The tank can be modelled from first principles with the provision of certain assumptions. For the sake of simplicity, it is possible to consider the top section of the tank with parallel sides and then extend this model to deal with the whole tank. The prismatic section of the tank can be considered to be a simple rectangular tank with an inflow Q and an outflow, QL as in Figure. By considering conservation of matter, we can say that the flow into the tank must be equal to the flow out plus the flow converted into a change in level.

Figure Prismatic tank

but if then

So this is a 1st order model for the tank.

The tapered section can be dealt with in much the same way, with the understanding that the cross sectional area A is now varying and dependant on the level h. If we are interested in a particular operating point in the 'V' section, we could assume that the tank had vertical sides at this operating point. This would be OK provided we did not move too far from this operating condition. In other words, small changes in h would produce small changes in A, which our controller may be able to cope with. This is not very practical, as any significant disturbance would make the system go unstable.