The Simple Numbers Behind Heat Pumps

Air source heat pump maths really comes down to three simple questions:

Maths for Air Source Heat Pumps

The Simple Numbers Behind Heat Pumps

Air source heat pump maths really comes down to three simple questions:

How much heat will the heat pump give?
How much electricity will it use?
How much will it cost to run?

At first, this may sound technical. However, the maths is much simpler than many people expect.

In fact, you do not need advanced maths or physics. Instead, most heat pump calculations use simple multiplication, division, and temperature differences.

This guide explains the key numbers step by step.

1. Start with the Basic Units

Before looking at heat pumps, it helps to understand two basic units.

kW (Kilowatts)

kW measures power.

In simple terms, power tells you how quickly heat or electricity is being delivered.

kWh (Kilowatt-hours)

kWh measures energy used over time.

This tells you how much energy you actually used.

Simple Example

Imagine a 2 kW heater running for 3 hours.

The energy used would be:

Energy = Power × Time
2 × 3 = 6 kWh

So, the key idea is:

kW × hours = kWh

You will see this idea again and again.

2. The Most Important Number: COP

The main heat pump number is COP, which stands for Coefficient of Performance.

COP tells you how much heat you get for each unit of electricity you put in.

Formula

COP = Heat Output ÷ Electricity Input

Example

Suppose a heat pump gives 6 kW of heat while using 2 kW of electricity.

The COP would be:

6 ÷ 2 = 3

So, for every 1 unit of electricity, you get 3 units of heat.

Therefore, a higher COP means better efficiency.

3. SCOP: Efficiency Across the Year

COP only shows efficiency at one moment.

However, outdoor temperatures change throughout the year. Because of this, heat pump efficiency also changes.

That is why we use SCOP, or Seasonal Coefficient of Performance.

SCOP shows average efficiency over a whole heating season or year.

Formula

SCOP = Total Heat Output ÷ Total Electricity Used

Example

Suppose your home needs 12,000 kWh of heat in a year, and the heat pump uses 4,000 kWh of electricity.

The SCOP would be:

12,000 ÷ 4,000 = 3

This means the system gives 3 units of heat for every 1 unit of electricity, on average across the year.

4. Why Efficiency Falls in Cold Weather

Heat pumps still work in cold weather.

However, when outdoor temperatures fall, efficiency usually drops.

This happens because the heat pump has to work harder to pull heat from colder air.

Typical COP Examples

Outdoor Temperature	Typical COP
+7°C	Around 4–5
0°C	Around 3
-7°C	Around 2–3

The main point is simple:

Colder weather usually means lower efficiency.

Even so, heat pumps still work below freezing.

They simply use more electricity to produce the same amount of heat.

5. Heat Loss Drives Everything

A heat pump must replace the heat your home loses.

So, the first thing to understand is heat loss.

There are two main ways homes lose heat.

Fabric Heat Loss

This is heat escaping through:

Walls
Roof
Floors
Windows

Ventilation Heat Loss

This happens when warm indoor air leaves and colder air comes in.

Together, these tell you how much heating your home needs.

6. A Simple Heat Loss Example

Let’s use a simple example.

Suppose:

Indoor temperature = 21°C
Outdoor temperature = −3°C

That gives a temperature difference of:

21 − (−3) = 24°C

Now imagine the home loses:

1.62 kW through walls, roof, floor, and windows
0.80 kW through ventilation

Total heat loss:

1.62 + 0.80 = 2.42 kW

So, on a very cold day, this home needs about 2.4 kW of heat to stay warm.

7. Choosing Heat Pump Size

Once you know heat loss, choosing size becomes much easier.

A simple rule is:

Heat pump output should roughly match heat loss in cold weather

So, if a home loses 2.4 kW, you might look at a heat pump around 3 kW.

In practice, installers also allow for:

Hot water needs
Defrost cycles
Safety margins

8. Running Cost Maths

This is usually the number people care about most.

Once you know heat demand and SCOP, estimating running cost is quite simple.

Step 1: Estimate Electricity Use

Formula:

Electricity Used = Heat Needed ÷ SCOP

Example:

Heat needed = 12,000 kWh
SCOP = 3

Electricity used:

12,000 ÷ 3 = 4,000 kWh

Step 2: Work Out Cost

Formula:

Cost = Electricity × Unit Price

If electricity costs 25p per kWh:

4,000 × £0.25 = £1,000 per year

So, this gives you a simple estimate of yearly running cost.

9. Why Flow Temperature Matters

Heat pumps work best when they heat water to lower temperatures.

Because of this, they often work well with:

Larger radiators
Underfloor heating
Better insulation

A simple rule is:

Lower flow temperature = Better efficiency

Higher flow temperature = Lower efficiency

This is why system design matters so much.

10. A Quick Reality Check

Suppose your home needs 6 kW of heat on a cold day.

If COP = 3

Electricity needed:

6 ÷ 3 = 2 kW

If COP = 2

Electricity needed:

6 ÷ 2 = 3 kW

The heat demand stays the same.

However, electricity use rises by 50% when COP falls from 3 to 2.

That shows why outdoor temperature and system setup matter so much.

The Five Numbers That Matter Most

Almost every heat pump calculation comes back to five key numbers:

Heat loss (kW)
COP or SCOP
Heat demand (kWh)
Electricity use (kWh)
Running cost (£)

That is really all you need.

Once you understand these numbers, heat pump maths becomes much easier.