An evacuated tank is filled with gas from a constant-pressure line. Develop an expression relating the temperature of the gas in the tank to the temperature T? of the gas in the line. Assume the gas is ideal with constant heat capacities, and ignore heat transfer between the gas and the tank.

Question

What will be an ideal response?

Answer 1

The mass balance on the tank is



and the energy balance on the tank is



where m is the total mass in the tank and U is the specific internal energy of the gas in the tank. Q and W are both zero because the tank is both rigid (no non-flow work) and insulated (no heat flow through its boundaries). We can combine the two balance equations to get



Integrating these, and using the fact that the enthalpy of the gas coming into the tank is constant, we get



where the subscript t denotes the value at some arbitrary time t and the subscript t = 0 denotes the value at t = 0. Since m is zero at t = 0, this is simply



or U = H' for any time. So, the specific internal energy of the gas in the tank is equal to the enthalpy of the gas entering the tank. The specific enthalpy of the gas in the tank is

H = U + PV = U + RT = H' + RT

where T is the temperature of the gas in the tank and H is the specific enthalpy of the gas in the tank. The enthalpy change when the gas goes from the temperature of the inlet line to the temperature of the tank is H - H' = Cp(T - T').

Substituting this into the above equation gives

H - H' = Cp(T - T') = RT

Or

CpT' = (Cp - R)T = CvT



Or

T = CpT'/Cv = ?T'

The temperature of the gas inside the tank will be higher than that of the gas entering the tank.