Gas-liquid reactions in well-mixed reactors—A fresh perspective

Pergamon PII:

Chemical Engineering Science, Vol. 51, No. 20, pp. 4561 4577, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved S0009-2509(96)00294-1 0009 2509/96 $15.00+ 0.00

GAS-LIQUID REACTIONS IN WELL-MIXED REACTORS-A FRESH PERSPECTIVE SATISH J. P A R U L E K A R * and N O R AISHAH SAIDINA A M I N Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL 60616, U.S.A. (Received 7 August 1995; accepted 22 March 1996) Abstract--Classificationof gas-liquid reactions in the past has been based on comparison of characteristic times for diffusional mass transfer and chemical reaction in liquid film near the gas-liquid interface. The fast chemical reactions occur almost entirely in liquid film and the very slow reactions are confined to bulk liquid. Gas-liquid reactions that do not belong to either class have received less attention and have been in general assumed to be confined to either liquid film or bulk liquid for simplification of modeling and design of reactors. A unified approach for modeling of gas-liquid reactors is provided here considering a single reaction occurring over the entire liquid phase, with at least the reactant or the product being volatile. The two-way linkages between liquid film and bulk liquid, largely ignored previously, are properly accounted for in the present approach. The dispositions of concentration profiles for reactant and product in the liquid film are presented and discussed. A first-order reaction is considered as a specific example. Numerical illustrations for a perfectly mixed reactor demonstrate that an unwarranted confinement of liquid-phase reaction to either liquid film or bulk liquid can lead to incorrect design of and erroneous prediction of performance of gas-liquid reactors. Variation in relative importance of reaction in liquid film with respect to reaction in bulk liquid with variations in process parameters is investigated. The effectiveness of the liquid phase reaction in the presence of mass transfer resistances is examined. Copyright © 1996 Elsevier Science Ltd Keywords: Augmentation factor, effectiveness of absorption, desorption, and liquid-phase reaction, enhancement factor, gas liquid reactions, linkage between liquid film and bulk liquid, perfectly mixed isothermal reactor, single reversible reaction, two-film theory. 1. INTRODUCTION G a s - l i q u i d reactions are of considerable importance in several gas purification and liquid-phase production processes (Astarita, 1967; Danckwerts, 1970; Astarita et al., 1983; Doraiswamy and Sharma, 1984; Westerterp et al., 1984; Froment and Bischoff, 1990). The extensive literature on interphase mass transfer accompanied by chemical reactions is devoted almost entirely to absorption with liquid-phase reaction(s). The mathematical description and representation of the steady state operation of gas-liquid reactors has been largely based on the two-film theory, with molecular diffusion being considered to be the dominant transport mechanism in the gas and liquid films on either side of the gas-liquid interface. Depending on the characteristic times for diffusional mass transfer and chemical reaction steps in the liquid-phase, the gas-liquid reactions belong to different regimes (Danckwerts, 1970). The so-called 'fast reactions' are considered to be completed predominantly in the liquid film, result in substantial enhancement in rates of interphase mass transfer, but result in very low extents of liquid-phase utilization. These reactions are attractive in gas purification and separation processes, but are not as desired in liquid-phase production processes due to very low liquid-phase

* Corresponding author.

utilization. The reactions at the other end of the spectrum, the so-called 'very slow reactions', are considered to occur almost entirely in the bulk liquid phase. The enhancement in rates of mass transfer across the gas-liquid interface due to these reactions is negligible. Between these two limiting cases of the liquid-phase reactions are the so-called 'slow reactions' and reactions that are neither slow nor fast. These reactions are frequently encountered in liquidphase production processes. The extensive literature on gas-liquid reactions has, with a few exceptions (Kulkarni and Doraiswamy, 1975; Mann, 1983; Westerterp et al., 1984; DeLeye and Froment, 1986a, b; Froment and Bischoff, 1990; Landau, 1992; Kastanek et al., 1993; Parulekar and Saidina Amin, 1996), been devoted to reactions that are considered to be completed entirely in liquid film (fast reactions) and reactions that are relegated to bulk liquid (reactions that are not fast). Confining the reaction to either the liquid film or the bulk liquid leads to simpler mathematical models for gas-liquid reactors and the design of these reactors becomes less cumbersome. For proper design of gas-liquid reactors, the two-way linkages among bulk gas, gas-side film, liquid-side film, and bulk liquid must be incorporated in mathematical models for these reactors. The linkage between liquid film and bulk liquid has largely been ignored. The illustrations provided in this article will demonstrate that an unwarranted confinement of liquid-phase reaction to

4561

S. J. PARULEKARand N. A. S. AMIN

4562

either liquid film or bulk liquid can lead to serious pitfalls in the design of gas-liquid reactors. A unified approach for modeling of gas-liquid reactors is provided in this article considering that liquidphase reaction occurs over the entire liquid phase. A single irreversible/reversible chemical reaction is considered in the present work. The principal reactant and the reaction product are considered to be volatile or non-volatile, at least one of these species being volatile. The bulk gas and bulk liquid-phases are considered to be perfectly mixed. After accounting for the relevant processes in different parts of the two phases, the dispositions of concentration profiles of reactant and product in the liquid film are presented and discussed. The implications of these dispositions on parameters such as augmentation and enhancement factors and distribution of the extent of reaction between liquid film and bulk liquid are discussed. As a specific example, a first-order reaction is considered. Expressions for concentrations of reactant and product in different portions of the two phases are obtained analytically. Numerical illustrations reveal that relegation of reaction to either liquid film or bulk liquid can result in incorrect design of and erroneous prediction of performance of gas-liquid reactors in certain situations and the effectiveness of the reaction is reduced in the presence of mass transfer resistances.

product (B) is present in both gas (g) and liquid (1) phases. A(I) ~ nB(l) (1) A(g) ~ A(I) B(I) ~

(2)

B(g).

(3)

The three situations to be considered here are as follows [with the processes among eqs 0)-(3) that apply for a particular situation being indicated in the parentheses]: (a) volatile A and non-volatile B [eqs (1) and (2)], (b) volatile A and volatile B [eqs (1)-(3)], (c) non-volatile A and volatile B [-eqs (1) and (3)]. The representation of the reaction in eq. (1) does not imply that only A and B participate in this reaction. Only these two species are shown in eq. (1) since the focus here is on a reactant (A) and a product (B). For example, when A is volatile, it must react with a nonvolatile coreactant (species C), not shown in the description of reaction (1). When A is non-volatile, the transformation of A may or may not require a coreactant. The conservation equations for A and B for the liquid-side film are expressed as d2Cj

DjL--~X2 =rj,

J=A,B,

O
(4) r

-nr

if J = A if J = B "

2. PROBLEM FORMULATION

A mathematical model, in both dimensional and dimensionless forms, for a perfectly mixed gas-liquid reactor that accounts for appropriate linkages among bulk gas, gas-side film, liquid-side film, and bulk liquid is presented first (Fig. 1). This is followed by definitions of various indicators of the performance of gas-liquid reactor. Novel among these are the definitions of augmentation factor for transport of a species across the gas-liquid interface, effectiveness factor for reaction, and effectiveness of absorption/desorption of a species participating in liquid-phase reaction. For the sake of illustration, we consider a single liquidphase reaction in which at least the reactant (A) or the

Liquid in

Gas out

The appropriate boundary condition gas-liquid interface for species J is dCj =0 dx

atx=0,

J=AorB

the

(5)

when species J is non-volatile or

-DjLda~=kj~(pj-pJt) Psl= HjCjI,

at Djo

x=0,

(6)

kj~ = 6'

when species J is volatile. It must be realized that Hj is trivial if the species J is non-volatile. When the

Gas-liquid interface

00o00

Liquid out

at

vection, Reaction

Gas in Diffusion - -

__ Diffusion, Reaction

Fig. 1. A continuous flow perfectly mixed gas-liquid reactor.

Gas-liquid reactions in well-mixed reactors gas-side mass transfer resistance is negligible (i.e., when kjc, ~ oo ), condition (6) reduces to PJ = PJI,

eqs (4)-(10) can be restated as d2A @2 - 9 2

(7)

PJI = H j C j I .

In this article, both bulk gas and bulk liquid phases are considered to be perfectly mixed. The reactor operation is isothermal, with the feed and effluent flow rates being the same for each phase. The conservation equations for species J in the bulk liquid and bulk gas (where appropriate) phases are then expressed as CJF -- CJb -- O~L'CLrjb -- a-SL'gL DjL d x i x =

~

-

r dx. a

(10)

fo'

z~ dA mA(Ac,e -- A6) + ~L rl d y ).=0 = 0,

CjI

= ~/--, qL

~[ = ~ 9 l dy.

(17)

It must be noticed that condition (14) reduces to condition (5) when H j = 0, condition (6) when Hs > 0 and kjG < oC, and condition (7) when H j > 0 and kjG ~ oc. Elimination ofgl in eqs (13) and integration of the resulting equation reveals that

B+ocoA=fly+f2,

0~
(18)

f2 = Bt +

~oA~. (19)

with - o~oAi],

One then obtains the following overall balances for bulk gas and bulk liquid-phases from eqs (15) and (16)

CJF

CJb

JLe-vjC.,

JL--vjC.,

PJF

(11)

PJ

vjHjC~4

if H j > 0 ,

DAL

vjHjC]

VA= 1, v B = n .

32r 9l --

O~LZLrb ,

DaLC*

a£LrLDAL

tps

rna(A~e--AG)+mB(B6v-B~)+raqfl=o.

(21)

The formulation presented thus far also applies for the situation where the extent of reaction in liquid film is negligible (92 = 0) with

and J a - - -

An appropriate choice for C* may be pAp~HA when A is volatile and Car when A is non-volatile. The definitions in eqs (11) ensure continuity of A and B at the gas-liquid interface. In terms of these and the additional dimensionless parameters defined below,

(~

(20)

~X0

~;L ~X0

Jc~v---

--

(16) rG r/ dB = 0 mn(Bo~. - Be) + z--~Lot--oodyl,,=o

and

vjC* '

rl

~-y y=l = 0 ,

if Hs > 0 gas phase

vjHjC*

DBL

(14)

r/dA

ALe + Bcr -- AL -- BL -- qfl = 0

PJ

,

at y = 0

liquid phase

v j C Y~

j,-

(13)

as

Cj

J =

J=A,B

(15)

f~ = [BL + ~XoAL-- BI

Let the dimensionless concentrations of A and B be defined as (J = A, B)

J(l)=Jc,

_r/dB =0 B L F - - B L + q L ctodyy=l

,9,

(

Cto92, 0 < y < l ;

ALF--AL--qL--

and

(

d2B dy2

1 dJ - ms(Jc, - Jr) tpj dy

(8)

The quantity (1 - ~) is the reciprocal of the 'Hinterland ratio' (Westerterp et al., 1984). Conditions (8) provide a linkage between the bulk liquid and the liquid-side film. With the exception of few studies (Kulkarni and Doraiswamy, 1975; Mann, 1983; Westerterp et al., 1984; DeLeye and Froment, 1986a, b; Froment and Bischoff, 1990; Landau, 1992; Kastanek et al., 1993; Parulekar and Saidina Amin, 1996), this linkage has largely been ignored. In this regard, it is instructive to examine the ratio of total rate of reaction in liquid film to total rate of reaction in bulk liquid, viz.,

and

J(0)=J,,

6 = O,

= 1 -- a3

4563

'

Y

=

ksaRT6 - - , DjL

qL -X -~ '

C* Hj

ms

RT'

J = A,B,

,

(12)

dJ dy

--=dL-dt,

0~
J=A,B.

(22)

The 'enhancement factor' for transport of a species across the gas-liquid interface is defined as the ratio of the rate of transport considering the reaction in the liquid film and the rate of transport in the absence of reaction in the liquid film, viz., 1

dJ

Tyy ,=o

if J is volatile.

(23)

In the definition above, the same values of Ji and JL are considered to persist whether or not reaction

4564

S. J. PARULEKAR a n d N. A. S. AMIN

occurs in the liquid film. As has been reported elsewhere (Cornelisse et al., 1980; Winkelman and Beenackers, 1993; Parulekar and Saidina Amin, 1996) and will be illustrated later, the definition above can lead to negative or unbounded enhancement factors in certain situations. An alternate and better measure of the effect of chemical reaction on transport of a species across the gas-liquid interface is provided by 'augmentation factor', which is defined as 1 {d~JyJ } i f J is volatile. Ey = (JL - JI)NR y=O R

t/

dA

(25)

(qL)~=oT}y y=o"

The reaction rate in the absence of liquid-side diffusional resistance (a = 1) is provided by the solution of the following material balances ALF -- AL -- qL + PA = O,

l

_aDjL~ x s ~:~ = Oh.

The volume-specific gas-liquid interfacial area, a, being based on total liquid-phase volume, rather than volume of bulk liquid-phase, the correct form of the equation above is -- a_DjLdC j dx x=~ = aOb.

(24)

The liquid-phase utilization factor is defined as the ratio of the rate of absorption of A (if volatile) to the rate of consumption of A in the absence of diffusional resistance on the liquid side, viz.,

'7~

are considered to be

A comparison of eq. (28) with eqs (8) reveals that the effectiveness factor defined by Kulkarni and Doraiswamy (1975) applies only to operations that do not involve addition or withdrawal of liquid to or from the reactor (batch operation with respect to liquidphase), since transport effects in the bulk liquid are not accounted for in estimating effectiveness factor. The fractional conversion of A in the gas-liquid reactor is expressed as

fA :

[mA(AGF -- A a ) + Z~ (ALj: -- A L ) ] "eL

BLF -- BL + qL + PB = O

JI=JL

(26b)

with pj = mjOj(J~ -- JL), Oj = a e L Z L R T k j a , J =

A,B. (26c)

The variable t/L is the effectiveness factor for the liquid-phase reaction only under the presumption that all of the volatile reactant absorbed is utilized for the reaction. The effectiveness factor for the liquid-phase reaction, defined as the ratio of the total rate of reaction and the total rate of reaction in the absence of diffusional resistance in liquid phase, assumes the form #=(1 +~)

qz

(qD~=o

.

It must be noted that the effectiveness factor defined in eq. (27) is also the ratio of fA in the presence of diffusional resistances and fa in the absence of diffusional resistance in liquid-phase. Two additional indicators of the performance of the well-mixed gas-liquid reactor are the fraction of A absorbed (if A is volatile) that is consumed in the reaction (XA) and the fraction of B generated that is desorbed (if B is volatile) (XB), viz., X a = 1--

Ze ( A L - - A L r ~ "CLmA \ A a r -- AG,]

and

(30) XB 1+

27G

"eLmB

[(BL -- BLF)/(Ba -- BaF)]

(27)

The effectiveness factor with negligible liquid-side mass transfer resistance alone (6' > 0) as the basis will be referred to as liquid-phase effectiveness factor (~/L) and that with negligible gas- and liquid-side mass transfer resistances (6' = 0, Ja = JL, J = A, B) as the basis will be referred to as overall effectiveness factor (qo)- The relations to be solved simultaneously to estimate reaction rate when 3 = 0 are eqs (26) when 3' > 0 and eqs (26a) and (26b) when 6' = 0. Kulkarni and Doraiswamy (1975) have extended the definition of effectiveness factor used for gas-solid catalytic reactions to gas-liquid reactions. In their analysis, pertaining to the reaction in (1) being irreversible and first-order with respect to A (ra kCA), the conservation equations for species J in the bulk liquid phase =

(29)

mAAGF + - - ALE TL

(26a) mj(JGF - Ja) - - -"ca pj=O, EL

(28)

3. DISPOSITIONS O F C O N C E N T R A T I O N P R O F I L E S F O R A AND B IN LIQUID F I L M

Dispositions of concentration profiles for A and B in liquid film for single irreversible liquid-phase reaction are discussed first. This is followed by some thoughts on possible dispositions of concentration profiles for a reversible liquid-phase reaction. No restrictions are placed on the nature of rate expression for the liquid-phase reaction. In practical operations of gas-liquid reactors, for the three situations under consideration, the following scenarios are appropriate: (i) A6F>>max(At.F, Bar, BLr) if A is volatile, and (ii) ALr>>max(BaF, BLF) if A is non-volatile. When the reaction in (1) is irreversible, both ffl and qL are positive. The concentration profile for A cannot therefore exhibit a local maximum in 0 < y < 1, while the

Gas-liquid reactions in well-mixed reactors

(a)

(b)

(a)

Y

(t)

(h)

Fig. 2. Dispositions of profiles of A [J = A in (a) and (h)] and B [J = B in (b)-(g)] in the liquid film (0 < y < 1) for positive 9/(0 < y < 1) and non-negative qL. J = Jt at y = 0 a n d J = J L a t y = l (J = A, B). In (b), dB/dy = 0 at y = 0; dB/dy = 0 at y = l in (d); and dA/dy = 0 at y = 0 in (h). The dashed lines in (a) and (c)-(g) represent profiles of A and B in 0 ~
profile for B cannot exhibit a local minimum in this interval. Further, since 9l > 0, no inflection points are permissible for the concentration profiles for A and B in the liquid film [eqs (13)]. It follows from the conservation equations in the liquid film that dA/dy increases and dB/dy decreases with increasing y for 0 ALF and therefore dA/dy is negative at y = 0 and 1 [eqs (15) and (16)]. The concentration profile for A therefore has the disposition shown in Fig. 2(a). It is evident that Ea ~> 1 [eq. (23)], i.e., the liquid-phase reaction enhances the rate of absorption of A. Since B is the product of reaction (1) and it is anticipated that BL > BLr in practical reactor operations, it follows from eqs (15) that t/ dB qL

>

- - - -

cto dy ly= 1

(31)

4565

y = 0 for BG > Boe. The dispositions of the concentration profiles for B that are admissible are presented in Fig. 2(c)-(g). For the profiles in Fig. 2(c)-(e), BL > Bx and one can deduce that EB >/ 1, i.e., desorption of B is enhanced by the liquid-phase reaction. The enhancement factor for desorption of B becomes unbounded as ( B L - B x ) ~ O + or 0 - ( E B ~ oo or -- oo, respectively) [Fig. 2(f)]. The profile in Fig. 2(g) represents the situation where EB is negative (Bx > BL). In this situation, by neglecting the reaction in the liquid film, one predicts absorption of B instead of desorption of B. Negative enhancement factors have been reported previously by Cornelisse et al. (1980), A1-Ghawas and Sandall (1988), Winkelman and Beenackers (1993), and Parulekar and Saidina Amin (1996). If A is non-volatile, then it is supplied to the reactor through the liquid feed. In view of eqs (13) and the constraint dA/dy = 0 at y = 0, one can deduce that dA/dy is positive for y > 0 and increases monotonically with an increase in y. Therefore, the only admissible disposition of concentration profile for A is that shown in Fig. 2(h). Since it is expected that Bo > Boy in practical reactor operations, one can deduce from eqs (16) that dB/dy must be positive at y = 0. The concentration profiles for B that are admissible are available in Fig. 2(c)-(g). As discussed earlier, En is (i) finite and ~> 1 for the profiles in parts (c) le), (ii) unbounded for the profile in part (f), and (iii) negative for the profile in part (g). It must be noted that in the situations involving non-volatile A and non-volatile B, from the overall bulk gas-phase balance, eq. (21), it follows that ( i ) f l > 0 since BG > Bc,,v when A is non-volatile (ma = 0) arid (ii) fl < 0 since Ao < AGF when B is non-volatile (ms = 0). When B is non-volatile, from the applicable profiles in Fig. 2(a) and (b), it follows that f l is indeed negative. When A is non-volatile, positivity of fl requires that BI -- BL < O:o(AL -- AI).

(32)

F r o m the profiles in Fig. 2(c)-(f) and (h), it should be evident thatf~ is indeed positive. When the profile for B is as in Fig. 2(g), eq. (32) provides the positive upper limit on ( B t - BL). Some thoughts on the possible dispositions of concentration profiles for A and B in the liquid film when the reaction in (1) is reversible are provided next. When the reaction in (1) is reversible, the possibility ofr being negative (A being generated from B) in some portion of the liquid-phase cannot be ruled out. In practical reactor operations, one would anticipate a net generation of B from A. In view of eqs (13), (15), and (16), the following constraints must be satisfied: "CG

ma(Aov - Aa) + - - ( A L r - - A L ) "CL

When B is non-volatile (dB/dy = 0 at y = 0), the monotonic decrease in dB/dy with increasing y implies that the only admissible disposition of concentration profile for B is that in Fig. 2(b). When B is volatile, from eqs (16) it is evident that dB/dy > 0 at

= mB(BG -- Bee) + - - (BL -- BLv) "~L

= "re (1 + ~)qz > 0. 1"L

(33)

4566

S.J. PARULEKARand N. A. S. AMIN

When 9~ > 0 in 0 < y < 1 and qL t> 0, it is not difficult to observe that the concentration profiles for A and B in the liquid film have the dispositions shown in Fig. 2. In view of eqs (33) and the constraint sgn {91(1)} = sgn(qL), it follows that negativity of 9l in some portion(s) of the liquid film requires positivity of the same in the remainder of the film. Since the concentration profiles for A and B are continuous functions of the parameters of a gas-liquid reaction system, transition from the dispositions shown in Fig. 2 to dispositions that allow for negative 91 in a portion of liquid film require admissibility of inflection point(s) for concentration profiles for A and B at certain points in the multidimensional space of system parameters. If 91 is negative in an interior portion of the liquid film, the concentration profiles for A and B must admit two inflection points, one each at the boundaries of this portion. A necessary condition for this can be deduced to be the existence of a negative local minimum in 91 in 0 < y < 1. It must be realized that the observations in this section have been made without placing any restrictions on the dependence of the rate of reaction in (1) on concentrations of A and B. In what follows, the issues considered here are pursued further considering a first-order or pseudofirst-order reversible reaction as a specific example. 4. SPECIFIC EXAMPLE--LINEAR KINETICS

Expressions for concentrations of A and B in the liquid film are obtained analytically for a first-order or pseudo-first-order reversible reaction. This is followed by a discussion of dispositions of concentration profiles for A and B in liquid film when the reaction in (1) is reversible. Simplifications in analytical expressions that result when the reaction is irreversible are discussed after this. Analytical expressions for effectiveness factors in the two limiting cases of the reaction in (1), viz., an extremely slow reaction (k ~ 0) and an instantaneous reaction (k ~ ~ ) , are provided at the end of the section. The kinetics of reaction in (1) is described as

with the solution for B(y) being provided by eqs (18), (19), and (35). In this case, qL and ( assume the form q_.~L

A L -

Da

,

Da = O~8L'CLk;

(36) c~= 1 if 6 = 0 and

(1 - a) {cosh(v) - 1) ~,

sinh(~,) (37)

It can be deduced from eqs (35) that at the most one inflection point is admissible for concentration profiles for A and B in the liquid film where A1 + Bt tanh(yy) = 0. For finite k, a unique inflection point is admissible (i) at y = 0 if AI = B f f K and AL # Br./K, and (ii) at one y in the interval 0 < y ~< 1 if AIB~ < 0. Admissibility of the unique inflection point in the interior of the liquid film requires of course that (AI - BI/K)(AL - BL/K) < 0. If At = Bt/ K and A t = BL/K, then it follows from eqs (18), (19), and (35) that A and B are in equilibrium in the entire film and the concentration profiles of A and B in the liquid film are linear. This situation arises only when the reversible reaction is instantaneous. Net generation of B from A requires that [eqs (33), (36) and (37)] (At _ B._/B_I)> (l+dpo(9O)(BL\__~-- A L ) .

(38)

When concentration profiles for A and B admit a unique inflection point yt (0 < y~ < 1), satisfaction of constraint (38) requires applicability of one of the following two scenarios ( v = A - B / K , 9l, or d2A/dy2). v >OforO<

y < yx,

v<0fory~
qL<0; (39a)

or

v
v>0fory~
In view of relation (18), the solution to eqs (13) can be represented as A(y) = Aa cosh(vy) + B~ sinh(Ty) + O(f~y + A ) (35a) with y:

~(1+ V DAr.

O = ( K +ao) -1, nl

DAL ~ DnLK J '

K = - -K'

At=(At-Of2),

[AL -- (At -- Of2)cosh(7) - O(fl +f2)] sinh(v)

(39b) The sketches of the dispositions that are candidates for the two scenarios described in eqs (39) were obtained for the three situations under consideration here. When A is non-volatile and B is volatile, it is anticipated that [eqs (15), (16), and (21)] dA

d-~ = 0 and

n

(35b)

qL>0.

> 0 (since B~ > B~F) at y = O;f~ > 0

and dA -->Oaty=lifqLAL. dy

4567

Gas-liquid reactions in well-mixed reactors

Following simplifications result for an irreversible reaction when A is non-volatile.

AL AI - cosh(7~ '

(7

J

At. =

ALF [1 + Da + t/ytanh(y)] '

(f)

(40) (1 - ~) tanh(7) 7 Since a6<
Y Fig. 3.Dispositions of profiles of A [J = A in (a) and (c)] and B [J = B in (b) and (d)-(i)] in the liquid film (0 < y < 1) admissible for a first-order reversible reaction in addition to dispositions shown in Fig. 1. I denotes the unique inflection point. The profiles in (a) and (b) are applicable when A is non-volatile and B is volatile; those in (c) and (d) are applicable when A is volatile and B is non-volatile; and those in (c) and (e)-(i) are applicable when both A and B are volatile.

Subject to the constraints stated above and those stated in eqs (39), it was deduced that the only dispositions that are admissible for a reversible reaction are those in Fig. 3 (a) and (b) (qr < 0). When A is volatile and B is not, it is anticipated that [eqs (15), (16), and (21)]

+ tl7Affsinh(y)] AL-- [1[ALr + Da + qT/tanh(7)]'

(A~LL ~ = qS°

) + 1 , (41)

7 [AI cosh(y) -- ALl EA = sinh(7 ) (At - At) The expression for AL stated above is a vivid illustration of how the events in the bulk liquid influence the events in the liquid film and vice versa. The 'feedback' from bulk liquid to liquid-side film evident in this expression has been almost entirely ignored in the extensive literature on gas-liquid reactions. In the special case where ALF = 0, the expressions for ~ and EA in eqs (41) assume the form

~=~bo 1 +

dB dA < 0 (since AG < A~r) at y = 0; f l < 0 dy _ 0 and ~-y

l+Da+

, t42)

and y dB --<0aty=lifqr<0sinceBL>BLv. dy Subject to the constraints stated above and those stated in eqs (39), it was deduced that the only dispositions that are admissible in this situation are those in Fig. 3(c) and (d) (qr < 0). When both A and B are volatile, it is expected that [eqs (16) and (21)] dA --<0(AG
and

dB ~>0(B~>BGr)

at y = 0; and fl < 0. Subject to the constraints stated above and those stated in eqs (39), it was deduced that the only dispositions that are admissible in this situation are those in Fig. 3(c) and (e}-(i) (qr < 0). F o r none of the three situations, dispositions that are subject to the constraints in eqs (39) and permit qr to be positive are admissible.

Ea

-

sinh(y)

si~-(y) qy r/7 1 + Da + tanh(7~) sinh(y

,J

Both ~ and Ea provided by expressions (42) can be deduced to be monotonically increasing functions of k, all other parameters remaining unaltered. The reaction in (1) is therefore increasingly shifted to liquid film (from bulk liquid) as k is increased. The expression for AL in eqs (41) also applies if the extent of reaction in the liquid film is considered to be negligible (91 = y = 0) for simplification of reactor design. The fractional conversion of A, fA, lies in the interval 0 ~fA ~
4568

S. J. PARULEKARand N. A. S. AMIN

details of which are spared here:

(4) Finite

liquid-side

mass

transfer

resistance

- m~ +

(47)

(,~ > 0) )cA,=

[~I~AAGF + ~2ALF - - (1/K){fl3mBBGF +

+K

/~4BLF}]

2j,

F ( m A A G F -I- (TG/'CL)ALF)

The expressions for flfs (i = 1, 2, 3, 4) and F are provided below, reaction (1) occurring in the entire liquid phase unless indicated otherwise. (1) (a) Negligible overall mass transfer resistance

/~, = ~

(5 = ,~' = 0);

+ ~(~

+ ,,54)

LZL 1.

(PA

(b) Non-volatile A or B (trivial mA or m~),

%'a

TL ,/ ~2

(

mB @

53

mA +

54

mA "~L

(44)

fl3=zLfl*=za ma+~-~ ,

r/t = r / ( 1 + K ) ,

fa3 =fa~.

1

F=~l "~~]~3, fAl =fAu. (2) Negligible liquid-side mass transfer resistance (,S = 0), ~1

fl,

~G TL

TG ~L

= -za - 53a,,

F =

~G ~L

5~a2

It should be evident from eqs (43) and (44) that ~/L = ~/o = 1 at very large values of k ( ~ ~ ) when either A or B is non-volatile. When both A and B are volatile, it is obvious that flL=fa3/fA2 and r/o = fa3/fa i • For extremely small ( ~ 0) k, working with appropriate mass balances [eqs (14)-(16)], one can deduce that

~1a ~ a , ,

+

A o rna+ZG+ TL

(

~L ~

~1= m~ + - - +

~3 =

TL

, ~2 =

+

0~4 =

+

m A + "ca + "eL

TL ]

, (45,

[mAA~r+{~+~alALrlrL)l

,

r/o =

,

(48)

[ mAAGF+zaALF1,LA

fA2 ~fAu" (3) Reaction in liquid film neglected ( ~ = 0 0
[

~r Ao =

ZG

Za

#~ = n - - ~ l ,

~ Za

#~ = - - 5 1 ~ : ,

TL

P3 = - - - - ~ 3 ,

~L

ZG

~L

~0 ~L

TL

51 =

[ m B + - -1

O~

J

The following relations can be obtained from eqs (48) for k ~ 0 .

1

~4

{rG ] -~L+ (I+rnA
mAAGF +

r/o~<~/L< 1 f o r A a r > A L r ; r l O = 0 L = +

(Ds

~o ( m B + ra')l , ZL,/A

( 1 a2 =

~-~L)

m A +--

q>A+

+r/ m a + - -

,46)

tl

,

, 54=

ot3 =

[(1)

mB+

mA +

for

AaF = ALF;

1 (49)

0o ~> r/L > 1 for AGF < ALF ; flL = O0 when Aav # ALF only if 3 -, 0. When the reaction in eq. (1) is irreversible, eqs (43)-(47) reduce to (Parulekar and Saidina Amin, 1996)

4569

Gasqiquid reactions in well-mixed reactors fA=

0.2

1

[2mAAGF + ('~G/'~L)ALF]

I

A

[mAAGv + (Z~/ZL)ALF]

(50)

B

0.1s

0.5

if m A = 0

1

0.1 l; L

1 + ,--~a Z=

0

ifr~>0

rL~l+ma] l + TG[OA n J

0.0015

if 5 > 0

0.5

and ~ = 0 0

for 0 < y <

1.

0

In view of eqs (50), it should be evident that for an irreversible reaction, f/z = 1 and #o =fA, as k ~ oo. 5. NUMERICAL ILLUSTRATIONS

For a first-order reaction, the relevant material balances represent a set of linear algebraic relations among At, AL, BI, BL, AG (volatile A), and BG (volatile B), the solution to material balances in the liquid film being provided by eqs (18), (19), and (35). The zero gradient condition that applies for a species (A or B) when it is non-volatile generates an additional relation among At, AL, BI, and BL as follows: yB1 + 0./'1 = 0

if A is non-volatile. (51)

~oB17 = (1 - c%0)fl

if B is non-volatile. (52)

For the three situations under consideration, the relations to be solved simultaneously are as follows: (a) volatile A and non-volatile B (ms = 0) [eqs (14) and (15) for A, (20), (21), and (52)], (b) volatile A and volatile B [eqs (14) (J = A, B), (15) and (16) for A, (20), and (21)], (c) non-volatile A (mA = 0) and volatile B [eqs (14) (J = B), (15) for A, (20), (21), and (51)]. When the extent of reaction in liquid film is assumed to be negligible, the appropriate equations are solved using relations (22) in eqs (14)-(16). The concentration ratios for species J in phase P (Zje) defined below allow one to examine the impact of neglecting the reaction in liquid film on the performance of gas-liquid reactor. Zse

(J~)~

(JJ')NR

J = A, B;

P = G, L, I. (53)

A significant deviation of Zse from unity when concentration of species J in phase P is significantly non-trivial implies that confining the reaction solely to bulk liquid will lead to erroneous prediction of performance of gas-liquid reactors and incorrect design of these. The parameter values used in numerical illustrations to be discussed next are based on recommendations in prior literature (Doraiswamy and Sharma, 1984; Westerterp et al., 1984; Landau, 1992; Winkelman and Beenaekers, 1993) and are representative of typical experimental gas-liquid reaction systems. Illustrations for irreversible reaction are provided first. Unless mentioned otherwise, the

0.5 Y

1 0

0.5 Y

1

Fig. 4. Profiles of A and B in liquid film for y = 1.0 (i and ii) and 7 = 10.0 (iii and iv) for a first-order irreversible reaction when A is non-volatile and B is volatile. The parameter values employed are provided in eqs (54). The solid and dashed curves denote profiles obtained by considering the reaction in liquid film and neglecting the reaction in liquid film, respectively. [-These almost overlap each other in (ii) and (iv).] ZAL, ZBL, and ZBG were 0.994, 0.999, and 1.143, respectively, for 7 = 1 and 0.995, 0.999, and 1.004, respectively, for 7 = 10. The reaction in liquid film accounted for, ~, Be, and fa were 0.00769, 0.027, and 0.833, respectively, at ~, = 1 and 0.001, 0.029, and 0.998, respectively, at "/= 10. kinetic coefficient k is considered to be the variable parameter. Profiles of concentrations of A and B in liquid film are presented in Fig. 4 for the situation where A is non-volatile and B is volatile. The parameter values employed for generating these profiles were n = I, ALF = I, BLv = BGr = O, mA = O, mB = 0.066666, So = 2, q = 0.05, ~oa = 60, = 0.99, za = 0.1 ZL, and Da = 4.95 72.

(54)

As discussed in the preceding section, it is expected that (<< 1 and ( is a monotonically decreasing function of 7. Promotion of reaction in bulk liquid vis-a-vis reaction in liquid film with increasing k leads to decreases in AL and A~/AL, an increase in BL, increasing nonlinearity of the profile of B in the liquid film and gradual approach of ZBG to unity. The insignificance of reaction in liquid film vis-a-vis reaction in bulk liquid is evident from the closeness of ZAL, ZBL, and the effectiveness factors ~/L and #o (data not shown) to unity at both values of 7- For 7 = 1, ZB~ is significantly different from unity; BG however is much lower than BL. Most of B produced is retained in the liquid phase [XB -~ 0.02]. For the situations where A is volatile (B may or may not be volatile), profiles of ~ vs 7 are presented in Fig. 5 for two different liquid space times. The parameter values employed to obtain these profiles were n = l, A~p = I, ALp= BLr = Bar (if B volatile) = 0, ~ = 0.99, s0 = 2, and r / = 0.05, Da = 4.95 72 (profile 1) or q = 2, Da = 198 72 (profile 2).

(55)

4570

S. J. PARULEKARand N. A. S. AMIN

100

3B2

0.8

A 0.4

0 0.8

0.01

i

0.1

i

,

,

, i l l |

i

,

i

,

2

10

Fig. 5. Portraits of ( and 7 for the situations where A is volatile for two different liquid space times. The parameter values employed to obtain these portraits for a first-order irreversible reaction are provided in eqs (55). Profiles l and 2 share all parameters in common with the exception of q and Da/7 z.

The contrast in the two profiles demonstrates very clearly that the extents of reaction in the two portions of liquid phase depend not just on k and parameters associated with liquid film, as has been considered to be the case almost entirely in prior literature, but also on parameters, such as space time, associated with bulk liquid. The dependence of ( on ~,for pseudo-firstorder irreversible reactions is therefore not unique as widely purported to be in prior literature. The lower limit on ~ is attained as ~,--*0 and can be deduced, from eqs (42), to be (1 - c0[-I + 1/{2q}]/~. It has been reported that the reaction is completed entirely in the liquid film when ~ > 2 and in bulk liquid when 7 < 0.3 (Westerterp et al., 1984). The results in Fig. 5 provide an example where these criteria are not necessarily valid. For example, in the immediate neighborhood of = 0.3 (~, < 0.3), the extent of reaction in liquid film is not negligible vis-a-vis the extent of reaction in bulk liquid. By the same token, in the immediate neighborhood of ~ = 2 (y > 2), the extent of reaction in bulk liquid is not negligible vis-h-vis the extent of reaction in liquid film. Caution must therefore be exercised in relegating the reaction entirely to either liquid film or bulk liquid. When A is volatile and B is not (rn8 = 0), concentration profiles for A and B in liquid film are presented in Fig. 6 for two values of k. The parameters used for generating profile 2 in Fig. 5 were used for these illustrations. Additional parameters were assigned the following values. mA = 0.66666, ~0A = 30, re = 0.1 zL.

(iii)

0.4

ill

1 Y

,.

1 3

(56)

For both values of 3', AL was much less than A I (and therefore A t ) (AL = 0.00579 for ~ = 1.0 and AL = 0.00296 for 7 = 1.3). Despite the low values of AL, the extent of reaction in bulk liquid was greater than that in liquid film at both ~, = 1.0 (~ = 0.558) and y = 1.3 (( = 0.987). By observing the concentration profiles for A in Fig. 6, one may be tempted to consider the liquid-phase reaction to be confined essentially to liquid film. Such temptation would certainly lead to

0

0.5

1 0

0.5

Y

1

y

Fig. 6. Profiles of A and B in liquid film for ~/= 1.0 (i and ii) and 7 = 1.3 (iii and iv) for a first-order irreversible reaction when A is volatile and B is not. The parameter values employed are provided in eqs (55) and (56). The solid and dashed curves denote profiles obtained by considering the reaction in liquid film and neglecting the reaction in liquid film, respectively.

erroneous results. The concentration ratios ZAL , ZBL and Zao were 0.7835, 1.2260, and 0.9380, respectively, at 7 = 1 and 0.6742, 1.3401, and 0.9037, respectively, at y = 1.3. The fractional conversions of A, fa, predicted by considering reaction in liquid film and by neglecting reaction in liquid film were 0.27 and 0.219, respectively, at y = 1.0, and 0.3 and 0.221, respectively, at 7 = 1.3. These results provide ample evidence of the serious pitfalls that can result in prediction of performance of and design of gas-liquid reactors when the extent of reaction in liquid film is assumed to be negligible, in order to obtain simpler mathematical models, when such assumption is not warranted. In the absence of liquid-side diffusional resistance alone (6 = 0, 6' > 0), values offA for k values corresponding to 7 = 1.0 and 7 = 1.3 are 0.8293 and 0.8404, respectively, with the corresponding ~/L being 0.3232 and 0.3514, respectively. When the overall diffusional resistance is negligible (3 = 3 ' = 0), values of fa for k values corresponding to V = 1 and ~, = 1.3 are 0.9631 and 0.9778, respectively, with the corresponding F/o being 0.2783 and 0.302, respectively. Conversion of A to B is therefore limited by absorption of A. Indeed, at both values of k, XA was close to unity and qL was close to ~/L. The potential for increase in the extent of reaction via reduction in mass transfer resistances is evident from these results. Concentration profiles for A and B (both species volatile) are presented in Fig. 7 for two values of k. The parameters used for generation of profile 2 in Fig. 5 were used for these illustrations. Additional parameters were assigned the following values.

m A = 0.1, mB = 0.01, ~9A 150, ~0R = 300, and z~ = 0.1 zL. =

(57)

Just like for the results presented in Fig. 6, AL is much less than At (and therefore A6) for both values of y. When reaction in liquid film is fully accounted for, AL

Ga~liquid reactions in well-mixed reactors 0.4

A 0.2

<

0 0.4

0.8

4571

10

0.7

1=_ EA/ I

1

Zjp

...

I

5

0.6

0.1

0.5 0.9

0.01

E o

-5 0.1

B

1

Y

10

0.1

1

10

Y

5

0.7

0.2

B 2.5 0.5

0 0

0.5 Y

1

0

0.5 y

0

I

0

o.B3

0.74

B 0.72

0.8

I 0.2 Y

0.4 0

0.2 y

04

Fig. 7. Profiles of A and B in liquid film for 7 = 1.0 (i and ii) and 7 = 1.3 (iii and iv) for a first-order irreversible reaction when both A and B are volatile. The parameter values employed are provided in eqs (55) and (57). The solid and dashed curves denote profiles obtained by considering the reaction in liquid film and neglecting the reaction in liquid film, respectively. Parts v and vi, which are inserts for parts ii and iv, respectively (reaction in liquid film being accounted for) show that B undergoes a maximum in the neighborhood of y = 0 (y > 0).

is 0.00228 at 7 = 1.0 and 0.0011 at y = 1.3. The insignificance of AL in comparison to At should not be misconstrued to be the insignificance of extent of reaction in bulk liquid. In fact, the extent of reaction in bulk liquid was greater than that in liquid film at both Y = 1.0 (( = 0.558) and 7 = 1.3 (( = 0.987). The concentration ratios ZaL, ZnL, ZaG, and ZBa were 0.697, 1.066, 0.84, and 1.35, respectively, at Y = 1.0; and 0.567, 1.09, 0.768, and 1.51, respectively, at y = 1.3. The fractional conversions of A, fa, predicted by considering reaction in liquid film and by neglecting reaction in liquid film were 0.7 and 0.647, respectively, at 7 = 1.0; and 0.73 and 0.648, respectively, at 7 = 1.3. It is evident that assuming the liquid-phase reaction to occur solely in bulk liquid, when unwarranted, can lead to incorrect design of and erroneous predictions of performance of gas-liquid reactors. The predictive capability of a mathematical model that does not consider reaction in liquid film becomes poorer as the extent of reaction in liquid film is increased, i.e., as ( is increased, which in turn requires an increase in 7 and Da. When the liquid-side diffusional resistance alone is negligible (6 = 0, 6 ' > 0), values of fa for k values corresponding to 7 = 1 and Y = 1.3 are 0.9583 and 0.9621, respectively, with the corresponding ~/L being 0.7344 and 0.76, respectively. In the absence of gasand liquid-side resistances (3 = 6' = 0), values offa for k values corresponding to y = 1.0 and 7 = 1.3 are 0.9901 and 0.9941, respectively, with the correspond-

0.5

1

Y

Fig. 8. Portraits of Zje (JP = AG, AL, BG, BL) and y (a) and E (E = Ea, EB, E*) and y (b) for a first-order irreversible reaction when both A and B are volatile. The parameter values employed are defined in eqs (55) and (57) (q = 2, Da = 19872). The profiles of B in liquid film in (c) were obtained at Y= 10 and za = 0.001 zL, all other parameters being fixed at the values employed in (a) and (b). The solid and dashed curves in (c) denote profiles obtained by considering the reaction in liquid film and neglecting the reaction in liquid film, respectively.

ing f/o being 0.7109 and 0.7355, respectively. Absorption of A is therefore the rate limiting step in conversion of A to B. Indeed, XA was close to unity and qL was close to ~/L at both values of k. At the two values of k, 10% of B produced was desorbed. As the profiles of concentration ratios presented in Fig. 8 reveal, the various concentration ratios are equal to or close to unity at low y (or low (). These ratios deviate increasingly from unity as Yis increased. While EA increases with an increase in 7 and is a continuous function of 7, Ea exhibits discontinuity at a critical value of 7 (7*) for the system parameters under consideration (Fig. 8). For 7 < 7", BL > Bt; at 7 = Y*, BL = Bt; and BL < Bt for 7 > Y*. [E~ is positive for 7 < 7", negative for 7 > 7", and unbounded for 7=7* (EB~ as 7 - + 7 " - and E B ~ - - ~ as 7 ~ Y*+).] For the results presented in Fig. 7, E~ is negative at both 7 = 1 and 7 = 1.3. As k is increased, the transition in dispositions of concentration profiles for B in liquid film occurs as per the following sequence (alphabets refer to parts of Fig. 2): (c) (e), (f) (7 = 7"), and (g). Negative enhancement factors have been reported by Cornelisse et al. (1980) and A1Ghawas and Sandall (1988) for multiple complex reversible reactions, by Parulekar and Saidina Amin (1996) for multiple irreversible reactions, and by Winkelman and Beenackers (t993) for a single first-order reversible reaction at much higher 7 values than the ones where negative Es is predicted in Fig. 8. To the best of our knowledge, negative enhancement factors have not been reported for a single irreversible reaction thus far. Another illustration of negative enhancement factor is provided in Fig. 8(c). It should be evident from eq. (23) that a necessary condition for admission of negative enhancement

S. J. PARULEKARand N. A. S. AMIN

4572

1F

NOMR~

0.8

NR

0 1

~

0 . 4

fA 0.2

(d)

(c)

~L

0 1

5o

~' @0.8

0.8

0.8 1

XA

0.5 ~/s/ '

0~ 0.01

/

1

100 Da'

0.6 0.18

XB z

(e) 10000 0,01

I 1

I 100

~/ 0.12

(0

0.06 10000

Da'

Fig. 9. Profiles of(a, b)fA, (c) t/L, (d) ~/o,(e) X a, and (f) Xn as a function of k (or Da', Da' = eLzLk). The parameter values are listed in eqs (55), (57) and (58) (r/= 2). The legends NMR, NOMR, NR and R denote (i) negligible liquid-side diffusional resistance, (ii) negligible overall diffusional resistance, (iii)neglect of reaction in liquid film, and (iv) consideration of reaction in the entire liquid, respectively. K ~ ~ (irreversible reaction) in (a) and K = 1 in (b). In (c)-(f), the curves denoted by (- - -), (-.-.-.),and ( ) represent profiles for K = 1, 10 and ~ , respectively. factor for a species is the existence of local extremum in concentration profile of that species. It must be realized that the definition of enhancement factor Es is appropriate as long as dJ/dy at y = 0 and (JL - - J r ) have the same sign. The bulk liquid and interfacial concentrations of species J used in the definition of E j are based on concentration profile of species J in the presence of chemical reaction in liquid film. These concentrations may or may not be established if the reaction in liquid film were negligible. Thus, based on the definition of EB, one may conclude erroneously that for the concentration profiles for B in Figs 7 and 8, B (product of liquid-phase reaction) would be absorbed in the absence of reaction in liquid film (BI > BL), which is an unrealistic scenario. This controversial situation can be circumvented by obtaining concentration profiles for B by neglecting reaction in liquid film and solving the appropriate material balances. The profiles thus obtained (denoted by dashed lines in Figs 7 and 8) predict desorption of B just like the profiles obtained by accounting for reaction in liquid film do. Negative enhancement factors do not have any physical significance beyond indicating existence of at least one extremum in concentration profile for a species in liquid film. Unlike the enhancement factor, the augmentation factor [eq. (24)] will be positive for all 7 and a continuous function of 7. For the results presented in Fig. 8, E~ increases with increasing y just like EA does.

As anticipated, the fractional conversion of A, fA, increases with increasing k for each of the four scenarios under consideration (referred to by legends NMR, NOMR, NR and R in Fig. 9). At very low and very high values of k, fA, ~L, and ~/o approach their respective asymptotic values provided in eqs (48) and (50) and the statement immediately following eqs (50). It is evident that diffusional resistances (especially the resistance in the liquid phase for the parameters under consideration) lead to lower fA. Both effectiveness factors (OL and 9o) are significantly lower than unity at intermediate values of k with both exhibiting minima with respect to k (Fig. 9). The considerable potential for better utilization of A by reducing the mass transfer resistances is evident from these results. Relegation of the reaction to bulk liquid leads to underprediction of fa beyond a critical k, the extent of underprediction increasing with increasing k (Fig. 9). Both the fraction of A absorbed that is consumed in the reaction and the fraction of B generated that is desorbed increase with increasing k, the profiles of Xa and Xn being sigmoidal shaped curves. The increase in XA is more dramatic than the increase in Xn. Acceleration of reaction (1) leads therefore to better utilization of absorbed A and increased effectiveness of desorption of B (Fig. 9). Illustrations for reversible reaction are provided next. The variations in performance of a well-mixed gas liquid reactor when reaction in liquid film is accounted for or neglected with variation in k (all other parameters being fixed) are presented in Tables 1-3 for a first-order reversible reaction for the three situations under consideration in this article. The resuits in Table 1 are for non-volatile A and volatile B, K' being unity, with the remaining parameters employed being specified in eqs (54). As k (or 7) is increased, the fractional conversion of A is increased, as anticipated. An increase in 7 leads to decline in both (AL -- B L / K ) and (A~ - B I / K ) , the decline in the former being more rapid than that in the latter. Over the broad range of y investigated (1.0-100.0), ~ remains much less than unity. This was also observed to be the case for other values of K'. The concentration ratios ZaL and ZnL are therefore very close to unity over the entire range of 7 under consideration. Although ZBG deviates significantly from unity, it must be realized that this deviation is of no significance since B~ is much lower than AL and BL. For the entire range of 7 considered here, as was also the case with an irreversible reaction, the extent of reaction in liquid film can be considered to be negligible. For the parameters under consideration, a neglect of liquid-side diffusional resistance alone (6 = 0, 6' > 0) leads to marginally higherfA values than those presented in Table 1 (data not shown, ~/L is less than and very close to unity). As the entries in the last column of Table 1 reveal, the values of fA in the absence of overall mass transfer resistance (~ = 6' = 0) are significantly higher than the corresponding values offa in the presence of diffusional resistances on both sides. (0o is significantly less than unity.) The higher

4573

Gas-liquid reactions in well-mixed reactors Table 1. Results for a first-order reversible reaction with non-volatile A and volatile B (K = 1)

7

AL

BL

B~

~

ZAL

ZBL

Zs~

1.0 1.3 2.0 10.0 20.0 100.0

0.6110 0.5700 0.5290 0.4920 0.4910 0.4900

0.3790 0.4190 0.4580 0.4890 0.4900 0.4900

0.0144 0.0165 0.0196 0.0268 0.0281 0.0293

0.0136 0.0152 0.0187 0.0251 0.0257 0.0263

0.9966 0.9962 0.9952 0.9916 0.9908 0.9900

0.9996 0.9984 0.9965 0.9917 0.9908 0.9900

1.3044 1.3336 1.4637 1.8647 1.9520 2.0310

fa 0.3890 0.4300 0.4710 0.5074 0.5090 0.5098

A1

BI

fA*

0.5540 0.4960 0.4210 0.2771 0.2532 0.2324

0.1100 0.1270 0.1500 0.2056 0.2157 0.2246

0.4545 0.5115 0.5714 0.6227 0.6244 0.6250

* Negligible gas-side and liquid-side diffusional resistances.

Table 2. Results for a first-order reversible reaction with volatile A and non-volatile B 7'

AL

BL

A6

(

fA

1.0 1.3 2.0 10.0

0.4555 0.4629 0.4734 0.4912

0.4543 0.4623 0.4732 0.4912

0.8635 0.8612 0.8580 0.8527

0.7909 1.1292 1.6753 2.0018

1.0 1.3 2.0 10.0

0.1428 0.1509 0.1705 0.2348

1.3844 1.4874 1.6993 2.3474

0.7709 0.7543 0.7195 0.6127

0.5897 1.0933 2.5176 11.007

1.0 1.3 2.0 10.0

0.0392 0.0398 0.0451 0.0751

1.6887 1.8491 2.2184 3.7565

0.7408 0.5643 0.7167 0.9926 0.6605 2.7228 0.4253 50.857

ZBL

~('A)NR

K 0.0681 0.0694 0.0710 0.0737

ZAG

At

BI

,fA*

= 1 0.0636 0.0637 0.0638 0.0638

1.0714 1.0886 1.1130 1.1543

0.9897 0.9872 0.9835 0.9774

0.8408 0.8381 0.8343 0.8281

0.5936 0.6370 0.6980 0.7997

0.1149 0.1151 0.1153 0.1154

K = 10 0.2077 0.1763 0.2231 0 . 1 7 7 0 0.2548 0.1776 0.3521 0.1780

1.1781 1.2606 1.4352 2.0984

0.9574 0.9372 0.8945 0.7619

0.7327 0.7133 0.6728 0.5481

1.7319 2.0010 2.5646 4.3027

0.5540 0.5589 0.5630 0.5659

K 0.2533 0.2774 0.3328 0.5635

1.2107 1.3194 1.5764 2.6617

0.9430 0.9130 0.8420 0.5420

0.6976 0.6695 0.6039 0.3295

2.0998 2.4786 3.3643 7.0796

0.8392 0.8503 0.8599 0.8668

= 50 0.2092 0.2102 0.2111 0.2117

* Negligible gas-side and liquid-side diffusional resistances.

Table 3. Results for a first-order reversible reaction with both A and B being volatile

7'

AL

BL

A~

BG

~

1.0 1.3 2.0 10.0

0.2460 0.2478 0.2505 0.2552

0.2452 0.2474 0.2504 0.2552

0.4793 0.4731 0.4643 0.4497

0.2947 0.3167 0.3477 0.3998

0.8690 1.2734 1.9778 2.4722

1.0 1.3 2.0 10.0

0.0566 0.0570 0.0588 0.064l

0.5470 0.5607 0.5852 0.6411

0.3345 0.3134 0.2741 0.1821

0.6188 0.6885 0.8194 1.1272

1.0 1.3 2.0 10.0

0.0145 0.0137 0.0135 0.0150

0.6137 0.6313 0.6645 0.7478

0.3030 0.2780 0.2287 0.1021

0.6884 0.7703 0.9325 1.3510

fa

(fA)NR

ZBL

ZA~

ZBG

At

BI

fA*

K=I 0.2747 0.2581 0.2791 0.2585 0.2852 0.2588 0.2952 0.2590

1.0339 1.0418 1.0529 1.0722

0.9522 0.9401 0.9229 0.8939

1.4081 1.5111 1.6573 1.904l

0.4620 0.4555 0.4465 0.4313

0.3045 0.3273 0.3593 0.4132

0.3536 0.3541 0.3545 0.3548

0.5982 1.0335 2.6185 13.1170

K=10 0.6089 0.5621 0.6296 0.5638 0.6671 0.5652 0.7538 0.5662

1.0590 1.0823 1.1267 1.2322

0.8720 0.8190 0.7170 0.4770

1.3580 1.5061 1.7880 2.4550

0.3123 0.2905 0.2499 0.1548

0.6394 0.7114 0.8467 1.1648

0.8391 0.8419 0.8444 0.8461

0.5660 0.9968 2.7473 60.1830

K= 0.6826 0.7084 0.7577 0.8829

1.0640 1.0906 1.1450 1.2860

0.8477 0.7790 0.6430 0.2870

1.3523 1.5081 1.8207 2.6327

0.2797 0.2539 0.2030 0.0722

0.7114 0.7960 0.9636 1.3961

0.9557 0.9594 0.9626 0.9648

50 0.6279 0.6299 0.6317 0.6329

* Negligible gas-side and liquid-side diffusional resistances.

conversion is attributable to p r o m o t i o n in desorption of B that accompanies the reduction in gas-side diffusional resistance. The results presented in Table 2 pertain to volatile A and non-volatile B, with the parameter values being specified in eqs (55) and (56) ( r / = 2) and the relation between 7 and Da being

Da = 198 7'2,

/ 7 = 7'X/ 1 + 2 , K = K'.

7' = ~/DA L , (58)

The results presented in Table 3 and Fig. 10 pertain to volatile A and volatile B, with the p a r a m e t e r values

4574

S. J. PARULEKARand N. A. S. AMIN 3.5

(i) 2

ZAL 0.5 1 0.9

0.8

1.5

(ii)

0.7 1.45 1.3

EA" 1.15 1 3

(c)

ZAL

t

1 0.5

0

1

I

I

1

10

100

1000

l

I

10000 100000

K EB* 2

1 0.01

0.1

1 K

10

100

Fig. 10. Portraits of (a) (, (b) t/o, (c) E*, (d) E~ and K for a first-order reversible reaction when both A and B are volatile. Profiles 1, 2, 3, and 4 are for y' = 1.0, 1.3, 2.0, and 10.0, respectively. The parameter values employed are provided in eqs (55), (57), and (58).

being specified in eqs (55) and (57) (q = 2) and the relation between y and Da being provided by eqs (58). Comparison of variations in ~ with variation in k (or 7') observed in Tables 2 and 3 and Figs 5 and 10 reveals that the increase in ( with an increase in y' is much less prominent for a reversible reaction vis-h-vis an irreversible reaction. Further, the lower is the equilibrium coefficient for the reversible reaction, the lesser is the variation in ( with variation in y' (i.e., lower is the d(/d7'). With the reaction in liquid film being properly accounted for, fa increases with increasing y' at lower values of the same and saturates to an equilibrium value at very high y' (see also Fig. 9). A comparison of the fractional conversions of A attained when B is non-volatile and B is volatile for fixed y' and K reveals that a larger fraction of A fed is consumed in the reactor when some of the B desorbs into gas phase (volatile B) since such removal of B increases the driving force for the liquid phase reaction. As argued earlier, an increase in K' (or K since n = 1) facilitates conversion of A to B. For the results presented in Tables 2 and 3 and Figs 6 and 7 therefore, the concentration ratios for the same k (or 7') deviate increasingly from unity at higher values of K ' (Fig. 11). Neglecting the extent of reaction in liquid film thus becomes increasingly risky as the transformation of A into B is facilitated. The substantial differences observed in

Fig. 11. Portraits of ZaL and K (or K') for different k (or ~,') for the situations involving (i) volatile A and non-volatile B and (ii) volatile A and volatile B. The parameter values employed for (i) are provided in eqs (55), (56), and (58) and those for (ii) are provided in eqs (55), (57), and (58). Profiles 1, 2, 3, and 4 are for y' = 1.0, 1.3, 2.0, and 10.0, respectively.

fractional conversions estimated by accounting for reaction in liquid film and by neglecting it for K' = 10, 50, and ~ are clear indicators of this. For fixed k, while ZsL, Zso and Za~ vary monotonically as K ' is varied, the variation in ZaL is non-monotonic (Fig. 11). Substantial deviations of ZaL from unity on both sides of unity are observed for higher values of k. A comparison of the fractional conversions of A when the gas-side and liquid-side diffusional resistances are accounted for (fA) and when these are neglected (f~a) [6 = ~' = 0] in Tables 2 and 3 reveals that ~/o is significantly less than unity (see also Figs 9 and 10). It is interesting to observe that while for lower k's, ~/o and ~/L decrease with increasing K, the trend is reversed at higher k values (Figs 9 and 10). The augmentation factors for absorption of A and desorption of B both increase with increasing k (or ~,') (Fig. 10). For the results presented in Table 3, both XA and X~ [eqs (30)] increase with an increase in k (for fixed K), the increases being higher, the higher the equilibrium coefficient (Fig. 9). When both A and B are volatile, fa increases with increasing k for each of the four scenarios under consideration (designated by legends NMR, NOMR, NR and R in Fig. 9). The results presented in Tables 1-3 share certain common characteristics. The driving force for the reaction in bulk liquid, ( A L - BL/K), is much less than that at the gas liquid interface, (At - BI/K). In fact, for each entry in these tables, the driving force ( A - B/K) was non-negative and decreased monotonically with increasing y. In each of the three situations considered here, (AL -- BL/ K) decreased faster

4575

Gas liquid reactions in well-mixedreactors 0.4

(ii) with decreasing K when A is non-volatile. When both A and B are volatile, the presence of mass transfer resistances leads to lower equilibrium conversion of A (Table 4).

\\ DF 0.2

6. CONCLUSIONS

o 0.01

\\\\\

I

0.1

I

y'

10

Fig. 12. Portraits of the driving force for reaction (1) in bulk liquid, DF ( = A L - BL/K), and k (or 7') for volatile A and volatile B. The dashed and solid curves represent profiles for K = l and 50, respectively.The parameter values employed are listed in eqs (55), (57), and (58).

to zero than (At - B I / K ) as k was increased. Beyond certain k, A and B would be nearly at equilibrium in bulk liquid (AL ~--BL/K) (Fig. 12). An increase in k from a lower value to this critical value is accompanied by a rapid increase in (. Any increase in k beyond the critical value leads to, for the parameter values considered here, establishment of near equilibrium between A and B in the liquid film (A ~ B / K ) in a neighborhood ofy = 1. As k is increased beyond the critical value, the reaction is confined to a shrinking portion of liquid film in the neighborhood of the gas-liquid interface. In the limiting case of extremely high k (k ~ oo ), A and B are in equilibrium in the entire liquid phase, a situation that corresponds to an instantaneous reversible reaction. The profiles of A and B in liquid film are linear in this special case. In consistency with the observations made above and the results in Fig. 12 [(A L - - B L / K ) > 0 and (AL -- B L / K ) ~ 0 as k --, oo ], for the parameter values considered in obtaining the results presented in Tables 1-3 and Figs 9-11, inflection points in the interior of liquid film were not observed in any of the three situations. All other parameters being fixed, it is anticipated that the equilibrium conversion of A (k ~ ~ ) will be lower, the lower the equilibrium coefficient for the reaction owing to a reduction in the driving force for the reaction, (A - B / K ) , that accompanies a reduction in K' (Table 4). Neglect of reaction in liquid film leads to consistent underprediction of equilibrium conversion of A. On a percent basis, the underprediction becomes more severe (i) with increasing K when A is volatile and supplied solely through gas feed and

A fresh perspective on modeling of gas-liquid reactors has been provided here considering the liquidphase reaction to occur in the entire liquid phase. Three different situations were considered depending on whether or not the reactant and the product are volatile, at least one of these species being constrained to be volatile. A single liquid-phase reaction was considered and the bulk gas and bulk liquid-phases were considered to be perfectly mixed. Dispositions of concentration profiles for reactant and product in liquid film were obtained for an irreversible reaction with arbitrary kinetics. Some thoughts on admissibility of additional dispositions for a single reversible reaction were presented. A first-order irreversible/reversible reaction was considered as a specific example for further analysis. Both analytical and numerical results presented in the article demonstrate that the relative importance of extent of reaction in liquid film vis-i~-vis extent of reaction in bulk liquid depends not only on parameters associated with liquid film (as has been considered to be the case in the extensive body of prior literature on gas-liquid reactions) but also on parameters associated with bulk liquid. Solutions of mathematical models that account for chemical reaction in liquid film and neglect the reaction in liquid film for identical process parameters revealed that the concentrations of reactant and product in the bulk gas and bulk liquid and at the gas-liquid interface and fractional conversion of the reactant predicted by the two models can be significantly different for reactions that are neither very slow nor fast. Negative enhancement factors for product species (ER) were predicted in certain cases for even irreversible reaction. The phase portraits of E, and kinetic coefficient k (or 7) were observed to be discontinuous in these cases. Phase portraits of augmentation factor, which was proposed as an alternative to enhancement factor in such cases, and 7 did not exhibit any discontinuity. The numerical illustrations revealed that assumption of chemical reaction to be confined solely to liquid film or bulk liquid can lead to serious pitfalls in design of and prediction of performance of gas--liquid reactors. The effectiveness of the positive ordered liquidphase reaction is reduced by resistance to diffusional mass transfer in the two phases. While a perfectly

Table 4. Equilibrium conversions [k ~ ~ , eqs (43)-(47)] for the results presented in Tables 1-3 ma = 0

1.0 10.0 50.0

0.625 0.943 0.988

ms = 0

0.505 0.911 0.981

0.115 0.566 0.867

ma > 0, mB> 0

0.064 0.178 0.212

0.305 0.804 0.952

0.355 0.846 0.965

0.259 0.566 0.633

4576

S. J. PARULEKARand N. A. S. AMIN

mixed reactor was considered in this study for simplicity, one can anticipate much more deviation from reality in predictions of mathematical models that represent gas-liquid reactors that are not perfectly mixed (such as plug flow and axial dispersion models) and confine a priori the reaction to bulk liquid or liquid film without adequate justification.

Zjp

Greek letters

0~0, ~ ,

Acknowledgment--The authors gratefully acknowledge the financial support received by one of us (N.A.S.A.)in the form of a federal fellowship from the Universiti Teknologi Malaysia.

NOTATION

A A1,BI

concentration ratio for species J in phase P, defined in eq. (53)

/~, F Z Ao

fraction of total liquid in the reactor that makes the bulk liquid, defined in eq. (8) dimensionless parameters defined in eqs (12) defined in eqs (45)-(47), i = 1, 2, 3, 4 defined in eqs (44)-(47), i = 1, 2, 3, 4 defined in eqs (50) defined in eqs (48) thickness of liquid-side film, m thickness of gas-side film, m liquid-phase holdup liquid-phase utilization factor, defined in eq. (25) defined in eqs (47) effectiveness factor for liquid phase reaction, defined in eq. (27) effectiveness factors with negligible liquidside mass transfer resistance and negligible overall mass transfer resistance, respectively, as the basis defined in eqs (35b) defined in eqs (58) defined in eqs (11), J = A, B defined in eqs (37) defined in eqs (26c) defined in eqs (17) space times for gas and liquid phases, respectively, (ratio of volume of empty reactor to volumetric flow rate of gas or liquidphase, as appropriate), s ratio of total rate of reaction in liquid film to total rate of reaction in bulk liquid, defined in eqs (10), (17) and (37)

reactant ~L defined in eqs (35b) qL a specific gas-liquid inteffacial area, m 2 / m 3 liquid A, B dimensionless concentrations of A and B, respectively, defined in eqs (11) B product ?~L, ?]o C non-volatile coreactant for reaction (1) c~ basis concentration, kmol/m 3 concentration of species J in liquid-phase, G kmole/m 3 7,0 Da defined in eqs (36) 7' Dj~ diffusivity of J in gas phase, m2/s ~j DjL diffusivity of species J in liquid phase, mZ/s Ej enhancement factor for transport of species p j, Oj J across the gas-liquid interface, defined in eq. (23) ~G, "eL E~ augmentation factor for species J, defined in eq. (24) fractional conversion of A, defined in eq. (29) f~ upper limit on fA, defined in eqs (43)-(47) f~ and (50) defined in eqs (43)-(45) and (47), j = 1, 2, 3 fnj defined in eqs (19) A,f~ Hs Henry's law constant for species J, Subscripts m 3 kPa/kmol b bulk liquid (x = 6, y = 1) J A,B F feed K defined in eqs (35b) G, g gas phase K' equilibrium constant for first-order reversI gas-liquid interface (x = y = 0) ible reaction J AorB kinetic coefficient for the forward step in L, l liquid phase reaction (1), s-1 NR reaction in liquid film neglected kjG gas-side mass transfer coefficient for species R reaction in liquid film considered J, kmol/kPa.m2.s 6 = 0 negligible liquid-side diffusional resistance mj defined in eqs (12) n stoichiometric coefficient REFERENCES partial pressure of species J in bulk gas, kPa PJ A1-Ghawas, H. A. and Sandall, O. C., 1988, Modeling the qL defined in eqs (12) and (36) simultaneous transport of two acid gases in tertiary R universal gas law constant, kPa m3/kmol K amines with reversible reactions. Sep. Sci. Technol. 23, r net rate of transformation of A into B, 1523-1540. kmol/m 3 s Astarita, G., 1967, Mass Transfer with Chemical Reaction. rj Elsevier, Amsterdam. rate of consumption of species J, kmol/m 3 s Astarita, G., Savage, D. W. and Bisio, A., 1983, Gas Treating T reactor temperature, K with Chemical Solvents. Wiley, New York. X distance from gas-liquid interface, m Cornelisse, R., Beenackers, A. A. C. M., van Beckum, F. P. H. XA, Xn defined in eqs (30) and van Swaaij, W. P. M., 1980, Numerical calculation of y dimensionless parameter, defined in eqs (12) simultaneous mass transfer of two gases accompanied by

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Kulkarni, B. D. and Doraiswamy, L. K., 1975, Effectiveness factors in gas-liquid reactions. AIChE J. 21, 501-506. Landau, J., 1992, Desorption with a chemical reaction. Chem. Engng Sci. 47, 1601-1606. Mann, R., 1983, Absorption with complex reaction in gas-liquid reactors, In Mass Transfer with Chemical Reaction in Multiphase Systems. Vol. I: Two-Phase Systems. (Edited by Alper, E.), pp. 223 289. NATO ASI Series, Martinus Nijhoff Publishers, Hague, Netherlands. Parulekar, S. J. and Saidina Amin, N. A., 1996, Complex gas liquid reactions: feedback from bulk liquid to liquidside film. Chem. Engng Sci. 51, 2079-2088. Westerterp, K. R., van Swaaij, W. P. M. and Beenackers, A.A.C.M., 1984, Chemical Reactor Design and Operation. Wiley, Chichester. Winkelman, J.G.M. and Beenackers, A.A.C.M., 1993, Simultaneous absorption and desorption with reversible first-order chemical reaction: analytical solution and negative enhancement factors. Chem. Engng Sci. 48, 2951 2955.