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Chapter 2: Problem 159

A solar heat flux \(\dot{q}_{s}\) is incident on a sidewalk whose thermal conductivity is \(k\), solar absorptivity is \(\alpha_{s}\), and convective heat transfer coefficient is \(h\). Taking the positive \(x\) direction to be towards the sky and disregarding radiation exchange with the surroundings surfaces, the correct boundary condition for this sidewalk surface is (a) \(-k \frac{d T}{d x}=\alpha_{s} \dot{q}_{s}\) (b) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)\) (c) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)-\alpha_{s} \dot{q}_{s}\) (d) \(h\left(T-T_{\infty}\right)=\alpha_{s} \dot{q}_{s}\) (e) None of them

Short Answer

Expert verified
Answer: The correct boundary condition for the surface of the sidewalk in this exercise is: -k (dT/dx) = h(T-T∞) - αs q̇s.

Step by step solution

01

Analyze Option (a)

Assume the boundary condition as given in option (a): \(-k \frac{d T}{d x}=\alpha_{s} \dot{q}_{s}\) This equation represents the balance between the conductive heat transfer in the sidewalk and the absorbed solar radiation. However, it does not account for the convective heat transfer that happens between the sidewalk surface and the surrounding air. Hence, this option is incorrect.
02

Analyze Option (b)

Now, let's analyze the boundary condition given in option (b): \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)\) This equation represents the balance between the conductive heat transfer in the sidewalk and the convective heat transfer with the surrounding air. However, it does not take into account the absorbed solar radiation. Hence, this option is also incorrect.
03

Analyze Option (c)

Consider the boundary condition given in option (c): \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)-\alpha_{s} \dot{q}_{s}\) This equation represents the balance between the conductive heat transfer, convective heat transfer, and absorbed solar radiation, which accounts for all necessary factors in the problem. Therefore, this option is correct.
04

Analyze Option (d)

For completion, let's analyze the boundary condition given in option (d): \(h\left(T-T_{\infty}\right)=\alpha_{s} \dot{q}_{s}\) This equation represents the balance between the convective heat transfer and the absorbed solar radiation. However, it does not include the conductive heat transfer in the sidewalk which is needed for a complete boundary condition. Hence, this option is incorrect.
05

Conclusion

Based on our step-by-step analysis, we can conclude that the correct boundary condition for the sidewalk surface in this exercise is given by option (c): \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)-\alpha_{s} \dot{q}_{s}\)

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Most popular questions from this chapter

How do differential equations with constant coefficients differ from those with variable coefficients? Give an example for each type.

In subsea oil and natural gas production, hydrocarbon fluids may leave the reservoir with a temperature of \(70^{\circ} \mathrm{C}\) and flow in subsea surrounding of \(5^{\circ} \mathrm{C}\). As a result of the temperature difference between the reservoir and the subsea surrounding, the knowledge of heat transfer is critical to prevent gas hydrate and wax deposition blockages. Consider a subsea pipeline with inner diameter of \(0.5 \mathrm{~m}\) and wall thickness of \(8 \mathrm{~mm}\) is used for transporting liquid hydrocarbon at an average temperature of \(70^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner pipeline surface is estimated to be \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The subsea surrounding has a temperature of \(5^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient on the outer pipeline surface is estimated to be \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the pipeline is made of material with thermal conductivity of \(60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), by using the heat conduction equation (a) obtain the temperature variation in the pipeline wall, \((b)\) determine the inner surface temperature of the pipeline, \((c)\) obtain the mathematical expression for the rate of heat loss from the liquid hydrocarbon in the pipeline, and \((d)\) determine the heat flux through the outer pipeline surface.

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