You are asked to design a heating system for a swimming pool

You are asked to design a heating system for a swimming
pool that is 2 m deep, 25 m long, and 25 m wide. Your
client desires that the heating system be large enough to raise
the water temperature from 20°C to 30°C in 3 h. The heater
must also be able to maintain the pool at 30°C at the outdoor
design conditions of 15°C, 1 atm, 35 percent relative humidity,
40 mph winds, and effective sky temperature of 10°C.
Heat losses to the ground are expected to be small and can
be disregarded. The heater considered is a natural gas furnace
whose efficiency is 80 percent. What heater size (in Btu/h
input) would you recommend that your client buy?

Solution

The heater size is found using the amount of heat required to power the swimming pool under relative humidity condition,

total heat transfer, dot q = convective heat transfer of water + forced heat transfer of air.

dot q = h_water * A * dT + dot m * Cp * dT ----------- eqn 1

h_water - 50 W / m2K

Cp of mixed gas - Cp of water vapor + Cp of dry air

Cp of dry air at 288 K - 1.047 kJ / kg K

Cp of water vapor at 288K - 1.864 kJ / kg K

The mass flow rate of air, dot m_{mixed gas} = rho_air* A * V + dot m_water

The humidity ratio of the mixed air is 6 (approx) at 15 deg C and 35% RH.

dot m_water = 0.006 * dot m_air = 0.006 * 1.225 * 1 * 40 * 0.44 = 0.129 kg /s

The total mass flow rate is dot m_{mixed gas} = 1.225 * 1 * 40 * 0.44 + 6.468 = 21. 68 kg / s

Now the total heat transfer, dot q is found using the above eqn 1

dot q = h_water * A * dT + dot m_{mixed gas} * Cp * dT

dot q = 50 * 25 * 25 * (300 - 290) + 21.68 * 3 * 3600 * (1.047+ 1.864) * (300 - 285)

dot q = 312500 + 10209849.12 = 10531349.12 W or 10.53 MW or 35.14 * 10^3 BTU / hr.

The size of the heater is 35.14*10^3 BTU/hr to heat the poole size of 652 m2.