How efficient is a heat pump?

An electric heater turns electricity into heat. Give it 1 unit of electricity, and you get 1 unit of heat.

A heat pump can do much better: from the same electricity, it can give you 3 or 4 units of heat. How is that possible, and how do we measure it? Let’s find out.

Measuring energy in kilowatt-hours

To compare heaters, we need to measure energy. Electricity bills measure it in kilowatt-hours, written kWh. One kilowatt-hour is the energy that a 1,000-watt heater uses when it runs for one hour. (Watts measure power: how fast energy is used.)

Heat is energy too, so we can measure it in kilowatt-hours as well. That lets us compare the electricity going in with the heat coming out.

What is COP?

The coefficient of performance, or COP, compares these two amounts:

$$\text{COP} = \frac{\text{heat delivered}}{\text{electricity used}}$$

Heat delivered is the heat that the heat pump puts into your home. Electricity used is the electrical energy it needs to run. Both are measured in the same unit, such as kWh, so the COP is a plain number without a unit.

Let’s try it. On a mild winter day, a heat pump delivers 3.5 kWh of heat into the house for every 1 kWh of electricity it uses:

$$\text{COP} = \frac{3.5\ \text{kWh}}{1\ \text{kWh}} = 3.5$$

An electric heater turns 1 kWh of electricity into exactly 1 kWh of heat, so its COP is 1. The heat pump gives three and a half times as much heat for the same electricity. That means a much smaller electricity bill.

Where does the extra heat come from?

Wait — 3.5 kWh of heat from only 1 kWh of electricity? Energy can never appear from nothing. This rule is called conservation of energy. So where do the other 2.5 kWh come from?

From outside! A heat pump does not make heat. It moves heat, the way a sponge carries water from one place to another. It collects heat from the outdoor air and carries it into your home.

Even cold winter air holds some heat. Inside the heat pump flows a refrigerant that is even colder than the outdoor air, so heat flows from the air into it. The electricity mostly runs the compressor, a pump that squeezes the refrigerant so it can deliver that heat indoors. You can follow the whole trip in how heat pumps work.

So the energy adds up perfectly: 1 kWh of electricity + 2.5 kWh of heat from outside = 3.5 kWh of heat for your home. No energy is created. It is only moved and added together.

Why does the COP change?

Heat flows by itself only from warm to cold, like water flowing downhill. A heat pump makes heat go the other way: from the cold outdoors into your warm home. That is like carrying water uphill.

So the COP falls when the hill gets higher:

• It gets colder outside. The heat pump must collect heat from colder air.
• The home needs hotter water. Most heat pumps heat water that flows through radiators or through pipes under the floor. The hotter this water must be, the higher the hill.

This is why heat pumps love underfloor heating. Warm water flows through pipes under the whole floor. Because the floor is so big, water at about 30–35 °C is enough to warm the room. Small radiators often need water at 50 °C or more.

Here are typical values for a heat pump that takes heat from the outdoor air and heats water for underfloor heating:

Outdoor temperature	COP (roughly)	Heat from 1 kWh of electricity
+7 °C	about 4	about 4 kWh
−7 °C	about 2.5–3	about 2.5–3 kWh
Any weather (electric heater)	1	1 kWh

Even on a frosty day, the heat pump still gives two to three times as much heat as an electric heater.

How much heat from 1 kWh of electricity?

Change the weather and the kind of heating. Yellow is electricity; blue is heat gathered from the outside air.

Temperature outside

7 °C

The house is heated by

Underfloor heating (35 °C water) Radiators (55 °C water)

COP 4.1

Heat from 1 kWh 4.1 kWh

Because the COP changes with the weather, makers also give a seasonal COP (SCOP): the average over a whole heating season.

Go deeper: The Carnot limit: the best COP possible for grown-ups and the extra curious

Even a perfect heat pump, with no losses at all, has a highest possible COP. It follows from the work of the French engineer Sadi Carnot (1824), so it is called the Carnot limit:

$$\text{COP}_{\max} = \frac{T_{\text{hot}}}{T_{\text{hot}} - T_{\text{cold}}}$$

• $T_{\text{hot}}$ is the temperature at which the heat is delivered, for example the water for the floor.
• $T_{\text{cold}}$ is the temperature from which the heat is collected, for example the outdoor air.
• Both must be in kelvin (K): the temperature in °C plus 273.15. The kelvin scale starts at absolute zero (−273.15 °C), the coldest temperature possible.

Worked example: water for underfloor heating at 35 °C, outdoor air at 0 °C.

$$\begin{aligned} T_{\text{hot}} &= 35 + 273.15 = 308.15\ \text{K} \\ T_{\text{cold}} &= 0 + 273.15 = 273.15\ \text{K} \\ \text{COP}_{\max} &= \frac{308.15}{308.15 - 273.15} = \frac{308.15}{35} \approx 8.8 \end{aligned}$$

Real heat pumps reach roughly 40–55% of this limit, so about 3.5 to 4.8 here.

Now try radiators that need water at 55 °C, with the same 0 °C outside:

$$\text{COP}_{\max} = \frac{328.15}{328.15 - 273.15} = \frac{328.15}{55} \approx 6.0$$

A real machine then reaches only about 2.4 to 3.3. The formula shows why: a bigger temperature difference at the bottom of the fraction makes the COP smaller.

Real machines fall short of the limit for several reasons. The refrigerant has to be colder than the outdoor air and hotter than the water, so the real “hill” is higher than the one in the formula. Fans, pumps and friction use electricity too, and in winter the outdoor coil has to be defrosted from time to time.

Go deeper: Cooling: the EER for grown-ups and the extra curious

When an air conditioner or a heat pump cools a room, what matters is the heat it takes out of the room. That ratio is called the energy efficiency ratio (EER):

$$\text{EER} = \frac{\text{heat removed from the room}}{\text{electricity used}}$$

In Europe the EER is a plain number, like the COP, and typical air conditioners reach about 3 to 4. Its seasonal average is called the SEER. In the USA the EER is given in other units (British thermal units per hour for each watt), so the numbers there look about 3.4 times bigger.

For the same machine working between the same two temperatures, the heat it gives off equals the heat it collects plus the electricity it uses. So its heating COP is about 1 more than its cooling EER.

Go deeper: Why COP is not called an efficiency for grown-ups and the extra curious

An efficiency compares the useful energy that comes out with the energy that goes in, so it can never be more than 100%. A heat pump’s COP is above 1 because its output includes heat it did not make: heat collected from outside. That is why engineers call it a coefficient of performance, not an efficiency.

Check yourself

A heat pump uses 2 kWh of electricity and delivers 8 kWh of heat. What is its COP?

2 4 6 16

Show the answer

4 — COP = heat delivered ÷ electricity used = 8 kWh ÷ 2 kWh = 4. The other 6 kWh of heat were collected from outside.

Check yourself

Why does a heat pump’s COP fall on a very cold day?

Electricity is weaker in winter It has to lift heat from much colder air, which takes more electricity Cold air holds no heat at all

Show the answer

It has to lift heat from much colder air, which takes more electricity — The colder the outdoor air, the higher the hill the heat has to be carried up. Each kilowatt-hour of heat then needs more electricity.