In this exercise we'll be producing a simple overall mass balance for the tank shown in the diagram. The tank is a simple vertical cylinder with a cross-sectional area equal to 'A' square metres. Liquid flows into the tank at volumetric flowrate 'Fin' (cubic metres per hour) and flows out at volumetric flow rate 'Fout'. The liquid level in the tank is 'h' metres.
We want to build a model than tells us how the level in the tank changes when either the inlet flow, or the outlet flow changes. For the moment we'll assume that the outlet flowrate is independent of the level in the tank (as it approximately would be if there were a pump at the tank outlet).
We'll also assume that the density of the liquid streams and the liquid held up in the tank is constant and equal.
The only balance we can do on this system is a total mass balance.
The first thing we need to sort out is the accumulation term. The easiest way to do this is to write down an equation which describes how much mass is held up in the tank and then to surround this equation with a d/dt(). Try writing the accumulation term yourself before revealing the next step.
Now that we have the accumulation term, we need to form the right-hand side of the balance. In this problem we have one mass flowrate in, one out and no generation of total mass in the tank. Have a go at completing the balance before revealing the next step.
Note that, in this case, we have ended up with a balance where the terms have units of volume/time. This is NOT a general result and only arises because we have assumed constant and equal densities across all the streams.
In some cases the flow out of a tank of this sort will not be independent of the level in the tank. The most extreme case of this is when we have a free draining gravity discharge. In this case the exit from the tank discharges into the open air (i.e. there is no back pressure). The mass balance for this case can be rewritten (if a discharge-coefficient flow equation is assumed) as:
