Question

Consider a flat-plate solar collector placed horizontally on the flat roof of a house. The collector is \(1.5 \mathrm{~m}\) wide and \(4.5 \mathrm{~m}\) long, and the average temperature of the exposed surface of the collector is \(42^{\circ} \mathrm{C}\). Determine the rate of heat loss from the collector by natural convection during a calm day when the ambient air temperature is \(8^{\circ} \mathrm{C}\). Also, determine the heat loss by radiation by taking the emissivity of the collector surface to be \(0.85\) and the effective sky temperature to be \(-15^{\circ} \mathrm{C}\). Answers: $1314 \mathrm{~W}, 1762 \mathrm{~W}$

Short answer

Answer: The total heat loss from the flat-plate solar collector during a calm day is the sum of the heat loss due to natural convection and the heat loss due to radiation. This can be calculated as follows: Total heat loss = Natural convection heat loss + Radiation heat loss Total heat loss = 1314 W + 1762 W Total heat loss = 3076 W Therefore, the total heat loss from the flat-plate solar collector during a calm day is 3076 W.

Step-by-step solution

Calculate the heat loss due to natural convection.

To determine the rate of heat loss from the collector due to natural convection, we can use the formula for heat loss due to natural convection from a relatively flat horizontal surface: \(Q_{conv} = h_{c} A (T_s - T_{\infty}) \) Where \(Q_{conv}\) is the rate of heat loss due to convection, \(h_c\) is the convection heat transfer coefficient, \(A\) is the surface area of the collector, \(T_s\) is the surface temperature of the collector, and \(T_{\infty}\) is the ambient air temperature. For this problem, we can estimate \(h_c\) for natural convection for a heated horizontal plate is: \(h_{c} = 1.32[(T_s - T_{\infty}) /L]^{1/4}\) Where \(L\) is the characteristic length (the average of the width and length of the collector). We have \(T_s = 42^{\circ} \mathrm{C}\) (surface temperature of the collector) and \(T_{\infty} = 8^{\circ} \mathrm{C}\) (ambient air temperature). We will first need to determine the characteristic length \(L\).

Calculate the characteristic length of the collector.

The characteristic length can be found by taking the average of the width and length of the collector: \(L = (1.5 + 4.5)/2 = 3 \mathrm{~m}\) Now, we can calculate the natural convection heat transfer coefficient, \(h_{c}\).

Calculate the natural convection heat transfer coefficient.

Using the formula for \(h_{c}\), we can calculate the heat transfer coefficient: \(h_{c} = 1.32[(T_s - T_{\infty}) /L]^{1/4} = 1.32[(42-8)/3]^{1/4} = 5.1 \mathrm{~W/m^2K}\) Now, we can determine the rate of heat loss due to natural convection.

Calculate the natural convection heat loss.

Using the formula for \(Q_{conv}\) and our calculated values: \(Q_{conv} = h_{c} A (T_s - T_{\infty}) = 5.1\cdot(1.5\cdot4.5)(42-8) = 1314 \mathrm{~W}\) Therefore, the rate of heat loss from the collector due to natural convection is 1314 W.

Calculate the heat loss due to radiation.

For the radiation part, we can use the formula for heat loss due to radiation from a flat surface: \(Q_r= \varepsilon \sigma A(T_s^4-T_{sky}^4)\) Where \(\varepsilon\) is the emissivity of the collector surface (\(0.85\)), \(\sigma\) is the Stefan-Boltzmann constant (\(5.67\times10^{-8} \mathrm{~W/m^2K^4}\)), and \(T_{sky}\) is the effective sky temperature (\(-15^{\circ} \mathrm{C}\)). First, we need to convert all the temperatures to Kelvin scale. \(T_s = 42 +273.15 = 315.15 \mathrm{~K}\), \(T_{sky} = -15 +273.15 = 258.15 \mathrm{~K}\) Now, we can determine the rate of heat loss due to radiation.

Calculate the radiation heat loss.

Using the formula for \(Q_r\) and our calculated values: \(Q_r= \varepsilon \sigma A(T_s^4-T_{sky}^4) = 0.85 \cdot 5.67\times10^{-8} \cdot (1.5\cdot4.5)(315.15^4-258.15^4) = 1762 \mathrm{~W}\) Therefore, the rate of heat loss from the collector due to radiation is 1762 W. In conclusion, the total heat loss from the collector during a calm day is: Natural convection heat loss: \(1314 \mathrm{~W}\) Radiation heat loss: \(1762 \mathrm{~W}\)