Principles of Process Control Exercises

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PRINCIPLES OF PROCESS

CONTROL EXERCISESMUSTAFA IQBAL – 1323293

This report aimed to investigate the dynamics behind a step change response in oxygen

partial pressure for a stirred vessel of deionised water. From experimental data, the nature of the

system was determined as second order with dead time, from which appropriate methods of

estimation were researched and applied to determine model parameters for each configuration of

impeller speed and gas flowrate used. Hence, the effect of energy input on model parameters was

quantified, and an empirical model of the system was derived. Next, an unsteady state mass balance

was applied to the system. Combined with a probe dynamic model this formed analytical and

numerical solutions to a theoretical model of the system. separate model considering the mass

transfer process alone was used, and, comparing all four responses showed the probe was of a slow

nature, and that the theoretical models were accurate. To preclude, a !"# closed loop controller

was formed using the empirical model, which was tuned using both the $iegler%Nichols method and

the tune function of &imulin', the latter of which proved superior. Thus, a disturbance in the form of

oxygen upta'e from cell culture in the tan' was considered, showing a decreased set point of ()*.

Finally, methods of #+ measurement were investigated in practice. "t was found that direct

reference adaptive and nonlinear predictive controllers are in use, and a caste study of #+ control

for a complex water networ' in Chicago was investigated.

UNIVERSITY OF BIRMINGHAMSCHOOL OF CHEMICAL ENGINEERING

Lab Group 22



!rinciples of !rocess Control xercises

Contents

1. Introduction.............................................................................................................................................................1

2. Experimenta !pparatus..........................................................................................................................................1

". Empirica #ode o$ %x&gen #ass 'rans$er (&namics............................................................................................2

".1. )arameter Estimation *ased on the In$ection )oint o$ the Step +esponse Cure %ne )oint.......................2

".2. 'hree )oints....................................................................................................................................................."

".". +esuts o$ Estimation......................................................................................................................................."

".. E$$ects o$ Energ& Input on S&stem +esponse.................................................................................................."./. Seection o$ #anipuated 0ariabe.................................................................................................................../

. 'heoretica #ode o$ %x&gen #ass 'rans$er (&namics........................................................................................./

.1. Soing $or a input o$ step change ................................................................................................................

.2. %pen Loop +esponse.......................................................................................................................................3

.2.1. 'hree modes o$ probe and mass trans$er d&namics................................................................................3

.2.2. %pen oop response o$ three modes........................................................................................................4

.2.". #ass 'rans$er )rocess What is actua& happening in the s&stem s. 5hat 5as measured....................6

.". Cosed Loop 7eedbac8 Contro S&stem 95ith )I(:.......................................................................................1;

.".1. Simuin8 'une $unction..........................................................................................................................11

.".2. ieger-<ichos method $or )I( caibration..........................................................................................12

.. Cosed Loop 7eedbac8 Contro S&stem 95ith )I( and %2 =pta8e:...............................................................1..1. ! note on dead time...............................................................................................................................1

..2. Comparison o$ modes 5ith and 5ithout disturbance............................................................................1/

/. (issoed %x&gen contro in practice....................................................................................................................1

. +e$erences...............................................................................................................................................................1

7igure 1> !pparatus Con$iguration..................................................................................................................................1

7igure 2> %era s&stem response data...........................................................................................................................2

7igure "> '&pica response o$ the s&stem, < ? 3;; +)#, Q ? L@min...........................................................................2

7igure > In$ection point o$ a t&pica second order pus dead time response..................................................................2

7igure /> Simuin8 boc8 diagram o$ empirica mode...................................................................................................."

7igure > <? 3;; +)#, Q? L@min Comparison o$ Experimenta and Empirica #ode...............................................

7igure 3> <?";; +)#, Q?2 L@min Comparison o$ Experimenta and Empirica #ode................................................7igure 4> +esponse o$ the s&stem $or an identica step change under con$igurations used..............................................

7igure 6> Comparison o$ A and ' $or di$$erent impeer speeds and air $o5rates............................................................/

7igure 1;> %pen Loop +esponse (ata $or Q? L@min.....................................................................................................4

7igure 11> %pen Loop +esponse (ata $or Q?2 L@min.....................................................................................................4

7igure 12> %pen Loop +esponse 5ith #ass 'rans$er )rocess incuded, Q? [email protected]

7igure 1"> %pen Loop +esponse 5ith #ass 'rans$er )rocess incuded, Q?2 [email protected]

7igure 1> Cosed Loop 7eedbac8 Contro S&stem 95ith )I(:.....................................................................................1;

7igure 1/> !ternate $orm o$ Cosed Loop 7eedbac8 Contro S&stem 9With )I(:.........................................................11

7igure 1> Soution o$ tune $unction $or )I( controer, 3;; +)#, L@min.................................................................11

7igure 13> Empirica )I( Cosed Loop 7eedbac8 Contro S&stem...............................................................................12

7igure 14> )I( response $rom ieger-<ichos tuning method......................................................................................1"7igure 16> Scope o$ step into ce gro5th trans$er $unction...........................................................................................1

7igure 2;> Simuin8 mode o$ step change be$ore ce gro5th trans$er $unction...........................................................1

7igure 21> Scope o$ ramp into ce gro5th trans$er $unction.........................................................................................1

7igure 22> Simuin8 #ode With %x&gen =pta8e (isturbance.....................................................................................1

7igure 2"> S&stem response 5ith disturbance Q? L@min.............................................................................................1/

7igure 2> S&stem response 5ith disturbance, Q?2 L@min............................................................................................1/

Y

'abe 1> 'an8 )arameters................................................................................................................................................1

'abe 2> 0aues o$ yI ,A, XI,tI and ' based on point o$ in$ection 9one point method:........................................"

'abe "> 0aues o$ α , A and ' based on three point method...................................................................................... "

'abe > #ode )arameters $or use in theoretica and ana&tica mode...........................................................................4

1

#usta$a IBba 1"2"26"




'abe /> Contro )arameters and +esponse )arameters o$ )I( Controer....................................................................12

'abe > +esuts o$ ieger-<ichos method..................................................................................................................1"

2





<omencature

Ite

m(escription

< Impeer Speed 9+)#:

Q !ir 7o5rate 9L@min:

ξ (amping Coe$$icient

θ (ead 'ime 9s:

T %era 'ime Constant 9s:

T p #ass 'rans$er )rocess 'ime Constant 9s:

T m )robe #easurement )rocess 'ime Constant

9s:

y I 7ractiona Step +esponse at In$ection )oint

X I

'ime o$ In$ection )oint in Step +esponse

t X

t x 'ime $or Step +esponse to reach 1; x o$

stead& state aue

C ∞ Concentration at eBuiibrium ox&gen tension

saturation 91;;:

C Concentration o$ ox&gen at instantaneous

ox&gen tension

K %era Gain

K p #ass 'rans$er )rocess Gain

K m )robe #easurement )rocess Gain

N C 7iter Coe$$icient

V 0oume 9 m3

:

) )roportiona Gain

I Integra 'ime Constant 9s:

( (eriatie 'ime Constant 9s:

1. Introduction

(issoed ox&gen has man& appications. Whist in

engineering contexts main& bioreactors are

considered, in reait& other processes such as

5aste5ater $rom se5age treatment pants, aBuatic

animas and chemica reactions aso consume ox&gen,

in its dissoed $ormD (%. I$ (% ees in such s&stems

drop beo5 a reBuired ee, sensitie aBuatic i$e can

be a$$ected, the decomposition o$ organic materias in

5aste5ater 5i not occur, and a desired chemica

reaction 5i not per$orm as sought a$ter b& the

engineer. 'hus, the abiit& to monitor $urthermore

contro (% has critica bene$its in arious scenarios.

9E)!, <.(.:

2. Experimenta !pparatus

! gass esse $ied 5ith deionised 5ater and

$itted 5ith a ba$$e cage 5as used. 'an8 speci$ications

are gien in 'abe 1. #echanica agitation 5as

achieed through the use o$ a bade +ushton turbine

impeer, 5hich can be modeed as 7*' in iterature.

Table - Tan' !arameters

! )oarographic dissoed ox&gen probe 5as used

to measure (%2 ees. 'he unit proided a reading in

terms o$ percentage saturation, 5ith 1;; saturationcacuated at a standard temperature and pressure 5ith

air as a sparged gas. In order to maintain constant

conditions o$ temperature and pressure a cooing $inger

and eectric band heater 5ere used in conunction 5ith

a contro s&stem maintaining the reBuired temperature.

!ir and nitrogen inputs coud be interchanged, and

5ere introduced through a sparger distributing the gas

$o5 in the bottom o$ the tan8. 'he $o5rate o$ air 5as

controed ia a gas $o5meter, the $o5rate o$ nitrogen

ho5eer, 5as not.

'he combined temperature and (%2 ees 5ere

ogged using Eectroab eLogger and eGrapher

so$t5are on a connected computer. 'he so$t5are

ogged data at / second interas.

'he con$iguration o$ the experimenta apparatus is

gien in 7igure 1.

3


)arameter (e$inition 0aue

' 'an8 (iameter 9m: ;.12

( Impeer (iameter 9m: ;.;42

C Impeer Cearance $rom 'an8

*ottom 9m:

;.;

F 'an8 7i Feight 9m: ;.12

W Impeer *ade Width 9m: ;.;26




Figure - pparatus Configuration

". Empirica #ode o$ %x&gen #ass 'rans$er

(&namics

<itrogen 5as used to purge the soution o$ ox&gen,

thus reducing the measured (%2 ee to a minimum

o$ ;.3. 'his represented a step do5n o$ ox&gen partia pressure in the s&stem. ! step up o$ ox&gen

partia pressure 5as achieed b& exchanging the

nitrogen input 5ith that o$ air, at a 8no5n $o5rate.

'he s&stem responses to these disturbances can be

used to Buanti$& process parameters such as time

constant, τ and the gain, K o$ the s&stem under

di$$erent operating conditions 9impeer +)# and air

$o5rate:. Fence a decision can be made on 5hich

condition is more signi$icant in the nature o$ contro

parameters, and thus the condition to use as amanipuated ariabe in a $eedbac8 contro oop to

contro dissoed ox&gen tension can be determined.

In tota step up and step do5n operations 5ere

per$ormed under " impeer +)#s and 2 gas $o5rates.

'he oera s&stem response is sho5n in 7igure 2.

-1;;;.;;

1;;;.;;

";;;.;;

;

2;

;

;

4;

1;;

1;;

";;

/;;

3;;

6;;

(%2 9: Stirrer Speed 9rpm:

'ime 9s:

(%2 9: Impeer Speed 9+)#:

Figure ( +verall system response data

%n coser obseration o$ the data the nature o$

the measurement s&stem response can be determined.

In the experimenta procedure, ogging 5as started as

soon as the step change 5as made. I.e. 5hen a constant$o5rate o$ air 5as introduced. 'he response o$ the

s&stem $rom this point $or each run is noted, in order to

account $or an& dead time.

'hus, a t&pica response o$ the s&stem is gien

in 7igure ". +ecaing that each point represents an

intera o$ / seconds, it is deduced that a / second

dead time ensues, a$ter 5hich a second order oer

damped response occurs. 'his is usti$ied as there is no

osciation, hence damping is su$$icient. Fo5eer it

appears that the mode is inhibited in reaching stead&

state in good time due to oer damping b& the so5

response o$ the s&stem.

26".;; "1".;; """.;; "/".;; "3".;;;

2;

;

;

4;

1;;

< ? 3;; +)#, Q ? L@min

'ime 9s:

(%2 9:

Figure / Typical response of the system, N 0 1))

2!3, 4 0 5 67min

Fuang 91642: proposed methods o$ estimation$or such s&stems 5ithout computer usage. 'he

a$orementioned methods achiee estimation o$ mode

4





parameters $rom one, t5o or three points o$ the step-

response data in conunction 5ith correations. 'he

second order pus dead time $ound in this experiment

is not dissimiar to man& contro s&stems used in rea

chemica processes. Fence its inestigation is o$ great

use.

".1.)arameter Estimation *ased on the In$ection)oint o$ the Step +esponse Cure %ne )oint

(i$$erent methods o$ parameter estimation are

proposed, the& are per$ormed as $oo5s. Figher order

responses hae in$ection points, these hae been

$ound to be in8ed to the mode parameters T and

ξ .

'he nature o$ the in$ection point $or such a

s&stem is sho5n in 7igure , courtes& o$ 'u$a 9<.(.:.

Figure 5 "nflection point of a typical second order

plus dead time response

Fence, as the s&stem considered is oer damped,

ξ>1 >

t I

T =

1

(ξ2−1 )0.5 tanh

−1

( (ξ2−1)0.5

ξ

) y I =1−

2 ξ

ξ+ (ξ2−1 )0.5 (ξ−(ξ2−1)0.5

ξ+(ξ2−1 )0.5 )exp(ξ−(ξ2

2 (ξ2

'he soution o$ ξ $or a 8no5n aue o$

y I is di$$icut due to the nature o$ the eBuation.

Where y I = output at inflection

final steady state value of output .

'hus an empirica inerse $unction can be used>

ξ=0.8637 y I −0.578−0.865

'he aboe correation is aid $or

0.0365 ≤ y I ≤0.358 .

<ote that in a cases θ=5 s . 'hus

t I = X I −θ . !s expected ξ>1 as the s&stem is

oerdamped. Fence, the $oo5ing eBuation can be

soed to gie T .

T = t I (ξ

2−1 )0.5

tanh−1( (ξ

2−1 )0.5

ξ )".2.'hree )oints(ue to the nature o$ the in$ection point, the one

point method can proe to &ied signi$icant error. 'hus,

the t5o or three point method based on the aues o$

step response data can be used. 'he three point method

is the most accurate o$ the estimations, as it considers

the most points.

'he t5o@three point methods re& on the duration

o$ time eapsed $or the s&stem response to reach 1;,

/; and 6; o$ its stead& state aue 9past the dead

time:, denoted t 1 , t 5 and t 9 respectie&.

Correations hae been estabished reating these

parameters to ξ and in turn aues o$T t

1 ,

T t 5 andT t

9 can be obtained and then aeraged

to obtain T . 'he methodoog& is outined as

$oo5s.

α =(t 9−t 1)/t 5

ξ=0.464610−8exp (6.4075α )+0.605α −0.2

0aid $or 1.5≤α ≤3.0 and 0.7≤ ξ ≤3.0 .

'hen>

T t 1= t 1

0.0137 ξ2+0.07267ξ+0.4445

T t 5= t 5

0.03922ξ2+1.09678 ξ+0.548

5





T t 9= t 9

−0.0469 ξ2+5ξ−0.983

".".+esuts o$ Estimation

Cacuated mode parameters $or each

con$iguration are gien in 'abe 2 and 'abe ", based

on the one and three point method respectie&.

Table ( 8alues of y I ,9, X I ,t I and T based on

point of inflection :one point method;

Impe

er

Speed

9+)#:

!ir

7o5rat

e

9L@min:

y I ξ X I ( t I ( T (s−1

";; ;.2

3

1.;

3"; 2/ 2/.

";; 2 ;.21 1.1; "/ "; "1

/;; ;.21

1

1.2

2; 1/ 1."

/;; 2;.1

2

1.

12; 1/ 13.6

3;; ;.21."

213./ 12./ 1".4

3;; 2;.2"

"

1.1

2; 1/ 1/.3

Table / 8alues of α , 9 and T based on three point

method

Impeer Speed

9+)#:

!ir 7o5rate

9L@min:α ξ T ( s

−1

";; 22.2

1."

23.2

";; 1.6

4

1.2

;2./4

/;; 21.4

3

1.1

"14.3;

/;; 1.3

2

1.;

13.24

3;; 21./

4

;.6

/1.61

3;; 1./

;

;.6

11/.4"

'hus using Simuin8 the deried empirica

mode can be tested b& potting a simuated response to

the respectie experimenta response data.

'he $orm o$ the empirica trans$er $unction,

based on a second order %(E 5ith dead time is sho5n

beo5. 'he Simuin8 mode is aso sho5n in 7igure /.

T and ξ 9gien as d in the boc8 diagram

trans$er $unction: $or each con$iguration can be

substituted in to gie the respectie empirica response.

! ( s )=" (s )

X (s )=

e−θs

T 2

s2+2Tξs+1

Figure < &imulin' bloc' diagram of empirical model

'his experimenta response@empirica mode

encompasses a s&stem that considers mass trans$er and

(%2 probe d&namics.

<ote that the gain o$ the s&stem is 1, as>

C ∞−C 0

C i=100−0

100[ ]=1

'he t5o extremities o$ s&stem conditions 5ere

compared as $oo5s. <ote that mode parameters $rom

the three point method as seen in 'abe " 5ere used, as

the& are o$ greater accurac&.

; 2; ; ; 4; 1;; 12; 1;;

1;

2;

";

;

/;

;

3;4;

6;

1;;

Experimenta Empirica

'ime 9s:

(%2 9:

Figure = N0 1)) 2!3, 405 67min Comparison of

xperimental and mpirical 3odel

6





; /; 1;; 1/; 2;; 2/; ";;;

1;

2;

";;

/;

;

3;

4;

6;

1;;

Experimenta Empirica

'ime 9s:

(%2 9:

Figure 1 N0/)) 2!3, 40( 67min Comparison of xperimental and mpirical 3odel

7rom 7igure and 7igure 3 it is cear that the

empirica response o$ a s&stem 5ith ess energ& input

9so5er impeer +)# and air $o5rate: is modeed

5ith a greater accurac& than the s&stem 5ith greater

energ& input, noting that sma error is expected

regardess due to the nature o$ the estimation used.

'his is due to the so5er s&stem changing at a esser

rate in a gien time intera 9/ seconds in the case o$

the data ogger:. 'here$ore 5hen considering an

in$ection point or t x , the nature o$ the s&stem

around the a$orementioned point is in greater

uncertaint& 5hen ess data measurements are ta8en and

the ox&gen tension experiences a drastic change in the

considered time intera. Fence the cause o$ error.

"..E$$ects o$ Energ& Input on S&stem +esponse

'he t5o $orms o$ energ& input 5ere

mechanica agitation $rom the impeer and $o5 o$ air

into the s&stem.

7igure 4 compares the responses o$ the s&stem

$or an identica step change in ox&gen partia pressure

under the conditions tested.

; /; 1;; 1/; 2;; 2/; ";; "/;;

1;

2;

";

;

/;

;

3;

4;

6;

1;;

3;; +)#, L@min 3;; +)#, 2 L@min

/;; +)#, L@min /;; +)#, 2 L@min

";; +)#, 2 L@min ";; +)#, L@min

'ime 9s:

(%2 9:

Figure > 2esponse of the system for an identical stepchange under configurations used

7rom the $igure it is cear that the time

reBuired $or the s&stem to reach stead& state is

inerse& proportiona to the impeer speed and air

$o5rate.

It is obsered that $or an identica impeer

speed, doubing the air $o5rate reduces the transient

time. ! decreased transient time suggests the s&stems

damping coe$$icient, ξ , is moing coser to a

critica& damped circumstance, 5here ξ=1 . In

conunction 5ith the a$orementioned change, the

s&stems 5ith greater energ& input respond $aster, hence

their aues o$ time constant, T , can be expected to

be smaer. 7igure 6 assesses the a$orementioned

expectations.

7





";; "/; ;; /; /;; //; ;; /; 3;;;.4;

;.6;

1.;;

1.1;

1.2;

1.";

1.;

1;.;;

12.;;

1.;;1.;;

14.;;

2;.;;

22.;;

2.;;

2.;;

24.;;

";.;;

A 92 L@min: A 9 L@min:

' 92 L@min: ' 9 L@min:

Impeer Speed 9+)#:

A ' 9s-1:

Figure ? Comparison of 9 and T for different impeller speeds and air flowrates

Fence, both T and ξ are sho5n to

decrease as energ& input increases. Fo5eer ξ

$as beo5 1 $or an impeer speed at 3;; +)#.

'hus the oer damping condition is not maintained,

and the empirica mode experiences sight oershoot.

With regards to 5hich $orm o$ energ& input is

more signi$icant in a$$ecting mode parameters, 7igure6 sho5s that impeer speed has a greater e$$ect on the

aues o$ ξ and T , i$ the anoma& o$

ξ=1.6 is omitted. In terms o$ the transient period,

7igure 4 sho5s that impeer speed decreases the

transient period to a greater extent than doubing the

air $o5rate.

"./.Seection o$ #anipuated 0ariabe

'he seection o$ the manipuated ariabe in this

scenario shoud be impeer speed, as it has a greatersigni$icance in disturbance to the s&stem. Fence,

controing dissoed ox&gen tension 5ith such a

s&stem 5oud be more sensitie to a change in input,

i.e. a greater gain. 'he remaining ariabe, air $o5rate,

5oud remain as a disturbance ariabe. 'his 5oud

input some disturbance to the s&stem as the aue o$ air

$o5rate natura& $uctuated, ho5eer it 5oud be

$aourabe oer the reerse scenario. !ternatie& a

mutipe input s&stem 9#IS% or #I#%: coud be

empo&ed to use both inputs as manipuated ariabes.

9+obbins, 2;1/:

'he reasoning $or impeer speed haing a greater

impication to the s&stem response is that as

determined $urther on, the s&stem time constant,

T p= 1

# $ α , a propert& o$ the s&stem. !s mechanica

agitation increases bubbe siHe decreases and so the

inter$acia area aaiabe $or mass trans$er, α ,

increasesD resuting in a decreased time constant $or a

s&stem 5ith higher impeer speed. 'he impact o$ air

$o5rate a$$ects the rate at 5hich a $ixed concentration

o$ ox&gen 921: is introduced to the s&stem,

technica& it 5oud create a greater concentration

gradient $or mass trans$er, ho5eer $rom the data this

is sho5n to be ess signi$icant than mechanica

agitation. 9Simmons, 2;1:

. 'heoretica #ode o$ %x&gen #ass 'rans$er(&namics

!ssuming constant iBuid densit& and temperature

in$ers good stirring and constant oume. !dditiona&

regarding the mass trans$er d&namics the gas side mass

trans$er coe$$icient signi$icant& greater than the iBuid

side mass trans$er coe$$icient, i.e. # !≫# $ . 'hus,

gas side mass trans$er d&namics can be ignored.

'hus ta8ing a mass baance on the bubbe inter$ace

in the tan8, $or basis %t >

Input +!ene&ation='utput + (eaction+ )ccumu

Input =V# $ α (C ∞−C ) %t

Chec8ing units o$ input>

m3

s−1

#*m−3

s + # *

'utput =0 , (eaction=0 ,

!ene&ation=0

)ccumulation=V%C

Chec8ing units o$ output>

m3

#* m−3

+ #*

+earranging and canceing V >

%C

%t =# $ α (C ∞−C )





Let %t 0 ,%C

%t

dC

dt so the soution

becomes exact>

dC

dt =# $ α (C ∞−C )

Changing to deiation ariabes>

C =C ¿−C s

C ∞=C ∞¿−C ∞s

d C s

dt =0

C ∞s=0

d C ¿

dt =# $ α (C ∞

¿−C ¿)

+earranging to obtain the aboe eBuation in the

$orm o$ a $irst order %(E>

1

# $ α

d C ¿

dt +C

¿=C ∞¿

Gies>

T pd C

¿

dt +C

¿= K p C ∞¿

(1)

Where>

T p= 1

# $ α , K p=1

Considering the mode $or the (%2 probe

9+obbins, 2;1/:>

T -

d C m

dt +C m= K m C

¿

Changing to deiation ariabes>

C m=C m¿−C ms

d C ms

dt

=0

T md C m

¿

dt +C m

¿ = K m C ¿

(2)

!pp&ing a Lapace trans$ormation to (1)>

T p s [C ¿ ( s )−C ¿ t =o

¿ ]+C ¿(s)= K pC ∞

¿

Initia& there is no deiation in the s&stem, so>

C ¿ t =0

¿+C

0

¿=0

T p s C ¿ ( s )+C

¿(s)= K pC ∞¿

(3)

!pp&ing a Lapace trans$ormation to (2)>

T m [ s C ¿ (s )−C m

¿¿t =0 ]+C m

¿ (s )= K m C ¿( s)

!ssume C m¿ ¿t =0=0 , as there is no deiation in

the s&stem initia&.

T m s C m¿ ( s)+C m

¿ (s )= K m C ¿(s) (4)

+ecaing that the trans$er $unction is eBuiaent to

the output oer the input, (3) and (4) can be rearranged

to the $oo5ing $orms.

!1( s )=

" (s )

X ( s) =

C ¿ (s )

C ∞¿

( s ) fo& (3)

!2( s )=

" (s )

X ( s )=

C m¿ ( s)

C ¿ (s )

fo& (4)

Fence rearranging (3) and (4)>

C ¿

C ∞¿ =

K p

T p s+1 (5)

C m¿ (s )C ¿

( s) = K m

T m s+1 (6)

'hen the oera trans$er $unction o$ the s&stem,

!(s) is represented b&>

! ( s )=!1( s) !

2(s)

'hus mutip&ing (5) and (6)>

! ( s )= K p K m

( T m s+1 ) (T p s+1)

9





T p+T

T mT p s2+(¿¿ m) s+1

! (s )=C m

¿

C ¿=

K p K m¿

(7)

+ecaing the $orm o$ a second order trans$er $unction>

" (s ) X (s )

= K

T 2

s2+2ξT s+1

It is sho5n>

T 2=T m T p , (T p+T m)=2ξT

, K = K p K m

∴T =√ T mT p

T p+T m=2ξ √ T m T p

∴ξ= T p+T m

2√ T m T p

K p=1 $rom be$ore, the probe gain, K m=1

aso as the change in input o$ ox&gen concentration

saturation percentage 5as eBuiaent to the change in

measured ox&gen saturation percentage 9; to 1;;:.

∴ K =1

.1.Soing $or a input o$ step change

(7) is the d&namic mode o$ the s&stem deried

theoretica&, 5hich can no5 be soed $or a step

change o$.

s in the input, C ¿ (s ) , to gie the

output to the step change C m¿ (s) .

'hus, substituting C ¿ (s ) $or . s >

C m¿ (s).

s

= 1

(T m s+1 ) (T p s+1 )

C m¿ (s)=

.

1

s (T m s+1 ) ( T p s+1 )

!pp&ing partia $ractions>

.

s (T m s +1 ) (T p s +1 )+

)

s +

/

T m s+1+

C

T p s+1

. = ) ( T m s+1) (T p s+1)+/s (T p s+1 )+Cs(T m s+1)

(8)

7or (8)>

Let s=0 , )=. .

Lets=

−1

T m>

. =/(−1

T m )(−T p

T m+1)

/= . T -

2

T p−T m

Lets=

−1

T p>

. =C (−1

T p )(−T m

T p+1)

C = . T p

2

T m−T p

'hus the response o$ the s&stem $or a step

change is gien b& the $oo5ing>

C m¿ (s)=

.

s +

. T m2

T p−T mT m s+1

+

. T p2

T m−T pT p s+1

In order to use this eBuation it needs to be

conerted bac8 to the time domain, this is achieed b&

rearranging the terms into $orms o$ 5hich inerse

Lapace trans$ormations can be in$erred $rom the

$or5ard Lapace trans$ormation tabe.

C m¿ ( s)=.

s +

. T m2

T p−T m ( 1

T m )( 1

s+ 1

T m)+

. T p2

T m−T p ( 1

T p )( 1

s+ 1

T p)

1!





C m¿ ( s)=

.

s +

. T mT p−T m (

1

s + 1

T m)+

. T pT m−T p (

1

s + 1

T p)

C m¿ ( t )=. +( . T m

T p−T m )e

−1

T mt

+( . T p

T m−T p )e

−1

T pt

(9)

Chec8ing units o$ (9)>

#* m−3=#* m

−3+( #* m−3

s

s )e

s

s+( #* m−3

s

s )e

s

s + #*

Chec8ing extreme behaiour o$ (9)>

t =0 0 C m¿ =0 as expected $rom initia gradient o$ ;.

t ∞ 0 C m¿=. as expected $rom $ina stead& state

aue o$ 1;; (%2. Fence the ana&tica eBuation is

sho5n to behae as expected.

.2.%pen Loop +esponse

.2.1. 'hree modes o$ probe and mass trans$er

d&namics

'hree modes o$ step response data can no5 begenerated. Experimenta data as sho5n in 7igure 4, a

theoretica& deried mode to be soed numerica&

9through Simuin8: as sho5n in eBuation (7) and an

ana&tica& soed theoretica mode as sho5n in

eBuation (9). 'hese can be denoted as Experimenta,

'heoretica 9e$$ectie& numerica due to the 5a&

#atL!* soes in the Lapace domain: and !na&tica

modes respectie&. !dditiona data is reBuired $or the

theoretica and ana&tica modes, speci$ica& the

mode parametersT m, T p, ξ∧# $ α

. 'hese are

obtained $rom experimenta data.

!s noted be$ore, T p=

1

# $ α .

T m can be obtained as the oera time

constant, T is 8no5n. +ecaing>

T =√ T m T p

∴T m= T

2

T p

'he reBuired data 5as cacuated and sho5n beo5 in

'abe .

Table 5 3odel !arameters for use in theoretical and

analytical model

.2.2. %pen oop response o$ three modes

'he open oop response o$ the three modes

can no5 be ana&sed, as $oo5s. <ote that a dead time

o$ / seconds has been incuded in a cases. In order to

mode a more accurate dead time the intera duration

o$ the data ogger needed to be o$ a esser aue in

order to more accurate& obsere 5hen the s&stem

responded to the step change. It 5oud be expected $or

a s&stem o$ higher energ& input to respond $aster as

T ∝1

Impelle& 1peed ( (2- )1

)i& 3lo4&ate( $/min)

, thus higher energ& input s&stems can be expected to

hae o5er dead times in reait&.

11


mpee

Speed

+)#:

!ir

7o5rat

e

9L@min:

# $ α (s−1

ξ T (s) T p ( s T m(s

";; 2 ;.;2 1." 23.2 /.2/ 1.2

";; ;.;" 1.2; 2./4 "/.;6 13.22

/;; 2 ;.; 1.1" 14.3; 2.21 1.

/;; ;.;/ 1.; 13.24 14."/ 1.24

3;; 2 ;.;/ ;.6/ 1.61 21.// 1".23

3;; ;.;3 ;.61 1/.4" 1".3 14.21




Figure -) +pen 6oop 2esponse #ata for 405 67min

Figure -- +pen 6oop 2esponse #ata for 40( 67min

It is assumed that the probe 5as $ast, i.e.

accurate& represents the true nature o$ the s&stem

$rom the experimenta resuts. Fence, $rom 7igure 1;7igure 11 it is sho5n that the theoretica and ana&tica

soutions to the s&stem response behaiour are

accurate to an acceptabe degree. 'he most critica

regimes o$ the response behaiour is matched, being

the transient duration in the beginning o$ the step

response 9second order pus dead time:, and the $ina

stead& state aue reached 91;;:. With contro

s&stems in mind, being abe to measure and understand

a s&stem as cose to the exact nature occurring is

essentia in $urther contro procedures such asmanipuated ariabes and set points.

Whist the three s&stems o$ response coincide

5e at the beginning o$ their respectie transient

periods and to5ards the occurrence o$ stead& state,

bet5een these t5o ocations the di$$erences bet5een

the three modes is obious. 'his error is sho5n to

increase $or a esser air $o5rate and increased impeer

speed. 'his is due to the d&namics o$ the s&stem, and

thus mode parameters changing. 7or a s&stem that

changes at a greater rate in a gien time basis, adecreased time constant and increased s&stem gain is

reBuired to sustain accurac&. 'hese trends are sho5n in

7igure 6, ho5eer s&stems 5ith greater disturbance

9increased impeer speed: become more erratic and

thus harder to mode accurate&. (ue to this, $or air

$o5rates o$ 4 $ /min the experimenta s&stem

reaches stead& state sight& earier than the theoretica

or ana&tica modes. ! simiar trend is obsered $or an

air $o5rate o$ 2 $/min .

.2.". #ass 'rans$er )rocess What is actua&

happening in the s&stem s. 5hat 5as

measured

'he preious discussion concerned a s&stem

consisting o$ the mass trans$er process and probe

d&namics. In order to tru& Buanti$& the importance o$

probe d&namics, a mode o$ the mass trans$er process

aone is reBuired. 'his is in the $oo5ing $orm o$ the

soed $irst order %(E obtained $rom the mass

baance>

C =C ∞−C ∞ e−# $ αt (A)

Chec8ing units>

#* m−3=#* m

−3−#* m3

es−1

s+#*m

−3

Chec8ing extreme behaiour>

t =0 0 C =0 , as expected.

t ∞ 0 C =100 , as expected.

12





'hus (A) is dimensiona& consistent and

behaes as expected.

'his $orms the $ourth mode 5hich can be used

in the open oop response. 'he di$$erence is that this

mode represents the mass trans$er process on&, and so

i$ assumed accurate, it represents 5hat is actua&

happening in the ph&sica process. Its comparison tothe other three modes, 5hich account $or the mass

trans$er process and probe d&namicsD 5i determine

the importance o$ probe d&namics in this case. 'he

aboe #') 9mass trans$er process: mode is expected

to be o$ a $irst order response acting instant& 9at t?;:,

as this 5as the exact point the step change 5as made.

I$ the $our modes coincide 5ith each other, the probe

is su$$icient& $ast to mode the process accurate&,

ho5eer, i$ signi$icant ag occurs, it can be deduced

that the probe 5as so5. 'he resuts o$ the ana&sis are

sho5n as $oo5s.

Figure -( +pen 6oop 2esponse with 3ass Transfer

!rocess included, 405 67min

Figure -/ +pen 6oop 2esponse with 3ass Transfer

!rocess included, 40( 67min

%n obseration o$ 7igure 12 and 7igure 1" a cear deduction can be made. 'he probe 5as so5. 'he s&stem

behaes as expected, 5here greater energ& input resuts in a reduced transient and setting time due to increased mass

trans$er, as expained in section "./. 'he probe is sho5n to ag behind the mass trans$er process, responding on& a$ter

its initia dead time o$ / seconds. !$ter 5hich it $oo5s the predetermined second order response cure. (epending on

the s&stem con$iguration the probe is sho5n to reach stead& state bet5een /; and 3/ seconds behind the probe,

responding reatie& so5er to s&stems 5ith higher energ& input, een though rise time and setting time are

decreased. In terms o$ using the s&stem to contro (%2, 5ith a so5 probe as used in the experiment issues coud

13





arise in stabiit&. Fo5eer as a simpe step change 5as made, and ph&sica imitations are in pace 9maximum

soubiit& o$ ox&gen in 5ater:, a simpe measurement s&stem is acceptabe. Fo5eer in a $uctuating s&stem $aster

measurement s&stems are reBuired to achiee the desired set point and remain 5ithin acceptabe range o$ it.

.".Cosed Loop 7eedbac8 Contro S&stem 95ith )I(:

In )I( contro, a process ariabe and a set point can be speci$ied. In this case the process ariabe 5as seected as

the manipuated ariabe, impeer speed, and the set point is the desired aue o$ C setpoint , 5hich in this case 5as

C ∞(100 5' 2 ) . 'he )I( controer determines the output aue o$ impeer speed reBuired to drie the process

coser to the set point, based on the error input to the controer.

'he three eements o$ a )I( controer gie the $oo5ing outputs>

• ) eement> )roportiona to the error at the instant, t .

o )resent error

• I eement> )roportiona to the integra o$ the error up to the instant, t .

o !ccumuation o$ past error

• ( eement> )roportiona to the deriatie o$ the error at the instant, t .

o )rediction o$ $uture error

Fence, a )I( controer uses the present, past and $uture aues o$ an input error into consideration in determining

its output. It is $or this reason )I( controers are $ar superior to simper proportiona methods, as seen be$ore.

9heera, 2;11:

'a8ing the empirica mode obtained in 7igure /, a )I( controer can be added, subtracting the output C m¿

$rom the input 5esi&ed 5'2 to gie an error. !s sho5n in 7igure 1. 'his assumes the purpose o$ the s&stem is

to reach a (%2 o$ 1;;, hence the speci$ied set point.

Figure -5 Closed 6oop Feedbac' Control &ystem :with !"#;

'he aboe Simuin8 mode actua& has t5o $orms, the one aboe 5ith has an oera trans$er $unction 5ith a

dead time be$ore it, and the $oo5ing $orm 5hich separates the second order trans$er $unction into t5o $irst order

$unctions. %ne o$ the $irst order $unctions is that o$ the mass trans$er process, 5hich is assumed to happen

instantaneous& and thus has no dead time. 'he second $irst order $unction is that o$ the probe measurement process,

5hich is assumed to encompass the entiret& o$ the dead time, as is represented in the Simuin8 mode in 7igure 1/.

14





Figure -< lternate form of Closed 6oop Feedbac' Control &ystem :@ith !"#;

.".1. Simuin8 'une $unction

5./.-.-. +vershoot A an issue

'&pica& correations or mathematica soutions can be used to obtain the parameters o$ the )I( controer,

being proportiona gain, integrator time constant and deriatie time constant. Fo5eer, Simuin8 incudes a tune

$unction, 5hich automates the process. !n initia tune procedure $or <?3;; +)#, Q? L@min &ieds the $oo5ing

response>

Figure -= &olution of tune function for !"# controller, 1)) 2!3, 5 67min

'he response in 7igure 1 seems $ast and controed, 5ith a $aster setting time o$ 77.6 seconds. Fo5eer

an oershoot occurs due to underdamping, /."6 to be exact.

'he aboe 5oud seem a reasonabe con$iguration o$ the s&stem. Fo5eer considering the impication o$

reait&, speci$ica& ph&sicsD is it possibe to hae a (%2 oer 1;; in the ph&sica s&stemJ 'he ans5er is no.

%bsering the $igure again sho5s the oershoot surpasses this maxima aue, and thus the contro s&stem is

attempting to generate a aue o$ (%2 in the s&stem that is in$easibe. 'hus in reait& the measured aue 5i pea8

at 1;;, and the contro s&stem 5i attempt to continue increasing the manipuated ariabe to cause an increase in

the contro ariabe. 'his 5i neer happen, and so the s&stem 5i become unstabe. In regards to process

optimisation this obious& is not acceptabe. Sa$et& considerations shoud aso be critica in process contro and so a

s&stem constant& increasing impeer speed@air $o5rate 5i cause sa$et& issues, not to mention damage to eBuipment

and thus s&stem stabiit&D resuting in een $urther sa$et& issues.

In order to $ix this issue, $urther tuning coud be done. Fo5eer this is on& a temporar& $ix o$ the issue. 'he

error occurs due to imitations in the $ormuation o$ the mode used. 'hus $or eer& speci$ic con$iguration this manua

tuning adustment 5oud be reBuired to preent the s&stem $rom surpassing 1;;. What is instead reBuired is a

modi$ied mode that cannot surpass 1;;. +egardess, the resuts o$ the )I( tuning are sho5n in 7igure 13.

15





5./.-.(. 2esults of &imulin' tuning for all configurations

Figure -1 mpirical !"# Closed 6oop Feedbac' Control &ystem

'he mode parameters are gien in 'abe /.

Table < Control !arameters and 2esponse !arameters of !"# Controller

<

9+)#:

Q

9L@min:

)roportiona

9):

Integra

9I:

(eriatie

9(:

7iter

Coe$$icient

9<C:

+ise

'ime 9s:

Setting

time 9s:

%ershoot

9:

)ea8

9(%2:

";; 2 1.4 ;.;" ;.;; 1;;.;; 1.2; 1.;; 3.4 1;;.4;

";; 2.;3 ;.;" ;.;; 1;;.;; "1.6; 1;.;; .1" 1;;.;

/;; 2 1.4/ ;.; /.3 ;.;3 22.3; 1;6.;; 3.1/ 1;;.3;

/;; 1.63 ;.;/ 6.4; ;.1" 13./; 6;.2; 3.23 1;;.3;

3;; 2 1.41 ;.;/ 6.4; ;.13 13.2; 46.3; .41 1;;.3;

3;; 2.;3 ;.; 1"."; ;.2 1".; 33.; /."6 1;;./;

7rom 'abe / and 7igure 13 it is deduced that con$igurations 5ith higher energ& inputs respond $aster. I.e. the

energ& input to the s&stem is inerse& proportiona to setting time and rise time. %ershoot is around 3 in a cases

except $or the con$iguration 5ith the highest energ& input, in 5hich oershoot /.. !n interesting behaiour is

obsered 5ith the aue o$ 7iter Coe$$icient, <C, and the (eriatie contro parameter, (. 7or +)#s o$ ";; (?; and

<C?1;;..".2. ieger-<ichos method $or )I( caibration

!ternatie&, the ieger-<ichos method can be appied to the second order s&stem 5ith dead time in use to

estimate sensibe parameters $or the )I( controer. 7irst&, K u , the utimate gain needs to be determined b& setting

I and ( to ;, and increasing, proportiona gain, 2 , unti osciations o$ a constant magnitude occur. 7rom this

aue, and the osciation period,T u , parameters can be de$ined as $oo5s>

2=0.6 K u , I = 2

T u, 5=

T u

8

16





'his procedure 5as underta8en $or each con$iguration. 'he nature o$ K u can be determined as $or 2< K u ,

osciations 5i diminish. Whereas $or 2> K u , osciation magnitude 5i increase, and the s&stem 5i be

unstabe. !t 2= K u , osciation magnitude is constant, and the s&stem is critica& stabe. 'he resuts o$ the

process $oo5. 9ieger, 162:

Table = 2esults of $iegler%Nichols method

< 9+)#: Q 9L@min: u 'u 9s: ) I 9s: ( 9s:

";; 2 1.3; /;.;; ".6" ;.;/ /.;;

";; 1".2/ /;.;; .6" ;.; /."

/;; 2 6.3/ ";.;; 3.6/ ;.; .2/

/;; 4.21 /.;; 1;.;2 ;.; .2/

3;; 2 3.; ;.;; /.4/ ;.;3 ".3/

3;; .// ;.;; . ;.;/ /.;;

Figure -> !"# response from $iegler%Nichols tuning method

Comparing 7igure 13 and 7igure 14 it is cear that the tune method o$ Simuin8 is $ar superior, $or

signi$icant& ess e$$ort. 'he ieger-<ichos method sho5s signi$icant osciation, up to ; $or 3;; +)#, L@min.'he t&pe o$ tuning method used depends on the purpose o$ the contro s&stem. ! baance must be made bet5een a $ast

responding, underdamped, oershooting response and a more conseratie, so5er responseD 5ith no oershoot.

..Cosed Loop 7eedbac8 Contro S&stem 95ith

)I( and %2 =pta8e:

7rom the origina mass baance, another term can

no5 be considered in the s&stem, due to ox&gen upta8e

b& ces in the tan8. 'his 5oud reBuire $urther

in$ormation and assumptions to understand the

d&namics under 5hich the ces operate, 5hich

depends on the number o$ ces in the tan8. !s the ce

cuture gro5s exponentia& under su$$icient ox&gen

supp&, the ox&gen upta8e 5i increase according&.

Fo5eer, i$ the assumption o$ ox&gen respiration in

the ces being eBuiaent to a ramp input to the

eBuiaent trans$er $unctionD accounting $or the

increase in ce number, a sensibe resut can be

obtained. In essence, assuming the ox&gen respiration

in ces is imited b& the rate o$ mass trans$er o$

ox&gen $rom the gas phase to the iBuid phase, a$ter5hich the ce number increases and so ox&gen upta8e

5i increase. In order to consider such a process in

17





Simuin8, the mass trans$er $unction obtained in (A)

can be used 5hen conerted to the Lapace domain.

I.e. f ( t )=eat

+ 3 ( s )= 1

s−a , 5ith eBuation (5).

9+obbins, 2;1/: !pp&ing a ramp be$ore the trans$er

$unction accounts $or the exponentia nature o$ ce

gro5th, opposed to the $irst order response that 5oudoccur i$ a step change 5as used. ! step change 5as

inestigated in 7igure 16 and 7igure 2;, sho5ing the

expected $irst order response, 5hich is dissimiar to

ce gro5th 8ineticsD i.e. not exponentia.

Figure -? &cope of step into cell growth transfer

function

Figure () &imulin' model of step change before cell

growth transfer function

Figure (- &cope of ramp into cell growth transfer

function

7igure 21 sho5s the scope 5hen a ramp input is used

5ith the ce gro5th trans$er $unction. 'his is more

simiar to the exponentia behaiour obsered in

gro5th 8inetics. Fence the $ina Simuin8 mode is

sho5n beo5 in 7igure 22. Where $or the ramp, initia

output is 1;; and the sope is 1.

Figure (( &imulin' 3odel @ith +xygen Bpta'e

#isturbance

..1. ! note on dead time

<ote that the dead time o$ / seconds has been

appied a$ter the process and be$ore the measurement

s&stem in 7igure 22D this assumes that a o$ the dead

time is accounted $or in the probe. Fo5eer, in reait&,

the mass trans$er process o$ ox&gen dissoing into the

iBuid 5i reBuire a short duration o$ time. 'he same is

$or the ox&gen upta8e b& the ces. 'he nature o$ these

t5o dead times is un8no5n, as the dead time obtained

5as $rom experimenta resuts, 5here the s&stemconsisted o$ the mass trans$er process and the probe

measurement process. I$ the dead times o$ the mass

1





trans$er process and ce upta8e process 5ere o$

signi$icant aue, the& coud be added as t5o separate

dead times be$ore their trans$er boc8s in the Simuin8

boc8 diagram.

..2. Comparison o$ modes 5ith and 5ithout

disturbance

Figure (/ &ystem response with disturbance 405

67min

Figure (5 &ystem response with disturbance, 40(

67min

7rom 7igure 2" and 7igure 2 it is cear that

the contro s&stem has success$u& detected theconsumption o$ ox&gen occurring. 'his is sho5n b&

the response rising ess, and setting at around 4; (%

in a con$igurations. Fence, the desired set point o$

1;; (% is unobtainabe due to the disturbance. 'he

next step 5oud be to introduce an input o$ the

manipuated ariabe, impeer speed, 5ith a

compimentar& trans$er $unction and gain to enabe the

contro s&stem to process it and thus contro it in order

to ater the nature o$ the output, C ¿

.

'he nature o$ ox&gen upta8e is di$$erent in acases, as it 5as modeed using the mass trans$er

process 5hich reies on T p , i.e.1

# $ α , 5hich is

8no5n to depend on impeer speed and air $o5rate.

'hus, $or higher energ& input s&stems the oershoot is

much greater, because the modeed ce gro5th 9and

hence ox&gen upta8e: is more signi$icant. 'his coud

be an incorrect assumption to ma8e, such that the case

o$ ce gro5th rate coud be constant regardess o$

impeer speed. Fo5eer this does not sound sensibe,i$ $or a higher impeer speed more ox&gen mass

19





trans$er is aaiabe, and thus through ce reproduction

higher upta8es o$ ox&gen can be achieed.

Comparing air $o5rates o$ 2 L@min and

L@min, increased ox&gen presence increases both the

mass trans$er process and thus the ox&gen upta8e

process, as seen b& the decreased rise time and thus

decreased setting time. 7urther increasing this$o5rate coud be inestigated in $uture to see 5here

mass trans$er becomes imiting in reait&, i.e. 5hen an

increase in air $o5rate does not a$$ect rise and setting

time due to mass trans$er or ox&gen upta8e

mechanisms being $u& saturated.

/. (issoed %x&gen contro in practice

%ne exampe o$ (%2 contro in practice 5as gien

b& Chot8o5s8i et. a. 92;;/:, in 5hich actiated sudge

processes reBuire dissoed ox&gen re$erence

traector& trac8ing. 'his 5as achieed through the use

o$ a noninear predictie controer mode and a direct

re$erence adaptie controer. '5o time scaes o$

dissoed ox&gen d&namics 5ere considered, $ast and

so5. It 5as discoered that the predictie controers

suppied good trac8ing per$ormance and robustness,

5hereas the direct mode 5as much simper to

impement. Such methods 5ere used due to the

uncertain nature o$ the s&stem used, consisting o$

mutipe time scaes, and a #I#% structure. Fence,

measurements during pant operation are scarce, and

thus mathematica modes 5ere essentia in the design

o$ the controer, abeit 5ith great uncertaint&.

'here$ore the ne5, more accurate methods aboe 5ere

proposed 5ith signi$icant success.

!nother exampe is gien b& #eching et. a.

92;1":, in 5hich the Chicago Water5a& S&stem 5as

reBuired to meet proposed (% standards $or 5ater-

Buait& management and poution contro. 'hus, $o5

augmentation and aeration stations 5ere introduced as

a cost e$$ectie soution. Fo5eer, in practice

di$$icuties 5ere met in 5hen to turn on the aeration

station and ocaised hea& oads o$ poutions duringstorms coud ead to ioations o$ the ega

reBuirement. 'here$ore, a ne5 s&stem denoted Wet

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Principles of Process Control Exercises

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