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I need help in calculating how much a constant flow of CO2, being released into a small pipeline from a pressurised gas bottle, cools down the pipeline (in watts), to figure out if I'd need to heat up the pipes.

At the moment I think I know how to calculate the temperature of the released gas using an isentropic flow equation:

$P/P_t = (T/T_t)^{(k/(k-1))}$

I know both pressures ($P/P_t$), the temperature of the pressurised gas ($T_t$) and the ratio of specific heats of CO2 ($k$), but I'd like to know how the flow rate affects the cooling of the pipes.

Do i need to get into fluid mechanics and Heat transfer mechanisms to get an answer or do i use some kind of equation based on the Joule-Thomson effect?

Sorry if I'm being incomprehensible, this is my 1st post and english is my 2nd language.

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The gas remaining inside the tank, to a good approximation, expands adiabatically and reversibly as it expels the gas ahead of it through the valve. So you can use your adiabatic expansion equation for the gas remaining in the tank, provided the lower pressure in your equation is equal to the current pressure in the tank, and the higher pressure is equal to the initial pressure. The gas being expelled through the valve experiences Joule-Thompson, so its exit temperature is, to a good approximation, the current temperature in the tank. The rate of gas leaving depends on the pressure drop flow rate relationship for the valve.

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Pressurized CO2 at room temperature, as in a fire extinguisher, is a liquid. That lets the cylinder store a lot more CO2.

The latent heated needed to transform it to a gas cools it to the atmospheric-pressure freezing point at about -80C.

So your gas will include some amount of “dry snow”. As that sublimes, it provides additional cooling.

Bottom line: from the starting pressure, temperature, and mass flow, you need to include the phase transition(s) in your calculation to get the energy needed to get the gas up to whatever minimum temperature you require.

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