Liquid Level Control System

Basically, these systems are analyzed on the basis of nature of the fluid flowing through the pipes. And this classification is done on the basis of a number called Reynolds number.

So, if the value of the Reynolds number falls below 2000 then the flow is assumed to be laminar. This means that the fluid, in this case, travels smoothly on a regular path. So, turbulence is not present in this type of flow.

While if the Reynolds number is more than 3000 to 4000 then, in this case, the flow is not laminar but turbulent.As we have already discussed in the beginning that these systems possess two

 for the purpose of analysis, the two majorly concerned variables are the level of fluid in the tank and the rate of flow of fluid through the pipes.

 So, resistance, capacitance and ihertance are regarded as the fundamental parameters of a liquid level control system.

 The term inertance corresponds to the inertia forces which comes into action in order to

 cause acceleration of the fluid through the pipes. It is regarded as an energy-storing element but the amount of stored energy due to inertance is quite small thus is neglected at the time of analysis.

 So, the liquid level control system is defined on the basis of only resistance and capacitance associated with the system.

 Consider the figure we have shown above.And as we have mentioned that at steady state,

  Q, =Qo

 However, when the fluid outflows from the tank then there will be some resistance in its

 flow. The amount of resistance offered depends on the outlet through which the fluid

 Comes out.

 In case of a hole as an outlet, the fluid will not flow out easily that causes the generation of

 resistance. While in the case of the pipe as an outlet, the nature of flow which we have

 discussed above i.e., laminar or turbulent and the frictional force between liquid and pipe are the deciding factors for the resistance.

 Also, the amount of offered resistance increases with the presence of the valve with the amount of offered resistance increases with th the pipe at the outlet.

 So, generally, the resistance of the liquid level system is expressed as:

$$R=\frac{Variation\;in\;level\;difference\;(m)}{Change\;in\;rate\;of\;flow\;(m³/s)}$$

 This is defined on the basis of change noticed in the difference in levels of two tanks needed for the unit change in the rate of flow of fluid.This means the ratio depends on the nature of the flow.

 Transfer Function of Liquid Level System

Suppose the flow is linearly turbulent where:

 • Qo denotes steady-state outflow rate

 • q represents a small change in rate with the fluid is entering from the steady state value

 •qo shows the small change in the rate with which the fluid is coming out from steady-state

 • H is the steady-state level of the fluid inside the tank

 • h is the small variation in fluid level from the steady-state value

 For liquid level system, the equation of capacitance is given as:

$$C\frac{dh}{dt}=q_i-q_o$$

Also,

$$R=\frac h{q_o}$$

$$q_o=\frac hR$$

 On substituting the value of qo. we will get,

$$C\;dh=\left(q_i-\frac hR\right)dt$$

 Further,

$$RC\;dh=\left(R\;q_i-h\right)dt$$

 On transforming

$$RC\;\frac{dh}{dt}=R\;q_i-h$$

 Therefore,

$$RC\;\frac{dh}{dt}+h=Rq_i$$

 Now. takina l anlace transform. we will get.

While if q, is considered as the output for input q, then the Laplace transform of the equation shown above i.e.,

 $$q_o=\frac hR$$

 Will be

 $$Q_o\left(s\right)=\frac{h\left(s\right)}R$$

 So, on substituting, H(s) from the transfer function obtained above, we will get,

$$Q_o\left(s\right)=\frac{Q_{i\left(s\right)}}{1+{}_sRC}$$

 Therefore,

$${\textstyle\frac{Q_o}{Q_i}}=\frac1{1+{}_sRC}$$

 : RC corresponds the time constant i.e., T, of the liquid level control system.