The filling level control system in the experimental state consists of containers connected via two pipes and a pump, which are located at different height levels (see Picture 1). The water is conveyed from the lower to the upper container via the pump and pipe (1) and, from there, can freely flow back to the lower container again via pipe (2) by means of gravitational force. The purpose of the automation task is to regulate the water level in the upper container.
Picture 1: Schematic representation of level control system
Theoretical model configuration
Model for the outflowing process from the upper container
This outflowing process can be represented in a very simplified form (see Picture 2).
Picture 2: Physical model of outflow process
The following abbreviations mean:
>> p0 – Atmospheric pressure;
>> pB = p0 + ρgHH – Pressure at the bottom;
>> ∆p = kR2Qab^2 – Pressure drop via the pipe;
>> Qab – Volumetric flow rate;
and ∆p = pB - p0 = ρgHh
becomes the volumetric outward flow Qab = k * √ {ρgHH}
Dynamic processes in piping systems, which could lead to delays of the processes, are disregarded in this instance, since the dynamic behaviour is determined mainly by the storage processes in the two containers.
Model for inflow process into the upper container
On the basis of the same deliberations as already described in detail, a balance must again exist between the partial pressures in the system, for which the following applies:
0 = ∆p0 − ∆pstat − ∆pR − ∆pi
Using
∆pstat = ρg (HA + HH − HU)
∆pR = kR1Qin^2
∆pi = kiQin^2
results in the condition:
0 = kpN^2 − ρg (HA + HH − HU) − (ki + kR) Qin^2
Here it should be noted that if the containers are identical in design, the total of the filling levels
HW = HU + HH
remains constant. This finally results in the correlation:
Qin = √ { kpN^2 − ρg*[HA + 2*HH − HW]/(ki + kR) }
between the pump speed N, level HH in the upper container and inflow Qin in the upper container. The filling level HH only changes if a difference ∆Q between inflow Qin and outflow Qout,
∆Q = Qin − Qout’ occurs.
Per time interval ∆t, the volume of water V in the upper container would then change by:
∆V = A * ∆HH = ∆Q * ∆t
If we use the inflow and outflow rates according to the equations quoted above are used here, this results in the following correlation in respect of the level change ∆HH per time interval ∆t:
∆HH/∆t = (1/A) * √ { kpN^2 − ρg*(HA − HW + 2HH)/(ki + kR) } − k * √ (ρg*HH) }
Steady-state model of the controlled system
In the stationary state (HH = const.), the inflow and outflow in the upper container must coincide,
Qin = Qout
and the correlation:
√ { kpN^2 − ρg*(HA − HW + 2HH)/(ki + kR) } = k * √ (ρg*HH)
must therefore apply. If this equation is resolved with respect to the filling level HH, this results in the following correlation for the interdependence between pump speed N and height level HH in the stationary state:
HH(N) = { kpN^2 − ρg*(HA − HW) } / { ρg*(2 + k^2 * [ki + kR]) }
Picture 3 illustrates the corresponding steady-state characteristic curve of the controlled system.
Picture 3: Qualitative process of steady-state characteristic curve of controlled system
Linear dynamic controlled system model
The above mentioned correlation, which describes the connection between the status change ∆HH per time interval ∆t and the speed N or the height level HH, applies in principle also to small process variable changes by their operating point values:
HH = HHO + hH
N = N0 + n
and results in the linear correlation:
∆hH/∆t = −a * hH + b∆n
with the two controlled system parameters a and b, which are system and operating point dependent.
Starting with an initial status (e. g. hH=0) and specified time-related course of speed variation n (e. g. step-type change at time t=0), this equation can be resolved step by step and then results in the system step response shown on Picture 4, if a sufficiently short time interval t is used.
Picture 4: Step response of level control system
This controlled system therefore also exhibits proportional behaviour with delay of the first order at the operating point and its characteristic values need to be experimentally determined according to the same considerations set out in the section Flow control system.
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