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Chapter 5: Problem 55

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?

Short Answer

Expert verified
(a) The heat transfer in the given medium is steady, as there is no time component in the equation. (b) The heat transfer in this medium is two-dimensional, as it involves temperatures in the horizontal (left and right) and vertical (top and bottom) directions. (c) There is no heat generation in the medium, as no additional term representing heat generation is in the equation. (d) We cannot determine if the nodal spacing is constant or variable based on the given information. (e) We cannot determine if the thermal conductivity of the medium is constant or variable based on the given equation.

Step by step solution

01

(a) Steady or transient heat transfer

To determine if the heat transfer is steady or transient, we should examine if there's any dependence on time in the given equation. The equation states that temperature at the node is the average of its neighboring left, top, right, and bottom temperatures: $$T_{\text {node }}=\left(T_{\text {left }}+T_{\text {top }}+T_{\text {right }}+T_{\text {bottom }}\right) / 4$$ Since there is no time component in the equation, we can conclude that the heat transfer in this medium is steady.
02

(b) Dimensionality of heat transfer

The given equation considers temperatures around the node i.e., left, top, right, and bottom. This suggests that heat is transferred in two dimensions—horizontal (left and right) and vertical (top and bottom). Hence, heat transfer in this medium is two-dimensional.
03

(c) Heat generation in the medium

In case of heat generation, there would be an additional term in the equation representing the effect of heat generation. However, we do not see any such term in the given equation: $$T_{\text {node }}=\left(T_{\text {left }}+T_{\text {top }}+T_{\text {right }}+T_{\text {bottom }}\right) / 4$$ Consequently, we can infer that there is no heat generation in the medium.
04

(d) Nodal spacing

The given equation does not provide any information regarding nodal spacing; it only expresses the relationship between the temperatures at the node and its neighboring points. As a result, we cannot make a definitive statement about whether the nodal spacing is constant or variable based on the given information.
05

(e) Thermal conductivity of the medium

Similar to nodal spacing, the given equation does not include any information about the thermal conductivity of the medium. It only provides the relationship between the temperatures at the node and its neighboring points. Therefore, we cannot determine whether the thermal conductivity is constant or variable based on the given equation.

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Most popular questions from this chapter

What are the limitations of the analytical solution methods?

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 \(\Delta x\). The temperature at the right boundary (node 5) is specified. Using the energy balance approach, obtain the finite difference formulation of the boundary node 0 on the left boundary 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 \(\dot{q}_{0}\). Also, obtain the finite difference formulation for the rate of heat transfer at the right boundary.

Consider a 5 -m-long constantan block \((k=23 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) \(30 \mathrm{~cm}\) high and \(50 \mathrm{~cm}\) wide (Fig. P5-70). The block is completely submerged in iced water at \(0^{\circ} \mathrm{C}\) that is well stirred, and the heat transfer coefficient is so high that the temperatures on both sides of the block can be taken to be \(0^{\circ} \mathrm{C}\). The bottom surface of the bar is covered with a low-conductivity material so that heat transfer through the bottom surface is negligible. The top surface of the block is heated uniformly by a 8-kW resistance heater. Using the finite difference method with a mesh size of \(\Delta x=\Delta y=10 \mathrm{~cm}\) and taking advantage of symmetry, (a) obtain the finite difference formulation of this problem for steady twodimensional heat transfer, \((b)\) determine the unknown nodal temperatures by solving those equations, and \((c)\) determine the rate of heat transfer from the block to the iced water.

The wall of a heat exchanger separates hot water at \(T_{A}=90^{\circ} \mathrm{C}\) from cold water at \(T_{B}=10^{\circ} \mathrm{C}\). To extend the heat transfer area, two-dimensional ridges are machined on the cold side of the wall, as shown in Fig. P5-76. This geometry causes non-uniform thermal stresses, which may become critical for crack initiation along the lines between two ridges. To predict thermal stresses, the temperature field inside the wall must be determined. Convection coefficients are high enough so that the surface temperature is equal to that of the water on each side of the wall. (a) Identify the smallest section of the wall that can be analyzed in order to find the temperature field in the whole wall. (b) For the domain found in part \((a)\), construct a twodimensional grid with \(\Delta x=\Delta y=5 \mathrm{~mm}\) and write the matrix equation \(A T=C\) (elements of matrices \(A\) and \(C\) must be numbers). Do not solve for \(T\). (c) A thermocouple mounted at point \(M\) reads \(46.9^{\circ} \mathrm{C}\). Determine the other unknown temperatures in the grid defined in part (b).

What is the basis of the energy balance method? How does it differ from the formal finite difference method? For a specified nodal network, will these two methods result in the same or a different set of equations?
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