SimApp Tank Level Model

7/30/2019 SimApp Tank Level Model

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VentiMar, LLC 1

SimApp IntroductionTank Level Model

Peter Waywww.ventimar.com


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VentiMar, LLC 2

Outline This application note is a tank level model that

can be used in a variety of chemical processes. The nonlinear model for turbulent flow is

developed first

The model is then linearized about an operatingpoint to allow frequency analysis.

Finally, the tank example is extended to two

tanks in series.


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VentiMar, LLC 3

SimApp Process Example

H [m]

Qi

Qo

Control valve flow rate [m^3/s or l/s]

Load valve flow rate

Tank Capacitance C [m^2] (change in volume per change in H) Not tank capacity [m^3]

Units shown in [ ]

Model the level and flow of a water tank that supplies adownstream process.

Volume: 10 m3

Height: 5 m

Radius 0.8 m

Example


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Turbulent vs. Laminar Flow The Reynolds number (Re) is a dimensionless quantity that

determines the type of flow: turbulent or laminar

Re = Dynamic pressure/ Shear stress

Laminar: Re4000, otherwise Transitional

Reference: http://www.engineeringtoolbox.com/reynolds-number-d_237.html

fluidofviscositydynamic-

diameterLpipesection-crosscircularafor

duct)ofeterduct/PerimofArea*(4lengthsticcharacteri-L

speedfluid

densityfluid

Re

=

=

u

uL


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Write the equationsTwo types of flow:

HKQ

KHQ

o

o

=

=

:Turbulent

allyexperimentmeasuredtypicallyisK:Laminar

Flow equation (examine the turbulent case in this example):

Increase of volume as head (level) H changes equals inflow-outflow over time

change. Capacitance of the tank is the cross-sectional area

i

oi

QHKdt

dHC

dtQQCdH

=+

=

iQandHbetweeniprelationshgettoRearrange

)(

Note: the terminology of resistance and capacitance borrowed

from electrical engineering is no accident! Also H Voltage, Q Current


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Make the SimApp model

C

HKQH

QHKHC

i

i

=

=+

.

:SimAppinequationtheimplementtogRearrangin

.

An easy way to make the block diagram model is to solve for the highest derivative.Then the equation is built as a block diagram as shown belowAlso we will use the dot notation for the derivative.

H

Reset

Hold

H

Ti 2 s

I

K(m^2.5/s)

K 0.008944

P

Qo(l/s)

K 1000

SQRT

Qi(l/s)

K 1000

Qi (m^3/s)

H 0.01

TD 0 s

H(m)

.


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Model features

C

HKQH

HKQ

i

o

=

=

.First, integrate thederivative

Each blocks nameis the output

C is the timeconstant of theintegrator. K, C are

example values

This implementsequation from H toQo

The summationcalculates thederivative

For now, we candrive the systemwith 10 l/s flow

These probes are for plotting andscaling the output. In this casefrom m^3/s to l/s

Reset

Hold

H

C 2 s

I

K(m^2.5/s)

K 0.008944

P

Qo(l/s)

K 1000

SQRT

Qi(l/s)

K 1000

Qi (m^3/s)

Qi 0.01

TD 0 s

H(m)

H.


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Time response You can now make a time response plot to the Qi input. Qo eventually reaches Qi (tank does not overflow or empty) At this flow rate the tank fills to 1.24 meters

-200 0 200 400 600 800 1200 1600 2000 2400 2800 3200 3600 3900

0

1

2

3

4

5

6

7

8

9

10

-0.5

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

Source group 0

H(m) 1.2444

Qi(l/s) 10

Qo(l/s) 9.9771


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VentiMar, LLC 9

Simple proportional control

We want to control H (process variable) to 3 m. H is measured, compared

with the Hsp (set point) which drives the proportional valve.

A simple proportional valve is used but other controllers could be applied

The ramp of Hsp (set point) or gain can be selected to determine the time todesired H, and the maximum Qi required to achieve the goal.

Reset

Hold

H

C 2 s

I

K(m^2.5/s)

K 0.008944

P

Qo(l/s)

K 1000

SQRT

Qi(l/s)

K 1000ProportionalValve

K 0.1

P

Ramp

A 0.01 s-1

Ymax 3

Hsp (x10m)

K 10

H(x10 m)

K 10

Controller: Proportional control oftank Height H

Plant


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Controller time responses H values are scaled by 10x to show dynamics better Qi needs to reach over 33 l/s to fill the tank to 3m in 400 seconds Proportional control has some error from the setpoint. Gain could be increased but

then the flow rate to fill the tank would be greater.

-50 0 50 100 150 200 250 300 350 400 450 500-25 25 75 525

0

5

10

15

20

25

30

35

-2.5

2.5

7.5

12.5

17.5

22.5

27.5

32.5

Source group 0

H(x10 m) 28.485

Hsp (x10m) 30

Qi(l/s) 15.154

Qo(l/s) 15.095


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Linearizing the turbulent model

point.setaindicatessubscripttheWhere2

Kforngsubstituti2

valveandpipetheacross

changewChange/FloEffortResistance-RDefine

:rearrange:Turbulent

0

0

0

2

2

Q

HR

K

Q

dQ

dHR

K

Q

HHKQ

=

==

=

==


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VentiMar, LLC 12

Linearizing the turbulent model

2

resistanceforequationtheusingThen

setpointthe

fromanddeviationssmallconsiderNow

0

0

00

Q

H

q

hR

, HQ

qh

==

Nonlinear vs. Linearized

-4.000

-2.000

0.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

0 5 10 15 20 25 30

Q

H

H Nonlinear [m]

h Linearized [m]

Q0,H0

qh

C

Rhqh

HKh/R

C

HKQH

i

i

/.

:)forngsubstituti(afterbecomes

onsperturbatismallforequationThe

.equation

headtoflowinputthememberingRe

=

=


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Linearized system block diagram Now the nonlinear element has been replaced by a linear resistor.

This makes it possible to calculate a frequency response.

The Height of the tank needs to be biased to 3m since that is the

operating point of the non-linear model. This is only to match timeresponses, and does not affect frequency response.

Reset

Hold

H

C 2 s

I

1/R

K 0.002631

P

Qo(l/s)

K 1000

Qi(l/s)

K 1000ProportionalValve

K 0.1

P

Ramp

A 0.01 s-1Ymax 3

Hsp (x10 m)

K 10

H ( x10 m)

K 10


Linearized Plant

Frequency

f

H Bias

3


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Time response of linearized system

-50 0 50 100 150 200 250 300 350 400 450 500-25 25 75 525

0

5

10

15

20

25

30

35

-2.5

2.5

7.5

12.5

17.5

22.5

27.5

32.5

37.5

Source group 0

H ( x10 m) 0

Hsp (x10 m) 0

Qi(l/s) 0

Qo(l/s) 7.893

Linearized model agrees quite closely

with the nonlinear model except in startup transient


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Frequency response oflinearized system

-100

-80

-60

-40

-90

-70

-50

[dB] Amplitude

0.01 0.10.02 0.03 0.05 10.2 0.3 0.5 0.7 102 3 4 5 6 7

-80

-60

-40

-20

-70

-50

-30

-10

[] Phase

Frequency

1/R

Amplitude [dB] -55.088

Phase -46.632


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Two tank example

H1 [m]

Qi

Q1

Control valve flow rate [m^3/s or l/s]

Units shown in [ ]

Model the level and flow of a water tank that supplies asecond tank. Control the height of the second tankthrough input Qi. Q2 supplies a downstream process.

H2 [m]Q2


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Two tank modelNote the correspondence between the tank illustration in prior slide and new connections

Reset

Hold

H1

C 2 s

I

K(m^2.5/s)

K 0.008944

PQ1(l/s)

K 1000

SQRT

Qi(l/s)

K 1000Proportional

Valve

K 0.08

P

Ramp

A 0.01 s-1Ymax 3

Hsp (x10m)

K 10

H1 (x10 m)

K 10


Tank 1

Reset

Hold

H2

C 2 s

I

K(m^2.5/s)

K 0.008944

P

Q2(l/s)

K 1000

SQRT

H2 (x10 m)

K 10

Tank 2

Process variable feedback

H1 - H2 drives the flow Q1 between tanks.

Q1 supplies flowinto tank 2


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VentiMar, LLC 18

Two tank time response Since there are now two lags to control, the simple gain control is no longer sufficient

for good performance.

Note that the height of liquid in tank 1 gets unacceptably large.

-200 0 200 400 600 800 1000 1200 1400 1600 1800 2000100

-60

-40

-20

0

20

40

60

80

100

120

140

-70

-50

-30

-10

10

30

50

70

90

110

130

150

Source group 0

H2 (x10 m) 0

Q2(l/s) 0

H1 (x10 m) 0

Hsp (x10m) 0

Qi(l/s) 0

Q1(l/s) 0


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SimApp Summary You can build simple models for understanding including nonlinear

effects. In this example, the model works for any level and flow

situation. Then you can simulate controllers of increasing sophistication

In the tank example, you could fill the tank to the desired H first, thenmake a linearized version of the model

You could then perform frequency and stability analysis on linearmodels and...

Design better controllers.

The block diagram method promotes understanding and lets youextend the model to new situations.

See the tutorial and design spreadsheet atwww.simapp.com/simulation-tutorials

You can build simple models for understanding including nonlineareffects. In this example, the model works for any level and flow

situation. Then you can simulate controllers of increasing sophistication

In the tank example, you could fill the tank to the desired H first, thenmake a linearized version of the model

You could then perform frequency and stability analysis on linearmodels and...

Design better controllers.

The block diagram method promotes understanding and lets youextend the model to new situations.

See the tutorial and design spreadsheet atwww.simapp.com/simulation-tutorials

Documents

SimApp Tank Level Model