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A novel approach for describing mixing effects in homogeneous reactors
Chemical Engineering Science 58 (2003) 1053 – 1061
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A novel approach for describing mixing e%ects in homogeneous reactors Saikat Chakraborty, Vemuri Balakotaiah∗ Department of Chemical Engineering, University of Houston, 4800 Calhoun, Houston, TX 77204-4004, USA
Abstract The Liapunov–Schmidt technique of classical bifurcation theory is used to spatially average the convection–di%usion–reaction (CDR) equations over smaller time/length scales to obtain low-dimensional two-mode models for describing mixing e%ects due to local di%usion, velocity gradients and reactions. For the cases of isothermal homogeneous tubular, loop/recycle and tank reactors, the two-mode models are described by a pair of coupled balance equations for the mixing-cup (Cm ) and spatial average (C) concentrations. The global equation describes the variation of Cm with residence time (or position) in the reactor, while the local equation expresses the coupling between local di%usion, velocity gradients and reaction at the local scales, in terms of the di%erence between Cm and C. It is shown that the two-mode models have many similarities with the classical two-phase models of heterogeneous catalytic reactors with the concept of transfer between phases being replaced by that of exchange between the two-modes. It is also shown that when the local Damk4ohler number (ratio of local di%usion to reaction time) is small, the solution of two-mode models approaches the exact solution of full CDR equations, while for fast reactions the two-mode models retain all the qualitative features of the latter. Examples are provided to illustrate the usefulness of these two-mode models in predicting micromixing e%ects on homogeneous reactions. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Mixing; Micromixing; Homogeneous reactors; Spatial averaging; Liapunov–Schmidt method; Convective–di%usion equation
1. Introduction The study of mixing e%ects on chemical reactions has been an active area of research since the pioneering works of Danckwerts (1958) and Zweitering (1959). This topic has become a part of classical chemical reaction engineering and has been discussed in several textbooks (Levenspiel, 1972; Froment & Bishcho%, 1990; Westerterp, van Swaaij, & Beenackers, 1984) and review articles (Villermaux, 1991). In the last decade, many computaional ?uid dynamics (CFD) codes have been developed to describe reactive mixing, but the task of solving the three-dimenisonal convection–di%usion–reaction (CDR) equations numerically is very expensive, especially for the case of fast reactions. Moreover, due to strong coupling between transport and reaction rate processes, the model equations for reacting ?ows are highly non-linear and are known to exhibit a variety of complex spatio-temporal behaviors. For most cases of practical interest, even with the present day computational power, it is impractical to explore the di%erent types of solutions and bifurcation behaviors that exist in the multi-dimensional parameter space, using the CFD ∗
Corresponding author. Tel.:+1-713-743-4318; fax:+1-713-743-4323. E-mail address: [email protected] (V. Balakotaiah).
codes. Traditional low-dimensional mixing models like the axial dispersion, recycle and tanks in series models, use a single parameter (like axial Peclet number Pe, recycle ratio , or number of tanks N ), and a single concentration variable for each species to describe macromixing. However, it is well known that these classical models cannot describe micromixing e%ects caused by local di%usion, local velocity gradients and reaction at the small scales. The main purpose of this article is to show that in a low-dimensional approach to mixing, we require a second concentration variable for each species and a local balance equation that captures micromixing in terms of the di%erence between the two concentrations. The two concentration variables are called the two modes of the system and these models are called two-mode models (TMMs). In this work, we present the TMMs for various homogeneous reactors and illustrate their application with examples. 2. Formulation of TMMs 2.1. Tubular reactors To illustrate the spatial averaging procedure, the derivation of the TMMs using Liapunov–Schmidt (L–S) technique
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(02)00647-4
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
of bifurcation theory and other main concepts of our approach, we consider the classical example of laminar ?ow in a tube with homogeneous reaction. We assume that the scalar concentration C( ; ; x; t ) in a tubular reactor obeys the CDR equation @C @C + ux ( ) = ∇ · (D⊥ ∇C) @t @x @C @ Dx − R(C); + @x @x
=p
@c @
+
1 @2 c 2 @ 2
(1)
(2)
with boundary and initial conditions given by 1 @c = u()[c − cin ] Pe @z @c =0 @z
at z = 1;
@c =0 @
at = 0; 1;
c(; ; z; t = 0) = c0 :
a2 ux t⊥ = ; C LD⊥
Da =
Pe =
tx ux L = ; C Dx
C LR(CR ) = ; tR ux CR
t⊥ a2 R(CR ) = = pDa; tR D⊥ C R t⊥ Dx a 2 p 2 = = = : tX D⊥ L Pe
(4)
We note that when p = = 2 = 0, or equivalently, when the transverse (local) di%usion time is vanishingly small compared to convection, reaction and macromixing time scales, transverse gradients vanish, and Eqs. (2) and (3) simplify to the classical one-mode axial dispersion model. If local di%usion time is small (but Fnite) as compared to the other three time scales, or in other words, a time/length scale separation exists in the system, we can write c(; ; z; t) = c(z; t) + c (; ; z; t);
(5a)
where c is the transverse averaged concentration, and c is the deviation of the concentration from the mean value c, which goes to zero, as p → 0. (Also, by deFnition, c = 0 for all values of p.) Multiplying Eq. (5a) by u() = u + u (), and averaging over the cross-section gives
1 @2 c @c @c − + Da r(c) + u() @t Pe @z 2 @z
≡ pg(c)
p=
2 =
where ux ( ) is the fully developed velocity Feld, D⊥ and Dx are the e%ective di%usivities of the species in the transverse and the axial directions, respectively, R(C) is the reaction rate, and ; ; x are the radial, azimuthal and axial coordinates of the tube, respectively. (We could take D⊥ =Dx =Dm (molecular di%usivity), but for generality and later use, we have assumed D⊥ = Dx .) Eq. (1) is subject to no-?ux, periodic, and Danckwerts boundary conditions in the radial, azimuthal and axial directions, respectively. In dimensionless form, Eq. (1) is given by 1 @ @
groups:
cm = c(z) + u c ;
(5b)
where cm is the mixing-cup (velocity weighted) concentration. For the case of a tubular reactor 1 1 2u()c(; z) d 2c(; z) d 0 cm = ; c = 0 1 : 1 2u() d 2 d 0 0
at z = 0;
c(; ; z; t) = c(; + 2; z; t); (3)
Here, z and are the dimensionless axial and radial coordinates, non-dimensionalized with respect to the length L and the radius a of the tube, respectively, t is the dimensionless time (t = t ux =L); c is the dimensionless concentration (c = C=CR , where CR is the reference concentration), u() is the dimensionless axial velocity proFle [u() = 2(1 − 2 ) for fully developed laminar ?ow in a tube], and r(c) is the dimensionless reaction rate [r(c) = R(C)=R(CR )]. The four characteristic time scales present in the model, namely the transverse di%usion time, t⊥ (=a2 =D⊥ ), the convection time, C (=ux =L), the reaction time, tR (=CR =R(CR )), and the axial di%usion (or macromixing) time, tx (=L2 =Dx ), lead to the following dimensionless
It may be mentioned that cm is representative of the convection (global) scale of the system, while c is representative of the reaction–di%usion (local) scale of the system. Eq. (5b) is the local equation, which shows that the di%erence between cm and c depends on the local velocity gradients (u ) and the local concentration gradients (c ) caused by molecular di%usion and reaction at the local scales. Micromixing is captured by the local equation as an exchange between the two modes (scales), cm and c. In order to determine c (and hence the term u c ), we substitute Eq. (5b) in Eq. (2) to obtain ∇2⊥ c = pg(c + c ):
(6a)
The L–S technique solves Eq. (6a) for c by expanding it in the parameter p as c =
∞ i=1
p i ci :
(6b)
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
Such an expansion is possible, since for p=0, the transverse di%usion operator in Eq. (2) has a zero eigenvalue with a constant eigenfunction. Thus, u c could be determined to any order in p. In practice, the leading term (that is of order p) is suJcient to retain all the qualitative features of the full CDR equation. (For details of L–S procedure, see Golubitsky & Schae%er, 1984; Balakotaiah & Chang, 2003, and Chakraborty & Balakotaiah, 2002). Here, we present the averaged form of the CDR equation to O(p): @c @cm 1 @2 c + Da r(c) = 0; + − @t @z Pe @z 2 cm − c = − 1 p
@cm @z
(7) (8)
1 @c = cm − cm; in ; Pe @z @c = 0; @z c = c0 ;
at z = 0;
(9)
at z = 1;
(10)
ux
at t = 0;
dcm cm − c =− ; dz 1p
(11)
cm = cm; in
1 pDa r(c):
at z = 0;
Cm − C dCm =− ; dx tmix
(12) (13)
We shall refer to this model as the two-mode convection model. In the limit of p → 0, this model reduces to the ideal plug ?ow reactor model.
Cm = Cm; in
Cm − C = tmix R(C):
at x = 0;
(14) (15)
Eq. (14) is referred to as the global equation, which gives the evolution of the mixing-cup concentration with axial position, while Eq. (15) is the local equation. Here, tmix is the local mixing time of the system, which is the characteristic time scale for exchange between the local and global scales, and is given by 1
a2 : D⊥
(16)
It is interesting to note that the two-mode convection model has a striking similarity with the two-phase model for heterogeneous reactions. This could be seen more clearly when Eqs. (14) and (15) are rewritten as ux
where 1 is constant that depends on the type and the geometry of the reactor and the ?ow proFle. For a tubular reactor with fully developed laminar velocity proFle, 1 is obtained as 1/48. We shall refer to this averaged model (Eqs. (7)– (11)) as the two-mode axial dispersion model. It may be noted that L–S averaging technique reduces the inFnite number of local (radial) modes present in the CDR equation to two radial modes cm and c, while retaining all the parameters of the CDR equation. As a result, the two-mode axial dispersion model, like the CDR equation, is capable of describing both macro- and micro-mixing e9ects through the parameters Pe and p, respectively. It is only in the limit of micromixing e%ects being very small, i.e. p → 0, that the two-parameter two-mode axial dispersion model (Eqs. (7)–(11)) could be simpliFed to the traditional one-parameter one-mode axial dispersion model, with an e%ective Taylor–Aris dispersion coeJcient (Taylor, 1953; Aris, 1956). For small but Fnite values of p, lumping of the two-parameters p and Pe into a single parameter and use of a single concentration variable is not valid. For the limiting case of Pe → ∞ (i.e. no macromixing is present in the reactor), Eqs. (7)–(11) under steady-state conditions, reduce to
cm − c =
In dimensional form, the two-mode convection model is given by
tmix =
with boundary and initial conditions given by
1055
Cm − C dCm =− = −R(C) dx tmix
(17)
with Cm = Cm; in at x = 0. The two-phase model for a heterogeneous wall-catalyzed reaction is given by ux
C m − CS dCm =− = −R(CS ) dx tTP
(18)
with Cm = Cm; in at x = 0. The spatially averaged concentration, C, in the TMM is replaced by the surface (wall) concentration, Cs , in the two-phase model, while the local mixing time tmix of the TMM is replaced by the characteristic mass-transfer time between the two phases, tTP in the latter. Just as the two-phase model can capture the mass-transfer limited asymptote in a heterogeneous reaction (which is missed by the pseudo-homogeneous model), so can the two-mode model capture the mixing-limited asymptote in homogeneous reactions, which is rendered inaccessible by the traditional one-mode models. The extension of the two-mode axial dispersion model to the case of fully developed turbulent ?ow in a pipe could be achieved by starting with the time-smoothed (Reynolds averaged) CDR equation, given by Eqs. (2) and (3), where the reaction rate term r(c) in the above equations is replaced by the Reynolds averaged reaction rate term rav (c). While in case of laminar ?ows, the e%ective di%usion term D⊥ (or Dx ) in Eq. (1) is same as the molecular di%usivity of the species Dm , in case of turbulent ?ows, it is given by D⊥ = Dm + DT ;
(19)
where DT is the turbulent di%usivity, which could be obtained on the basis of turbulent shear stress and expressed in terms of Reynolds numbers using a formula similar to that presented by Wen and Fan (1975). Also, the Reynolds averaged reaction rate rav (c) could be evaluated by using
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
simple closure models of Bourne and Toor (1977), Li and Toor (1986), etc. We skip the details of these methods, since the closure of the Reynolds’ averaged transport equation is not the focus of this paper. It should also be pointed out that the spatial averaging presented here is independent of the methodology using which rav (c) and DT are evaluated, or in other words, spatial averaging follows time averaging. We use an universal velocity distribution obtained by Churchill (2001) to approximate the fully developed velocity proFle u() across the turbulent core, given by
f f u() = 5:5 + 2:5 ln (1 − ) Re 2 8 15 10 + (1 − )2 − (1 − )3 ; 4 3
(20)
where f is the Fanning friction factor and Re is the Reynolds number. The important result is that the TMMs for a turbulent :ow tubular reactor are the same as those for laminar :ow tubular reactors. The two-mode axial dispersion model for turbulent ?ow tubular reactors is again given by Eqs. (7)–(11), while the two-mode convection model for the same is given by Eqs. (12) and (13), where the reaction rate term r(c) is replaced by the Reynolds averaged reaction rate term rav (c). The local mixing time tmix for turbulent ?ows is again given by Eq. (16) as in the case of laminar ?ow, D⊥ is given by Eq. (19), and 1 is given by
f 2 − : (21) 1 = 0:1f 2:05 + 2:5 ln Re + 8 f We note that in turbulent ?ows, typically Dm DT , as a result of which, the local mixing time tmix is independent of the molecular di%usivity or the molecular Schmidt number. 2.2. Loop and recycle reactors In this section, we present the TMMs for loop and recycle reactors. In a steady-state loop reactor of loop length L, a ?ow-rate of q enters and leaves the reactor at points s = 0 and l, respectively (where s is the length coordinate along the loop), and the total ?ow-rate in the loop is Q+q between points s = 0 and l, and is Q between points s = l and L, due to a recycle rate of Q. The recycle ratio is the ratio of the volume of ?uid returned to the reactor entrance per unit time to the volume of ?uid leaving the system per unit time, and is given by = Q=q. The two-mode model for a loop reactor is 1 − R(C); 0 6 s ¡ l; dCm 1 + us = (22) ds − 1 R(C); l 6 s 6 L; Cm − C = tmix R(C);
06s6L
(23)
with the boundary conditions Cm (s = 0) =
Cm; in + Cm (s = L) ; 1+
C(s = l− ) = C(s = l+ );
(24)
where the average velocity us is deFned as us = q=AC , and AC is the cross-sectional area of the loop reactor. For the special case when no reaction occurs between s=l and L, i.e., Cm (s=l)=Cm (s=L), the loop reactor reduces to a recycle reactor of length l, the two-mode model for which is given by us
dCm 1 R(C); =− ds 1+
Cm − C = tmix R(C);
06s6l
(25) (26)
with the boundary condition being given by Cm (s = 0) =
Cm; in + Cm (s = l) : 1+
(27)
The two-mode loop and recycle reactor models, like the two-mode axial dispersion model, are two-parameter TMMs. Here, the two parameters are the recycle ratio , and the local mixing time tmix , which describe macro- and micro-mixing e%ects in the system, respectively. 2.3. Tank reactors (CSTRs) It is well known that as the recycle ratio of a recycle reactor is increased the behavior shifts from tubular plug ?ow reactor at =0 (no macromixing) to a CSTR at =∞ (perfect macromixing). We use this idea to obtain the two-mode model for a perfectly macromixed CSTR, by integrating Eq. (25) along the length of the reactor s and simplifying the resulting equation for 1. This gives the two-mode model for a CSTR as Cm − Cm; in C − Cm = ; C tmix
(28)
Cm − C = tmix R(C);
(29)
where C is the residence time of the tank and tmix is the local mixing time. Eqs. (28) and (29), which are the global and local equations, respectively, constitute a two-mode one-parameter model for a perfectly macromixed CSTR. Micromixing effects are captured through the local mixing time tmix , and in the limit of complete micromixing (i.e. tmix → 0), it reduces to the ideal one-mode zero-parameter CSTR model. It is interesting to note that the local equations for the di9erent reactor types (Eqs. (15), (23), (26) and (29)) are the same. The reactor geometry or the ?ow Feld only changes the local mixing time of system. Alternate methods of obtaining Eqs. (28) and (29) include integrating the two-mode axial dispersion model
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
(Eqs. (7)–(11)) along the length of the reactor z, in the limit of complete macro-mixing i.e. Pe → 0. The TMM for tank reactors for the case of multiple reactions involving m species is given by −1 Cm − Cm; in = C Tmix (C − Cm ); −1 Tmix (C − Cm ) = R(C);
q2
q1 Zone A
QE Zone A
q1
where Tmix is an m×m matrix of the local mixing times. For the case of laminar ?ows with unequal species di%usivities, Tmix =tmix D−1 R where DR is the matrix of relative di%usivities of the species. For turbulent ?ows, where the local mixing times are nearly independent of molecular Schmidt numbers, DR ≈ I. The physical system equivalent to the two-mode model of a CSTR is a tank reactor, consisting of two zones, each of size V , namely, a non-reacting convection zone (A), represented by Cm , and a reaction zone (B), represented by C. The interaction between the zones A and B is quantiFed by an exchange of reactants and products from the convection scale (zone A) to the reaction–di%usion scale (zone B), which is represented by qE . This exchange of material between zones A and B, occurs only through local di9usion, and tmix (=qE =V ), which is the characteristic time scale for this exchange, scales as the local di%usion time of the system. In general, any inFnitesimal volume dV inside the tank could be so imagined to consist of two zones/ scales, and a corresponding two-mode model could be written (Eqs. (28) and (29)) for a CSTR of volume dV . If macromixing in the CSTR is complete, the two-mode model for any control volume dV could be integrated over the entire volume of the tank V to generate a single two-mode model (Eqs. (28) and (29)) for the whole tank. Macromixing e%ects are however not negligible in real tanks, and are in?uenced by several factors including the type and speed of impellers (turbines) and the manner of feed distribution. Macromixing e%ects in tanks have often been modeled by using compartment models in the mixing literature (Baldyga & Bourne, 1992). The two-compartment model, for example, divides the tank into two-compartments, namely the circulation zone and the impeller zone, which are then modeled as two interacting CSTRs, with macromixing being described as an exchange of material at rate QE between the compartments. Here, we use the two-compartment model to describe macromixing, with each compartment itself being described by a two-mode model that accounts for micromixing. The resulting model is a two-mode two-compartment model that describes both macro- and micro-mixing e%ects in tanks. A schematic of this is shown in Fig. 1. It consists of two interacting CSTRs of sizes V1 and V2 , each of which is described by a two-mode model. Materials are exchanged between zones A and B (micromixing) of each CSTR at rates of qE1 and qE2 , respectively, and between CSTRs (macromixing) at a rate of QE . The unsteady state two-mode two-compartment
q2
q E2
q E1
(30)
1057
QE Zone B
Zone B
CSTR 1 of Volume V1
CSTR 2 of Volume V2
Fig. 1. Schematic of a two-mode two-compartment model that describes both macro- and micro-mixing e%ects in tanks.
model is given by dC1 Cm1 − C1 C2 − C1 = + + R(C1 ); dt tmix1 tG1
(31)
Cin − Cm1 C1 − Cm1 Cm2 − Cm1 + + = 0; 1 tmix1 tG1
(32)
dC2 Cm2 − C2 C1 − C2 + − R(C2 ); = dt tmix2 tG2
(33)
Cin − Cm2 C2 − Cm2 Cm1 − Cm2 + + = 0; 2 tmix2 tG2
(34)
where C1 and C2 are the spatially averaged concentrations in compartments 1 and 2, respectively, Cm1 and Cm2 are the mixing-cup concentrations in the two compartments, 1 (=V1 =q1 ) and 2 (=V2 =q2 ) are the residence times of respective compartments, tmix1 (=V1 =qE1 ) and tmix2 (=V2 =qE2 ) are the local (micro-) mixing times, while tG1 (=V1 =QE ) and tG2 (=V2 =QE ) are the global (macro-) mixing times between the compartments. Needless to mention, the macromixing times tG1 and tG2 depend on the impeller type, tank geometry, and manner of feeding, while the micromixing times tmix1 and tmix2 depend on the local di%usional times and the local shear rates. We now attempt to obtain a one-parameter two-mode model that describes both macro- and micro-mixing e%ects in a tank reactor. This could be achieved by employing L–S technique in Fnite dimensions (Balakotaiah & Chang, 2003) if the macromixing times tG1 and tG2 are small as compared to the residence times 1 and 2 , respectively, i.e. macromixing is fast as compared to convection. We deFne a parameter -, which represents the deviation from the ideal CSTR model. Ratios of macro-/micro-mixing times and respective residence times (i.e. tmix1 =1 ; tmix2 =2 ; tG1 =1 ; tG2 =2 ) all scale as the parameter -, in case of small deviations from the perfectly macro- and micro-mixed ideal CSTR state. L–S reduction in Fnite dimensions is possible if there exists a zero eigenvalue for the system described by Eqs. (31)– (34), corresponding to - = 0 (i.e. ideal CSTR state). In this case, the eigenvector corresponding to - = 0 is given by
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
ˆ 1 1 1]T , where Cˆ is the concentration in a sin= C[1 gle ideal CSTR. L–S reduction of Eqs. (31)–(34) about the base state 0 (ideal CSTR state) results after some algebraic manipulations in a single two-mode model involving one mixing-cup concentration Cm and one spatially averaged concentration C, given by dC Cm − C − R(C); (35a) = dt tM
100
0
3. Application of TMMs In this section, we present three examples to illustrate the usefulness of TMMs in predicting micromixing e%ects on isothermal non-linear reactions occurring in tank reactors. For simplicity, we assume all species di%usivities to be equal in these examples.
η=0 80
Exit Conversion, X (%)
Cm − C Cm; in − Cm = ; (35b) tM C where C is the residence time of the whole tank, and tM is the e%ective mixing time, which incorporates both macroand micro-mixing e%ects. The e%ective mixing time tM depends intricately on the tank geometry, type and number of impellers, baPe positions, power dissipation in the system and feed distribution, as well as the local di%usion time. More complicated ?ow Felds like the one resulting from multiple/stacked impellers could be modeled by extending the same idea of compartments to n compartments or n interacting loops. It needs to be mentioned that this two-mode model (Eq. (35)) has been derived to O(-), and remains unaltered even for the general case of n interacting loops, in which case 2n equations are reduced to 2 equations using L–S technique. Eq. (35) could be derived only if macromixing times are small as compared to the residence time C , or in other words, exchange of materials between di%erent ?ow loops in the tank is fast as compared to convective transport. Otherwise, a two mode two-parameter model is necessary to capture both macro- and micro-mixing e%ects in tank reactors.
Ideal CSTR
k
A + B→P;
rate = kCA CB ;
where k is the second-order reaction rate constant. The balance equations in dimensionless form are given by 3 + cA 43 + cB cA; m = ; cB; m = ; 1+3 1+3 cA; m − cA = 3DacA cB = cB; m − cB ;
(36a)
η=0.1 η=0.5
η=1.0 40 η=2.0
20 η=10.0
0 0.01
0.1
1
10
100
Da
Fig. 2. Variation of exit conversion, X , with Damk4ohler number, Da, for a bimolecular second-order reaction in a CSTR, for di%erent values of the dimensionless local mixing time, 3.
where 4 is the feeding ratio of B to A (4 ¿ 1, if A is the limiting reactant), the Damk4ohler number, Da is given by Da = kCA; in C , and the dimensionless local mixing time 3 is given by 3 = tmix =C . The exit conversion X (=1 − cA; m ) could be obtained by solving the quadratic equation X = Da[1 − X (1 + 3)][4 − X (1 + 3)]:
(36b)
This is plotted in Fig. 2 as a function of Da for di%erent values of 3 and stoichiometric feeding (4 = 1). This Fgure shows the variation of the exit conversion X with Damk4ohler number Da, for di%erent values of the dimensionless local mixing time 3. The case of 3 = 0 corresponds to the ideal CSTR. For 3 ¿ 0 and Da → ∞, the local concentrations approach zero (cA = cB = 0), while the mixing cup concentrations approach the mixing limited asymptote of
3.1. Bimolecular second-order reactions Bimolecular second-order reactions provide the simplest example of non-linear kinetics, where micromixing has signiFcant e%ects. We use the TMM for a CSTR (Eqs. (28) and (29)) to examine the same in the case of a reaction of the type
60
A+B k>P CA,in= CB,in Da = k CA,inτC η= tmix / τC
cA; m =
3 ; 1+3
cB; m =
43 ; 1+3
X = cP; m =
1 : (37) 1+3
We note that this mixing-limited asymptote for homogeneous reactions is analogous to the mass-transfer limited asymptote for wall-catalyzed reactions. Just as the wall (surface) concentrations approach zero for the case of inFnitely fast surface reactions (while the bulk/mixing-cup concentrations remain Fnite), so do the local concentrations ci for inFnitely fast homogeneous reactions (i = A; B). Unlike in catalytic reactors, where exchange between the phases occurs at the solid–?uid boundary, the exchange between modes (scales) in homogeneous reactors occurs in the entire domain.
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
reaction, given by Da2 = k2 CB; in C . Fig. 3 corresponds to the case when the Frst reaction is inFnitely fast and A and B are fed in stoichiometric amounts [i.e. CA; in = CB; in , and CR; in = CS; in = 0]. While no S is formed for the case of 3 = 0 (ideal CSTR), a signiFcant increase in yield of S is obtained if Fnite micromixing limitations are present in the system. The maximum yield of S, obtained when the mixing limited asymptote is attained also for the second reaction, is 23 1 + 23 for 3 6 1; YS; max = (39) 2 for 3 ¿ 1: 1 + 23
60
A+B 50
Yield of S, YS (%)
B+R 40
k1
>R
k2
η = 0.05
>S
CA,in= CB,in Da2 = k2 τC CB,in η = tmix / τC
30
η = 0.1
20
η = 0.2
10
0 0.1
η = 0.5
1
10
100
1059
Thus, in this case, an optimal yield of S exists, and obtained for 3 = 1, which is obvious intuitively. 1000
Da2
3.3. Polymerization reactions
Fig. 3. Variation of the yield of S with Damk4ohler number for a competitive-consecutive reaction scheme, when the Frst reaction is inFnitely fast.
3.2. Competitive-consecutive reactions k
1 Competitive-consecutive reactions of the type A + B→R; k2 are prototype of many multistep reactions like niB + R→S tration of benzene and toluene, diazo coupling, bromination reactions, etc. Experimental observations (Li & Toor, 1986) show that if the Frst reaction is inFnitely fast as compared to the second one (i.e. k1 =k2 → ∞), under perfectly mixed conditions B is completely consumed by the Frst reaction and the yield of S is zero (if A and B are fed in stoichiometric amounts). However, it was observed that if the mixing of A and B is not attained down to the molecular scale, the Frst reaction is not complete and there remains a local excess of B, which can then react with R to produce S. The yield of S increases monotonically as the reaction rate of the second reaction increases, Fnally attaining a mixing limited asymptote. We use the TMM for a CSTR to verify this observation. If A and B are fed in stoichiometric amounts, the local excess of B that remains after the Frst reaction attains its mixing limited asymptote (in the limit of Da1 = k1 CB; in C → ∞), (1) is obtained from Eq. (37) as CB; m =CB; in = 3=(1 + 3), while the amount of R formed from the Frst reaction is given by (1) CR; m =CB; in = 1=(1 + 3), where 3 is the dimensionless mixing time of the system. The second reaction is then simply a bimolecular second-order reaction between B and R, as in example 3.1. The resultant yield of S is given by YS , where 2CS; m (38) YS = CR; m + 2CS; m
and CR; m and CS; m are the exit concentrations in a CSTR in which only the second reaction occurs and with a feed (1) (1) concentration of CB; m and CR; m . Fig. 3 shows the increase in the yield of S with Da2 , the Damk4ohler number of the second
It is well known that in polymerization reactions, mixing a%ects monomer conversion, copolymer distribution and molecular weight distribution (Villermaux, 1991). In linear polymerization systems, imperfect mixing is found to broaden the molecular weight distribution (MWD), while in non-linear polymerization with signiFcant branching, depending on reaction conditions, imperfect mixing can broaden or narrow the MWD (Zhang & Ray, 1997). Here, we verify the Frst of the two above-mentioned observations by examining the case of an anionic polymerization using the TMM for a CSTR. Anionic polymerization, often used industrially to produce polymers of narrow MWD, is typically characterized by the lack of a termination step. The kinetics is k
I I + M →P I;
k
P Pj + M →P j+1 ;
where the rate of the initiation (assuming a constant initiator concentration) is given by RI = kI M;
(40)
while the rate of propagation is given by RP = kP MPj ;
(41)
where kI and kP are the initiation and propagation rate constants, respectively. Application of the TMMs for a CSTR to the above kinetics results in a set of non-linear algebraic equations, which when solved gives the MWD and the poly-dispersity index. Fig. 4 shows the variation of the MWD with n, where the MWD is deFned as ∞ MWD = n2 Pn nPn (42) n=1
and n is the chain length of a polymer chain Pn . The MWDs shown in Fig. 4 correspond to the parameter values of Da = kI C =10−3 and kP =kI =2×105 , where Da is the Damk4ohler number and C is the residence time of the tank. While the 3 = 0 case shows that the MWD for a perfectly mixed CSTR
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
0.5
0.3
η=
0.2
5 η= 2 η= 1 η=
MWD
0.4
0
0.1
0 0
500
100 0
150 0
200 0
250 0
300 0
n Fig. 4. E%ects of mixing on polymer properties: variation of MWD with chain length n, for di%erent values of the dimensionless mixing time, 3.
is fairly narrow, a signiFcant broadening of the MWD is observed as the dimensionless mixing time of the system, 3 (=tmix =C ), is increased. 4. Discussion and conclusions As discussed in the Section 1, a complete quantiFcation of the mixing process by solving the CDR equations from the largest (reactor scale) all the way to the smallest scale (molecular di%usion and reaction scale) is impractical, necessitating the development of low-dimensional models, involving the two representative scales: global and local. Spatial averaging by the LS technique gives us a framework for formalizing such dimension reduction for reactive ?ows. The global equation of the two-mode model describes evolution of the mixing-cup concentration with residence time (or position) in the reactor, while the local equation characterizes the interaction between the global (convection) and local (reaction–di%usion) scales. Exchange of material between the two scales (i.e. C and Cm , respectively) describes micromixing present in the system, characterized by a local mixing time, which depends on the local di%usional time and the local shear rates. An additional parameter like the axial Peclet number (Pe) in the axial dispersion model, the recycle ratio () in the recycle model, the number of tanks (N ) in the tanks-in-series model or a dimensionless exchange time tG in the two-compartment model of a single tank, may be used to characterize macromixing in the system. However, these traditional models cannot describe micromixing e%ects. A two-parameter two-mode model is necessary to characterize both macro- and micromixing in the system. Whether two modes are su=cient to describe mixing in reactors remains a question that could also be answered by using the L–S technique. The local equation, which has been derived to O(tmix ) in this paper, could be derived to
all orders in tmix using Eq. (6). The region of validity of the two-mode model could then be speciFed by examining the radius of convergence of the inFnite series in the local equation (Eq. (5b)). For example, the region of convergence of the two-mode convection model for a Frst-order reaction is given by (Chakraborty & Balakotaiah, 2002) Daloc = ktmix 6 0:858, where k is the reaction rate constant and Daloc is the local Damk4ohler number. For the reaction schemes considered in Sections 3.1 and 3.2, it as been shown that the solution of the two-mode convection model for tubular reactor matches accurately (to 3 decimal places) with the numerical solution of the CDR equation for all Daloc ¡ 0:1, and retains the qualitative features of the latter for fast reactions (i.e. larger values of Daloc ). A similar higher order expansion of Eq. (6) for the case of tank reactors shows that the two-mode model holds good for all values of the local Damk4ohler number. As illustrated in Section 3 with examples, the TMMs have a striking similarity with the two-phase models of catalytic reactors, with the following one-to-one correspondence: mass-transfer coeJcient → exchange coeJcient, surface (wall) concentration → spatially averaged concentration C, mass-transfer limited reaction → mixing-limited reaction. The analogy could be carried further by noting that for all cases of well-deFned ?ow-Felds, where mass-transfer coeJcients can be estimated theoretically, we can also estimate the exchange coeJcient 1 or the local mixing time tmix of the TMMs. For more complex ?ow-Felds (e.g. packed bed), the local mixing time, like the mass-transfer coeJcient, could be correlated to Re, Sc and the geometrical characteristics of the system. Thus, the TMMs of homogeneous reactors are as general as the two-phase models of catalytic reactors and have a similar range of applicability. It has been shown elsewhere (Chakraborty & Balakotaiah, 2002) that the TMMs can predict multiple solutions in case of autocatalytic kinetics, capture e%ects of di%erences in species di%usivities and non-uniform reactant feeding on conversion and predict mixing-limited asymptotes in non-linear reactions. The TMMs could also be easily extended to the case of non-isothermal reactions. To summarize, the TMMs are the minimal models that provide a low-dimensional description of mixing, by coupling the interaction between chemical reaction, diffusion and velocity gradients at the local scales to the macro-scale reactor variables (such as mixing-cup concentration, ?ow-rate and reactor size or residence time). Due to their simplicity and generality, it is hoped that they will Fnd applications in the preliminary design and optimization of homogeneous chemical reactors. Acknowledgements This work was supported by grants from the Robert A. Welch Foundation, the Texas Advanced Technology Program and the Dow Chemical Company.
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
References Aris, R. (1956). On the dispersion of a solute in a ?uid through a tube. Proceedings of the Royal Society of London A, 235, 67. Balakotaiah, V., & Chang, H.-C. (2003). Hyperbolic homogenized models for thermal and solutal dispersion. SIAM Journal on Applied Mathematics, in press. Baldyga, J., & Bourne, J. R. (1992). Interactions between mixing on various scales in stirred mixing tank reactors. Chemical Engineering Science, 478, 1839. Bourne, J. R., & Toor, H. L. (1977). Simple criteria for mixing e%ects in complex reactions. A.I.Ch.E. Journal, 23, 602. Chakraborty, S., & Balakotaiah, V. (2002). Low-dimensional models for describing mixing e%ects in laminar ?ow tubular reactors. Chemical Engineering Science, 57, 2545. Churchill, S. W. (2001). Turbulent ?ow and convection: the prediction of turbulent ?ow and convection in a round tube. In J. P. Hartnett, T. F. Irvine, Y. I. Cho, & G. A. Greene (Eds.), Advances in heat transfer. New York: Academic Press. Danckwerts, P. V. (1958). The e%ect of incomplete mixing on homogeneous reactions. Chemical Engineering Science, 8, 93.
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Froment, G. F., & Bishcho%, K. B. (1990). Chemical reactor analysis and design. New York: Wiley. Golubitsky, M., & Schae%er, D. G. (1984). Singularities and groups in bifurcation theory, vol. 1. Berlin: Springer. Levenspiel, O. (1972). Chemical reaction engineering. New York: Wiley. Li, K. T., & Toor, H. L. (1986). Turbulent reactive mixing with a series-parallel reaction-e%ect of mixing on yield. A.I.Ch.E. Journal, 32, 1312. Taylor, G. I. (1953). Dispersion of soluble matter in solvent ?owing slowly through a tube. Proceedings of the Royal Society of London A, 235, 67. Villermaux, J. (1991). Mixing e%ects on complex chemical reactions in a stirred tank. Reviews of Chemical Engineering, 7(1), 51. Wen, C. Y., & Fan, L. T. (1975). Models for :ow systems and chemical reactors. New York: Marcel Dekker. Westerterp, K. R., van Swaaij, W. P. M., & Beenackers, A. A. C. M. (1984). Chemical reactor design and operation. New York: Wiley. Zhang, S. X., & Ray, W. H. (1997). Modeling of imperfect mixing and its e%ects on polymer properties. A.I.Ch.E. Journal, 43, 1265. Zweitering, T. N. (1959). The degree of mixing in continuous ?ow systems. Chemical Engineering Science, 11, 1.
www.elsevier.com/locate/ces
A novel approach for describing mixing e%ects in homogeneous reactors Saikat Chakraborty, Vemuri Balakotaiah∗ Department of Chemical Engineering, University of Houston, 4800 Calhoun, Houston, TX 77204-4004, USA
Abstract The Liapunov–Schmidt technique of classical bifurcation theory is used to spatially average the convection–di%usion–reaction (CDR) equations over smaller time/length scales to obtain low-dimensional two-mode models for describing mixing e%ects due to local di%usion, velocity gradients and reactions. For the cases of isothermal homogeneous tubular, loop/recycle and tank reactors, the two-mode models are described by a pair of coupled balance equations for the mixing-cup (Cm ) and spatial average (C) concentrations. The global equation describes the variation of Cm with residence time (or position) in the reactor, while the local equation expresses the coupling between local di%usion, velocity gradients and reaction at the local scales, in terms of the di%erence between Cm and C. It is shown that the two-mode models have many similarities with the classical two-phase models of heterogeneous catalytic reactors with the concept of transfer between phases being replaced by that of exchange between the two-modes. It is also shown that when the local Damk4ohler number (ratio of local di%usion to reaction time) is small, the solution of two-mode models approaches the exact solution of full CDR equations, while for fast reactions the two-mode models retain all the qualitative features of the latter. Examples are provided to illustrate the usefulness of these two-mode models in predicting micromixing e%ects on homogeneous reactions. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Mixing; Micromixing; Homogeneous reactors; Spatial averaging; Liapunov–Schmidt method; Convective–di%usion equation
1. Introduction The study of mixing e%ects on chemical reactions has been an active area of research since the pioneering works of Danckwerts (1958) and Zweitering (1959). This topic has become a part of classical chemical reaction engineering and has been discussed in several textbooks (Levenspiel, 1972; Froment & Bishcho%, 1990; Westerterp, van Swaaij, & Beenackers, 1984) and review articles (Villermaux, 1991). In the last decade, many computaional ?uid dynamics (CFD) codes have been developed to describe reactive mixing, but the task of solving the three-dimenisonal convection–di%usion–reaction (CDR) equations numerically is very expensive, especially for the case of fast reactions. Moreover, due to strong coupling between transport and reaction rate processes, the model equations for reacting ?ows are highly non-linear and are known to exhibit a variety of complex spatio-temporal behaviors. For most cases of practical interest, even with the present day computational power, it is impractical to explore the di%erent types of solutions and bifurcation behaviors that exist in the multi-dimensional parameter space, using the CFD ∗
Corresponding author. Tel.:+1-713-743-4318; fax:+1-713-743-4323. E-mail address: [email protected] (V. Balakotaiah).
codes. Traditional low-dimensional mixing models like the axial dispersion, recycle and tanks in series models, use a single parameter (like axial Peclet number Pe, recycle ratio , or number of tanks N ), and a single concentration variable for each species to describe macromixing. However, it is well known that these classical models cannot describe micromixing e%ects caused by local di%usion, local velocity gradients and reaction at the small scales. The main purpose of this article is to show that in a low-dimensional approach to mixing, we require a second concentration variable for each species and a local balance equation that captures micromixing in terms of the di%erence between the two concentrations. The two concentration variables are called the two modes of the system and these models are called two-mode models (TMMs). In this work, we present the TMMs for various homogeneous reactors and illustrate their application with examples. 2. Formulation of TMMs 2.1. Tubular reactors To illustrate the spatial averaging procedure, the derivation of the TMMs using Liapunov–Schmidt (L–S) technique
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(02)00647-4
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
of bifurcation theory and other main concepts of our approach, we consider the classical example of laminar ?ow in a tube with homogeneous reaction. We assume that the scalar concentration C( ; ; x; t ) in a tubular reactor obeys the CDR equation @C @C + ux ( ) = ∇ · (D⊥ ∇C) @t @x @C @ Dx − R(C); + @x @x
=p
@c @
+
1 @2 c 2 @ 2
(1)
(2)
with boundary and initial conditions given by 1 @c = u()[c − cin ] Pe @z @c =0 @z
at z = 1;
@c =0 @
at = 0; 1;
c(; ; z; t = 0) = c0 :
a2 ux t⊥ = ; C LD⊥
Da =
Pe =
tx ux L = ; C Dx
C LR(CR ) = ; tR ux CR
t⊥ a2 R(CR ) = = pDa; tR D⊥ C R t⊥ Dx a 2 p 2 = = = : tX D⊥ L Pe
(4)
We note that when p = = 2 = 0, or equivalently, when the transverse (local) di%usion time is vanishingly small compared to convection, reaction and macromixing time scales, transverse gradients vanish, and Eqs. (2) and (3) simplify to the classical one-mode axial dispersion model. If local di%usion time is small (but Fnite) as compared to the other three time scales, or in other words, a time/length scale separation exists in the system, we can write c(; ; z; t) = c(z; t) + c (; ; z; t);
(5a)
where c is the transverse averaged concentration, and c is the deviation of the concentration from the mean value c, which goes to zero, as p → 0. (Also, by deFnition, c = 0 for all values of p.) Multiplying Eq. (5a) by u() = u + u (), and averaging over the cross-section gives
1 @2 c @c @c − + Da r(c) + u() @t Pe @z 2 @z
≡ pg(c)
p=
2 =
where ux ( ) is the fully developed velocity Feld, D⊥ and Dx are the e%ective di%usivities of the species in the transverse and the axial directions, respectively, R(C) is the reaction rate, and ; ; x are the radial, azimuthal and axial coordinates of the tube, respectively. (We could take D⊥ =Dx =Dm (molecular di%usivity), but for generality and later use, we have assumed D⊥ = Dx .) Eq. (1) is subject to no-?ux, periodic, and Danckwerts boundary conditions in the radial, azimuthal and axial directions, respectively. In dimensionless form, Eq. (1) is given by 1 @ @
groups:
cm = c(z) + u c ;
(5b)
where cm is the mixing-cup (velocity weighted) concentration. For the case of a tubular reactor 1 1 2u()c(; z) d 2c(; z) d 0 cm = ; c = 0 1 : 1 2u() d 2 d 0 0
at z = 0;
c(; ; z; t) = c(; + 2; z; t); (3)
Here, z and are the dimensionless axial and radial coordinates, non-dimensionalized with respect to the length L and the radius a of the tube, respectively, t is the dimensionless time (t = t ux =L); c is the dimensionless concentration (c = C=CR , where CR is the reference concentration), u() is the dimensionless axial velocity proFle [u() = 2(1 − 2 ) for fully developed laminar ?ow in a tube], and r(c) is the dimensionless reaction rate [r(c) = R(C)=R(CR )]. The four characteristic time scales present in the model, namely the transverse di%usion time, t⊥ (=a2 =D⊥ ), the convection time, C (=ux =L), the reaction time, tR (=CR =R(CR )), and the axial di%usion (or macromixing) time, tx (=L2 =Dx ), lead to the following dimensionless
It may be mentioned that cm is representative of the convection (global) scale of the system, while c is representative of the reaction–di%usion (local) scale of the system. Eq. (5b) is the local equation, which shows that the di%erence between cm and c depends on the local velocity gradients (u ) and the local concentration gradients (c ) caused by molecular di%usion and reaction at the local scales. Micromixing is captured by the local equation as an exchange between the two modes (scales), cm and c. In order to determine c (and hence the term u c ), we substitute Eq. (5b) in Eq. (2) to obtain ∇2⊥ c = pg(c + c ):
(6a)
The L–S technique solves Eq. (6a) for c by expanding it in the parameter p as c =
∞ i=1
p i ci :
(6b)
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
Such an expansion is possible, since for p=0, the transverse di%usion operator in Eq. (2) has a zero eigenvalue with a constant eigenfunction. Thus, u c could be determined to any order in p. In practice, the leading term (that is of order p) is suJcient to retain all the qualitative features of the full CDR equation. (For details of L–S procedure, see Golubitsky & Schae%er, 1984; Balakotaiah & Chang, 2003, and Chakraborty & Balakotaiah, 2002). Here, we present the averaged form of the CDR equation to O(p): @c @cm 1 @2 c + Da r(c) = 0; + − @t @z Pe @z 2 cm − c = − 1 p
@cm @z
(7) (8)
1 @c = cm − cm; in ; Pe @z @c = 0; @z c = c0 ;
at z = 0;
(9)
at z = 1;
(10)
ux
at t = 0;
dcm cm − c =− ; dz 1p
(11)
cm = cm; in
1 pDa r(c):
at z = 0;
Cm − C dCm =− ; dx tmix
(12) (13)
We shall refer to this model as the two-mode convection model. In the limit of p → 0, this model reduces to the ideal plug ?ow reactor model.
Cm = Cm; in
Cm − C = tmix R(C):
at x = 0;
(14) (15)
Eq. (14) is referred to as the global equation, which gives the evolution of the mixing-cup concentration with axial position, while Eq. (15) is the local equation. Here, tmix is the local mixing time of the system, which is the characteristic time scale for exchange between the local and global scales, and is given by 1
a2 : D⊥
(16)
It is interesting to note that the two-mode convection model has a striking similarity with the two-phase model for heterogeneous reactions. This could be seen more clearly when Eqs. (14) and (15) are rewritten as ux
where 1 is constant that depends on the type and the geometry of the reactor and the ?ow proFle. For a tubular reactor with fully developed laminar velocity proFle, 1 is obtained as 1/48. We shall refer to this averaged model (Eqs. (7)– (11)) as the two-mode axial dispersion model. It may be noted that L–S averaging technique reduces the inFnite number of local (radial) modes present in the CDR equation to two radial modes cm and c, while retaining all the parameters of the CDR equation. As a result, the two-mode axial dispersion model, like the CDR equation, is capable of describing both macro- and micro-mixing e9ects through the parameters Pe and p, respectively. It is only in the limit of micromixing e%ects being very small, i.e. p → 0, that the two-parameter two-mode axial dispersion model (Eqs. (7)–(11)) could be simpliFed to the traditional one-parameter one-mode axial dispersion model, with an e%ective Taylor–Aris dispersion coeJcient (Taylor, 1953; Aris, 1956). For small but Fnite values of p, lumping of the two-parameters p and Pe into a single parameter and use of a single concentration variable is not valid. For the limiting case of Pe → ∞ (i.e. no macromixing is present in the reactor), Eqs. (7)–(11) under steady-state conditions, reduce to
cm − c =
In dimensional form, the two-mode convection model is given by
tmix =
with boundary and initial conditions given by
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Cm − C dCm =− = −R(C) dx tmix
(17)
with Cm = Cm; in at x = 0. The two-phase model for a heterogeneous wall-catalyzed reaction is given by ux
C m − CS dCm =− = −R(CS ) dx tTP
(18)
with Cm = Cm; in at x = 0. The spatially averaged concentration, C, in the TMM is replaced by the surface (wall) concentration, Cs , in the two-phase model, while the local mixing time tmix of the TMM is replaced by the characteristic mass-transfer time between the two phases, tTP in the latter. Just as the two-phase model can capture the mass-transfer limited asymptote in a heterogeneous reaction (which is missed by the pseudo-homogeneous model), so can the two-mode model capture the mixing-limited asymptote in homogeneous reactions, which is rendered inaccessible by the traditional one-mode models. The extension of the two-mode axial dispersion model to the case of fully developed turbulent ?ow in a pipe could be achieved by starting with the time-smoothed (Reynolds averaged) CDR equation, given by Eqs. (2) and (3), where the reaction rate term r(c) in the above equations is replaced by the Reynolds averaged reaction rate term rav (c). While in case of laminar ?ows, the e%ective di%usion term D⊥ (or Dx ) in Eq. (1) is same as the molecular di%usivity of the species Dm , in case of turbulent ?ows, it is given by D⊥ = Dm + DT ;
(19)
where DT is the turbulent di%usivity, which could be obtained on the basis of turbulent shear stress and expressed in terms of Reynolds numbers using a formula similar to that presented by Wen and Fan (1975). Also, the Reynolds averaged reaction rate rav (c) could be evaluated by using
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
simple closure models of Bourne and Toor (1977), Li and Toor (1986), etc. We skip the details of these methods, since the closure of the Reynolds’ averaged transport equation is not the focus of this paper. It should also be pointed out that the spatial averaging presented here is independent of the methodology using which rav (c) and DT are evaluated, or in other words, spatial averaging follows time averaging. We use an universal velocity distribution obtained by Churchill (2001) to approximate the fully developed velocity proFle u() across the turbulent core, given by
f f u() = 5:5 + 2:5 ln (1 − ) Re 2 8 15 10 + (1 − )2 − (1 − )3 ; 4 3
(20)
where f is the Fanning friction factor and Re is the Reynolds number. The important result is that the TMMs for a turbulent :ow tubular reactor are the same as those for laminar :ow tubular reactors. The two-mode axial dispersion model for turbulent ?ow tubular reactors is again given by Eqs. (7)–(11), while the two-mode convection model for the same is given by Eqs. (12) and (13), where the reaction rate term r(c) is replaced by the Reynolds averaged reaction rate term rav (c). The local mixing time tmix for turbulent ?ows is again given by Eq. (16) as in the case of laminar ?ow, D⊥ is given by Eq. (19), and 1 is given by
f 2 − : (21) 1 = 0:1f 2:05 + 2:5 ln Re + 8 f We note that in turbulent ?ows, typically Dm DT , as a result of which, the local mixing time tmix is independent of the molecular di%usivity or the molecular Schmidt number. 2.2. Loop and recycle reactors In this section, we present the TMMs for loop and recycle reactors. In a steady-state loop reactor of loop length L, a ?ow-rate of q enters and leaves the reactor at points s = 0 and l, respectively (where s is the length coordinate along the loop), and the total ?ow-rate in the loop is Q+q between points s = 0 and l, and is Q between points s = l and L, due to a recycle rate of Q. The recycle ratio is the ratio of the volume of ?uid returned to the reactor entrance per unit time to the volume of ?uid leaving the system per unit time, and is given by = Q=q. The two-mode model for a loop reactor is 1 − R(C); 0 6 s ¡ l; dCm 1 + us = (22) ds − 1 R(C); l 6 s 6 L; Cm − C = tmix R(C);
06s6L
(23)
with the boundary conditions Cm (s = 0) =
Cm; in + Cm (s = L) ; 1+
C(s = l− ) = C(s = l+ );
(24)
where the average velocity us is deFned as us = q=AC , and AC is the cross-sectional area of the loop reactor. For the special case when no reaction occurs between s=l and L, i.e., Cm (s=l)=Cm (s=L), the loop reactor reduces to a recycle reactor of length l, the two-mode model for which is given by us
dCm 1 R(C); =− ds 1+
Cm − C = tmix R(C);
06s6l
(25) (26)
with the boundary condition being given by Cm (s = 0) =
Cm; in + Cm (s = l) : 1+
(27)
The two-mode loop and recycle reactor models, like the two-mode axial dispersion model, are two-parameter TMMs. Here, the two parameters are the recycle ratio , and the local mixing time tmix , which describe macro- and micro-mixing e%ects in the system, respectively. 2.3. Tank reactors (CSTRs) It is well known that as the recycle ratio of a recycle reactor is increased the behavior shifts from tubular plug ?ow reactor at =0 (no macromixing) to a CSTR at =∞ (perfect macromixing). We use this idea to obtain the two-mode model for a perfectly macromixed CSTR, by integrating Eq. (25) along the length of the reactor s and simplifying the resulting equation for 1. This gives the two-mode model for a CSTR as Cm − Cm; in C − Cm = ; C tmix
(28)
Cm − C = tmix R(C);
(29)
where C is the residence time of the tank and tmix is the local mixing time. Eqs. (28) and (29), which are the global and local equations, respectively, constitute a two-mode one-parameter model for a perfectly macromixed CSTR. Micromixing effects are captured through the local mixing time tmix , and in the limit of complete micromixing (i.e. tmix → 0), it reduces to the ideal one-mode zero-parameter CSTR model. It is interesting to note that the local equations for the di9erent reactor types (Eqs. (15), (23), (26) and (29)) are the same. The reactor geometry or the ?ow Feld only changes the local mixing time of system. Alternate methods of obtaining Eqs. (28) and (29) include integrating the two-mode axial dispersion model
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
(Eqs. (7)–(11)) along the length of the reactor z, in the limit of complete macro-mixing i.e. Pe → 0. The TMM for tank reactors for the case of multiple reactions involving m species is given by −1 Cm − Cm; in = C Tmix (C − Cm ); −1 Tmix (C − Cm ) = R(C);
q2
q1 Zone A
QE Zone A
q1
where Tmix is an m×m matrix of the local mixing times. For the case of laminar ?ows with unequal species di%usivities, Tmix =tmix D−1 R where DR is the matrix of relative di%usivities of the species. For turbulent ?ows, where the local mixing times are nearly independent of molecular Schmidt numbers, DR ≈ I. The physical system equivalent to the two-mode model of a CSTR is a tank reactor, consisting of two zones, each of size V , namely, a non-reacting convection zone (A), represented by Cm , and a reaction zone (B), represented by C. The interaction between the zones A and B is quantiFed by an exchange of reactants and products from the convection scale (zone A) to the reaction–di%usion scale (zone B), which is represented by qE . This exchange of material between zones A and B, occurs only through local di9usion, and tmix (=qE =V ), which is the characteristic time scale for this exchange, scales as the local di%usion time of the system. In general, any inFnitesimal volume dV inside the tank could be so imagined to consist of two zones/ scales, and a corresponding two-mode model could be written (Eqs. (28) and (29)) for a CSTR of volume dV . If macromixing in the CSTR is complete, the two-mode model for any control volume dV could be integrated over the entire volume of the tank V to generate a single two-mode model (Eqs. (28) and (29)) for the whole tank. Macromixing e%ects are however not negligible in real tanks, and are in?uenced by several factors including the type and speed of impellers (turbines) and the manner of feed distribution. Macromixing e%ects in tanks have often been modeled by using compartment models in the mixing literature (Baldyga & Bourne, 1992). The two-compartment model, for example, divides the tank into two-compartments, namely the circulation zone and the impeller zone, which are then modeled as two interacting CSTRs, with macromixing being described as an exchange of material at rate QE between the compartments. Here, we use the two-compartment model to describe macromixing, with each compartment itself being described by a two-mode model that accounts for micromixing. The resulting model is a two-mode two-compartment model that describes both macro- and micro-mixing e%ects in tanks. A schematic of this is shown in Fig. 1. It consists of two interacting CSTRs of sizes V1 and V2 , each of which is described by a two-mode model. Materials are exchanged between zones A and B (micromixing) of each CSTR at rates of qE1 and qE2 , respectively, and between CSTRs (macromixing) at a rate of QE . The unsteady state two-mode two-compartment
q2
q E2
q E1
(30)
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QE Zone B
Zone B
CSTR 1 of Volume V1
CSTR 2 of Volume V2
Fig. 1. Schematic of a two-mode two-compartment model that describes both macro- and micro-mixing e%ects in tanks.
model is given by dC1 Cm1 − C1 C2 − C1 = + + R(C1 ); dt tmix1 tG1
(31)
Cin − Cm1 C1 − Cm1 Cm2 − Cm1 + + = 0; 1 tmix1 tG1
(32)
dC2 Cm2 − C2 C1 − C2 + − R(C2 ); = dt tmix2 tG2
(33)
Cin − Cm2 C2 − Cm2 Cm1 − Cm2 + + = 0; 2 tmix2 tG2
(34)
where C1 and C2 are the spatially averaged concentrations in compartments 1 and 2, respectively, Cm1 and Cm2 are the mixing-cup concentrations in the two compartments, 1 (=V1 =q1 ) and 2 (=V2 =q2 ) are the residence times of respective compartments, tmix1 (=V1 =qE1 ) and tmix2 (=V2 =qE2 ) are the local (micro-) mixing times, while tG1 (=V1 =QE ) and tG2 (=V2 =QE ) are the global (macro-) mixing times between the compartments. Needless to mention, the macromixing times tG1 and tG2 depend on the impeller type, tank geometry, and manner of feeding, while the micromixing times tmix1 and tmix2 depend on the local di%usional times and the local shear rates. We now attempt to obtain a one-parameter two-mode model that describes both macro- and micro-mixing e%ects in a tank reactor. This could be achieved by employing L–S technique in Fnite dimensions (Balakotaiah & Chang, 2003) if the macromixing times tG1 and tG2 are small as compared to the residence times 1 and 2 , respectively, i.e. macromixing is fast as compared to convection. We deFne a parameter -, which represents the deviation from the ideal CSTR model. Ratios of macro-/micro-mixing times and respective residence times (i.e. tmix1 =1 ; tmix2 =2 ; tG1 =1 ; tG2 =2 ) all scale as the parameter -, in case of small deviations from the perfectly macro- and micro-mixed ideal CSTR state. L–S reduction in Fnite dimensions is possible if there exists a zero eigenvalue for the system described by Eqs. (31)– (34), corresponding to - = 0 (i.e. ideal CSTR state). In this case, the eigenvector corresponding to - = 0 is given by
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
ˆ 1 1 1]T , where Cˆ is the concentration in a sin= C[1 gle ideal CSTR. L–S reduction of Eqs. (31)–(34) about the base state 0 (ideal CSTR state) results after some algebraic manipulations in a single two-mode model involving one mixing-cup concentration Cm and one spatially averaged concentration C, given by dC Cm − C − R(C); (35a) = dt tM
100
0
3. Application of TMMs In this section, we present three examples to illustrate the usefulness of TMMs in predicting micromixing e%ects on isothermal non-linear reactions occurring in tank reactors. For simplicity, we assume all species di%usivities to be equal in these examples.
η=0 80
Exit Conversion, X (%)
Cm − C Cm; in − Cm = ; (35b) tM C where C is the residence time of the whole tank, and tM is the e%ective mixing time, which incorporates both macroand micro-mixing e%ects. The e%ective mixing time tM depends intricately on the tank geometry, type and number of impellers, baPe positions, power dissipation in the system and feed distribution, as well as the local di%usion time. More complicated ?ow Felds like the one resulting from multiple/stacked impellers could be modeled by extending the same idea of compartments to n compartments or n interacting loops. It needs to be mentioned that this two-mode model (Eq. (35)) has been derived to O(-), and remains unaltered even for the general case of n interacting loops, in which case 2n equations are reduced to 2 equations using L–S technique. Eq. (35) could be derived only if macromixing times are small as compared to the residence time C , or in other words, exchange of materials between di%erent ?ow loops in the tank is fast as compared to convective transport. Otherwise, a two mode two-parameter model is necessary to capture both macro- and micro-mixing e%ects in tank reactors.
Ideal CSTR
k
A + B→P;
rate = kCA CB ;
where k is the second-order reaction rate constant. The balance equations in dimensionless form are given by 3 + cA 43 + cB cA; m = ; cB; m = ; 1+3 1+3 cA; m − cA = 3DacA cB = cB; m − cB ;
(36a)
η=0.1 η=0.5
η=1.0 40 η=2.0
20 η=10.0
0 0.01
0.1
1
10
100
Da
Fig. 2. Variation of exit conversion, X , with Damk4ohler number, Da, for a bimolecular second-order reaction in a CSTR, for di%erent values of the dimensionless local mixing time, 3.
where 4 is the feeding ratio of B to A (4 ¿ 1, if A is the limiting reactant), the Damk4ohler number, Da is given by Da = kCA; in C , and the dimensionless local mixing time 3 is given by 3 = tmix =C . The exit conversion X (=1 − cA; m ) could be obtained by solving the quadratic equation X = Da[1 − X (1 + 3)][4 − X (1 + 3)]:
(36b)
This is plotted in Fig. 2 as a function of Da for di%erent values of 3 and stoichiometric feeding (4 = 1). This Fgure shows the variation of the exit conversion X with Damk4ohler number Da, for di%erent values of the dimensionless local mixing time 3. The case of 3 = 0 corresponds to the ideal CSTR. For 3 ¿ 0 and Da → ∞, the local concentrations approach zero (cA = cB = 0), while the mixing cup concentrations approach the mixing limited asymptote of
3.1. Bimolecular second-order reactions Bimolecular second-order reactions provide the simplest example of non-linear kinetics, where micromixing has signiFcant e%ects. We use the TMM for a CSTR (Eqs. (28) and (29)) to examine the same in the case of a reaction of the type
60
A+B k>P CA,in= CB,in Da = k CA,inτC η= tmix / τC
cA; m =
3 ; 1+3
cB; m =
43 ; 1+3
X = cP; m =
1 : (37) 1+3
We note that this mixing-limited asymptote for homogeneous reactions is analogous to the mass-transfer limited asymptote for wall-catalyzed reactions. Just as the wall (surface) concentrations approach zero for the case of inFnitely fast surface reactions (while the bulk/mixing-cup concentrations remain Fnite), so do the local concentrations ci for inFnitely fast homogeneous reactions (i = A; B). Unlike in catalytic reactors, where exchange between the phases occurs at the solid–?uid boundary, the exchange between modes (scales) in homogeneous reactors occurs in the entire domain.
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
reaction, given by Da2 = k2 CB; in C . Fig. 3 corresponds to the case when the Frst reaction is inFnitely fast and A and B are fed in stoichiometric amounts [i.e. CA; in = CB; in , and CR; in = CS; in = 0]. While no S is formed for the case of 3 = 0 (ideal CSTR), a signiFcant increase in yield of S is obtained if Fnite micromixing limitations are present in the system. The maximum yield of S, obtained when the mixing limited asymptote is attained also for the second reaction, is 23 1 + 23 for 3 6 1; YS; max = (39) 2 for 3 ¿ 1: 1 + 23
60
A+B 50
Yield of S, YS (%)
B+R 40
k1
>R
k2
η = 0.05
>S
CA,in= CB,in Da2 = k2 τC CB,in η = tmix / τC
30
η = 0.1
20
η = 0.2
10
0 0.1
η = 0.5
1
10
100
1059
Thus, in this case, an optimal yield of S exists, and obtained for 3 = 1, which is obvious intuitively. 1000
Da2
3.3. Polymerization reactions
Fig. 3. Variation of the yield of S with Damk4ohler number for a competitive-consecutive reaction scheme, when the Frst reaction is inFnitely fast.
3.2. Competitive-consecutive reactions k
1 Competitive-consecutive reactions of the type A + B→R; k2 are prototype of many multistep reactions like niB + R→S tration of benzene and toluene, diazo coupling, bromination reactions, etc. Experimental observations (Li & Toor, 1986) show that if the Frst reaction is inFnitely fast as compared to the second one (i.e. k1 =k2 → ∞), under perfectly mixed conditions B is completely consumed by the Frst reaction and the yield of S is zero (if A and B are fed in stoichiometric amounts). However, it was observed that if the mixing of A and B is not attained down to the molecular scale, the Frst reaction is not complete and there remains a local excess of B, which can then react with R to produce S. The yield of S increases monotonically as the reaction rate of the second reaction increases, Fnally attaining a mixing limited asymptote. We use the TMM for a CSTR to verify this observation. If A and B are fed in stoichiometric amounts, the local excess of B that remains after the Frst reaction attains its mixing limited asymptote (in the limit of Da1 = k1 CB; in C → ∞), (1) is obtained from Eq. (37) as CB; m =CB; in = 3=(1 + 3), while the amount of R formed from the Frst reaction is given by (1) CR; m =CB; in = 1=(1 + 3), where 3 is the dimensionless mixing time of the system. The second reaction is then simply a bimolecular second-order reaction between B and R, as in example 3.1. The resultant yield of S is given by YS , where 2CS; m (38) YS = CR; m + 2CS; m
and CR; m and CS; m are the exit concentrations in a CSTR in which only the second reaction occurs and with a feed (1) (1) concentration of CB; m and CR; m . Fig. 3 shows the increase in the yield of S with Da2 , the Damk4ohler number of the second
It is well known that in polymerization reactions, mixing a%ects monomer conversion, copolymer distribution and molecular weight distribution (Villermaux, 1991). In linear polymerization systems, imperfect mixing is found to broaden the molecular weight distribution (MWD), while in non-linear polymerization with signiFcant branching, depending on reaction conditions, imperfect mixing can broaden or narrow the MWD (Zhang & Ray, 1997). Here, we verify the Frst of the two above-mentioned observations by examining the case of an anionic polymerization using the TMM for a CSTR. Anionic polymerization, often used industrially to produce polymers of narrow MWD, is typically characterized by the lack of a termination step. The kinetics is k
I I + M →P I;
k
P Pj + M →P j+1 ;
where the rate of the initiation (assuming a constant initiator concentration) is given by RI = kI M;
(40)
while the rate of propagation is given by RP = kP MPj ;
(41)
where kI and kP are the initiation and propagation rate constants, respectively. Application of the TMMs for a CSTR to the above kinetics results in a set of non-linear algebraic equations, which when solved gives the MWD and the poly-dispersity index. Fig. 4 shows the variation of the MWD with n, where the MWD is deFned as ∞ MWD = n2 Pn nPn (42) n=1
and n is the chain length of a polymer chain Pn . The MWDs shown in Fig. 4 correspond to the parameter values of Da = kI C =10−3 and kP =kI =2×105 , where Da is the Damk4ohler number and C is the residence time of the tank. While the 3 = 0 case shows that the MWD for a perfectly mixed CSTR
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S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
0.5
0.3
η=
0.2
5 η= 2 η= 1 η=
MWD
0.4
0
0.1
0 0
500
100 0
150 0
200 0
250 0
300 0
n Fig. 4. E%ects of mixing on polymer properties: variation of MWD with chain length n, for di%erent values of the dimensionless mixing time, 3.
is fairly narrow, a signiFcant broadening of the MWD is observed as the dimensionless mixing time of the system, 3 (=tmix =C ), is increased. 4. Discussion and conclusions As discussed in the Section 1, a complete quantiFcation of the mixing process by solving the CDR equations from the largest (reactor scale) all the way to the smallest scale (molecular di%usion and reaction scale) is impractical, necessitating the development of low-dimensional models, involving the two representative scales: global and local. Spatial averaging by the LS technique gives us a framework for formalizing such dimension reduction for reactive ?ows. The global equation of the two-mode model describes evolution of the mixing-cup concentration with residence time (or position) in the reactor, while the local equation characterizes the interaction between the global (convection) and local (reaction–di%usion) scales. Exchange of material between the two scales (i.e. C and Cm , respectively) describes micromixing present in the system, characterized by a local mixing time, which depends on the local di%usional time and the local shear rates. An additional parameter like the axial Peclet number (Pe) in the axial dispersion model, the recycle ratio () in the recycle model, the number of tanks (N ) in the tanks-in-series model or a dimensionless exchange time tG in the two-compartment model of a single tank, may be used to characterize macromixing in the system. However, these traditional models cannot describe micromixing e%ects. A two-parameter two-mode model is necessary to characterize both macro- and micromixing in the system. Whether two modes are su=cient to describe mixing in reactors remains a question that could also be answered by using the L–S technique. The local equation, which has been derived to O(tmix ) in this paper, could be derived to
all orders in tmix using Eq. (6). The region of validity of the two-mode model could then be speciFed by examining the radius of convergence of the inFnite series in the local equation (Eq. (5b)). For example, the region of convergence of the two-mode convection model for a Frst-order reaction is given by (Chakraborty & Balakotaiah, 2002) Daloc = ktmix 6 0:858, where k is the reaction rate constant and Daloc is the local Damk4ohler number. For the reaction schemes considered in Sections 3.1 and 3.2, it as been shown that the solution of the two-mode convection model for tubular reactor matches accurately (to 3 decimal places) with the numerical solution of the CDR equation for all Daloc ¡ 0:1, and retains the qualitative features of the latter for fast reactions (i.e. larger values of Daloc ). A similar higher order expansion of Eq. (6) for the case of tank reactors shows that the two-mode model holds good for all values of the local Damk4ohler number. As illustrated in Section 3 with examples, the TMMs have a striking similarity with the two-phase models of catalytic reactors, with the following one-to-one correspondence: mass-transfer coeJcient → exchange coeJcient, surface (wall) concentration → spatially averaged concentration C, mass-transfer limited reaction → mixing-limited reaction. The analogy could be carried further by noting that for all cases of well-deFned ?ow-Felds, where mass-transfer coeJcients can be estimated theoretically, we can also estimate the exchange coeJcient 1 or the local mixing time tmix of the TMMs. For more complex ?ow-Felds (e.g. packed bed), the local mixing time, like the mass-transfer coeJcient, could be correlated to Re, Sc and the geometrical characteristics of the system. Thus, the TMMs of homogeneous reactors are as general as the two-phase models of catalytic reactors and have a similar range of applicability. It has been shown elsewhere (Chakraborty & Balakotaiah, 2002) that the TMMs can predict multiple solutions in case of autocatalytic kinetics, capture e%ects of di%erences in species di%usivities and non-uniform reactant feeding on conversion and predict mixing-limited asymptotes in non-linear reactions. The TMMs could also be easily extended to the case of non-isothermal reactions. To summarize, the TMMs are the minimal models that provide a low-dimensional description of mixing, by coupling the interaction between chemical reaction, diffusion and velocity gradients at the local scales to the macro-scale reactor variables (such as mixing-cup concentration, ?ow-rate and reactor size or residence time). Due to their simplicity and generality, it is hoped that they will Fnd applications in the preliminary design and optimization of homogeneous chemical reactors. Acknowledgements This work was supported by grants from the Robert A. Welch Foundation, the Texas Advanced Technology Program and the Dow Chemical Company.
S. Chakraborty, V. Balakotaiah / Chemical Engineering Science 58 (2003) 1053 – 1061
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