A thin metal plate is insulated on the back and exposed to...

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 0.8 for solar radiation. If solar radiation is incident on the plate at a rate of $450 \mathrm{W} / \mathrm{m}^{2}$ and the surrounding air temperature is $25^{\circ} \mathrm{C}$, determine the surface temperature of the plate when the heat loss by convection equals the solar energy absorbed by the plate. Assume the convection heat transfer coefficient to be $50 \mathrm{W} / \mathrm{m}^{2} \cdot^{\circ} \mathrm{C}$ and disregard heat loss by radiation.

Key Concepts

Steady-State Temperature

The steady-state temperature is reached when the rate of energy input into a system equals the rate of energy loss, meaning that the temperature remains constant over time. Determining this temperature involves setting up an energy balance equation where absorbed energy (e.g., from radiation) is exactly balanced by the energy losses (e.g., through convection), allowing for the calculation of the equilibrium temperature of the system.

Thermal Energy Balance

Thermal energy balance is a fundamental concept in heat transfer where, at steady state, the energy absorbed by a system is equal to the energy lost. In the context of a heated plate exposed to a radiation source and cooled by convection, the absorbed solar energy is balanced by the convective heat loss to the surroundings, which determines the steady-state temperature of the plate.

Absorptivity

Absorptivity is a measure of how effectively a surface absorbs incident radiation. It is defined as the fraction of incoming radiation that is absorbed by the surface rather than reflected or transmitted. In energy balance problems, absorptivity is crucial because it directly influences the amount of solar energy actually converted into thermal energy in the material.

Convective Heat Transfer

Convective heat transfer involves the transfer of heat from a surface to a fluid (or vice versa) due to the fluid’s motion. This process is quantified by the convection heat transfer coefficient, which determines how efficiently heat is removed (or supplied) by the fluid. The concept is critical in situations where heat loss by convection plays a major role in setting the temperature of objects exposed to fluid environments.

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