Process Dynamics and Control

(1)

UNIVERSITY OF LAGOS

Chemical Engineering Department

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Question 1

A process furnace heats a process stream from near ambient temperature to a desired temperature of A process furnace heats a process stream from near ambient temperature to a desired temperature of 300

300oo_{C. The process stream outlet temperature is regulated, by manipulating the flow of fuel gas to the}_{C. The process stream outlet temperature is regulated, by manipulating the flow of fuel gas to the}

furnace, as shown below. furnace, as shown below.

a.

a. WhWhat at arare te the he obobjejectctivives es of of ththis is cocontntrorol l ststraratetegygy b

b.. WWhhaat t iis s tthhe e mmeeaassuurreed d oouuttppuutt c

c.. WWhhaat t iis s tthhe e mmaanniippuullaatteed d iinnppuutt d

d.. wwhhaat at arre te thhe pe poossssiibblle de diissttuurrbbaanncceess e.

e. !s !s ththis is a fa feeeed"d"foforwrwaard rd or or fefeededbabac# c# cconontrtrol ol sysyststemem ff.. $i$iscscususs ss safafetetyy, , enenviviroronmnmenentatal al and nd ececononomomic ic isissusueses..

(2)

Question 2

The process in the figure below

The process in the figure below vapourises li%uid butane and mi&es the vapour with compressed airvapourises li%uid butane and mi&es the vapour with compressed air.. The mi&ture flows to a pac#ed bed reactor after it has been cooled in a heat e&changer. !n order to The mi&ture flows to a pac#ed bed reactor after it has been cooled in a heat e&changer. !n order to meet our control objectives of safety and %uality control, the following needs to be

meet our control objectives of safety and %uality control, the following needs to be implemented'implemented'

The pressure in the vapouriser (P1), which is important for

The pressure in the vapouriser (P1), which is important for safety, should be controlled by adjustingsafety, should be controlled by adjusting the vapo

the vapour releaur release from the se from the vavapoupourisriser (v3); er (v3); i!i!uid leveuid level l in the in the vapvapourouriseiser r (1(1), shoul), should d not not overflow the vessel or drain empty, and this can be controlled by adjusting the flow of

overflow the vessel or drain empty, and this can be controlled by adjusting the flow of li!uid butaneli!uid butane from storage to the vapori"er (v1); the percentage of butane in the mi#ed stream ($1), which is from storage to the vapori"er (v1); the percentage of butane in the mi#ed stream ($1), which is important in order to avoid an e#plosive concentration, can be controlled by adjusting the flow of important in order to avoid an e#plosive concentration, can be controlled by adjusting the flow of air through the valve in the

air through the valve in the compressor suctioncompressor suction, v%&, v%&

a

a.. ((iisst t tthhe e oouuttppuut t ))ccoonnttrroolllleedd* * aannd d iinnppuut t ))mmaanniippuullaatteedd* * vvaarriiaabblleess..

b.

(3)

Question 3

$ra

$raw w the procethe process ss insinstrutrumenment t diadiagragram m fofor r thethe ControlControl of theof the SugaSugar-Rer-Refininfining g ProceProcessss discusseddiscussed below.

below.

'se either $uto$, *icrosoft +ffice isio, the draw toolbar of *icrosoft -ord or PowerPoint or any 'se either $uto$, *icrosoft +ffice isio, the draw toolbar of *icrosoft -ord or PowerPoint or any other flow.sheeting /oftware

other flow.sheeting /oftware..

The process units, shown below, form part of the process to refine sugar. aw sugar is fed to the The process units, shown below, form part of the process to refine sugar. aw sugar is fed to the process through a screw conveyor. Water is sprayed over it to form sugar syrup. The syrup is process through a screw conveyor. Water is sprayed over it to form sugar syrup. The syrup is heated in the dilution tan#. /rom the dilution tan# the syrup flows to the preparation tan# where heated in the dilution tan#. /rom the dilution tan# the syrup flows to the preparation tan# where mor

more e heheatiating ng and mi&inand mi&ing g is is accaccompomplislishedhed. . /ro/rom m the the prepreparparatation ion tantan# # the the syrsyrup up floflows ws to to thethe blending tan#. hosphoric acid is added to the syrup as it flows to the blending tan#. !n the blending tan#. hosphoric acid is added to the syrup as it flows to the blending tan#. !n the blending tan# lime is

blending tan# lime is added. This treatment with acid, lime, and added. This treatment with acid, lime, and heat serves two purposes. The firstheat serves two purposes. The first is that of clarification, that is, the treatment causes the coagulation and precipitation of the no" is that of clarification, that is, the treatment causes the coagulation and precipitation of the no" suga

sugar r orgaorganicsnics. . The second purposThe second purpose e is to is to elimineliminate the ate the colorcoloratioation n of the of the raw sugarraw sugar. /rom the. /rom the blending tan# the syrup continues to other processes in the refinery plant.

blending tan# the syrup continues to other processes in the refinery plant.

Sugar refining process Sugar refining process

The following variables are thought to be important to control. The following variables are thought to be important to control. ii.. TTeemmppeerraattuurre ie in tn thhe de diilluuttiioon tn taann##..

ii

ii.. TeTempmpereraatuture re in in tthe he pprrepepaarraattioion n tatann#.#. ii

iii.i. $e$ensnsitity oy of thf the se syryrup up leleavavining thg the pe prerepapararatition ton tanan#.#. iivv.. ((eevveel il in pn prreeppaarraattiioon tn taann##..

v

v.. (ev(evel iel in 0n 01 a1 acicid tad tan#n#. Th. The lee levevel in l in ththe 2e 21 a1 acicid tad tan# cn# can ban be ase assusumemed cod consnstatantnt..

vi.

vi. The The strstrengength oth of tf the he 01 a01 acidcid. Th. The ste strengrength oth of tf the he 21 a21 acid cid tantan# ca# can bn be ae assumssumed ed consconstantant.t.

vi

vii.i. ThThe fe flolow ow of sf syryrup up anand d 01 01 acacid id to to ththe be blelendndining tg tanan## vi

viiiii.. ThThe pe p o of tf the he sosolulutition on in in ththe be blelendndining tg tanan#.#. i&

i&.. TTeempmpereraatuture re in in tthe he bblelendndining tg taann#.#. &.

&. ThThe be blelendndining tg tanan# r# re%e%uiuireres os onlnly a y a hihighgh"l"levevel el alalararm.m.

Question 4

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Consider the conical water tan# shown below. 4btain the model of the process if the flowrate out of the Consider the conical water tan# shown below. 4btain the model of the process if the flowrate out of the tan# is a function of the

tan# is a function of the s%uare root of height of water in the s%uare root of height of water in the tan# )tan# ) F F 00 ==β β hh*. (ist state *. (ist state variablesvariables, input, input

variables and parameters

variables and parameters. )int' 5se . )int' 5se height as a height as a state variable.*state variable.*

Question 5

6odel a mi&ing tan# with two feed

6odel a mi&ing tan# with two feed streams, as shown below. Assstreams, as shown below. Assume that there are two components, Aume that there are two components, A and 7. C represents the concentration of A. )C

and 7. C represents the concentration of A. )C88 is the mass concentration of A in stream 8 is the mass concentration of A in stream 8 and Cand C99 is the is the

mass concentration of A in stream 9*' mass concentration of A in stream 9*' 6odel the following cases'

6odel the following cases' a.

a. ConConstastant vont volumlume, coe, constnstanant dent densitsityy b.

b. ConsConstant votant volume, denlume, density vsity variearies lineas linearly witrly with conch concentraentrationtion c.

c. :a:ariablriable volue volume, denme, density vsity variaries linees linearly warly with coith concentncentratirationon

Question 6

Consider two tan#s in series where the ;ow out of

Consider two tan#s in series where the ;ow out of the <rst tan# enters the second tan#. 4urthe <rst tan# enters the second tan#. 4ur objective is to develop a model to

objective is to develop a model to describe how the height of li%uid describe how the height of li%uid in tan# 9 changes with time,in tan# 9 changes with time, given the input

given the input ;owrate;owrate F F oo

((

t t

))

. Assume that the ;ow out of . Assume that the ;ow out of each tan# is a linear function of theeach tan# is a linear function of the

height of li%uid in the tan# )

F

₁_{1 =}=

β

₁₁

h

₁₁ and and F F ₂_{2 =}=

β

₂₂hh₂₂* and each tan# has a constant cross"* and each tan# has a constant cross" sectional area.

sectional area.

Question 7

Tw

Two li%uid o li%uid surge tan#s )with surge tan#s )with constant cross"sectionaconstant cross"sectional area* are l area* are placed in placed in series.series. Write the modeling e%uations for the height of li%uid

Write the modeling e%uations for the height of li%uid in the tan#s assuming that the ;owrate fromin the tan#s assuming that the ;owrate from the first tan# is a function of

the first tan# is a function of the difference in levels of the tan#s and the the difference in levels of the tan#s and the ;owrate from the second;owrate from the second tan# is a function of the

tan# is a function of the level in the second tan#. Consider two cases'level in the second tan#. Consider two cases' ))ii** tthhe fe fuunnccttiioon in is ls liinneeaar ar anndd

)i

)ii*i* ththe fue funcnctition ion is a s%s a s%uauare rre roooot ret relalatitiononshshipip.. +tate all other

+tate all other assumptionsassumptions..

Question 8

(i%uid surge tan#s, containing hydrocarbons, usually have a gas =blan#et> of

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dio&ide to prevent the

dio&ide to prevent the accumulation of e&plosive vapours forming above the li%uid, as accumulation of e&plosive vapours forming above the li%uid, as shown belowshown below

$evelop the modeling e%uation with gas pressure and li%uid volume as the

$evelop the modeling e%uation with gas pressure and li%uid volume as the state variablesstate variables. (et. (et

q

_f_f and

and

q

represent the inlet and outlet gas molar flowrates,represent the inlet and outlet gas molar flowrates,

F

_f_f and and

_F

the the li%uid volumetric flowrates,li%uid volumetric flowrates, 

 the constant )total* the constant )total* volume,volume,   11 the li%uid volume, and the li%uid volume, and P P the gas pressure. Assume the ideal gas the gas pressure. Assume the ideal gas

law. +how that the modeling e%uations are' law. +how that the modeling e%uations are'

1 1 11

(

)

(

))

f f f f f f

dV

F

dt

d

dP

P

R

RT

T

F

q

d

dt

t V

V V

V

V V

V

= = −− = = −− + + −− − − −−

and state any other assumptions made. and state any other assumptions made.

Question 9

6ost chemical process plants have a

6ost chemical process plants have a natural gas header that circulates through the process plant. natural gas header that circulates through the process plant. AA simplified version of such a header is shown below.

simplified version of such a header is shown below.

The natural gas enters the

The natural gas enters the process plant from a source )the process plant from a source )the natural gas pipeline* through a controlnatural gas pipeline* through a control valve. !t flows through the plant piping, which for simplicity has been represented as a perfectly valve. !t flows through the plant piping, which for simplicity has been represented as a perfectly mi&ed drum. Another valve connects the plant piping to the gas drum for a

mi&ed drum. Another valve connects the plant piping to the gas drum for a boilerhouse unit, fromboilerhouse unit, from where it passes through another valve to

where it passes through another valve to the furnaces.the furnaces.

Write modeling e%uations assuming that the pressures in drums 8 and 9 are the state variables. (et Write modeling e%uations assuming that the pressures in drums 8 and 9 are the state variables. (et the input variables be

the input variables be hh11)valve position 8*,)valve position 8*, hh00 )valve position 9*, and )valve position 9*, and P P i i )source pressure*.)source pressure*.

Question 1

What is the (aplace transform of the function

(6)

Question 11

a.

What is the What is the (aplace transform(aplace transform u u ss

(( ))

of the following input functionof the following input function

b.

?valuate?valuate u(s)u(s) for A @8,  @8, b @ 8, and c @ 9 for A @8,  @8, b @ 8, and c @ 9

Question 12

4btain the partial"fraction e&pansion of the following

4btain the partial"fraction e&pansion of the following function using 6AT(A7'function using 6AT(A7'

2 2

1

10 0(

(

2 2))(

(

4 4))

(

( ))

(

1 1))(

(

3 3))(

(

5 5))

s

ss

F

F ss

s

ss

+ + ++ = = + + + + ++

Then, obtain the inverse the

Then, obtain the inverse the inverse (aplace transform ofinverse (aplace transform of   )s*)s*

Bint' you can also use the synta&

Bint' you can also use the synta&convconv or orpolypoly for the for the numerator and denominatonumerator and denominator DD..r DD..

Question 13

A process input has

A process input has the following (aplace transform'the following (aplace transform' u u

₍₍

ss

₎₎

₌₌

2

s s33 −−

6

s s22 ee − −33ss

What is the time domain input,

What is the time domain input, u(t)u(t) +#etch the time domain input. +#etch the time domain input.

Question 14

/or the given

/or the given transfer functiontransfer function

2 2

(

( )

)

2

2 (

( ))

₍

₂

₂₎₎₍

₍

₉

₉₎₎

y

y ss

u

u ss

==

_s

+ +

_ss

++ $erive

$erive

y

y t

(( ))

t

with respect to a unit with respect to a unit impulse input and s#etch this response.impulse input and s#etch this response.

Question 15

5se the initial and final value theorems of (aplace transforms to s#etch the response of the process 5se the initial and final value theorems of (aplace transforms to s#etch the response of the process output for a unit step

output for a unit step input change to the following transfer functioninput change to the following transfer function

a+b a+b aa t t 0 0 1 1 f(t) f(t)

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5

12

12 (( ))

7

4

s s g g ss s s + + = = + +

Question 16

/or the given ordinary differential e%uation /or the given ordinary differential e%uation

d

22

y

d t

22 + +

4

4 d d y y

d t

++

5

5 y

y

==

u u

((

t t

),

with

y

(0)

== ′′

y

(0)

==

0

E ,,,m, E ,,,m,

5sing any plotting software, s#etch the time"domain response for a step input of magnitude 3. 5sing any plotting software, s#etch the time"domain response for a step input of magnitude 3.

Question 17

a*

a* A caA car tyr tyre hre has a as a sloslow lew lea#. a#. The The ;ow;owrarate of te of air air out out of tof the tyhe tyre ire is prs propooportirtionaonal to tl to the prhe pressessureure of air in the tyre. The initial

of air in the tyre. The initial pressure is 30 psig, and after <ve days the pressure is down topressure is 30 psig, and after <ve days the pressure is down to 90 psig. ow long will it

90 psig. ow long will it ta#e to reach 80 psigta#e to reach 80 psig b*

b* eepeapeat a* t a* but but thithis tis time ame assussuminming thg the ;oe ;owrwratate of e of air air out out of tof the the tyryre is e is proproporportiotionanal to l to thethe s%uare root of the pressure of air in the

s%uare root of the pressure of air in the tyretyre Compare your results

Compare your results

..

Question 18

Write, in terms of

Write, in terms of deviation variables, the linear appro&imation to the deviation variables, the linear appro&imation to the nonlinear functionnonlinear function

f

((

x

,,

y

,,

z

))==22

x

22 ++

x

xy

y

22 −−33

y

z

at the point

at the point x x _s_s==

1 1,,

y y _s_s ==

2 2,

, zz

_s_s==

3 3,,

Question 19

The density of ideal gas is given by

The density of ideal gas is given by the following formula'the following formula'

MP

RT

ρ

==

where 6 is the molecular weight and 

where 6 is the molecular weight and  is the ideal gas constant.is the ideal gas constant.

/ind the linear appro&imation, in terms of deviation variables, of the density as a function of T

/ind the linear appro&imation, in terms of deviation variables, of the density as a function of T and and  and evaluate the coefficients for air )6 @ 9F* at 300G

and evaluate the coefficients for air )6 @ 9F* at 300G and atmospheric pressure )808,300 EHmand atmospheric pressure )808,300 EHm99_*._*.

!deal gas constant @

!deal gas constant @ I,38J E"mH#gmole"GI,38J E"mH#gmole"G

Question 2

!n +ummary, what

!n +ummary, what !no"le#ge!no"le#ge have you gained in class under the topic =!ntroductory Concepts and have you gained in class under the topic =!ntroductory Concepts and eview of mathematical tools> This tutorial is

eview of mathematical tools> This tutorial is to help your learning process andto help your learning process and un#erstan#ingun#erstan#ing of of what was taught in class, and note that this understanding will be tested as

what was taught in class, and note that this understanding will be tested as soon as possible.soon as possible. Therefore, be wise )

Therefore, be wise )"is#o$"is#o$ is the application of is the application of understood #nowledge*understood #nowledge*

Process Dynamics and Control

UNIVERSITY OF LAGOS

UNIVERSITY OF LAGOS

Chemical Engineering Department

Chemical Engineering Department

20!"20# SESSION$ %t

20!"20# SESSION$ %t SE&ESTER

SE&ESTER

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C'G !0( )r*ce%% D+namic% an, C*ntr*l

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Question 1

Question 1

Question 2

Question 2

b.

Question 3

Question 3

Question 4

Question 4

Question 5

Question 5

Question 6

Question 6

((

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F

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