Dynamic Simulation of a Gravity-Flow Tank

The dynamic behavior of a tank and pipe system is described by the following coupled ordinary differential equations:

,

,

where is the pipe length, is the acceleration of gravity, is the inlet flow rate to the tank, is the outlet flow rate from the tank, is the tank's height, is the fluid's density, is the fluid's velocity in the pipe, is a unit conversion factor, and are the cross-sectional area of the pipe and the tank, respectively, and is a proportionality coefficient that appears in the frictional force .

The first equation represents a force balance on the outlet line and the second equation is the continuity equation on the liquid in the tank. We have the following relation: . If one neglects the dynamic behavior of the pipe, then we get the classic equation: .

This Demonstration shows that it is important to take into account the dynamic behavior in the pipe (see blue curve) since in some cases it allows the prediction of tank overflow, a phenomenon that will not be observed otherwise (see red curve). All numerical values used in the present Demonstration can be found in Luyben's book.