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Chapter 5: Problem 19

Consider steady heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(30^{\circ} \mathrm{C}\) while the right surface (node 8 ) is subjected to a heat flux of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heat generation. Also obtain the finite difference formulation for the rate of heat transfer at the left boundary.

Short Answer

Expert verified
Answer: The finite difference formulation for the rate of heat transfer at the left boundary of the plane wall is given by: $$q''_{0} = -k \left(\frac{T_{1} - 30}{\Delta x}\right)$$

Step by step solution

01

Understand the given information and notation

We have a plane wall with left surface (node 0) at a temperature of \(30^{\circ} \mathrm{C}\) and the right surface (node 8) subjected to a heat flux of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). The heat conduction is steady, and there is no heat generation.
02

Apply the Finite Difference Method to the Boundary Nodes

The finite difference method can be applied to the boundary nodes by considering the governing equation for a steady heat conduction process without heat generation, which can be given by the Laplace equation: $$\nabla^{2} T = 0$$ Applying this to Node 0 and Node 8, we have: $$\frac{T_{1}-2 T_{0}}{\Delta x^{2}} = 0$$ and $$\frac{T_{7}-2 T_{8}}{\Delta x^{2}} = 0$$ For the boundary condition at the left surface (Node 0): $$T_{0} = 30$$ For the boundary condition at the right surface (Node 8): $$q'' = -k \frac{dT}{dx}\big|_{x=L} = 1200 \mathrm{~W}/\mathrm{m}^{2}$$
03

Express the Finite Difference Formulation for Boundary Nodes 0 and 8

Using the boundary conditions and the governing equations, we arrive at the expressions for the boundary nodes 0 and 8: For Node 0: $$T_{0} = 30^{\circ}C$$ For Node 8: $$T_{7} - 2 T_{8} = -1200 \Delta x \frac{1}{k \Delta x^2}$$
04

Obtain the Finite Difference Formulation for the Rate of Heat Transfer at the Left Boundary

Using the governing equation at Node 0 and the boundary condition for the left surface: $$\frac{T_{1} - 2T_{0}}{\Delta x^2} = 0$$ Solve for the rate of heat transfer at the left boundary from Node 0: $$q''_{0} = -k \left(\frac{T_{1} - T_{0}}{\Delta x}\right)$$ Replacing the temperature values at Node 0, we have: $$q''_{0} = -k \left(\frac{T_{1} - 30}{\Delta x}\right)$$

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Most popular questions from this chapter

Consider one-dimensional transient heat conduction in a composite plane wall that consists of two layers \(A\) and \(B\) with perfect contact at the interface. The wall involves no heat generation and initially is at a specified temperature. The nodal network of the medium consists of nodes 0,1 (at the interface), 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 insulation at the left boundary (node 0 ) and radiation at the right boundary (node 2 ) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\).

A hot surface at \(100^{\circ} \mathrm{C}\) is to be cooled by attaching 3 -cm- long, \(0.25\)-cm-diameter aluminum pin fins \((k=\) \(237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) with a center-to-center distance of \(0.6 \mathrm{~cm}\). The temperature of the surrounding medium is \(30^{\circ} \mathrm{C}\), and the combined heat transfer coefficient on the surfaces is \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be \(0.5 \mathrm{~cm}\), determine \((a)\) the finite difference formulation of this problem, \((b)\) the nodal temperatures along the fin by solving these equations, \((c)\) the rate of heat transfer from a single fin, and \((d)\) the rate of heat transfer from a \(1-\mathrm{m} \times 1-\mathrm{m}\) section of the plate.

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

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 \(T_{\text {surr }}\).

Suggest some practical ways of reducing the roundoff error.
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