I'm trying to replicate the model presented in this [paper](http://www.sciencedirect.com/science/article/pii/S1359431103000474), which is basically to model heat and mass transfer along a one-dimensional duct.(adsbygoogle = window.adsbygoogle || []).push({});

There are four characteristic equations for this problem :

Momentum conservation

$$\frac{\partial Y}{\partial t} +V\frac{\partial Y}{\partial z}+\omega_1\frac{\partial W}{\partial t}=0 $$

Mass transfer:

$$ \frac{\partial W}{\partial t} + \omega_2 (Y_w - Y) = 0 $$

Energy conservation:

$$ \frac{\partial T}{\partial t} + V\frac{\partial T}{\partial z} + \omega_3\frac{\partial T^*}{\partial t} = \omega_4 (Y-Y_w) $$

and Heat Transfer

$$\frac{\partial T^*}{\partial t} + \omega_5 (T^* - T) + \omega_6 (Y_w - Y) + \omega_7 (Y_w - Y)(T-T^*) = 0$$

I'm attempting to use a forward-difference approximation method, i.e.

$$ \frac{\partial Y}{\partial z} = \frac{ Y_{i+1}^m - Y_i^m }{\Delta z}$$

and so on for the other variables, where i is the spacial index in z and m is the time index. The $\omega$s and $V$ are constants.

I have initial and boundary values for all the parameters, so that leaves me needing to find $$Y_i^{m+1}, Y_{i+1}^m, W_i^{m+1}, T_i^{m+1}, T_{i+1}^m, T_i^{*m+1} $$

I have four equations and six unknowns. The paper states that an 'implicit up-wind difference form' and Gauss-Jordan elimination is used, but I can't see how my approach differs from that. I do acknowledge that my approach is an explicit method, but I believe the issue of indeterminacy remains. Am I missing something in my approach?

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# Finite difference method to solve first-order, multivariable

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