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Concept explainers
To calculate: The temperature on the heated plate with fixed temperature at the boundaries and round corners at upper right and lower left corners; considering the value of weighing factor for overrelaxation as 1.5 and performing iteration up to
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Answer to Problem 7P
Solution:
Heated plate having rounded upper right and lower left corners has the temperature distribution with the desired level of accuracy as,
Explanation of Solution
Given Information:
The fixed temperature at top and right boundary is
For overrelaxation the weighing factor is 1.5.
Limiting condition for iteration is
Formula used:
For a square grid, that is,
With
Overrelaxation is applied using the following expression after each iteration,
In the above expression,
The percent relative error is calculated using the following expression,
Laplacian equation for a node adjacent to an irregular boundary is solved for temperature distribution as,
Calculation:
Consider,
Consider the value of the parameters as,
The temperature at the node
Thus,
Further, the temperature at the node
Thus,
Liebmann’s method is used to solve the temperature on the heated plate iteratively in MATLAB, until the percent relative error falls below 1%.
Use the following MATLAB code to implement the Liebmann’s method and the equation at the nodes
Execute the above code to obtain the temperature distribution and the number of iterations performed, in the output as shown below,
Hence, the temperature on the heated plate with desired accuracy is achieved after 9 iterations as obtained in the above MATLAB output.
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Chapter 29 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
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