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Gibbs iteration equations

In order to solve for corrections to initial estimate of composition $ n_j$, Lagrangian multiplier $ \lambda_i$, moles n, and temperature T, we will use a descent Newton-Raphson method. This method involves a Taylor series expansion of the equation and use the terms of the first order. The correction variables are $ \Delta \ln{n_j}$ for the gases, $ \Delta n_j$ for the condensed species, $ \Delta \ln{n}$, $ \pi_i =
-\frac{\lambda_i}{RT}$ the dimensionless Lagrangian multipliers and $ \Delta \ln{T}$.

$\displaystyle \Delta \ln{n_j} - \sum_{i=1}^{l}{a_{ij}\pi_i} - \Delta \ln{n} - \...
... \frac{(H_T^0)_j}{RT}\right]\Delta \ln{T} = -\frac{\mu_j}{RT} \quad j = 1,...,m$ (12)

$\displaystyle -\sum_{i=1}^{l}{a_{ij}\pi_i}-\left[\frac{(H_T^0)_j}{RT}\right]\Delta \ln{T} = -\frac{\mu_j}{RT} \quad j = m+1,...,n$ (13)

$\displaystyle \sum_{j=1}^{m}{a_{kj}n_j}\Delta\ln{n_j} + \sum_{j=m+1}^{n}{a_{kj}}\Delta n_j = k_k^0 - b_k \quad k = 1,...,l$ (14)

$\displaystyle \sum_{j=1}^{m}{n_j}\Delta \ln{n_j} - n\Delta \ln{n} =n-\sum_{j=1}^{m}{n_j}$ (15)

$\displaystyle \sum_{j=1}^{m} {\left[\frac{n_j(H_T^0)_j}{RT}\right]}\Delta \ln{n...
...eft[\sum_{j=1}^{n}{\frac{n_j(C_p^0)_j}{R}}\right]\Delta \ln{T}=\frac{h_0-h}{RT}$ (16)

$\displaystyle \sum_{j=1}^{m} {\left(\frac{n_jS_j}{R}\right)}\Delta \ln{n_j}+\su...
...rac{n_j(C_p^0)_j}{R}}\right]\Delta \ln{T}=\frac{s_0-s}{R}+n-\sum_{j=1}^{m}{n_j}$ (17)


next up previous
Next: Reduced Gibbs iteration equations Up: cpropep Previous: Equations describing chemical equilibria
Antoine Lefebvre 2000-03-26