robust adaptive control for multicomponent distillation column via

1 downloads 0 Views 410KB Size Report
ticomponent nonideal distillation column seems to be very important to design an advanced controller. The objectives of the distillation columns control are.
+N5A #AWV, A+w 6+ NwAW,A #W5AWwwAW wN VW #vW ,N+w ,A`+k `i? vc *i @?_ih 5 L3?)@!c a@4i *@hi3 5iUUL? _i L?|hL* |L4@|ULc WV,5AVWc W 2DfHc e.efc i UL #6c f. Sfc i UL 6@ G nD2D.e..fHbc i4@*G )9U|h*U?it|@4

Mt|h@U|

dq rswlpdo vhw0srlqw> wkdw qhhgv d rswlpdo rshudwlrq frqwuroohu wr surylgh wkh rshudwlrqv ri d glvwloodwlrq froxpq forvh wr wkh hfrqrplf rswlpxp +hqhuj| vdylqj dqg kljkhu surgxfw |lhog,1

Lq wklv sdshu wkh urexvw dgdswlyh frqwuro ri d pxowlfrpsr0 qhqw qrqlghdo glvwloodwlrq froxpq lv frqvlghuhg1 D vlpsoh g|qdplf qhxudo qhwzrun lv xvhg wr hvwlpdwh wkh glvwloodwlrq froxpq/ wkhq d dgdswlyh frqwuroohu lv sursrvhg1 Ehfdxvh wkh dssur{lpdwlrq fdsdelolw| ri g|qdplf qhxudo qhwzrun lv olplwhg/ d volglqj prgh frpshqvdwru lv dgguhvvhg1 Lw lv vkrzq wkdw wkh hhfw ri v|vwhp xqfhuwdlqwlhv fdq eh holplqdwh dqg wkh erxqghgqhvv ri wudfnlqj huuru fdq eh jxdudqwhhg1 Nh|zrugv= Glvwloodwlrq Froxpq/ Qhxudo Qhwzrunv/ Ur0 exvw Dgdswlyh Frqwuro Frpshqvdwlrqv



Wkh uvw rqh ohdgv wr wzr nlqgv ri frqwuro= txdolw| frq0 Wkh vhfrqg rqh lv edvhg rq dq rswlpl}dwlrq whfkqltxh wr qg rswlpdo wud0 mhfwrulhv ^7`1 Hyhq iru lghdo elqdu| glvwloodwlrq froxpqv wkh sodqw vkrzv dq %loo0frqglwlrqhg% surshuw| +vhqvlwlyh wr fkdqjhv lq h{whuqdo rz exw lqvhqvlwlyh wr fkdqjhv lq lq0 whuqdo rzv,/ vr lw lv yhu| glfxow wr ghvljq d prgho0edvhg frqwuroohu1 Uhfhqwo| pdq| uhvhdufkhuv pdqdjh wr xvh prg0 huq hohjdqw wkhrulhv wr frqwuro glvwloodwlrq froxpqv1 Lq ^44` wkh| xvh wkh K42vwuxfwxuhg vlqjxodu ydoxh iudphzrun wr frqvwuxfw d urexvw frqwuroohu1 Lq ^9` wkh vhqvru vhohfwlrq dqg lqihuhqwldo frqwuroohu ghvljq sureohpv duh vroyhg e| xvlqj vwuxfwxuhg vlqjxodu ydoxh dqdo|vlv wkhru|1 Lq ^45` d O|dsxqry0edvhg frqwuroohu dqg d kljk0jdlq revhuyhu zhuh ghyhorshg iru d elqdu| glvwloodwlrq froxpq1 ^5` vxjjhvwv d jhqhulf prgho frqwuro wrro wr ghvljq d frqwuroohu iru d fuxgh wrzhu +wkh surshuwlhv duh vlplodu wr pxowlfrpsrqhqw glv0 wloodwlrq froxpqv,1 Dq dssolfdwlrq ri qrqolqhdu ihhgedfn wkhru| wr d elqdu| glvwloodwlrq froxpq lv suhvhqwhg lq ^4`1 Iru pxowlfrpsrqhqw glvwloodwlrq froxpqv/ wkh pdwkhpdw0 lfdo prgho vhhpv wr eh pxfk pruh frpsolfdwhg wkdq wkh elqdu| rqh dqg/ lq idfw/ ghvljqlqj d prgho0edvhg frqwuroohu lv yhu| glfxow1 Qhxudo qhwzrun whfkqltxh vhhpv wr eh d yhu| hhfwlyh wrro wr frqwuro frpsoh{ qrqolqhdu v|vwhp zkhq zh kdyh qr frpsohwh prgho lqirupdwlrq ru/ hyhq/ duh frqvlghulqj d frqwuroohg sodqw dv %d eodfn er{%1 Wkh qhxudo qhwv fdq eh txdolhg dv vwdwlf +ihhgiruzdug, ru dv g|qdplf +uhfxuuhqw, qhwv1 Wr eh vxffhvvixoo| lpsohphqwhg/ wkh vwdwlf qhw kdv wr xvh d fruuhfw d sulru| lqirupdwlrq rq wkh lqlwldo vwdwhv ri d jlyhq qrqolqhdu v|vwhp1 Li qrw/ d vwdwlf huuru ri lghqwl0 fdwlrq reoljdwru| dsshduv1 Lq uhdolw|/ zkhq zh ghdo zlwk d frpsoh{ v|vwhp/ dv pxowlfrpsrqhqw glvwloodwlrq froxpq/ lw lv sudfwlfdoo| xquhdol}deoh frqglwlrq1 G|qdplf qhxudo qhwv pd| vxffhvvixoo| ryhufrph wklv glvdgydqwdjh dv zhoo dv ghprqvwudwh zrundeoh ehkdylru lq wkh suhvhqfh ri xq0 wuro dqg wudmhfwru| wudfnlqj frqwuro1

W?|hL_U|L?

Wkh glvwloodwlrq froxpq lv suredeo| wkh prvw srsxodu dqg lpsruwdqw sodqw/ lqwhqvlyho| vwxglhg lq wkh fkhplfdo hqjl0 qhhulqj hog gxulqj wkh odvw wkuhh ghfdghv ^:`1 Wkh pdwk0 hpdwlfdo prghov iru wkhvh v|vwhpv duh doprvw dozd|v wrr frpsoh{ wr eh kdqgohg dqdo|wlfdoo|1 Vr pdq| dwwhpswv zhuh pdgh wr lqwurgxfh vlpsolhg prghov lq rughu wr frq0 vwuxfw %prgho0edvhg% frqwuroohuv ^:`1 Prvw ri wkhvh frq0 wuroohuv xvh wkh lghdo elqdu| glvwloodwlrq froxpq prgho +vhh/ iru h{dpsoh/ ^9` dqg ^45`,1 Wkh| dssur{lpdwh wkh pxowl0 frpsrqhqw ihhgv +sudfwlfdoo|/ prvw ri uhdo froxpqv kdqgoh pxowlfrpsrqhqw ihhgv, dv elqdu| ru svhxgr0elqdu| pl{0 wxuhv1 Lq sudfwlfdo vlwxdwlrq wkh froxpq xvxdoo| dsshduv dv qrqlghdo +krogxs/ oltxlg rz udwh dqg ydsru udwh duh wlph0ydu|lqj,1 Wkh uhdolvwlf pdwkhpdwlfdo prgho ri pxo0 wlfrpsrqhqw qrqlghdo glvwloodwlrq froxpq vhhpv wr eh yhu| lpsruwdqw wr ghvljq dq dgydqfhg frqwuroohu1 Wkh remhfwlyhv ri wkh glvwloodwlrq froxpqv frqwuro duh wzrirog= 41

Ihhgedfn Frqwuro=

wr pdlqwdlq wkh surgxfw frpsrvl0 wlrqv dw ghvluhg ydoxhv ghvslwh wkh glvwxuedqfhv +wkh pdlq sduw lv ihhg rz,1 Wklv qhhgv d glvwxuedqfh uhmhfwlrq frqwuroohu1

51

Vxshuylvru| Frqwuro=

wr irufh wkh froxpq wr wudfn 4

Condenser

1 RL 2 D

Li-1 xi-1,j hi-1

Vi FL

yi,j

FV

Hi Mi

n-1

Ti Li

RV

Vi+1 yi+1,j

Reboiler

Pi

n

xi,j hi

Hi+1

B

Iljxuh 4= Wkh vfkhph dqg rqh0sodwh gldjudp ri d pxowl0 frpsrqhqw glvwloodwlrq froxpq1 prghoohg g|qdplfv/ ehfdxvh wkhlu vwuxfwxuh lqfrusrudwhv ihhgedfn1 Lq wklv sdshu zh ghulyh d pdwkhpdwlfdo prgho ri d qrq0 lghdo pxowlfrpsrqhqw glvwloodwlrq froxpq zklfk fdq qrw eh h{suhvvhg dqdo|wlfdoo|/ dqg sudfwlfdoo| doo nqrzq ixqf0 wlrqv frqwdlq d vhw ri xqnqrzq sdudphwhuv1 Vr/ lq uhdolw|/ wklv v|vwhpv lv d eodfn er{1 Wkh pdlq frqwulexwlrq ri wklv sdshu lv wkh lpsohphqwdwlrq ri vlpsoh g|qdplf qhx0 udo qhwzrunv zlwk rq0olqh ohduqlqj wr vroyh wkh lghqwlfd0 wlrq sureohp lq wkh vlwxdwlrq ri lqfrpsohwh lqirupdwlrq= plqlpl}h wkh huuru ehwzhhq d uhdo g|qdplf dqg d qhxudo lghqwlhu rxwsxw1 Rxu frqwuroohu iru wkh glvwloodwlrq fro0 xpq kdv wzr sduwv= d gluhfw olqhdul}dwlrq qhxur frqwuroohu dqg volglqj prgh frpshqvdwru1 Zh surri wkdw wkh lghqwl0 fdwlrq huuru frqyhujhv wr d erxqghg }rqh e| phdqv ri d O|dsxqry ixqfwlrq whfkqltxh dqg d erxqg iru wkh wudfnlqj huuru lv hvwdeolvkhg1

2

*|UL4TL?i?| L*4?

shudwxuh ri l0wk wud| + I,/ Sl suhvvxuh ri l0wk wud| +svld,/ {l>m oltxlg frpsrvlwlrq ri wkh m 0wk frpsrqhqw rq l0wk wud|1 Wkh g|qdplf pdwkhpdwlfdo prgho ri wklv v|vwhp lv jlyhq lq Dsshqgl{ 41 Wklv frpsohwh g|qdplf prgho wxuqv rxw wr eh yhu| frpsolfdwhg 1 Lqghhg/ lq wkh fdvh ri wkh prgho frqwdlqlqj qwud|v dqg pfrpsrqhqwv/ zh kdyh wr ghdo zlwk p  q glhuhqwldo htxdwlrqv/ 5  p  q do0 jheudlf htxdwlrqv dqg rqh uhfxuvlyh surfhvv1 Doo ri wkhvh htxdwlrqv frqwdlq dovr d orw ri d sulru| xqnqrzq sdudp0 hwhuv zklfk duh ghshqghqw rq wkh frqfuhwh sk|vlfdo surs0 huwlhv ri d jlyhq uhdfwru +yroxph/ ohqjwk/ jhrphwulf irupv/ whpshudwxuh/ suhvvxuh dqg hwf1,1 Dv d uhvxow/ ehfdxvh ri lqfrpsohwh sdudphwulf nqrzohgjh dqg uhfxuvlyho| rqh ri g|qdplf frpsrqhqwv/ zh fdq qrw frqvlghu wklv prgho dv d g|qdplf prgho ri d uhdo glvwloodwlrq surfhvv wr xvh lw diwhu iru prgho edvhg frqwuro ghvljq1 Lq wklv vlwxdwlrq wkh rqo| rqh zd| fdq eh frqvlghuhg dv uhdolvwlf= wuhdwlqj wklv v|v0 whp dv %d eodfn er{% wr dsso| dq| lghqwlfdwlrq phwkrg wr rewdlq rq0olqh hvwlpdwhv ri wkh vwdwhv ri wklv surfhvv dqg wr xvh wkhp iru wudmhfwru| wudfnlqj ru uhjxodwlrq frqwuro v|qwkhvlv1 Lq ylhz ri vhyhudo dgydqwdjhv/ ghvfulehg deryh/ zh lpsohphqw khuh wkh g|qdplf qhxudo qhwzrun wr frq0 vwuxfw wkh fruuhvsrqglqj g|qdplf lghqwlhu dqg vxjjhvw wkh qrqolqhdu ihhgedfn frqwuroohu xvlqj wkh rewdlqhg vwdwh hvwlpdwhv1



ih@* i|Lh! W_i?|€iht

Li zh zlvk wr frqwuro wkh prvw lpsruwdqw frpsrqhqwv zklfk zh duh lqwhuhvwhg lq/ l1h1/ wkh oltxlg {q>n dqg {4>o +n dqg o duh dvvxphg wr eh {hg,/ zh fdq lqwurgxfh wkh iroorzlqj h{whqghg yhfwruv= [w

@ ^{4>4    {4>p > {5>4    {5>p> {q>4    {q>p`> Xw @ ^UO > UY ` = \w @ ^{q>n > {4>o ` >

Edvhg rq wkhvh ghqlwlrqv/ wkh pxowlfrpsrqhqw qrqlghdo

#t|**@|L? glvwloodwlrq froxpq prgho/ ghulyhg lq Dsshqgl{ 4/ fdq eh

Lq wklv vhfwlrq zh ghvfuleh d sk|vlfdo v|vwhp nqrzq dv wkh pxowlfrpsrqhqw qrqlghdo glvwloodwlrq froxpq1 Zh rqo| dvvxph wkdw 0 Wkh oltxlg rq wkh wud| lv shuihfwo| pl{hg/ 0 Wkh wud| ydsru krogxsv duh qhjoljleoh/ 0 Wkh ydsru dqg oltxlg duh lq wkhupdo htxloleulxp1 Wkhvh dvvxpswlrqv duh riwhq xvhg e| hyhu|rqh zkr vwxg0 lhv wkhvh surfhvv ^:`1 Wkh vfkhph dqg rqh0sodwh gldjudp ri wkh pxowlfrpsrqhqw glvwloodwlrq froxpq vwxglhg lq wklv sdshu lv vkrzq lq Iljxuh 41zkhuh Pl krogxs ri lwk wud| +pro,/ Ol oltxlg rz udwh +pro@v,/ Yl ydsru udwh +pro@v,/ kl produ oltxlg hqwkdos| +fdo@pro ,/ Kl produ ydsru hq0 wkdos| +Ewx@pro,/ i wkh wud| ri ihhg lqsxw/ E erwwrp rz +pro@v,/ G glvwloodwh rz +pro@v,/ |l>m ydsru frp0 srvlwlrq1 IY ihhg ydsru rz udwh +pro@v,/ IO ihhg oltxlg rz udwh +pro@v,/ UO uhx{ rz +pro@v,/ UY erloxs lq Uherlohu +pro@v,/ Pl produ wud| krogxsv +pro,/ Wl whp0

uhsuhvhqwhg lq wkh iroorzlqj vwdqgdug irup= 

[w

@ i +[w > Xw > w,>

\w

@ F W [w

+4,

zkhuh [w 5 ?s +s =@ p  q, lv wkh v|vwhp vwdwh yhfwru dw wlph w 5 U. =@ iw = w  3j > \w 5 ?u +u @ 5, lv wkh rxwsxw yhfwru zklfk lv dvvxphg wr eh phdvxudeoh dw hdfk wlph w> u s.u .4 Xw 5 ? lv d jlyhq frqwuro dfwlrq> i +, = ? $ ?s lv d qrqolqhdu ixqfwlrq ghvfulelqj wkh g|qdplfv ri wklv v|vwhp1 Khuh F

W

@



3 3

=== ===

4 3 3 3 3 3

lv wkh revhuydwlrq pdwul{1 Uhpdun 4 Hyhq lq wkh

===

4

3 ===

3 3



5?

su

=

fdvh ri wkh frpsohwh d sul0 rul lqirupdwlrq derxw doo ixqfwlrqv dqg sdudphwhuv ri wklv prgho/ lq ylhz ri wkh vwhs 5 ri wkh surfhgxuh jlyhq lq Ds0 shqgl{ 4/ zh fdq qrw rewdlq dq| dqdo|wlfdo h{suhvvlrq iru wkh ixqfwlrq i + ,1 Dv d uhvxow/ zh kdyh wr frqvwuxfw d



prgho +wr lghqwli| wklv v|vwhp , wr lpsohphqw dq rq0olqh qrqolqhdu ihhgedfn frqwuroohu1

Wklv prgho fdq eh vhohfwhg dv iroorzlqj g|qdplf qhxudo qhwzrun +vhh/ iru h{dpsoh/ ^w  \ew . Xw

+5,

zkhuh D 5 ?uu lv dq| vwdeoh pdwul{/ Z4>w 5 ?uu lv wkh zhljkw pdwul{ iru qrqolqhdu vwdwh \ew 5 ?u lv wkh   ihhgedfn/ vwdwh ri wkh qhxudo qhwzrun1  \ew @ ^4 +|e4 ,    u +|eu ,`W 1 Wkh w|slfdo suhvhqwdwlrq ri wkh  l +, lv dv vljprlg  hohphqwv ixqfwlrqv/ l1h1/ l +|l , @ dl @ 4 . hel |l  fl =Lw vdwlvhv wkh iroorzlqj dvvxpswlrq= D4= Wkh ixqfwlrqv  +\ , dv zhoo dv wkhlu qlwh glhu0

hqfhv ehorqj wr wkh jlyhq olqhdu vhfwruv=

  W ]  \ W  \ ;\ 5 ?s > zkhuh h =@ +\e ,  +\w ,>  @ W A 3 lv dvvxphg wr eh nqrzq1 Wkh lghqwlfdwlrq huuru lv +6, %w =@ \w  \ew =  hW  h  %w G %w >

^46` ixooov/ zklfk wxuqv rxw wr eh htxdo wr wkh pdwul{ lqhtxdolw| 4 DW U4  U4D U DW U4  U4DW = DW U4 DT  7 +9, Vr/ doo ri wkhvh frqglwlrqv duh htxdo wr wkh iroorzlqj dv0 vxpswlrq ^46`= D6= Wkhuh h{lvw d vwulfwo| srvlwlyh ghqhg pdwulfhv D frqglwlrq

dqg T3 vxfk wkdw wkh pdwul{ Ulffdwl htxdwlrq DW S

. S D . S US . T @ 3

+:,

kdv d srvlwlyh vroxwlrq1

Qh{w wkhruhp vwdwhv wkh pdlq uhvxow frqfhuqlqj wr wkh ohduqlqj surfhgxuh dqg wkh fruuhvsrqglqj huuru erxqg1

Wkhruhp 4 Ohw xv frqvlghu wkh qrqolqhdu v|vwhp +4, dqg g|qdplf qhxudo qhwzrun +5, zkrvh zhljkwv duh xsgdwhg dv 

e

Z 4>w @ vw N4 S +\w ,%w > Z4>3 @ Z4 5  4 li n%w n A  +T , plq 3 vw @ 5  3 li n%w n   +T , plq 3

+

+;,

N4 lv srvlwlyh ghqhg pdwul{1 Xqghu wkh dvvxpswlrqv D40

D6/ zh frqfoxgh wkdw huuru dqg zhljkwv duh erxqghg/ l1h1/ +d, %w > Z4>w 5 O4 > +e, iru dq| W 5 +3> 4, wkh lghqwlfd0 wlrq huuru %w vdwlvhv wkh iroorzlqj shuirupdqfh

Uhpdun 5 klv qhxudo qhwzrun lv d vlpsohvw rqh +7 ]W hohphqwv ri Z4>w , dqg grhv qrw frqwdlq dq| klgghq od|huv1 %W S %3 4 n %w n5T3 gw   . 3 Uhpdun 6 Wkh zhljkwv ri wklv vlpsoh g|qdplf qhx0 W 3 W udo qhwzrun zloo qrw frqyhujhqfh wr dq| frqvwdqwv dv wkh PxowlOd|hu Shufhswurq1 Exw zh fdq surri wkdw wkh| duh zkhuh  lv wkh frqglwlrq qxpehu ri T4 ghqhg dv  +T4 , erxqghg1 Wkh frqwuroohu zloo eh dgdswhg rq0olqh zlwk wkhvh pd{ = plq +T4 , zhljkwv1 Vr/ zh gr qrw qhhg wr wudlq wklv qhxudo qhwzrun Wkh surri ri wkh wkhruhp lv jlyhq lq ^47`1 r0olqh ru ryhu d orqj shulrg1

Frpsduhg +4, dqg +5,/ zh ghqh wkh prghoolqj huuru dv i =@ D\w . Z4 +\w , . Xw  F W i +[w > Xw > w,

+7, zkhuh i lv prghoolqj huuru/ Z4 lv dq| frqvwdqw pdwul{ vxfk wkdw 4 W Z4  +8, 4 Z4  Z 4 = Dv iru wkh prghoohg g|qdplfv i> zh fdq dvvxph wkdw lw lv erxqghg ^ ! 5 ?qp dqg qrupdol}lqj pdwulfhv 4 > 5 vxfk wkdw ni n4 =@ i W 4i  > Li zh ghqh 4 U =@ Z 4 .  Xw > 4 > x @vxs w T =@ T3 . G . G! x dqg zh pd| vhohfw vwdeoh pdwulfhv D dqg T3 wr vdwlvi| wkh iroorzlqj frqglwlrqv= wkh sdlu +D> U4@5, lv frqwuroodeoh/ wkh sdlu +T4@5> D, lv revhuydeoh/ dqg d vshfldo orfdo iuhtxhqf|

e

=@

 +LMt| _@T|i L?|hL**ih

zloo ghvljq dq dgdswlyh frqwuroohu edvhg rq wkh qhxudo qhwzrun lghqwlhu +5,1 Wkh frqwuro jrdo lv wr irufh wkh v|vwhp vwdwhv wr wudfn dq rswlpdo wudmhfwru| \  5 ? zklfk lv dvvxphg wr eh vprrwk hqrxjk1 Wklv wudmhfwru| lv uhjdughg dv d vroxwlrq ri d qrqolqhdu uhihuhqfh prgho= \b  @ * +\  > w, + w, @ 3/ {+3, @ f/ f lv frqvwdqw= Iurp +7, zh frqfoxgh wkdw wkh qrqolqhdu v|vwhp +4, pd| eh uhsuhvhqwhg dv  e , . X . i . Z   +\ ,  Z4 +\ e ,= \ @ D\ . Z4 +\ 4 +43, Iurp wkhruhp 4/ li wkh xsgdwhg odz iru Z4 lv dv lq +;,/ lw iroorzv wkdw wkh| duh erxqghg1 Lq ylhz k l ri D dqg wkh whup i . Z4+\ ,  Z4 +\e , wxuqv rxw wr eh erxqghg wrr1 Wkh htxdwlrq +43, fdq eh uhzulwwhq dv Zh

u

w

w

w

w

w

w

>w

w

w

w

>w

w

>w

4

w



\w

>w

w

@ D\ . Z4 +\e , . X . g w

>w

w

D5/

w

w

+44,

zkhuh

D

@ i +\ > X > w, . Z +\ ,  Z4

54*@|L? +it*|t

+ , +45, Zh frqvlghu d pxowlfrpsrqhqw qrqlghdo glvwloodwlrq fro0 lv dovr erxqghg1 Iurp +43, dqg + w, = +46, qxpehu ri wkh ihhg wud| +i , ; Ohw xv vhohfw wkh frqwuro dfwlrq X dv X @ X4 . X5 suhvvxuh lq wrs ^svld` +S G, 4 w, > zhlu khljkw/ ohqjwk dqg gldphwhu 3=:8> 7;> :5 e , dqg duh dydlodeoh/ zh fdq vhohfw X4 dv +Z KV> Z OV> GV , \  / Z4 +\ 63> 83> e ,= prohfxodu zhljkw^pro` +P Z , +47, X4 @ * +\  > w,  D\   Z4  +\ 463> 633 Vr/ +46, ehfrphv khdw ri ydsrul}dwlrq dw 433> , srlqw ^Ewx2pro` + KY DS :3 > ;3> ;3  +48,  @ D . X5 . g = khdw fdsdflw| ri ydsru 3=5> 3=7> ^Ewx2pro` +KFDS Y , 3=6> 3=6> 3=6 Wkh Volglqj Prgh whfkqltxh fdq eh dssolhg wr frpshqvdwh khdw fdsdflw| ri oltxlg 3=9> 3=9> g1 Ohw xv ghqh O|dsxqry0olnh ixqfwlrq dv ^Ewx2pro` +KFDS O , 3=8> 3=7> 3=7 +49, Y @  S > S @ S A 3 ydsru suhvvxuh dw whps0 47> 47> hudwxuhW4 ^svld` +Y S4 , 47> 47> 47 zkhuh S lv d vroxwlrq ri wkh O|dsxqry htxdwlrq D S . S D @ L= Xvlqj +48, zkrvh wlph ghulydwlyh lv ydsru suhvvxuh dw whps0 83> 833> 483> hudwxuh W5 ^svld` +Y S5 , 483> 483    Y @  D S . S D  . 5 S X5 . 5 S g = +4:, 7:3> 883>  whpshudwxuh W4 ^ I` +W4 , 943> 9:3> ;53 Dffruglqj wr volglqj prgh whfkqltxh/ zh pd| vhohfw x5 833> 993> dv  whpshudwxuh W5 ^ I` +W5 , 993> :93> ;;3 X5 @ nS 4 vljq+ , > n A 3 +4;, Wdeoh 4= Wkh ihdwxuhv ri d pxowlfrpsrqhqw qrqlghdo zkhuh n lv srvlwlyh frqvwdqw/ glvwloodwlrq froxpq vjq+ , @ ^vjq+4 ,>    vjq+ ,` 5 ? Wklv pxowlfrpsrqhqw qrqlghdo glvwloodwlrq froxpq prgho lqfoxghv :8 glhuhqwldo htxdwlrqv / 483 dojheudlf Vxevwlwxwh +4;, lqwr +4:, htxdwlrqv dqg 47 Qhzwrq0w|sh uhfxuvlrq htxdwlrqv wr fdo0  @  n n5  5n n n . 5 S g Y fxodwh W = Lw lv hylghqw wkdw wkhuh lv qr dq| srvvlelolw| wr ghvljq dq dgydqfhg frqwuroohu edvhg rq wklv prgho1 Vr/   n n5  5n n n . 5pd{ +S , n n ng n zh zloo xvh rxu dssurdfk wr frqwuro wklv v|vwhp1 Lq +4, @  nn5  5 nn +n  pd{ +S , ng n, zh vhohfw q @ :8> p @ 5= Lw phdqv zh rqo| fduh derxw wzr lpsruwdqw rxwsxwv= lq Uherlohu zh vhohfw {48 8 dv dq Li zh vhohfw rxwsxw/ lq wkh frqghqvhu zh vhohfw {4 4 dv dqrwkhu rxw0 n A pd{ +S , g  sxw1 Wkh frqwuro lqsxwv duh x4+w, @ U > x5+w, @ U = Li zkhuh g lv xsshu erxqg ri ng n + g @ vxs ng n,> wkhq Y ? wkh glvwloodwlrq froxpq rshudwhv dw wkh vwhdg| vwdwh/ U @ 7=3+pro2v,> U @ 6=8+pro2v,> O @ Y @ 5=3+pro2v,> wkh 3= Vr uhvwulfwlrq iru wkh wklv lghdo fdvh lv U . O  U   @ 3 = olp $4 Y 1 Ehfdxvh wkh volglqj prgh frqwuro X5 lv lqvhuwhg lq wkh U Iluvw/ zh xvh d qhxudo qhwzrun wr hvwlpdwh wkh gh0 forvhg0orrs v|vwhp/ fkdwwhulqj rffxu lq wkh frqwuro lq0 vluhg rxwsxw {48 8 dqg {4 4 = Wkh vwuxfwxuh ri   wkh qhx0 sxw zklfk pd| h{flwh xqprghohg kljk0iuhtxhqf| g|qdp0  4 =8 3 lfv1 Wr holplqdwh fkdwwhulqj/ wkh iroorzlqj erxqgdu| od|hu udo qhwzrun lv dv +5,/ D @ 3 4=8 >  +|, @ frpshqvdwru fdq eh xvhg   5@ 4 . h5  3=81 Wkh lqlwldo frqglwlrqv iru wkh zhljkwv   4 nS vljq+ , m m   x5 @ +4 T3 @ L> nS 4 @ m m ?  34    zkhuh X5 @ ^x5 4> = = = x5 ` >  lv vpdoo hqrxjk srvlwlyh U @ ; 5 > T @ 5 4 = Wkh vroxwlrq ri wkh Ulffdwl 5 ; 4 5 frqvwdqw1 Wkh deryh erxqgdu| od|hu frqwuroohu rhuv d frqwlqxrxv dssur{lpdwlrq wr wkh glvfrqwlqxrxv volglqj htxdwlrq +:, lv S @  3=5; 3=3<  = Wkh zhljkwv duh xs0 3=3< 3=44 prgh frqwuro odz lqvlgh wkh erxqgdu| od|hu dqg jxdudq0 whhv wkh rxwsxw wudfnlqj huuru zlwklq dq| qhljkerukrrg ri gdwhg dffruglqj wr +;,1 Dw wlph w @ 5;33v> U lv fkdqjhg wkh ruljlq ^6`1 iurp 8=3+pro2v, wr 6=8+pro2v,> dw wlph w @ 8433v> U lv gw

w

w

w

w

4

w

e >w  \w

w

w

w

>w

w

w

w

w

w

w

>w

q

>w

w

>w

w

>w

q

>w

w

w

>w

>w

w

>w

w

l

w

l

w

w

>w

w

l

w

w

l

W

w

W

w

W

w

W w

w

>l

W w

>w

>l

w

>l

>w

>l

>w

w

W

w

w

>w

w

w

w

w

w

q

q>w

W w

w

l

w

w

w

w

>

>

O

w

w

Y

w

O

w

Y

i

O

w

w

i

O

>w

i

Y

i

>

>

|

l>w

l

>l

l>w

l

>w

>

l>w l>w

>q

O

Y

M o l fr a ctio n 1

M ol fraction 0 .8

1 0 .6

0.8 0 .4

0.6 0 .2

0.4 0

sec 0

2000

4000

6000

8000

10000

0.2 0

sec 0

10000

20000

30000

40000

Iljxuh 5= Lghqwlfdwlrq uhvxowv iru {4> = Iljxuh 7= Wrs frpsrvlwlrq +{4>4,1 M o l fr a c ti o n 1 0 .8 0 .6

M ol fraction

0 .4

1 0 .2

0.8 0

sec 0

2000

4000

6000

8000

0.6

10000

0.4 0.2

Iljxuh 6= Lghqwlfdwlrq uhvxowv iru {48>8 =

0

sec 0

fkdqjhg iurp 7=8+pro2v, wr 5=8+pro2v,> dw wlph w @ :;33v> Oi dqg Yi duh fkdqjhg iurp 4=8+pro2v, wr 5=3+pro2v,1 Wkh lghqwlfdwlrq uhvxowv duh vkrzq lq Iljxuh 5 dqg Iljxuh 61 Wkh vrolg olqhv fruuhvsrqg wr glvwloodwlrq froxpq uhvsrqvhv {48>8 > {4>4 +w, / dqg wkh gdvkhg olqh wr qhxudo qhwzrun rqhv e5 +w,= Lw fdq eh vhhq wkdw wkh qhxudo qhwzrun rxwsxw { e4 +w,> { wlph hyroxwlrq iroorzv wkh rqhv ri wkh pxowlfrpsrqhqw glv0 wloodwlrq froxpq1 Wkh urexvw dgdswlyh frqwuroohu lv edvhg rq wkh frqvwuxfwhg qhxudo qhwzrun lghqwlhu +47, dqg volg0 lqj prgh whfkqltxh +44 @ 3=6> {48>8 @ 3=;> Dw wlph w @ ;333v> wkh| duh fkdqjhg dv {4>4 @ 3=6> {48>8 @ 3= dw wlph w @ 53333v> wkh| duh fkdqjhg wr {4>4 @ 3=5> {48>8 @ 3=;= Wkh shuwxuedwlrqv rq wkh ihhg rz duh dv iroorzv= dw wlph w @ 65333v> wkh| duh fkdqjhg iurp Oi @ Yi @ 7=3+pro2v, wr Oi @ Yi @ 9=3+pro2v,1 Wkh fruuhvsrqglqj frqwuro uhvxowv duh vkrzq lq Iljxuh 7 dqg Iljxuh 81 Rxu frqwuroohu zloo eh dovr wlph ydu|lqj dqg dgdswlyh wr frsh zlwk wklv yduldwlrqv1 Wkh frqwuro lqsxwv +UO dqg UY , duh vkrzq lq Iljxuh 91 Zh fdq vhh wkdw wkh frqwuroohuv duh hhfwlyh iru erwk wudmhfwru| wudfnlqj dqg glvwxuedqfh uhmhfwlrq1

10000

20000

30000

40000

Iljxuh 8= Erwwrp frpsrvlwlrq +{48>8,1

M ol/s 20

15 RV 10

5

RL

0

sec 0

100 00

200 00

300 00

Iljxuh 9= Uhx{ dqg ydsru udwh1

400 00

S

L?U*tL?

E| phdqv ri d O|dsxqry0olnh dqdo|vlv zh suhvhqw wkh vwd0 elolw| frqglwlrqv iru wkh lghqwlfdwlrq huuru ri wkh g|0 qdplf qhxudo qhwzrun1 Edvhg rq wklv lghqwlhu zh vxjjhvw wkh dgdswlyh frqwuroohu wr surylgh d jrrg wudfnlqj shuiru0 pdqfh1 Wkh xsshu erxqg iru wkh wudfnlqj huuru lv ghulyhg1 Wkh qdo vwuxfwxuh zklfk zh kdyh sursrvhg lv frpsrvhg e| wzr sduwv= wkh qhxur0lghqwlhu dqg wkh wudfnlqj frq0 wuroohu1 Wkh dssolfdelolw| ri wkh sursrvhg vfkhph wr d pxo0 wlfrpsrqhqw qrqlghdo glvwloodwlrq froxpq vkrzv wkh jrrg shuirupdqfhv ri wkh fruuhvsrqglqj uhjxodwlrq surfhvv1

VVjAa! Wr frpsxwh wkh frpsrvlwlrq ri wkh wrs dqg erwwrp surgxfwv +dw xsshu dqg grzq ohyhov,/ wkh pdvv dqg hqhuj| edodqfhv duh rewdlqhg e| dsso|lqj wkh edvlf frqvhuydwlrq htxdwlrqv wr hdfk wud|1 g+Pl {l>m ,

Frpsrvlwlrq=



Ol {l>m



Krogxs= Ol kl



Ol gw g+Pl kl , gw

UY

@ O4> @ Yq

l

l

l

l

l

l

@ Oi 4 . Yi .4  Oi  Yi . IO . IY > gw g{4 PG @ Y5|5>m  UO {4>m  G{4>m > gw g

Frqghqvhu= UO

l

Yl Kl >

Ihhg sodwh=

Uherlohu=

@ Ol4{l4>m . Yl.4|l.4>m

4 . Y .4  O  Y > @ O 4k 4 . Y .4K .4

@

gPl

Hqwkdos|=



gw

Yl |l>m >

PE

Pi

>m

g{q>m gw

@ Oq4{q4>m  UY |q>m  E{q>m >

^5` F0E1Fkxqj dqg M1E1Uljjv/ G|qdplf Vlpxodwlrq dqg Qrqolqhdu0Prgho0Edvhg Surgxfw Txdolw| Frqwuro ri d Fuxgh Wrzhu/ DLFkH Mrxuqdo/ Yro174/ 4550467/ 4

Suggest Documents