Energy Calculations for Water Heating

Water heating is just that – the energy required to heat water.

The basic equation for the energy required to heat water is

\(Q = m c \Delta T \)

where \( Q \) is the energy required (J), \( m \) is the mass of water to be heated (kg), \( c \) is the specific heat of water (J.kg^-1.K^-1), and \( \Delta T \) is the temperature rise (K or °C). For water, \( c \) = 4184 J kg^-1 K^-1.

Simplifying the Equation

While the above equation works fine, the amount of water is normally measured by volume e.g. in litres, and electrical energy is sold in kilowatt-hours. The density of water is about 1 kg L^-1 i.e. the mass of water may be taken as being equal to the volume in litres. It is also noted that 1 kWh = 3.6 MJ.

The equation for heat can be simplified to

\( Q = \frac{4184}{3600000} V  \Delta T \)

where \( Q \) is the energy required in kWh, \( V \) is the volume of water in litres, and \( c \) is the specific heat per litre of water. Because 1 L = 1 kg for water, \( c \) = 4184 in this equation too.

Making those substitutions, we get

\( Q = 0.00116222 V  \Delta T \)

This is the amount of energy (in kWh) required to heat a volume \( V \) of water (in liters) a given temperature rise (\( \Delta T \)).

Note that the amount of electrical energy required may be more - the amount of electrical energy required depends on the efficiency of the water heater.

Temperature Rise - \( \Delta T \)

The temperature rise \( \Delta T \) is simply the difference between the initial and final temperatures. For example, if your initial temperature is 20°C, and your final temperature is 65°C, your temperature rise is 45°C.

Effect of Efficiency

The equation for \( Q \) is only the amount of energy required to heat the water itself. A hot water cylinder uses electrical energy to do this. While heaters are nearly always 100% efficient, there are still energy losses through heat losses to the surroundings. The electrical energy required is given by:

\( E = \frac{Q}{\eta} \)

where \( E \) is the electrical energy (kWh), \( Q \) is the energy to heat the water (kWh), and \( \eta \) is the efficiency of the hot water heater. Note that in this equation \( \eta \) is in the range of 0-1, 0 being 0% efficient and 1 being 100% efficient.

Cost of Running

The cost of running is given by

\( C = r E \)

where \( C \) is the total cost (c), \( r \) is the tariff (c.kWh^-1), and \( E \) is the energy used (kWh).