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Reduced Gibbs iteration equations

The large number of equation that were presented in the last section could be reduced to a much smaller number by algebric substitution. We substitute the expression for $ \Delta \ln{n_j}$ of equation 12 into equation 14 to 17. The resulting equations are

\begin{multline}
\sum_{i=1}^{l}\sum_{j=1}^{m}{a_{kj}a_{ij}n_j\pi_i}+\sum_{j=m+1}...
...
+ \sum_{j=1}^{m}{ \frac{a_{kj}n_j\mu_j}{RT}} \quad (k = 1,...,l)
\end{multline}

$\displaystyle \sum_{i=1}^{l}{a_{ij}\pi_i} + \left[ \frac{(H_T^0)_j}{RT}\right]\Delta \ln{T} = \frac{\mu_j}{RT} \quad (j=m+1,...,n)$ (18)

$\displaystyle \sum_{i=1}^{l}\sum_{j=1}^{m}{a_{ij}n_j\pi_i}+\left( \sum_{j=1}^{m...
...ght]\Delta \ln{T} = n - \sum_{j=1}^{m}{n_j}+\sum_{j=1}^{m}{\frac{n_j\mu_j}{RT}}$ (19)

\begin{multline}
\sum_{i=1}^{l} \left[
\sum_{j=1}^{m}{\frac{a_{ij}n_j(H_T^0)_j}{...
...rac{h_o-h}{RT} + \sum_{j=1}^{m}{\frac{n_j(H_T^0)_j\mu_j}{R^2T^2}}
\end{multline}

\begin{multline}
\sum_{i=1}^{l} \left[
\sum_{j=1}^{m}{\frac{a_{ij}n_jS_j}{R}}
\r...
... - \sum_{j=1}^{m}{n_j} + \sum_{j=1}^{m}{\frac{n_jS_j\mu_j}{R^2T}}
\end{multline}

Those equation can be simulataneously solve to get a new estimate. Using those new symbols, a resume of those equations in the form of a matrix is presented in the second document.

$\displaystyle \EuScript{H}_j = \frac{(H_T^0)_j}{RT}$ $\displaystyle \qquad \EuScript{H} = \frac{h}{RT}$ (20)
$\displaystyle \EuScript{G}_j = \frac{\mu_j}{RT}$ $\displaystyle \qquad \EuScript{H}_0 = \frac{h_0}{RT}$ (21)
$\displaystyle \EuScript{S}_j = \frac{S_j}{R}$ $\displaystyle \qquad \EuScript{S} = \frac{s}{R}$ (22)
$\displaystyle \EuScript{C}_j = \frac{(C_p^0)_j}{R}$ $\displaystyle \qquad \EuScript{S}_0 = \frac{s_0}{R}$ (23)


next up previous
Next: Cpropep functions description Up: cpropep Previous: Gibbs iteration equations
Antoine Lefebvre 2000-03-26