Question

A thin metal plate is insulated on the back and exposed to solar radiation on the front surface. The exposed surface of the plate has an absorptivity of \(0.7\) for solar radiation. If solar radiation is incident on the plate at a rate of \(550 \mathrm{~W} / \mathrm{m}^{2}\) and the surrounding air temperature is \(10^{\circ} \mathrm{C}\), determine the surface temperature of the plate when the heat loss by convection equals the solar energy absorbed by the plate. Take the convection heat transfer coefficient to be \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and disregard any heat loss by radiation.

Short answer

Answer: The surface temperature of the plate is 25.4°C.

Step-by-step solution

Calculate the absorbed solar energy

To calculate the solar energy absorbed by the surface, we need to multiply the incident solar radiation (\(550 W/m^2\)) by the absorptivity (\(0.7\)), which will give us the absorbed energy rate per unit area: $$E_\text{absorbed} = 0.7 \cdot 550 W/m^2 = 385 W/m^2$$

Equation for the heat loss due to convection

The heat loss due to convection can be calculated using the following equation: $$q_\text{conv} = h_\text{conv} A (T_\text{plate} - T_\text{air})$$ In our case, we are given the convection heat transfer coefficient \(h_\text{conv} = 25 W/m^2 \cdot K\) and the air temperature \(T_\text{air} = 10^\circ C\). The terms q_conv and A cancel out, giving us: $$q_\text{loss} = 25 (T_\text{plate} - 10)$$

Equate the absorbed solar energy and the convection heat loss, then solve for the surface temperature of the plate

Now, we equate the absorbed solar energy and the convection heat loss: $$E_\text{absorbed} = q_\text{loss}$$ $$385 W/m^2 = 25 W/m^2 \cdot K \cdot (T_\text{plate} - 10^\circ\textbf{ C})$$ Now we solve for \(T_\text{plate}\):$$T_\text{plate} - 10^\circ \textbf{ C} = \frac{385}{25}$$ $$T_\text{plate} = 10^\circ \textbf{ C} + \frac{385}{25}$$ $$T_\text{plate} = 25.4^\circ \textbf{ C}$$ So the surface temperature of the plate when the heat loss by convection equals the solar energy absorbed by the plate is \(25.4^\circ C\).

Key concepts

Convection

Convection is the transfer of heat through a fluid (such as air or water) caused by molecular motion. It occurs when a fluid is heated and the warmer, less dense part of the fluid rises, while the cooler, denser part sinks. This process creates a cycle known as convection currents.

In the context of our exercise, convection plays a crucial role in transferring heat away from the metal plate to the surrounding air. As the metal plate heats up due to solar radiation, heat must be dissipated to maintain a balance. Convection is effective in this because it involves bulk movement of the fluid, which can transport significant amounts of heat.

Convection can be:

• Natural Convection: Occurs due to natural differences in temperature and density within a fluid. For example, warm air rising from a heated surface.
• Forced Convection: Occurs when external forces, such as fans or pumps, enhance the fluid flow, leading to increased heat transfer.

Solar Radiation

Solar radiation is the energy emitted by the sun, primarily in the form of visible light, with smaller amounts in the form of ultraviolet and infrared radiation. It is the primary natural source of energy for the Earth, powering weather systems, photosynthesis, and significantly impacting the temperature of objects exposed to it.

When the thin metal plate in our exercise is exposed to solar radiation, it absorbs a portion of this energy. This absorbed energy increases the plate's temperature until an equilibrium is reached between the incoming solar energy and the energy lost through convection.

The intensity of solar radiation can be measured in watts per square meter ( W/m² ), and factors such as time of day, atmospheric conditions, and geographical location can influence how much solar radiation reaches a specific surface.

Absorptivity

Absorptivity is a measure of how well a material can absorb radiation. It is the fraction of incident radiation that gets absorbed by a surface. In the context of our problem, absorptivity is paramount because it determines how much of the incident solar energy actually contributes to heating the plate.

For the metal plate, the absorptivity is given as 0.7. This means 70% of the solar radiation that hits the surface is absorbed, and 30% is reflected or scattered. The higher the absorptivity, the more effective a surface is at capturing and retaining heat from radiation.

Absorptivity can vary depending on:

• Material Properties: Different materials have varying absorptivity based on their composition and texture.
• Surface Condition: A polished surface may reflect more energy compared to a matte surface, which may have higher absorptivity.
• Wavelength: Different wavelengths of radiation are absorbed differently by materials.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient (\( h \) ) is a numerical value that quantifies the effectiveness of convection in transferring heat from a surface to a fluid or vice versa. It is a crucial parameter in understanding how efficiently heat is transferred during convection.

In our exercise, the convection heat transfer coefficient is given as 25 W/m²·K. This indicates the rate of heat transfer per unit area per unit temperature difference between the plate and the surrounding air.

Factors influencing this coefficient include:

• Type of Flow: Whether the fluid flow is laminar or turbulent can significantly alter the convection coefficient. Turbulent flows generally enhance heat transfer.
• Properties of Fluid: Viscosity, thermal conductivity, and specific heat capacity of the fluid play pivotal roles.
• Surface Characteristics: The surface roughness and shape can affect the flow pattern, thus influencing the heat transfer coefficient.
• Temperature Gradient: Larger temperature differences between the surface and the fluid may lead to increased heat transfer rates, affecting the coefficient.

Understanding this coefficient helps in designing systems for efficient thermal management, whether it's a cooling system or a heating application.