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Chapter 5: Problem 20

Consider steady 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 energy balance approach, obtain the finite difference formulation of the boundary nodes for the case of uniform heat flux \(\dot{q}_{0}\) 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 \(T_{\infty}\).

Short Answer

Expert verified
Answer: The finite difference formulations for the boundary nodes are as follows: 1. For node 0: \(T_0 = T_1 - \frac{\Delta x}{k} \cdot \dot{q}_0\) 2. For node 4: \(T_4 = \left(\frac{\Delta x}{k} \cdot h + \Delta x \cdot \frac{1}{k}\right) T_{3} - \Delta x \cdot \frac{1}{k} \cdot T_{\infty}\)

Step by step solution

01

Write energy balance equation for node 0

Write the energy balance equation for node 0: $$\dot{q}_0 - k \frac{T_1 - T_0}{\Delta x} = 0$$ Rearrange this equation to obtain the finite difference formulation for node 0: $$T_0 = T_1 - \frac{\Delta x}{k} \cdot \dot{q}_0$$
02

Write energy balance equation for node 4

Write the energy balance equation for node 4: $$k \frac{T_{3} - T_{4}}{\Delta x} - h \cdot (T_{4} - T_{\infty}) = 0$$ Rearrange this equation to obtain the finite difference formulation for node 4: $$T_4 = \left(\frac{\Delta x}{k} \cdot h + \Delta x \cdot \frac{1}{k}\right) T_{3} - \Delta x \cdot \frac{1}{k} \cdot T_{\infty}$$

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Most popular questions from this chapter

What happens to the discretization and the round-off errors as the step size is decreased?

Consider transient heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(50^{\circ} \mathrm{C}\) while the right surface (node 6) is subjected to a solar heat flux of \(600 \mathrm{~W} / \mathrm{m}^{2}\). The wall is initially at a uniform temperature of \(50^{\circ} \mathrm{C}\). Express the explicit finite difference formulation of the boundary nodes 0 and 6 for the case of no heat generation. Also, obtain the finite difference formulation for the total amount of heat transfer at the left boundary during the first three time steps.

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

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.

How can a node on an insulated boundary be treated as an interior node in the finite difference formulation of a plane wall? Explain.
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