Evan Selin & Terrance Hess
Find temperature at points throughout asquare plate subject to several types ofboundary 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
Required Properties:
◦Temperatures at eachboundary
◦Conductivity, k
◦Heat flux, q” (W/m2)
Positive flux enteringplate
Equations used:
Determine (x,y) positionof each node
Create finite differenceequations for desired setof boundary conditions
Build augmented matrixfor solution
Solve matrices fortemperatures at eachnode (matrix inversion)
Build algorithm toautomatically generatesolution matrix
12
-4
2
0
0
1
0
0
0
0
0
0
0
1
-4
1
0
0
1
0
0
0
0
0
0
0
1
-4
1
0
0
1
0
0
0
0
0
0
0
1
-4
0
0
0
1
0
0
0
0
1
0
0
0
-4
2
0
0
1
0
0
0
0
1
0
0
1
-4
1
0
0
1
0
0
0
0
1
0
0
1
-4
1
0
0
1
0
0
0
0
1
0
0
1
-4
0
0
0
1
0
0
0
0
1
0
0
0
-4
2
0
0
0
0
0
0
0
1
0
0
1
-4
1
0
0
0
0
0
0
0
1
0
0
1
-4
1
0
0
0
0
0
0
0
1
0
0
1
-4
Coefficient Matrix for 1heat flux
T1 = 35 ℃, T2 = 50 ℃, T3 = 100 ℃, T4 = 50 ℃
4 divisions
5 divisions
9 divisions
T1 = 0 ℃, T2 = 50 ℃, T3 = 100 ℃, q”4 = 50 W/m2,k = 15.1 W/m*K
4 divisions
5 divisions
9 divisions
T1 = 100 ℃, q”2 = 75 W/m2, T3 = 50 ℃, q”4 = -25 W/m2,k = 15.1 W/m*K
4 divisions
5 divisions
9 divisions
Numerical Solution Software is very complex
Setting up equations is the hard part
Matrix increases size on order of divisionssquared
Calculations take a long time for large veryfine mesh
