The reasons for this post is that I have seen several comments about whether the heat from power plants contributes to global warming.

The answer is that it does, but not significantly compared with the effect of greenhouse gas emissions.

Is The Earth A Sphere Or A Circle?

The mathematics get a bit complicated if we think of the Earth as a sphere – different amounts hit the poles compared to the equator etc.

What we want to do is to work out the total energy that we get from the Sun. This is the amount of irradiance that passes through a circle the same radius as the Earth (6371Km). This makes the mathematics a lot easier.

However, at the poles, this irradiance is spread over a larger (red) area. That is why it is warmer at the equator than at the poles.

The reverse of this is seen when we look at the moon. The amount of light hitting the ‘poles’ of the moon is a lot less than that hitting the ‘equator’. However, since we are looking at less light being reflected from a larger area the two cancel out and the brightness of the moon looks uniform.

Amount Of Energy Emitted By The Sun

This can be calculated by the Stefan-Boltzman law, which states that energy flux density (the energy radiated per unit surface energy per unit time) is:

The total power of the sun is, therefore:

where is the Stefan-Boltzman constant (5.67×10^-8Js^-1m-2K⁴)

Energy Reaching The Earth

This energy is spread over a sphere with a radius equal to the distance between the Earth and the Sun: . Where x is the distance between the Earth and the Sun.

However, only a small proportion of this goes through our circle. This is the area of the circle.

Therefore, the energy from the sun going through our circle is:

The temperature of the Sun (T_sun) is about 5770K

The radius of the Sun (r_sun)is 6.98×10⁸m

The distance between the Earth and the Sun (x) is 1.50×10¹¹m (about 150 million km – 93 million miles).

This gives a figure of 6.98×10¹⁷ Watts.

If we divide this by the area of the Earth (the big circle in the top diagram)  it gives a figure of 1361 W/m². This is about the same as the measured value.

It is important to note that this is the power per unit area going through our two-dimensional circle, not that hitting our three-dimensional Earth. Since the Earth is curved and for half the time it is night, the average is about a quarter of this.

World Energy Production

World energy production of 160,000TWh¹ which is 5.76×10²⁰J per year which equates to 1.83×10¹³W or 0.036W/m².

This is very much lower than the 1361 W/m² we get from the Sun.

Heat Generation Vs Greenhouse Effect

What we really want to do is compare the figure for the forcing due to thermal emissions with that due to greenhouse gasses. The diagram below² show that the forcing due to greenhouse gasses is currently well over 3W/m².

Therefore, the heating due to emissions from power plants etc is several orders of magnitude lower than that due to the greenhouse effect.

With continually increasing emissions, the effect of greenhouse gas emissions is rising. The IPCC represent this by giving Representative Concentration Pathways (RCPs).

Various pathways are given with the number after giving the amount of forcing that would be expected by 2100. For example, RCP8.5 would result in 8.5W/m² extra forcing  and RCP4.5 would be 4.5W/m² forcing by 2100.

These are significantly higher than the 0.036W/m² we might expect from thermal emissions.

Not All Energy Sources Are Equal

There are two types of energy sources.

The first type is where that energy would have heated the Earth anyway, but we just capture it to make use of it before. This includes: solar, wind, wave and tidal.

The second are sources that would not have heated the Earth if it had not been for human activity. These include fossil fuels and nuclear.

Local Effects

Although the average heat emitted is small, it can have significant effects locally. For example, a large power plant can cause significant heating if it discharges into a sea or river.

For example, during the heatwave in 2018 France had to shut down some nuclear reactors because they overheated rivers³.

The effects of local heating have not yet been fully integrated (if at all) into various climate models⁴.

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¹ Our World In Data (https://ourworldindata.org/energy-production-consumption)

² Climate Change Indicators: Climate Forcing, United States Environmental Protection Agency (https://www.epa.gov/climate-indicators/climate-change-indicators-climate-forcing)

³ Heatwave forces France to shut down four nuclear reactors, The Independent, 06 August 2018 (https://www.independent.co.uk/news/world/europe/france-nuclear-reactors-shut-down-edf-europe-heat-wave-a8477776.html)

⁴ Integrating anthropogenic heat flux with global climate models, Flanner, M. G. (2009), Geophys. Res. Lett., 36, L02801, doi:10.1029/2008GL036465. (https://agupubs.onlinelibrary.wiley.com/action/showCitFormats?doi=10.1029%2F2008GL036465)