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 at, rectangular metal plate is often derived.

The temperature is held constant at T₁ on three sides and at T₂ on the fourth side (see Figure P38). The temperature T(x, y) as a function of the xy coordinates shown is given by

 

Where

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 this fact for n = 1, …, 19 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.

Figure P38

Tx, ) 3D (, )w(, ) + Ti