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HOT PLATE CONDUCTION NUMERICAL SOLVER AND VISUALIZERKurt Hinkle and Ivan Yorgason
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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
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METHODMesh
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METHOD
1000ºC
500ºC
0ºC
100ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
Initial Values
500ºC 500ºC
1000ºC
1000ºC
0ºC
0ºC
100ºC 100ºC
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METHOD
1000ºC
500ºC
0ºC
100ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
0ºC
375ºC
0ºC
Calculate FirstElement Temperature
500ºC 500ºC
1000ºC
1000ºC
0ºC
0ºC
(1000ºC + 500ºC + 0ºC + 0ºC)/4 = 375ºC
?
100ºC 100ºC
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METHOD
1000ºC
500ºC
0ºC
100ºC
80.1ºC
179.7ºC
82.6ºC
140.6ºC
218.8ºC
150.4ºC
343.8ºC
375ºC
360.9ºC
1st Iteration Complete
500ºC 500ºC
1000ºC
1000ºC
0ºC
0ºC
100ºC 100ºC
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METHOD
1000ºC
500ºC
0ºC
100ºC
144.6ºC
228.5ºC
116.7ºC
267.2ºC
333.9ºC
222.1ºC
504.3ºC
515.6ºC
438.7ºC
2nd Iteration Complete
500ºC 500ºC
1000ºC
1000ºC
0ºC
0ºC
100ºC 100ºC
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METHOD
1000ºC
500ºC
0ºC
100ºC
177.5ºC
259.9ºC
133.4ºC
333.6ºC
395.1ºC
255.9ºC
572.6ºC
584.6ºC
473.7ºC
3rd Iteration Complete
500ºC 500ºC
1000ºC
1000ºC
0ºC
0ºC
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 (N^2 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
Mesh Size:The number of elementsbetween opposite walls.
Temperature:The temperature of thewall.
Calculate:Calculates the elementtemperatures and displaysthem colorfully.
Close:Closes the program.
Print:Calculates the elementtemperatures and once the algorithm is complete, itprints the resulting elementtemperatures to results.datin a matrix format along withthe wall temperatures.
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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?