Convergence Tear Streams

8/20/2019 Convergence Tear Streams

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CHE654 Supplementary Notes

FlowsheetFlowsheet ConvergenceConvergence –– Tear StreamsTear Streams

1

Prepared y!r" Hong#m$ng %u

Chem$cal Eng$neer$ng Pract$ce School Program

%$ng &ong'ut(s )n$vers$ty o* Technology Thonur$

Copyr$ght +pr$l, -../#-.1/ – use w$th perm$ss$on *rom the author only


2/46

Solut$on +pproaches to Process S$mulat$on

There are - as$c approaches to process s$mulat$on0

1" Seuent$al &odular +pproach 2S&+3

-" Euat$on#r$ented +pproach 2E+3

-

Seuent$al &odular +pproach

Process un$t #### &athemat$cal model #### FT+N surout$nes

e"g" to model a reactor ## wh$ch model to use7

sto$ch$ometry7

plug *low7

CST7


3/46


4/46

Seuent$al &odular +pproach 2Cont’d3

Conseuence0 &ult$ple#pass calculat$ons and must solve a system o*

nonl$near euat$ons to converge the tear stream"

+dvantages o* S&+0

4

" oncep ua s mp c y

-" Correspondence to phys$cal structure

/" eu$res l$ttle storage and computer memory

!$sadvantage o* S&+0# :ne**$c$ent, nested loops

ma'$ng $t d$**$cult to solve opt$m$


5/46


6/46

Euat$on#8ased +pproach 2Cont’d3

-" eu$res good est$mates

/" eu$res large storage and computer memory

4" No correspondence to phys$cal structure

6

5" eu$res stale, rel$ale N?E solvers

@ +SPEN P?)S $s a Seuent$al &odular s$mulator

@ SPEE!)P $s an Euat$on#r$ented s$mulator


7/46

Part$t$on$ng and Tear$ng a Flowsheet

&ost commerc$al steady#state s$mulators use the

seuent$al modular approach 2S&+3"

A

modular s$mulator"

Two as$c prolems ar$se $n the S&+"

1" Part$t$on$ng a *lowsheet

-"Tear$ng a *lowsheet

B$ll descr$e and de*$ne these

terms $n more deta$ls later


8/46


9/46

Part$t$on$ng and Tear$ng a Flowsheet 2Cont(d3

Now, cons$der a sl$ghtly d$**erent *lowsheet wh$ch $s a mod$*$cat$on

to the prev$ous one"S6S8

S7

D S6 $s now a recycle stream $nstead o* a process product stream"

D !o you see any compl$cat$ons th$s t$me7 an $mpasse>

D Bhat $s the computat$onal seuence *or th$s *lowsheet7

S2

MIXER REACTOR HEATX FLASH

S9


10/46

Compl$cat$ons w$th ecycle Streams

To wor' around the prolem, we must per*orm tr$al#and#error"

The wor'around $s to tear a stream, say S6"

“ ”

1.

.

9.

; stream compos$t$on, T, and P

S6

x0

(S6) x1

(S6)

Convergence loc' D :* 9

.

2S63 ; 91

2S63 w$th$n acceptale

tolerance, then we are done"

D therw$se, must update 91

2S63 somehow"


11/46

Compl$cat$ons w$th ecycle Streams 2Cont’d3

Convergence loc' mechan$sm *or updat$ng a tear stream

Numer$cal methods mathemat$cal method=algor$thm *or

updat$ng a tear stream"

11

E9ample o* a s$mple numer$cal method $s !$rect Sust$tut$on"

ther numer$cal methods commonly used are0

1" Begste$n(s method

-" Newton#aphson(s method

/" 8royden(s method


12/46


:* S6 $s the tear stream, the computat$onal seuence o* the recycled

*lowsheet $s0 Tear S6 ### &:E ### E+CT ### HE+T ###

F?+SH ### )pdate S6

1-

:nterest$ngly, S6 $s not the only val$d tear stream, $"e" a tear stream

$s not un$ue" Can also tear S/, s4, or S5"

:* S/ $s the tear stream, what $s the computat$onal seuence7

Tear S/ ### E+CT ### HE+T ### F?+SH ### &:E ###

)pdate S/


13/46


Bhen a model $s comple9 w$th many recycle streams, $t $s not

poss$le to Geyeall the *lowsheet and come up w$th tear streams"

So - cr$t$cal $ssues *ac$ng the S&+ s$mulat$on

1/

1" &$n$mum numer o* tear streams and the$r locat$ons

-" Computat$onal seuence

7I o* recycle streams ; m$n$mum I o* tear streams


14/46


S6S7

B1 B2 B3 B4

S1 S3 S4 S5

14

S2

S8

&$n$mum I o* tear streams ; -, namely S6 and SA"


15/46


S6

S7

Bhat aout th$s one7

15

B1 B2 B3 B4

S2

S8

&$n$mum I o* tear streams ; 7

Computat$onal seuence ; 7

:n conclus$on0 I o* recycle streams m$n$mum I o* tear streams


16/46

+ Somewhat Comple9 Flowsheet

A B C D E

16

F G H I J

K ML

O

N

P Bhat $s the m$n$mum I o* tear streams7

The answer $s 5"


17/46


Summary0

1" How many tear streams7 2necessary ecause o* recycles3-" Bh$ch ones7

/" Convergence method7

1A

&any pul$cat$ons related to tear stream determ$nat$on" The

$mportant ones are as *ollows0

Sargent and Bestererg 164

Forder and Hutch$son 16

8ar'ley and &otard 1A-

4" :n wh$ch order should one converge 2part$t$on$ng37


18/46

+nother Type o* Convergence Prolem

Bhen a *eedac' controller $s present

FLASHHEATX

T ; 7

D Called des$gn spec$*$cat$on $n +J

1

S8

FC

Bant 9C1

; .".1

D ecycle o* $n*ormat$on ecause

guess H outlet temperature,

calculate 9C1

$n S $* 9C1

; .".1

stopK otherw$se update THguess

!es$gn spec$*$cat$on $s a lot eas$er to converge than tear streams,

ecause $t $nvolves only 1 var$ale"


19/46

+lgor$thms *or !eterm$n$ng Tear Streams

1


20/46


21/46

S$mple E9ample o* Part$t$on$ng and Precedence rder$ng

A B C D E

FS1

-1

D Part$t$on$ng0 / un$t groups, namely +8C!, E, and F

D Procedence rder$ng0 +8C!, then E, then F

D +ctual computat$onal seuence0

Tear S1 ## C ## ! ## + ## 8 ## )pdate S1 ## E ## F


22/46

Part$t$on$ng +lgor$thm

Path Trac$ng algor$thm y Sargent and Bestererg 21643

# + s$mple algor$thm *or trac$ng un$t outputs# 8as$cally, one traces *rom one un$t to the ne9t through the un$t

output streams, *orm$ng a Gstr$ng o* un$ts"

--

Th$s trac$ng cont$nues unt$l2a3 + un$t $n the str$ng reappears"

+ll un$ts etween the repeated un$t, together w$th the repeated un$t, ecome a group, wh$ch $s collapsed together and treated as a s$ngle

un$t, and the trac$ng cont$nues *rom $t"

23 + un$t or group o* un$ts w$th no more outputs $s encountered"

The un$t or group o* un$ts $s placed at the top o* a l$st o* groups and $sdeleted ent$rely *rom the prolem"


23/46

Part$t$on$ng +lgor$thm 2Cont’d3

+lgor$thm0

1" Select a un$t=group

-" Trace outputs downstream unt$l

-/

2a3 a un$t or a group on the path reappears" Mo to step /"23 a un$t or a group $s reached w$th no e9ternal outputs" Mo to

step 4"

/" ?ael all un$ts $nto a group" Mo to step -"

4" !elete the un$t or group" ecord $t $n a l$st" Mo to step -"

Seuence $s *rom ottom to top o* l$st>


24/46

E9ample o* S B’s Part$t$on$ng +lgor$thm

A B C M EF G H D L

-4

IJK1" Start w$th un$t +

A B C M E I J K

!elete % and !elete O, s$nce no output?$st

%

O


25/46

E9ample o* S B 2Cont’d3

A B C M ED LF G H

-5

-" Start w$th un$t + aga$n

A B C M E I L E

?oop ; E:? $s a group

IJK


26/46

E9ample o* S B 2Cont’d3

E:?! w$ll e a group

3. A

?$st

B C M EIL D EIL

-6

B C M

4" !elete E:?! s$nce $t has no more outputs O

E:?!

&

C

8

+

5. A

!elete &

6. A B C

!elete C, 8, and then +


27/46

E9ample o* S B 2Cont’d3?$st

% O

E:?!

F G H

-A

!elete MH, and then delete F

&C

8

+MH

F

Computat$onal seuence $s0

F GH A B C M EILD

K J


28/46

Conclus$ons

1" There are two convergence loops $n th$s *lowsheet, and we

'now the$r the$r relat$ve order"

(

-

"

streams $n each loop and what the$r locat$ons are"

/" +ll we 'now $s that the tear streams $n each loop

must e converged s$multaneously>


29/46

Tear$ng an :rreduc$le Mroup

M$ven an $rreduc$le group0

&ust determ$ne the m$n$mum I o* tear streams and the$r

locat$ons"

-

1" F$nd m$n$mum I o* tear streams us$ng

8ar'ley and &otard(s 28 &3 algor$thm

-" F$nd all loops us$ng

?oop F$nder algor$thm y Forder#Hutch$son


30/46


31/46

8ar'ley &otard(s +lgor$thm 2Cont(d3

A B C D E1 2 3 4

6

78E9ample0

/1

5Trans*ormat$on0

# Nodes ecome arcs"

# +rcs ecome nodes"

# !$rect$on o* arc $s *rom

$nput to output"

Note that all process $nputs and

outputs have een deleted"

21 3 4

5 6 7 8


32/46


1" Mraph educt$on0

# &erge nodes w$th s$ngle precusor

precursor 0 all nodes prov$d$ng $nput to a g$ven node are precursors

/-

" " "

The node w$th a s$ngle precursor $s to e represented y that

precursor e"g" Node - has a s$ngle precursor 1" So erase node - and

represent $t w$th node 1"

# &erge parallel arcs 2same d$rect$on3


33/46


-" Node El$m$nat$on 2may see *unny patterns a*ter graph reduct$ons3

a3 El$m$nate nodes w$th sel*#loops

#

//

"

El$m$nate common node o* a o$nt two#way edge pa$r Q""

Two#way edge pa$r

Oo$nt two#way edge pa$r

El$m$nate common node to ecome


34/46


!$so$nt pa$rs

/4

Q" " ,

choose ar$trar$ly" eturn to Step 1 2graph reduct$on3 a*ter each

el$m$nat$on $n 2a3 or 23" Every el$m$nat$on $s a tear stream"

/" :* no progress poss$le, el$m$nate node w$th ma9$mum I o* outputedeges" :n case o* t$e, choose ar$trar$ly" Mo to step 1"


35/46


Node#Precursor ?$st There $s no need to draw the *low d$agramevery t$me you mod$*y or apply the

procedure"

Node Precursors

/5

1 A- 1,

/ -, 5

4 /

5 4, 6

6 -, 5

A -, 5

/


36/46


Node Precursors

1 A

- A, /

/6

,

4 /5 /, 6

6 -, 5

A -, 5

/


37/46


2 3 5

/A

7 6

-, A5, 6-, /

/, 5

-, /, 5A, -, /

/, 5, 6

Two#way edge pa$rs

Oo$nt two#way edge pa$rs

So the common nodes

are -, /, and 5 >>>


38/46


# 8ut node - and 5 have the largest numer o* output streams"

# So el$m$nate node - and delete node - *rom the tale"

# Stream - $s a tear stream"

/

Node Precursors

- A, /

/ -, 5

5 /, 66 -, 5

A -, 5

so that nodes /, 6, and A have s$ngle precursor"


39/46


Node Precursors

/ 5

5 /, 6 5 sel*#loop

/

A 5

So stream 5 $s another tear stream"

Tear streams are Stream - and Stream 5"

The computat$onal seuence $s0 C ## ! ## E ## + ## 8


40/46

Forder#Hutch$son’s ?oop F$nder +lgor$thm

8ased on path trac$ng also, ut records oth streams and loc's

encountered"

5 4E9ample

4.

A B C D6

F$rst, some de*$n$t$ons0

Full str$ng # seuence o* un$ts and streams on a path, e"g" +, S1, 8, S-, C, S/Stream str$ng # *ull str$ng m$nus un$ts, e"g" S1, S-, S/

Str$ng loop # a *ull str$ng that *orms a loop, e"g" C, S/, !, S6, C

Stream loop # str$ng loop m$nus un$ts, e"g" S/, S6


41/46

?oop F$nder +lgor$thm 2Cont’d3

+lgor$thm0

1" M$ven a un$t , trace outputs downstream unt$l a un$t reappears"

ecord the str$ng loop *ound" Mo to Step -"

41

+, S1, 8, S-, C, S/, !, S6, C

loop

-" eturn to -nd to the last un$t and resume trac$ng unt$l another

un$t $s repeated"

+, S1, 8, S-, C, S/, !, S4, + loop

eturn to Step - and repeat"


42/46


/" :* the last un$t has no more outputs rema$n$ng to e traced, s'$p to

the ne9t upstream un$t and go to Step -"+, S1, 8, S-, C, S5, + loop

So0 C, S/, !, S6, C

4-

+, S1, 8, S-, C, S/, !, S4, ++, S1, 8, S-, C, S5, +

Construct a ?oop :nc$dence &atr$90

?oop S1 S- S/ S4 S5 S61 1 1

- 1 1 1 1

/ 1 1 1


43/46


emar's0

1" The algor$thm g$ves all val$d sets o* tear streams, not ust one"-" Not all val$d sets o* tear streams are eually des$rale"

Some val$d tear sets are0

4/

RS/, S5w$ll rea' all the loops0Computat$on order ; Tear /,5 ## ! ## + ## 8 ## C ##)pdate tears

RS-, S6 order ; Tear -,6 ## C ## ! ## + ## 8 ## )pdate tears

RS-, S/$s a val$d tear set too08ut the troule $s we are rea'$ng ?oop - tw$ce"

order ; Tear -,/ ## ! ## C ## + ## 8 ## C ## )pdate tears


44/46


8loc' C $s calculated tw$ceK not des$rale ecause o* unneccessary

calculat$ons"

!e*$ne

44

&ult$pl$c$ty o* a tear set ; ma9$mum I o* t$mes a loop $s ro'en y

a tear set"

&ult$pl$c$ty ; 1 *or RS/, S5, RS-,S6 ; 21,1,13

; - *or RS-,S/ ; 21,-,13

E9clus$ve tear set ; tear set w$th a mult$pl$c$ty o* 1


45/46

+nother E9ample o* ?oop F$nder

A B

C D

5

1

62

7

48

How aout th$s one7 How many loops7 6

loops" They are01 8, S-, !, S6, 8- +, S1, 8, S5, +/ +, S, C, S4, +

45

3

, , , ,

5 +, S1, 8, S-, !, S/, C, S4 +6 8, S5, +, S, C, SA, !, S6 8

?oop S1 S- S/ S4 S5 S6 SA S

1 1 1

- 1 1/ 1 1

4 1 1

5 1 1 1 1

6 1 1 1 1


46/46

+nother E9ample o* ?oop F$nder 2Cont’d3

Th$s *lowsheet does not conta$n any e9clus$ve tear set"

Some val$d tear sets are0

1" RS4, S5, S6, SA ## 21,1,1,1,1,/3K mult$ l$c$t ; /

46

-" RS/, S, S1, S6 ## 21,1,1,1,-,-3K mult$pl$c$ty ; -

However, can’t say *or sure wh$ch set $s more des$rale ecause all

the loops are ro'en t$mes $n oth tear sets"

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Convergence Tear Streams