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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$on *rom the author only
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Solut$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$ometry7
plug *low7
CST7
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Seuent$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$
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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
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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
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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$s *lowsheet7
S2
MIXER REACTOR HEATX FLASH
S9
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Compl$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"
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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
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Compl$cat$ons w$th ecycle Streams 2Cont’d3
:* 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/
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Compl$cat$ons w$th ecycle Streams 2Cont’d3
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$mum I o* tear streams
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Compl$cat$ons w$th ecycle Streams 2Cont’d3
S6S7
B1 B2 B3 B4
S1 S3 S4 S5
14
S2
S8
&$n$mum I o* tear streams ; -, namely S6 and SA"
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Compl$cat$ons w$th ecycle Streams 2Cont’d3
S6
S7
Bhat aout th$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
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+ 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 $s 5"
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Compl$cat$ons w$th ecycle Streams 2Cont’d3
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
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+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"
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+lgor$thms *or !eterm$n$ng Tear Streams
1
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S$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
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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"
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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>
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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
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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:? $s a group
IJK
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E9ample o* S B 2Cont’d3
E:?! w$ll e a group
3. A
?$st
B C M EIL D EIL
-6
B C M
4" !elete E:?! s$nce $t has no more outputs O
E:?!
&
C
8
+
5. A
!elete &
6. A B C
!elete C, 8, and then +
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E9ample o* S B 2Cont’d3?$st
% O
E:?!
F G H
-A
!elete MH, and then delete F
&C
8
+MH
F
Computat$onal seuence $s0
F GH A B C M EILD
K J
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Conclus$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>
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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$son
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8ar'ley &otard(s +lgor$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 $nputs and
outputs have een deleted"
21 3 4
5 6 7 8
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8ar'ley &otard(s +lgor$thm 2Cont(d3
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
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8ar'ley &otard(s +lgor$thm 2Cont(d3
-" Node El$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$nate common node to ecome
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8ar'ley &otard(s +lgor$thm 2Cont(d3
!$so$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$ly" Mo to step 1"
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8ar'ley &otard(s +lgor$thm 2Cont(d3
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
/
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8ar'ley &otard(s +lgor$thm 2Cont(d3
Node Precursors
1 A
- A, /
/6
,
4 /5 /, 6
6 -, 5
A -, 5
/
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8ar'ley &otard(s +lgor$thm 2Cont(d3
2 3 5
/A
7 6
-, A5, 6-, /
/, 5
-, /, 5A, -, /
/, 5, 6
Two#way edge pa$rs
Oo$nt two#way edge pa$rs
So the common nodes
are -, /, and 5 >>>
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8ar'ley &otard(s +lgor$thm 2Cont(d3
# 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"
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8ar'ley &otard(s +lgor$thm 2Cont(d3
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 $s0 C ## ! ## E ## + ## 8
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Forder#Hutch$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
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?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 $s repeated"
+, S1, 8, S-, C, S/, !, S4, + loop
eturn to Step - and repeat"
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?oop F$nder +lgor$thm 2Cont’d3
/" :* the last un$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$90
?oop S1 S- S/ S4 S5 S61 1 1
- 1 1 1 1
/ 1 1 1
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?oop F$nder +lgor$thm 2Cont’d3
emar's0
1" The algor$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$ce"
order ; Tear -,/ ## ! ## C ## + ## 8 ## C ## )pdate tears
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?oop F$nder +lgor$thm 2Cont’d3
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$ty o* 1
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+nother E9ample o* ?oop F$nder
A B
C D
5
1
62
7
48
How aout th$s one7 How many loops7 6
loops" They are01 8, S-, !, S6, 8- +, S1, 8, S5, +/ +, S, C, S4, +
45
3
, , , ,
5 +, S1, 8, S-, !, S/, C, S4 +6 8, S5, +, S, C, SA, !, S6 8
?oop S1 S- S/ S4 S5 S6 SA S
1 1 1
- 1 1/ 1 1
4 1 1
5 1 1 1 1
6 1 1 1 1
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+nother E9ample o* ?oop F$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"