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Heat is conducted along a metal rod positioned between two fixed temperature walls. Aside from conduction, heat is transferred between the rod and the surrounding air by convection. Based on a heat balance, the distribution of temperature along the rod is described by the following second-order differential equation. d²T +h' (Ta-T) dx² (1) where T = temperature (°C), h' = a bulk heat transfer coefficient reflecting the relative importance of convection to conduction (0.01 m¹), x = distance along the rod (10 m), and Ta = temperature of the surrounding fluid (20 °C). a) Given values for the parameters, forcing functions, and boundary conditions, calculus can be used to develop an analytical solution. if T(0) = 40 °C and T(10) = 200 °C, obtain analytical solution (i.e. solve differential equation (1)). Write out all the detailed procedure and steps you have taken. b) Above heat transfer equation can be transformed into a set of linear algebraic equations by conceptualizing the rod as consisting of a series of nodes. For example, the rod in Figure is divided into six equi-spaced nodes. Since the rod has a length of 10, the spacing between nodes is Ax = 2. Finite-difference approximations provide a means to transform derivatives into algebraic form. For example, the second derivative at each node can be approximated as d²T dx² Ti+1-2T; + Ti-1 Ax² where Ti designates the temperature at node i. This approximation can be substituted into Eq. (1) to give - Ti+1 2T+Ti-1 Ax² +h' (Ta − T;) = 0 T₁ = 40 T₁ = 20 Ax-T=20 x = 0 x = 10 Ts=200 Develop linear equations applied to each of the nodes (for i = 1, 2, 3, 4). Write out the linear equations you developed.

Heat is conducted along a metal rod positioned between two fixed temperature walls. Aside from conduction, heat is transferred between the rod and the surrounding air by convection. Based on a heat balance, the distribution of temperature along the rod is described by the following second-order differential equation. d²T +h' (Ta-T) dx² (1) where T = temperature (°C), h' = a bulk heat transfer coefficient reflecting the relative importance of convection to conduction (0.01 m¹), x = distance along the rod (10 m), and Ta = temperature of the surrounding fluid (20 °C). a) Given values for the parameters, forcing functions, and boundary conditions, calculus can be used to develop an analytical solution. if T(0) = 40 °C and T(10) = 200 °C, obtain analytical solution (i.e. solve differential equation (1)). Write out all the detailed procedure and steps you have taken. b) Above heat transfer equation can be transformed into a set of linear algebraic equations by conceptualizing the rod as consisting of a series of nodes. For example, the rod in Figure is divided into six equi-spaced nodes. Since the rod has a length of 10, the spacing between nodes is Ax = 2. Finite-difference approximations provide a means to transform derivatives into algebraic form. For example, the second derivative at each node can be approximated as d²T dx² Ti+1-2T; + Ti-1 Ax² where Ti designates the temperature at node i. This approximation can be substituted into Eq. (1) to give - Ti+1 2T+Ti-1 Ax² +h' (Ta − T;) = 0 T₁ = 40 T₁ = 20 Ax-T=20 x = 0 x = 10 Ts=200 Develop linear equations applied to each of the nodes (for i = 1, 2, 3, 4). Write out the linear equations you developed.

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Kreith, Frank; Manglik, Raj M.
Chapter4: Numerical Analysis Of Heat Conduction
Section: Chapter Questions
Problem 4.21P
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Please handwrite and solve part a and b, thanks.

Heat is conducted along a metal rod positioned between two fixed temperature walls. Aside from conduction, heat is transferred between the rod and the surrounding air by convection. Based on a heat balance, the distribution of temperature along the rod is described by the following second-order differential equation. d²T dx² +h' (Ta − T) (1) = where T = temperature (°C), h' = a bulk heat transfer coefficient reflecting the relative importance of convection to conduction (0.01 m²¹), x = distance along the rod (10 m), and Ta temperature of the surrounding fluid (20 °C). a) Given values for the parameters, forcing functions, and boundary conditions, calculus can be used to develop an analytical solution. if T(0) = 40 °C and T(10) = 200 °C, obtain analytical solution (i.e. solve differential equation (1)). Write out all the detailed procedure and steps you have taken. b) Above heat transfer equation can be transformed into a set of linear algebraic equations by conceptualizing the rod as consisting of a series of nodes. For example, the rod in Figure is divided into six equi-spaced nodes. Since the rod has a length of 10, the spacing between nodes is Ax = 2. Finite-difference approximations provide a means to transform derivatives into algebraic form. For example, the second derivative at each node can be approximated as - d²T Ti+1 − 2T¿ + Ti−1 dx² = Δχ2 where Ti designates the temperature at node i. This approximation can be substituted into Eq. (1) to give - Ti+1 − 2T¿ + Ti−1 Δχ2 + h' (Ta − T;) = 0 To = 40 2 T₁ = 20 Ax T = 20 x = 0 4 T5 = 200 x = 10 Develop linear equations applied to each of the nodes (for i = 1, 2, 3, 4). Write out the linear equations you developed.
Transcribed Image Text:Heat is conducted along a metal rod positioned between two fixed temperature walls. Aside from conduction, heat is transferred between the rod and the surrounding air by convection. Based on a heat balance, the distribution of temperature along the rod is described by the following second-order differential equation. d²T dx² +h' (Ta − T) (1) = where T = temperature (°C), h' = a bulk heat transfer coefficient reflecting the relative importance of convection to conduction (0.01 m²¹), x = distance along the rod (10 m), and Ta temperature of the surrounding fluid (20 °C). a) Given values for the parameters, forcing functions, and boundary conditions, calculus can be used to develop an analytical solution. if T(0) = 40 °C and T(10) = 200 °C, obtain analytical solution (i.e. solve differential equation (1)). Write out all the detailed procedure and steps you have taken. b) Above heat transfer equation can be transformed into a set of linear algebraic equations by conceptualizing the rod as consisting of a series of nodes. For example, the rod in Figure is divided into six equi-spaced nodes. Since the rod has a length of 10, the spacing between nodes is Ax = 2. Finite-difference approximations provide a means to transform derivatives into algebraic form. For example, the second derivative at each node can be approximated as - d²T Ti+1 − 2T¿ + Ti−1 dx² = Δχ2 where Ti designates the temperature at node i. This approximation can be substituted into Eq. (1) to give - Ti+1 − 2T¿ + Ti−1 Δχ2 + h' (Ta − T;) = 0 To = 40 2 T₁ = 20 Ax T = 20 x = 0 4 T5 = 200 x = 10 Develop linear equations applied to each of the nodes (for i = 1, 2, 3, 4). Write out the linear equations you developed.
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