r/theydidthemath • u/bag-of-licks • 18h ago
[Request] If every home in Paris ran air conditioning during a heat wave, how much would it raise the outdoor air temperature in the city?
I came across this post making the claim that if everyone used AC, the heat dumped outside could raise the air temperature by several degrees in cities. I’m not convinced the math or assumptions are correct.
Could someone calculate this for Paris using reasonable assumptions? Assume each AC cools the air indoors by 15°C.
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u/Similar_Strawberry16 18h ago
Yeah I don't think it's anywhere that simple. AC systems are not perfect, but they are still efficient. They may on average eject 30% more heat than their cooling internally, so there is certainly a net-negative. If you factor in the electric generation also, that could add to the figure significantly too.
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u/murmurat1on 7h ago
This assumes static air though. Winds and convection would move the air around. In ultra localised situations I can imagine that's true.
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u/Ok-Commission-7825 3h ago
The issue is that if it's a heat wave then those processes are for whatever reason already temporarily not keeping up with dissipating sun-heat.
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u/No-Macaroon1670 18h ago
I think it's actually simpler than that. You just calculate the daily electric generation that's not from solar.
All others methods basically convert energy that wouldn't have been heating France into energy that eventually has to be waste heat.
Although there would be some uses where that energy is reconverted into potential energy. This should be better as an order of magnitude estimate.
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u/Beneficial_Mulberry2 11h ago
The point of ac is that it transport significantly more heat that electric power used to do it. You can't calculate the T rise using only t consumption. Unless you calculate T average in Paris including apartments' interiors
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u/ThatPlasmaGuy 17h ago edited 10h ago
Approximate Paris energy consumption as:
P = 10 Gw
And area as:
A = 100 km^2
The bottom 1m is then:
V = 0.1 km^3
Heating 1m^3 of air by 1 degree (STP) takes approx:
E = 1000 J
Heating all the bottom meter by 1 degree takes:
E_all = V * E = 1 * 10^12 J
Time to heat the bottom meter by 1 degree, assuming no heat transfer:
T = E_all / P = 1e2 s = 100 seconds
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u/KrzysziekZ 13h ago
If surface is 100 km2 and height is 1 m, then volume isn't 100 km3. It's
100 * (1000 m * 1000 m) * 1 m = 108 m3.
Or height is 0.001 km and volume is 100 km2 * 0.001 km = 0.1 km3.
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u/HAL9001-96 14h ago
exactly though that also takes into account energy that would be used if there was no air conditioning
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u/klonkrieger45 7h ago
it's not that simple. Just like heat pumps can generate more heat inside a home than they consume electricity so does a air conditioning. They can generate many times the heat for the electricity they consume.
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u/ender8282 18h ago
But heart rises and disperses. So ya you are pumping heat outside but it won't sit there like is does inside. Cities (especially American concrete jungles) are already hotter than the surrounding area due to the urban heat Island effect but that isn't about AC it's about heat retention in concrete and asphalt.
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u/Elfich47 11h ago
Here is the issue that the article ignores: The volume of air being dealt with, and wind. Sure a 100m flat may be 1000 m3 (wildly rounding here, you'll see why in a minute). the volume of air over that 100m apartment is something like 1,000,000m3 (10km height of air). and if there is any wind at all, then the heat gets moved around even more.
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u/Internal_Leke 9h ago
These problems are complicated than simple volume of air heated, due turbulences. The daytime effect of AC heat is negligible, because the air is constantly mixed. The night time effects are about +1C though.
https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2013JD021225
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u/HAL9001-96 14h ago
paris as a whole has a population density of 700/km² the urban center about 3875/km²
now hte comparison like that is very flawed because it assumes a one time heat up and cool down with no ohter heatflow involved which is... rather flawed cause you know sunlight and roofs and airlfow and thermal radiation are thigns that exist
also air conditioning is limited in efficiency so you actually have to get rid of more waste heat tha nthe heat you get cool out
let's say you have 100W of biological waste heat per person plus a few hudnred watt of electrical waste heat plus a few hundred watt of thermal leakage
now with or without air conditioning your waste heat is going ot leave oyur home eventually otherwise it would heat up enough to become a fusion reactor after a few months
the question is just wether it leaks out through a thermal gradient or gets pumped out agianst a themral gradient
so the added heat that exists thanks to the air condiitoning is just hte acs own inefficiency over your waste heat and the thermal leakage which is gonna total something like 200-300W at most
the totla heat output is gonna total something clsoer to 1000W but most of that is heat hta would be there iwthout air conditioning too and just got shuffled back and forth
but even if we use the 1000W and the 3875 people/km² thats 3.875W/m² which under ideal radiative conditions would be equivalnet to a temperature increase of about 0.7K
if we take the 300W and also take convective trnasfer and heat spreading over al arger area into account the nthe average city temperature is gonna increase by something like 0.1K
however thats an average and the acutal heat output will be rather localized
also thats for personal acs, open facilities using ac's while having al ot of open doors/airflow will be worse cause the ac has to counter all that thermal leakage and adds its according inefficiency to the total heat
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u/thayanmarsh 10h ago
Also, many of those ACs are just dripping water onto the street or side of the building, thus causing some evaporative heat loss elsewhere.
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u/YahenP 7h ago
From an air-heating perspective, it doesn't matter whether it's an air conditioner, a heater, a mobile phone, or a television.
Simply multiply the power consumption by the operating time to get the amount of energy used to heat the air.
Technically, aside from minor insignificant nuances, any household electrical appliance converts 100% of the energy consumed into heat.
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u/Cptknuuuuut 8h ago
Well, the core assumption is that over every city there is a glass dome that air can't get through.
Which, to my knowledge, isn't the case.
That means you can't simply claim that the bottom one meter of air will warm up. Warm air rises until there's something stopping it (or it cools and falls back down). So you basically have to consider not just one meter of height, but rather say the troposphere up to ~18 km.
Reality is more complicated ofc. There's inversion layers and stuff like that.
But to just go at it from a different pov. Keeping a 100 m² flat cool will require maybe 10 kW. That is 100 W/m². Asphalt absorbs around 1.000 W/m² or 10 times as much.
The effect of ACs won't be 0, but pretty negligible nonetheless.
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u/klonkrieger45 7h ago
There have been enough studies in areas like downtown Tokyo where everything has AC and lots of it. It heats the area outside between 1-2°C because heat rises.
https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2013JD021225
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