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Chapter 2: Problem 164

The variation of temperature in a plane wall is determined to be \(T(x)=110-60 x\) where \(x\) is in \(\mathrm{m}\) and \(T\) is in \({ }^{\circ} \mathrm{C}\). If the thickness of the wall is \(0.75 \mathrm{~m}\), the temperature difference between the inner and outer surfaces of the wall is (a) \(30^{\circ} \mathrm{C}\) (b) \(45^{\circ} \mathrm{C}\) (c) \(60^{\circ} \mathrm{C}\) (d) \(75^{\circ} \mathrm{C}\) (e) \(84^{\circ} \mathrm{C}\)

Short Answer

Expert verified
Answer: The temperature difference between the inner and outer surfaces of the wall is 45°C.

Step by step solution

01

Identify the values for the inner and outer surfaces.

The inner and outer surfaces of the wall are located at \(x = 0\) and \(x = 0.75\), respectively.
02

Calculate the temperature at the inner surface.

Substitute \(x = 0\) into the temperature function: \(T(0) = 110 - 60(0) = 110^\circ \mathrm{C}\).
03

Calculate the temperature at the outer surface.

Substitute \(x = 0.75\) into the temperature function: \(T(0.75) = 110 - 60(0.75) = 110 - 45 = 65^\circ \mathrm{C}\).
04

Calculate the temperature difference between the inner and outer surfaces.

Subtract the temperature at the outer surface from the temperature at the inner surface: Temperature difference = \(110^\circ \mathrm{C} - 65^\circ \mathrm{C} = 45^\circ\mathrm{C}\). The temperature difference between the inner and outer surfaces of the wall is (b) \(45^{\circ} \mathrm{C}\).

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Most popular questions from this chapter

A stainless steel spherical container, with \(k=\) \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is used for storing chemicals undergoing exothermic reaction. The reaction provides a uniform heat flux of \(60 \mathrm{~kW} / \mathrm{m}^{2}\) to the container's inner surface. The container has an inner radius of \(50 \mathrm{~cm}\) and a wall thickness of \(5 \mathrm{~cm}\) and is situated in a surrounding with an ambient temperature of \(23^{\circ} \mathrm{C}\). The container's outer surface is subjected to convection heat transfer with a coefficient of \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). For safety reasons to prevent thermal burn to individuals working around the container, it is necessary to keep the container's outer surface temperature below \(50^{\circ} \mathrm{C}\). Determine the variation of temperature in the container wall and the temperatures of the inner and outer surfaces of the container. Is the outer surface temperature of the container safe to prevent thermal burn?

What is the difference between an algebraic equation and a differential equation?

In a food processing facility, a spherical container of inner radius \(r_{1}=40 \mathrm{~cm}\), outer radius \(r_{2}=41 \mathrm{~cm}\), and thermal conductivity \(k=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is used to store hot water and to keep it at \(100^{\circ} \mathrm{C}\) at all times. To accomplish this, the outer surface of the container is wrapped with a 800 -W electric strip heater and then insulated. The temperature of the inner surface of the container is observed to be nearly \(120^{\circ} \mathrm{C}\) at all times. Assuming 10 percent of the heat generated in the heater is lost through the insulation, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the container, \((b)\) obtain a relation for the variation of temperature in the container material by solving the differential equation, and \((c)\) evaluate the outer surface temperature of the container. Also determine how much water at \(100^{\circ} \mathrm{C}\) this tank can supply steadily if the cold water enters at \(20^{\circ} \mathrm{C}\).

A spherical container, with an inner radius of \(1 \mathrm{~m}\) and a wall thickness of \(5 \mathrm{~mm}\), has its inner surface subjected to a uniform heat flux of \(7 \mathrm{~kW} / \mathrm{m}^{2}\). The outer surface of the container is maintained at \(20^{\circ} \mathrm{C}\). The container wall is made of a material with a thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.33 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.0023 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). Determine the temperature drop across the container wall thickness.

Consider steady one-dimensional heat conduction through a plane wall, a cylindrical shell, and a spherical shell of uniform thickness with constant thermophysical properties and no thermal energy generation. The geometry in which the variation of temperature in the direction of heat transfer will be linear is (a) plane wall (b) cylindrical shell (c) spherical shell (d) all of them (e) none of them
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