Thermodynamics swimming pool question

[stevex23]

Homework Statement

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 rate of
heat loss from the water to the air at the outdoor design conditions
is determined to be 960 W/m2, and the heater must also
be able to maintain the pool at 30°C at those conditions. 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 to your client?

Homework Equations

The Attempt at a Solution

 

[jack action]

E_water = m * c_p * T

P = E / t

P_needed = P_heated + P_losses

 

[stevex23]

Do i calculate the Energy at each temperature then take the difference?

 

[jack action]

Yes.

 

[DeeNos]

I would first calculate the volume of the swimming pool using the given dimensions (2 m x 25 m x 25 m) to be 1,250 m3. Next, I would use the specific heat capacity of water (4.186 J/g°C) to calculate the amount of energy needed to raise the temperature of the pool from 20°C to 30°C, which is approximately 1,250,000,000 Joules.

To determine the heater size, I would use the equation Q = mcΔT, where Q is the amount of energy needed, m is the mass of water (1,250,000 kg), c is the specific heat capacity of water, and ΔT is the change in temperature (10°C). This gives me a required energy input of 5,232,500,000 Joules.

Since the heater considered is a natural gas furnace with an efficiency of 80%, I would divide the required energy input by 0.80 to account for the efficiency, giving me a final energy input of 6,540,625,000 Joules.

To convert this to Btu/h, I would use the conversion factor of 1 Btu = 1055.06 Joules and divide the energy input by 3 hours to get the heater size of approximately 2,074,238 Btu/h. This would be the recommended heater size to meet the client's requirements.