Comparison between the dispersion and tanks in series models

Shorter Commumcations However, usual phase temperature

we can offer a qualitative explanation for the unbehavlour of these systems The muumum m the liquid composition graphs corresponds, within the expenmental error which IS appreciable for these coloured and VISCOUSsystems, to the muumum m the qolubllity-hqtud composdton graph for all three systems (see for example Fe 3) To the left of the muumum both the mcreasmg concentration of alcohol and the correspondmg lower solubtity of salt m the mixture contibute to a decreasing b01hng pomt but to the right of the muumum the rapldly rncreasmg soluhhty of the salt mves nse to an elevation of bohg pomt which IS far m excess of any small change m boding point of the salt-free solvent mixture Systems which have this deep soluhhty mnumum would be expected to exlubd this unusual phase behavrour The majonty of salt-saturated two-solvent systems can IX conveniently represented as bmanes on a salt-free basis However, those systems which exhibEt a deep solublllty mmlmum present problems (see F@ 2) which are induced by the bmary representation When a tnangular diagram IS employed no such dticultles anse and the only difference between deep mmlmum systems and true azeotropic systems IS that only the latter have a tie-hne wluch Joins hqutd and vapour compositions of equal value

Acknowledgements-The authors are indebted to Dr M B King (University of Bummgham, U K ) and Drs J Bueno and J R Alvarez for helpful dlscusslon Lkpartamento

de QuCmwa

TPcntca

D JAQUESt MAGALAN

fig 3 Solublltty of cobalt(II) chlonde at the boding point vs liquid composition on a salt-free we&t percent basis for ethanol-water nuxtures

Facultad de C~encas Unrversuiad de Salamanca Salamanca, *ain

contradict the Gibbs-Konovolov theorem but, on closer exammabon, we have a restncted system and so we cannot ckum a poslhve breakdown of the theorem

111 Kmg M B , Phase Equdabnum m Mutures, p 128 Pergamon Press, Oxford I%9 12lAlvarczJ R,GalanM A andMatillaM C,An Quint 1979 75 82 [3] Alvarez J R and Galan M A, An! Quim 1974 70 271 [4] Galan M A, Labrador M D and Alvarez J R, Advan

REFERJNCES’

tOn leave from the Department of Apphed Chenustry, Royal Melbourne Institute of Technology, Melbourne 3oo0, Austraba, and to whom correspondence should be addressed

fIz%alIlcaJ E2l#mmg Pea’wmm Press Lid

Chem Ser 1976 I55 85 [51 Jaques D, Ind Engng Chem Proc Des Deu 1977 16 I29

.scwmceVol 35 pp mulam 1980 Pnnted m Great Bntam

Comparison

between the dispersion and tanks in series models (Accepted 6 February 1980)

Flow pattern within reactors seldom correspond to Ideal flow (plug or backmlx flow) and several types of models have been developed m order to charactenze them The one parameter models most used are the dlsperslon and tanks m series models and several authors have studied thear mterrelatlon [ l-31 In the present commumcatlon the relations for a first order reversible reaction are analyzed on the basis of gettmg the same converslon and the same start-up time paUAL CONVERSION

CRITERION

When the ObJectwe of the comparison IS to find equivalent performances, rt IS sound to choose the cnterton of equal conversion The conversion of a reactant undergoing a tirst order reversible reactton m a dispersion model reactor can be found as an extension of Danckwerts eqn (4) for an Irreversible reactIon

4aK exp (Pe/2) 1-xA=(l+K~(1+a)Zexp(aPe/2)-(1-a)*exp(-aPe/2)}

where a = Vl+4/3lPe

Slmdarly,

for the tank m series model It can be found that

(2) The relationship between the Peclet number and the number of tanks IS represented m Fig 1 for different converstons and equihbrmm constants When the conversion at e-qmhbnum IS much larger than the exit conversion (e g XA = 0 495 for K = NO), so that the reaction can be consldered to be u-reverable, the curves obtamed are comcident with the ureverslble reactIon case [I, 31 For lower values of K or higher values of conversron, lower values of Pe m the dispersion model are eqmvalent to the same number of tanks m the tanks In series model At the same ttme, dserences within this cntenon are small at low converslons and high numbers of tanks EQUAL SFART-UP TIME CRITERION

In many cases the transient behavlour of a reactor can be Important Therefore, rt seems useful to analyze the relation between the two models on the basis of getting equal start-up

1805 and therefore

the ttme r, IS ohtamed

from

xew +nV99/7)n-’ ,=I

01 Ftg

1 Relation

I

1

I

I

02

04

06

06

between parameters of the models the equal converston crtterton

I

’ o ‘/PC

accordmg

to

tune or equal time to reach 59% approach of the steady state concentrabon In order to get an expression for the start-up ttme for a first order reversible reactton m a dtsperston model reactor, an analysts similar to that by Sawtnsky and Hunek[S] ts carried out The residence ttme dtstnbutton functton for the dtsperston model IS given by d?(t) =

2ePc'*2

K,

,)1+1

where =

(-

y,' (I,

and y,

IS

K, e-QJ’lr

(3)

v: +Pe +F

2 Pe =&+T

the root of the equatron cotg ‘y, = $

Pe

4v,

For Peclet numbers lower than IO the above distnbutlon function IS accurate enough if only one term in the summation 1s considered Therefore It follows that[6]

I +j& exp f-03 +a,Vh) 1

1-&d~)=t1-xAw)

and the time t,

(1 -x,4@))

2 exp (Pe/2) ,+,lK

can be obtained

m~$e,2)=z

c

KI Kexp(-nit/7)

(n - 1)’

The value for the steady state conversion xa(m) IS given by eqn (2) The relattonshtp between Peclet number and the number of tanks IS represented m Fig 2, for both reversible and ureverslble reactrons In the last case as the Damkohler number increases, the Peclet number m the dtsperston model, equivalent to a fixed number of tanks m the tanks m series model, decreases The curve for Dul = 0 (equal transient period in non-reacting systems) represents also the hmd for low equthbrmm constants at all Damkohler numbers For Dar = I, the equivalence curves for different values of K he between the bounds correspondmg to K +O and K-+= and they shift to the rtght as the equdtbnum constant increases However, as the Damkohler number increases a more complex behavtour IS found[6] We have represented the curves for Da, = 10 and It can be seen that for K < 10 the curves are corncldent However, for values of K > 10 the curves shift first to the left (K = 100) but they tend later to the Irreversible reaction curve (K = 1000) It follows then that the sensltlvdy of the curves to the eqmhbrmm constant value IS almost ml for K < 10 but increases and reaches a maximum for values of K between 100 and IO00 As the Damkohler number Increases, the equivalence curves are comctdent over a wider range of eqmhbnum constants Ftgure 3 presents the equtvalence of these modeIs based on the two mentioned criteria and the equal vartance cntenon[ I] A reasonable agreement between all the methods of comparison IS found at high values of the eqmhbnum constant and low values of the Damkohler number At converslons of approxtmately 0 5 the equal variance and equal conversion criteria become almost equivalent However they differ mcreastngly with an increase m the conversion or a decrease m the equdrbrmm constant In all these cases, for a fixed number of tanks m the tanks m senes model, lower Peclet numbers m the dlsperston model are found for the equal conversion criterion On the other hand, for very low Dal, equal start-up time cnterron gives values of Peclet numbers, for a fixed number of tanks, that are higher than those arrsmg from the equal vanance cnterlon At high Da, these cnterla differ widely It can also be seen that as the equdlbrmm constant Increases the curves for the equal conversion cnterlon (Fig 1) shift to the left, while for the

from

exp(-at b5.h)

The expression for the steady state converslon x,+(m) IS gtven m eqn (1) For the tanks m senes model the expression for the transient concentratton at the exit 1s aven by 1 -x,&)=

1 -x”(D1)-i-&

[+_)”

e-CB+“),lr 01

02

04

06

06

0

11% Fig 2 Relation between parameters of the models the equal start-up time cnterlon CESVd3SNobI

accordmg

to

MO6 -

Shorter Commumcations

1

n

f O-

----

Equal varlancc Equal start -up Equal conversion

Departamento de Industnas Fact&ad de Ciennas Eractas y Naturafes Unlversrdad de Buenos Aires &dad Unlversrtana Buenos Aires Argentina

ttme Ix* = 0 495)

08-

N 0 LEMCOFF

NOTATION

a

c _ Da=10

1

0

Ftg 3 Companson

I

02

I

of parameters

I

I

I

04

06

00

accordrng

10 1 /Pe

to different

cnterla

(K = IO)

parameter defined m eqn (1) parameter defined m eqn (3) 2 dtsperston coeffictent Dar Damkohler number (= kr) E(t) restdence ttme drstnbutlon function K, parameter defined m eqn (3) K equrhbrtum constant k rate constant L reactor length number of tanks A Peclet number (= u LjD) t99 start-up time I( mean velocity conversion XA

Greek symbols equal start-up time cntenon (Ftg 2) two opposite behavrours are found At low Da,, the curves shrft to the rtght and at high Damkohler numbers they first shift to the left but finally behave as the low Dar curves Centro de Investrgacrdn y Desarrollo err Procesos CataItItcos (CINDECA) Calle 47, No 257, 1900 La Plata

to whom correspondence

r

number (=&(l+$))

parameter defined m eqn (3) resrdence trme ItJJFERE?NCBS

E N PONZI*

M G GONZALEZ

[I]

Levensptel 0 , Chem Engng Scr 1962 11 576

[2] Kramers H and Alberda G , Chem Engng SC: 1953 2 173 [31 Ntshtwakt A and Kato Y ,Can J Chem Engng 1974 52 276 [4] Danckwerts P V , Chem Engng Scr 1953 2 1

Argentina

*Author

t3 modified Damkohler y,

should be addressed

On the reference

IS] Sawmsky 161 Go&.lez published

time in the multipiicity

J and Hunek J , Chem Engg Scr 1977 32 1265 M G , Ponzt E N and Lemcoff. N 0, to be (1980)

analysis

(Received 11 July 1979, accepted 10 December

Recently, Uppai et al [l] pubhshed a semmal &per on the influence of reactor residence time for a contrnuous stared tank reactor (CSTR) They found that the evolutton of multtphctty and hmtt cycles IS much more bizarre as the reactor restdence time vanes than that obtained by changmg Damkohler number atone An Important startmg point and basis of then work IS the reference time. which they defined but for whrch no exphclt expression was provided In thts note an expresston for the reference time IS developed and some practical analytic cnterta are denved therefrom, whtch it IS hoped wtll help further apphcattons of their Important work R&F&B&NCETIME ANALYSIS It should first be noted that the work of Uppal et al [I] cannot be applied to adiabatic reactors For adiabatic CSTRs, there IS no difference in the multtphcity pattern between varymg reactor residence time and changing Damkohler number, and thus the earher work of Uppal et a/ [2] may be applied In that case For a first-order. nreverstble, exothernnc reactton A+ B occumng m a CSTR, Uppal et al [I] found that for y values not satisfying

for CSTRs

1979)

untqueness IS gnaranteed can be snnpllfied to

We note that for x,, B - (1 + g)/&

whmh m the case x2, = 0, slmphfies

further to (3)

r>4 For y vaIues satrsfymg (1). the necessary existence of multiplicity IS [1] 4 1+/3+?

BZ

(

*+B-$(l+B+~ If we define fl=&r,

(1)

)

condttton

for the

z (4) Y >

eqn (4) with the equality sign wtll have betng real and the larger one posmve, Uppal et al [l] define the reference tune III such a way that c = I at the larger root By domg so and deflmng #le= ,9&,

two roots for r For the roots