Some pool owners use a heat pump to warm their pool water. A certain pool contai

Question

Some pool owners use a heat pump to warm their pool water. A certain pool contains 20,000 gallons of water and is at an initial temperature of 10.0 C. If the temperature of the water is to be increased to 14.0 C in 48.0 hours, what must be the continuous power input to the heat pump? Assume that all of the heat output from the heat pump is used to heat the water, that no heat is lost to the surroundings, and that the heat pump has a COP of 2.54 (NOTE: ANSWER: 2.89kW) NEED HELP figuring out how to get 2.89kW.

Explanation / Answer

First, use the equation of heat: Q = m*cp*(Tf-Ti)

Now, let's calculate first the mass of water in grams (assuming a density of 1 g/mL):
m = 20000 gal * 3.7854 L/gal * 1 1kg/L * 1000 g/kg = 75708000 g

Now the heat, assuming a cp for water of 4.184 J/g°C
Q = 75708000 * 4.184 * (14-10) = 1,267,049,088 J

This value between it's COP: 1,267,049,088 / 2.54 = 498,838,223.6 J

Finally, divide this by the time to get the power input:
P = 498,838,223.6 / (48*3600)
P = 2886.795 J/s

1 W is 1 J/s so, to get the kW:
P = 2886.795 W / 1000 = 2.887 kW or 2.89 kW

Hope this helps

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