Non-uniform adiabatic gas compression

[Margalit]

I would like some insight on a problem I have been working on. I have a seal container of air where the outside chamber has a slight temperature differential between top and bottom. I understand this causes a density variation in the air between the top and bottom of the chamber. Now I am looking to expand the chamber to a larger volume and get the new pressure inside the chamber relative to the previous pressure. I assume there is no heat loss during this process, so I believe in the uniform case I can use the formula : (P1/P2) =(V2/V1)^1.4 for air.

My question then is this formula valid for the case where the density and temperature is not uniform? Does it matter if the expansion happens from the top (lower density air) or bottom (higher density air)?

I appreciate any insight of how to think about this problem. Thank you.

 

[voko]

I do not think this problem is fully specified. Nothing is stated about the cause of the temperature gradient, and it is not clear how that cause will affect the state of the system after expansion. Assuming that cause disappears, the end state will have uniform temperature. Given the original temperature/density profile and the assumption of the uniformity after the expansion, conservation of energy and the ideal gas laws should be sufficient.

 

[Margalit]

More specificity

Thank for the response. I will try to be more specific in defining the problem by using a example. Suppose we have a 1 m cube box with fully rigid walls. Air is trapped in the chamber at 100 kPa. The bottom of the cube is externally forces to 24C and the top is forced to 26C. The average temperature inside the box is 25C. A piston is then pushed into the chamber reducing its volume by 1 Liters. This is done in a rapid fashion <0.1 seconds. This causes a slight internal temperature rise of the gas because work has been done on the gas. Given the size of the box and speed of sound in air, a uniform pressure(ignoring gravity or convection) is established in the chamber rapidly <0.2 seconds. Thermal equilibrium in such a situation will take much longer 30+ seconds. I am looking for the pressure in the vessel before there is heat loss to the walls (not-isothermal). This is obviously a classical problem when the temperature is uniform, but I could not find anything that deals with non-uniform temperature.

My question is under these non-uniform temperature conditions for the start, are the basic formulas still usable for the relationship between volume compression and pressure rise?