Implicit Finite difference method to solve a Heat Transfer problem

fogel1497

I'm working on this project where I am trying to determine how long it will take a ceramic plate, with heat generating wire embedded in it and one side exposed to open convection, to reach %95 of its steady state condition.

This is not what I need help with, I have set up the equations and rearranged them so they are simple polynomials. By creating two different control volumes, (one around the surface exposed to the free convection and one containing the heat generating wire), and writing out energy balances for those control volumes I obtained the following two equations:

0 = 2(T₂^p) + (1/6)T₁^p+1 + (1/6)T₃^p+1 - (5.633)T₂^p+1 + (1/6)T₃^p+1 + (3)T₅^p+1 + 90.9

and

0 = (4)T₇^p + (10.67)T₇^p+1 + (2/3)T₈^p+1 + (3)T₄^p+1 + (3)T₁₀^p+1

Where I am confused is I am not sure what to do with all these terms coefficients. I know I need to make a coefficient matrix A out of all of the coefficients I have but I am not sure of its exact structure. Also once I have the coefficient matrix I know it needs to be multiplied by a matrix C which contains the values of the temperature for all my nodes at time = p.

Where I run into a problem is trying to figure out how from these matrices I can determine the amount of time it will take to reach %95 of steady state conditions. If anyone is familiar with this method and can shed some light I would really appreciate it.

Thanks in advance