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1. 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, Ta 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 +T)/4 or 4T₁ T₂-T₁ = 30 20⁰ 10' 10' 20 40⁰ 30' 30' a. Write a system of four equations whose solution gives estimates for the temperatures T₁,..., T4 b. Solve the system of equations using Naïve-Gauss Elimination

1. 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, Ta 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 +T)/4 or 4T₁ T₂-T₁ = 30 20⁰ 10' 10' 20 40⁰ 30' 30' a. Write a system of four equations whose solution gives estimates for the temperatures T₁,..., T4 b. Solve the system of equations using Naïve-Gauss Elimination

Advanced Engineering Mathematics
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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1. 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,..., T. 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+T₁)/4 or 4T₁ T₂-T₁4 = 30 20⁰ 10' 10' 20 30 40' 30' a. Write a system of four equations whose solution gives estimates for the temperatures T₁,..., T4 b. Solve the system of equations using Naïve-Gauss Elimination
Transcribed Image Text:1. 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,..., T. 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+T₁)/4 or 4T₁ T₂-T₁4 = 30 20⁰ 10' 10' 20 30 40' 30' a. Write a system of four equations whose solution gives estimates for the temperatures T₁,..., T4 b. Solve the system of equations using Naïve-Gauss Elimination
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