System Dynamics

System Dynamics

Prof. Davide Manca – Politecnico di Milano

Dynamics and Control of Chemical Processes

Solution to Lab #2

SE2

E1 – Dynamics of a tank

Given the tank in the picture:

F

F

A

h

1. Evaluate the dynamics of the level of the tank considering a step increase on the inlet flowrate, so that it results half of its initial value.

2. Evaluate the level dynamics considering a linear decrease of the inlet flowrate in 30 seconds (ramp variation) up to the half of the initial value.

Remark : the tank is at steady state before the disturbances occur.

Model of the system

i

dh h

A F

dt   r

F

F

A

h

2

3

2

30 m

7.5 m s 0.4 s m

i

A F r







  ⁰

h  h

Data: I.C.:

Solution procedure question (1)

1. Assessment of the steady state solution:

2. Assignment of the step disturbance on the inlet flowrate:

3. Evaluation of the system dynamics by integrating the differential equation of the model. The steady state represents the initial conditions.

0

3 m

dh h

A F h r F

dt    r    

    3

2 3.75 m s

new old

i i

F  F 

MATLAB implementation

h0 = r * Fi0;

Fi = Fi0/2;

[t,h] = ode45(@(t,y)Sisdif(t,y,A,Fi,r),tSpan,h0,options);

function dy = Sisdif(t,y,A,Fi,r)

dy(1) = (Fi – h/r)/A

Main

Sisdif

Dynamics in case of a step disturbance

0 20 40 60 80 100

1 1.5 2 2.5 3 3.5

Tempo [s]

A lte zza li q u id o [ m ]

Time [s]

0 20 40 60 80 100 3

4 5 6 7 8

Tempo [s]

Portata [m3 /s]

Resolution procedure question (2)

• Evaluation of system dynamics by integration of the model differential equation.

The inlet flowrate varies with time:

   ⁰

0

2

30

i

i i

F  F  F t

 ⁰

i i

2

F  F

If t  30 s

Else

Time [s]

Flowrate [m3 /s]

MATLAB implementation

h0 = r * Fi0;

[t,h] = ode45(@(t,y)Sisdif(t,y,A,Fi0,r),tSpan,h0,options);

function dy = Sisdif(t,y,A,Fi0,r) h = y(1);

if(t < 30)

Fi = Fi0 - (Fi0/2)/30 * t;

else

Fi = Fi0/2;

end

dy(1) = (Fi – h/r)/A

Main

Sisdif

Dynamics in case of a ramp disturbance

0 20 40 60 80 100

1 1.5 2 2.5 3 3.5

Tempo [s]

A lte zza li q u id o [ m ]

Time [s]

Liquid level [m]

E2a – Dynamics of two non interacting tanks

Given the two tanks in the picture:

F

r

A

h

A

h

r

Evaluate the dynamics of the level of the two tanks in case of a step disturbance on the inlet flowrate, such that halves the initial flowrate.

Remark: the tanks are in steady state before the disturbance occurs

System model

Fi

r1

A2

h2

A1

h1

r2

1 1

1

1

2 1 2

2

1 2

i

dh h

A F

dt r

dh h h A dt r r

  

 

  



Data: I.C.:

2 2

2 2

50 m 0.7 s m A

r





Tank 2:

2 1

2 1

30 m 1.2 s m A

r





Tank 1:

9.4 m s

F 

Steady state conditions

Resolution procedure

1. Assessment of the steady state conditions:

2. Assignment of the step disturbance on the inlet flowrate:

3. Evaluation of the system dynamics by integrating the differential equation of the model. The steady state represents the initial conditions.

 

 

1 1

1

1 1

1

2 1 2 2 2

2

1 2

0 11.28 m

6.58 m 0

i s

i s

i

dh h

A F

h r F

dt r

dh h h h r F

A dt r r

   

   

  

 

 

     



    3

2 4.7 m s

new old

i i

F  F 

MATLAB implementation

h0 = [r1*Fi0 r2*Fi0];

Fi = Fi0/2;

[t,h] = ...

ode45(@(t,y)Sisdif(t,y,A1,A2,r1,r2,Fi),tSpan,h0,options);

function dy = Sisdif(t,y,A1,A2,r1,r2,Fi) dy = zeros(2,1);

h1 = y(1);

h2 = y(2);

dy(1) = (Fi – h1/r1)/A1;

dy(2) = (h1/r1-h2/r2)/A2;

Main

Sisdif

Dynamics of two non interacting tanks

0 100 200 300 400 500

2 4 6 8 10 12

Tempo [s]

A lte zza li q u id o [ m ]

Serbatoi non interagenti

Serbatoio 1 Serbatoio 2

Time [s]

Non interacting tanks

Tank 1 Tank 2

Liquid level [m]

E2b - Dynamics of two interacting tanks

Given the two tanks in the picture:

F

r

A

h

A

h

r

Evaluate the dynamics of the level of the two tanks in case of a step disturbance on the inlet flowrate, such that it halves the initial flowrate.

Remark: the tanks are in steady state before the disturbance occurs

System model

1 1 2

1

1

2 1 2 2

2

1 2

i

dh h h

A F

dt r

dh h h h

A dt r r

   

  

  



Data:

2 2

2 2

50 m 0.7 s m A

r





Tank 2:

2 1

2 1

30 m 1.2 s m A

r





Tank 1:

9.4 m s

F 

Fi

r1

A2

h2

A1

h1 r₂

I.C.: Steady state conditions

Resolution procedure

1. Determination of steady state conditions:

2. Assignment of the step disturbance on the inlet flowrate:

3. Evaluation of the system dynamics by integration of the differential equation of the model. The steady state represents the initial conditions.

 

 

 

1 1 2

1

1 1 2

1

2 1 2 2 2 2

2

1 2

0 17.86 m

6.58 m 0

i s

i s

i

dh h h

A F

h r r F

dt r

dh h h h h r F

A dt r r

    

    

  

    

    



    3

2 4.7 m s

new old

i i

F  F 

MATLAB implementation

h0 = [(r1+r2)*Fi0 r2*Fi0];

Fi = Fi0/2;

[t,h] =

ode45(@(t,y)Sisdif(t,y,A1,A2,r1,r2,Fi),tSpan,h0,options);

function dy = Sisdif(t,y,A1,A2,r1,r2,Fi) dy = zeros(2,1);

h1 = y(1);

h2 = y(2);

dy(1) = (Fi – (h1-h2)/r1)/A1;

dy(2) = ((h1-h2)/r1-h2/r2)/A2;

Main

Sisdif

Dynamics of interacting tanks

0 100 200 300 400 500

0 5 10 15 20

Tempo [s]

A lte zza li q u id o [ m ]

Serbatoi interagenti

Serbatoio 1 Serbatoio 2

Time [s]

Liquid level [m]

Interacting tanks

Tank 1 Tank 2

Comparison

0 100 200 300 400 500

0 5 10 15 20

Tempo [s]

Altezza liquido [m]

Tank 1, case: non interacting tanks Tank 1, case: interacting tanks Tank 2, case: non interacting tanks Tank 2, case: interacting tanks

Time [s]

Liquid level [m]