1. Evan Selin & Terrance Hess

2.  Find temperature at points throughout a square plate subject to several types of boundary conditions  Boundary Conditions: ◦ 4 Constant Temperature surfaces ◦ 3 Constant Temperatures and 1 heat flux surface ◦ 2 Constant Temperatures and 2 heat flux surfaces  Automate the construction of solver matrices

3.  Required Properties: ◦ Temperatures at each boundary ◦ Conductivity, k ◦ Heat flux, q” (W/m 2 )  Positive flux entering plate

4.  Equations used:

5.  Determine (x,y) position of each node  Create finite difference equations for desired set of boundary conditions  Build augmented matrix for solution  Solve matrices for temperatures at each node (matrix inversion)  Build algorithm to automatically generate solution matrix 12 -420010000000 1 1001000000 01 100100000 001 00010000 1000 2001000 01001 100100 001001 10010 0001001 0001 00001000 200 000001001 10 0000001001 1 00000001001 Coefficient Matrix for 1 heat flux

6. T1 = 35 ℃, T2 = 50 ℃, T3 = 100 ℃, T4 = 50 ℃ 4 divisions5 divisions 9 divisions

7. T1 = 0 ℃, T2 = 50 ℃, T3 = 100 ℃, q” 4 = 50 W/m 2, k = 15.1 W/m*K 4 divisions 5 divisions 9 divisions

8. T1 = 100 ℃, q” 2 = 75 W/m 2, T3 = 50 ℃, q” 4 = -25 W/m 2, k = 15.1 W/m*K 4 divisions 5 divisions9 divisions

9.  Numerical Solution Software is very complex  Setting up equations is the hard part  Matrix increases size on order of divisions squared  Calculations take a long time for large very fine mesh