Question

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).

Short answer

The system of equations whose solution gives the estimates for the temperatures \({T_1},...,{T_4}\)is shown below:

\(\begin{array}{c}4{T_1} - {T_2} - {T_4} = 30\\ - {T_1} + 4{T_2} - {T_3} = 60\\ - {T_2} + 4{T_3} - {T_4} = 70\\ - {T_1} - {T_3} + 4{T_4} = 40\end{array}\)

Step-by-step solution

Write the second equation for temperature 2

From the figure, it is observed that for temperature 2, i.e., \({T_2}\) (lies at the interior nodes of the mesh), the nearest temperatures are \({T_1}\), \(20^\circ \), \(40^\circ \), and \({T_3}\).

It is given thatthe temperature at a node is approximately equal to the average of the four nearest nodes. So, temperature \({T_2}\) at a node is approximately equal to the average of the four temperatures at the nearest nodes \({T_1}\), \(20^\circ \), \(40^\circ \), and \({T_3}\) as shown below:

\({T_2} = \left( {{T_1} + 20 + 40 + {T_3}} \right)/4\)

Simplify the above expression.

\(4{T_2} - {T_1} - {T_3} = 60\)

Write the second equation for temperature 3

From the figure, it is observed that for temperature 3, i.e., \({T_3}\) (lies at the interior nodes of the mesh), the nearest temperatures are \({T_4}\), \(30^\circ \), \(40^\circ \), and \({T_2}\).

It is given thatthe temperature at a node is approximately equal to the average of the four nearest nodes. So, the temperature \({T_3}\) at a node is approximately equal to the average of the four temperatures at the nearest nodes \({T_4}\), \(30^\circ \), \(40^\circ \), and \({T_2}\) as shown below:

\({T_3} = \left( {{T_4} + 30 + 40 + {T_2}} \right)/4\)

Simplify the above expression.

\(4{T_3} - {T_4} - {T_2} = 70\)

Write the second equation for temperature 4

From the figure, it is observed that for temperature 4, i.e., \({T_4}\) (lies at the interior nodes of the mesh), the nearest temperatures are \({T_1}\), \(10^\circ \), \(30^\circ \), and \({T_3}\).

It is given thatthe temperature at a node is approximately equal to the average of the four nearest nodes. So, temperature \({T_4}\) at a node is approximately equal to the average of the four temperatures at the nearest nodes \({T_1}\), \(10^\circ \), \(30^\circ \), and \({T_3}\) as shown below:

\({T_4} = \left( {{T_1} + 10 + 30 + {T_3}} \right)/4\)

Simplify the above expression.

\(4{T_4} - {T_1} - {T_3} = 40\)

Rearrange the obtained system of equations

The obtained system of equations is \(4{T_2} - {T_1} - {T_3} = 60\), \(4{T_3} - {T_4} - {T_2} = 70\),and \(4{T_4} - {T_1} - {T_3} = 40\). Also, the given equation is \(4{T_1} - {T_2} - {T_4} = 30\).

Rearrange all the equations as shown below:

\(\begin{array}{c}4{T_1} - {T_2} - {T_4} = 30\\ - {T_1} + 4{T_2} - {T_3} = 60\\ - {T_2} + 4{T_3} - {T_4} = 70\\ - {T_1} - {T_3} + 4{T_4} = 40\end{array}\)

Thus, the system of four equations whose solution gives the estimates for the temperatures \({T_1},...,{T_4}\)is shown below:

\(\begin{array}{c}4{T_1} - {T_2} - {T_4} = 30\\ - {T_1} + 4{T_2} - {T_3} = 60\\ - {T_2} + 4{T_3} - {T_4} = 70\\ - {T_1} - {T_3} + 4{T_4} = 40\end{array}\)