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 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

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

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.

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.

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.

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.

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): \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)-\alpha_{s} \dot{q}_{s}\)

Key concepts

Solar Radiation

Solar radiation is the energy emitted by the sun and transmitted through space. It reaches the Earth in the form of sunlight, carrying significant amounts of energy. The amount of energy received by a surface is described by the solar heat flux, denoted as \(\dot{q}_{s}\).

Solar radiation is crucial in many heat transfer problems, especially when analyzing surfaces exposed to sunlight, like sidewalks. It impacts the temperature of such surfaces.

Material properties, such as solar absorptivity (\(\alpha_{s}\)), determine how much of this radiation is absorbed by the surface. A high solar absorptivity means a large fraction of the solar radiation is absorbed, causing the surface temperature to rise quickly. Understanding how solar radiation interacts with different surfaces helps in efficiently designing outdoor structures and managing heat.

Thermal Conductivity

Thermal conductivity is a material property that measures a substance's ability to conduct heat. It is denoted by the symbol \(k\) and is expressed in units of watts per meter-kelvin (W/m·K).

High thermal conductivity means that a material can easily transfer heat, which is essential in structures that require quick dissipation or distribution of heat. In the context of a sidewalk, the thermal conductivity influences how heat, absorbed from solar radiation, moves through the material.

The equation \(-k \frac{d T}{d x}\) represents the rate of heat transfer through a material by conduction. Here, \(-k \) signifies that heat flows in the direction of decreasing temperature. Effective thermal management in any construction depends on choosing the right materials with suitable thermal conductivities.

Convective Heat Transfer

Convective heat transfer is the process of heat transfer between a surface and a fluid (such as air), depending on fluid motion. This motion can be driven by natural forces (natural convection) or external means, like fans (forced convection).

In our example, convective heat transfer is represented by the term \(h(T-T_{\infty})\). Here, \(h\) is the convective heat transfer coefficient, and \(T-T_{\infty}\) is the temperature difference between the sidewalk surface and the surrounding air.

Convective heat transfer helps cool or warm a surface by transporting heat away or towards it via fluid movement. The efficiency of this process largely depends on the convective heat transfer coefficient. A higher coefficient can mean more effective heat transfer, impacting how quickly a surface returns to equilibrium temperature. This principle is vital in maintaining comfortable conditions in outdoor spaces.

Boundary Conditions

Boundary conditions describe how heat is transferred or retained at the edges or surfaces of materials in a heat transfer analysis. They detail what happens at the boundaries of a system.

In the original exercise, the boundary condition for a sidewalk involves considering all forms of heat transfer in play: conduction, convection, and solar radiation. The correct boundary condition balances these by incorporating the effects of each into a single equation:

\[-k \frac{d T}{d x} = h(T - T_{\infty}) - \alpha_{s} \dot{q}_{s}\]

This equation takes into account:

• Conductive heat transfer through the sidewalk (\(-k \frac{d T}{d x}\))
• Convective heat transfer with the surrounding air (\(h(T - T_{\infty})\))
• Absorbed solar radiation (\(\alpha_{s} \dot{q}_{s}\))

By understanding and applying the correct boundary conditions, engineers can predict temperature distributions, ensuring safety, comfort, and energy efficiency in various construction scenarios.