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.2 For instance, T₁ = (10+20 + T₂+ T4)/4 or 4T₁ - T₂ - T₁ = 30 Write a system of four equations whose solution gives estimates for the temperatures T1,..., T4. 10° 20° 10° 1 4 20° 2 3 30° 30° 40° 40° Solve the system of equations. [Hint: To speed up the calculations, interchange rows 1 and 4 before starting "replace" operations.]
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.2 For instance, T₁ = (10+20 + T₂+ T4)/4 or 4T₁ - T₂ - T₁ = 30 Write a system of four equations whose solution gives estimates for the temperatures T1,..., T4. 10° 20° 10° 1 4 20° 2 3 30° 30° 40° 40° Solve the system of equations. [Hint: To speed up the calculations, interchange rows 1 and 4 before starting "replace" operations.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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.2
For instance, T₁ = (10 + 20 + T₂ + T₁)/4 or 4T₁ - T₂ - T₁ = 30
Write a system of four equations whose solution gives estimates for the temperatures
T1,..., T4.
Solve the system of equations. [Hint: To speed up the calculations, interchange rows 1 and 4 before
starting "replace" operations.]
10°
20° 20°
10°
1 2
4
3
30° 30°
40°
40°](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7103eedd-f21d-40e2-861a-0d6fe16888aa%2F011d77da-03db-4dba-9b75-70d9d30df5a2%2Fknxjk9f_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.2
For instance, T₁ = (10 + 20 + T₂ + T₁)/4 or 4T₁ - T₂ - T₁ = 30
Write a system of four equations whose solution gives estimates for the temperatures
T1,..., T4.
Solve the system of equations. [Hint: To speed up the calculations, interchange rows 1 and 4 before
starting "replace" operations.]
10°
20° 20°
10°
1 2
4
3
30° 30°
40°
40°
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