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A FUNDAMENTAL MODEL FOR PREDICTION OF SIEVE TRAY EFFICIENCY S. R. Syeda, A. Afacan, and K. T. Chuang Ã Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada.

Abstract: A phenomenological model for froth structure is proposed based on the analysis of

froth images of an active sieve tray taken from a 0.153 m distillation column. Froth is deﬁned as a com combin binati ation on of bub bubble bles s and continuo continuous us jet jets s tha thatt bre break ak the sur surfac face e of fro froth th pro projec jectin ting g liquid liq uid spl splash ashes es and dro drops ps abo above ve the sur surfac face. e. To est estima imate te the fra fracti ction on of sma smallll bub bubble bles s in froth, a fundamentally sound theoretical expression is derived from turbulent break-up theory. A new mod model el for pre predic dictin ting g poi point nt ef efﬁci ﬁcienc ency y of cro crossss-ﬂow ﬂow sie sieve ve tra trays ys has bee been n dev develo eloped ped based on the hydrodynamics of an operating sieve tray represented by the proposed froth structure model. This efﬁciency model is applicable for both froth and spray regime. Fraction of by-passed or uninterrupted gas jet is considered as the determining factor for froth to spray transition. The net efﬁciency is estimated by adding up the contributions of both bubbles and jets present in the dispersion. The model is tested against the efﬁciency data of cyclo-hexane/ cyclo-hexane / n-heptane and i-butane/ i-butane /n-butane mixtures. Keywords: distillation; tray efﬁciency; froth; turbulent break-up; bubble size

INTRODUCTION

Ã Correspondence to: Dr K.T. Chuang, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2G6. E-mail: [email protected] ualberta.ca DOI: 10.1205/cherd06111 0263–8762/07/ $30.00 0.00

þ

Chemical Engineering Research Researc h and Design

Trans IChemE, Part A, February 2007 # 2007 Institution

of Chemical Engineers

the na the natu ture re of tw twoo-ph phas ase e mi mixt xtur ure e in th the e trans tra nsit itio ion n zo zone ne an and d as ask k fo forr tw two o se sepa para rate te expr ex pres essi sion ons s of in inte terfa rfaci cial al ar area ea to pr pred edic ictt the th e tr tray ay ef efﬁc ﬁcie ienc ncy y in th thes ese e tw two o re regi gime mes. s. Zuid Zu ider erwe weg g (1 (198 982) 2) an and d St Stic ichl hlma mair ir (1 (197 978) 8) developed their tray efﬁciency models based on this approach. The FRI efﬁciency data of commercial sieve trays, on the other hand, show smooth transition of tray efﬁciency from the weeping to ﬂoo ﬂo odi din ng poi oint nt.. Th This is co comp mpe ell lle ed ma many ny resear res earche chers rs to res resort ort to a sin single gle ef efﬁci ﬁcienc ency y mode mo dell fo forr bo both th fr frot oth h an and d sp spra ray y re regi gime mes. s. Mostt of the exi Mos existi sting ng tra tray y ef efﬁci ﬁcienc ency y mod models els (AIChE (AI ChE,, 195 1958; 8; Cha Chan n and Fai Fair, r, 198 1984; 4; Che Chen n and Chauang, 1993) are of this type. None of the abovementioned models took into account the struc structure ture of the two-phase two-phase mixture that is generated on the tray in different regimes. The only major attempt that considers sid ers the dis disper persio sion n str struct ucture ure in the fro froth th regime was made by Prado and Fair (1990) for the air /water system. They treated the dispersion as three regions: a region near the tray where the gas can either be jetting or bubbling, a bulk froth region which contains bubbles bubbl es with bimo bimodal dal distr distributi ibution on dispe dispersed rsed in the liquid and a spray region at the top. Howeve How everr, the they y ign ignore ored d the spr spray ay reg region ion in their the ir det detail ailed ed mas mass s tra transf nsfer er mod model. el. Lat Later er,, Garc Ga rcia ia an and d Fa Fair ir (2 (200 000a 0a,, b) ex exte tend nded ed th this is model mod el to oth other er sys system tems. s. The Their ir mod model el was show sh own n to ag agre ree e fa favo vour urab ably ly wi with th a wi wide de range of data. However, However, sever several al adjus adjustable table

The sim simult ultane aneous ous mas mass s and hea heatt tra transf nsfer er combin com bined ed wit with h the com compli plicat cated ed two two-ph -phase ase ﬂuid dynamics make distillation formidable to conduct any fundamental analysis of distillation. Furthermore, distillation became a wellestablishe estab lished d indus industry try long before the theor theory y of tra transp nsport ort phe phenom nomena ena was est establ ablish ished. ed. Thus Th us,, th the e co comm mmon on tr tren end d of di dist stililla lati tion on research resea rch to date mostly remai remains ns empir empirical, ical, semi se mi em empi piri rica call or me mech chan anis isti tic c in na natu ture re.. Mass Ma ss tr tran ansfe sferr ef efﬁc ﬁcie ienc ncy y in di dist stililla lati tion on is asso as soci ciat ated ed wi with th th the e ﬂu ﬂuid id dy dyna nami mics cs on a sieve sie ve tra tray y tha thatt det determ ermine ines s the dis disper persio sion n struct str ucture ure or the con contac tactt are area a bet betwee ween n the gas and liquid phases. The ﬂow regimes on a sieve tray inﬂuence the efﬁciency directly by affe affecting cting the inter interfacia faciall area. Numerous Numerous studies on ﬂow regimes have been done to understan unde rstand d the hydro hydrodyna dynamic mic beha behaviour viour of sieve trays. Most of these studies are mainly focussed on the transition from froth to spray regime reg ime.. The deﬁnitio deﬁnition n of fro froth th its itself elf is sti stillll very vague in the literature. In froth regime, the th e pr pres esen ence ce of pu puls lsat atin ing g je jets ts ra rang nges es of bubble bub bles, s, liq liquid uid spl splash ashes es and dro drople plets ts giv give e rise to a highly complex dispersion structure. The tra tradit dition ionall ally y per percei ceived ved pic pictur ture e of the froth regime consists of bubbles in a liquid cont co ntin inuo uous us ph phas ase e an and d th that at of th the e sp spra ray y regime consists of droplets in a gas continuous pha phase. se. The These se deﬁ deﬁnit nition ions s of fro froth th and spray spr ay reg regime ime sug sugges gestt a sud sudden den change change in 269

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parameters needed to be introduced at different stages of the model to emphasize its mechanistic nature. Bennett et al. (1997) (19 97),, dev develo eloped ped a poi point nt ef efﬁci ﬁcienc ency y mod model el bas based ed on the mechanistic analysis of sieve tray froth height. The model considers the ﬂuid on the tray to be contained in a liquid-continuous region near the tray deck and a vapour-continuous region on top of the liquid-continuous region. The ﬁnial simpliﬁed model takes into account the mass transfer of the liquid continuous region only and thus has limited applicability in spray regime. A recent study of van Sinderen et al. (2003), thatt dea tha deals ls wit with h ent entrai rainme nment nt and max maximu imum m vap vapour our loa load d of tray tr ays s pr pres esen ente ted d a tw two o or th thre ree e la laye yerr mo mode dell of th the e tw twoophase mixture on the tray. This study, although provides a detailed detai led insight into the dynamics of the froth, partic particular ularly ly the mechanisms of entrainment formation but was unrelated to mass transfer efﬁciency. From the above discussions it is evident that most existing correlations for point efﬁciency are highly empirical and do not deal with froth dynamics on a sieve tray. The very few studies, which consider the nature of the dispersion structure in their models generally, ignore the contribution of drops and sprays. These models agree with a wide range of data when using adjustable parameters, but are less applicable in spray regime on theoretical ground. In this study, the froth regime is modelled based on the analysis of froth images taken from a 0.153 m diameter distillation column. The model describes the froth as a combination of bubbles and continuous jets. At higher gas load, the jet jettin ting g fra fractio ction n dom domina inates tes and gives rise to the spray regime reg ime.. Thi This s fro froth th mod model el is fur furthe therr ado adopte pted d to dev develo elop p a fundamental model for predicting sieve tray efﬁciency. The efﬁciency model takes into account the contribution of both bubbles and jets to the net mass transfer.

MODEL STRUCTURE Froth images taken in a 0.153 m diameter distillation column are shown in Figures 1 and 2. Based on a careful study of these kinds of froth images, a froth structure has been schematically presented in Figure 3, where froth is shown as a combi com bina natio tion n of je jets, ts, bu bubb bble les s an and d liliqu quid id sp spla lash shes. es. Th The e images (Figures 1 and 2) show that the liquid droplets and

Figure 1. Froth image of pure methanol on a sieve tray in a 0.153 m

distillation column.

Figure 2. Froth image of 67 wt% methanol/water mixture on a sieve

tray in a 0.153 m distillation column.

Figure 3. Schematic representation of froth on an operating sieve

tray.

splashes constitute a major part of the froth. A portion of the droplets is formed when bubbles break out of the surface of the froth. However, the presence of liquid splashes conﬁrms that some of the gas jets manage to penetrate through the froth without forming bubbles and generates liquid splashes at the end of liliqu quid id con contin tinuo uous us zon zone. e. Fig Figur ure e 4 is a mo more re detailed representation of the froth model, showing both jetting and bubbling zones. The jetting zone elaborates how some of the gas jets formed at the sieve tray holes, cross the froth uninterrupted and throw liquid splashes above by tearing up the liquid surface. The bubbling zone shows the process of large and small bubble formation in the froth. Both zones are present and remain intimately mixed with each other in real froth fro th.. No je jetti tting ng is ach achiev ieved ed at a rel relati ative vely ly lo low w liliqu quid id ﬂow rate.. This regime is call rate called ed bub bubblin bling g reg regime, ime, which occ occurs urs close to the weeping limit and is of limited signiﬁcance signiﬁcance for commercial sieve tray operation. As the gas load is increased, an increasingly increa singly greater proportion proportion of gas passes the disper dispersion sion in the form of jets. The spray regime occurs when most of the gas jets formed at the oriﬁce, reach the liquid surface uninterrupted and project the liquid up to form small drops. Unlike in froth regime, where bubbles bubbles form a major part of the interfa interfacial cial area, in spray regime drops are the only contributor to the interfacial interfa cial area. The point efﬁc efﬁciency iency is estimate estimated d by combini combining ng

Trans IChemE, Part A, Chemical Engineering Research and Design , 2007, 85(A2): 269–277

FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y

271 27 1

MODEL DEVELOPMENT In the following sections a method to estimate point efﬁciency E OG OG from equations (1) and (2) has been discussed.

Bubbling Zone Bubbli Bub bling ng zon zone e is con conside sidered red to hav have e bimo bimodal dal size distribution of bubbles as reported in many studies (Porter et al., 1967; 196 7; Ash Ashley ley and Hase Haselden lden,, 197 1972; 2; Loc Lockett kett et al al.., 19 1979 79;; Kaltenbacher, 1982; Hofer, 1983; Klug and Vogelpohl, 1983). The small bubbles are the secondary bubbles formed by the turbulent break-up of the primary bubbles originated from the oriﬁce. The large bubbles are the unbroken primary bubbles that remain in the froth due to incomplete break-up. The speci speciﬁc ﬁc inter interfacia faciall area, aiG and residence time, t GLB GLB for the lar large ge bub bubble bles s in fro froth th can be est estima imated ted from the following equations, respectively:

Figure 4. Froth structure model on an operating sieve tray.

aiG

the con contrib tributio utions ns from both bubb bubbling ling and jett jetting ing zon zones es that exist on a tray. E OG OG

¼ (1 À f i )E B þ f i E j

(1)

where f j is the volume fraction of the gas that bypasses the bubbles as continuous jets, E B and E j are contributions of bubbling and jetting zone, respectively, to the net point efﬁciency. Due to incomplete break up of the large (primary) bubbles both large (primary) and small (secondary) bubbles coexist in bubbling zone. Thus E B has contributions from both large and small bubble bubbles, s,

¼ d 6

t GLB GLB

(7)

32L 32L

f ¼ U hLB LB

(8)

Due to complex nature of the process, there are few analytical expression expre ssions s for any design in distil distillatio lation n liter literature ature.. The generall tre era trend nd is to use cor correl relati ations ons,, whi which ch are supporte supported d by reliab rel iable le exp experi erimen mental tal dat data. a. The fol follow lowing ing equ equati ations ons are used to estimate the Sauter mean diameter and raise velocity of the large bubbles formed at the oriﬁce.

¼ 0:887D0H 846u 0H 21 1 6 U LB þ u a LB ¼ 2:5(V LB LB ) :

d 32L 32L

:

=

E B

¼ (1 À FSB)E LB LB þ FSB Ã E SB SB

(2)

where FSB is the fraction of small bubbles.

THEORY OF MASS TRANSFER Following Follow ing exp expres ressio sions ns can be obt obtain ained ed fro from m two two-ﬁl -ﬁlm m theory, N G

¼ aiGk Gt G N L ¼ aiL k L t L

(3) (4)

where aiL t L

¼

r L Gf r G Lf

aiG t G

(5)

Here aiG and aiL represent the geometrical interfacial area per unit volume of gas and liquid phases, respectively. Assuming that the liquid composition does not change vertically and vapo va pour ur pa pass sses es as pl plug ug ﬂo ﬂow w wi with thou outt mi mixi xing ng,, th the e ov over eral alll mass transfer unit can be related to point efﬁciency as follows: E OG OG

¼ 1 À exp( À N OG OG )

(6)

In the present study, E OG OG is obtained from the published Murphree efﬁciency, E mv , mv data as outlined by Garcia and Fair (2000a).

(9) (10)

Where DH and u H are the hole diameter and velocity; V LB LB is the large bubble volume and u a is the gas velocity based on the tray active (bubbling) area. Equation (9) based on the bub bubble ble siz size e dat data a mea measur sured ed by ele electr ctroni onic c pro probes bes jus justt above the sieve tray (Prado et al., 1987). Thus the equation estimates the unbroken primary bubbles in froth. Three different liquid systems with nine different tray geometries were used to generate the bubble size data. This is by far the only correlation for primary bubbles on a sieve tray. Equation (10) was originally developed for estimating rise velocity of bubble swarms through a porous bed (Nicklin, 1962). Later Burgess and Calderbank (1975) showed that this equation adequately predicts rise velocity of large bubbles in froth on sieve trays. This is again the only study done on this topic. The mass transfer coefﬁcient for the liquid phase, k LLB LLB, is modelled with Higbie penetration theory (Higbie, 1935), k LLB LLB

¼ 1:13

DL t GLB GLB

0:5

(11)

This is a well-established model used previously by numerous stu studie dies. s. The mas mass s tra transf nsfer er coe coefﬁ fﬁcie cient nt for gas pha phase, se, k GLB GLB, of the large bubbles is estimated from the numerical soluti sol ution on pre presen sented ted by Zar Zaritz itzky ky and Cal Calvel velo o (Za (Zarit ritsky sky and Calveio, 1979). This solution was developed for mass transport models in distillation. It was tested against experimental data and was applied in efﬁciency models such as those by Prado Pra do and Fair (19 (1990) 90) and Gar Garcia cia and Fai Fairr (20 (2000b 00b). ). The solution is presented as a plot of Peclet number ( PeG) of

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SYEDA et al.

the gas pha phase se ver versus sus the asy asympt mptoti otic c She Sherwo rwood od num number ber (Sh ). Within the range 40 , PeG , 200 the following polynomiall provi nomia provides des an excel excellent lent ﬁt for the exper experimen imental tal data: /

Sh

/

¼ À11:878 þ 25:879 879(( log PeG ) À 5:64(log PeG)2

(12)

For the ran range ge PeG . 20 200, 0, It wa was s fo foun unff th that at Sh ha had d an essentially constant value of 17.9. Froth height, hf , is estimated from Bennett et al.’s (1983) correlation for effective froth height, 1

hf

¼ hw þ C W QL

ae

¼ exp

12:55 u s

r L

r G

À

r G

(14)

0:91

and C

¼ 0:5 þ 0:438exp( À 137:8 hw)

(15)

There are a number of correlations available in literature to estimate the froth height on a sieve tray. The unique characteristic of equation (13), proposed by Bennett et al. (1983), is that unlike any other correlations it gives effective froth height i.e., the height of the liquid continuous region. Since in the present model, froth height is used to estimate the residence time of bubbles in froth, the height of liquid continuous region calcul cal culate ated d by equ equati ation on (13 (13)) giv gives es the app approp ropria riate te val value. ue. Other correlations, which give total froth height i.e., the combined height of liquid and vapour continuous region, would over estimate the residence time of bubbles. Using the above information, N GLB GLB and N LLB LLB can be calculated from equations (3) and (4). Equation (4) is then used to get the overall mass transfer unit, N OGLB OGLB, from which the contribution of the large bubbles, E LB LB, to the net efﬁciency is obtained by using the equation (6). The portion of small bubbles in froth is considered to reach equili equ ilibri brium um whe when n mas mass s tra transf nsfer er rat rate e is hig high h (Lo (Locke ckett tt and Plaka, 1983). Kaltenbacher (1982) also suggested that the small bubbles get trapped in the froth and leave the froth practically saturated. In this case, because equilibrium prevails vai ls bet betwee ween n the vapour vapour and the liq liquid uid phase of sma smallll bubbles, the efﬁciency of small bubbles becomes unity, i.e., E SB SB

¼1

Here k is the breakage rate constant and N is the number of large bubbles. Two additional assumptions are made to keep the calculation simple.

Let us consider that the number of large bubbles entering the froth at t ¼ 0 is N i. Due to turbulent break-up, N i reduces to N f f at t ¼ Dt . Here Dt is the residence time of large bubbles in the ﬂow ﬁeld. Therefore, by integrating equation (17) from N i at t ¼ 0 to N f f at t ¼ Dt , the following expression is obtained:

where 0:5

(17)

¼ ÀkN

(13)

ae

! 35

dN dt

(1) All large bubbles are bigger than the maximum stable bubble size and are equally susceptible to the break-up process. (2) The number number of large and small bubbles bubbles at any parti particular cular cross section of the froth is constant.

0:67

2 4À

used thi used this s con concep ceptt for bub bubble ble bre breakak-up up in pip pipeli elines nes.. The same sam e app approa roach ch is app applie lied d her here e for sie sieve ve tra tray y ana analys lysis, is, where a ﬁrst order bubble breakage rate is assumed. The breakage rate of large bubbles in froth is given by

(16)

In order to estimate the contribution of small bubbles to the total efﬁciency, we need to determine the fraction of small bubbles, FSB, in froth. Due to lac lack k of exp experi erimen mental tal data and reliable method to estimate this parameter, expression for FSB has been derived from turbulent break-up theory of bubbles. In any ﬂow ﬁeld, the FSB is governed by the bubble breakage rate and the bubble residence time in turbulent zone. Previous theoretical studies (Valentas et al., 1966; Valentas and Amu Amunds ndson, on, 196 1966) 6) dea dealin ling g wit with h dro drop p siz size e dis distri tribut bution ion assumed that the breakage rate of a drop is of ﬁrst order with respect to the number of drops. Later Hesketh et al . (1991)

N f f

¼ N i eÀ

k Dt

(18)

Let us consider that the fractions of large and small bubbles at t Dt represent the average fraction of large and small bubbles in the froth. The number of unbroken large bubbles at t Dt is given by

¼ ¼

N f f

¼ N i eÀ

k Dt

(19)

For binary breakage, N s

¼ 2(N i À N f f )

(20)

where N s is the number of small bubbles formed at t Dt . Thus the volume fraction of small bubbles in froth can be estimated as follows:

¼

FSB

2(N i À N f f ) s ¼ N V N sþV N ¼ V 2(N À N ) þ N ðV =V Þ s

s

f f L

i

f f

f f

L

(21)

s

Here V S and V L are the volumes of small and large bubbles, respectively. Assuming bubbles have spherical shapes, we get the follo following wing expression expression for FSB from equa equations tions (19) and (21): FSB

¼

2(1

À eÀ

k Dt

À eÀ ) 3 À ) þ (d 32L 32L =d 32S 32S ) e

2(1 k Dt

k Dt

(22)

The ratio of large bubble diameter to small bubble diameter, d 32L obta tain ined ed fr from om th the e ex exis isti ting ng lilite tera ratu ture re.. Th The e 32L/d 32S 32S, is ob reported diameter ratios are summarized in Table 1. From the above table, we ﬁnd that the most probable value of the ratio d 32L 32L/d 32S 32S is 5. The breakage rate constant k is a function of the turbulent ﬂow ﬁeld and the ﬂuid physical properties. Hesketh et al. (1991) showed that the measured deformati ma tion on ti time mes s an and d br brea eaka kage ge ti time me of bu bubb bble les s ca can n be characterized by the natural mode of oscillation of a sphere given by Lamb (1932) and proposed the following functionality of the rate constant k ,

Trans IChemE, Part A, Chemical Engineering Research and Design , 2007, 85(A2): 269–277

FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y tray. Table 1. Reported bubble size distribution on an operating sieve tray. Source

Small bubble

Large bubble

Hofer (1983) Ashley and Haselden (1972) Kaltenbacher (1982) Porter et al. (1967) Lockett et al. (1979)

5 mm 5 – 1 0 mm

25 mm 40 – 80 mm

5 8

4 mm 5 mm 5 mm

25 mm 20 mm 25 mm

6 4 5

k

3:8

¼

0:1 0:3

0:6

r L r G v

We0cr :9

Ratio

(23)

s 0:4

Here, v is the rate of energy dissipation in unit mass; Wecr is the critical Weber number given as Wecr

¼

r u u 2 d max max

(24)

¼

k

3:8

¼

0:1 0:3

r L r G

We0cr :9

s 0:4

(u s g )0:6

(25)

Dt can

be expressed as (26)

¼ nt GLB GLB

here, n is any value between 0 and 1. Since both n and Wecr are unknowns, we can combine them into single constant: C 00

¼ 0:13 À 0:065 2 G G À 5 2:6 Â 10 k Lj Lj ¼ k Gj Gj

r

r

(1 , r G

,

80kgmÀ3 )

(29) (30)

0:25

m L

E j

À

¼ 1 À exp

ahf

40

¼ F 0 3 :

ahf K OGj OGj u s

2 F bba hL FP s

(31)

0:37

(32)

where, F bba bba is vapour rate based on active area, F is the ratio of hole area to active area, hL is the clear liquid height and expressed as hL

¼ 0:6 hW

p FP b

0:25

(33)

and at total reﬂux FP

0:5

¼ r G

(34)

r L

The breakage time Dt

small tray by using free trajectory model. However, Raper et al. (1979) showed that Fane et al.’s model under-predicts the tra tray y ef efﬁci ﬁcienc ency y whe when n app applie lied d for ind indust ustria riall siz size e tra tray y. Another important attempt to predict mass transfer efﬁciency in spray regime was made by Zuiderweg (1982). His semiempiri emp irical cal mod model el is bas based ed on the FRI exp experi erimen mental tal dat data. a. This is the only model so far that is not case sensitive and is readily applicable for spray regime. In this study, we have chos ch osen en Zu Zuid ider erwe weg’ g’s s sp spra ray y re regi gime me mo mode dell [e [equ quat atio ions ns (29–34)] to estimate the contribution of jetting zone to the total mass transfer efﬁciency in froth regime;

s

where u 2 is the mean square square velocity of turbulent turbulent ﬂow ﬁeld and d max maximum mum sta stable ble bub bubble ble size aga against inst tur turbule bulent nt max is the maxi break-up; r and s are the density and surface tension of the liquid phase, respectively. The values of reported Wecr range over an order of magnitude depending on the ﬂow pattern responsible for the deformation of the bubble. In distillation, there is no reporte reported d value for Wecr . The rate of energy dissipation, however however,, is appro approximately ximately estimate estimated d by v u s g (Kawase and Moo-Young, 1990); thus the rate constant becomes

273 27 3

¼ Wen0 9

(27)

:

cr

The experimental data obtained by Raper et al. (1982) are used to evaluate the volume fraction of gas that bypasses the bubbles formation and forms jets, f j and to estimate the net con contri tribut bution ion of jet jettin ting g zon zone. e. Fol Follow lowing ing equ equati ation on is an excellent ﬁt for the average value of jetting fraction, f j as a function of F -factor, -factor, F bba bba. f j

¼ À0:1786 þ 0:9857(1 À eÀ1 43 :

F bba bba

)

(35)

By multiplying equations (25) and (26) we get 3:8r 0L:1 r 0G:3 k Dt ¼ C 00 (u s g )0:6 0:4

s

(28)

The constant C 00 will be estimated by comparing the model with the measured efﬁciency data.

Jetting Zone In fro froth th reg regime ime,, it is dif difﬁcu ﬁcult lt to inv invest estiga igate te jet jettin ting g zon zone e separately as jets are intimately mixed with bubbles. No information is available in literature on the size of jets or droplets present prese nt in froth. In this study study,, we will treat the jetting zone as spray and use the correlations of spray regime to estimate the contribution of jets in froth regime. Although numerous studies have been done to determine the onset of spray, very few studies have been focussed exclusively on mass transfer efﬁciency in this regime. Fane et al. (1977) achieved some success in predicting efﬁciency in spray regime on a

DETERMINING CONSTANT C 00 Constant C 00 is determined by comparing the model with seven sets of FRI data (Sakata and Yanagi, 1979; Yanagi and Sakata, 1982). These data sets cover two hydrocarbon systems, system s, cyclo cyclo-hexa -hexane ne//n-hep n-heptane tane and i-but i-butane ane//n-butane, at ﬁve different pressures in two different tray geometries. The cyclo-hexane/ cyclo-hexane/n-hep n-heptane tane system is widel widely y used for testing distillation tray performance. The properties of this system are representative of many hydrocarbon systems operated at 400 kPa pressure or below. The data sets for this system are taken at two different pressures, 34 kPa and 165 kPa. The data sets for i-butane/ i-butane/n-butane cover three different pressure leve le vels ls.. Th The e me meas asur ured ed ef efﬁc ﬁcie ienc ncie ies s at hi high gh pr pres essu sure res s (2068 kPa and 2758 kPa) have been corrected for vapou vapour r entrainmen entra inmentt with the down ﬂow liqui liquid d (Hock and Zuide Zuiderweg rweg,, 1982). Figure 5 presents the effect of different values of constant C 00 on the estimated point efﬁciency for the seven sets of FRI data. The average absolute error was calculated by

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SYEDA et al.

efﬁciency; ciency; expressed expressed as Figure Figur e 5. Effect of constant C 00 on point efﬁ average absolute error error..

Figure 7. Comparison of measured and predicted point efﬁciencies

for the cyclo-hexane/n-heptane system at 165 kPa (open hole-area 14%).

the following equation, Error%

Estimated

P ¼ j

À Experimentalj=Experimental nnumber of data

(36)

The minimum error was obtained at C 00 ¼ 0.16. The reported theoretical values of Wecr range from 1 to 4.7 (Hinze, 1955; Lewis Lew is and Dav Davids idson, on, 198 1982). 2). Wit Within hin thi this s ran range, ge, Dt varies 00 from 0.16 t GLB GLB to 0.644 t GLB GLB at C ¼ 0.16. The values are reasonable for obtaining the average bubble size distribution within withi n the froth froth..

PREDICTION OF POINT EFFICIENCY The present model introduces a new method to estimate sieve tray efﬁciency based on a froth structure that describes the hydrodynamics of an operating sieve tray. The predicted point efﬁc efﬁciencie iencies, s, E OG from m the propos proposed ed mo mode dell ar are e co commOG, fro pared with the FRI data in Figures 6–12. In all cases, predictions from two earlier models of Chen and Chuang (1993) and Garcia and Fair (2000b) are also compared with the proposed model. The unique characteristics of the new model is that unlike the two other models it predicts the trend of efﬁciency change from weeping to ﬂooding point more closely (Figures 7–9). The steady decrease in both fraction of small bubbles and bypassed jets results in gradual decrease of the point efﬁciency, E OG -factor approach approaches es the weeping weeping poin point. t. The OG as the F -factor

Figure 8. Comparison of measured and predicted point efﬁciencies for

the iso-butane/n-butane system at 1138 kPa (open hole-area 14%).

model also predicts the smooth transition of E OG OG from froth to spray regime. Under high operating pressures (Figures 10 and 11), the breakage rate constant k is high enough to cause breakag bre akage e of all large bub bubble bles. s. This makes the frac fractio tion n of small bubbles FSB unity and gives rise to high point efﬁciency under such operating condition. The experimentally measured fraction of bypassed gas is 0.8 at F -factor -factor 2 (Fane et al. 1977). Beyond this point froth is dominated by spray and the model reduces red uces to Zuid Zuiderwe erweg’s g’s model for spr spray ay reg regime ime.. Thu Thus s any

Figure 6. Comparison of measured and predicted point efﬁciencies

Figure 9. Comparison of measured and predicted point efﬁciencies

for the cyclo-hexane/n-heptane system at 34 kPa (open hole-area 14%).

for the iso iso-but -butane ane/n-bu n-butane tane syst system em at 1138 kPa (ope (open n hole hole-are -area a 8.3%).

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FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y

Figure 10. Comparison of measured and predicted point efﬁciencies

for the isoiso-buta butane ne/n-bu n-butane tane sys system tem at 2068 kPa (ope (open n hole hole-are -area a 8.3%).

error in predic predicting ting E OG -factor 2 is inh inherit erited ed from OG beyond F -factor Zuiderweg’s model. The prediction of Chen and Chuang (1993) model is satisfacto fa ctory ry fo forr al alll six sets of da data. ta. Th The e in inter terfa facia ciall are area a in th this is model is estimate estimated d from the bubble size distrib distribution. ution. However, However, since the vapour /liquid dispersion dispersion in spray regime mostly consists of drops, the model is applicable only to froth regime. The Gar Garcia cia and Fair (20 (2000b 00b)) mod model el pre predic dicts ts the low low-pressure tray efﬁciency data adequately. However, it predicts signiﬁcantly lower tray efﬁciency than the measured values at high hig h pre pressu ssures res.. Thi This s dis discre crepan pancy cy res result ults s fro from m the hig highly hly empi em piri rica call na natu ture re of th the e mo mode del. l. Th The e mo mode dell in invo volv lves es a number of equations and at least four adjustable parameters thatt mat tha match ch abo about ut 22 set sets s of tra tray y ef efﬁci ﬁcienc ency y dat data, a, mos mostly tly measured under low or moderate pressures. The under prediction of three sets of data out of 22 sets, did not affect the ﬁnal ﬁn al form of th the e mo mode del. l. Th Thus us,, th the e mo mode dell is fo foun und d to be suitable at low and moderate pressures only. Figure 13 compares the overall performance of the three models. The proposed model predicts within +10% for all the sys system tems s and shows better better per perfor forman mance ce tha than n the two other oth er mod models els.. The agr agreem eement ent bet betwee ween n the exp experi erimen mental tal data and predictions of the new model proves the validity of the proposed approach.

275 27 5

Figure 12. Comparison of measured and predicted point efﬁciencies

for cyclo-h cyclo-hexane exane/n-he n-heptan ptane e syst system em at 165 kPa (ope (open n hole hole-are -area a 8.3%).

Tray hydrodynamics is considered to be the key factor in determining the nature of two-phase mixture in distillation.

The un The uniq ique ue fe feat atur ure e of th the e pr prop opos osed ed mo mode dell is th that at it is based on the analysis of tray hydrodynamics (Figures 1–4) that describes the real situation on a sieve tray. The model includes both bubble and jet contribution to the total point efﬁciency. The often reported bimodal distribution of bubbles in froth is explained as the result of incomplete break-up of primary mar y bub bubble bles s in tur turbul bulent ent ﬂow ﬁel ﬁeld. d. The fraction fraction of sma smallll bubbles, FSB, is directly estimated by theoretical analysis of the rate of bubble breakage in froth. The only other similar effort to estimate FSB was done by Garcia and Fair (2000b). Although their ﬁnal model agreed with the database favourably ab ly,, th the e st stud udy y fa faililed ed to id iden enti tify fy th the e so sour urce ce of bi bimo moda dall bubble size distribution observed in froth, which made their semi-theoretically obtained FSB expression rather arbitrary. The present model has been developed to incorporate both the froth and spray regimes. The fraction of gas that forms continuous jets, f j, is the determining factor of the contribution from fro m eac each h of the regimes regimes.. For example, example, in fro froth th reg regime ime,, 0 , f j , 1. As f j increases with higher a gas load, transition to spray regime occurs gradually and f j becomes unity as spray regime is reached. No sudden change in dispersion stru st ruct ctur ure e oc occu curs rs du duri ring ng th this is tr tran ansi siti tion on,, an and d th ther ere e is a smooth transition of FRI efﬁciency data from froth to spray regime. Thus the effect of the present approach of considering in g th the e ef effe fect ct of th the e ﬂo ﬂow w re regi gime mes s on th the e tra tray y ef efﬁc ﬁcie ienc ncy y adopte ado pted d in the pro propos posed ed mod model el dif differ fers s fro from m tha thatt res result ulting ing from the two previous approaches of the existing models. One On e of th the e ap appr proa oach ches es is to ap appl ply y th the e sa same me ef efﬁc ﬁcie ienc ncy y model for both froth and spray regimes without considering

Comparis parison on of mea measured sured and pred predicted icted point ef efﬁcien ﬁciencies cies for Figure 11. Com

Figure 13. Overall comparison of the proposed model with two other

the iso-butane/n-bu n-butane tane system at 2758 kPa (open hole-area hole-area 8.3% 8.3%). ).

existing models.

DISCUSSION

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the effect of change of the dispersion structure (AICHE, 1958; Chan and Fair, 1984; Chen and Chuang, 1993) The other approach is to use two completely different models for froth and spray regime (Zuiderweg, 1982). Since the dispersion struc str uctu ture re in fro froth th re regi gime me is ju just st in inve vers rse e to th that at of sp spra ray y regime, applying the same efﬁciency model for both regimes without considering the change in the dispersion structure is the incorrect way to estimate the tray efﬁciency. On the other hand ha nd,, wh when en tw two o se sepa para rate te mo mode dels ls ar are e us used ed fo forr th the e tw two o regimes regim es dif difﬁcult ﬁculties ies arise in iden identifyin tifying g the exact transition point. By including the fraction of jetting, dependent on gas ﬂow rate, the new model takes into account the difference in dis disper persio sion n str struct ucture ure bet betwee ween n the coe coexis xistin ting g fro froth th and spray regimes. Thus the model provides a logical solution thatt can be app tha applie lied d con contin tinuou uously sly over the range of ﬂow rates, without resorting to an arbitrary selection of the use of th the e sa same me or se sepa para rate te mo mode dels ls fo forr bo both th th the e fo fort rth h an and d spray spr ay reg regime imes, s, and the thereb reby y ful fully ly des descri cribes bes the smo smooth oth transition between the regimes. The inclusion of physical properties considered in the estimation mat ion of fra fracti ction on of sma smallll bub bubble bles s FSB [equ [equation ation (28)] makes the model applicable to systems with wide range of physic phy sical al pro proper pertie ties s and und under er dif differ ferent ent pre pressu ssure re lev levels els,, where physical properties of the same systems can vary signiﬁcantly niﬁca ntly.. More Moreover over,, the calcu calculatio lation n steps of the propo proposed sed model mod el are much sim simple plerr and less rig rigoro orous us tha than n tho those se of other similar models (Garcia and Fair, 2000a, b). The present model fully incorporates the jetting fraction of the dispersion as spray. Due to lack of deﬁnitive data on the structur stru cture e of the spr spray ay reg regime ime,, this study utilized utilized the sem semiiempirical empiric al spray regime model of Zuiderw Zuiderweg eg to estima estimate te the jetting contribution. Thus the current level accuracy of predicting using usi ng the pro propos posed ed mod model el is lim limited ited by the semi semi-emp -empiric irical al nature from Zuiderweg’s model and is not applicable for systems with vapour density less than unity. More fundamental studies of drop dynamics and quantiﬁcation of point efﬁciency in spray regime will improve the model and enhance the correlation between the model and experimental data.

CONCLUSIONS A fundamental model to predict point efﬁciency has been propos pro posed ed bas based ed on the hyd hydrod rodyna ynamic mics s of an ope operat rating ing sieve tray. The new model predicts the FRI efﬁciency data of hydrocarbon systems within +10%. It is also able to predict the trend of tray efﬁciency from weeping to the ﬂooding poin po intt mo more re cl clos osel ely y th than an an any y ot othe herr mo mode del. l. Th The e pr pres esen entt mode mo dell is ba base sed d on th the e an anal alys ysis is of re real al fr frot oth, h, an and d so is based on sound empirical data, and so the model is more adoptable to the diversiﬁed conditions than any other existing models. The model can be used throughout the froth and spray regime reg imes s and the tra transi nsitio tion n bet betwee ween n the them, m, and so wil willl be more applicable for the prediction of distillation tray efﬁciency. Furtherr funda Furthe fundamenta mentall resea research rch on poin pointt ef efﬁcien ﬁciency cy in spray regime, however, would make the model more universal.

NOMENCLATURE a aiG aiL b C

interfacial area per volume of two-phase mixture, m2 m23 geometrical interfacial area per volume of gas, m2 m23 geometrical interfacial area per volume of liquid, m2 m23 weir length per unit bubbling area, m21 constant deﬁned by equation (13)

d max max d 32L 32L d 32S 32S DG DH DL E B E j E LB LB E OG OG E SB SB f j F F bba bba FP FSB g Gf hf hL hW k k G k Gj Gj k GLB GLB k L k LLB LLB k Lj Lj K OGj OGj Lf N N i N f f N f f N G N GLB GLB N L N LLB LLB N OG OG N OGLB OGLB N s p PeG QL Sh t G t GLB GLB t L Dt /

u a u H u s U LB LB u 2 V LB LB W Wecr

maximum stable bubble diameter in turbulent ﬂow ﬁeld, m sauter mean bubble diameter of large bubbles, m sauter mean bubble diameter of small bubbles, m molecularr diffu molecula diffusion sion coefﬁcient coefﬁcient for gas, m2 s21 oriﬁce diameter, m molecularr diffu molecula diffusion sion coefﬁcient coefﬁcient for liquid, m2 s21 overall point efﬁciency for bubbling zone overall point efﬁciency for jetting zone overall point efﬁciency for large bubbles overall point efﬁciency (gas composition basis) overall point efﬁciency for small bubbles volume fraction of gas bypasses the froth bubbles as continuous continuou s jet ratio of hole to active (bubbling) area vapour rate based on active area (u a r G ), (kg m23)0.5m s 21 ﬂow parameter, (r G =r L )0:5 at total reﬂux fraction of small bubbles gravitational gravitati onal constant, 9.8 m s22 gas mass ﬂow rate, kg s21 froth height, m clear liquid height, m weir height, m ﬁrst order bubble breakage rate constant, s21 gas-phase mass transfer coefﬁcient, m s21 k G for jetting zone k G for large bubbles liquid-phase liquid-ph ase mass transfer coefﬁcient, coefﬁcient, m s21 k L for large bubbles k L for jetting zone K OG OG for jetting zone liquid mass ﬂow rate, kg s21 the number of large bubbles the number of large bubbles formed at the oriﬁce at any instant the number of unbroken large bubbles leaving the froth at any instant number of unbroken large bubbles remained from N i at t Dt number of gas-phase mass-transfer units N G for large bubbles number of liquid-p liquid-phase hase mass-transfer units N L for large bubbles number of overall gas-phase mass-transfer units N OG OG for large bubbles number of secondary bubbles formed from N i at t Dt pitch of holes on sieve plate, m Peclet number (d 32L 32LU LB LB/DG) liquid ﬂow rate, m3 s2 1 asymptotic Sherwood number (k GLB GLBd 32L 32L/DG) mean residence time of gas in dispersion, s mean residence time of large bubbles in dispersion, s mean residence time of liquid in dispersion, s the time when half of the total secondary bubbles are formed in froth from the initial N i number of bubbles, s gas velocity based on active(bubbling) area, m s21 gas velocity based on total open hole area, m s21 gas velocity based on total column cross-sectional area, m s21 rise velocity of large bubbles, m s21 2/3 mean square velocity of turbulent ﬂow ﬁeld, (v d ,m d max max) 3 volume of large bubbles, m weir height, m critical Weber number

p

ﬃﬃ ﬃ ﬃ

¼

¼

Greek symbols ae m L r G r L s

froth density deﬁned by equation (14) liquid viscosity, viscosity, Pa s21 gas density, kg m23 liquid density, kg m23 surface tension, N m21

Trans IChemE, Part A, Chemical Engineering Research and Design , 2007, 85(A2): 269–277

FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y

REFERENCES Ameican Institute of Chemical Engineers (AIChE), 1958,Bubble Tray Design Manual (AICHE, New York, USA). Ashley Ashley,, M.J M.J.. and Hase Haselden lden,, G.G G.G., ., 1972 1972,, Ef Effect fectiven iveness ess of vapo vaporrliquid liqu id cont contact acting ing on a siev sieve e pla plate, te, Tran rans s Ins Instt Che Chem m Eng , 50: 119–124. Bennett, D.L., Agarwal, R. and Cook, P.J., 1983, New pressure drop correlat corr elations ions for sie sieve ve tray dist distill illatio ation n colu columns, mns, AI 29:: AICh ChE E J , 29 434–442. Bennett, L.D., Watson, N.D. and Wiescinski, M.A., 1997, New correlation for sieve-tray point efﬁciency, entrainment, and section efﬁciency. AIChE J , 43(6): 1611–1625. Burgess Burg ess,, J.M. and Cald Calderba erbank, nk, P.H., 1975 1975,, The mea measure suremen mentt of bubble parameters in two-phas two-phase e dispersio dispersion-II, n-II, Che Chem m Eng Sci , 30: 1107–1121. Chan, H. and Fair, J.R., 1984, Prediction of point efﬁciencies on sieve trays, Ind Eng Chem Process Des Dev , 23: 814–819. Chen, G.X. and Chuang, K.T., K.T., 1993, Predictio Prediction n of point efﬁciency efﬁciency for sieve tray in distillation, Ind Eng Chem Res , 32(4): 701–708. Fane, A.G., Lindsey, J.K. and Sawistowski, H., 1977, Operation of a sieve plate in the spray regime of column operation, Indian Chem Engr , Jan: 45. Garcia, J.A. and Fair, J.R., 2000a, A fundamental model for the prediction of distillation sieve tray efﬁciency. 1. Database development, Ind Eng Chem Res , 39: 1809–1817. Garcia, J.A. and Fair, J.R., 2000b, A fundamental model for the prediction of distillation sieve tray efﬁciency. 2. Model development and validati validation, on, Ind Eng Chem Res , 39: 1818–1825. Hesketh, R.P., Etchells, A.W. and Fraser Russell, T.W., 1991, Bubble breakage in pipeline ﬂow, Chem Eng Sci , 46(1): 1–9. Hinze, J.O., 1955, Fundamentals of the hydrodynamics mechanism of splitting in dispersion processes, AIChE J , 1(3): 289–295. Higbie, R., 1935, The rate of absorption of a pure gas into a still liquid during short periods of exposure, Trans Am Inst Chem Eng , 31: 365–388. Hoek, P.J. and Zuiderweg, F.J., 1982, Inﬂuence of vapour entrainment on dist distilla illation tion tray ef efﬁci ﬁciency ency at hig high h pres pressure sure,, AIC AIChE hE J , 28(4): 28(4 ): 535–539. Hofer, H., 1983, Inﬂuence of gas-phase dispersion on plate column efﬁciency, Ger Chem Eng , 6: 113–118. Kaltenbacher, E., 1982, On the effect of the bubble size distribution and the gasgas-phas phase e dif diffus fusion ion on the sele selectiv ctivity ity of siev sieve e tray trays, s, Chem Eng Fund , 1(1): 47–68. Kawase, Kawa se, Y. and Moo Moo-Y -Young oung,, M., 1990 1990,, Mat Mathem hematic atical al mod models els for design of bioreactors: applications of kolmogoroff’s theory of isotropic turbulenc turbulence, e, The Chem Eng , 43: B19–B36. Klug, P. and Vogelpohl, A. 1983, Bubble formation with superimposed liquid motion at single-hole plates and sieve plates, Ger Chem Eng , 6: 311–317.

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Lamb, H., 1932, Hydrodynamics, 6th edition (Cambridge University Press, Cambridge, UK). Lewis, D.A. and Davidson, J.F., 1982, Bubble splitting in shear ﬂow, Trans I Chem Eng , 60: 283–291. Lockett, M.J., Kirkpatrick, R.D. and Uddin, M.S., 1979, Froth regime pointt ef poin efﬁci ﬁciency ency for gasgas-ﬁlm ﬁlm controlled controlled mass transfer transfer on a twodimensional sieve tray, Trans IChemE , 57: 25–34. Lockett, M.J. and Plaka, T., 1983, Effect on non-uniform bubbles in the frot froth h on the corr correlat elation ion and pred predicti iction on of poin pointt ef efﬁci ﬁcienci encies, es, Chem Eng Res Des 61: 119–124. Nicklin,, D.J., 1962, Two Nicklin Two-phase -phase bubbleﬂow,Ch 17:: 69 693– 3– 70 702. 2. Chem em EngSci , 17 Porter, K.E., Davis, B.T. and Wong, P.F.Y., 1967, Mass transfer and bubble sizes in cellular foams and froths, Trans Inst Chem Eng , 45: T265– T27 T273. 3. Prado, M., Johnson, K.L. and Fair, J.R., 1987, Bubble-to-spray transition on sieve trays, Chem Eng Prog , 83(3): 32–40. Prado, M. and Fair Fair,, J.R., 1990, Fundamental Fundamental model for the prediction of sieve tray efﬁciency. Ind Eng Chem Res , 29: 1031–1042. Raper, J.A., Hai, N.T., Pinczewski, W.V. and Fell, C.J.D., 1979, Mass transfer efﬁciency on simulated industrial sieve trays operating in the spray regime Inst Chem Engrs Symp Series No 56 , 2.2/57 57– – 2.2/74. Raper, J.A., Kearney, M.S., Burgess, J.M. and Fell, C.J.D., 1982, The structure structur e of industrial sieve tray froths,Chem Eng Sci , 37(4): 501– 506. Sakata, M. and Yanagi, Y., 1979, Performance of a commercial-scale sieve tray, Inst Chem Eng Symp Ser No 56 , 3.2/21–3.2/30. Stichlmair, J., 1978, Die Grundlagen des Ga-Flussig-Kontaktapparates Bodenkolonne (Verlag Chemie, Weinheim, Germany). Valentas, K.J. and Amundson, N.R., 1966, Breakage and coalescence in dispersed phase systems, Ind Engng Chem Fundam , 5: 533–544. Valentas, K.J., Bilous, O. and Amundson, N.R., 1966, Analysis of break bre akage age in di dispe sperse rsed d pha phase se sys syste tems ms,, In Ind d En Engn gng g Ch Chem em Fundam , 5(2): 271–279. van Sinderen, A.H., Wijn, E.F. E.F. and Zanting. W.R., 2003, Entrainm Entrainment ent and maximum vapour ﬂow rate of trays. Trans IChemE , 81: 94– 106. Yanagi, Y. and Sakata, M., 1982, Performance of a commercial-scale 14% hole area sieve tray, Ind Eng Chem Process Des Dev , 21(4): 712–717. Zaritzky, N. and Calvelo, A., 1979, Internal mass transfer coefﬁcient within single bubbles. Theory and experiment, Can J Chem Eng , 57: 58–64. Zuiderweg, F.J., 1982, Sieve trays: A view on the state of the art, 37(10): 0): 1441– 1464 1464.. Chem Eng Sci , 37(1 The manuscript manuscript was received 30 March 2005 and accep accepted ted for publication after revision 1 September 2006.

Trans IChemE, Part A, Chemical Engineering Research and Design , 2007, 85(A2): 269–277

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Abstract: A phenomenological model for froth structure is proposed based on the analysis of

froth images of an active sieve tray taken from a 0.153 m distillation column. Froth is deﬁned as a com combin binati ation on of bub bubble bles s and continuo continuous us jet jets s tha thatt bre break ak the sur surfac face e of fro froth th pro projec jectin ting g liquid liq uid spl splash ashes es and dro drops ps abo above ve the sur surfac face. e. To est estima imate te the fra fracti ction on of sma smallll bub bubble bles s in froth, a fundamentally sound theoretical expression is derived from turbulent break-up theory. A new mod model el for pre predic dictin ting g poi point nt ef efﬁci ﬁcienc ency y of cro crossss-ﬂow ﬂow sie sieve ve tra trays ys has bee been n dev develo eloped ped based on the hydrodynamics of an operating sieve tray represented by the proposed froth structure model. This efﬁciency model is applicable for both froth and spray regime. Fraction of by-passed or uninterrupted gas jet is considered as the determining factor for froth to spray transition. The net efﬁciency is estimated by adding up the contributions of both bubbles and jets present in the dispersion. The model is tested against the efﬁciency data of cyclo-hexane/ cyclo-hexane / n-heptane and i-butane/ i-butane /n-butane mixtures. Keywords: distillation; tray efﬁciency; froth; turbulent break-up; bubble size

INTRODUCTION

Ã Correspondence to: Dr K.T. Chuang, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2G6. E-mail: [email protected] ualberta.ca DOI: 10.1205/cherd06111 0263–8762/07/ $30.00 0.00

þ

Chemical Engineering Research Researc h and Design

Trans IChemE, Part A, February 2007 # 2007 Institution

of Chemical Engineers

the na the natu ture re of tw twoo-ph phas ase e mi mixt xtur ure e in th the e trans tra nsit itio ion n zo zone ne an and d as ask k fo forr tw two o se sepa para rate te expr ex pres essi sion ons s of in inte terfa rfaci cial al ar area ea to pr pred edic ictt the th e tr tray ay ef efﬁc ﬁcie ienc ncy y in th thes ese e tw two o re regi gime mes. s. Zuid Zu ider erwe weg g (1 (198 982) 2) an and d St Stic ichl hlma mair ir (1 (197 978) 8) developed their tray efﬁciency models based on this approach. The FRI efﬁciency data of commercial sieve trays, on the other hand, show smooth transition of tray efﬁciency from the weeping to ﬂoo ﬂo odi din ng poi oint nt.. Th This is co comp mpe ell lle ed ma many ny resear res earche chers rs to res resort ort to a sin single gle ef efﬁci ﬁcienc ency y mode mo dell fo forr bo both th fr frot oth h an and d sp spra ray y re regi gime mes. s. Mostt of the exi Mos existi sting ng tra tray y ef efﬁci ﬁcienc ency y mod models els (AIChE (AI ChE,, 195 1958; 8; Cha Chan n and Fai Fair, r, 198 1984; 4; Che Chen n and Chauang, 1993) are of this type. None of the abovementioned models took into account the struc structure ture of the two-phase two-phase mixture that is generated on the tray in different regimes. The only major attempt that considers sid ers the dis disper persio sion n str struct ucture ure in the fro froth th regime was made by Prado and Fair (1990) for the air /water system. They treated the dispersion as three regions: a region near the tray where the gas can either be jetting or bubbling, a bulk froth region which contains bubbles bubbl es with bimo bimodal dal distr distributi ibution on dispe dispersed rsed in the liquid and a spray region at the top. Howeve How everr, the they y ign ignore ored d the spr spray ay reg region ion in their the ir det detail ailed ed mas mass s tra transf nsfer er mod model. el. Lat Later er,, Garc Ga rcia ia an and d Fa Fair ir (2 (200 000a 0a,, b) ex exte tend nded ed th this is model mod el to oth other er sys system tems. s. The Their ir mod model el was show sh own n to ag agre ree e fa favo vour urab ably ly wi with th a wi wide de range of data. However, However, sever several al adjus adjustable table

The sim simult ultane aneous ous mas mass s and hea heatt tra transf nsfer er combin com bined ed wit with h the com compli plicat cated ed two two-ph -phase ase ﬂuid dynamics make distillation formidable to conduct any fundamental analysis of distillation. Furthermore, distillation became a wellestablishe estab lished d indus industry try long before the theor theory y of tra transp nsport ort phe phenom nomena ena was est establ ablish ished. ed. Thus Th us,, th the e co comm mmon on tr tren end d of di dist stililla lati tion on research resea rch to date mostly remai remains ns empir empirical, ical, semi se mi em empi piri rica call or me mech chan anis isti tic c in na natu ture re.. Mass Ma ss tr tran ansfe sferr ef efﬁc ﬁcie ienc ncy y in di dist stililla lati tion on is asso as soci ciat ated ed wi with th th the e ﬂu ﬂuid id dy dyna nami mics cs on a sieve sie ve tra tray y tha thatt det determ ermine ines s the dis disper persio sion n struct str ucture ure or the con contac tactt are area a bet betwee ween n the gas and liquid phases. The ﬂow regimes on a sieve tray inﬂuence the efﬁciency directly by affe affecting cting the inter interfacia faciall area. Numerous Numerous studies on ﬂow regimes have been done to understan unde rstand d the hydro hydrodyna dynamic mic beha behaviour viour of sieve trays. Most of these studies are mainly focussed on the transition from froth to spray regime reg ime.. The deﬁnitio deﬁnition n of fro froth th its itself elf is sti stillll very vague in the literature. In froth regime, the th e pr pres esen ence ce of pu puls lsat atin ing g je jets ts ra rang nges es of bubble bub bles, s, liq liquid uid spl splash ashes es and dro drople plets ts giv give e rise to a highly complex dispersion structure. The tra tradit dition ionall ally y per percei ceived ved pic pictur ture e of the froth regime consists of bubbles in a liquid cont co ntin inuo uous us ph phas ase e an and d th that at of th the e sp spra ray y regime consists of droplets in a gas continuous pha phase. se. The These se deﬁ deﬁnit nition ions s of fro froth th and spray spr ay reg regime ime sug sugges gestt a sud sudden den change change in 269

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parameters needed to be introduced at different stages of the model to emphasize its mechanistic nature. Bennett et al. (1997) (19 97),, dev develo eloped ped a poi point nt ef efﬁci ﬁcienc ency y mod model el bas based ed on the mechanistic analysis of sieve tray froth height. The model considers the ﬂuid on the tray to be contained in a liquid-continuous region near the tray deck and a vapour-continuous region on top of the liquid-continuous region. The ﬁnial simpliﬁed model takes into account the mass transfer of the liquid continuous region only and thus has limited applicability in spray regime. A recent study of van Sinderen et al. (2003), thatt dea tha deals ls wit with h ent entrai rainme nment nt and max maximu imum m vap vapour our loa load d of tray tr ays s pr pres esen ente ted d a tw two o or th thre ree e la laye yerr mo mode dell of th the e tw twoophase mixture on the tray. This study, although provides a detailed detai led insight into the dynamics of the froth, partic particular ularly ly the mechanisms of entrainment formation but was unrelated to mass transfer efﬁciency. From the above discussions it is evident that most existing correlations for point efﬁciency are highly empirical and do not deal with froth dynamics on a sieve tray. The very few studies, which consider the nature of the dispersion structure in their models generally, ignore the contribution of drops and sprays. These models agree with a wide range of data when using adjustable parameters, but are less applicable in spray regime on theoretical ground. In this study, the froth regime is modelled based on the analysis of froth images taken from a 0.153 m diameter distillation column. The model describes the froth as a combination of bubbles and continuous jets. At higher gas load, the jet jettin ting g fra fractio ction n dom domina inates tes and gives rise to the spray regime reg ime.. Thi This s fro froth th mod model el is fur furthe therr ado adopte pted d to dev develo elop p a fundamental model for predicting sieve tray efﬁciency. The efﬁciency model takes into account the contribution of both bubbles and jets to the net mass transfer.

MODEL STRUCTURE Froth images taken in a 0.153 m diameter distillation column are shown in Figures 1 and 2. Based on a careful study of these kinds of froth images, a froth structure has been schematically presented in Figure 3, where froth is shown as a combi com bina natio tion n of je jets, ts, bu bubb bble les s an and d liliqu quid id sp spla lash shes. es. Th The e images (Figures 1 and 2) show that the liquid droplets and

Figure 1. Froth image of pure methanol on a sieve tray in a 0.153 m

distillation column.

Figure 2. Froth image of 67 wt% methanol/water mixture on a sieve

tray in a 0.153 m distillation column.

Figure 3. Schematic representation of froth on an operating sieve

tray.

splashes constitute a major part of the froth. A portion of the droplets is formed when bubbles break out of the surface of the froth. However, the presence of liquid splashes conﬁrms that some of the gas jets manage to penetrate through the froth without forming bubbles and generates liquid splashes at the end of liliqu quid id con contin tinuo uous us zon zone. e. Fig Figur ure e 4 is a mo more re detailed representation of the froth model, showing both jetting and bubbling zones. The jetting zone elaborates how some of the gas jets formed at the sieve tray holes, cross the froth uninterrupted and throw liquid splashes above by tearing up the liquid surface. The bubbling zone shows the process of large and small bubble formation in the froth. Both zones are present and remain intimately mixed with each other in real froth fro th.. No je jetti tting ng is ach achiev ieved ed at a rel relati ative vely ly lo low w liliqu quid id ﬂow rate.. This regime is call rate called ed bub bubblin bling g reg regime, ime, which occ occurs urs close to the weeping limit and is of limited signiﬁcance signiﬁcance for commercial sieve tray operation. As the gas load is increased, an increasingly increa singly greater proportion proportion of gas passes the disper dispersion sion in the form of jets. The spray regime occurs when most of the gas jets formed at the oriﬁce, reach the liquid surface uninterrupted and project the liquid up to form small drops. Unlike in froth regime, where bubbles bubbles form a major part of the interfa interfacial cial area, in spray regime drops are the only contributor to the interfacial interfa cial area. The point efﬁc efﬁciency iency is estimate estimated d by combini combining ng

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271 27 1

MODEL DEVELOPMENT In the following sections a method to estimate point efﬁciency E OG OG from equations (1) and (2) has been discussed.

Bubbling Zone Bubbli Bub bling ng zon zone e is con conside sidered red to hav have e bimo bimodal dal size distribution of bubbles as reported in many studies (Porter et al., 1967; 196 7; Ash Ashley ley and Hase Haselden lden,, 197 1972; 2; Loc Lockett kett et al al.., 19 1979 79;; Kaltenbacher, 1982; Hofer, 1983; Klug and Vogelpohl, 1983). The small bubbles are the secondary bubbles formed by the turbulent break-up of the primary bubbles originated from the oriﬁce. The large bubbles are the unbroken primary bubbles that remain in the froth due to incomplete break-up. The speci speciﬁc ﬁc inter interfacia faciall area, aiG and residence time, t GLB GLB for the lar large ge bub bubble bles s in fro froth th can be est estima imated ted from the following equations, respectively:

Figure 4. Froth structure model on an operating sieve tray.

aiG

the con contrib tributio utions ns from both bubb bubbling ling and jett jetting ing zon zones es that exist on a tray. E OG OG

¼ (1 À f i )E B þ f i E j

(1)

where f j is the volume fraction of the gas that bypasses the bubbles as continuous jets, E B and E j are contributions of bubbling and jetting zone, respectively, to the net point efﬁciency. Due to incomplete break up of the large (primary) bubbles both large (primary) and small (secondary) bubbles coexist in bubbling zone. Thus E B has contributions from both large and small bubble bubbles, s,

¼ d 6

t GLB GLB

(7)

32L 32L

f ¼ U hLB LB

(8)

Due to complex nature of the process, there are few analytical expression expre ssions s for any design in distil distillatio lation n liter literature ature.. The generall tre era trend nd is to use cor correl relati ations ons,, whi which ch are supporte supported d by reliab rel iable le exp experi erimen mental tal dat data. a. The fol follow lowing ing equ equati ations ons are used to estimate the Sauter mean diameter and raise velocity of the large bubbles formed at the oriﬁce.

¼ 0:887D0H 846u 0H 21 1 6 U LB þ u a LB ¼ 2:5(V LB LB ) :

d 32L 32L

:

=

E B

¼ (1 À FSB)E LB LB þ FSB Ã E SB SB

(2)

where FSB is the fraction of small bubbles.

THEORY OF MASS TRANSFER Following Follow ing exp expres ressio sions ns can be obt obtain ained ed fro from m two two-ﬁl -ﬁlm m theory, N G

¼ aiGk Gt G N L ¼ aiL k L t L

(3) (4)

where aiL t L

¼

r L Gf r G Lf

aiG t G

(5)

Here aiG and aiL represent the geometrical interfacial area per unit volume of gas and liquid phases, respectively. Assuming that the liquid composition does not change vertically and vapo va pour ur pa pass sses es as pl plug ug ﬂo ﬂow w wi with thou outt mi mixi xing ng,, th the e ov over eral alll mass transfer unit can be related to point efﬁciency as follows: E OG OG

¼ 1 À exp( À N OG OG )

(6)

In the present study, E OG OG is obtained from the published Murphree efﬁciency, E mv , mv data as outlined by Garcia and Fair (2000a).

(9) (10)

Where DH and u H are the hole diameter and velocity; V LB LB is the large bubble volume and u a is the gas velocity based on the tray active (bubbling) area. Equation (9) based on the bub bubble ble siz size e dat data a mea measur sured ed by ele electr ctroni onic c pro probes bes jus justt above the sieve tray (Prado et al., 1987). Thus the equation estimates the unbroken primary bubbles in froth. Three different liquid systems with nine different tray geometries were used to generate the bubble size data. This is by far the only correlation for primary bubbles on a sieve tray. Equation (10) was originally developed for estimating rise velocity of bubble swarms through a porous bed (Nicklin, 1962). Later Burgess and Calderbank (1975) showed that this equation adequately predicts rise velocity of large bubbles in froth on sieve trays. This is again the only study done on this topic. The mass transfer coefﬁcient for the liquid phase, k LLB LLB, is modelled with Higbie penetration theory (Higbie, 1935), k LLB LLB

¼ 1:13

DL t GLB GLB

0:5

(11)

This is a well-established model used previously by numerous stu studie dies. s. The mas mass s tra transf nsfer er coe coefﬁ fﬁcie cient nt for gas pha phase, se, k GLB GLB, of the large bubbles is estimated from the numerical soluti sol ution on pre presen sented ted by Zar Zaritz itzky ky and Cal Calvel velo o (Za (Zarit ritsky sky and Calveio, 1979). This solution was developed for mass transport models in distillation. It was tested against experimental data and was applied in efﬁciency models such as those by Prado Pra do and Fair (19 (1990) 90) and Gar Garcia cia and Fai Fairr (20 (2000b 00b). ). The solution is presented as a plot of Peclet number ( PeG) of

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SYEDA et al.

the gas pha phase se ver versus sus the asy asympt mptoti otic c She Sherwo rwood od num number ber (Sh ). Within the range 40 , PeG , 200 the following polynomiall provi nomia provides des an excel excellent lent ﬁt for the exper experimen imental tal data: /

Sh

/

¼ À11:878 þ 25:879 879(( log PeG ) À 5:64(log PeG)2

(12)

For the ran range ge PeG . 20 200, 0, It wa was s fo foun unff th that at Sh ha had d an essentially constant value of 17.9. Froth height, hf , is estimated from Bennett et al.’s (1983) correlation for effective froth height, 1

hf

¼ hw þ C W QL

ae

¼ exp

12:55 u s

r L

r G

À

r G

(14)

0:91

and C

¼ 0:5 þ 0:438exp( À 137:8 hw)

(15)

There are a number of correlations available in literature to estimate the froth height on a sieve tray. The unique characteristic of equation (13), proposed by Bennett et al. (1983), is that unlike any other correlations it gives effective froth height i.e., the height of the liquid continuous region. Since in the present model, froth height is used to estimate the residence time of bubbles in froth, the height of liquid continuous region calcul cal culate ated d by equ equati ation on (13 (13)) giv gives es the app approp ropria riate te val value. ue. Other correlations, which give total froth height i.e., the combined height of liquid and vapour continuous region, would over estimate the residence time of bubbles. Using the above information, N GLB GLB and N LLB LLB can be calculated from equations (3) and (4). Equation (4) is then used to get the overall mass transfer unit, N OGLB OGLB, from which the contribution of the large bubbles, E LB LB, to the net efﬁciency is obtained by using the equation (6). The portion of small bubbles in froth is considered to reach equili equ ilibri brium um whe when n mas mass s tra transf nsfer er rat rate e is hig high h (Lo (Locke ckett tt and Plaka, 1983). Kaltenbacher (1982) also suggested that the small bubbles get trapped in the froth and leave the froth practically saturated. In this case, because equilibrium prevails vai ls bet betwee ween n the vapour vapour and the liq liquid uid phase of sma smallll bubbles, the efﬁciency of small bubbles becomes unity, i.e., E SB SB

¼1

Here k is the breakage rate constant and N is the number of large bubbles. Two additional assumptions are made to keep the calculation simple.

Let us consider that the number of large bubbles entering the froth at t ¼ 0 is N i. Due to turbulent break-up, N i reduces to N f f at t ¼ Dt . Here Dt is the residence time of large bubbles in the ﬂow ﬁeld. Therefore, by integrating equation (17) from N i at t ¼ 0 to N f f at t ¼ Dt , the following expression is obtained:

where 0:5

(17)

¼ ÀkN

(13)

ae

! 35

dN dt

(1) All large bubbles are bigger than the maximum stable bubble size and are equally susceptible to the break-up process. (2) The number number of large and small bubbles bubbles at any parti particular cular cross section of the froth is constant.

0:67

2 4À

used thi used this s con concep ceptt for bub bubble ble bre breakak-up up in pip pipeli elines nes.. The same sam e app approa roach ch is app applie lied d her here e for sie sieve ve tra tray y ana analys lysis, is, where a ﬁrst order bubble breakage rate is assumed. The breakage rate of large bubbles in froth is given by

(16)

In order to estimate the contribution of small bubbles to the total efﬁciency, we need to determine the fraction of small bubbles, FSB, in froth. Due to lac lack k of exp experi erimen mental tal data and reliable method to estimate this parameter, expression for FSB has been derived from turbulent break-up theory of bubbles. In any ﬂow ﬁeld, the FSB is governed by the bubble breakage rate and the bubble residence time in turbulent zone. Previous theoretical studies (Valentas et al., 1966; Valentas and Amu Amunds ndson, on, 196 1966) 6) dea dealin ling g wit with h dro drop p siz size e dis distri tribut bution ion assumed that the breakage rate of a drop is of ﬁrst order with respect to the number of drops. Later Hesketh et al . (1991)

N f f

¼ N i eÀ

k Dt

(18)

Let us consider that the fractions of large and small bubbles at t Dt represent the average fraction of large and small bubbles in the froth. The number of unbroken large bubbles at t Dt is given by

¼ ¼

N f f

¼ N i eÀ

k Dt

(19)

For binary breakage, N s

¼ 2(N i À N f f )

(20)

where N s is the number of small bubbles formed at t Dt . Thus the volume fraction of small bubbles in froth can be estimated as follows:

¼

FSB

2(N i À N f f ) s ¼ N V N sþV N ¼ V 2(N À N ) þ N ðV =V Þ s

s

f f L

i

f f

f f

L

(21)

s

Here V S and V L are the volumes of small and large bubbles, respectively. Assuming bubbles have spherical shapes, we get the follo following wing expression expression for FSB from equa equations tions (19) and (21): FSB

¼

2(1

À eÀ

k Dt

À eÀ ) 3 À ) þ (d 32L 32L =d 32S 32S ) e

2(1 k Dt

k Dt

(22)

The ratio of large bubble diameter to small bubble diameter, d 32L obta tain ined ed fr from om th the e ex exis isti ting ng lilite tera ratu ture re.. Th The e 32L/d 32S 32S, is ob reported diameter ratios are summarized in Table 1. From the above table, we ﬁnd that the most probable value of the ratio d 32L 32L/d 32S 32S is 5. The breakage rate constant k is a function of the turbulent ﬂow ﬁeld and the ﬂuid physical properties. Hesketh et al. (1991) showed that the measured deformati ma tion on ti time mes s an and d br brea eaka kage ge ti time me of bu bubb bble les s ca can n be characterized by the natural mode of oscillation of a sphere given by Lamb (1932) and proposed the following functionality of the rate constant k ,

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FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y tray. Table 1. Reported bubble size distribution on an operating sieve tray. Source

Small bubble

Large bubble

Hofer (1983) Ashley and Haselden (1972) Kaltenbacher (1982) Porter et al. (1967) Lockett et al. (1979)

5 mm 5 – 1 0 mm

25 mm 40 – 80 mm

5 8

4 mm 5 mm 5 mm

25 mm 20 mm 25 mm

6 4 5

k

3:8

¼

0:1 0:3

0:6

r L r G v

We0cr :9

Ratio

(23)

s 0:4

Here, v is the rate of energy dissipation in unit mass; Wecr is the critical Weber number given as Wecr

¼

r u u 2 d max max

(24)

¼

k

3:8

¼

0:1 0:3

r L r G

We0cr :9

s 0:4

(u s g )0:6

(25)

Dt can

be expressed as (26)

¼ nt GLB GLB

here, n is any value between 0 and 1. Since both n and Wecr are unknowns, we can combine them into single constant: C 00

¼ 0:13 À 0:065 2 G G À 5 2:6 Â 10 k Lj Lj ¼ k Gj Gj

r

r

(1 , r G

,

80kgmÀ3 )

(29) (30)

0:25

m L

E j

À

¼ 1 À exp

ahf

40

¼ F 0 3 :

ahf K OGj OGj u s

2 F bba hL FP s

(31)

0:37

(32)

where, F bba bba is vapour rate based on active area, F is the ratio of hole area to active area, hL is the clear liquid height and expressed as hL

¼ 0:6 hW

p FP b

0:25

(33)

and at total reﬂux FP

0:5

¼ r G

(34)

r L

The breakage time Dt

small tray by using free trajectory model. However, Raper et al. (1979) showed that Fane et al.’s model under-predicts the tra tray y ef efﬁci ﬁcienc ency y whe when n app applie lied d for ind indust ustria riall siz size e tra tray y. Another important attempt to predict mass transfer efﬁciency in spray regime was made by Zuiderweg (1982). His semiempiri emp irical cal mod model el is bas based ed on the FRI exp experi erimen mental tal dat data. a. This is the only model so far that is not case sensitive and is readily applicable for spray regime. In this study, we have chos ch osen en Zu Zuid ider erwe weg’ g’s s sp spra ray y re regi gime me mo mode dell [e [equ quat atio ions ns (29–34)] to estimate the contribution of jetting zone to the total mass transfer efﬁciency in froth regime;

s

where u 2 is the mean square square velocity of turbulent turbulent ﬂow ﬁeld and d max maximum mum sta stable ble bub bubble ble size aga against inst tur turbule bulent nt max is the maxi break-up; r and s are the density and surface tension of the liquid phase, respectively. The values of reported Wecr range over an order of magnitude depending on the ﬂow pattern responsible for the deformation of the bubble. In distillation, there is no reporte reported d value for Wecr . The rate of energy dissipation, however however,, is appro approximately ximately estimate estimated d by v u s g (Kawase and Moo-Young, 1990); thus the rate constant becomes

273 27 3

¼ Wen0 9

(27)

:

cr

The experimental data obtained by Raper et al. (1982) are used to evaluate the volume fraction of gas that bypasses the bubbles formation and forms jets, f j and to estimate the net con contri tribut bution ion of jet jettin ting g zon zone. e. Fol Follow lowing ing equ equati ation on is an excellent ﬁt for the average value of jetting fraction, f j as a function of F -factor, -factor, F bba bba. f j

¼ À0:1786 þ 0:9857(1 À eÀ1 43 :

F bba bba

)

(35)

By multiplying equations (25) and (26) we get 3:8r 0L:1 r 0G:3 k Dt ¼ C 00 (u s g )0:6 0:4

s

(28)

The constant C 00 will be estimated by comparing the model with the measured efﬁciency data.

Jetting Zone In fro froth th reg regime ime,, it is dif difﬁcu ﬁcult lt to inv invest estiga igate te jet jettin ting g zon zone e separately as jets are intimately mixed with bubbles. No information is available in literature on the size of jets or droplets present prese nt in froth. In this study study,, we will treat the jetting zone as spray and use the correlations of spray regime to estimate the contribution of jets in froth regime. Although numerous studies have been done to determine the onset of spray, very few studies have been focussed exclusively on mass transfer efﬁciency in this regime. Fane et al. (1977) achieved some success in predicting efﬁciency in spray regime on a

DETERMINING CONSTANT C 00 Constant C 00 is determined by comparing the model with seven sets of FRI data (Sakata and Yanagi, 1979; Yanagi and Sakata, 1982). These data sets cover two hydrocarbon systems, system s, cyclo cyclo-hexa -hexane ne//n-hep n-heptane tane and i-but i-butane ane//n-butane, at ﬁve different pressures in two different tray geometries. The cyclo-hexane/ cyclo-hexane/n-hep n-heptane tane system is widel widely y used for testing distillation tray performance. The properties of this system are representative of many hydrocarbon systems operated at 400 kPa pressure or below. The data sets for this system are taken at two different pressures, 34 kPa and 165 kPa. The data sets for i-butane/ i-butane/n-butane cover three different pressure leve le vels ls.. Th The e me meas asur ured ed ef efﬁc ﬁcie ienc ncie ies s at hi high gh pr pres essu sure res s (2068 kPa and 2758 kPa) have been corrected for vapou vapour r entrainmen entra inmentt with the down ﬂow liqui liquid d (Hock and Zuide Zuiderweg rweg,, 1982). Figure 5 presents the effect of different values of constant C 00 on the estimated point efﬁciency for the seven sets of FRI data. The average absolute error was calculated by

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SYEDA et al.

efﬁciency; ciency; expressed expressed as Figure Figur e 5. Effect of constant C 00 on point efﬁ average absolute error error..

Figure 7. Comparison of measured and predicted point efﬁciencies

for the cyclo-hexane/n-heptane system at 165 kPa (open hole-area 14%).

the following equation, Error%

Estimated

P ¼ j

À Experimentalj=Experimental nnumber of data

(36)

The minimum error was obtained at C 00 ¼ 0.16. The reported theoretical values of Wecr range from 1 to 4.7 (Hinze, 1955; Lewis Lew is and Dav Davids idson, on, 198 1982). 2). Wit Within hin thi this s ran range, ge, Dt varies 00 from 0.16 t GLB GLB to 0.644 t GLB GLB at C ¼ 0.16. The values are reasonable for obtaining the average bubble size distribution within withi n the froth froth..

PREDICTION OF POINT EFFICIENCY The present model introduces a new method to estimate sieve tray efﬁciency based on a froth structure that describes the hydrodynamics of an operating sieve tray. The predicted point efﬁc efﬁciencie iencies, s, E OG from m the propos proposed ed mo mode dell ar are e co commOG, fro pared with the FRI data in Figures 6–12. In all cases, predictions from two earlier models of Chen and Chuang (1993) and Garcia and Fair (2000b) are also compared with the proposed model. The unique characteristics of the new model is that unlike the two other models it predicts the trend of efﬁciency change from weeping to ﬂooding point more closely (Figures 7–9). The steady decrease in both fraction of small bubbles and bypassed jets results in gradual decrease of the point efﬁciency, E OG -factor approach approaches es the weeping weeping poin point. t. The OG as the F -factor

Figure 8. Comparison of measured and predicted point efﬁciencies for

the iso-butane/n-butane system at 1138 kPa (open hole-area 14%).

model also predicts the smooth transition of E OG OG from froth to spray regime. Under high operating pressures (Figures 10 and 11), the breakage rate constant k is high enough to cause breakag bre akage e of all large bub bubble bles. s. This makes the frac fractio tion n of small bubbles FSB unity and gives rise to high point efﬁciency under such operating condition. The experimentally measured fraction of bypassed gas is 0.8 at F -factor -factor 2 (Fane et al. 1977). Beyond this point froth is dominated by spray and the model reduces red uces to Zuid Zuiderwe erweg’s g’s model for spr spray ay reg regime ime.. Thu Thus s any

Figure 6. Comparison of measured and predicted point efﬁciencies

Figure 9. Comparison of measured and predicted point efﬁciencies

for the cyclo-hexane/n-heptane system at 34 kPa (open hole-area 14%).

for the iso iso-but -butane ane/n-bu n-butane tane syst system em at 1138 kPa (ope (open n hole hole-are -area a 8.3%).

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FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y

Figure 10. Comparison of measured and predicted point efﬁciencies

for the isoiso-buta butane ne/n-bu n-butane tane sys system tem at 2068 kPa (ope (open n hole hole-are -area a 8.3%).

error in predic predicting ting E OG -factor 2 is inh inherit erited ed from OG beyond F -factor Zuiderweg’s model. The prediction of Chen and Chuang (1993) model is satisfacto fa ctory ry fo forr al alll six sets of da data. ta. Th The e in inter terfa facia ciall are area a in th this is model is estimate estimated d from the bubble size distrib distribution. ution. However, However, since the vapour /liquid dispersion dispersion in spray regime mostly consists of drops, the model is applicable only to froth regime. The Gar Garcia cia and Fair (20 (2000b 00b)) mod model el pre predic dicts ts the low low-pressure tray efﬁciency data adequately. However, it predicts signiﬁcantly lower tray efﬁciency than the measured values at high hig h pre pressu ssures res.. Thi This s dis discre crepan pancy cy res result ults s fro from m the hig highly hly empi em piri rica call na natu ture re of th the e mo mode del. l. Th The e mo mode dell in invo volv lves es a number of equations and at least four adjustable parameters thatt mat tha match ch abo about ut 22 set sets s of tra tray y ef efﬁci ﬁcienc ency y dat data, a, mos mostly tly measured under low or moderate pressures. The under prediction of three sets of data out of 22 sets, did not affect the ﬁnal ﬁn al form of th the e mo mode del. l. Th Thus us,, th the e mo mode dell is fo foun und d to be suitable at low and moderate pressures only. Figure 13 compares the overall performance of the three models. The proposed model predicts within +10% for all the sys system tems s and shows better better per perfor forman mance ce tha than n the two other oth er mod models els.. The agr agreem eement ent bet betwee ween n the exp experi erimen mental tal data and predictions of the new model proves the validity of the proposed approach.

275 27 5

Figure 12. Comparison of measured and predicted point efﬁciencies

for cyclo-h cyclo-hexane exane/n-he n-heptan ptane e syst system em at 165 kPa (ope (open n hole hole-are -area a 8.3%).

Tray hydrodynamics is considered to be the key factor in determining the nature of two-phase mixture in distillation.

The un The uniq ique ue fe feat atur ure e of th the e pr prop opos osed ed mo mode dell is th that at it is based on the analysis of tray hydrodynamics (Figures 1–4) that describes the real situation on a sieve tray. The model includes both bubble and jet contribution to the total point efﬁciency. The often reported bimodal distribution of bubbles in froth is explained as the result of incomplete break-up of primary mar y bub bubble bles s in tur turbul bulent ent ﬂow ﬁel ﬁeld. d. The fraction fraction of sma smallll bubbles, FSB, is directly estimated by theoretical analysis of the rate of bubble breakage in froth. The only other similar effort to estimate FSB was done by Garcia and Fair (2000b). Although their ﬁnal model agreed with the database favourably ab ly,, th the e st stud udy y fa faililed ed to id iden enti tify fy th the e so sour urce ce of bi bimo moda dall bubble size distribution observed in froth, which made their semi-theoretically obtained FSB expression rather arbitrary. The present model has been developed to incorporate both the froth and spray regimes. The fraction of gas that forms continuous jets, f j, is the determining factor of the contribution from fro m eac each h of the regimes regimes.. For example, example, in fro froth th reg regime ime,, 0 , f j , 1. As f j increases with higher a gas load, transition to spray regime occurs gradually and f j becomes unity as spray regime is reached. No sudden change in dispersion stru st ruct ctur ure e oc occu curs rs du duri ring ng th this is tr tran ansi siti tion on,, an and d th ther ere e is a smooth transition of FRI efﬁciency data from froth to spray regime. Thus the effect of the present approach of considering in g th the e ef effe fect ct of th the e ﬂo ﬂow w re regi gime mes s on th the e tra tray y ef efﬁc ﬁcie ienc ncy y adopte ado pted d in the pro propos posed ed mod model el dif differ fers s fro from m tha thatt res result ulting ing from the two previous approaches of the existing models. One On e of th the e ap appr proa oach ches es is to ap appl ply y th the e sa same me ef efﬁc ﬁcie ienc ncy y model for both froth and spray regimes without considering

Comparis parison on of mea measured sured and pred predicted icted point ef efﬁcien ﬁciencies cies for Figure 11. Com

Figure 13. Overall comparison of the proposed model with two other

the iso-butane/n-bu n-butane tane system at 2758 kPa (open hole-area hole-area 8.3% 8.3%). ).

existing models.

DISCUSSION

Trans IChemE, Part A, Chemical Engineering Research and Design , 2007, 85(A2): 269–277

276

SYEDA et al.

the effect of change of the dispersion structure (AICHE, 1958; Chan and Fair, 1984; Chen and Chuang, 1993) The other approach is to use two completely different models for froth and spray regime (Zuiderweg, 1982). Since the dispersion struc str uctu ture re in fro froth th re regi gime me is ju just st in inve vers rse e to th that at of sp spra ray y regime, applying the same efﬁciency model for both regimes without considering the change in the dispersion structure is the incorrect way to estimate the tray efﬁciency. On the other hand ha nd,, wh when en tw two o se sepa para rate te mo mode dels ls ar are e us used ed fo forr th the e tw two o regimes regim es dif difﬁcult ﬁculties ies arise in iden identifyin tifying g the exact transition point. By including the fraction of jetting, dependent on gas ﬂow rate, the new model takes into account the difference in dis disper persio sion n str struct ucture ure bet betwee ween n the coe coexis xistin ting g fro froth th and spray regimes. Thus the model provides a logical solution thatt can be app tha applie lied d con contin tinuou uously sly over the range of ﬂow rates, without resorting to an arbitrary selection of the use of th the e sa same me or se sepa para rate te mo mode dels ls fo forr bo both th th the e fo fort rth h an and d spray spr ay reg regime imes, s, and the thereb reby y ful fully ly des descri cribes bes the smo smooth oth transition between the regimes. The inclusion of physical properties considered in the estimation mat ion of fra fracti ction on of sma smallll bub bubble bles s FSB [equ [equation ation (28)] makes the model applicable to systems with wide range of physic phy sical al pro proper pertie ties s and und under er dif differ ferent ent pre pressu ssure re lev levels els,, where physical properties of the same systems can vary signiﬁcantly niﬁca ntly.. More Moreover over,, the calcu calculatio lation n steps of the propo proposed sed model mod el are much sim simple plerr and less rig rigoro orous us tha than n tho those se of other similar models (Garcia and Fair, 2000a, b). The present model fully incorporates the jetting fraction of the dispersion as spray. Due to lack of deﬁnitive data on the structur stru cture e of the spr spray ay reg regime ime,, this study utilized utilized the sem semiiempirical empiric al spray regime model of Zuiderw Zuiderweg eg to estima estimate te the jetting contribution. Thus the current level accuracy of predicting using usi ng the pro propos posed ed mod model el is lim limited ited by the semi semi-emp -empiric irical al nature from Zuiderweg’s model and is not applicable for systems with vapour density less than unity. More fundamental studies of drop dynamics and quantiﬁcation of point efﬁciency in spray regime will improve the model and enhance the correlation between the model and experimental data.

CONCLUSIONS A fundamental model to predict point efﬁciency has been propos pro posed ed bas based ed on the hyd hydrod rodyna ynamic mics s of an ope operat rating ing sieve tray. The new model predicts the FRI efﬁciency data of hydrocarbon systems within +10%. It is also able to predict the trend of tray efﬁciency from weeping to the ﬂooding poin po intt mo more re cl clos osel ely y th than an an any y ot othe herr mo mode del. l. Th The e pr pres esen entt mode mo dell is ba base sed d on th the e an anal alys ysis is of re real al fr frot oth, h, an and d so is based on sound empirical data, and so the model is more adoptable to the diversiﬁed conditions than any other existing models. The model can be used throughout the froth and spray regime reg imes s and the tra transi nsitio tion n bet betwee ween n the them, m, and so wil willl be more applicable for the prediction of distillation tray efﬁciency. Furtherr funda Furthe fundamenta mentall resea research rch on poin pointt ef efﬁcien ﬁciency cy in spray regime, however, would make the model more universal.

NOMENCLATURE a aiG aiL b C

interfacial area per volume of two-phase mixture, m2 m23 geometrical interfacial area per volume of gas, m2 m23 geometrical interfacial area per volume of liquid, m2 m23 weir length per unit bubbling area, m21 constant deﬁned by equation (13)

d max max d 32L 32L d 32S 32S DG DH DL E B E j E LB LB E OG OG E SB SB f j F F bba bba FP FSB g Gf hf hL hW k k G k Gj Gj k GLB GLB k L k LLB LLB k Lj Lj K OGj OGj Lf N N i N f f N f f N G N GLB GLB N L N LLB LLB N OG OG N OGLB OGLB N s p PeG QL Sh t G t GLB GLB t L Dt /

u a u H u s U LB LB u 2 V LB LB W Wecr

maximum stable bubble diameter in turbulent ﬂow ﬁeld, m sauter mean bubble diameter of large bubbles, m sauter mean bubble diameter of small bubbles, m molecularr diffu molecula diffusion sion coefﬁcient coefﬁcient for gas, m2 s21 oriﬁce diameter, m molecularr diffu molecula diffusion sion coefﬁcient coefﬁcient for liquid, m2 s21 overall point efﬁciency for bubbling zone overall point efﬁciency for jetting zone overall point efﬁciency for large bubbles overall point efﬁciency (gas composition basis) overall point efﬁciency for small bubbles volume fraction of gas bypasses the froth bubbles as continuous continuou s jet ratio of hole to active (bubbling) area vapour rate based on active area (u a r G ), (kg m23)0.5m s 21 ﬂow parameter, (r G =r L )0:5 at total reﬂux fraction of small bubbles gravitational gravitati onal constant, 9.8 m s22 gas mass ﬂow rate, kg s21 froth height, m clear liquid height, m weir height, m ﬁrst order bubble breakage rate constant, s21 gas-phase mass transfer coefﬁcient, m s21 k G for jetting zone k G for large bubbles liquid-phase liquid-ph ase mass transfer coefﬁcient, coefﬁcient, m s21 k L for large bubbles k L for jetting zone K OG OG for jetting zone liquid mass ﬂow rate, kg s21 the number of large bubbles the number of large bubbles formed at the oriﬁce at any instant the number of unbroken large bubbles leaving the froth at any instant number of unbroken large bubbles remained from N i at t Dt number of gas-phase mass-transfer units N G for large bubbles number of liquid-p liquid-phase hase mass-transfer units N L for large bubbles number of overall gas-phase mass-transfer units N OG OG for large bubbles number of secondary bubbles formed from N i at t Dt pitch of holes on sieve plate, m Peclet number (d 32L 32LU LB LB/DG) liquid ﬂow rate, m3 s2 1 asymptotic Sherwood number (k GLB GLBd 32L 32L/DG) mean residence time of gas in dispersion, s mean residence time of large bubbles in dispersion, s mean residence time of liquid in dispersion, s the time when half of the total secondary bubbles are formed in froth from the initial N i number of bubbles, s gas velocity based on active(bubbling) area, m s21 gas velocity based on total open hole area, m s21 gas velocity based on total column cross-sectional area, m s21 rise velocity of large bubbles, m s21 2/3 mean square velocity of turbulent ﬂow ﬁeld, (v d ,m d max max) 3 volume of large bubbles, m weir height, m critical Weber number

p

ﬃﬃ ﬃ ﬃ

¼

¼

Greek symbols ae m L r G r L s

froth density deﬁned by equation (14) liquid viscosity, viscosity, Pa s21 gas density, kg m23 liquid density, kg m23 surface tension, N m21

Trans IChemE, Part A, Chemical Engineering Research and Design , 2007, 85(A2): 269–277

FUND FU NDAM AMEN ENT TAL MO MODE DEL L FO FOR R PR PRED EDIC ICTI TION ON OF SI SIEV EVE E TRA RAY Y EF EFF FIC ICIE IENC NCY Y

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