 # Calculate Temperature Distribution Through an Object Using MATLAB

by admin in on December 1, 2019

### Overview:

Many applications require us to know the temperature distribution in an object. For example, this information is important for controlling the material properties, such as hardness, when cooling an object formed from molten metal. In a heat-transfer course, the following description of the temperature distribution in a flat, rectangular metal plate is often derived. The temperature is held constant at T1 on three sides and at T2 on the fourth side. The temperature T(x,y) as a function of the x,y coordinates shown is given by: Where: ### Example:

Example of using the past mentioned equations. Use the following data: T1 = 70 F, T2 = 200 F and W = L = 2 ft:

a) The terms in the preceding series become smaller in magnitude as n increases. Write a MATLAB program to verify n = 1,…,19, this fact for the center of the plate (x = y = 1).
b) Using x = y = 1, write a MATLAB program to determine how many terms are required in the series to produce a temperature calculation that is accurate to within 1 percent. (That is, for what value of n will the addition of the next term in the series produce a change in T of less than 1 percent?) Use your physical insight to determine whether this answer gives the correct temperature at the center of the plate.
c) Modify the program from part b to compute the temperatures in the plate; use a spacing of 0.2 for both x and y.

Solution of the equations will give such charts:  ### References:

1. Geankoplis, Christie John (2003). Transport Processes and Separation Principles(4th ed.). Prentice Hall. ISBN 0-13-101367-X.
2. B.S. Chemical Engineering”. New Jersey Institute of Technology, Chemical Engineering Departement. Archived from the original on 10 December 2010. Retrieved 9 April 2011.
3. Lienhard, John H. IV; Lienhard, John H. V (2019). A Heat Transfer Textbook (5th ed.). Mineola, NY: Dover Pub. p. 3.
4. Welty, James R.; Wicks, Charles E.; Wilson, Robert Elliott (1976). Fundamentals of momentum, heat, and mass transfer (2nd ed.). New York: Wiley. ISBN 978-0-471-93354-0. OCLC 2213384.
5. Faghri, Amir; Zhang, Yuwen; Howell, John (2010). Advanced Heat and Mass Transfer. Columbia, MO: Global Digital Press. ISBN 978-0-9842760-0-4.
6. Taylor, R. A. (2012). “Socioeconomic impacts of heat transfer research”. International Communications in Heat and Mass Transfer39 (10): 1467–1473. doi:10.1016/j.icheatmasstransfer.2012.09.007.
7. “Mass transfer”. Thermal-FluidsPedia. Thermal Fluids Central.
8. Abbott, J.M.; Smith, H.C.; Van Ness, M.M. (2005). Introduction to Chemical Engineering Thermodynamics (7th ed.). Boston, Montreal: McGraw-Hill. ISBN 0-07-310445-0.
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10. Çengel, Yunus (2003). Heat Transfer: A practical approach (2nd ed.). Boston: McGraw-Hill. ISBN 978-0-07-245893-0.

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December 1, 2019

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December 1, 2019