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July 21

How much heat comes out of 5000 L air being compressed to 500 L?

If I understand what I have read correctly, when 5000 L of air is compressed to 500 L (i.e. from 1 atm to 10 atm), if it's starting temperature is 273 K, it's end temperature, assuming it doesn't loose anything to the environment yet is 2730 K. How can I work out the amount of heat energy that is going to come out of the compressed air as it cools down to, say, an ambient temperature of 273 K, ideally in kWh? I figured it's about 3 kWh. 78.148.95.111 (talk) 16:01, 21 July 2022 (UTC)[reply]

The conversion factor between kelvins and joules is either the Boltzmann constant (per molecule) or the ideal gas constant (per mole). For the ideal gas constant, use the form 8.314 joules per kelvin per mole to get joules, then convert joules to kWh; being 3.6E6 J per kWh. Air is 22.4 liters per mole at 1 atm and 273K. That should be enough information for you to work out the rest via dimensional analysis. --Jayron 32 17:00, 21 July 2022 (UTC)[reply]

Thanks. Is that about 1.3 kWh then? Compressing air seems like a rubbish way to get heat! I guess that's why heat pumps are using gasses they can liquify. 78.148.95.111 (talk) 17:33, 21 July 2022 (UTC)[reply]

Jayron32 has supplied some excellent information. I can approach the problem from a different direction and hopefully throw a little extra light.

The question begins with a ten-fold reduction in volume, but then suggests this will bring about a ten-fold increase in pressure from 1 to 10 atmospheres! This suggestion implies an isothermal compression, but a reduction in volume won’t be isothermal until the compressed air has time to cool to its original temperature 273K. An isothermal compression is described by the equation:

PV

is constant.

For an adiabatic compression of an ideal gas, the equivalent relationship is:

PV^{\gamma }

is constant where

\gamma

is the heat capacity ratio. See Adiabatic process.

Assuming air is an ideal gas with a heat capacity ratio of 1.4 leads to the conclusion that following a ten-fold reduction in volume, the pressure has increased twenty-five fold! So we have 500 litres of compressed air at a pressure of 25 atmospheres.

To determine the temperature of the compressed air before it begins losing heat to the surroundings, we can use the relationship:

TV^{\gamma -1}

is constant.

Again, assuming $\gamma$ is 1.4 leads to the conclusion that following a ten-fold reduction in volume, the thermodynamic temperature has increased 2.5 fold. So the temperature has increased from 273K to 685K (or 413 degrees Celsius.)

The amount of heat that will be released as this body of gas cools from 413 to 0 Celsius, and the pressure falls from 25 atmospheres to 10 atmospheres, is the subject of a separate calculation. Dolphin (t) 08:13, 22 July 2022 (UTC)[reply]

Dolphin's answer is much more comprehensive. The spherical cow my expression works out assumes all energy transfer as heat, and ignores work, which is a gas is commonly called "PV work" because the mechanical energy is in the form of changing pressures and volumes. That sort of complicating factor is what makes gas thermodynamics so messy, but really, makes my answer less than useful. --Jayron 32 11:50, 22 July 2022 (UTC)[reply]