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Chapter 3: Problem 14

Consider steady one-dimensional heat transfer through a multilayer medium. If the rate of heat transfer \(Q\) is known, explain how you would determine the temperature drop across each layer.

Short Answer

Expert verified
Answer: To find the temperature drop across each layer, first calculate the thermal resistance (R_th) of each layer using the formula R_th = L/(kA), where L is the thickness, k is the thermal conductivity, and A is the area of heat transfer. Then, use the formula ΔT = QR_th to determine the temperature drop across each layer.

Step by step solution

01

Understand the problem

Consider a multilayer medium in which heat is being transferred at a steady one-dimensional rate. We are given that the rate of heat transfer is Q. The objective is to determine the temperature drop across each layer of the medium.
02

Fourier's Law of Heat Conduction

Fourier's Law of heat conduction states that the rate of heat transfer through a material or layer is proportional to the temperature difference across the layer and the area of heat transfer, and inversely proportional to the layer's thickness. Mathematically, it is given by \[-Q = kA\frac{\Delta T}{L},\] where \(Q\) is the heat transfer rate, \(k\) is the thermal conductivity of the material, \(A\) is the area of heat transfer, \(\Delta T\) is the temperature difference across the layer, and \(L\) is the thickness of the layer.
03

Thermal Resistance

The concept of thermal resistance (\(R_\text{th}\)) relates the heat transfer rate to the temperature drop across the layer. The total thermal resistance for a multilayer medium can be defined as the sum of the thermal resistance of each individual layer. To find the thermal resistance of a layer, we can rearrange Fourier's Law like so: \[R_\text{th} = \frac{\Delta T}{Q} = \frac{L}{kA},\] where \(R_\text{th}\) is the thermal resistance of the layer.
04

Find the Thermal Resistance for Each Layer

Using the formula for thermal resistance, calculate the thermal resistance for each layer of the multilayer medium. You will need to know the thermal conductivity of each layer, as well as the area of heat transfer and the thickness of each layer.
05

Determine the Temperature Drop across Each Layer

Given the thermal resistance of each layer and the known heat transfer rate, we can now determine the temperature drop across each layer. Simply rearrange the formula for thermal resistance to solve for the temperature difference: \[\Delta T = Q R_\text{th},\] Calculate this value for each layer to find the temperature drop across it.

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Most popular questions from this chapter

An 8-m-internal-diameter spherical tank made of \(1.5\)-cm-thick stainless steel \((k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is used to store iced water at \(0^{\circ} \mathrm{C}\). The tank is located in a room whose temperature is \(25^{\circ} \mathrm{C}\). The walls of the room are also at \(25^{\circ} \mathrm{C}\). The outer surface of the tank is black (emissivity \(\varepsilon=1\) ), and heat transfer between the outer surface of the tank and the surroundings is by natural convection and radiation. The convection heat transfer coefficients at the inner and the outer surfaces of the tank are \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Determine \((a)\) the rate of heat transfer to the iced water in the tank and \((b)\) the amount of ice at \(0^{\circ} \mathrm{C}\) that melts during a 24 -h period. The heat of fusion of water at atmospheric pressure is \(h_{i f}=333.7 \mathrm{~kJ} / \mathrm{kg}\).

Hot water at an average temperature of \(53^{\circ} \mathrm{C}\) and an average velocity of \(0.4 \mathrm{~m} / \mathrm{s}\) is flowing through a \(5-\mathrm{m}\) section of a thin-walled hot-water pipe that has an outer diameter of \(2.5 \mathrm{~cm}\). The pipe passes through the center of a \(14-\mathrm{cm}\)-thick wall filled with fiberglass insulation \((k=0.035 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). If the surfaces of the wall are at \(18^{\circ} \mathrm{C}\), determine \((a)\) the rate of heat transfer from the pipe to the air in the rooms and \((b)\) the temperature drop of the hot water as it flows through this 5 -m-long section of the wall. Answers: \(19.6 \mathrm{~W}, 0.024^{\circ} \mathrm{C}\)

A \(0.2\)-cm-thick, 10-cm-high, and 15 -cm-long circuit board houses electronic components on one side that dissipate a total of \(15 \mathrm{~W}\) of heat uniformly. The board is impregnated with conducting metal fillings and has an effective thermal conductivity of \(12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). All the heat generated in the components is conducted across the circuit board and is dissipated from the back side of the board to a medium at \(37^{\circ} \mathrm{C}\), with a heat transfer coefficient of \(45 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Determine the surface temperatures on the two sides of the circuit board. (b) Now a 0.1-cm-thick, 10-cm-high, and 15 -cm-long aluminum plate \((k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) with \(200.2\)-cm-thick, 2-cm-long, and \(15-\mathrm{cm}\)-wide aluminum fins of rectangular profile are attached to the back side of the circuit board with a \(0.03-\mathrm{cm}-\) thick epoxy adhesive \((k=1.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Determine the new temperatures on the two sides of the circuit board.

Consider a very long rectangular fin attached to a flat surface such that the temperature at the end of the fin is essentially that of the surrounding air, i.e. \(20^{\circ} \mathrm{C}\). Its width is \(5.0 \mathrm{~cm}\); thickness is \(1.0 \mathrm{~mm}\); thermal conductivity is \(200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\); and base temperature is \(40^{\circ} \mathrm{C}\). The heat transfer coefficient is \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Estimate the fin temperature at a distance of \(5.0 \mathrm{~cm}\) from the base and the rate of heat loss from the entire fin.

Consider a \(1.5\)-m-high and 2 -m-wide triple pane window. The thickness of each glass layer \((k=0.80 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is \(0.5 \mathrm{~cm}\), and the thickness of each air space \((k=0.025 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is \(1 \mathrm{~cm}\). If the inner and outer surface temperatures of the window are \(10^{\circ} \mathrm{C}\) and \(0^{\circ} \mathrm{C}\), respectively, the rate of heat loss through the window is (a) \(75 \mathrm{~W}\) (b) \(12 \mathrm{~W}\) (c) \(46 \mathrm{~W}\) (d) \(25 \mathrm{~W}\) (e) \(37 \mathrm{~W}\)
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