A two-dimensional rectangular plate is subjected to prescribed boundary conditions. Using the results of the exact solution for the heat equation presented in Section 4.2, calculate the temperatures along the mid-plane of the plate (x = 1 m) at y = 0.25, 0.5, and 0.75 m by considering the first five nonzero terms of the infinite series. Assess the error resulting from using only the first three terms of the infinite series.

The temperature calculated with 5 terms at (1,0.25),(1,0.5) and (1,0.75) are 71.24 C, 94.53 C and 121.14 C respectively.

The temperature calculated with 3 terms at (1,0.25),(1,0.5) and (1,0.75) are 71.24 C, 94.59 C and 121.89 C respectively.

The error percentage between two temperatures for (1,0.25),(1,0.5) and (1,0.75) is 0%, 0.06% and 0.62% respectively.

Explanation:

From the given section, the equation of Temperature distribution is given as

This is also given by the exact solution as

For point 1 x=1, y=0.25

From the attached diagram

L=2

W=1

x/L=1/2

y/L=1/8

W/L=1/2

So the equation becomes

Now as seen the value of terms for n=2,4,6,8.... is zero so thus only first five odd terms (n=1,n=3,n=5,n=7,n=9) will be considered as

From the diagram

T_2=150

T_1=50

Now the equation becomes

Using the 1st 3 terms the solution is given as

The error is 0%.

Similarly for point (1,0.5)

x/L=1/2

y/L=1/4

W/L=1/2

The solution is given as

With using 1st 3 terms the value is given as

Ther error is given as

Similarly for point (1,0.75)

x/L=1/2

y/L=3/8

W/L=1/2

The solution is given as

With using 1st 3 terms the value is given as

Ther error is given as

So

The temperature calculated with 5 terms at (1,0.25),(1,0.5) and (1,0.75) are 71.24 C, 94.53 C and 121.14 C respectively.

The temperature calculated with 3 terms at (1,0.25),(1,0.5) and (1,0.75) are 71.24 C, 94.59 C and 121.89 C respectively.

The error percentage between two temperatures for (1,0.25),(1,0.5) and (1,0.75) is 0%, 0.06% and 0.62% respectively.