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Question 3 Based on the same simplified ODE obtained from question (1), find the analytical solution for T(x), subject to the following boundary conditions. Again, let a² response.) Ch ка when formulating your solution. (Use T, to represent T fluid and Tw to represent T wall in your At x = 0, T = T₁ wall' At x = L, T = Tfluid" T(x) = Note, these bondary conditions are not unreasonable as it is assumed that at the tip of the fin, x = L, the temperature is the same as the ambient fluid T = T fluid That is, in making this assumption, it is assumed that the length of the fin L is "long enough" such that at the tip of the fin, the temperature is the same at the ambient fluid. Question 1 From the following equation, assume that the cross-sectional area A is a constant. d dx dT ка dx KAT) = hC(T - T fluid) Also assume that all other physical parameters k, h, A, C, L, Tw wall' and Tfluid are well known and assumed to be positive constants. (a) Based on these assumptions, the above ODE can be simplified to which of the follow? hc(T-Tfluid) T ка dx2 dA dk dT + A. dx dx dx hc(T-T fluid) dT Ока = (hC(T Tfluid)) dx dx 이은 dA dk d²T + A. = dx dx dx² hc(T-T fluid) dx -(KA) = (hC(T-Tfluid)) dT (b) Solve the ODE based on the following two boundary conditions. For simplicity, let a² Ch for when formulating your solution. (Use T, to represent T fluid and Tw to represent T ка wall in your response.) At x = 0, T = T¸ wall' dT At x = L, = 0. dx T T T(x) @ αχ 2aL 1+e e + (1-T)eal -αx e + 1+e 2aL - If - (c) Define a "dimensionless" variable (x) T(x) T₂ T wall-fluid fluid Based on the T(x) solution, rewrite the solution in terms of (x) and the hyperbolic cosine function, where cosh(y) 8(x) = cosh(x) + 1-e2aL ox 1+e 2aL 1+cosh(x) ey + e˜y = Again, let a² Ch α = 2 for when formulating your solution. ка

Question 3 Based on the same simplified ODE obtained from question (1), find the analytical solution for T(x), subject to the following boundary conditions. Again, let a² response.) Ch ка when formulating your solution. (Use T, to represent T fluid and Tw to represent T wall in your At x = 0, T = T₁ wall' At x = L, T = Tfluid" T(x) = Note, these bondary conditions are not unreasonable as it is assumed that at the tip of the fin, x = L, the temperature is the same as the ambient fluid T = T fluid That is, in making this assumption, it is assumed that the length of the fin L is "long enough" such that at the tip of the fin, the temperature is the same at the ambient fluid. Question 1 From the following equation, assume that the cross-sectional area A is a constant. d dx dT ка dx KAT) = hC(T - T fluid) Also assume that all other physical parameters k, h, A, C, L, Tw wall' and Tfluid are well known and assumed to be positive constants. (a) Based on these assumptions, the above ODE can be simplified to which of the follow? hc(T-Tfluid) T ка dx2 dA dk dT + A. dx dx dx hc(T-T fluid) dT Ока = (hC(T Tfluid)) dx dx 이은 dA dk d²T + A. = dx dx dx² hc(T-T fluid) dx -(KA) = (hC(T-Tfluid)) dT (b) Solve the ODE based on the following two boundary conditions. For simplicity, let a² Ch for when formulating your solution. (Use T, to represent T fluid and Tw to represent T ка wall in your response.) At x = 0, T = T¸ wall' dT At x = L, = 0. dx T T T(x) @ αχ 2aL 1+e e + (1-T)eal -αx e + 1+e 2aL - If - (c) Define a "dimensionless" variable (x) T(x) T₂ T wall-fluid fluid Based on the T(x) solution, rewrite the solution in terms of (x) and the hyperbolic cosine function, where cosh(y) 8(x) = cosh(x) + 1-e2aL ox 1+e 2aL 1+cosh(x) ey + e˜y = Again, let a² Ch α = 2 for when formulating your solution. ка

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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I need help with Question 3 described below and an explanation for the solution. (Differential Equations)

Question 3 Based on the same simplified ODE obtained from question (1), find the analytical solution for T(x), subject to the following boundary conditions. Again, let a² response.) Ch ка when formulating your solution. (Use T, to represent T fluid and Tw to represent T wall in your At x = 0, T = T₁ wall' At x = L, T = Tfluid" T(x) = Note, these bondary conditions are not unreasonable as it is assumed that at the tip of the fin, x = L, the temperature is the same as the ambient fluid T = T fluid That is, in making this assumption, it is assumed that the length of the fin L is "long enough" such that at the tip of the fin, the temperature is the same at the ambient fluid.
Transcribed Image Text:Question 3 Based on the same simplified ODE obtained from question (1), find the analytical solution for T(x), subject to the following boundary conditions. Again, let a² response.) Ch ка when formulating your solution. (Use T, to represent T fluid and Tw to represent T wall in your At x = 0, T = T₁ wall' At x = L, T = Tfluid" T(x) = Note, these bondary conditions are not unreasonable as it is assumed that at the tip of the fin, x = L, the temperature is the same as the ambient fluid T = T fluid That is, in making this assumption, it is assumed that the length of the fin L is "long enough" such that at the tip of the fin, the temperature is the same at the ambient fluid.
Question 1 From the following equation, assume that the cross-sectional area A is a constant. d dx dT ка dx KAT) = hC(T - T fluid) Also assume that all other physical parameters k, h, A, C, L, Tw wall' and Tfluid are well known and assumed to be positive constants. (a) Based on these assumptions, the above ODE can be simplified to which of the follow? hc(T-Tfluid) T ка dx2 dA dk dT + A. dx dx dx hc(T-T fluid) dT Ока = (hC(T Tfluid)) dx dx 이은 dA dk d²T + A. = dx dx dx² hc(T-T fluid) dx -(KA) = (hC(T-Tfluid)) dT (b) Solve the ODE based on the following two boundary conditions. For simplicity, let a² Ch for when formulating your solution. (Use T, to represent T fluid and Tw to represent T ка wall in your response.) At x = 0, T = T¸ wall' dT At x = L, = 0. dx T T T(x) @ αχ 2aL 1+e e + (1-T)eal -αx e + 1+e 2aL - If - (c) Define a "dimensionless" variable (x) T(x) T₂ T wall-fluid fluid Based on the T(x) solution, rewrite the solution in terms of (x) and the hyperbolic cosine function, where cosh(y) 8(x) = cosh(x) + 1-e2aL ox 1+e 2aL 1+cosh(x) ey + e˜y = Again, let a² Ch α = 2 for when formulating your solution. ка
Transcribed Image Text:Question 1 From the following equation, assume that the cross-sectional area A is a constant. d dx dT ка dx KAT) = hC(T - T fluid) Also assume that all other physical parameters k, h, A, C, L, Tw wall' and Tfluid are well known and assumed to be positive constants. (a) Based on these assumptions, the above ODE can be simplified to which of the follow? hc(T-Tfluid) T ка dx2 dA dk dT + A. dx dx dx hc(T-T fluid) dT Ока = (hC(T Tfluid)) dx dx 이은 dA dk d²T + A. = dx dx dx² hc(T-T fluid) dx -(KA) = (hC(T-Tfluid)) dT (b) Solve the ODE based on the following two boundary conditions. For simplicity, let a² Ch for when formulating your solution. (Use T, to represent T fluid and Tw to represent T ка wall in your response.) At x = 0, T = T¸ wall' dT At x = L, = 0. dx T T T(x) @ αχ 2aL 1+e e + (1-T)eal -αx e + 1+e 2aL - If - (c) Define a "dimensionless" variable (x) T(x) T₂ T wall-fluid fluid Based on the T(x) solution, rewrite the solution in terms of (x) and the hyperbolic cosine function, where cosh(y) 8(x) = cosh(x) + 1-e2aL ox 1+e 2aL 1+cosh(x) ey + e˜y = Again, let a² Ch α = 2 for when formulating your solution. ка
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