Question

A 60 -gallon water heater is initially filled with water at $50^{\circ} \mathrm{F}$. Determine how much energy (in Btu) needs to be transferred to the water to raise its temperature to \(120^{\circ} \mathrm{F}\). Evaluate the water properties at an average water temperature of \(85^{\circ} \mathrm{F}\).

Short answer

Answer: The energy required is approximately 33662 Btu.

Step-by-step solution

Calculate the mass of water

The volume of the water is given as 60 gallons. To convert this to mass, we need to know the density of water. At an average temperature of 85°F, the density of water is approximately 0.960 kg/L. We can use this to find the mass of the water. 60 gallons = 60 * 3.78541 L = 227.1246 L Now, we can find the mass of water by multiplying the volume by the density: Mass = Volume x Density Mass = 227.1246 L * 0.960 kg/L = 217.8396 kg

Calculate the change in temperature

The initial temperature is given as 50°F, and the final temperature is 120°F. To find the change in temperature, we need to subtract the initial temperature from the final temperature: ΔT = T_final - T_initial ΔT = 120°F - 50°F = 70°F

Find the specific heat of water at 85°F

The specific heat of water is not constant and varies with temperature. At an average temperature of 85°F, which is approximately 29.4°C, the specific heat of water is approximately 4.182 kJ/(kg·K).

Use the specific heat equation to calculate the energy needed

The specific heat equation is given by: Q = mcΔT where Q is the energy needed, m is the mass of the water, c is the specific heat, and ΔT is the change in temperature. Using the values we found in the previous steps: Q = 217.8396 kg * 4.182 kJ/(kg·K) * 70°F To convert the change in temperature from Fahrenheit to Kelvin (or Celsius), we use the conversion factor: 1°F = 5/9 K, so 70°F = 38.9 K Q = 217.8396 kg * 4.182 kJ/(kg·K) * 38.9 K ≈ 35514 kJ Since 1 Btu is approximately equal to 1.055 kJ, we need to convert the energy in kJ to Btu: Energy = 35514 kJ / 1.055 kJ/Btu ≈ 33662 Btu

Final Answer

The amount of energy needed to raise the temperature of 60 gallons of water from 50°F to 120°F, evaluated at an average water temperature of 85°F, is about 33662 Btu.