You own a swimming pool that must be heated to Tpool = 80°F using the system shown below.
The system uses both a conventional, natural gas-fired boiler as well as a low-temperature, unglazed solar collector. On a typical day, the outdoor air temperature is To = 70°F and the solar flux is SF = 850 W/m^2. The total heat loss from the pool under these conditions is Qpool = 6.5 kW. The collector has a total surface area Acol = 4 ft x 8 ft. A pump pulls water out of the pool and pumps it through the solar collector where it is heated to Tc, out and returned to the pool. The collector absorbs all of the incident solar radiation and transfers all of it to the water passing through it. The pump runs for 8 hours each day.
The pressure drop associated with flow through the collector has been characterized by the manufacturer and is given by the equation: ΔPcol = 5 x 10^12V^2 where V is the volumetric flow rate of water through the collector (in m^3/s) and ΔPcol is the pressure drop across the collector (in Pa). The dead-head pressure rise produced by the pump (i.e., the pressure rise with no flow) is ΔPdh, pump = 150 x 10^3 Pa. The open-circuit volumetric flow rate produced by the pump (i.e., the flow rate produced with no pressure rise) is Voc, pump = 5 x 10^-4 (m^3/s). You may assume that the pump curve is linear between ΔPdh, pump at zero flow and Voc, pump at zero pressure rise. That is, the pump pressure rise as a function of flow rate is given by: ΔPpump = ΔPdh, pump (1-((V)/(Voc, pump))). where ΔPpump is the pressure rise produced by the pump and V is the volumetric flow rate produced by the pump. The pump efficiency is relatively constant and equal to ηp = 0.42. The cost of the electricity required to run the pump is ec = 0.12 $/kW-hr. The portion of the total pool heating load that is not met by the solar collector must be met using a natural gas-fired furnace (Qfurnace). The cost of natural gas is ngc = 1.85 $/therm. You may model the pool water as an incompressible substance with vw = 0.001 m^3/kg and cw = 4200 J/kg-K.
a.) Determine the volumetric flow rate of water pumped through the collector, V (gpm). You will need to do this by determining where the pump curve and the system resistance curve (for the collector) intersect.
b.) Determine the pump power required, Wp (W).
c.) Determine the temperature of the water leaving the pump, Tp, out (°F).
d.) Determine the temperature of the water leaving the solar collector, Tc, out (°F).
e.) What is the rate at which energy is provided to the pool from the solar collector system (kW)?
f.) What is the cost ($/day) associated with running the pump?
g.) What is the savings ($/day) associated with avoided natural gas cost?
