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Chapter 5: Problem 3

Consider a heat conduction problem that can be solved both analytically, by solving the governing differential equation and applying the boundary conditions, and numerically, by a software package available on your computer. Which approach would you use to solve this problem? Explain your reasoning.

Short Answer

Expert verified
Answer: When deciding between an analytical or numerical approach for solving a heat conduction problem, the factors to consider include the problem's complexity, the resources and computation time required for each method, the accuracy needed for the problem, and prior knowledge or expertise available for each approach. The final decision should be based on balancing these factors and their importance for the specific problem. For example, a simple problem with high accuracy requirements and knowledge of governing equations may be best suited for an analytical approach, while a complex problem with irregular geometries and inhomogeneous materials may be better suited for a numerical method using specialized software.

Step by step solution

01

Analyze the problem's complexity

Before deciding which approach to use, it is crucial to analyze the complexity of the problem. A highly complex problem may require a numerical solution to get a practical answer, while a less complex problem may be easily solved analytically.
02

Evaluate the resources and computation time required for each method

Analytical methods generally require less computational resources, as they involve solving the governing equations directly. However, their weakness is that they struggle with complex geometries and inhomogeneous materials. On the other hand, numerical methods (e.g., finite element, finite volume methods) can handle complex geometries and inhomogeneous materials but may require significant computational resources and time.
03

Assess the accuracy needed for the problem

Analytical methods can provide exact solutions to the governing equations under specific conditions. If high accuracy is required (and provided the problem is relatively simple), analytical methods may be preferred. Numerical methods, however, may lead to approximate solutions depending on the discretization and tolerances used, even for complex problems.
04

Determine if prior knowledge or expertise is available for each approach

It is important to consider personal knowledge, expertise, or available resources when deciding on an approach. Familiarity with numerical software or the governing equations (e.g., Heat Equation) may influence the choice between analytical and numerical methods.
05

Make the decision and justify the reasoning

After analyzing the problem's complexity, the resources required, the need for accuracy, and personal experience, we can decide on the best approach. For example, if the problem has a simple geometry and homogeneous material properties, with a high accuracy requirement, and prior knowledge of the governing equations, an analytical approach may be best. However, if the problem is highly complex, involving irregular geometries and/or inhomogeneous materials, then a numerical method using specialized software may be more appropriate. The final decision should be based on a balance of these factors, taking into consideration the importance of each for the specific problem.

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Most popular questions from this chapter

The implicit method is unconditionally stable and thus any value of time step \(\Delta t\) can be used in the solution of transient heat conduction problems. To minimize the computation time, someone suggests using a very large value of \(\Delta t\) since there is no danger of instability. Do you agree with this suggestion? Explain.

In the energy balance formulation of the finite difference method, it is recommended that all heat transfer at the boundaries of the volume element be assumed to be into the volume element even for steady heat conduction. Is this a valid recommendation even though it seems to violate the conservation of energy principle?

A circular fin \((k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) 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 tip has a temperature of \(200^{\circ} \mathrm{C}\), and it is exposed to 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, and (b) determine the nodal temperatures along the fin by solving those equations and compare the results with the analytical solution.

Consider one-dimensional transient heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The nodal network of the medium consists of nodes 0,1 , and 2 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the explicit finite difference formulation of this problem for the case of specified heat flux \(\dot{q}_{0}\) and convection at the left boundary (node 0 ) with a convection coefficient of \(h\) and ambient temperature of \(T_{\infty}\), and radiation at the right boundary (node 2 ) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\).

Two 3-m-long and 0.4-cm-thick cast iron \((k=\) \(52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \varepsilon=0.8)\) steam pipes of outer diameter \(10 \mathrm{~cm}\) are connected to each other through two \(1-\mathrm{cm}\)-thick flanges of outer diameter \(20 \mathrm{~cm}\). The steam flows inside the pipe at an average temperature of \(200^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The outer surface of the pipe is exposed to convection with ambient air at \(8^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) as well as radiation with the surrounding surfaces at an average temperature of \(T_{\text {surr }}=290 \mathrm{~K}\). Assuming steady one-dimensional heat conduction along the flanges and taking the nodal spacing to be \(1 \mathrm{~cm}\) along the flange \((a)\) obtain the finite difference formulation for all nodes, \((b)\) determine the temperature at the tip of the flange by solving those equations, and \((c)\) determine the rate of heat transfer from the exposed surfaces of the flange.
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