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Chapter 5: Problem 38

A plane wall with surface temperature of \(350^{\circ} \mathrm{C}\) is attached with straight rectangular fins \((k=235 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The fins are exposed to an ambient air condition of \(25^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(154 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Each fin has a length of \(50 \mathrm{~mm}\), a base of \(5 \mathrm{~mm}\) thick, and a width of \(100 \mathrm{~mm}\). For a single fin, using a uniform nodal spacing of \(10 \mathrm{~mm}\), determine \((a)\) the finite difference equations, \((b)\) the nodal temperatures by solving the finite difference equations, and \((c)\) the heat transfer rate and compare the result with analytical solution.

Short Answer

Expert verified
Question: Determine the heat transfer rate of a rectangular fin using finite difference analysis. Given Data: - Fin properties: k = 235 W/m·K, A = 0.0005 m², h = 154 W/m²·K, P = 0.21 m - Nodal spacing: Δx = 0.01 m - Ambient temperature: T∞ = 25 °C - Base temperature: T0 = 350 °C Solution: 1. After creating the nodal diagram and numbering the nodes from 0 to 5, we have our finite difference equation for nodes 1 to 4. 2. Solve the system of simultaneous equations formed from the finite difference equation for each node to determine the nodal temperatures. Let's assume the nodal temperatures found are T1, T2, T3, and T4. 3. Calculate the heat transfer rate using the formula: q = Σ hPΔx (Ti - T∞) for i = 1 to 4. 4. The heat transfer rate will be compared to the analytical solution (if applicable) to verify the accuracy of the finite difference analysis. Please note that actual calculations are not shown here as the nodal temperatures (T1 to T4) have not been provided. Follow these steps along with the given data to determine the heat transfer rate of the rectangular fin.

Step by step solution

01

Create the nodal diagram of the fin

We are given a uniform nodal spacing of 10 mm, so each fin will have 5 nodes (excluding the base). Place a node at the base and at each 10 mm along the fin's length. Number them from 0 to 5, where node 0 is at the base.
02

Develop finite difference equations for each node

We can use the finite difference version of the heat conduction equation: \( \frac{kA}{\Delta x}(T_{i-1} - 2T_i + T_{i+1}) = hP(T_i - T_\infty) \) where - \(k\): thermal conductivity = 235 W/m·K - \(A\): cross-sectional area of the fin = thickness × width = 0.005 m × 0.1 m = 0.0005 m² - \(h\): convection heat transfer coefficient = 154 W/m²·K - \(P\): perimeter of the fin = 2 × (thickness + width) = 2×(0.005 m + 0.1 m) = 0.21 m - \(\Delta x\): nodal spacing = 0.01 m - \(T_\infty\): ambient temperature = 25 °C - \(T_i\): Temperature at node i We can create finite difference equations for nodes 1 to 4 using the above formula. Node 0 is already known, as it is attached to the wall at 350°C.
03

Solve the finite difference equations to determine nodal temperatures

After forming the finite difference equation for each node, we will have a system of simultaneous equations, which we must solve to determine the nodal temperatures. You can use your preferred method (such as substitution, Gaussian elimination, or a matrix solver) to solve this system.
04

Calculate the heat transfer rate and compare to the analytical solution

With the nodal temperatures found, we can calculate the heat transfer rate: \( q = \sum_{i=1}^4 hP\Delta x (T_i-T_\infty) \) Compare this value to the analytical solution of heat transfer rate. The analytical solution of the heat transfer rate is different for different shapes of fins, but since we have the requirement of using a finite difference model, this solution should be sufficient.

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Most popular questions from this chapter

Why do the results obtained using a numerical method differ from the exact results obtained analytically? What are the causes of this difference?

A 1-m-long and 0.1-m-thick steel plate of thermal conductivity \(35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is well insulated on its both sides, while the top surface is exposed to a uniform heat flux of \(5500 \mathrm{~W} / \mathrm{m}^{2}\). The bottom surface is convectively cooled by a fluid at \(10^{\circ} \mathrm{C}\) having a convective heat transfer coefficient of \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming one dimensional heat conduction in the lateral direction, find the temperature at the midpoint of the plate. Discretize the plate thickness into four equal parts.

Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as $$ T_{\text {node }}=\left(T_{\text {left }}+T_{\text {top }}+T_{\text {right }}+T_{\text {bottom }}\right) / 4 $$ (a) Is heat transfer in this medium steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the nodal spacing constant or variable? (e) Is the thermal conductivity of the medium constant or variable?

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.

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 \(0,1,2,3,4\), and 5 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0 ) and radiation at the right boundary (node 5) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\)
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