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Solving system of non-linear partial differential and algebraic equations.

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Hi guys!
I already had a couple of questions about my fluid-bed drying topic. It is almost over, and I understand pretty much all of it, but I cannot figure out how to solve these equations, because there is a differential equation which is non-linear, and the model is a system of partial differential and algebraic equations. The main object of these equations is to get a result for T_b (last equation), and T_e which are in a system of equations.
The ’z’ coordinate is solved with the explicit forward Euler (FE) method to keep these internal calculations as simple as possible.
I copied the equations from the article, with the texts between them, so you guys can understand it easier.
I wrote the value of constants here, but some of them changes in the function of the final result, so they are probably not very exact, but you can ignore the constants, I’m just curoius about the method solving the main object, how to get T_b and T_e (equations 49, 50, 51 and 54).
Thank you in advance, hope someone can help me out of it.

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