B.5.FM:5.2 - Include the air when modeling liquid discharge

Consider a rigid vessel with a liquid outlet near its bottom and trapped air above the liquid. The question is whether enlarging the outlet will allow most of the liquid to drain. A model based only on liquid height can miss what changes when liquid leaves.

For a first conditional account, assume that no air enters and the liquid is incompressible. Treat the trapped air as an ideal gas at constant temperature and fixed amount. Consider slow flow whose inertia is negligible and whose motion is dissipated by resistance at the outlet. Include the gas volume and absolute pressure, the liquid height above the outlet and the outside pressure. Liquid leaving increases gas volume and lowers gas pressure.

At the no-flow equilibrium, the internal pressure at the outlet equals the outside pressure. For the following rough calculation take liquid density as 1,000 kg/m³ and gravitational acceleration as 10 m/s². Initial liquid height is 20 cm, trapped gas volume is 100 cm³, vessel cross-section is 100 cm² and initial gas and outside pressures are both 100 kPa.

Let q be the discharged volume in cm³. The gas occupies 100+q cm³; liquid height is 20−q/100 cm. The isothermal gas relation and hydrostatic head give the equilibrium equation:

10000/(100+q) + 0.1*(20−q/100) = 100   [kPa]

The positive solution is about 2.039 cm³, with gas pressure about 98.002 kPa and liquid height about 19.980 cm. In this slow-flow account, the discharge approaches a stop after a very small volume. Changing outlet size changes resistance and the approach to equilibrium; this same equilibrium balance applies while air entry and inertial effects remain negligible.

Allowing ambient air to reach the gas space changes the model: gas pressure can remain near outside pressure as liquid leaves. Air entering through the outlet, gas-temperature change, vessel deformation or appreciable capillary pressure requires another account. The immediate useful result is the distinction between an outlet-flow restriction and a gas-replacement restriction. Investigate the air path before treating outlet enlargement as the answer.