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!)
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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