Question

In a grid of wires, the temperature at exterior mesh points is maintained at constant values, $\mathbf{\left(}{\mathbf{in}}^{\mathbf{°}}\mathbf{C}\mathbf{\right)}$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,

${\mathit{T}}_{\mathbf{2}}\mathbf{=}\frac{{\mathbf{T}}_{\mathbf{3}}\mathbf{+}{\mathbf{T}}_{\mathbf{1}}\mathbf{+}\mathbf{200}\mathbf{+}\mathbf{0}}{\mathbf{4}}$

Find the temperatures ${\mathbf{T}}_{\mathbf{1}}\mathbf{,}{\mathbf{T}}_{\mathbf{2}}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{T}}_{\mathbf{3}}$andwhen the grid is in thermal equilibrium.

Short answer

The temperatures${\mathrm{T}}_1,{\mathrm{T}}_2,\mathrm{and}{\mathrm{T}}_3$when the grid is in thermal equilibrium are${\mathrm{T}}_1=75\mathrm{T},{\mathrm{T}}_2=100\mathrm{T}\mathrm{and}{\mathrm{T}}_3=125\mathrm{T}.$

Step-by-step solution

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.

$$\begin{gathered} T_2=\frac{T_3+T_1+200+0}{4} \\ {T}_1=\frac{0+T_2+200+0}{4} \\ {T}_3=\frac{T_2+0+0+400}{4} \end{gathered}$$

Rearrange the terms of the above equations

Consider the simplified equations.

$$\begin{gathered} 4T_2-T_3-T_1=200......\left(1\right) \\ 4T_1-T_2=200......\left(2\right) \\ 4T_3-T_2=400.....\left(3\right) \end{gathered}$$

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

Consider the matrices.

$\left(\begin{array}{ccc}-1& 4& -1\\ 4& -1& 0\\ 0& -1& 4\end{array}\right)\left(\begin{array}{c}{T}_1\\ T_2\\ T_3\end{array}\right)=\left(\begin{array}{c}200\\ 200\\ 400\end{array}\right)$

Perform the multiplication operation.

$\left(\left.\begin{array}{c}{T}_1\\ T_2\\ T_3\end{array}\right)=\left(\begin{array}{c}75\\ 125\\ 100\end{array}\right.\right)$

The values are obtained as$T_1=75,T_3=125,T_2=100$.

Thus,${\mathrm{T}}_1=75\mathrm{T},{\mathrm{T}}_2=100\mathrm{T}{\mathrm{andT}}_3=125\mathrm{T}$are the values of the temperatures,${\mathrm{T}}_1,{\mathrm{T}}_2,{\mathrm{andT}}_3$