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Chapter 5: Problem 40

A circular fin of uniform cross section, with diameter of \(10 \mathrm{~mm}\) and length of \(50 \mathrm{~mm}\), is attached to a wall with surface temperature of \(350^{\circ} \mathrm{C}\). The fin is made of material with thermal conductivity of \(240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and it is exposed to an ambient air condition of \(25^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assume steady one-dimensional heat transfer along the fin and the nodal spacing to be uniformly \(10 \mathrm{~mm},(a)\) using the energy balance approach, obtain the finite difference equations to determine the nodal temperatures, \((b)\) determine the nodal temperatures along the fin by solving those equations and compare the results with the analytical solution, and (c) calculate the heat transfer rate and compare the result with the analytical solution.

Short Answer

Expert verified
#Question: Determine and compare the nodal temperatures and heat transfer rate for a circular fin of uniform cross-section using the finite difference method, given the following parameters: - Diameter \(D = 10\mathrm{~mm}\) - Length \(L = 50\mathrm{~mm}\) - Wall temperature \(T_w = 350^\circ\mathrm{C}\) - Ambient air temperature \(T_\infty = 25^\circ\mathrm{C}\) - Convection heat transfer coefficient \(h = 250\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) - Thermal conductivity \(k = 240\mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) #Answer: To determine and compare the nodal temperatures and heat transfer rate for the circular fin, perform the following steps: 1. Define parameters and calculate the area of the fin. 2. Derive the finite difference equations for the energy balance at each node. 3. Solve the finite difference equations for the nodal temperatures using numerical methods like Gauss-Seidel or Jacobi method. Compare the numerical results with the analytical solution. 4. Calculate and compare the heat transfer rate using the convective heat transfer equation. Note: The actual nodal temperature values and heat transfer rate will depend on the numerical method and software used for solving the finite difference equations.

Step by step solution

01

Define parameters and calculate area

First, define the given parameters: - Diameter \(D = 10\mathrm{~mm} = 0.01\mathrm{~m}\) - Length \(L = 50\mathrm{~mm} = 0.05\mathrm{~m}\) - Wall temperature \(T_w = 350^\circ\mathrm{C}\) - Ambient air temperature \(T_\infty = 25^\circ\mathrm{C}\) - Convection heat transfer coefficient \(h = 250\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) - Thermal conductivity \(k = 240\mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) Calculate the cross-sectional area of the fin: \(A = \pi(\frac{D}{2})^{2} = \pi (\frac{0.01}{2})^{2} = 7.854\mathrm{x}10^{-5} \mathrm{~m}^2\)
02

Derive finite difference equations

We are considering a steady-state one-dimensional heat transfer along the fin. The energy balance approach would involve calculating the energy balance at each node. Divide the fin into equal segments of length \(0.01\mathrm{~m}\). The energy balance between node \(i-1\), node \(i\), and node \(i+1\) would be: \(Q_{i-1} + Q_\text{conv} = Q_{i+1}\) Where: - \(Q_{i-1}\) is the heat exchange with node \(i-1\) - \(Q_\text{conv}\) is the convective heat exchange at node \(i\) - \(Q_{i+1}\) is the heat exchange with the node \(i+1\) Rewrite the equation in terms of temperatures, using thermal conductivity: \(\frac{kA(T_{i-1}-T_i)}{\Delta x} + hP(T_{\infty}-T_i)\Delta x = \frac{kA(T_i-T_{i+1})}{\Delta x}\) Where: - \(P\) is the perimeter of the fin at each node - \(\Delta x\) is the nodal spacing (equal to \(0.01\mathrm{~m}\)) Solve for the temperatures at each node using numerical methods, such as the Gauss-Seidel or Jacobi method.
03

Solve for nodal temperatures

Using a numerical method, solve the finite difference equations at each node. Since a comparison needs to be made with the analytical solution, it would be best to use specialized software or a scripting language (like Python or MATLAB). Once numerical solutions have been obtained for the nodal temperatures, compare them with the analytical solution. The analytical solution can be found in standard heat transfer textbooks or research papers. The comparison ensures the accuracy and validity of the numerical solutions.
04

Calculate the heat transfer rate

To calculate the total heat transfer rate, use the convective heat transfer equation: \(Q = hA(T_\text{base} - T_\infty)\) Where \(T_\text{base}\) is the temperature at the base of the fin. Calculate the heat transfer rate and compare the result with the analytical solution. The comparison validates the accuracy of the heat transfer rate calculated in this exercise. In conclusion, the finite difference equations can be used to determine the nodal temperatures along the fin, the heat transfer rate can be calculated, and the results can be validated by comparing them with the analytical solutions.

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Most popular questions from this chapter

Consider steady two-dimensional heat transfer in a long solid bar \((k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of square cross section \((3 \mathrm{~cm} \times 3 \mathrm{~cm})\) with the prescribed temperatures at the top, right, bottom, and left surfaces to be \(100^{\circ} \mathrm{C}, 200^{\circ} \mathrm{C}, 300^{\circ} \mathrm{C}\), and \(500^{\circ} \mathrm{C}\), respectively. Heat is generated in the bar uniformly at a rate of \(\dot{e}=5 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\). Using a uniform mesh size \(\Delta x=\Delta y=1 \mathrm{~cm}\) determine \((a)\) the finite difference equations and \((b)\) the nodal temperatures with the Gauss-Seidel iterative method.

Consider steady two-dimensional heat conduction in a square cross section \((3 \mathrm{~cm} \times 3 \mathrm{~cm}, k=20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=\) \(6.694 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) with constant prescribed temperature of \(100^{\circ} \mathrm{C}\) and \(300^{\circ} \mathrm{C}\) at the top and bottom surfaces, respectively. The left surface is exposed to a constant heat flux of \(1000 \mathrm{~W} / \mathrm{m}^{2}\) while the right surface is in contact with a convective environment \(\left(h=45 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\) at \(20^{\circ} \mathrm{C}\). Using a uniform mesh size of \(\Delta x=\Delta y\), determine \((a)\) finite difference equations and \((b)\) the nodal temperatures using Gauss-Seidel iteration method.

Consider a long solid bar \((k=28 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=\) \(12 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) of square cross section that is initially at a uniform temperature of \(32^{\circ} \mathrm{C}\). The cross section of the bar is \(20 \mathrm{~cm} \times 20 \mathrm{~cm}\) in size, and heat is generated in it uniformly at a rate of \(\dot{e}=8 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). All four sides of the bar are subjected to convection to the ambient air at \(T_{\infty}=30^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=45 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Using the explicit finite difference method with a mesh size of \(\Delta x=\Delta y=10 \mathrm{~cm}\), determine the centerline temperature of the bar \((a)\) after \(20 \mathrm{~min}\) and \((b)\) after steady conditions are established.

Consider steady one-dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes \(0,1,2,3\), and 4 with a uniform nodal spacing of \(\Delta x\). Using the finite difference form of the first derivative (not the energy balance approach), obtain the finite difference formulation of the boundary nodes for the case of uniform heat flux \(q_{0}\) at the left boundary (node 0 ) and convection at the right boundary (node 4) with a convection coefficient of \(h\) and an ambient temperature of \(T_{\infty}\).

Consider transient heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes \(0,1,2,3\), and 4 with a uniform nodal spacing of \(\Delta x\). The wall is initially at a specified temperature. The temperature at the right boundary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary node 0 for the case of combined convection, radiation, and heat flux at the left boundary with an emissivity of \(\varepsilon\), convection coefficient of \(h\), ambient temperature of \(T_{\infty}\), surrounding temperature of \(T_{\text {surr }}\), and uniform heat flux of \(\dot{q}_{0}\) toward the wall. Also, obtain the finite difference formulation for the total amount of heat transfer at the right boundary for the first 20 time steps.
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