Adiabatic expansion of CO2 - rate of cooling

[mattbanham90]

Hi Physics forum,

I have a pipe which is of length L, inner diameter D. It is packed with CO2 at pressure P (assume it's still a gas with none of this funny supercritical business!) and temperature T. The density of the CO2 is rho and the joule-kelvin coefficient is mu. the pipe is initially sealed at both ends. However, it suddenly breaks at one end (in an infinitesimally small time). The gas is set to emerge into the atmosphere at a high rate. However, JT cooling of the gas causes some of it to freeze.

My questions are:

1.) what is the rate of cooling of the CO2 (ignoring freezing for now) in terms of the parameters above?

2.) what is the rate of freezing of the gas?

3.) will the pipe freeze up and seal preventing further CO2 from escaping?

4.) if so, how much gas escapes before it freezes up and how long does it take before it seals?

If anyone has any pointers to this then fantastic.

Matt

 

[Jhero]

Hello Matt,

Thank you for your interesting post. I would like to provide some insights into your questions.

1. The rate of cooling of CO2 can be determined using the Joule-Thomson coefficient, which is the change in temperature with respect to pressure at constant enthalpy. It is given by the following equation:

α = (1/Cp)(∂T/∂P)H

Where:
α - Joule-Thomson coefficient
Cp - Specific heat at constant pressure
T - Temperature
P - Pressure
H - Enthalpy

The rate of cooling can be calculated by multiplying the Joule-Thomson coefficient with the pressure gradient, which is the change in pressure over time. This can be expressed as:

Rate of cooling = α * (∆P/∆t)

2. The rate of freezing of the gas will depend on the temperature and pressure conditions at the point of release and the surrounding environment. As the gas expands and cools, it may reach its freezing point, causing some of it to freeze. The rate of freezing can be calculated by considering the change in enthalpy of the gas as it freezes.

3. It is possible that the pipe may freeze up and seal, especially if the temperature and pressure conditions are favorable for freezing. However, this will also depend on the rate of cooling and the rate of freezing.

4. The amount of gas that escapes before the pipe freezes up and seals will depend on the initial conditions and the rate of cooling and freezing. The time it takes for the pipe to freeze up and seal can be calculated by considering the rate of cooling and the freezing point of CO2.

I hope this helps answer your questions. If you need further clarification or have additional questions, please don't hesitate to ask. Happy experimenting!