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Feedback Control Systems ( FCS )

Feedback Control Systems ( FCS ). Lecture-13 Mathematical Modelling of Liquid Level Systems . Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL : http://imtiazhussainkalwar.weebly.com/. Laminar vs Turbulent Flow. Laminar Flow

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Feedback Control Systems ( FCS )

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  1. Feedback Control Systems (FCS) Lecture-13 Mathematical Modelling of Liquid Level Systems Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/

  2. Laminar vs Turbulent Flow • Laminar Flow • Flow dominated by viscosity forces is called laminar flow and is characterized by a smooth, parallel line motion of the fluid • Turbulent Flow • When inertia forces dominate, the flow is called turbulent flow and is characterized by an irregular motion of the fluid.

  3. Resistance of Liquid-Level Systems • Consider the flow through a short pipe connecting two tanks as shown in Figure. • Where H1 is the height (or level) of first tank, H2 is the height of second tank, R is the resistance in flow of liquid and Q is the flow rate.

  4. Resistance of Liquid-Level Systems • The resistance for liquid flow in such a pipe is defined as the change in the level difference necessary to cause a unit change inflow rate.

  5. Resistance in Laminar Flow • For laminar flow, the relationship between the steady-state flow rate and steady state height at the restriction is given by: • Where Q = steady-state liquid flow rate in m/s3 • Kl= constant in m/s2 • and H = steady-state height in m. • The resistance Rl is

  6. Capacitance of Liquid-Level Systems • The capacitance of a tank is defined to be the change in quantity of stored liquid necessary to cause a unity change in the height. • Capacitance (C) is cross sectional area (A) of the tank. h

  7. Capacitance of Liquid-Level Systems h

  8. Capacitance of Liquid-Level Systems h

  9. Modelling Example#1

  10. Modelling Example#1 • The rate of change in liquid stored in the tank is equal to the flow in minus flow out. • The resistance R may be written as • Rearranging equation (2) (1) (2) (3)

  11. Modelling Example#1 • Substitute qo in equation (3) • After simplifying above equation • Taking Laplace transform considering initial conditions to zero (1) (4)

  12. Modelling Example#1 • The transfer function can be obtained as

  13. Modelling Example#1 • The liquid level system considered here is analogous to the electrical and mechanical systems shown below.

  14. Modelling Example#2 • Consider the liquid level system shown in following Figure. In this system, two tanks interact. Find transfer function Q2(s)/Q(s).

  15. Modelling Example#2 • Tank 1 Pipe 1 • Tank 2 Pipe 2

  16. Modelling Example#2 • Tank 1 Pipe 1 • Tank 2 Pipe 2 • Re-arranging above equation

  17. Modelling Example#2 • Taking LT of both equations considering initial conditions to zero [i.e. h1(0)=h2(0)=0]. (1) (2)

  18. Modelling Example#2 (1) (2) • From Equation (1) • Substitute the expression of H1(s) into Equation (2), we get

  19. Modelling Example#2 • Using H2(s) = R2Q2 (s) in the above equation

  20. Modelling Example#3 • Write down the system differential equations.

  21. To download this lecture visit http://imtiazhussainkalwar.weebly.com/ End of Lectures-13

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