numerical methods for distributed sysytems

8/17/2019 Numerical Methods for Distributed Sysytems

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Federal University of Rio de Janeiro

Program of Chemical Engineering (COPPE)

Course Name:Numerical methods for distriuted systems

!itle of Pro"ect:

On the solution of #o#ulation alance e$uations (P%E)

y discreti&ation ' Fied Pivot !echni$ue (ethod of

Class)

By

Ali khajehesamedini

Supervised by

Prof. Argimiro R. Secchi

January 20!

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Index pag

e. Popula"ion Balance #$ua"ion .. %e&ni"ions .2. %imensions .'. Applica"ion 2.(. )eneral %i*eren"ial +orm, -% Popula"ion 2.. PB# Solu"ion /echni$ue 2.!. omen"s of PB# '.1. /runca"ion of "he in&ni"e domain '

2. o"iva"ion of "he projec" ('. e"hodology e"hod of class3 (. 4ase s"udies 5. 4onclusion and %iscussion 'Appendi6 A 7 a"lab code (Reference

*+ Po#ulation %alance E$uation

*+*+ ,e-nition:

/he popula"ion balance e6"ends "he idea of mass and energy balances "o

coun"able objec"s dis"ribu"ed in some proper"y i" holds "he general la8of chemical engineering 8hich is7

In - Out + Net Generation = Accumulation

*+.+ ,imensions:

A general popula"ion balance e$ua"ion PB#3 has e6"ernal and in"ernaldimensions. /he e6"ernal dimensions of a PB# are dimensions of "he


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environmen" 8hich are '-% space x,y,z or r,z,q or r,q,f 3 and "ime. /hein"ernal dimensions of a PB# are dimensions of "he popula"ion such asdiame"er, volume, surface area, concen"ra"ion, age, molecular 8eigh",number of branches, etc.

*+/+ 0##lication:

PB#s have been in"roduced in several branches of modern science,mainly in branches 8i"h par"icula"e en"i"ies. /his includes "opics like

- Par"icles granula"ion, 9occula"ion, crys"alli:a"ion, mechanical

alloying, aerosol reac"ors, combus"ion, crushing, grinding, 9uid

beds3- %rople"s li$uid-li$uid e6"rac"ion, emulsi&ca"ion3- Bubbles 9uid beds, bubble columns, reac"ors3

- Polymerspolymeri:es, e6"ruders3- cells fermen"a"ion, bio"rea"men"3

Popula"ion balances describe ho8 dis"ribu"ions evolve;

*+1+ 2eneral ,i3erential Form4 *5, Po#ulation:

( , , )[ ( , , ) ( , , )] [ ( , , ) ( , , )] [ ( , , ) ( , , )] ( , , ) p p n V x t u V x t n V x t D V x t n V x t G V x t n V x t S V x t V t

∂ ∂−∇ + ∇ ∇ − + =

∂ ∂ 3

?n "his e$ua"ion, "he &rs" "8o "erms on "he righ" side of "he e$ua"ion arerela"ed "o conven"ion and difuusion 8hich are @?n - u" e6"ernalcoordina"es of PB#. /he "hird "erm on "he righ" side of "he e$ua"ion is

rela"ed "o "he gro8"h 8hich is @?n - u" in"ernal coordina"e3. @Srepresen"s "he sources and sinks 8hich is ne" genera"ion. And +inaly "helef" hand side "erm of "he e$ua"ion is accumula"ion. ?" should be no"ed

"ha" objec"Cs veloci"y may di*er from 9uidCs veloci"y o8ing "o ei"her slip

or ac"ion of e6"ernal forces. ?n "he general form of PB# n(V,x,t 3represen"s "he number-based densi"y dis"ribu"ion func"ion.

*+6+ P%E 7olution !echni$ue

Several ma"hema"ical me"hods have been o*ered "o solve "he PB#.

Some of "he main ma"hema"ical "echni$ues "o solve PB# are7

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- Similari"y Solu"ions- Daplace /ransforms- Simple %iscre"i:a"ion e"hods- Sec"ional %iscre"i:a"ion e"hods- r"hogonal Polynomial e"hods

- on"e 4arlo e"hods- omen" e"hods

Based on "he ma"hema"ical and physical charac"eris"ics of PB# and i"sini"ial and boundary condi"ions, "he proper Solu"ion /echni$ue is chosen.


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m& and m represent the total numer and total olume of particles in the system$ %he

second moment (m') has een shown to e useful in predicting the onset of gelation

[smith]$

*+9+ !runcation of the in-nite domain /he follo8ing e$ua"ion sho8s "he di*eren"ial-in"egral form of a generalPB# Ramkrishna>7

&

&

[ ( , )] $[ ( , )] $[ ( , )] ( , ) ( , ) ( , )

'

( , ) ( , ) ( , ) ( ) ( , ) ( , ) (

V

V V

V

n V t un V t G n V t V n t n V t d t

n V t p V n t d S p V n t d S V

β φ φ φ φ φ

φ φ φ φ φ φ φ ∞ ∞

∂+ ∇ +∇ = − −

∂

− + −

∫

∫ ∫ '3

%he infinite domain of a PBE must e truncated to a finite upper limit, so that it may e

spanned y a finite numer of elements$ %his truncation results in an underestimation of theintegrals of PBE terms$ *t can also e anticipated that the ith moment of the solution will e

underestimated y an amount

ma+

( )dte ii

v

m V n V dV ∞

= ∫

(3

Ghere mid"e is "he error incurred in "he i"h momen" due "o domain"runca"ion and vma6 is "he upper limi" of "he &ni"e domain. ?n mos"prac"ical applica"ions "he densi"y dis"ribu"ion nv3 asymp"o"es "o8ards:ero a" suFcien"ly large par"icle volumes, so vma6 can be selec"ed "o besuFcien"ly large "ha" such underes"ima"ion is negligibly small.are must e ta#en to aoid selection of unnecessarily large alues of vma6 since tailregions can e difficult andor computationally e+pensie to conerge, ecause of the ery

small alues they can attain at large particle olumes$.elard and /einfeld define 0uantities M i

ma+

&( )

vi

i

i

V n V dV M

m= ∫

3

and selec" vma6 such "ha" 0 and do no" di*er appreciably from uni"y.)enerally vma6 is selec"ed so "ha" appro6ima"ely sa"is&es "he cri"erion2H0.555. ?f 2H0.555 "hen "he addi"ional cri"eria 0H0.555 andH0.555 8ill also be sa"is&ed =Iicmanis>.

(


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.+ otivation of the #ro"ect%uring oil produc"ion, "he mi6"ure of 8a"er and oil is submi""ed "o largeshear ra"es genera"ing 8a"eroil emulsions 8hich have a large sal"con"en". #lec"ros"a"ic demulsi&ca"ion of 8a"er-in-oil emulsions is largelyapplied in "he oil indus"ry "o desal" "he crude dead oil prior "o i"s

re&ning. %rople" si:es and sal" concen"ra"ion are key variables inde&ning "he separa"ion eFciency.

/he emulsion 9o8 is a polydisperse mul"iphase 9o8 "ha" can besimula"ed by coupling a popula"ion balance model =Ramkrishna> 8i"h

compu"a"ional 9uid dynamics archisio e" al., 200'3. /he accuracy of "his approach heavily relies on "he ade$uacy of "he employed breakageand coalescence models, 8hich are no" generally applicable and have

adjus"ed parame"ers "ha" usually depend on "he charac"eris"ics of "hemul"iphase sys"em.

/he formula"ion of breakage and coalescence models are based on "hecharac"eris"ics of "he process. /he parame"ers of "he breakage and

coalescence models are de"ermined using "he e6perimen"al da"a. /heini"ial dis"ribu"ion of "he disperse 9o8 is "he inpu" of "he PB#. /henparame"er es"ima"ion me"hods are used "o calcula"e "he coeFcien"s of

"he models =i"re>.

Among "he several numerical me"hods "ha" have been developed for "hesolu"ion of "he popula"ion balance e$ua"ion, "his 8ork used "he met"odof classes of Kumar and Ramkrishna 55!3 enforcing "he conserva"ionof number and volume of "he par"icle popula"ion. /his me"hod 8as

chosen because i" allo8s "he direc" usage of "he par"icle si:e classesde&ned by "he par"icle si:e analy:er. /herefore, "he e6perimen"al da"a isused "o de&ne "he discre"e drop si:e dis"ribu"ions 8i"hou" any

manipula"ion.

/herefore, in "his 8ork "he me"hod of class for solu"ion of PB# 8ill bediscussed and e6amined using several kno8n case s"udies.


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'+ ethodology4onsider "he PB# for a popula"ion of par"icles 8hich undergo break-up as8ell as aggrega"ion.

'

&

&

( , )( , ) ( , ) ( , ) ( ) ( , )

( , ) ( , ) ( , ) ( , ) ( ) ( , )

v

v

n v t n v v t n v t Q v v v dv v n v t

t

n v t n v t Q v v dv v v v n v t dvβ ∞ ∞

∂′ ′ ′ ′ ′= − − − Γ

∂

′ ′ ′ ′ ′ ′ ′− + Γ

∫

∫ ∫

!3

/he momen"s L of "he number densi"y func"ion nv,"3 de&ned as

( , )v

M v n v t dv µ

µ

∞= ∫

13

are "hen ob"ained from e$.!3 as

'

& &

& &

[ ( ) ] ( , ) ( , ) ( , )

1 ( , ) ( ) [ ( ) ( , ) ]v

dM dv dv v v v v Q v v n v t n v t

dt

vdv n v t v v v v dv

v

µ µ µ µ

µ

µ β

∞ ∞−

′∞

′ ′ ′ ′ ′= + − + ×

′ ′ ′ ′ ′Γ × −

′

∫ ∫

∫ ∫

M3

ur in"eres" here is in formula"ing popula"ion balances in discre"epar"icle s"a"e space. /he $ues" for a discre"e formula"ion of e$.!3 maybe likened "o macroscopic balances in "he analysis of "ranspor" problems

8here one seeks conserva"ion e$ua"ions for an en"i"y in a chosen &ni"evolume of ma"erial or space. ?n"egra"ing "he con"inuous e$ua"ion =e$.!3> over a discre"e si:e in"erval, say vi "o viN,

' & &

( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( ) ( , ) ( ) ( , )

i i

i i

i i

i i

v v vi

v v

v v

v v v

dN t dv n v v t n v t Q v v v dv n v t dv n v t Q v v dv

dt

dv v v v n v t dv dv v n v t β

+ +

+ +

∞

∞

′ ′ ′ ′ ′ ′ ′ ′= − − −

′ ′ ′ ′+ Γ − Γ

∫ ∫ ∫ ∫

∫ ∫ ∫ 53

8here

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( ) ( , )i

i

v

iv

N t n v t dv+

= ∫

03

/he numerical "echni$ue proposed here divides "he en"ire si:e range in"osmall sec"ions. /he si:e range con"ained be"8een "8o si:es v i and viN is

called "he i"h sec"ion class3. /he par"icle popula"ion in "his si:e range isrepresen"ed by a si:e 6i also called grid poin"3, such "ha" vi O 6 i OviN. A "ypical grid along 8i"h i"s represen"a"ive volumes grid poin"s3 issho8n in +ig..

+ig.. A general grid 8hich can be used 8i"h "he proposed numerical"echni$ue. is given by

( , ) ( ) ( , )i

i

v

Bbv v

R dv v v v n v t dvβ + ∞

′ ′ ′ ′= Γ ∫ ∫

3

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Since "he par"icle popula"ion is assumed "o be concen"ra"ed a"represen"a"ive si:e, 6i s, "he number densi"y nv,"3 can be e6pressed as

( , ) ( ) ( ) M

k k

k

n v t N t v xδ =

= −∑

23

Subs"i"u"ing for nv,"3 from e$ua"ion '3 8e ob"ain

,

( ) ( ) M

Bb i k k k

k

R i n N t =

= Γ ∑

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'

&( , ) ( , ) ( , )

i

i

v v

Bav

R dv n v v t n v t Q v v v dv+

′ ′ ′ ′ ′= − −∫ ∫

13

is also modi&ed. /he popula"ion a" represen"a"ive volume 6i ge"s a

frac"ional par"icle for every par"icle "ha" is born in si:e range 6 i,6iN3 or6i-,6i3.

Subs"i"u"ing for nv,"3 from e$. '3, and af"er some algebraicmanipula"ions, 8e ob"ain

, , , ,

,( )

( ) ( ) ( ) ( )

i j k i

j k

Ba j k j k i j k j k

j k x x x x

R i Q N t N t δ η

− +

≥

≤ + ≤

= −∑

M3

Ghere

, ( , ) j k j k Q Q x x=

53

+or preserva"ion of numbers and mass, U is given by "he simplee6pressions

, ,

( ) ( )

( ) ( )

i j k

i j k i

i i

j k i

j k i

i j k i

i i

x x x x x x x

x x

x x x x x x x x x

η

++

+

−−

−

− +≤ + ≤ −

=

+ − ≤ + ≤ −

203

,eath aggregation:

Subs"i"u"ing for nv,"3 from e$. 2(3 in "he dea"h "erm =four"h "erm on "her.h.s of e$. 53> given by

&( , ) ( , ) ( , )

i

i

v

Da v R n v t dv n v t Q v v dv

+ ∞

′ ′ ′= ∫ ∫ 23

Ge ob"ain

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,

( ) ( ) ( ) M

Da i i k k

k

R i N t Q N t =

= ∑

223

/herefore "he &nal e$ua"ion for any of "he classes 8ill be

, , , ,

,( )

, ,

( ) ( ) ( ) ( )

'

( ) ( ) ( ) ( )

i j k i

j k

i j k j k i j k j k

j k x x x x

M M

i i k k i k k k i i

k k

dN t Q N t N t

dt

N t Q N t n N t N t

δ η

− +

≥

≤ + ≤

= =

= −

− + Γ − Γ

∑

∑ ∑

2'3

1+ Case studies

+or case s"udies "he pure breakage and pure aggrega"ion cases 8ere

s"udied. Since "he me"hod of class is 8orking 8i"h "he number ofdis"ribu"ion no" "he number densi"y3 "he &gures presen"ed in "his 8orkare all &gured by number par"icles versus volume par"icle.

/o e6amine "he correc"ness of "he solu"ion "he second momen" 8hichsho8s "he "o"al volume is calcula"ed via "his appro6ima"ion7

&

total i i

i

V nv dv x N ∞∞

=

= = ∑∫

?n case of e6ac" solu"ion "he second momen" "o"al volume of "he

par"icles3 should be cons"an" "hrough breakage andQor coalescence.Al"hough for mos" of "he prac"ical cases a small change in secondmomen" is accep"able.

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Pure %reaage

?n "his case "he "erms of aggrega"ion 8ere made e$ual "o :ero and "hefunc"ions of breakage 8ere considered as

/v3 v2

Vv,vC3 2QvC

+ina "ime 0s

+or "he ini"ial densi"y dis"ribu"ion of a normal dis"ribu"ion is considered.a"hema"ical formula for a normal dis"ribu"ion is7

'

( )

'

'

e

'

X µ

σ

πσ

− −

8here

µ

is "he mean,

σ

is s"andard devia"ion and W can "ake on anyvalue from minus in&ni"y "o plus in&ni"y.

/he discre"i:a"ion considered in "his case is geome"rical and "hegeome"ry ra"io is S. "he value of S based of sensi"ivi"y of dis"ribu"ion can

vary be"8een .0 and ' for ma6. ?n "his 8ork "he simula"ions 8ere doneby S be"8een .0 and .2. Since "he discre"i:a"ion is geome"rical verysmall change in "he value of S 8ill change "he ini"ial dis"ribu"ion.

/he follo8ing &gure sho8s "he ini"ial, &nal dis"ribu"ions and "he second

volume consis"ency &gure for "he above pure breakage case.


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+ig2. ?ni"ial %rople" number dis"ribu"ion Pure Breakage case3

+ig'. +inal %rople" number dis"ribu"ion Pure Breakage case3

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+ig(. Pure Breakage case3

+igure ( sho8s "he conserva"ion of "o"al volume 8hich veri&es "hecorrec"ness of "he solu"ion.

Pure 0ggregation

?n "his case "he "erms of breakage 8ere made e$ual "o :ero and "hefunc"ions of aggrega"ion fre$uency is considered as

Case a)

X6,y3

+ina "ime 0s

?n "his case logari"hmic discre"i:a"ion 8i"h e6ponen"ial ini"ial dis"ribu"ion8as considered.

/he follo8ing &gure sho8s "he ini"ial, &nal dis"ribu"ions and "he second

volume consis"ency &gure for "he above pure aggrega"ion case.

'


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(


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Case )

X6,y3

+ina "ime 0s

?n "his case geome"ric discre"i:a"ion 8i"h e6ponen"ial ini"ial dis"ribu"ion8as considered.

/he follo8ing &gure sho8s "he ini"ial, &nal dis"ribu"ions and "he secondvolume consis"ency &gure for "he above pure aggrega"ion case.


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!


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Case c)

X6,y3 6Ny

+ina "ime 0s



1


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Case c)

X6,y3 6Ny

+ina "ime 0s


M


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Case d)

X6,y3 6Yy

+ina "ime 0s

?n "his case geome"ric discre"i:a"ion 8i"h e6ponen"ial ini"ial dis"ribu"ion

8as considered.


5


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6+ Conclusion and ,iscussions

?n "his 8ork "he me"hod of class as a robus" "echni$ue "o solve "hePopula"ion Balance #$ua"ion 8as in"roduced.

4ase s"udies for Pure Breakage and Pure Aggrega"ion case 8as solved8i"h "he me"hod. /he Accuracy of "he solu"ion 8as sho8n using "hesecond momen".

Al"hough "he resul"s sho8n in "his 8ork had good degree of sa"isfac"ory,bu" some cau"ions should be considered "o have a good ans8er in PB#problem. )ood discre"i:a"ion is of high value of impor"ance. Also, asui"able ini"ial normal dis"ribu"ion is impor"an". %ue "o charac"eris"ics of

numerical "echni$ue "he ini"ial dis"ribu"ion is very impor"an" "o havegood resul".As a sugges"ion for fu"ure 8orks, "he moving pivo" "echni$ue andgro8"h, nuclea"ion problems can be s"udied.

0cno;ledgment /he au"hor of repor" gra"efully ackno8ledge Professor Argimiro+RJ3,Professor Paulo+RJ3, Professor Ramikrishnaniversi"y of perdue3,

Professor Kumarniversi"y of Aberdeen3 and %r. Ziviane #ngepol3 for"heir "echnical suppor"s.

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0##endi 0: atla code

?n "his 8ork, a"lab sof"8are 8as used for programming. Seven m-&lescomprise "he programs 8hich are7

- Be"a- calcula"e[agg- me"hodofclass- me"hodofclass[di* - ncalcula"or[br- Phi- /e"a

/he "ask of each of "he m &les are described in follo8

5 %eta:

/his func"ion calcula"es "he value of be"a, 8hich is appeared in e$ua"ion3 rela"ed "o bir"h breakage "erm.

5 calculate


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5 ncalculator


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Ramkrishna, %.7 @n "he solu"ion of popula"ion balance e$ua"ions bydiscre"i:a"ion-?. A &6ed pivo" "echni$ue, hem. %n&. S'i.$ Zol. , Io. M, pp.'-''2 55M3 (77@)$

?itre$ 9 Paulo $$ 3Droplet rea#age and coalescence models for the flow of water-in-

oil emulsions through a ale-li#e element4, chemical engineering research and design 7''A7=-'8&< ('&A)

archisio, %.D., Zigil, R.%., +o6, R.., 200'. ?mplemen"a"ion of "he$uadra"ure me"hod of momen"s in 4+% codes for aggrega"ion-breakageproblems. 4hem. #ng. Sci. M, '''1''.

2(

numerical methods for distributed sysytems

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