Gas is bled from a tank. Neglecting heat transfer between the gas and the tank, show that mass and energy balances produce the differential equation:

Question

Here, U and m refer to the gas remaining in the tank; H? is the specific enthalpy of the gas leaving the tank. Under what conditions can one assume H? = H?

Answer 1

If we take the tank as our control volume, then the mass balance gives us



where m is the mass of fluid still in the tank and is the mass flow rate of the stream leaving the tank.

Likewise, the energy balance gives



where H' is the specific enthalpy of the fluid leaving the tank and U is the specific internal energy of the fluid still in the tank. In the energy balance we have neglected contributions of kinetic and gravitational potential energy. Q, the rate of heat transfer in the energy balance is zero because the tank is insulated. W is zero because no work is being done on or by the system other than the flow work that is already included in H'.

We can expand the derivative in the energy balance to get



We can use the mass balance to eliminate from this equation to get



Rearranging this gives



From which we can eliminate time as a variable and get the desired relationship



If conditions within the tank are uniform (no gradients in pressure, temperature, etc. between the main part of the tank and the tank exit) then H' = H.