Question

A manufacturer makes absorber plates that are $1.2 \mathrm{~m} \times 0.8 \mathrm{~m}$ in size for use in solar collectors. The back side of the plate is heavily insulated, while its front surface is coated with black chrome, which has an absorptivity of \(0.87\) for solar radiation and an emissivity of \(0.09\). Consider such a plate placed horizontally outdoors in calm air at \(25^{\circ} \mathrm{C}\). Solar radiation is incident on the plate at a rate of \(600 \mathrm{~W} / \mathrm{m}^{2}\). Taking the effective sky temperature to be \(10^{\circ} \mathrm{C}\), determine the equilibrium temperature of the absorber plate. What would your answer be if the absorber plate is made of ordinary aluminum plate that has a solar absorptivity of \(0.28\) and an emissivity of \(0.07\) ? Evaluate air properties at a film temperature of $70^{\circ} \mathrm{C}$ and 1 atm pressure. Is this a good assumption?

Short answer

Question: Determine the equilibrium temperature of an absorber plate with black chrome surface (absorptivity of 0.87 and emissivity of 0.1). The solar radiation incident on the absorber plate is 600 W/m². The plate has dimensions of 1.2 m by 0.8 m. The effective sky temperature is 10°C, and the air temperature is 25°C. The convective heat transfer coefficient is evaluated at a film temperature of 70°C and 1 atm.

Step-by-step solution

Calculate the Solar radiation absorbed

First, we need to find the solar radiation absorbed by the absorber plate. The formula for this is: \(q_{absorbed} = G * A * \alpha\) Where \(G\) is the solar radiation incident on the plate (600 W/m²), \(A\) is the area of the plate, and \(\alpha\) is the absorptivity of the plate material. The area of the plate is found by multiplying its length and width, so: \(A = 1.2 m \times 0.8 m = 0.96 m^2\) For the first plate with black chrome, the absorptivity is 0.87. Thus, the solar radiation absorbed will be: \(q_{absorbed(black~chrome)} = 600 * 0.96 * 0.87 = 505.44 W\)

Consider the energy balance for the absorber plate

The energy balance for the absorber plate will be as follows: - Solar radiation absorbed by the plate - Heat transfer by radiation between the plate and sky - Heat transfer by convection between the plate and air The equilibrium temperature will be achieved when the heat absorbed equals the heat lost to the surroundings, so: \(q_{absorbed} = q_{radiative} + q_{convective}\)

Calculate the radiative heat transfer

To find the radiative heat transfer between the plate and sky, we can use the following formula: \(q_{radiative} = \epsilon * A_{plate} *\sigma * (T_{plate}^4 - T_{sky}^4)\) Where \(\epsilon\) is the emissivity of the plate, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67\times10^{-8} W/ m^2.K^4\)), \(T_{plate}\) is the temperature of the absorber plate, and \(T_{sky}\) is the effective sky temperature (10°C, which should be converted to Kelvin).

Calculate the convective heat transfer

To find the convective heat transfer between the plate and air, we can use the following formula: \(q_{convective} = h * A_{plate} * (T_{plate} - T_{air})\) Where \(h\) is the convective heat transfer coefficient (which can be found using empirical relationships for natural convection with air properties evaluated at the film temperature of 70°C and 1 atm) and \(T_{air}\) is the air temperature (25°C, which should be converted to Kelvin).

Solve the energy balance equation for the equilibrium temperature

Now, we have all components needed to solve the energy balance equation. The equation for the absorber plate with black chrome should look like this: \(505.44 W = \epsilon_{chrome} * A_{plate}*\sigma * (T_{plate}^4 - T_{sky}^4) + h_{chrome} * A_{plate} * (T_{plate} - T_{air})\) The equilibrium temperature of the absorber plate can be found by iteratively solving this equation for \(T_{plate}\). Similar calculations can be done for the absorber plate made of ordinary aluminum with a solar absorptivity of 0.28 and an emissivity of 0.07. Lastly, to verify if the assumption of evaluating air properties at a film temperature of \(70^{\circ} \mathrm{C}\) and 1 atm pressure is good, compare the calculated equilibrium temperature to the assumed film temperature. If the calculated temperature is close to the assumed value, the assumption can be considered valid.