The liquid level system consists of the storage tanks and the connecting pipes. The analysis of such systems depends on the nature of fluid flow through the pipes. The flow is divided based on a number called Reynolds number.
1. If Reynolds number is less than 2000, the flow is treated to be laminar. There is no turbulence in such a flow.
2. If Reynolds number is greater than 3000 to 4000, the low is turbulent.
The laminar flow is governed by linear differential equations while the turbulent flow is governed by nonlinear differential equations.
The height in fluid in tanks and the flow rate of fluid through the connecting pipes are the important variables in the liquid level systems.
The three basic parameters in any liquid level system are resistance, capacitance and inductance. The inductance comes into existence due to the inertia forces required to accelerate the fluid through the pipes. Thus inductance is an energy storing element like capacitance. But energy storage due to inductance is practically negligible hence inductance is neglected in the practical analysis. Let us define resistance and capacitance related to the liquid level system.
1.1 Resistance and Capacitance
Consider a liquid level system shown in the Fig.1.
For the system shown in the Fig.1.
At the steady state, the volume-balance gives,
The fluid when flows out of the tank, can meet the resistance in many ways. If the outlet is a hole then liquid cannot flow easily through a hole and a hole creates a resistance. If the outlet pipe then the nature of liquid flow i.e. laminar or turbulent and friction of liquid with pipe together decide the resistance. If the pipe has a valve, there is further increase in the resistance to the flow of a liquid.
Note : The resistance of a liquid level system is defined as the change in the level difference between the liquid levels of the two tanks necessary to cause a unit change in the flow rate.
Mathematically it is expressed as,
The ratio changes, depending on whether the flow is laminar or turbulent.
For a laminar flow of liquid from the pipe connected to the outlet, the steady state flow rate and the steady state head are directly proportional to each other.
If Q = Steady state flow rate in m3/sec
H = Steady state head in m
Where K = Coefficient in m/sec
This is the law which governs the laminar flow of liquid. This is analogous to Ohm's law which says that voltage and current are proportional to each other. The head analogous to potential difference while Q is analogous to current. Thus the resistance for the laminar flow can be defined as,
So RL is constant similar to the resistance in the electrical circuits.
While if the flow is turbulent, Q and H related by the equation,
Hence resistance for the turbulent flow can be obtained as,
Thus Rt is not constant but depends on the flow rate and the head. Practically most of the systems have a turbulent flow.
If changes in steady state values of Q and H are small then Rt can be assumed constant and we can write,
Practically if K is unknown then the graph of H against Q is obtained experimentally, as shown in the Fig.2. The steady state operating point is P given by Q and N. the tangent line is drawn at point P which intersects the Y axis at the point T(0, -H). Thus slope of the line can be obtained from points T(0, -H) and P(Q, H).
Thus slope of the tangent line at the operation point P gives the turbulent resistance Rt .
If practically there is a small variation in head shown as h and small change in flow shown as q, near the operating point then assuming curve to be a tangent line, the resistance can be obtained as,
Note : This is the linear approximation used in practice where actual curve does not differ much from the tangent straight line.
Symbolically the constant laminar resistance and variable turbulent resistance of a liquid level system are denoted as shown in Fig.3. (a) and (b).
Note : The capacitance of a liquid level system is defined as the change in the liquid stored measured in m necessary to cause a unit change in head measured in m.
Such a capacitance is also called hydraulic capacitance. Now the change in the liquid stored is the difference between input flow rate and output flow rate while change in head is dh/dt. Hence the capacitance can be expressed as,
The equation in analogous to the capacitor equation in an electrical circuit which is,
1.2 Transfer Function of Simple Liquid Level System
Consider a simple liquid system with a single tank as shown in the Fig.4.
For this system, the various variable are,
Assumed linearised turbulent flow, behaving according to the tangent line to the actual head-flow curve.
From the equation of capacitance of the liquid level system we can write,
This is the linear differential equation governing the system.
From the definition of resistance,
Substituting in equation ( 8)
Taking Laplace transform of both sides,
Neglect the initial conditions.
Thus for a qi as the input and h as the output, the transfer function of system is H(s)/Qi(s).
If qo is assumed to be output then from Laplace transform of equation ( 9 ) we can write,
RC is the time constant of the liquid level system.
1.3 Tranfer Function of Liquid System with Interaction
If one tank connected with another tank in the process, then the liquid level system is said to have an interaction. But the transfer function of such a system is not the product of two individual first order transfer function. Consider such a system with interaction as shown in the Fig.5.
From the definition of capacitance we can write,
From the definition of resistance we can write,
Taking Laplace transform of all the equations and neglecting initial conditions we get,
Simulating the above equations using block diagram we get,
Using block diagram reduction rules, the final transfer function can be obtained as,
The term R2 C2 appearing in the denominator is the indication of the interaction between the two tanks.