Hot plate Conduction Numerical Solver and Visualizer

1
Hot plate Conduction Numerical Solver and
Visualizer

• Kurt Hinkle and Ivan Yorgason

2
Introduction

• There are analytical methods that, in certain
  cases, can produce exact mathematical solutions
  to 2D steady state conduction problems.
• There are even solutions that are available for
  simple geometries with specific boundary
  conditions that can be used simply by plugging in
  numbers.
• Sometimes, however, there are geometries and/or
  boundary conditions that are not covered by the
  aforementioned solutions.
• When this occurs, numerical techniques, such as
  finite-difference, finite-element, and
  boundary-element methods are used to provide
  approximate solutions.
• This project uses the finite-difference form of
  the heat equation to solve for the temperatures
  across a square plate.

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Limitations and assumptions

• 2D steady state conduction
• Constant wall temperatures
• No convection
• Square plate
• Square elements
• Temperatures ranging 0ºC - 1000ºC
• Mesh size ranging 3 - 80

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Method
5
Method
Mesh
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Method
500ºC
500ºC
500ºC
Initial Values
0ºC
0ºC
0ºC
1000ºC
0ºC
1000ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
1000ºC
0ºC
100ºC
100ºC
100ºC
7
Method
500ºC
500ºC
500ºC
0ºC
0ºC
375ºC
1000ºC
0ºC
?
Calculate First Element Temperature
(1000ºC 500ºC 0ºC 0ºC)/4 375ºC
1000ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
1000ºC
0ºC
100ºC
100ºC
100ºC
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Method
500ºC
500ºC
500ºC
179.7ºC
218.8ºC
375ºC
1000ºC
0ºC
1st Iteration Complete
1000ºC
0ºC
80.1ºC
140.6ºC
343.8ºC
82.6ºC
150.4ºC
360.9ºC
1000ºC
0ºC
100ºC
100ºC
100ºC
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Method
500ºC
500ºC
500ºC
228.5ºC
333.9ºC
515.6ºC
1000ºC
0ºC
2nd Iteration Complete
1000ºC
0ºC
144.6ºC
267.2ºC
504.3ºC
116.7ºC
222.1ºC
438.7ºC
1000ºC
0ºC
100ºC
100ºC
100ºC
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Method
500ºC
500ºC
500ºC
259.9ºC
395.1ºC
584.6ºC
1000ºC
0ºC
3rd Iteration Complete
1000ºC
0ºC
177.5ºC
333.6ºC
572.6ºC
133.4ºC
255.9ºC
473.7ºC
1000ºC
0ºC
100ºC
100ºC
100ºC
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Method

• Differences with finite-difference method
• Instead of setting up a matrix and inverting it
  to solve for all temperatures at once, the
  temperatures are solved for through an iterative
  process.
• This iterative process (N2 algorithm) is limited
  by a time which is calculated based on the mesh
  size. Larger mesh sizes are allowed more time to
  iteratively solve for the element temperatures.

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Functionality
13
Functionality

• Live Demo
• 14.exe

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Post processing
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Future Work

• Allow for other shapes and holes in the geometry
• Allow for different mesh element types
  (tetrahedral, etc.)
• Stop the iterative solver based on a tolerance
  instead of a time limit
• Export .jpg of visualized results with
  results.dat file
• Have the color scheme be relative to the maximum
  and minimum temperatures instead of the scale
  being absolute (1000ºC red and 0ºC blue).

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Conclusion

• Provides quick and accurate results for the given
  assumptions
• Graphically displays the results in an
  understandable and pleasing manner
• With the option to print the results to a file,
  further analysis is easily accomplished
• The finite-difference form of the heat equation
  is easy to implement programmatically

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Questions?