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Chapter 1: Q30E (page 6)

In a grid of wires, the temperature at exterior mesh points is maintained at constant values, (in掳C)as shown in the accompanying figure. When the grid is in thermal equilibrium, the temperature Tat each interior mesh point is the average of the temperatures at the four adjacent points. For example,

T2=T3+T1+200+04

Find the temperatures T1,T2,andT3andwhen the grid is in thermal equilibrium.

Short Answer

Expert verified

The temperaturesT1,T2,andT3when the grid is in thermal equilibrium areT1=75T,T2=100TandT3=125T.

Step by step solution

01

Consider the diagram and extract equations

The temperature T at each interior mesh point is the average of the temperatures at the four adjacent points.

Using the diagram given, create the equations as given.

T2=T3+T1+200+04T1=0+T2+200+04T3=T2+0+0+4004

02

Rearrange the terms of the above equations

Consider the simplified equations.

4T2-T3-T1=200......(1)4T1-T2=200......(2)4T3-T2=400.....(3)

03

Solve the above equations (1), (2) and (3).

Consider the matrices.

-14-14-100-14T1T2T3=200200400

Perform the multiplication operation.

T1T2T3=75125100

The values are obtained asT1=75,T3=125,T2=100.

Thus,T1=75T,T2=100TandT3=125Tare the values of the temperatures,T1,T2,andT3

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