The recovery time of a hot water heater is the time required to heat all the water in the unit to the desired temperature. Suppose that a 62-gal (1.00 gal = 3.79 x 10-3 m3) unit starts with cold water at 11 °C and delivers hot water at 53 °C. The unit is electric and utilizes a resistance heater (120 V ac, 3.4 Ù) to heat the water. Assuming that no heat is lost to the environment, determine the recovery time (in hours) of the unit.
To determine the recovery time of the unit, we need to find the amount of heat energy required to heat the water from the initial temperature to the desired temperature.
First, let's convert the volume of the water from gallons to cubic meters:
62 gal * (3.79 x 10^-3 m^3/gal) = 0.23498 m^3
Next, we need to find the mass of water in the unit. We can use the density of water, which is approximately 1000 kg/m^3:
Mass = Density * Volume
Mass = 1000 kg/m^3 * 0.23498 m^3 = 234.98 kg
Now, let's calculate the change in temperature:
ΔT = Final temperature - Initial temperature
ΔT = 53 °C - 11 °C = 42 °C
To find the amount of heat energy required, we can use the specific heat capacity of water, which is approximately 4.18 kJ/kg°C:
Heat energy = Mass * specific heat capacity * ΔT
Heat energy = 234.98 kg * 4.18 kJ/kg°C * 42 °C
Heat energy = 41463.885 kJ
Next, let's calculate the power of the electric heater. We can use Ohm's law:
Power = Voltage^2 / Resistance
Power = (120 V)^2 / 3.4 Ω
Power = 4800 W
Now, we can determine the time required to transfer the heat energy using the equation:
Time = Heat energy / Power
Time = 41463.885 kJ / 4800 W
Now, let's convert the units:
1 kJ = 1000 J, 1 W = 1 J/s
Time = (41463.885 kJ * 1000) / (4800 J/s)
Time = 8.63580708 hours
Therefore, the recovery time of the unit is approximately 8.64 hours.
To determine the recovery time of the unit, we need to calculate the amount of heat required to raise the temperature of the water from 11 °C to 53 °C and then convert it into the time required.
Step 1: Calculate the mass of water in the unit
Given: 1.00 gal = 3.79 x 10^(-3) m^3
Volume of water in the unit = 62 gal = 62 * 3.79 x 10^(-3) m^3
Density of water = 1000 kg/m^3 (approximately)
Mass of water = density * volume
Mass of water = 1000 kg/m^3 * 62 * 3.79 x 10^(-3) m^3
Step 2: Calculate the amount of heat required
Specific heat capacity of water (c) = 4.18 kJ/kg°C (approximate)
Heat required = mass of water * specific heat capacity * change in temperature
Heat required = (1000 kg/m^3 * 62 * 3.79 x 10^(-3) m^3) * 4.18 kJ/kg°C * (53 °C - 11 °C)
Step 3: Convert the heat required into TeraJoules (10^12 Joules)
Heat required = (1000 kg/m^3 * 62 * 3.79 x 10^(-3) m^3) * 4.18 kJ/kg°C * (53 °C - 11 °C) * 10^9 J/kJ
Step 4: Calculate the power (P) consumed by the resistance heater
Given: Voltage (V) = 120 V, Resistance (R) = 3.4 Ω
Power (P) = V^2 / R
Power (P) = (120 V)^2 / 3.4 Ω
Step 5: Calculate the time required
Time (t) = Heat required / Power
Time (t) = ((1000 kg/m^3 * 62 * 3.79 x 10^(-3) m^3) * 4.18 kJ/kg°C * (53 °C - 11 °C) * 10^9 J/kJ) / ((120 V)^2 / 3.4 Ω)
Finally, you can simplify and calculate the value of "t" to find the recovery time in hours.