Question

The hot water needs of a household are to be met by heating water at \(55^{\circ} \mathrm{F}\) to \(180^{\circ} \mathrm{F}\) by a parabolic solar collector at a rate of \(5 \mathrm{lbm} / \mathrm{s}\). Water flows through a \(1.25\)-in-diameter thin aluminum tube whose outer surface is anodized black in order to maximize its solar absorption ability. The centerline of the tube coincides with the focal line of the collector, and a glass sleeve is placed outside the tube to minimize the heat losses. If solar energy is transferred to water at a net rate of \(350 \mathrm{Btu} / \mathrm{h}\) per \(\mathrm{ft}\) length of the tube, determine the required length of the parabolic collector to meet the hot water requirements of this house. Also, determine the surface temperature of the tube at the exit.

Short answer

The required length of the parabolic solar collector is 6,430 feet, and the surface temperature of the tube at the exit is approximately 180°F.

Step-by-step solution

Calculate the energy needed to heat the water

First, we need to find the energy required to heat the water from its initial temperature of 55°F to the desired temperature of 180°F. For this, we can use the formula: Energy = mass flow rate × specific heat × (Final temperature - Initial temperature) The specific heat of water can be approximated to \(1 \thinspace Btu / (lbm \thinspace °F)\). Energy = \((5 \thinspace lbm/s) × (1 \thinspace Btu/(lbm \thinspace °F)) × (180 °F - 55 °F)\) Energy = \((5 \thinspace lbm/s) × (1 \thinspace Btu/(lbm \thinspace °F)) × (125 °F)\) Energy = \(625 \thinspace Btu/s\)

Calculate the energy that the solar collector can provide

We are given that the solar collector can provide 350 Btu/h per foot of the tube. First, we need to convert the energy rate from Btu/h to Btu/s: Energy_rate = \(350 \thinspace Btu/h \times (1 \thinspace h / 3600 \thinspace s)\) Energy_rate = \(0.0972 \thinspace Btu/s \thinspace ft\)

Determine the required length of the collector

Now, we need to find the length of the collector required to provide the needed energy to heat the water. We can use the formula: Length = Energy / Energy_rate Length = \(625 \thinspace Btu/s / 0.0972 \thinspace Btu/s \thinspace ft\) Length = \(6430 \thinspace ft\) So, the required length of the parabolic collector to meet the hot water requirements of the house is 6,430 feet.

Calculate the surface temperature of the tube at the exit

To calculate the surface temperature of the tube at the exit, we need to assume that the temperature difference between the tube surface and the water inside is negligible. This assumption is valid because of the good thermal conductivity of aluminum and the negligible heat losses due to the glass sleeve. Thus, we can assume: Exit surface temperature of the tube ≈ the final temperature of the water The final temperature of the water is 180°F; therefore, the surface temperature of the tube at the exit is also approximately 180°F. In conclusion, the required length of the parabolic collector is 6,430 feet, and the surface temperature of the tube at the exit is approximately 180°F.