Finite Difference Analysis of a flat plate

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Finite Difference Analysis of a flat plate

• Evan Selin Terrance Hess

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Goals

• 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

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Problem Set Up

• Required Properties
• Temperatures at each boundary
• Conductivity, k
• Heat flux, q (W/m2)
• Positive flux entering plate

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Problem Set Up

• Equations used

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Process
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-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

• 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

Coefficient Matrix for 1 heat flux
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Solution 1st boundary condition

• T1 35 ?, T2 50 ?, T3 100 ?, T4 50 ?

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Solution 2nd boundary condition

• T1 0 ?, T2 50 ?, T3 100 ?, q4 50 W/m2,
  k 15.1 W/mK

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Solution 3rd boundary condition

• T1 100 ?, q2 75 W/m2, T3 50 ?, q4 -25
  W/m2, k 15.1 W/mK

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Conclusions

• 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

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Appendix Hand Work
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Appendix Hand Work