mccabethiele - graphical design of fractionating columns

7/28/2019 McCabeThiele - Graphical Design of Fractionating Columns

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Graphical Design of Fractionating Columns'By W . L . McCabe and E. W . Thiele

111.: ixtiu~i:iI(hip f r n lw i i i >ipj):ir:it~isor . scpnra t i i ig Rodc i~ i~s l i , '~i i t l Savitrit.Ii l I o ~ v c v c r , it is fclt tha t tlicfollon.iiig mctliod is siinplcr, csliibits its rcsults in a plninermanner thnn n ny metliod, annlytirnl or gnph ic n l , so fn rproposed, nnd is acciirntc cnough for :dl practic:il I ISC.This pnpcr is to consider the dctcrrninntiou of th e nuni bcr ofthcorcticnl plates only, qiid it is nssumctl tlirougliout tllnt theplntc efficiency, o r tlic 13. E. T. I) . , fo r the tyjx of coliimn aridth e mistiire undcr consitlcmtion is approximately known.

t h e conipn~ieii ts f i i i in ry niist ,urcs of vcdatilc, misciblcvoiiijm~iciitsdates back to Sorcl.2 Hc nssumcd thntequi1ib;iuin is est~nhlishcdbctwcen t l ic liquid on cntnli platc: i n t i t h c vapor rising fixmi it, n i i t l cnlculntecl tlic cnricliinentf i m i plttte to plntc by licnt ant1 mnt crial bnlanccs. Sorel'si i i e t l iod, t hough s o u n d , is tedious wheii npplicd annlytically.hinny writers h n ~ c iinplificd or modified Sorel's rncthod,among thcm being Le\ris,3 Lcslie,4I ' c t e r ~ , ~ an r \ ; y ~ , ~1id Gay,' whohavc all used nndyticnl methods.I aw i s ' s mcthod is typical of mniiyc.iirichmcrit per plntc \VI IS sm:ill, n-rotcc n t i n ] l y , lit,cgrntctl t,, ~o (Ictrr-l i l i , l c ( l , llC l l u r l , ] , c r () E l ' t l C] : t rKrr tllC l u r n b c r f l,]ntcs tiiCmOrcRcc,lrntc is tIliS I i l c t ] l o (~ , ,ut fo rco]u ll lns o f fC\" P]r,tCs I lr errori I l t r o ( ] u c e ( ~by r ~ s s u n , i n g co l i t i l l~ lousfo r s t , c p \ \ i s c colldit ions is apprc(:i-

Preliminary AssumptionsFor the purpose of initial simplifica-

innde , which are commonly rcnlized(, f t he m . L,,\,.is nssilInet~+Jlnt tile tion ccrtnin nssuinptions will becIiric]lnlcl,t (]iffcr- or npproaclrctl ii i ortlinnry practicewitliin tlic precision of t h e dntn .

swy fo r an y prtrticular cwc, tlic rc-etrirt ions so irnposctl enn 1x3 rcmovctlI t \vi11 I)c ~1ion.n n ter tha t , i f nc(*cs-

n u m b c r of rnols of vnporble.111 al l tire proccclurcu ~iiciitio~ietl

ii 110y e , pc rf cc t tl i ccoliiinn! and licncc complcte cquilib-riLiiii bctwccii the vapor a n d liq~iitlof cncli plate, is nssuincd, nntl th eiiumhcr of plntcs $0 dctcrmined is t h e~ iun ibc r of thcorcticxl or pcrfcctplntcs ncccssnry to cffcct th e tlcsircdscpnrntion. If nil actrinl colurnn isone of th e plntc typ e, tlic i i u i n l m of

i1.111 o11 i :I g 0 E

oiling pooiiits of tlic com-

thcnrctical p1:itcs is tlivitlcd by n n nvcrngc plntc cficicncy to~f tIic co ]u l n n is .Yofr-Troutori 'Y ri l le s t n t e s t h a t for normill l i q u i d s t h e n i o l n l entropyof vnr~ariz.ntirm b ircnriy cons tnn t, or ( J f r l T ) - c o n s t n n t , w h c r c .\f = niolcc.

u n i t Wright., tc it into i l c , l , l l l l ~ ~ c r>f nct,li,]plntcs,

(,I Ijn(:+eti tylx t l l c n r l l n ~ , c r f tllcorctical plnt ,cs is n l u l t i -plicd by tlic lrciglit of coliiinn equivalcnt to on c tlicoreticnlplate (11. E. T. P . ) 8 f t.hc height of thc COlUlnii iS desirC(1.Both factors nre cxpcrimcntnliy detcrmillcd mid bot.11 Yarywith tlic design of tlic coiumn iiiid uritli tllc misturc distilled.sinccilcy l i n v c no tbeen nccurntcly tletcrrnirlcd or rill tyPsin determining theoreticnl plntes with grcnt precision and amcthcxl such ns hu ' i s ! s offers sufficicnt nccurn cy i n spite of theapproximation inhcrent in i t . H o w c v c r , sincc thc equationsusctl in nl l mcthods are linear, they n d n p t themselves to

lihr wcif i11t8T = I io ilinf i t c mpe r ntur c (nbso l t l lc ) , rind 7 - l a t e n t he at pc r2--l'llc Iced c r l t c r s t h e column at a t c i npe r a t u r c

prclieatcd w r y nearly to t h i s tcnipcrnturc by theeed o ts usrlallyastc.boilirlg 1,oint of th e liquid on th c fcctl platc.3--Thc on ly ccindcriscr is n siniplc or total cotrtirnsrr, so th ccompositiori of t l ic product is t h e samc a s t h a t of thc vapor fronit h e top platc.

toTh e

'If O r f o r nlisturcs~ there l i t t l e N o [ r - , j silrlple comdenscr colldeo scs ( h e v u p o r f c d to i t , b u t Tcmo,.tlof l, e sens i l r l e4 - - ~ 1 ~ ~l c a t is

condensrd liq,,id.a t th c basc of colulnt l by n l c ~ i ~ sof "closed stcam," so th e coridc:isirig s t cam ~ W Sio t t l i lu tc thew a s t e .stepwisc gmphical mnnipulstions, with conscqucnt sim- . V o f c - O bv ious I y , o p e n s t e u n i rnn be used only a.!icri t h e less volatiic

p l j c i t ,y r o n ! w n e n t is wntrr, i f tlie m i x t u r e is to rrninin ot ic or t w o c o n r p n r n t s only.the nRsumption f cont j l l , , i ty f rom p]Rtc toNomenclaturelatc. aGrnphicnl methods I invc bccn prcscntcd by P o i i c h o n , ~

pr e Rc nltd I , c f n r e , h e D i v i , i o n o rRc r c ive t l 1 2 eb ru n ryChcrnicnl Society, Slalt imorc, X f c l . , Aprll 0 to I O , 1025.

Tlic nr~~ncnc lnturcs n n~otlif irntioii f t l i n t uw d by Lewis,fcrrct! to ns thc rectifying columi l ~ t n d tm t part below th cfeed plntc ns tlic exlinusting caluiiin. . * \ I1 coiircii trl \ t ioiisarc cxprcsscd ns mol frnct.ioris of tlic 1 1 1 0 ~ ~o1:ltilc coni~x)i ici i t .Liquit! coinpositioiis nr e dcnotcd by r , i i d v n l m coillimositionsby 2/. Al l qunntiti cs pcrtnining to tlic cshnust iug coIu~nn PC( ~ iR t i l l g , l iR ~ lC ~ fro ln c o r n ? s i w n ~ i n g c,r,nlltjties crtnirljng to tile

I n d u s t r i v l E n d I: .np$ncrrlng Chemistry n t t h e 0 0 t h M e c t l n x of the Anie r ic nn Tlint PHrt O f the np[)flrnt\IR RbOVC t l l C fN d p h t C Will k C-' C o m p f . r t n d . , 68 , 1 1 2 8 , 1 2 0 4 , 1317 (1,389); M, 1 2 1 3 (1804).'T i i i a J O U Y N A I . , 14 , 1 b 2 ( lWZ) ,' M o t o r fuel^," 1911, p. 70 . Cbemical C a t a l o g C o . , N ew Y o r k .* Tlfr!, J O U R N A I , , 11, 4 0 2 (IY23!.I Ckr m. MCI . nu., 10, 207, 2.55, 311, 3 AQ , 408 (1023) .C h i m i c b i n d u s f r i t , 8 , 4 , 178, 7~ ( 1 8 2 0 ) : 6,

I ' r t r f S , ' h 5 J O I I R N A I , . 14 , 47 f i ( ] !122) .7 ' r r l i . m o d r r n r , 18 , 20, ,A r l l r 2 1 ) .

(1921) : 10,Y l I , 1026 (1023).10 Tiiir JOUMXAI.. 14 , 103U (1022) .11 Chl 'm i r b ndMsfri , , Spc r iul Nu. W T - 8 6 , hlity. IO28


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a n d y3 cnlculatccl. The

INDUSTRIAL AND BNGINEERIiYG CHEMISTRY Vol. 17, No. 6rcct i fying colr imn by n l iuc Ivri t tc i i over thcm. The p ln tcstire numbcred from t l ic top plate of thc rect i fying colui i ind o w n t o t l ie bott i rr i i plntc of th e cxhnu st ing colu mn . T h efccd p l s t c , v h i i ~ l i s t,hc t o p p l n tc of t he cx i inus ting column.will simply l x a plnte wit~lin def in i t e number . =\ gcnernlpltit,c \vi11 bc d e n o t e d b y 71 , f i n t he rec t i fy ing colum n, or Iiyn i f in tlic csl i i ius t ing column. W i c r c nny q u n n t i t y is writ-t e n wi th n s u l w - r i p t , t h c s u b s c r i p t r e fc r s t o the point oforigin of d i n t q u s i i t i ty ; t h u s y,, iiictiiis t l ic composition of thevapor r is ing from pintc 7i tirid ?,- rcprescnts the coniposi-t ion of th e liquid clcsrcriding frorii plntc ii+1 O t h e r symI)olsn r c :P = riiols of distillntc \v i? l idrawi i a s ~ ~ r o d u c tn a giver1 t i m e.?.? = coinposition of p r o d u c tF = mols of m i s t u r e fed to thc apparatus i r i a giveii t i m e.I-,, = composition of fecd niixturcI ( = rrio!s or wastc withdrawti f r o m c o l u m n in a g i v c n timex v = composition of waste0 = rnols overflow from a n y plate i n tjic r e c t i f y i n g c o l u mr i0 = rnols overflow from a n y p l a t e i n t h e c x h a u s t i n g columiiL- =mols 01 vapor r i s ing from any p l a t e i n th c r e c t i f y i n gI. = rriols of vapor rising f r o m :iny p l a t e i n t l i c exhausting. . t , A , E , B = cocfficicnts i r i t h e e n r i c h m e n t cquations17 1 tire u-liolc disrz i . \ . s io t i , the bclsia u t i 1 1 be 1 mo l of product-

Derivation of EquationsThe fol lowing cqunt ions Iin1.c I we n de vc lo~ xdm id pub-l ishcd innny t imes .Figure 1 rcprcscnts a p c r f c c t c o l u i n n prfor r i i i i ig i n RC -corc!jtricc with th e nssuinpt ion l is ted nbovc. By Iiicnns ofbnlancca of total rnntcrial ar id n io rc voln t i l c component i thas been repentct i ly shown t h n t , for the rcctifjing colwnn

-

-CQlumIlcolum 1

-

i. e . , 1 = !.

0 .Y Jr:1 = O f l and B =-+1then >*, = ,1s, +B ( 3 )

which, wlicr. plottctl w i t h y as ordi i intcs l c rms 5 ns nbscissns,is t he cquxt ion of r s t ra ig ht l inc l iav ing n slope of A nr id ILy-a s i s intercept of I j . This ccjunt ion !vi11 Ix enllcd t hc en-r i c h m e n t c q u n t i n n of t he rec t i fy ing column, or s imply t l i crec t i fy ing equnt ion , and t he s t rn ight l ine rcprcscnting it t h cenr ichm ent l ine of t lic rec t i fyi ng column o r the rect i fying l inc.Cons ider the ent i re nppnrntus I d o w n .scctiori drawn bc-tween p la tes fi f 1 niid n. By rnntcrinl balnnccs, it c nnbe shown t h t

1e t 0 t I F -, I , = 0+ : i ! l d /3 =-C F Y (9)-Then x . = A y n i , r M (lWiwhich is the eclucitioil of nnothw s traight I i t ic hnvirlg a slop0

of 1 /A nnt l nn s -nxi s in te rccpt of B. This equ a t ion will IXrcfcrrcd to ns t he enr i chm ent cqunt ion of the cxhau s t ing col-uni i i o r t h e c x h n u s ti n g c q u n t i o n , nnd its s tm i g h t h e a.9tlie eiiriciiiricnt linc of t h o cxhnr i s t ing coluniii or thc cxliaust-in g line.

Use of Enrichment LinesF o r c v c r y binary mixture the re is I% dcfini te rclnt ionship h-t w w n 2 , the composi t ion of th e l iquid phnsc, nnd I /, t h e com-posi t ion of t he vapor phnsc in equi l ibrium wi t h it . This r eInt ionship is bes t exhibi ted in the form of IL r e c t a n g u l a rplot of y ns ord ina tes versus s as absc i s sas . The curve soformed is k n o w n ns t h c cquil ibr iurn curve , an d g ives the rc-Ia t ion bc twccn th e compos it ion of t he l iquid o n nny p la tea n d t h n t o f t.hc vnpor r i s ing f rom the same p la te .The cqui l ibrium curve n n d E q u n t i o n s 3 a n d 10 are allt h a t i s ricccssnry t o ca lcula te nna ly t i ca l ly the compos i t ionsof liquid nnd vapor pcr ta in ing to cnch p la te , nnd hence thenumbcr of theorct icnl plates , once th e compos i t ion of feed,wastc, nnd produc t , and rnngni tudcs of fecd and overflowhnvo bccn chosen.A KA LYTIc A I, M E W D-PI ot t h e equ 1 bri u 111 curve on

an x versus y plot , ns in Figurc 2. Sincc the coridenscr usedis n simple onc , y I , t l ie conceritrntion of the vnpor from t h et op p ln te , i s cqusl to xp, the composi t ion of the product ,an d thc rc fore i s known. Accordingly , the va lue of z l l t h ecomposibion of t he l iquid on the top p la te , en n be read d i rec t lyf rom the equi l ibr ium curve , s ince it i s rcprcsented by the a bscissn of t h e p o i n t on t h e curve wliicll has t h e o r d i n n t c yl. T h ecoefficients A nnd B of Eqiint ion 3 nrc cnlculntcd from zp n n d0. If t hc equnt ion i swr i t t en ns). = A s , +Ran d i f t l ic known vnlueso f A , 13, nnd xi n re s u bs t i tu tcd thc re in , y2, t h ccomposition of the vapor

r is ing from plate 2 , canbe calculatet i . Since y2is dctcrrnined, z?ca n berend from thc cqui l ih- n+lr ium curvc by f indingt h e p o i n t t he reon wi than ordir intc y2, n d rend-ing i ts nbscissn. Th cva luc of z2 is subs t i tu tedi n t h e e q u a t io n

n. l

pintc it is ncccssnry torcnd on c va luc f rom thearlc7-C wequi l ibrium curve , nndc n r ry o u t t h e one cnlcriln-t ion b y Equnt ion 3 . Sup-pose this prorc.ss lins I ~ c ncrrrriccl o u t fo r n nrirnlxrof plntcs , wid .TI, V I , ZZ , 2 , 3 , 3 ,tc . , Iinvc bcmi so detcrni incd.Suppow t h e following ~ x i r i t snrc plottcd with t l ic roiirrfi-nntcq indicntcd: a , ro6rdinntcs rl , ~ 1 : , coi i rdinntca 11, yt ;c, (Q. yn); d , (12. ~ 3 ) : nt l e , (zs, yr). Tlic po in t s so p l o t t e dcnii tx: r-onncctccl by sho rt h orizontal nrirl vertical litiw, s u c h11soh 4k, d , nnd d e , sitire t l ic two p u i t i b coi inrctcd by aiiyline l invc ci t l icr nhacisws nr ordinates in co m m o n . Fur-thcrrnore, the wink G n n d d nrc on n s t m i g h t line, t h e cqua-tion of whic h is

.V - A x + /3


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J u n e , 1925 I ND U ST I AL A N D EXGI N E ER I A'C CH EM IS T RY 607

since thcir ordinntcs were cnlculntcd from Equation 3 , whichis rcprcsentcd by this ,same strni ght line. Th e foregoing con-struct ion can. thcrcforc, be mndc much more simply withoutnnalyticril cnlculntions nt ni l by drnwing the linc rcprescnt,ingICquntion 3 in thc first plncc, stnrting witti thc point a, anddrawing n wries of stcps composed of nltcrnntc vcrtical nndhor izcntnl segments , cnch s tep reprcscnting n pcrfcct plntc,nnd defining tlic compositions of tlic liquid on cnch plntc nndt h c v n p r rising ttr tlint plritc from tlic plntc hclow.Similnrly, the cnrirlirncrit from plntc to plntc in th c cxtinust-ing colurnri cnri he tlctcrminctf by plotting thc cxhniiating linct o conform with ICqriiitiori 10 n n t l moving u p or tiown i n tlicmrnc Htcpwise rrinniicr-nnmcly, h y tlrnw*irig nlt crii ntc hori-zontnl nnd vcrticnl scgmcrits bctwccn tho cqriilibrium curvenntl tlic cxtintistirig linc. Tlirifc, f thc coijrcliiiiiks of point fnrc tliosc of ha r c ?", 2/; nrid those rrf i nrc in ,w+. 1 7Gn - : , t h o w of point q will lx t n - l , in ;12 V a n N y r , i n t h e c o t i r w of his nia the mnt ic nl [ ir ewntnt ior i of th e the or y

of rectif icat ion. u ~ e i diuKrnrn w c h Y S P i x i i r e 2. C ' h r m . M r / . E m f . , U,793 ( 1 9 2 3 ) .

/ I

I / I

.

Th e diagonal linc, z = vi is drawn for convcnicncc, sincethe point a is cnsily locnted by drawing th e vcr t icnl l inc z =T,,, finding its intersection with thc diagonal (point j ) , an ddrawiiig R Iiorizoiitnl linc ja . The coiirdinntcs of point jm ay bc considered to be xp, I , ns zp = VI .

Plotting the Enrichment LinesIiispcctioii of Equntions 1 nn d S diows tha t the o r icn tat ionof tlic criric!inicnt linc s, i f plot ted on r cch np i ln r co i id inn tcs ,dcixlnds oiily on tlic rcflux, 0, for nnygivcn viiliics of F , ,, nn dThc following nlRcbrnic propcrtics of thc cnr ichm cnt l incsfncilitntc plottiiig thcm:If tlic sulmcripts nrc droppctl f r m IS qu ati on 1, wid i fz s sct cqrinl to y, 0 vnnishcs, nnd it f o l l o w t h a t '

y = .v = .vp ( 1 1 )sfiowiiig thot thc rcctifying linc c m w he dingonal ( x = ylinc) a t tlic composition of tlie pnxluct, wha tc vc r the rcfluxmny bc .

X w.


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(111 n plnte wlicrc the coinjmitioii (If the liquid is between cnntl b, if the column is to funct ion . Economy demands th a tth e feed plntc be chosen such thnt t l ic total numbe r of plntcsis a minimum fo r tlie reflus clioscn, which m ea ns th at tlic en-ricIiment per plate, nnd tliercforc tlic nveragc length of thesegmcnts , mu st be asgrcntns possible. I t can be seen t l int therectifying line slioultl bc used f is fa r as point C nnd the ex-hausting line utilized from there on . Tlicreforc, tlic feedshould be introduced on that plate on ~vh i ch he coinposi- .tion of the liquid is just less t l i nn t h ~ tf the feed. In Figure3 the fccd plate is th e fourtli on e from the top . The s tepsare then ciirricd bcloiv C, using the eshnusting line, until oneof the horizontal segments crosses the line x = xu, nd eachstep rcpresents n pcrfcct plate. In Figure 3 the to ta l numberof pintcs is O?/,.

Infinite or Minimum Over f l ow-4s pointed out by LewisIJ here are two limiting coiiditionsin the dcsign of a frnctionnting column, depending on theoverflow. On one hand there is a theoret ical minimumoverflow which will effect the required separation in a p c r -fect column containing a n infinite number of plates. T h i s

miniinurn overflow for an y given mixture depcnds upon thexpnrnt io i i t l c i r c t l . 011 tlic other h s n d , i f n n irifinitelylnrgc rcfliis is nssriirictl, th e scprn t ion wi l l be effected in nc~oliimn cont:tiiiing n rniniinum number of perfect plates.In the f iw t cnse tlic cost of coluinn, and the fixed chargeson it, will be infinite, h i t tlic cost of he at is n minimum sincen miiiimurn qriantity of vnpor i s genera ted a t th e bo t tom oft h e column. In t l ic sccontl cnsc th e number of plates in thecolumn is IL miriiniuin, but tlie cost of heat is infinite, as aninfinite nrnouiit of vnpor must be generntcd. Tlic dcsigncrshould choose 1111 o v c r f o w somcwhcre between these twolimits sucli t h n t t h e sun1 of the fixed costs nnd the heat costis 11 miniinrim. Tlic overflow so dctcrmined is known ns th e"optimum oi.crflon.," nricl tlic! number of pcrfcct plates cor-responding to the opti in~iin verflow is cnlled the "optimumnumber of jhi tw ," n n d must tie determined ljy tr ial.

Minimum Overf lowSuppose tlic coiiccntrations of waste, feed, niid product areplottcd on th e z versus y tiingrrim (Figure 4 ) . It ]ins beenshown thnt tlir cnriclimerit lines par^ through t lw points d

172 / F/G. .


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an d e , respccti\.ely, nriti intersect on the line .x =I/. Theirslope depends o n the overflow only, since F is dctcrmincd.From Figure 4 i t is nppnrer i t t h a t , if the lirics intcrscct ribovcthe equilibrium ciir\ 'e as at, a , the desired -scpnrntion cannot,be effected; if tlic lines intcrscct a t the intersectioii of the.equilibrium curve n n d the line .c = 21 , s nt point 6 , the ~ c p n -r a t i o n c i i n be a t -tnincd, h i t only withan infinite number ofperfect plates; antlfinally, if the lines in-ter sec t a t any po in tabove z = y line andbelow the equilibriumcurve, ns point c , thes e p a r a t i o n can beeffected with a finitenum ber of plate s. Ast h e o n l y m e an s ofmoving the intcrscc-tion from a to c is byincreasing the over-flow, by definition theminimum overflow ist h a t o v e r f l o w f o rwhich the rect ifyingline will intcrscc t thez =z line nt point b.,The mag nitiitic of th e rninirnurii overflow is found by mcnsiir-, ing the distance '4 nnd computing 0 f rom the equation

-wh ic h is obtained froiii l ~q u n t i o t i ..Vo!r--Inspection of F i g u r e 4 s h o w s t h : t t l o r n o r m a l m i x t u r e s ( e q u i -l ibr ium cr irvcs concnv c downw, ; i rd at ii!l points) t h c leryest, or c o n t r o l l i n x

vr t iu r ol ~ ' I C rniriirnrrrn r r f l ux IF t h n t corrcsrmnding to th e c o m p o s i t i o n of t h e. f r c r l

Infinite O v e r f l o wInspection of Equntioiis I nnd 8 shows t h a t ns 0 increasesindef ini tely, thc slopcs of the enrichment lines approachunity, and fo r nn infinite overf low, the lines coincide with the. z= y liric, nr i t i tlic rnininiiirn riuniher of p1nt.w is deter min ed.by stcps, n s i n Figurc 5 .

Constant Roiling MixtuicsSincc n cons tniit hoilirig riiixtii~.e ehav es like n jxirc coriipo-pe nt , an infinite r i ~ i n i l x r of jilntcs worild tic ricccssaryto rcncli thc conccntrntioli corrcsj)oiiding to sucli n mixtureby distillrrtiori Itlone. The dingrnrn of sucli ii systcrn i l lus-. t rn ks this , s i iicc i f nn a t t e m p t is rnndc to attRin the coriccii-t r n t i o n of tlic constant boiiin,o mixture, cithcr with tlrc prod-

uct o r t h e a n s t c , tlrc cq~i i l i l~r iu rnu rv e , the ciiriclirricnt l i n ta n t l the . = T liric all iiiterscct a t t,lic poin t wlicrc L equals tlic,above conccritrntiun, arid tlic s t eps must bccoinc infiiiitely.,smnll to rench tlic apex of the :iriglc i n which they nrc confined.Technic of Using the Plots

l , i ,wnlly th c product is to t)c riciirly piirc nntl the wntcr p rw -t.ically cxh:iiistccl, spcrtki::g ir i tcrrns of tlic inurc volntilccornpr icn t . 'Tliiic rncrtiis tlint :I single dingrim will be to ocorideriscd a t tlic crids of tlic cqrrilil~riuin urve to uw it hprccisiun, nriti .wp aritt c, cnInrKct1 tlirigrnrrix of these tw o ROC-tom rn w t be rnnde. Ortlirinrily, tlic crluilibrium c u r ~ e illfollow I b o u l t ' s law iit the ttrp nnd Henry's Inw n t the liottorn,13i H a , p . 5 7 0 .11 W n l k c r . L e w k , n nd A I c A i l n m ~ . I 'dnc ip le . ol Chemicn i Bnxinceriop,"h f c C r a w . t l i l l nook Cam pnny .

an d so can be cnlculntcd and plotted with any dcsircd preci-sion in these ranges.As nn illustrt~tionof the npplication of the incthod to anRctunl case, the cxnrnplc cited hy LcwisJ is worked out.An alcohol-water mixture contniiiing 10 per ccrit alcohol byweight (4.17 mol per cent) is to be scp:tratctl into a productcontaining 94.5 per cent alcohol by w i g h t (57.05 rnol percent.) imd n waste contnining not more thnii 0.1 per centa lcoho l by wigh t (0.039 mol jjcr cent). The overflow i nthe rectifying colrimn is assumed to IIC 5 rnols p r mol ofproduct .When tlie equilibrium curve for alcoliol-~vatcrmixtures isplotted, as i n Figure 6 , and the values of z p ,zjra n d zu replot ted thereon, i t is evident tha t the entire operat ion cannotbe ind ic ak d on a single diagram of moderate size. Accord-ingly, t,hrce enlarged diagrams are drawn, as ir i Figures 7,8, an d 9, nd used with Figure 6, Figure 7 being an enlargeme ntbetween the l imits z = 70 a n d z = 90 mol pe r cent , andFigures 8 an d 9 from z = 5 to z = 0 mol per cent. Theenr ichment l ines are then drawn, the rect ifying line psssingthrough the point P (coordinates L,,, y l ) (Figure 7) , and wi thn slope of 5 / ( 5 +1) = 0.833, and the exhausting line pass-ing through the point 111 (the intersection of the line z =and the rectifying line) (Figure 6), arid the point N (coijrdi-na tes zw ,z) (Figure 9). The stepwisc procedure is s tar teda t p o i nt P (Figure 7 ) and carried through Figures 7 , 6, 5, an d9 i n turn , until th e composit ion of th e wnstc is reached with,in this case, tlic thirty-fifth st ep , nnd so the entire columncontains 34l/3 perfect plates. The cross-over from therect ifying l ine to the exhausting l ine occurs at the thir t iethstep, and so the fced plate is the thir t icth plate f rom the top.Most of the plates are in the rsiige represented by Figure7 , an d a small error in the equilibrium curve in this rnnge in-troduces a rclativcly lnrge error in the number of plates;bu t an y method of cnlculation is wb jc ct to the sn mel i ~ n ia tion.Fo r t h e sake of more open diagrams, t l ic scale of thc z-axistins bcen cspnndcd to fivc times $lint of the y -asis, in Fig-ures 5 and 9. This does not affec t the corist,ruction or t h eresult , except that i t changes the angle of slope of the l ines;thus, the z = 1/ line is no longer at 45 deflees.F o r a n y b i n a r ycoin h i ria t i o n , theequ I ib r i u in c u r v eneed be drawn onlyonce, in ink pcrtiaps;ttic enrichment lines 1L o $artd the steps rnny bedrawn in lightly, withpencil, for nny pos-sible combinntion ofoverflow, nnd feed,product , ntid w a d ecumposi tioils.

If Ixwis's rcsultsf o r t h e f o r eg o i n gproblctn nrc rnulti-plied by 0.8 to cliini-n a b h is assumptionof SO jxr cent plirtc

/ I

cficicricy, the resultsifre corni"xblc, ns i n the table; Lewis's fced plate comes e ta cornpositiori t = 2.05 mo l per cent.Number of Theoretical Plates

Lewis A u t h o r ,A ~ V Ceed 31 O 29U ~ I D W ce d 3 . 3 6.3

TOTAL 9 4 . 3 35.9- -


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610 I N D U S I ' R I A L .4 ND ENGIIVEERIXG CIIE.VISTRY Vol. 17, No. 6Tlir : i c c u~~ (~yf t . 1 ~ csrilts obtaincd by th e graphicalmcthod is tlicorcticnlly limitrd only by tlic error int~rotluccdby nssuitiing coiistari: 1ii01:d overflow, lvhcrcns t'lic nccurncyoi t h c I.civis t i i c t l ioo ns npplictl to tlic foregoing esamplcis liniitcd: not, oii ly b \ . tlic n .wi rn p t i o n giT.en above, but nlso

since the totnl aniount of more volntilc cornporrcnt lost in t iicwnste is, i n tlic cnse of closed stcam, fixed 8s Wz,.I t can he sliowii nlgebrnically t h a t i n the CILSC of open fitcamthe cshausti i ig I i i x still pnsscs tlirougli the point of inter-scctioti of th c r = ?/ inc and th e l ine z = x w , rid also crossesth c Z - Rxi s a t th c point where 1: = rut. Therefore, it is onlyncccssnry to carry th c steps, not only to \vhcrc 1: = z b u tto thc 2-nsis to nllow for tlic cffcct of open steam.,Yo(r-C)p~n s t c n~ i i u s t be i i s cd w h e r e o closcd coil w o i i l d bc c onic fouled

I I ~ il ? i w s t e , h u t when u s e d , t h c nttmlxr of p l n t e s i s increased ( t h o u g hv ~ r y iixIiIIy in I I I O L ~ n w s ) nrtd tlic w a s t e is d i l i r t c d .Feed N o t a t Boi l ing Po:?'

If thc iccd is n o t wliolly a liqiiid nt its bc~ilingpoint, th cnunihor of tnols of vapor rising from t.hc fccrl plxtr, is not thesmie ii s tiint rk ing to i t . If the fcctl is h c l o ~ v he boilingpoint, thc \.:ipor. rising t o thc fccti plate wi!l hea t the fccd t otlic iioiiiiig pnirit by condcnsation, thcrcby increasing theovcrflow below th e feed R I ~ecreasing the vapor above thefeed by the number of mols condensed. If the feed is partlyor wholly vapor , t hc vnpor above the feed will bc increased bythc nmouiit of vapor admitted, while t h e overflow below thefccd will be lcss than 0 +F. In cithcr case the effect onthe diagram is to cause the locus of the intersection of t h etw o enrichm ent Iines to incline from th e vertica l.Le t q = the total heat needed to convert 1 mol of feedi n t o sntumtcd vapor, dividcd by the molal lntcrit heat.Then i n thc cnsc of n liquid at th c boiling point, q = 1.I n the cnsc of a feed of wholly satur ated vapor, q =0.If th c feed is partly l iquid and partly vapor , q is t he frac-tionnl part wliich is liquid.If tlic fccd is below thc boiling point, t?icn q is grentcrt l in i i 1 nnd is cqurtl to I plus th e fractional part of R mol ofvapor nccdcd to mise OIIC mol of feed to th c boiling point.I n plncc of 15jiintion G JV C hnvc

I lcnec F;qunt,ion 8 liccomc.9-U = O + q F (20)

To find thc locus of th eintcrscction (for differentvnlries of 0 ) of this cn-richincnt linc with t h a trcprescntcti t)y Equntion1, c l i r n i i i x t c 0 f r o mEqua t ions 1 an d 21 rindas given Iiy Erjriation 10.\.Vht!ii this is t lnnc thcrcrcs!il tsThis is the cqii:itiori of Rstrnigli t lint.. If z = y,this line n l w a y ~Crossest h i z = ?/ liiic :it r f . If

. iylicti!!itC_I _ f:>r pitr, -;n!;;c

'is- ? - I ) ? = .Y (22) Ht i 1 ~ 1 1 2 =TI . ~ r c l l c s c

RG. /= 1, z = L I . This isthc resr~ l t lrcatly givenin Eountinn 17.

horiaontnl line pw.-".inRt ~ l ~ O l l K ~ i= ! = , r f .If q > 1, the elope ispositive-i . e . , u p niitl tothe r ight.If 9 < I , tlic slopc i R i icgntivr-up to tlic Icft.


7/7

Junc , 19 % I N D U S T R I A L d i V D ENGINEERING CHEM ISTRY 611Tlicsc fonr ( : I~PCP nrc re~~rcscntc t in Figure 10, wlicrc tlicnunibcr of iiiols of vapor ri sing to fccd i h t c pcr mol ofprot l -~ c ts tlic sniiic fo r n i l CRSCS.

Variation o n h lo la l Cl!, crflowTlic inolnl overflow miiy not, be coi.stcyit in a column, fortlircc r e n s o ~ i s . Tlic first c:iwc niny bc a Jific. -*;cc in tlic inolnlheats of \~:iporiz:\tiori of t l ic two co~i ipnr*:ite. i l l lownnce

niiiy bc nintlc for this by coniputing nl :lunnti ies, not i nmols, h i t i n latel i t 1ic:tt un its ,J s lint is , tlic c inntitics ofliquid cvnporntctl by a n y dcfinite quantity . ,t . Thus, iitlic liquids liiive heats of v:iporizatioii ( I m grain) of 969 anti305 cdorics, me inxy a d o p t as o ur base 96 9 calorics and theIat.cnt heat units will be N9/969 :tnd 969/305 grains of t,lit!tw o iiqiiids! rcspcct,ivcly. T h e y will t h e n bc trcatctl csactlyRS if t he y lint1 molcculnr weights of 1 and 3.177, rcspectively.Siricc no clitinicnl rcaction tiikcs place, tlie oidy reason fo rusing rnols :it :ill is t l ic f ilct th a t t h y corrcspoiitl iienrly toth c rnorc :iccur:itc Intcii t he at units ; if this corrcspond-ericc docs no t lioltl thcrc is n o object in using t i icm. IfIntent Iicat uriits iirc c ho sc n, t h y must be used tlirough-out-in tlic con stru ctio n of the equilibr ium curve as wcll asin cspressiiig tlic coinpositions on cnch plnte an d of the fccd,produc t , aiiti waste.1 ic scconti ~ C R S O I Iwhy tlic ovcrflow iiiiiy Ynry is tlic chnngcii i bo i l i ng ~ x i i i t f tlic liquids on the successive plntcs, andthe third cause m a y be iicat l osscs from tl ic column. To sllowfo r tlicsc cffccts rigorously would require the constructionof II tliffcrcnt cnricliincnt linc fo r every pln tc , a n d so would beted ious . J i ow vc r , t l ic c f fcc ts a rc usurilly sinal1 nnd tend tobnlniicc c w l i other! so t i icy rnrc ly rcquirc consideration.

r 7

ip1iic:il i n c r l i o d s of P V I I C ~ O ~ i i i < l S:i\,nriti i i r v rigorously: ipp l ic i tb lv to c : i s e \ n 1 1 c : c t I i c o v c r f l o w \miirks, liiit Ihcsc inethodu u rc r:atlierconi p l i c ; i red ,

Two FeedsSupposc it i y tlcsircti to rcctify in tlic min e column two mix-

tures of tlic s:tnic t ivo l iquids b u t of different compositioiis.Lct tlic i n o k of the tw o feeds (per mol of produc t) IIC / I nntlK . L et zh : i i id r k be tlic coinpositions of thc rnisturcs.Similiirly, lc t C; arid .zg lie the ainouiit and composition of tliem ix tu re t h t n%~i ldx produced if tlic tw o fectis were miscd.Thc nI ; = I i 4- K

CS, = I i S h -!- K.Si ( 2 3 )(2.1)h t xh > s: , PO t ha t tlic II fCCd W i l l liIltUKLlly b C p u ti ll lIi!;hr,r tt:c column. Tlic ct)!riniii i s tii:igrnmmnticallyrcprcscritc:ti hy Figurc 11 . Through ina te r idbnlanccs it ( w i 1x2 s l io~vt i , y dcriving t l ic cnricli~nc nt qun-tioiis f o r tlic tlircc ixirts of tlic colii inn (iibovc both feeds, be-t w c n the fccdr, i r r i t l bclon. both Iccds), znd dcteririining tlicloci of tlicir iritcrscctions hy clirninntiiig tlic ovcrflow, t ha t thccnriclirncnt l i t m fo r tlic top iind bottoin sections of tiic coluinucross tlic z = 2/ line n t z l ,~ i n d ., rcspcctivcly, and intcrsccton the vcrticnl line z = s,; and th at t l ic cnrichmcrit liric forthc m itidlc prirt of t,lic colurnn (bctwccn thc two feed plates)m us t p:rss thro ugh tw o poirits--namcly, tlic iiitcrscctioii ofthc r ippcr cnric l i rncnt liric w ith th e vcrt icnl line z = X Ann d t h c iritcrscction of th c lowcr cnrichinciit line with tlicvertical liric z = z l . For n two-fccd coluniii, tlicii, tlicscrulcs follow:

I--To coristriict thc cririclirricrit lirics fo r n two-Iced coltiriiri,co m p i i t c tlic cornpositiori tlr:it wotrltl restilt i f t l ic feeds weremixed. Th c v r i rich rncr it lilies fo r the crids oi t h e colirrnri willbe cxact ly th c ;wmc as fo r n siriglc-fcctl coliirtiii that is fe d wit11th i s m ix tu re . T hc cn r i c h ~ n r r i t iric for tlic rniddlc of the w l u m i ii.s f o u n d by joiniiig thc points whcrc the cririchmcrit 1iric.q for t l icCrith intcrscct the lirrcs x = X I arid x = XI, whcrc XI nrld x ka r c t h e comix~sitir~iisf t h c tw o ic tds .

Tl ic originiil nssiirii ptioiis will IJ C scd.

?-If t l icrc a rc t w o fccds, cnch h m t c d t o h i l i i i g , each f ccd ist o bc a d m i t t c d or1 t h c h i g h es t p l a te t h a t h a s n co m p o s i t i o n lowcrth3r i that fced.Thesc rulcs nrc illustrntcd in Figure 12 . W h e n th ctwo feeds arc very different in composition tlic column mayfunction wcll irlicn tlie fccds &re pu t i n a t the projxr points,t hough a t t he same oycrflow it would not function if th c fccdswere mixed.If fo r any rcnson part of tlic liquid (not R third consti tucnt)is witlitlrnmn regularly from tlic colurnn, t he calculations areqirnilar to those fo r two reedy, except that either N or Kwill be nc gn ve .Mai nsi4 has cxtcndetl Lewiss metho d t o the de,:,.. columns fo r th e scparntion of mixtures such as furfuralmd wi te r where the product, on cooling, forms two layers~vliichcan be separated in a continuous decanter and one ofthem returncd to tlic column. Th is is a special case of a

two-fcctl column wlierc thc second fccd is one of tlic layersformed in thc tlccnntcr. If this sccond Iced is n o t n t t h eboiling point the cnrichmcnt Iiiic for that pzrt of th e columnbclow the entrnncc of th c tlccrintcd fccti must be niodificd?E c!cscrIhi ~ I n d r r h c scctinn Fcctl Xo t a t I%oili iigPoint.

AcknowledgmentThe nritliors wish to csprcss thcir nppreciation to W. I

mccabethiele - graphical design of fractionating columns

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