HOT PLATE CONDUCTION NUMERICAL SOLVER AND VISUALIZER Kurt Hinkle and Ivan Yorgason.

HOT PLATE CONDUCTION NUMERICAL SOLVER AND VISUALIZERKurt Hinkle and Ivan Yorgason

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.

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

METHOD

METHODMesh

METHOD

1000ºC

500ºC

0ºC

100ºC

0ºC

0ºC

0ºC

0ºC

0ºC

0ºC

0ºC

0ºC

0ºC

Initial Values

500ºC 500ºC

1000ºC

1000ºC

0ºC

0ºC

100ºC 100ºC

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

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

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

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

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.

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.

FUNCTIONALITY

• Live Demo:• 14.exe

POST PROCESSING

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).

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

QUESTIONS?