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Chapter 13: Q24MP (page 664)

Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants, etc.T=A,鈥�鈥�T=B, on the four sides, and the rectangle covers the area 0<x<a,鈥�0<y<b.

Short Answer

Expert verified

The general solution for the steady-state temperatureisT=鈭�n=1鈭�bnsinn蟺x10 .

Step by step solution

01

Given information:

To find the general solution for the steady state temperature.

And the figure 2.2 is shown below:

02

Definition of Steady state temperature:

When a conductor reaches a point where no more heat can be absorbed by the rod, it is said to be at steady-state.

03

Step 3:Use Boundary conditions

Consider the boundary condition T=A0<x<aB0<y<b.

The given figure is:

Write the basis solution of the boundary condition.

T=ekysinkxe-kysinkxekycoskxe-kycoskx

The above basis solution satisfies the given boundary conditions.

Find the temperature for 0<x<a.

T=Ae-kasinka

Find the temperature for 0<y<b.

T=Be-kbsinkb

The solution is given by,

T=鈭�n=1鈭�bnsinn蟺xa

If,T=A0<x<aB0<y<b

Then

T=鈭�n=1鈭�en蟺yabnsinn蟺xa=B.

Hence the general solution for the steady-state temperatureis T=鈭�n=1鈭�bnsinn蟺x10.

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