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Chapter 2: Problem 48

Consider a spherical container of inner radius \(r_{1}\), outer radius \(r_{2}\), and thermal conductivity \(k\). Express the boundary condition on the inner surface of the container for steady onedimensional conduction for the following cases: \((a)\) specified temperature of \(50^{\circ} \mathrm{C},(b)\) specified heat flux of \(45 \mathrm{~W} / \mathrm{m}^{2}\) toward the center, (c) convection to a medium at \(T_{\infty}\) with a heat transfer coefficient of \(h\).

Short Answer

Expert verified
Question: Determine the boundary conditions for steady one-dimensional conduction on the inner surface of a spherical container in the given cases: (a) specified temperature of \(50^{\circ}\mathrm{C}\), (b) specified heat flux of \(45 \mathrm{W}/\mathrm{m}^2\) toward the center, and (c) convection to a medium at a temperature \(T_{\infty}\) with a heat transfer coefficient \(h\). Answer: (a) \(T(r_1) = 50^{\circ}\mathrm{C}\) (b) \(-k \frac{dT}{dr}\bigg|_{r=r_1} = 45 \mathrm{W}/\mathrm{m^2}\) (c) \(-k \frac{dT}{dr}\bigg|_{r=r_1} = h\big(T(r_1) - T_{\infty}\big)

Step by step solution

01

Case (a): Specified temperature of \(50^{\circ}\mathrm{C}\)

In this case, the boundary condition on the inner surface of the container is that the temperature is a fixed \(50^{\circ}\mathrm{C}\). Since the temperature is provided in degree Celsius, we can just state the boundary condition as follows: $$ T(r_1) = 50^{\circ}\mathrm{C} $$
02

Case (b): Specified heat flux of \(45 \mathrm{~W} / \mathrm{m}^{2}\) toward the center

In this case, the inner surface has a heat flux of \(45 \mathrm{W}/\mathrm{m}^2\) towards the center of the container. The heat flux \(q\) is related to the temperature gradient by Fourier's Law: $$ q = -k \frac{dT}{dr} $$ Since we know the heat flux \(q = 45 \mathrm{W}/\mathrm{m}^2\) toward the center, we must take into account the negative sign in Fourier's Law as the temperature gradient will be negative. The boundary condition on the inner surface can be expressed as: $$ -k \frac{dT}{dr}\bigg|_{r=r_1} = 45 \mathrm{W}/\mathrm{m^2} $$
03

Case (c): Convection to a medium at \(T_{\infty}\) with a heat transfer coefficient of \(h\)

In the last case, we have convection to a medium at a temperature \(T_{\infty}\) with a heat transfer coefficient \(h\). The heat being transferred from the inner surface to the medium can be expressed by Newton's Law of Cooling: $$ q = h\big(T(r_1) - T_{\infty}\big) $$ We can also relate the heat flux \(q\) to the temperature gradient using Fourier's Law, as shown in the previous case: $$ q = -k \frac{dT}{dr}\bigg|_{r=r_1} $$ Combining both these equations, we can express the boundary condition on the inner surface as: $$ -k \frac{dT}{dr}\bigg|_{r=r_1} = h\big(T(r_1) - T_{\infty}\big) $$

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Most popular questions from this chapter

A spherical container of inner radius \(r_{1}=2 \mathrm{~m}\), outer radius \(r_{2}=2.1 \mathrm{~m}\), and thermal conductivity \(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is filled with iced water at \(0^{\circ} \mathrm{C}\). The container is gaining heat by convection from the surrounding air at \(T_{\infty}=25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming the inner surface temperature of the container to be \(0^{\circ} \mathrm{C},(a)\) 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 by solving the differential equation, and \((c)\) evaluate the rate of heat gain to the iced water.

A long homogeneous resistance wire of radius \(r_{o}=\) \(5 \mathrm{~mm}\) is being used to heat the air in a room by the passage of electric current. Heat is generated in the wire uniformly at a rate of \(5 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\) as a result of resistance heating. If the temperature of the outer surface of the wire remains at \(180^{\circ} \mathrm{C}\), determine the temperature at \(r=3.5 \mathrm{~mm}\) after steady operation conditions are reached. Take the thermal conductivity of the wire to be \(k=6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).

Consider a large plane wall of thickness \(L=0.05 \mathrm{~m}\). The wall surface at \(x=0\) is insulated, while the surface at \(x=L\) is maintained at a temperature of \(30^{\circ} \mathrm{C}\). The thermal conductivity of the wall is \(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and heat is generated in the wall at a rate of \(\dot{e}_{\text {gen }}=\dot{e}_{0} e^{-0.5 x / L} \mathrm{~W} / \mathrm{m}^{3}\) where \(\dot{e}_{0}=8 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\). Assuming steady one-dimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the wall, \((b)\) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) determine the temperature of the insulated surface of the wall.

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.

Heat is generated uniformly at a rate of \(4.2 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\) in a spherical ball \((k=45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of diameter \(24 \mathrm{~cm}\). The ball is exposed to iced-water at \(0^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the temperatures at the center and the surface of the ball.
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