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Chapter 2: Problem 112

Consider a plane wall of thickness \(L\) whose thermal conductivity varies in a specified temperature range as \(k(T)=\) \(k_{0}\left(1+\beta T^{2}\right)\) where \(k_{0}\) and \(\beta\) are two specified constants. The wall surface at \(x=0\) is maintained at a constant temperature of \(T_{1}\), while the surface at \(x=L\) is maintained at \(T_{2}\). Assuming steady one-dimensional heat transfer, obtain a relation for the heat transfer rate through the wall.

Short Answer

Expert verified
Question: Determine the heat transfer rate through a plane wall of thickness L with a temperature-dependent thermal conductivity k(T) = k_0(1+βT²). The temperatures at the two surfaces of the wall are given as T₁ and T₂, assuming steady one-dimensional heat transfer. Answer: The heat transfer rate (q) through the wall can be determined using the following expression: $$q = -\frac{k_{0} \Big[(T_{2} - T_{1}) + \beta \frac{1}{3}(T_{2}^3 - T_{1}^3)\Big]}{L}$$

Step by step solution

01

Fourier's law of heat conduction

Fourier's law of heat conduction states that the heat transfer rate (\(q\)) is proportional to the temperature gradient and the material's thermal conductivity: $$q = -k(T) \frac{dT}{dx}$$ Since the thermal conductivity is temperature-dependent in this problem, we cannot directly integrate this equation to obtain the heat transfer rate.
02

Modify the Fourier's law equation

Let's rearrange the equation in step 1 to separate the \(x\) terms from the \(T\) terms: $$\frac{dx}{dT} = -\frac{k(T)}{q}$$ Now, substitute the expression for \(k(T)\): $$\frac{dx}{dT} = -\frac{k_{0}(1+\beta T^{2})}{q}$$ This equation represents a separable ordinary differential equation with variables \(x\) and \(T\).
03

Integrate the modified Fourier's law equation

Integrate both sides of the equation from step 2 to obtain the relationship between \(x\) and \(T\): $$\int_{x=0}^{x=L} dx = -\frac{k_{0}}{q} \int_{T=T_{1}}^{T=T_{2}} (1+\beta T^{2}) dT$$ Since the integral of \(dx\) from \(0\) to \(L\) is simply \(L\), the equation becomes: $$L = -\frac{k_{0}}{q} \Big[\int_{T=T_{1}}^{T=T_{2}}dT + \beta\int_{T=T_{1}}^{T=T_{2}} T^{2} dT\Big]$$
04

Evaluate the integrals and solve for the heat transfer rate

Evaluate the two integrals on the right-hand side of the equation: $$L = -\frac{k_{0}}{q} \Big[(T_{2} - T_{1}) + \beta \frac{1}{3}(T_{2}^3 - T_{1}^3)\Big]$$ Now, solve for the heat transfer rate \(q\) by rearranging the equation: $$q = -\frac{k_{0} \Big[(T_{2} - T_{1}) + \beta \frac{1}{3}(T_{2}^3 - T_{1}^3)\Big]}{L}$$ This is the final expression for the heat transfer rate through the wall with temperature-dependent thermal conductivity.

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Most popular questions from this chapter

A flat-plate solar collector is used to heat water by having water flow through tubes attached at the back of the thin solar absorber plate. The absorber plate has an emissivity and an absorptivity of \(0.9\). The top surface \((x=0)\) temperature of the absorber is \(T_{0}=35^{\circ} \mathrm{C}\), and solar radiation is incident on the absorber at \(500 \mathrm{~W} / \mathrm{m}^{2}\) with a surrounding temperature of \(0^{\circ} \mathrm{C}\). Convection heat transfer coefficient at the absorber surface is \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the ambient temperature is \(25^{\circ} \mathrm{C}\). Show that the variation of temperature in the absorber plate can be expressed as \(T(x)=-\left(\dot{q}_{0} / k\right) x+T_{0}\), and determine net heat flux \(\dot{q}_{0}\) absorbed by the solar collector.

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A spherical communication satellite with a diameter of \(2.5 \mathrm{~m}\) is orbiting around the earth. The outer surface of the satellite in space has an emissivity of \(0.75\) and a solar absorptivity of \(0.10\), while solar radiation is incident on the spacecraft at a rate of \(1000 \mathrm{~W} / \mathrm{m}^{2}\). If the satellite is made of material with an average thermal conductivity of \(5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and the midpoint temperature is \(0^{\circ} \mathrm{C}\), determine the heat generation rate and the surface temperature of the satellite.

Consider the uniform heating of a plate in an environment at a constant temperature. Is it possible for part of the heat generated in the left half of the plate to leave the plate through the right surface? Explain.

Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\text {gen. }}\). Heat is lost from the cylindrical surface at \(r=r_{o}\) by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). The bottom surface of the cylinder at \(z=0\) is insulated, while the top surface at \(z=H\) is subjected to uniform heat flux \(\dot{q}_{H}\). Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not solve.
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