- Email: [email protected]
Mass transfer with heterogeneous and first order homogeneous reaction using the film theory
MASS TRANSFER WITH HETEROGENEOUS AND FIRST ORDER HOMOGENEOUS REACTION USING THE FILM THEORY LAWRENCE
R. STOWEt
and JOSEPH
A. SHAEIWITZ*
Department of Chemical Engineering. Urdversity of Illinois, Urbana, IL 61801, U.S.A. (Received 5 February 1982; accepted 5 August l!%Z) Ah&act-The problem of mass transfer with both tirst order irreversible homogeneous and an arbitrary heterogeneous reaction has been analyzed using the film theory, subject to the assumption of a linear profile of reactant between the bulk phase and the heterogeneous reaction surface. The enhancement factor is shown to depend on the product of two dimensionless groups. One is analogous to the usual group arising from mass transfer and first order irreversible. reaction, and the other is a Damkiihler number which describe8 the transport of the reactant to and reaction on the surface. The result is discussed in the context of solubilization of an insoluble liquid amphiphile by an aqueous surfactant solution. A maximum in the flux enhancement is observed because of the resistances associated with transport of the reactant species to the reaction surface. The field of mass transfer with chemical reaction has traditionally been divided into two distinct subgroups: mass transfer with heterogeneous reaction and mass transfer with homogeneous reaction. In general, these two phenomena have been treated separately, probably because the physical situations of interest have involved only one type of reaction. Mass transfer with heterogeneous reaction arises most often in gas/solid non-catalytic reactions [ 11, although the solubilization of insoluble amphiphiles by a surfactant has also been modeled successfully using these concepts[24]. The resistance in series concept is usually associated with these problems; that is, the largest resistance or slowest step is rate determining. Heterogeneous reactions most often appear as boundary conditions in continuum descriptions [5]. Mass transfer with homogen’sous reaction arises most often when gas adsorption is augmented by reaction with a species in the liquid phase[6], although transport across membranes containing mobile carriers also involves this phenomenon[7]. There are several theories for mass transfer with chemical reaction[S]. The film theory, the simplest, assumes a stagnant layer near the interface between the two phases under consideration, within which all of the reaction occurs. The penetration theory is the unsteady state analog in which liquid elements are assumed to bc continuously brought to the interface where the species from the other phase may enter. The surface renewal theory extends this to allow for a distribution of lifetimes of surface elements near the interface. One can also envision boundary layer theory being used to describe this phenomenon. An important observation occurs for the case of mass transfer with irreversible first order homogeneous reaction. This is that the enhancement factor, the ratio of the mass *Author to whom correspondence should be addressed. tPre8ent address: Mobil Research and Development poration, Dallas, TX 75236, U.S.A.
Cor-
transfer coefficient with reaction to that without chemical reaction, is almost exactly equal for the film, penetration and surface renewal theories. Each of these theories predicts a different mass transfer coefficient, but the same enhancement. Therefore, the simplicity of the film theory makes it attractive for modeling these problems. In continuum models, homogeneous reactions appear as source terms in the differential equations [5]. In this paper, we treat the problem of mass transfer with both homogeneous and heterogeneous reaction. Specifically, we examine the case where a reactant diffuses to a surface where it undergoes a process that can be described as a heterogenous reaction, and as the product diffuses away from the interface, it reacts with the reactant by a fust order irreversible reaction. An analytical solution is obtained by making the assumption of a linear concentration profile of reactant. The physical situation which has given rise to this problem is the solubiliiation of an insoluble liquid amphiphile by an aqeuous surfactant solution at high ionic strength; although , one could certainly imagine this physical situation arising in a catalysis context where a product reacts with the reactant as the former leaves the catalyst surface. TliF.OKETtCALBACKGROUND A continuum statement of mass transfer with heterogeneous reaction might involve the differential equation
and boundary
conditions C, = C,,
j=-&%
at x = S,
I =- k'CAoCao X=”
at x = 0.
(2) (3)
Boundary condition (3) states that species B is consumed at the interface by a reaction with A, although for the 635
L. R. STOWEand J. A. SHAEMTZ
636
purposes of this discussion, A will be considered to be present in excess, and CA0 will be assumed constant. The resulting linear concentration profile is
in excess, and k* = kCB,, then the solution and either (lOa) or (lob) is the same
L/2 ke = $
(4)
and the steady state flux is obtained by solving for CBO at x = 0 in eqn (4) and plugging that value into eqn (3), yielding k’CnoCe..
-j=
1
+
(5)
k’C.408’ DB
If we define ka = 0,/S as a film theory mass transfer coe5cient, and k’C,, as a pseudo first order rate constant, then it can be seen that the pertinent dimensionless group is a Damkohler number, the rate of reaction divided by the rate of diffusion, k,lk’&. This same result can be obtained by equating the fluxes in series j = kB(C,,
- C&
= k’C,&&,
- k”C,,
= &&,,.
In this case, the ratio of the reverse to the forward rate constant appears as a partition coefficient in the series resistance expression. Equation (7) can also be obtained from a continuum description like eqns (l)-(5), but the present method is much simpler mathematically. Further examination of eqn (6) indicates that for a mven flux, as kB is increased, C,,C,, decreases. Therefore, for a given flux, a large mass transfer resistance means that Y = C,,/C,, will be very small, and a small mass transfer resistance means that Y -_, 1. A continuum statement of mass transfer with homogeneous reaction might be
EO
e = [k*&1”2 ~ kg
x=6,
(lOa)
dC,=Ratx=o
to be a constant,
group
for this
The essential features of this model include diffusion of reactant, B, to the interface. At the interface B reacts with an excess of A to form E. Then as E diffuses away from the the interface it reacts irreversibly with B to enhance the flux of E back into solution. Figure 1 illustrates this physical situation. A key assumption in this development is that the profile of B is always linear within the film. The equations for this situation are then d2C, &z-0.
_
(13)
D$$j-kC&=O, with boundary
(14)
conditions dCe=-R B dx
atx=O,
(15) (16)
~%z$z~~n*r~eX=o
Bulk
If CB is assumed
(12)
MASS TRANSFER WITE BOTH BOMOGENBOUS AND EETEROGENEOUSREAcllON
e---------_--fO_X.6 c0m
or
E dx
(11)
E
(9)
CE = Cso at x = 0,
-D
dimensionless
Ce=CsOatx=O,
conditions &=Oat
coth [w].
coth [w].
It is seen that the pertinent phenomenon is [k*De]“‘/kB
-D
with boundary
= [k*D,]“*
It is not obvious, a priori, that the solution would be the same for either (lOa) or (lob). However, this is seen to make intuitive sense in the following manner. For a given constant flux, R, increasing k* means that Ceo will drop. On the other hand, if C’,, were constant, increasing k* would mean that R would increase. A fixed surface reaction rate is typical of problems where the rate is controlled by the surface reaction, and a fixed surface concentration is typical of problems where the surface reaction is in rapid equilibrium. By usin8 the lilm theory definition for kg= D&S, where the superscript indicates a mass transfer coe5cient without reaction augmentation, an enhancement factor may be defined as 9 = kJk.& and this ratio becomes
(6)
Equation (5) is obtained by using the irreversible reaction only with the first diffusion step. If the reaction is assumed to be reversible, and product E is now diffusing back into solution containing no E,
to (8) and (9)
(lob)
CBm; that is, present
solution
E
Fig. 1. The physical situation of steady state mass transfer with homogeneous and heterogeneous reaction described by the film theory. The profile of reactant B is assumed to be linear with the film.
Mass transfer using the Blm theory Ce=Oat
x=6,
(17)
Ca=Ce-atx=S.
(18)
The term R in eqn (15) represents the surface reaction rate. A similar problem, with a zero flux condition in place of eqn (15), and with the reaction source term present in eqn (13) has been presented by van Krevelens and Hoftijzer [9], although only an approximate solution was available. The advantage of this formulation of the problem is that an analytical solution is available which should at least provide insight into the appropriate dimensionless groups for this physical situation. The other implications of the assumption of a linear profile of B will be discussed later. Equation (13) can be solved for Ce using conditions (15) and (I 8), yielding
This result can be inserted
Equation
(20) can be rearranged
3
=O.
variable
definitions,
eqn (25) can be
+C2ie113 I_ ,&$[g~)].
(27)
Applying the boundary conditions of eqns (16) and (lS), the constants C, and C2 are related through C,
i”” = -
C,j-
113
~-l/,(M) Im(Jw ’
(28)
where
Applyin the boundary condition in eqn (17) and using eqn (28), eqn (27) can be solved to produce
into eqn (14) to produce Cam+&(x-8)
D,f$f$-kCe
637
plus the previous converted to
c&‘/s
(20)
=
to obtain where
dZC, z-(PfQx)Ce=O,
(30) - F&&?i
Ca = Gee at x = 0,
(21)
where
GO _kceo ,,2 Ln(N) c DE 3
2
N=@ [
1
kC,, =, DB
and
F
=
IdM)
I-,,,(M)
Finally, eqn (27) can be written as p _ k(Cac.s - Rs/Dd DE
(22)
Rk Q= D,D,’
(23
and
Equation Equation (21) can be converted to a linear differential equation by the variable substitutions LY= P + Qx and C.&x) = W(a). With these substitutions, eqn (21) becomes d2W w-$=0. Following
Kamke[lO],
the solution to equation
(24) is
(31) can be differentiated
to produce
(32) At steady state, the flux at the interface is given by -D
!%
(33)
E dx I,_o=R-
So eqn (33) can be combined with eqn (32) to provide the product mass transfer coefficient, kE, where Z,,,s represents a modified Bessel function of the first hind and of order + l/3. Using the following relationship, obtained from eqns (19), (22) and (23), [P+Qx]=G
n_ -r
9
(26)
Lkc
1
l,,2
D
BO
E
,
(34) where in terms of the system parameters
L.R.
638
STOWI?
and J.A.
.%AElWlTZ
Cm,, in eqn (40) and rearranging to
and (36) DISCUSSION
Two observations are now apparent. First of all, eqn (34) is the same form as eqn (ll), with a different form for the multiplicative factor. The analogous form of the enhancement factor in eqn (12) is
I-,,&‘0
- Fl,n(N)
I
W&M”2 k_E
.
(37)
It is seen that the Bessel function ratio, with its dimensionless parameter, N, has replaced the coth term; and that a pseudo-first order rate constant, kCao, based on the surface concentration of B, appears both in N and in the other terms in eqn (37). The second observation concerns the form of the dimensionless parameter, N. Equation (35) can be rearranged as follows
NC2
y=
R 1-k&.2,,’
(41)
Tbe value of Y can be calculated from eqn (41) by knowing C’,,, knowing the flow geometry and hence k,, and measuring the steady state value of R. Figures 2 and 3 show the enhancement factor as a function of N for dilTerent values of Y. These graphs have substantially the same form as the one for mass transfer with 6rs.t order irreversible homogeneous reaction in that at low values of N, the enhancement factor is unity, and that as N increases the enhancement factor rises significantly. [81 For values of N which are not on these figures, a linear extrapolation is all that is needed. This is because above a value of N = 13.4, the Bessel function ratio approaches - 1, which again is completely analagous to the homogeneous reaction case since the coth term in eqn (12) approaches unity as its as its argument exceeds a value of 2.6. It can also be seen from Figs. 2 and 3, that a smaller
(kCmD,)"z D,JS
3 E
&IS
RIG
3’
For example, if R = k’C,,, then RI&, = k’, meaning that the ratio R/C,, is a measure of the rate of surface reaction. Therefore, the parameter N is seen to be a combination of the parameter associated with diffusion and homogeneous reaction, (kC,,DE)‘n/k& and the Damkohler number associated with diffusion and heterogeneous reaction, k,Jk’. The parameter M is merely the same group with the concentration evaluated at the bulk instead of surface value. Traditionally, mass transfer with first order reaction has been treated by plotting the enhancement factor, Ib: against the group equivalent to (kC&&)“‘/k& In this case, however, a second group is necessary to fully describe the physical situation. While the group k,J(R/CBo) may be a tempting choice, it is not a very practical one since the surface concentration of reactant, CaO, is not likely to be known; although, the surface reaction rate, R, is likely to be known from experiment and the assumption of steady state. If the first two resistances in series, diffusion of B to the surface and the surface reaction, are examined k,(C,,-
CA=
R.
(39)
and this is rearranged to
N
Pi. 2. Enhancement factors for mass transfer with homogeneous and heterogeneous reaction, low values of Y = C&C,,.
y=0.5 Y=o.4/y.a6Y=o.7
Y=O.8
lw: Y=O.9
3
2
k+l)=R/Go,
(40)
where Y= CaOICB,. it is seen that Y is a most appropriate representation of the Damkohler group. As discussed previously, for a given reaction rate at the surface, Y is also a measure of the the rate of diffusion to the surface to this rate. The advantage of the parameter Y is that it can be obtained from known quantities in an experimental situation by substituting for
.A 0
4
8
I2
16
N
Fig. 3. Enhancement factors for ma.ss transfer with homogeneous and heterogeneous reaction, high values of Y = C,,,/C,,.
Mass transfer using the film theory value of Y gives a larger enhancement factor for a given value of N. A small value of Y means that Cgg and CS, are very different, so at steady state, this corresponds to a small mass transfer coe5cient or a slow diffusion step. Finally, a brief comment on the assumption of a linear profile in B is in order. The major question is how good is this assumption. First of all, since it allows an analytical solution it is a valuable assumption if only to permit identification of the pertinent trends and dimensionless groups. Secondly, upon further examination, there are perfectly reasonable physical situations for which this assumption might be valid. One is if Y is large, meaning a large mass transfer coefficient of reactant toward the surface. The linear assumption is valid because as Y gets larger, the concentration profile between the edge of the lilm and the surface becomes flatter, and can more reasonably be approximated as linear. Another physical situation for which the linear assumption might be valid is if the diffusion process of reactant relaxes much faster than the homogeneous reaction proceeds. This corresponds lo reasonably slow homogeneous reaction rates and hence, lower values of N. This is particularly interesting since low values of N and Y can yield large enhancement factors. Either way, the trends shown in Figs. 2 and 3 do illustrate the correct qualitative behavior and should be very useful approximations. APPLICATION
TO SOLUBILIZATION
Our interest in this problem arose because of certain features that were observed pertaining to the solubilization dynamics of oleic acid by aqueous surfactant solutions of sodium taurodeoxycholate, especially at higher ionic strengths induced by either by added salt or large taurodeoxycholate concentrations. The major observation was that the flux could be enhanced up to one order of magnitude above a reasonable description of the diffusion limit as salt was added at constant taurodeoxycholate concentration, while the solubility only doubled over this range. Therefore, a reduction in one of the resistances with added salt was indicatedllll. Augmentation by a process describable as a homogeneous reaction can explain this phenomenon. If the overall mass transfer coefficient in eqn (7) is compared to an enhanced value, with the enhancement being in the diffusion of E away from the surface, we obtain an enhancement ratio
ilCL%w -= i”/Caco
1 $+-+ ts k’C,, 1
ka
I
k” 1 ( KC,, > b 1 + - k” _1 ’ ( k’C.m ) @kg k’C.m
639
1 l] suggest typical values for this physical situation to be DB = 5 X 1OP cm*/sec, Ds = 1 x 10-e cm%ec, Y= Re”= = 75, C,, = 0.02 g/cm’, 7.69 X lo-” cm’/sec, k’ C..,o w k”, with kB and kg being determined from a mass transfer solution from the hydrodynamics of the experimental apparatus[12]. The value of k’CAO, is obtained from experiment by measuring the flux -=R
1
CEl,
(43)
1 L+kB k’C,,+&
and then solving for (44)
Once PC,,, is known, it can be inserted into eqn (42) to calculate the enhancement ratio. Figure 4 illustrates the observed enhancement ratio as a function of the homogeneous rate constant, the only other variable which was not measured. Again the largest enhancement is observed for the smallest Y value and the largest homogeneous rate constant (essentially large N). A maximum enhancement is observed, which corresponds to $ becoming so large in eqn (42). and that particular resistance disappearing entirely. For this problem, this maximum enhancement ratio is five, but tbis value as well as the magnitude of the homogeneous rate constant necessary to achieve this maximum, is specific to our parameter values, only the trend toward a maximum enhancement is universal. We may now return to eqn (42), but consider the result which would be obtained if the linear assumption on reactant B were not made. In that case a reaction source term would appear in eqn (13), and an enhancement factor would be computed which would increase the value of I
t
1
I
_,-----Y=o.o5
5-
r
0
r
-
// Y=O.2
-
Y =0.4 Y=O.6
-
(42)
The first observation is that if the thud resistance is not rate determining, then there will be no flux enhancement. In this particular case, it is assumed that an emulsion droplet forms at the interface which decomposes into the micellar solution as it leaves the interface. Since this droplet is likely to be large compared to a micelle, it will probably diffuse rather slowly and the decomposition will enhance a slow step. Experiments which have been described elsewhere, 14,
-
.-
-
Y=O.8
Oh 0 k a I@,
4 Homogeneoos
8
12
Rote Cc.x+mf (cm’hmk
4. Ratio of the flux with homogeneous
16 set)
reaction augmentation to the flux with no homogeneous reaction for solubilization of oleic acid by sodium taurodeoxycholate. For the dashed line, multiply the rate constant on the abscissa by a factor of ten.
Fig.
L. R. SKJWEand J. A. S~asrwt~z
640
both S and kB. The resulting flux enhancement would be
ratio
(45)
Then, the maximum flux enhancement *+m
0i -P
I
=
would occur as
,+*+g.
max
Again, the trend toward a maximum enhancement arises because the homogeneous reaction does not enhance tbe hetergeneous reaction step at the interface, and this trend should exist regardless of the kinetics of either of the reactions. CONCLUSIONS
An approximate solution for mass transfer with homogeneous and heterogeneous reaction has been presented, subject to the assumption of a linear profile of reactant at the surface. An expression for the enhancement factor was derived, and was shown to depend upon the usual dimensionless group associated with mass transfer and first order irreversible reaction as well as a Damkohler number representing tbe transport of reactant to and reaction on the surface. For this physical situation, a maximum flux enhancement was predicted, due to the ultimate rate limitation on the surface reaction step. It was also shown how this solution could be applied to the problem of solubiization of an insoluble species by an aqueous surfactant solution. Acbowledgcment-Financial support from NSF Grant CPE 8@ 12431 is gratefully acknowledged. NOTATION
C
concentration Cl, CT constants of integration D diffusion coefficient F I-I/00/Z,/,(M) I” Modtied Bessel Function of First Kind and order II flux : homogeneous rate constant k* k’, k” ka k,
kc,,
heterogeneous rate constants, forward and reverse, respectively mass transfer coefficients of subscripted species
WQ) (k CmJD&3’2 (213Q) (k CXJ&)~‘* reaction rate at interface Reynolds number parameter defined in eqn (22) parameter defined in eqn (23) z W variable substituted for concentration distance from interface z G0lCS~ a P + Qx, independent variable for function W tilm thickness ; enhancement factor Y kinematic viscosity
M N R Re
Subscripts A refers to reactant
species present only at interface B refer to reactant species present initially in bulk phase E refers to product of heterogeneous reaction at interface 0 at interface - in hulk phase
SuperscrIpts
sat value at saturation O refers to flux or mass transfer without enhancement by reaction
coefficient
[l] Levenspiel O., 1972, Chemical Reaction Engineering, 2nd Edn. Chap. 12. New York, Wiley 1972. [2] Tao J. C., Cussler E. L. and Evans D. F., Proc. Nat. Acad. sci 1974 71 3917. [3] Ghan A. F.-C., Evans D. P. and Cussler E. L., A.1Ch.E. _I. 1976 22 1006. [41 Shaeiwitz J. A., Chan A. F.-C., Cussler E. L. and Evans D. F., .I CON.ht. Sci 1981 84 47. [S] Bird R. B., Stewart W.E. and Lightfoot E. N., Transport
Phenomena, p. 520 Wiley, New York 1960. P. V.. Gas-Liquid Reactions. McGraw-Hi, New York 1970, [7] Cussler E. L., Multicomponenf Diffusion, Chap. 8. Elsevier, [6] Danckwerts
New York 1975. [8] Sherwood T. K., Pigford R. L. and Wilke C. R., Moss Transfer, Chap. 8. McGraw-Hill, New York 1975. [9] van Krevelens D. W. and Hoftijzer P. J., Rec. Trac. CMm. 1948 67 563. [IO] Kamke E., Dijfrrentialgleichungen L.osungmerhoden und
Losungen & pp. 400, 440 Leipzig. Akademische Verlagsgesselschaft. Abramowitz M. and Stegun I. A. Handbook of Mathemarical Funcrions, p. 446 Dover New York (1972). [ 1 I] Stowe L. R. and Shaei.wi!z J. A., J. COILInk Sci. 1982 90 495. [12] .~IJ;;? R. and Shaetwtz I. A., Chem. Engng Commun. 11,
R. STOWEt
and JOSEPH
A. SHAEIWITZ*
Department of Chemical Engineering. Urdversity of Illinois, Urbana, IL 61801, U.S.A. (Received 5 February 1982; accepted 5 August l!%Z) Ah&act-The problem of mass transfer with both tirst order irreversible homogeneous and an arbitrary heterogeneous reaction has been analyzed using the film theory, subject to the assumption of a linear profile of reactant between the bulk phase and the heterogeneous reaction surface. The enhancement factor is shown to depend on the product of two dimensionless groups. One is analogous to the usual group arising from mass transfer and first order irreversible. reaction, and the other is a Damkiihler number which describe8 the transport of the reactant to and reaction on the surface. The result is discussed in the context of solubilization of an insoluble liquid amphiphile by an aqueous surfactant solution. A maximum in the flux enhancement is observed because of the resistances associated with transport of the reactant species to the reaction surface. The field of mass transfer with chemical reaction has traditionally been divided into two distinct subgroups: mass transfer with heterogeneous reaction and mass transfer with homogeneous reaction. In general, these two phenomena have been treated separately, probably because the physical situations of interest have involved only one type of reaction. Mass transfer with heterogeneous reaction arises most often in gas/solid non-catalytic reactions [ 11, although the solubilization of insoluble amphiphiles by a surfactant has also been modeled successfully using these concepts[24]. The resistance in series concept is usually associated with these problems; that is, the largest resistance or slowest step is rate determining. Heterogeneous reactions most often appear as boundary conditions in continuum descriptions [5]. Mass transfer with homogen’sous reaction arises most often when gas adsorption is augmented by reaction with a species in the liquid phase[6], although transport across membranes containing mobile carriers also involves this phenomenon[7]. There are several theories for mass transfer with chemical reaction[S]. The film theory, the simplest, assumes a stagnant layer near the interface between the two phases under consideration, within which all of the reaction occurs. The penetration theory is the unsteady state analog in which liquid elements are assumed to bc continuously brought to the interface where the species from the other phase may enter. The surface renewal theory extends this to allow for a distribution of lifetimes of surface elements near the interface. One can also envision boundary layer theory being used to describe this phenomenon. An important observation occurs for the case of mass transfer with irreversible first order homogeneous reaction. This is that the enhancement factor, the ratio of the mass *Author to whom correspondence should be addressed. tPre8ent address: Mobil Research and Development poration, Dallas, TX 75236, U.S.A.
Cor-
transfer coefficient with reaction to that without chemical reaction, is almost exactly equal for the film, penetration and surface renewal theories. Each of these theories predicts a different mass transfer coefficient, but the same enhancement. Therefore, the simplicity of the film theory makes it attractive for modeling these problems. In continuum models, homogeneous reactions appear as source terms in the differential equations [5]. In this paper, we treat the problem of mass transfer with both homogeneous and heterogeneous reaction. Specifically, we examine the case where a reactant diffuses to a surface where it undergoes a process that can be described as a heterogenous reaction, and as the product diffuses away from the interface, it reacts with the reactant by a fust order irreversible reaction. An analytical solution is obtained by making the assumption of a linear concentration profile of reactant. The physical situation which has given rise to this problem is the solubiliiation of an insoluble liquid amphiphile by an aqeuous surfactant solution at high ionic strength; although , one could certainly imagine this physical situation arising in a catalysis context where a product reacts with the reactant as the former leaves the catalyst surface. TliF.OKETtCALBACKGROUND A continuum statement of mass transfer with heterogeneous reaction might involve the differential equation
and boundary
conditions C, = C,,
j=-&%
at x = S,
I =- k'CAoCao X=”
at x = 0.
(2) (3)
Boundary condition (3) states that species B is consumed at the interface by a reaction with A, although for the 635
L. R. STOWEand J. A. SHAEMTZ
636
purposes of this discussion, A will be considered to be present in excess, and CA0 will be assumed constant. The resulting linear concentration profile is
in excess, and k* = kCB,, then the solution and either (lOa) or (lob) is the same
L/2 ke = $
(4)
and the steady state flux is obtained by solving for CBO at x = 0 in eqn (4) and plugging that value into eqn (3), yielding k’CnoCe..
-j=
1
+
(5)
k’C.408’ DB
If we define ka = 0,/S as a film theory mass transfer coe5cient, and k’C,, as a pseudo first order rate constant, then it can be seen that the pertinent dimensionless group is a Damkohler number, the rate of reaction divided by the rate of diffusion, k,lk’&. This same result can be obtained by equating the fluxes in series j = kB(C,,
- C&
= k’C,&&,
- k”C,,
= &&,,.
In this case, the ratio of the reverse to the forward rate constant appears as a partition coefficient in the series resistance expression. Equation (7) can also be obtained from a continuum description like eqns (l)-(5), but the present method is much simpler mathematically. Further examination of eqn (6) indicates that for a mven flux, as kB is increased, C,,C,, decreases. Therefore, for a given flux, a large mass transfer resistance means that Y = C,,/C,, will be very small, and a small mass transfer resistance means that Y -_, 1. A continuum statement of mass transfer with homogeneous reaction might be
EO
e = [k*&1”2 ~ kg
x=6,
(lOa)
dC,=Ratx=o
to be a constant,
group
for this
The essential features of this model include diffusion of reactant, B, to the interface. At the interface B reacts with an excess of A to form E. Then as E diffuses away from the the interface it reacts irreversibly with B to enhance the flux of E back into solution. Figure 1 illustrates this physical situation. A key assumption in this development is that the profile of B is always linear within the film. The equations for this situation are then d2C, &z-0.
_
(13)
D$$j-kC&=O, with boundary
(14)
conditions dCe=-R B dx
atx=O,
(15) (16)
~%z$z~~n*r~eX=o
Bulk
If CB is assumed
(12)
MASS TRANSFER WITE BOTH BOMOGENBOUS AND EETEROGENEOUSREAcllON
e---------_--fO_X.6 c0m
or
E dx
(11)
E
(9)
CE = Cso at x = 0,
-D
dimensionless
Ce=CsOatx=O,
conditions &=Oat
coth [w].
coth [w].
It is seen that the pertinent phenomenon is [k*De]“‘/kB
-D
with boundary
= [k*D,]“*
It is not obvious, a priori, that the solution would be the same for either (lOa) or (lob). However, this is seen to make intuitive sense in the following manner. For a given constant flux, R, increasing k* means that Ceo will drop. On the other hand, if C’,, were constant, increasing k* would mean that R would increase. A fixed surface reaction rate is typical of problems where the rate is controlled by the surface reaction, and a fixed surface concentration is typical of problems where the surface reaction is in rapid equilibrium. By usin8 the lilm theory definition for kg= D&S, where the superscript indicates a mass transfer coe5cient without reaction augmentation, an enhancement factor may be defined as 9 = kJk.& and this ratio becomes
(6)
Equation (5) is obtained by using the irreversible reaction only with the first diffusion step. If the reaction is assumed to be reversible, and product E is now diffusing back into solution containing no E,
to (8) and (9)
(lob)
CBm; that is, present
solution
E
Fig. 1. The physical situation of steady state mass transfer with homogeneous and heterogeneous reaction described by the film theory. The profile of reactant B is assumed to be linear with the film.
Mass transfer using the Blm theory Ce=Oat
x=6,
(17)
Ca=Ce-atx=S.
(18)
The term R in eqn (15) represents the surface reaction rate. A similar problem, with a zero flux condition in place of eqn (15), and with the reaction source term present in eqn (13) has been presented by van Krevelens and Hoftijzer [9], although only an approximate solution was available. The advantage of this formulation of the problem is that an analytical solution is available which should at least provide insight into the appropriate dimensionless groups for this physical situation. The other implications of the assumption of a linear profile of B will be discussed later. Equation (13) can be solved for Ce using conditions (15) and (I 8), yielding
This result can be inserted
Equation
(20) can be rearranged
3
=O.
variable
definitions,
eqn (25) can be
+C2ie113 I_ ,&$[g~)].
(27)
Applying the boundary conditions of eqns (16) and (lS), the constants C, and C2 are related through C,
i”” = -
C,j-
113
~-l/,(M) Im(Jw ’
(28)
where
Applyin the boundary condition in eqn (17) and using eqn (28), eqn (27) can be solved to produce
into eqn (14) to produce Cam+&(x-8)
D,f$f$-kCe
637
plus the previous converted to
c&‘/s
(20)
=
to obtain where
dZC, z-(PfQx)Ce=O,
(30) - F&&?i
Ca = Gee at x = 0,
(21)
where
GO _kceo ,,2 Ln(N) c DE 3
2
N=@ [
1
kC,, =, DB
and
F
=
IdM)
I-,,,(M)
Finally, eqn (27) can be written as p _ k(Cac.s - Rs/Dd DE
(22)
Rk Q= D,D,’
(23
and
Equation Equation (21) can be converted to a linear differential equation by the variable substitutions LY= P + Qx and C.&x) = W(a). With these substitutions, eqn (21) becomes d2W w-$=0. Following
Kamke[lO],
the solution to equation
(24) is
(31) can be differentiated
to produce
(32) At steady state, the flux at the interface is given by -D
!%
(33)
E dx I,_o=R-
So eqn (33) can be combined with eqn (32) to provide the product mass transfer coefficient, kE, where Z,,,s represents a modified Bessel function of the first hind and of order + l/3. Using the following relationship, obtained from eqns (19), (22) and (23), [P+Qx]=G
n_ -r
9
(26)
Lkc
1
l,,2
D
BO
E
,
(34) where in terms of the system parameters
L.R.
638
STOWI?
and J.A.
.%AElWlTZ
Cm,, in eqn (40) and rearranging to
and (36) DISCUSSION
Two observations are now apparent. First of all, eqn (34) is the same form as eqn (ll), with a different form for the multiplicative factor. The analogous form of the enhancement factor in eqn (12) is
I-,,&‘0
- Fl,n(N)
I
W&M”2 k_E
.
(37)
It is seen that the Bessel function ratio, with its dimensionless parameter, N, has replaced the coth term; and that a pseudo-first order rate constant, kCao, based on the surface concentration of B, appears both in N and in the other terms in eqn (37). The second observation concerns the form of the dimensionless parameter, N. Equation (35) can be rearranged as follows
NC2
y=
R 1-k&.2,,’
(41)
Tbe value of Y can be calculated from eqn (41) by knowing C’,,, knowing the flow geometry and hence k,, and measuring the steady state value of R. Figures 2 and 3 show the enhancement factor as a function of N for dilTerent values of Y. These graphs have substantially the same form as the one for mass transfer with 6rs.t order irreversible homogeneous reaction in that at low values of N, the enhancement factor is unity, and that as N increases the enhancement factor rises significantly. [81 For values of N which are not on these figures, a linear extrapolation is all that is needed. This is because above a value of N = 13.4, the Bessel function ratio approaches - 1, which again is completely analagous to the homogeneous reaction case since the coth term in eqn (12) approaches unity as its as its argument exceeds a value of 2.6. It can also be seen from Figs. 2 and 3, that a smaller
(kCmD,)"z D,JS
3 E
&IS
RIG
3’
For example, if R = k’C,,, then RI&, = k’, meaning that the ratio R/C,, is a measure of the rate of surface reaction. Therefore, the parameter N is seen to be a combination of the parameter associated with diffusion and homogeneous reaction, (kC,,DE)‘n/k& and the Damkohler number associated with diffusion and heterogeneous reaction, k,Jk’. The parameter M is merely the same group with the concentration evaluated at the bulk instead of surface value. Traditionally, mass transfer with first order reaction has been treated by plotting the enhancement factor, Ib: against the group equivalent to (kC&&)“‘/k& In this case, however, a second group is necessary to fully describe the physical situation. While the group k,J(R/CBo) may be a tempting choice, it is not a very practical one since the surface concentration of reactant, CaO, is not likely to be known; although, the surface reaction rate, R, is likely to be known from experiment and the assumption of steady state. If the first two resistances in series, diffusion of B to the surface and the surface reaction, are examined k,(C,,-
CA=
R.
(39)
and this is rearranged to
N
Pi. 2. Enhancement factors for mass transfer with homogeneous and heterogeneous reaction, low values of Y = C&C,,.
y=0.5 Y=o.4/y.a6Y=o.7
Y=O.8
lw: Y=O.9
3
2
k+l)=R/Go,
(40)
where Y= CaOICB,. it is seen that Y is a most appropriate representation of the Damkohler group. As discussed previously, for a given reaction rate at the surface, Y is also a measure of the the rate of diffusion to the surface to this rate. The advantage of the parameter Y is that it can be obtained from known quantities in an experimental situation by substituting for
.A 0
4
8
I2
16
N
Fig. 3. Enhancement factors for ma.ss transfer with homogeneous and heterogeneous reaction, high values of Y = C,,,/C,,.
Mass transfer using the film theory value of Y gives a larger enhancement factor for a given value of N. A small value of Y means that Cgg and CS, are very different, so at steady state, this corresponds to a small mass transfer coe5cient or a slow diffusion step. Finally, a brief comment on the assumption of a linear profile in B is in order. The major question is how good is this assumption. First of all, since it allows an analytical solution it is a valuable assumption if only to permit identification of the pertinent trends and dimensionless groups. Secondly, upon further examination, there are perfectly reasonable physical situations for which this assumption might be valid. One is if Y is large, meaning a large mass transfer coefficient of reactant toward the surface. The linear assumption is valid because as Y gets larger, the concentration profile between the edge of the lilm and the surface becomes flatter, and can more reasonably be approximated as linear. Another physical situation for which the linear assumption might be valid is if the diffusion process of reactant relaxes much faster than the homogeneous reaction proceeds. This corresponds lo reasonably slow homogeneous reaction rates and hence, lower values of N. This is particularly interesting since low values of N and Y can yield large enhancement factors. Either way, the trends shown in Figs. 2 and 3 do illustrate the correct qualitative behavior and should be very useful approximations. APPLICATION
TO SOLUBILIZATION
Our interest in this problem arose because of certain features that were observed pertaining to the solubilization dynamics of oleic acid by aqueous surfactant solutions of sodium taurodeoxycholate, especially at higher ionic strengths induced by either by added salt or large taurodeoxycholate concentrations. The major observation was that the flux could be enhanced up to one order of magnitude above a reasonable description of the diffusion limit as salt was added at constant taurodeoxycholate concentration, while the solubility only doubled over this range. Therefore, a reduction in one of the resistances with added salt was indicatedllll. Augmentation by a process describable as a homogeneous reaction can explain this phenomenon. If the overall mass transfer coefficient in eqn (7) is compared to an enhanced value, with the enhancement being in the diffusion of E away from the surface, we obtain an enhancement ratio
ilCL%w -= i”/Caco
1 $+-+ ts k’C,, 1
ka
I
k” 1 ( KC,, > b 1 + - k” _1 ’ ( k’C.m ) @kg k’C.m
639
1 l] suggest typical values for this physical situation to be DB = 5 X 1OP cm*/sec, Ds = 1 x 10-e cm%ec, Y= Re”= = 75, C,, = 0.02 g/cm’, 7.69 X lo-” cm’/sec, k’ C..,o w k”, with kB and kg being determined from a mass transfer solution from the hydrodynamics of the experimental apparatus[12]. The value of k’CAO, is obtained from experiment by measuring the flux -=R
1
CEl,
(43)
1 L+kB k’C,,+&
and then solving for (44)
Once PC,,, is known, it can be inserted into eqn (42) to calculate the enhancement ratio. Figure 4 illustrates the observed enhancement ratio as a function of the homogeneous rate constant, the only other variable which was not measured. Again the largest enhancement is observed for the smallest Y value and the largest homogeneous rate constant (essentially large N). A maximum enhancement is observed, which corresponds to $ becoming so large in eqn (42). and that particular resistance disappearing entirely. For this problem, this maximum enhancement ratio is five, but tbis value as well as the magnitude of the homogeneous rate constant necessary to achieve this maximum, is specific to our parameter values, only the trend toward a maximum enhancement is universal. We may now return to eqn (42), but consider the result which would be obtained if the linear assumption on reactant B were not made. In that case a reaction source term would appear in eqn (13), and an enhancement factor would be computed which would increase the value of I
t
1
I
_,-----Y=o.o5
5-
r
0
r
-
// Y=O.2
-
Y =0.4 Y=O.6
-
(42)
The first observation is that if the thud resistance is not rate determining, then there will be no flux enhancement. In this particular case, it is assumed that an emulsion droplet forms at the interface which decomposes into the micellar solution as it leaves the interface. Since this droplet is likely to be large compared to a micelle, it will probably diffuse rather slowly and the decomposition will enhance a slow step. Experiments which have been described elsewhere, 14,
-
.-
-
Y=O.8
Oh 0 k a I@,
4 Homogeneoos
8
12
Rote Cc.x+mf (cm’hmk
4. Ratio of the flux with homogeneous
16 set)
reaction augmentation to the flux with no homogeneous reaction for solubilization of oleic acid by sodium taurodeoxycholate. For the dashed line, multiply the rate constant on the abscissa by a factor of ten.
Fig.
L. R. SKJWEand J. A. S~asrwt~z
640
both S and kB. The resulting flux enhancement would be
ratio
(45)
Then, the maximum flux enhancement *+m
0i -P
I
=
would occur as
,+*+g.
max
Again, the trend toward a maximum enhancement arises because the homogeneous reaction does not enhance tbe hetergeneous reaction step at the interface, and this trend should exist regardless of the kinetics of either of the reactions. CONCLUSIONS
An approximate solution for mass transfer with homogeneous and heterogeneous reaction has been presented, subject to the assumption of a linear profile of reactant at the surface. An expression for the enhancement factor was derived, and was shown to depend upon the usual dimensionless group associated with mass transfer and first order irreversible reaction as well as a Damkohler number representing tbe transport of reactant to and reaction on the surface. For this physical situation, a maximum flux enhancement was predicted, due to the ultimate rate limitation on the surface reaction step. It was also shown how this solution could be applied to the problem of solubiization of an insoluble species by an aqueous surfactant solution. Acbowledgcment-Financial support from NSF Grant CPE 8@ 12431 is gratefully acknowledged. NOTATION
C
concentration Cl, CT constants of integration D diffusion coefficient F I-I/00/Z,/,(M) I” Modtied Bessel Function of First Kind and order II flux : homogeneous rate constant k* k’, k” ka k,
kc,,
heterogeneous rate constants, forward and reverse, respectively mass transfer coefficients of subscripted species
WQ) (k CmJD&3’2 (213Q) (k CXJ&)~‘* reaction rate at interface Reynolds number parameter defined in eqn (22) parameter defined in eqn (23) z W variable substituted for concentration distance from interface z G0lCS~ a P + Qx, independent variable for function W tilm thickness ; enhancement factor Y kinematic viscosity
M N R Re
Subscripts A refers to reactant
species present only at interface B refer to reactant species present initially in bulk phase E refers to product of heterogeneous reaction at interface 0 at interface - in hulk phase
SuperscrIpts
sat value at saturation O refers to flux or mass transfer without enhancement by reaction
coefficient
[l] Levenspiel O., 1972, Chemical Reaction Engineering, 2nd Edn. Chap. 12. New York, Wiley 1972. [2] Tao J. C., Cussler E. L. and Evans D. F., Proc. Nat. Acad. sci 1974 71 3917. [3] Ghan A. F.-C., Evans D. P. and Cussler E. L., A.1Ch.E. _I. 1976 22 1006. [41 Shaeiwitz J. A., Chan A. F.-C., Cussler E. L. and Evans D. F., .I CON.ht. Sci 1981 84 47. [S] Bird R. B., Stewart W.E. and Lightfoot E. N., Transport
Phenomena, p. 520 Wiley, New York 1960. P. V.. Gas-Liquid Reactions. McGraw-Hi, New York 1970, [7] Cussler E. L., Multicomponenf Diffusion, Chap. 8. Elsevier, [6] Danckwerts
New York 1975. [8] Sherwood T. K., Pigford R. L. and Wilke C. R., Moss Transfer, Chap. 8. McGraw-Hill, New York 1975. [9] van Krevelens D. W. and Hoftijzer P. J., Rec. Trac. CMm. 1948 67 563. [IO] Kamke E., Dijfrrentialgleichungen L.osungmerhoden und
Losungen & pp. 400, 440 Leipzig. Akademische Verlagsgesselschaft. Abramowitz M. and Stegun I. A. Handbook of Mathemarical Funcrions, p. 446 Dover New York (1972). [ 1 I] Stowe L. R. and Shaei.wi!z J. A., J. COILInk Sci. 1982 90 495. [12] .~IJ;;? R. and Shaetwtz I. A., Chem. Engng Commun. 11,