Maths for Solar Thermal

Maths for Solar Thermal

Solar thermal maths answers one simple question:

How much hot water heat can a solar thermal system produce from sunlight, and how do we estimate it through the year?

Solar thermal systems don’t make electricity. Instead, they collect heat and move it into a hot water tank. So the maths focuses on heat energy (kWh), collector area, sunlight, and losses.

Solar thermal

1. Power vs Energy (Same Idea as Solar PV)

Two units matter most:

  • kW (kilowatts) = heat power (how fast heat flows)
  • kWh (kilowatt-hours) = heat energy (how much heat you collect over time)

Key rule:

Energy (kWh) = Power (kW) × Time (hours)

Example:

  • If your system delivers 1.5 kW of heat for 2 hours:
Energy = 1.5 × 2 = 3 kWh

2. Sunlight on a Collector (The Input)

Solar thermal collectors rely on solar irradiance:

  • Irradiance (G) measured in W/m²
  • Collector size measured in m²

A simple “ideal heat in” calculation looks like:

Sun power hitting collector (W) = G (W/m²) × Area (m²)

Worked example

If sunlight is 600 W/m² and the collector area is 4 m²:

Power in = 600 × 4 = 2,400 W = 2.4 kW

That’s the sunlight power landing on the collector surface.

3. Collector Efficiency (Turning Sunlight into Useful Heat)

Collectors don’t turn all that sunlight into usable heat. They lose some to:

  • reflection from glass
  • heat leaking to outdoor air
  • piping heat loss

So we use efficiency (η):

Useful heat power (kW) = G × Area × η

Worked example

Using the same values:

  • G = 600 W/m²
  • Area = 4 m²
  • η = 50% (0.50)
Useful power = 600 × 4 × 0.50
Useful power = 1,200 W = 1.2 kW

So under those conditions, the system delivers about 1.2 kW of heat to the fluid.

Efficiency changes with:

  • sunlight level
  • outdoor temperature
  • how hot the tank already is

Still, this equation gives a very useful first estimate.

4. Turning Heat into Daily Energy (kWh per day)

Now use the power-to-energy rule:

If the system averages 1.2 kW for 4 hours of good collection time:

Energy per day = 1.2 × 4 = 4.8 kWh/day

That daily heat goes into your hot water tank.

5. How Much Water Can That Heat?

This is the most practical solar thermal calculation:

How much hot water can I make?

Water needs about 4.2 kJ to raise 1 kg by 1°C. A useful shortcut is:

To heat 1 litre of water by 1°C takes about 0.00116 kWh.

So:

Heat (kWh) ≈ Volume (litres) × Temperature rise (°C) × 0.00116

Worked example: heating a tank

Say you want to heat 150 litres of water by 35°C (e.g., from 15°C to 50°C):

Heat ≈ 150 × 35 × 0.00116
Heat ≈ 6.09 kWh

So you need about 6.1 kWh of heat to do that.

Now compare that to your solar thermal collection:

  • If you collected 4.8 kWh in a day, you would heat most of that tank, but not all.
  • On a sunnier day, you might cover it fully.

6. Solar Fraction (How Much You Cover)

Solar thermal systems usually supply a share of your hot water, not all of it year-round.

We can calculate the solar fraction:

Solar fraction (%) = (Solar heat delivered ÷ Total hot water heat needed) × 100

Worked example

If a home needs 2,000 kWh/year for hot water and solar thermal provides 1,000 kWh/year:

Solar fraction = (1,000 ÷ 2,000) × 100 = 50%

So solar thermal covers half of the annual hot water energy.

7. Annual Output: A Simple Estimate

A common way to estimate yearly solar thermal energy is:

Annual heat (kWh) ≈ Collector area (m²) × Annual solar energy on collector (kWh/m²) × System efficiency factor

Even without detailed weather data, the structure is clear:

  • bigger area → more heat
  • sunnier location → more heat
  • higher losses or shading → less heat

That’s why two similar systems can perform very differently on different roofs.

8. Why Output Changes Through the Year

Solar thermal output changes because:

  • Winter has fewer daylight hours and weaker sun
  • Collectors lose more heat to cold air
  • Tanks start colder, so the system “works harder”
  • Summer can produce more heat than you need (so the system may stop collecting)

In other words:

Summer often gives lots of heat. Winter gives less. Spring and autumn sit in between.

9. That’s All the Maths You Need

Most solar thermal calculations use just:

  1. Sunlight × area to find heat input
  2. Efficiency to estimate useful heat
  3. kW × hours = kWh to get energy
  4. Water heating formula to link kWh to litres and temperatures
  5. Solar fraction to see how much hot water you cover

No advanced maths required — just multiplication, division, and a couple of helpful constants.