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Tidied up some mixture derivative docs

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Ian Bell
2015-05-19 21:18:34 -06:00
parent 44bf43a65f
commit d0d41aab26
1 changed files with 9 additions and 9 deletions
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18
src/Backends/Helmholtz/MixtureDerivatives.h
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@@ -147,13 +147,13 @@ class MixtureDerivatives{
*
* The derivative term
* \f[
* n\left(\frac{\partial \phi^r}{\partial n_i} \right)_{T,V,n_j}
* n\left(\frac{\partial \alpha^r}{\partial n_i} \right)_{T,V,n_j}
* \f]
* which is equal to
* \f{eqnarray*}{
* n\left(\frac{\partial \phi^r}{\partial n_i} \right)_{T,V,n_j} &=& \delta \phi^r_{\delta}\left[ 1-\frac{1}{\rho_r}\left[\left(\frac{\partial \rho_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial \rho_r}{\partial x_k}\right)_{x_j} \right]\right]\\
* && +\tau \phi^r_{\tau}\frac{1}{T_r}\left[\left(\frac{\partial T_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial T_r}{\partial x_k}\right)_{x_j} \right]\\
* && +\phi^r_{x_i}-\sum_{k=1}^{N}x_k\phi^r_{x_k}
* n\left(\frac{\partial \alpha^r}{\partial n_i} \right)_{T,V,n_j} &=& \delta \alpha^r_{\delta}\left[ 1-\frac{1}{\rho_r}\left[\left(\frac{\partial \rho_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial \rho_r}{\partial x_k}\right)_{x_j} \right]\right]\\
* && +\tau \alpha^r_{\tau}\frac{1}{T_r}\left[\left(\frac{\partial T_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial T_r}{\partial x_k}\right)_{x_j} \right]\\
* && +\alpha^r_{x_i}-\sum_{k=1}^{N}x_k\alpha^r_{x_k}
* \f}
* @param HEOS The HelmholtzEOSMixtureBackend to be used
* @param i The index of interest
@@ -271,9 +271,9 @@ class MixtureDerivatives{
*
* The derivative term
* \f{eqnarray*}{
* \frac{\partial }{\partial \tau} \left( n\left(\frac{\partial \phi^r}{\partial n_i} \right)_{T,V,n_j} \right) &=& \delta \phi^r_{\delta\tau}\left[ 1-\frac{1}{\rho_r}\left[\left(\frac{\partial \rho_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial \rho_r}{\partial x_k}\right)_{x_j} \right]\right]\\
* && +(\tau \phi^r_{\tau\tau}+\phi^r_{\tau})\frac{1}{T_r}\left[\left(\frac{\partial T_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial T_r}{\partial x_k}\right)_{x_j} \right]\\
* && +\phi^r_{x_i\tau}-\sum_{k=1}^{N}x_k\phi^r_{x_k\tau}
* \frac{\partial }{\partial \tau} \left( n\left(\frac{\partial \alpha^r}{\partial n_i} \right)_{T,V,n_j} \right) &=& \delta \alpha^r_{\delta\tau}\left[ 1-\frac{1}{\rho_r}\left[\left(\frac{\partial \rho_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial \rho_r}{\partial x_k}\right)_{x_j} \right]\right]\\
* && +(\tau \alpha^r_{\tau\tau}+\alpha^r_{\tau})\frac{1}{T_r}\left[\left(\frac{\partial T_r}{\partial x_i}\right)_{x_j} - \sum_{k=1}^N x_k\left(\frac{\partial T_r}{\partial x_k}\right)_{x_j} \right]\\
* && +\alpha^r_{x_i\tau}-\sum_{k=1}^{N}x_k\alpha^r_{x_k\tau}
* \f}
* @param HEOS The HelmholtzEOSMixtureBackend to be used
* @param i The index of interest
@@ -285,9 +285,9 @@ class MixtureDerivatives{
*
* 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] \\
* \left(\frac{\partial }{\partial \delta} \left( n\left(\frac{\partial \alpha^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}\\
* &+&\phi^r_{\delta x_i}-\sum_{k=1}^{N}x_k\phi^r_{\delta x_k}
* &+&\alpha^r_{\delta x_i}-\sum_{k=1}^{N}x_k\alpha^r_{\delta x_k}
* \f}
* @param HEOS The HelmholtzEOSMixtureBackend to be used
* @param i The index of interest