Question

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 toward the sky and disregarding radiation exchange with the surrounding 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) \(-k \frac{d T}{d x}=h\left(T-T_{o c}\right)-\alpha_{s} \dot{q}_{s}\) (d) \(h\left(T-T_{\infty}\right)=\alpha_{s} \dot{q}_{s}\) (e) None of them

Short answer

Based on the step-by-step solution, the correct boundary condition for the surface of the sidewalk receiving solar heat flux is: (c) \(-k \frac{d T}{d x} = h\left(T-T_{\infty}\right) - \alpha_{s} \dot{q}_{s}\)

Step-by-step solution

1. Understand the boundary condition

The boundary condition will describe the heat balance on the sidewalk surface, which is the result of solar radiation absorption and convective heat loss.

2. Equation for absorbed solar radiation

The absorbed solar radiation per unit area of the sidewalk surface is given by: \(\alpha_{s^{+}} \cdot \dot{q}_{s}\)

3. Equation for convective heat transfer

The convective heat transfer per unit area between the sidewalk surface and ambient air is given by: \(h (T - T_\infty)\), where \(T\) is the temperature of the sidewalk surface and \(T_\infty\) is the ambient temperature.

4. Equation for conductive heat transfer

The conductive heat transfer is given by Fourier's law: \(q = -k \frac{d T}{d x}\), where \(-k \frac{d T}{d x}\) is the heat flux due to conduction.

5. Writing the boundary condition equation

At the surface of the sidewalk, the absorbed solar radiation and convective heat transfer must balance the conductive heat transfer. Therefore, the boundary condition equation is: $$-k \frac{d T}{d x} = \alpha_{s^{+}} \cdot \dot{q}_{s} - h (T - T_\infty)$$

6. Comparing the options and selecting the correct answer

Comparing our derived boundary condition with the given options, we find that the correct choice is: (c) \(-k \frac{d T}{d x} = h\left(T-T_{\infty}\right) - \alpha_{s} \dot{q}_{s}\)