial n_i}\right)_{T,V,n_j}\right)_{T,V,n_i}
\f]
@@ -431,24 +452,26 @@ public:
*/
long double mixderiv_nd2nalphardnidnj__constT_V(int i, int j);
/// GERG 2004 Monograph equation 7.48
/** The derivative term
/** \brief GERG 2004 Monograph equation 7.48
The derivative term
\f[
n\left(\frac{\partial \delta}{\partial n_i} \right)_{T,V,n_j} = \delta - \frac{\delta}{\rho_r}\cdot n\left(\frac{\partial \rho_r}{\partial n_i} \right)_{n_j}
\f]
*/
long double mixderiv_nddeltadni__constT_V_nj(int i);
/// GERG 2004 Monograph equation 7.49
/** The derivative term
/** \brief GERG 2004 Monograph equation 7.49
*
The derivative term
\f[
n\left(\frac{\partial \tau}{\partial n_i} \right)_{T,V,n_j} = \frac{\tau}{T_r}\cdot n\left(\frac{\partial T_r}{\partial n_i} \right)_{n_j}
\f]
*/
long double mixderiv_ndtaudni__constT_V_nj(int i);
/// \brief GERG 2004 Monograph equation 7.52
/** The derivative term
/** \brief GERG 2004 Monograph equation 7.52
*
The derivative term
\f{eqnarray*}{
\left( \frac{\partial}{\partial x_j}\left(n\left(\frac{\partial\alpha^r}{\partial n_i}\right)_{T,V,n_j}\right)\right)_{\delta,\tau,x_i} &=& \delta\alpha_{\delta x_j}^{r}\left[ 1-\frac{1}{\rho_r}\cdot n\left(\frac{\partial \rho_r}{\partial n_i}\right)_{n_j}\right] \\
&-& \delta\alpha_{\delta}^{r}\frac{1}{\rho_r}\left[ \left(\frac{\partial}{\partial x_j}\left(n\left(\frac{\partial \rho_r}{\partial n_i}\right)_{n_j}\right)\right)_{x_i}-\frac{1}{\rho_r}\left(\frac{\partial \rho_r}{\partial x_j}\right)_{x_i}\cdot n\left(\frac{\partial \rho_r}{\partial n_i}\right)_{n_j}\right] \\