Question

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: dU/H' - U = dm/m 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

The heat transfer between the gas and the tank is neglected. So, No heat transferred into system and no work transferred into the system. Also, the potential and kinetic energy will be negligible.

The mass balance and energy balance equations for the system(the tank) are as follows:

mass balance: dm/dt = -fm

where m is the mass of the gas and fm is the mass flow rate of the gas.

energy balance: dU/dt = d(m*u)/dt = -fm*H' = H' * dm/dt where, H' is the specific enthalpy of the gas leaving the tank.

The internal energy of the gas in the tank is the product of the mass and the specific internal energy.

U = m*u where, U is the total internal energy, u is the specific internal energy.

So we have, d(m*u)/dt = m*du/dt + u*dm/dt

rearranging the equation we get, m*du/dt = (H' - u)*dm/dt

Hence, du/H' - u = dm/m