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 \(4 \mathrm{lbm} / \mathrm{s}\). Water flows through a 1.25 -indiameter thin aluminum tube whose outer surface is blackanodized 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 \(400 \mathrm{Btu} / \mathrm{h}\) per ft length of the tube, determine the required length of the parabolic collector to meet the hotwater requirements of this house.

Short answer

Answer: The required length of the parabolic collector is approximately 4500 ft.

Step-by-step solution

Calculate the energy needed to heat the water

First, we need to calculate the energy required to heat 1 lb of water from 55°F to 180°F. To do this, we will use the specific heat capacity of water (c) which is approximately 1 Btu/(lbm °F). Energy to heat 1 lb of water (ΔQ) = m × c × ΔT where: ΔQ = Energy (Btu) m = mass of water (lbm) c = specific heat capacity of water (1 Btu/(lbm °F)) ΔT = Change in temperature (°F) ΔT = 180 - 55 = 125 °F ΔQ = 1 × 1 × 125 = 125 Btu

Calculate the energy absorbed per ft length of the collector

We are given the solar energy transferred to water at a net rate of 400 Btu/h per ft length of the tube. To find the energy absorbed per second, we need to convert the units. Energy absorbed per ft length = (400 Btu/h) × (1 h/3600 s) = 400/3600 Btu/s Energy absorbed per ft length ≈ 0.1111 Btu/s

Calculate the energy needed to heat the water at 4 lbm/s

The mass flow rate (m_dot) of the water is given as 4 lbm/s. We can calculate the energy needed to heat the water at this mass flow rate using the following formula: Total energy needed (Q_total) = m_dot × ΔQ Q_total = 4 lbm/s × 125 Btu/lbm = 500 Btu/s

Determine the required length of the parabolic collector

We will now calculate the required length of the parabolic collector by dividing the total energy needed by the energy absorbed per ft length of the collector. Required length (L) = Q_total / (Energy absorbed per ft length) L = 500 Btu/s / 0.1111 Btu/s L ≈ 4500 ft Hence, the required length of the parabolic collector to meet the hot water requirements of this house is approximately 4500 ft.