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 T1, . . . , T4 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, T1 = (10 + 20 + T2 + T4)/4, or 4T1 - T2 - T4 = 30 Write a system of four equations whose solution gives estimates for the temperatures T1, . . . , T4. Solve the system of equations from Problem 11.

          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 T1, . . . , T4 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,

T1 = (10 + 20 + T2 + T4)/4, or 4T1 - T2 - T4 = 30

Write a system of four equations whose solution gives estimates for the temperatures T1, . . . , T4.

Solve the system of equations from Problem 11.

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 T1, . . . , T4 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,

T1 = (10 + 20 + T2 + T4)/4, or 4T1 - T2 - T4 = 30

Write a system of four equations whose solution gives estimates for the temperatures T1, . . . , T4.

Solve the system of equations from Problem 11.

Added by Daniel D.

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