Question

You are asked to design a heating system for a swimming pool that is $2 \mathrm{~m}\( deep, \)25 \mathrm{~m}\( long, and \)25 \mathrm{~m}$ wide. Your client desires that the heating system be large enough to raise the water temperature from \(20^{\circ} \mathrm{C}\) to \(30^{\circ} \mathrm{C}\) in $3 \mathrm{~h}\(. The heater must also be able to maintain the pool at \)30^{\circ} \mathrm{C}\( at the outdoor design conditions of \)15^{\circ} \mathrm{C}, 1 \mathrm{~atm}, 35\( percent relative humidity, \)40 \mathrm{mph}$ winds, and effective sky temperature of \(10^{\circ} \mathrm{C}\). 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?

Short answer

Answer: 1. Calculate the volume of the pool: Volume = Depth × Length × Width = 2 m × 25 m × 25 m = 1250 m³ 2. Calculate the mass of water in the pool: Mass = Volume × Density = 1250 m³ × 1000 kg/m³ = 1,250,000 kg 3. Calculate the heat energy required to raise the temperature of the pool: Q = mcΔT = 1,250,000 kg × 4,186 J/kg°C × 10°C = 52,325,000,000 Joules 4. Convert Joules to Btu: Q (Btu) = Q (Joules) × 0.00094781 = 52,325,000,000 Joules × 0.00094781 = 49,542,177.5 Btu 5. Calculate the heat energy required per hour: Heat Energy per Hour = 49,542,177.5 Btu / 3 hours = 16,514,059.17 Btu/h 6. Account for heater efficiency: Heater Size (Btu/h input) = Heat Energy Required per Hour / Heater Efficiency = 16,514,059.17 Btu/h ÷ 0.8 = 20,642,573.96 Btu/h input A heater with a size of approximately 20,642,574 Btu/h input is required to achieve the desired temperature increase within the given time frame and heater efficiency.

Step-by-step solution

Calculate the volume of the pool

First, we need to find the volume of the pool. The pool dimensions are given as: Depth = 2 m Length = 25 m Width = 25 m The volume of the pool can be calculated using the formula: Volume = Depth × Length × Width

Calculate the mass of water in the pool

Next, we need to calculate the mass of water in the pool. We know that the density of water is approximately 1000 kg/m³. Therefore, we can calculate the mass of water using the formula: Mass = Volume × Density

Calculate the heat energy required to raise the temperature of the pool

To calculate the energy required to raise the temperature of the pool from 20°C to 30°C, we will use the formula: Q = mcΔT Where, Q = heat energy required (Joules) m = mass of water (kg) c = specific heat capacity of water (4,186 J/kg°C) ΔT = change in temperature (°C)

Convert Joules to Btu

Now, we need to convert the heat energy required (Q) from Joules to Btu. 1 Joule = 0.00094781 Btu. Therefore, Q (Btu) = Q (Joules) × 0.00094781

Calculate the heat energy required per hour

The problem requires us to raise the temperature in 3 hours. So, we will divide the total heat energy required (in Btu) by 3 to find the energy requirement per hour.

Account for heater efficiency

Since the heater efficiency is 80%, we need to account for the energy loss as well. We'll divide the energy requirement per hour by the heater's efficiency to get the required heater size in Btu/h input. Heater Size (Btu/h input) = Heat Energy Required per Hour / Heater Efficiency Now, we will calculate the heater size step by step.