an electric hot water heater takes in cold water at 13.0 degrees celsius and delivers hot water. the hot water has a constant temperature of 45 degrees celsius when the "hot" faucet is left open all the times, and the volume flow rate is 5.0 x 10 ^-6 m^3/s. what is the minimum power rating of the hot water heater
To find the minimum power rating of the hot water heater, we need to determine the amount of heat energy required to raise the temperature of the water from 13.0 degrees Celsius to 45 degrees Celsius, and then calculate the power using the formula:
Power = Energy / Time
Step 1: Calculate the energy required to raise the temperature of the water
The energy required to raise the temperature of a substance can be calculated using the formula:
Energy = mass * specific heat capacity * temperature change
In this case, we are given the flow rate of the water (volume per unit time), so we need to determine the mass of water that passes through the heater per unit time.
Step 2: Determine the mass of water passing through the heater per unit time
To calculate the mass of water, we can multiply the flow rate of the water by its density.
Density of water = 1000 kg/m^3 (at approximately room temperature and atmospheric pressure)
Mass flow rate = Volume flow rate * Density
Step 3: Calculate the energy required to raise the temperature of the water
Using the formula mentioned earlier:
Energy = mass * specific heat capacity * temperature change
The specific heat capacity of water is approximately 4186 J/(kg * °C).
Step 4: Calculate the power
Finally, we can use the formula:
Power = Energy / Time
Given the volume flow rate is 5.0 x 10^-6 m^3/s, we have the time as 1 second for this calculation.
Let's calculate the minimum power rating of the hot water heater:
Step 1: Calculate the mass flow rate:
Mass flow rate = (5.0 x 10^-6 m^3/s) * (1000 kg/m^3) = 0.005 kg/s
Step 2: Calculate the energy required to raise the temperature of the water:
Temperature change = 45°C - 13.0°C = 32.0°C
Energy = (0.005 kg/s) * (4186 J/(kg * °C)) * (32.0°C) = 670.4 J/s
Step 3: Calculate the power:
Power = Energy / Time = 670.4 J/s
Therefore, the minimum power rating of the hot water heater is 670.4 Watts.