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Chapter 2: Problem 110

Consider a \(1.5\)-m-high and \(0.6-\mathrm{m}\)-wide plate whose thickness is \(0.15 \mathrm{~m}\). One side of the plate is maintained at a constant temperature of \(500 \mathrm{~K}\) while the other side is maintained at \(350 \mathrm{~K}\). The thermal conductivity of the plate can be assumed to vary linearly in that temperature range as \(k(T)=\) \(k_{0}(1+\beta T)\) where \(k_{0}=18 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\beta=8.7 \times 10^{-4} \mathrm{~K}^{-1}\). Disregarding the edge effects and assuming steady onedimensional heat transfer, determine the rate of heat conduction through the plate. Answer: \(22.2 \mathrm{~kW}\)

Short Answer

Expert verified
Answer: The rate of heat conduction through the plate is approximately \(22.2\,\mathrm{kW}\).

Step by step solution

01

Write down the given quantities and desired output

We know the following: - Plate height (h) = \(1.5\,\mathrm{m}\) - Plate width (w) = \(0.6\,\mathrm{m}\) - Plate thickness (t) = \(0.15\,\mathrm{m}\) - Temperature on one side of the plate: \(T_1=500\,\mathrm{K}\) - Temperature on the other side of the plate: \(T_2=350\,\mathrm{K}\) - Thermal conductivity expression: \(k(T)=k_0(1+\beta T)\), with \(k_0=18\,\frac{\mathrm{W}}{\mathrm{m}\cdot \mathrm{K}}\) and \(\beta=8.7\times 10^{-4}\,\mathrm{K}^{-1}\) We want to find: Rate of heat conduction (Q) in \(\mathrm{kW}\).
02

Determine the average thermal conductivity of the plate

To simplify the problem, let's calculate the average thermal conductivity of the plate \((\bar{k})\): $$ \bar{k}=\frac{k(T_1)+k(T_2)}{2} $$ Plug in the values for \(k(T_1)\) and \(k(T_2)\) using the expression for \(k(T)\): $$ \bar{k}=\frac{k_0(1+\beta T_1)+k_0(1+\beta T_2)}{2} $$
03

Calculate the average thermal conductivity

Now, use the given values of \(k_0\), \(\beta\), \(T_1\), and \(T_2\) to find \(\bar{k}\): $$ \bar{k}=\frac{18(1+8.7\times 10^{-4}\times 500)+18(1+8.7\times 10^{-4}\times 350)}{2} $$ Calculate: $$ \bar{k}\approx20.57\,\frac{\text{W}}{\text{m}\cdot\text{K}} $$
04

Apply Fourier's Law of heat conduction

Using Fourier's Law in one dimension: $$ Q = \frac{(\bar{k})(h)(w)\Delta T}{t} $$ Insert values: $$ Q = \frac{(20.57)(1.5)(0.6)(500-350)}{0.15} $$
05

Calculate the rate of heat conduction

Now, compute Q: $$ Q\approx22,\!212\,\mathrm{W} $$ Convert watts to kilowatts: $$ Q\approx22.2\,\mathrm{kW} $$ The rate of heat conduction through the plate is approximately \(22.2 \,\mathrm{kW}\).

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Most popular questions from this chapter

Is the thermal conductivity of a medium, in general, constant or does it vary with temperature?

A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is \(5 \mathrm{~m}\) and its inner surface temperature is at \(120^{\circ} \mathrm{C}\). The wall of the vessel has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.01 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\beta=0.0018 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). The vessel is situated in a surrounding with an ambient temperature of \(15^{\circ} \mathrm{C}\), the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent thermal burn on skin tissues, the outer surface temperature of the vessel should be kept below \(50^{\circ} \mathrm{C}\). Determine the minimum wall thickness of the vessel so that the outer surface temperature is \(50^{\circ} \mathrm{C}\) or lower.

The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ Select the wrong statement below. (a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional.

What is a variable? How do you distinguish a dependent variable from an independent one in a problem?

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=27^{\circ} \mathrm{C}\) and \(h_{1}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.
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