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Mixing, reaction and precipitation: Environment micromixing models in continuous crystallizers—I. premixed feeds
Computers them.Engng,Vol.16,No.IO/II, pp.923-936, 1992 Printed in Great Britain. AU rights reserved
009th1354/92f5.00 +O.OO
Copyright 0 1992Pwgamon
Pms Ltd
MIXING, REACTION AND PRECIPITATION: ENVIRONMENT MICROMIXING MODELS IN CONTINUOUS CRYSTALLIZERS-I. PREMIXED FEEDS N. S. Department
TAVARE
of Chemical Engineering, University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88, Sackville Street, Manchester, U.K. (Received 22 February 1992;ji~l revision received 23 March 1992; received for publication 2 April 1992)
Abstract-The two-environment micromixing model of Ng and Rippin is extended to a process involving an elementary chemical reaction between two reactant species and subsequent crystallization of a product
in a continuous crystallizer. The model formulation in Part I of this paper deals with the premixed feeds case and assumes that the premixed reactants first enter a completely segregated entering environment and subsequently transfer to a maximum-mixedness leaving environment at a specified rate as defined by an environment transfer function. The sensitivity of this model to several process parameters like the DamkBhler number, micromixing parameter and dimensionless inlet concentration of excess reactant is explored. This computationally efficient two-environment model appears to characterize satisfactorily the micromixing effects in a continuous reactive precipitator with premixed feeds.
system. The scope of Part I of this paper is restricted to analysis of premixed feeds only. A variety of environment models have been proposed by many authors in the chemical reaction engineering literature because of their simplicity, versatility and computational economy (see e.g. Table 7; Tavare, 1986). Formulation of models for complete segregation and maximum mixedness for a continuous precipitator with any RTD (residence time distribution) function, as depicted by plug flow vessels with side exits and entrances respeztively, represents the intrinsic micromixing level in an environment (Danckwerts, 1958; Zwietering, 1959). In general, environment models for chemical reactors assume that the vessel volume consists of two or more environments having intrinsic, extreme states of micromixing. The reactors with premixed feeds may be modelled by either parallel (e.g. Villermaux and Zoulalian, 1969) or series (e.g. Ng and Rippin, 1965) environment models. In the parallel environment model the feed may be split into segregated and maximum-mixedness environments and the product is obtained by combining them at the reactor outlet with no interaction among the parallel environments. For the series’ two-environment model the entire feed stream may be fed to a segregated entering environment and subsequently transferred to a maximum-mixedness leaving environment at a prescribed rate, the effluents of these series environments h&g combi&. at the vesse1 exit. Several variations of these environment
INTRODUCTION
Characterization of intermediate micromixing or partial segregation for a reactive precipitation (or crystallization) system is important (Danckwerts, 1958; Becker and Larson, 1969; Garside and Tavare, 1984, 1985; Tavare, 1986, 1989). Pohorecki and Baldyga (1983) attempted the development of a model characterizing a micromixing level. Their single-parameter model was based on continuous mass transfer between a point and its environment to describe the molecular dissipation zone as originally proposed in IEM (Interaction by Exchange with the Mean) model by Costa and Trevissoi (1972). The model parameter was determined from the spectral interpretation of mixing in an isotropic homogeneous turbulent field. The influence of mixing intensity of the product precipitate and its crystal size distribution for a fast reactive system was evaluated. Tavare (1991) also extended an analysis of IEM micromixing model to a process of reactive precipitation. Two specific cases of feed conditions, viz. premixed and unpremixed feds were found to influence signiticantly both reaction and crystallization performance characteristics. The sensitivity of these two cases to several process and model parameters was explored and the model description for the premixed feeds case appeared better to represent a nearly segregated configuration. The purpose of this article is to extend the study of environment models to a reactive precipitation 923
N. S. TAVARB
924
Product E(B) %A EC
niLI
y=.
1
EE m-e.
,_,::I:
T
-
T
.
_I
T TA
B E(B)
+-I A=0
Fig. 1. Schematic representation of two-environment model for the premixed feeds case. models have been proposed differing either in structurei of environment interaction or in terms of transfer rates (see e.g. Weinstein and Alder, 1967; Rippin, 1967; Nishimura and Mats&am, 1970; Methot and Roy, 1971; Rao and Edwards, 1973; Goto and Matsubara, 1975; Takao and Murakami, 1976; Ritcbie, 1980). Environment models are essentially system specific and empirical in nature. Recently some mechanistic implications have been pointed out by establishing an analogy with the theory of isotropic turbulent mixing and its equivalence of other models. Baldyga and Pohorecki (1986) investigated the influence of mixing conditions on a reactive precipitation process for a premixed feeds case by employing atwo-en vironment mode1 under certain simpli5ed assumptions based on experimental results. They assumed that nucleation was a dominant process in the completely segregated entering enviromnent
whereas only crystal growth occurred in the maximally-mixed leaving environment. Effects of residence time, inlet concentration of reactants and micromixing parameter on the product precipitate were evaluated and compared with the experimental results obtained for precipitation of barium sulphate crystals in order to assess the adequacy of the model.
MODELDESCItIt’ilON For the present purpose the mode1 development for a continuous crystallizer will follow similar lines to that used for chemical reactors by Ng and Rippin (1965). A schematic representation of a continuous crystalhxer with premixed feed and its two-environment mode1 is shown in Fig. 1. The archetype overall reaction considered is: A+B+C,
rc=kc,,cs,
(1)
Mixing, reaction and ptipitation-I where the two species, A and B, react together homogeneously with first-order reaction kinetics with respect to each of the reactants; the component A being assumed limiting. Precipitation of the solid product C resulting from this liquid phase reaction occurs simultaneously when the fluid phase becomes supersaturated with respect to component C. Conventional power law expressions of the form: G = k,Ac8,
(2)
B=k,Acb=KxGi,
(3)
and
are used to represent the growth and nucleation kinetics of the precipitation process, respectively. All the product in both solid and liquid phases, together with tuneacted material, leaves the crystallizer through a single exit. In general the feed stream can be ascribed by its RTD, flowrate and compositions. In the model formulation the vessel is divided into two environments, the entering environment in which the fluid elements are completely segregated and the leaving environment which is always in the state of maximum mixedness. Material transfer from the entering environment is assumed to take place at a rate which is proportional to the amount of material remaining in the entering environment and with a specific rate of transfer parameter R which can be used as a micromixing parameter of the model. When R is zero, there is no transfer from the entering to the leaving environment and the whole vessel is completely segregated. When R is infinitely large the entire entering material is immediately transferred into the leaving environment so that the vessel is in the state of maximum mixedness. Intermediate values of R may thus provide degrees of micromixing. In order to provide a unified mathematical formulation of the model and evaluate the performance characteristics of a continuous reactive crystallizer the following dimensionless variables may be defined:
925
where g is the constant of proportionality and also the dimensionless micromixing parameter of the model. Reactants A and B are premixed at the entry of the entering environment where they reside reacting in a completely segregated state for their age + and subsequently transfer to the leaving environment where they spend their residual lifetime 1 reacting in a state of maximum mixedness. Of the material leaving the crystallixer the fraction that enters and remains in the crystallixer for ages between 0 and I,+ is P(+) and is related to the RTD function B(e) as:
E(@)=fg% Of the material of age + the fraction remaining entering environment from equation (4) is: F@(S ) = exp( - rlJI ) and the fraction
(6)
in the leaving environment
FL($) = 1 - exp(-q*).
is: (7)
Entering environment As the entering environment of the vessel is completely segregated each clump of reactive mixture can be considered as a small batch vessel. The concentration profiles of species A and C for this reactive clump can be given as: -&(8-1+x*),
where
(10) The pertinent
1 dW Q=~~,
in the
boundary
x,=1
and
conditions xc=0
are:
at @=O.
(11)
q=Rr.
The transfer of material between the environments is assumed to be at a rate proportional to the amount of material remaining in the entering enviromnent and thus first-order with respect to mass in the entering environment. This can be described by a transfer function which depends on the age rj as: (4)
The population balance equation for such a batch conIiguration with negligible breakage and agglomeration effects is:
d”++o, w with the boundary
conditions
n(L, 0) = 0;
(12) as:
n(0, #) = no.
(13)
N. S. TAVARB
926 The moment transformation respect to size yields:
%
w
=&
of equation
(12) with
The population balance equation for the solid precipitate C in the leaving environment is: &IL
&IL
-+yL;vI*]=o, 2T'"ar.
(14)
’
(27)
where (15) VII =
(16)
(17) with boundary
mn(L,#)exp(-rl$)Hti s0
The appropriate
boundary *=O drl
at * = 0, j = 0, 1,2,3.
(18)
The variation of crystal size in a batch clump may be represented as: dL -= dJI
conditions as
I-r00
are:
9
(29) (30)
The pertinent moment equations through moment transformation with respect to size as:
may be obtained of equation (27)
(31)
(19)
GT’
(28
n(0, II) = no.
conditions:
pj = 0
+l)d$.
with L = 0 at tj = 0 as the initial condition. (32)
Leaving environment Following Ng and Rippin (1965) [and see also Wen and Fan (1975)] the dimensionless concentration profiles for the limiting reactant A and product C in the leaving environment may be written as follows:
-t&xL,(B-l+x:)+q[““t;,
(33)
(34) IV],
(20) with the boundary
+-rx:(B-I+&)
conditions
dp; ==O
as
as: 1-rco.
(21)
+rl[Ilh;;VI]+B,
The weighted moment m, used equations may be evaluated as:
where II =
m [l
50
III = s0 IV= s VI=
- exti-tlJr)lW
(35)
+ A)W.
(22)
in the
moment
m VIILjdL.
mj=
(36)
s0
w exp(-?$)E($
+ A) ddr,
(23) Crystallizer outlet
om+(#)exp(-q$)E($
+ 1) d+,
(24)
x&)exp(-96)E($
+ A) d&
(25)
The appropriate boundary conditions are the same as that taken by Zwietering (1959) for maximum mixedness, i.e.
d”z=O and cu
&‘=O YE
atA+oO
(26)
The mean concentrations at the vessel outlet are obtained by mixing together, as specified by the RTD, the material in the leaving environment with a zero life expectation and the material still in the entering environment with a zero life expectation. The fractional quantity of reactant A and product C still in the entering environment is obtained from IV and VI, respectively, at 1= 0. Thus the mean exit dimensionless concentrations are: .% = [II& +Ivl1,-0.
(38)
&=P&+VIlI,_,.
(39)
Mixing, reaction and ptipitation-I Siilarly the mean population density of the product precipitate C can be obtained as:
A(L) = [IInL(I;) +vII)l*,o,
927
It is convenient to convert the integrals in equations (48) and (49) into diEcrential equations as:
WI
where VII and nL(L) are the population density functions evaluated at A= 0 for the entering and leaving environment from the population balance equations [equations (12, 2711, respectively.
(51) 2
= xcexp[-(q
+ I)*],
with and
&=O
.~?cc=O at $=O,
Application of RTDs Two specific cases of exit age distributions (i.e. RTDs) viz. an exponentially decaying RTD correspending to a backmixed or an MSMPR (mixedsuspension mixed-product removal) crystallizer as: E(@)=exp(-6’)
E(8) = 48 exp( -29).
JI1=l+?j III =
&
m,=exp(--r2)rii:,
Pm (55) For the other RTD of two stirred tanks-in-series [equation (4211the integrals in equations (22) and (23) yield:
exp(-l),
exe(--11,
&(n (44)
(46)
VII = @(L)exp(
-A),
z=A =
SE=
-xA s0
exp[-(rl + WI W,
0
+lMlW
1
exp(-U).
r1+2 >
(57)
+ RL],
(58)
VI =4exp(-21)[fi&
+ a&,],
(59)
where % 1ZE, and Z& are defined, respectively, as:
(48)
-E
(49)
and
1 +-
IV = 4exp(--U)[.Gt,
-
X.%2-
5axcew[-(~
(56)
The integrals in equations (24) and (25) require the numerical evaluations in two parts as:
(47)
where Rz, 2: and PI@(L)are the average dimensionless concentrations of A and C and the average population density function at the crystallizer outlet from the entering environment, respectively. These quantities are detined as:
4 r1+2 (
(45)
VI=feexp(-A),
+&)]exp(-2A)
and III =-
IV =Ziexp(-A),
(54)
where P$ is the average weighted moment at the crystallizer outlet from the entering environment defined as:
(42)
are considered for the present study. For a perfectly backmixed flow system with an exponentially decaying RTD [equation (4111 the evaluations of integrals in equations (25-28) at any residual lifetime 1 yield:
(53)
as the initial conditions. The values of 6 and Zc determined by the integration of equations (51) and (52) for $ + cc will equal Xi and a& respectively. The weighted moment m, in equation (36) for the single tank RTD results in:
(41)
and a function depicting the behaviour of two wellstirred vessels of equal volume in series as:
(52)
sow
XA+ewl-(2 + tlM1dtk
-6
(61)
(62)
xc1 =
and Ha(L) =
m x(L
#)exp[--(rl + l)$l dti.
(50)
_fg=
I
~xc~cxpI-_(2+sWlW.
0
(‘53)
N. S.
928
These integrals [equations @O-63)], as in the case of an exponentially decaying RTD, can be evaluated by transforming them into a set of differential equations and then integrating them numerically from # = 0 until $ + co. The integral in equation (28) can also be evaluated by splitting it into two parts as: VII = 4 exp(-21)[12nF(l)
+ A:(L)],
TAVAM segregation throughout the remainder of its stay within the entering environment. Each of these clumps thus acts as a small batch reactor and transfers the material from the entering environment to the leaving environment. This transfer rate between environments is assumed to be at a rate proportional to the amount of material remaining in the entering environment.
(64) Entering environment
where
A:(L) =
s
an(L, Ilr)exp[-(2
+sMl dr(r (65)
0
The weighted yields:
moments
in equation
(36) for this case
mi=4exp(-21)[Lr$,+fi@,
(67)
where m A$=
$(L)L’dL
(68)
s0 and
s m
lit;=
ii,e(L)LjdL.
(69)
0
MODEL
Computational
EVALUATION
details
In order to evaluate the performance characteristics of a reactive precipitation system described by the proposed model the specific set of parameters is given in Table 1. These physicochemical parameters are similar to those used in the previous studies (Garside and Tavare, 1984; Tavare, 1991). In the foregoing model description it is assumed that the two streams mix completely just at entry of the entering environment after which each clump of premixed feeds is assumed notionally to preserve its Table 1. Parameter vahes used in model calculations Molecular weigh of C, kg kmol~’ Solubility of C, c*=, km01 kg-’ Fad conoeatration of A, based on total flow, c&o. km01 kg-’ B (=%lJG) Y (=&r) rl (=R*) Crystal density of C, pc kg me3 Area shape factor Volume shape factor Mean rraidena time, 5, s Nuckation order, b Nuckation rate cxmstant kb, No./[s~kg(kmol Growth rate order, g Growth rate constant. k.. m/ls~kmol kc’Pl
100 0.001 0.002 I.5 10.0 0.1, 1, 10 2660 3.68 0.525 1000 kg-‘)‘]
The behaviour of a clump in the entering environment is described by the set of 11 equations [equations (8-18)]. Initially only reaction occurs in each of these clumps until the concentration of product C reaches saturation point. As soon as the clump gets supersaturated with respect to C, nucleation and subsequent particle growth begin and result in evolution of the crystal sire distribution. Consequently only those equations depicting the concentration profiles and evaluating average exit concentrations of the entering environment are integrated initially, after a having been set equal to zero, until the concentration of C in the liquid phase reaches the saturation value. Thus a set of four equations [equations (8,9), (51,52)] for the RTD of a single tank and a set of six equations [equations (8,9) and an additional four equations obtained from equations (60-63)] for the RTD of two stirred tanksin-series need to be integrated with the appropriate boundary conditions. Once the clump of the reaction mixture becomes supersaturated with respect to C all the differential equations describing concentrations and moments along with the partial differential equation [equation (12)] and the integrals evaluating the average population density [equation (SO)] or equations (65-66) and its moments [equation (55) or equations (68-69)] at the vessel exit from the entering environment need to be integrated simultaneously with the appropriate boundary conditions. A set of nine differential equations [equations (8-9) (14-l 7), (19), (5 l-52)] for the case of a single stirred-tank RTD and a set of 11 differential equations [equations (t&9), (14-17), (19), additional four equations corresponding to equations (60-63)] for the case of two stirred tanks-in-series RTD need to be considered. All the differential equations involved were integrated by the fourth-order Rung&Lutta method with an integration step length of A$ = 0.0001 starting from the appropriate initial conditions until $ = 10. The partial differential equation [equation (12)] was solved by the modified method of numerical integration along the characteristics with a specified grid length of sixe while the integrals in equations (50) or (65-66) and equation (55) or equations (68-69)
Mixing, reaction and were evaluated numerically using Euler’s formula. In the algorithms the set of differential equations depicting concentrations and moments was initially integrated with a step length of A@ = 0.001 until the increment in size was equal to the grid length of 2 pm used in the solution of the partial differential equation. The growth rate and hence nuclei population density no is delined at the end of the grid and the solution of the partial differential equation moved forward by the time required to increase the size by one grid length (2 pm). The step length for time in evaluating the integrals in equation (SO)or equations (65-66) was also the same, i.e. the time required to increase the size by one grid length (2 pm). The step length for size in evaluating the integrals in equation (55) or equations (68-69) was equal to one size grid length (2 pm). Leaving environment The concentration profiles in the leaving environment are described by equations (20-21), required integrals in equations (22-25) being evaluated analytically in terms of residual lifetime, 1 i.e. equations (43-46) for the RTD of a single stirred tank and equations (56-59) for the case of two stirred tanks-inseries. It is necessary to point out that an opposite sign to that used by Ng and Rippin (1965) [their equation (811 is used to describe the concentration profiles for A and C as in equations (20-21). As the life expectation increases in the opposite direction to teal time their sign convention in their equation (8) is correct for the analytical solutions for the leaving environment. In the present study the numerical integration process starts at A--, ODtherefore the concentrations profiles in equations (20-21) are described in the same sense as the age or real time and have the opposite sign to that of their equation (8). Numerical integration of all the equations for the leaving environment commences from a suitable high value of I, say L = IO,with the initial conditions derived from the concentration prosles [equations (20-21)] to satisfy equation (26) at the entry and finishes at the outlet where 1 = 0 to yield the exit values from the leaving environment. Initially equations depicting the concentration profiles in the leaving environment [equations (20-21)] are integrated with o = 0 until the concentration of C in the liquid phase reaches the saturation value. Once the reaction mixture becomes supersaturated with respect to C the set of seven differential equations [equations (20-21), (31-34), (3711 is integrated along with the partial dilYerential equation [equation (2711 depicting the population balance equation in the leaving environment. The other features used in numerical calculations were similar to those used for the entering environment.
precipitation-1
929
RESULTS AND DJSCUSSION
Using the physicochemical parameters listed in Table I calculations were performed to evaluate the crystallizer performance characteristics as predicted by the proposed model for the premixed feeds case. The resulting product size distributions from equation (40) as illustrated by the conventional population density plots for typical cases of this model are shown in Fig. 2. Also reported in Fig. 2 are the population density plots under otherwise similar conditions for the extremes of micromixing, viz. maximum mixedness (Mode1 I), MM(I) and complete segregation (Model II), CS(Il), from the previous studies (Garside and Tavare, 1985). Corresponding details of all these cases regarding dimensionless concentrations and product size distribution statistics are included in Table 2. According to the mode1 description when 11approaches zero the transfer from the entering to leaving environment reaches zero and the whole vessel therefore tends to be in the completely segregated entering environment. For a large value of n most of the material from the entering environment is transferred to the leaving environment occupying most of the vessel and then the whole reactor approaches the state of maximum mixedness. Thus with an increase in micromixing parameter, u there should be a gradual movement of the performance characteristics from the completely segregated to the maximum mixedness case. Both the reaction and crystallization performance characteristics reported in Fig. 2 and Table 2 appear to show this trend generally and approach these extreme limiting cases of micromixing over the range of micromixing parameter 9. Eflect of Damkiihler number y The dimensionless reaction group y, i.e. the DamkBhler number, can characterize the dimensionless concentration of A at the vessel exit and hence determines the reaction performance keeping all other parameters in Table 1 constant. The sensitivity of both the reaction and crystallization performance characteristics to the DamkGhler number was explored by varying y over the range 0.1-1000, covering a l@-fold range. The results of these calculations are reported in Figs 3 and 4. The variations of dimensionless concentrations of A and C at exit, LZ~and &, with the Damkiihler number y at a typical value of tbe micromixing parameter (u = IO)are shown in Fig. 3. Also included in Fig. 3 are t&se variations for extremes of micromixing levels and RTD of two stirred tanks-in-series. Although the difference between the dimensionless
N. S. TAV~
930
IC
c 10
0
2000
1000
3000
L (pm1
Fig. 2. Product population density plots (data as in Table 1). concentrations at two extreme micromixing levels is small, the calculated dimensionless concentration at any y lies within the small range and follows the correct trend, i.e. with an increase in q movement of concentration value from the completely segregated to the maximum mixedness case. Because of the small range of concentration between two extreme micromixing levels it appears that the dimensionless concentration changes are less sensitive to the micromixing parameter, 9 of this model. Low values of y result in lower reaction rates and yield lower conversion and hence higher _g*. With an increase in y, Z,, decreases and .Q increases rapidly at a lower y and then remains almost constant as a consequence of production of solid C. Similar variation of & has been reported by Ng and Rippin (1965) over the range of y from 0 to 100 in their original development.
Tabk n 0
0.1 1.0 10.0 co
2. Model
pd-nce
characterimks I;, (urn)
CVW (t/s)
(y = lo; Fig. 2) NT x 10-e (No. kc’)
5.
%
0.096
0.540
353
22.5
I.2
Ez 0.111
0.543 0.538 0.545
356 412 958
25.3 32.7 57.9
0.8 1.0 0.4
0.136
0.567
930
50
0.2
Figure 4 depicts the variation of crystallization performance characteristics at the vessel exit, viz. weight mean size, coefficient of variation and specific total number of crystals per unit of solvent mass. Also reproduced in Fig. 4 are these variations for extreme micromixing levels. The critical Damk%ler number, ytit at which the product solid C starts appearing depends on the micromixing parameter q. For a completely segregated system (q = 0) ysrit is lower (~0.1) and for a maximum-mixedness case (q + co) it is about 1. For an intermediate case the critical y lies between these two limits. At low values of y( < y&J only reaction occurs, the reaction rate is slow and controls the overall process. Increasing values of y (> yd) up to a value of about 5 produces significant changes in the sire distributions. The weight mean size increases rapidly and passes through a maximum. The variation of the weight mean size follows a correct trend. With an increase in q the peak in weight mean size increases and the curve moves towards that for the maximum mixedneas case. The coeSicient of variation passes through a slight maximum while the specific number of crystals per unit mass of solvent increases rapidly just after ypit and then mote slowly. Both these performance character&&s show a correct trend with the micromixing parameter q. Further increases in y then
931
Mixing, reaction and pmcipitation-I
0.6
0.6
04
Ix”
0.2
0 0.1
1
10
1000
100
Y
Fig. 3. Effect of DamkiihJernumber y on outlet concentrations of A and C.
produce comparatively small changes in the crystallization performance characteristics. Both weight mean size and coefficient of variation decrease with y, the decrease being more pronounced at higher micromixing parameter q. However, the specific numJxr of crystals per unit mass of solvent shows a slight increase. At higJrer y the reaction rate is fast resulting in local, sharper and earlier supersaturation peaks and thus a CSD having a higJrer number of crystals with smaller mean size and narrower distribution. At high y the results show the the performance characteristics of the CSD produced are closer to those of
a completely segregated system and less sensitive to the micromixing parameter q. EfJt
of micromixing parameter
q
Keeping aJ1 other parameters in Table 1 constant, q was varied over a range of 0. J-1000, thus covering a W-fold range of q. The resuJts of these calculations are reported in Fig. 5, the most sensitive range of q being l-1000. The variations of both zZ* and J& between the extreme micromixing limits are small (Fig. 3 and Tabk 2) and are not sensitive to q over the range. CrystaJJization performance
N. S. TAVARE
932
1200
800
400
0
1
10
100
Y
Fig. 4. Effect of DamkBhler number y on solid product characteristics.
characteristics, however, show substantial changes. The variation of the weight mean size with q shows a sigmoid curve, the point of in&ction being at about q = 10. At higher q the weight mean size exceeds the range predicted by extreme micromixing levels and it may perhaps be due to feeding of slurry from the entering environment to the leaving environment, operating at a higher level of supersaturation. The variation of the coeHicient of variation with q also shows a similar sigmoid curve with the point of inflection at about q =4 and exceeds the range of extreme micrommmg levels. The variation of the specific number of crystals per unit mass of solvent, however, shows an opposite sigmoid curve
showing a rapid decrease over the most sensitive range of q. Eflect of dimensionless
inlet concentration
of B, /I
The effect of the feed concentration for the case of premixed feeds was explored by changing dimensionless inlet concentrations of B, 4 over the range l-100, the rest of the parameters in Table 1 being kept constant. The results of these calculations are represented in Fig. 6. For this case Z* dccrroses because ofanincreascdreactionrateduetohigher~and~ remains practically constant as a consequence of two competitive procemes. Both these concentration profiles appear to lie within the range predicted in the
Mixing, reaction and precipitation-1
933
08
1200
06
600
04
400
0.2
E a 5 1-J
0
I
_I
0.1
10
1
100
’
0.1
looa
9
Fig. 5. Effect of micromixing
parameter q on performance characteristics
at crystallizer outlet.
ber of crystals per unit of mass increases thus producing many smaller particles with narrower distribution at higher /TX
extremes of micromixing (results not shown). With an increase in /3 both the weight mean size and the coefficient of variation decrease and the specific num-
25
1.2 20
Z 0
1.5
$
0.8
0 t e
%
‘0
I
I
I-?
10
z’
19 IX
;
04
-__0.5
/
C-...-..........._.,,(~.
_,.....,,...,....,.....
1
2
5
0
I
0 10
50
loo
B
Fig.
CACE
16/10-11-B
6. Effect of dimensionless
&let concentration crystalker
of B, /I, on performance exit.
characteristics
at
934
-__-
Single
tanks
v=lO
leq.41J
s + 2. 60
2.5
0.1
10
1
100
1000
Y
Fig. 7. Effect of Damkiihler numbery on solid product characteristicsfor the RTD of two stirred tanks-in-se&s.
RTD of two stirred tanks-in-series The performance characteristics of a crystallizer having RTD of two stirred tanks-in-series were evaluated over the range of y from 0.1 to 1000. The variations of & and Xc with y at a typical value of q = 10 are included in Fig. 3 and provide comparison with those for an RTD of a single stirred tank. The results of crystallization performance characteristics are reported in Fig. 7. Also reported in Fig. 7 for comparison are those characteristics of the two extremes of micromixing and micromixing parameter, q = 10 for an exponentially decaying single-tank
RTD. The product size distribution appears more sensitive to the micromixing parameter at higher values of y than that for the single-tank RTD case. For q = 10 both the weight mean size and the coefficient of variation pass through large peaks, thus producing a lesser number of large product crystals with wider distribution than those for the single-tank RTD case at higher values of y. MODEL In
eral
ASSESSMENT
foregoing analysis the effects of sevparameters (y. q. /I,RTD) on reaction and the
Miring, reaction and precipitation-I
crystallization performance characteristics of a reactive precipitation process in a continuous crystallizer are reported. Several other parameters like crystallization kinetics and other operating conditions may have a significant influence4 It is important to note that the present analysis is concerned with a global characterization of micromixing effects on the performance characteristics of a reactive precipitation system in a continuous crystallizer. Dimensionless concentration profiles of A are consistent with those reported in the literature (Ng and Rippin, 1965; Methot and Roy, 1971; Rao and Edwards, 1973). Studies concerning reactive precipitation systems are scarce. Only a study for the case of premixed feeds appears to be reported in the literature by Baldyga and Pohorecki (1986). They have simulated a fast reactive precipitation of barium sulphate crystals with its experimental verification. Although direct comparison with their study is difficult due to different physicochemical parameters used, the trends in the variations of mean size with micromixing parameters and residence time appear consistent with the present study. For their fast reactive precipitation system, the Sauter mean size increases with the micromixing parameter and decreases only slightly with the residence time (or the Damklihler number). Several authors (e.g. Rao and Edwards, 1973; Villermaux, 198 1, 1983, 1986; Villermaux and David, 1983) attempted to establish an equivalence relationship between all one-parameter micromixing models either on the basis of the turbulent mixing theory or by comparing the reaction performance. The micromixing parameter r) of the environment model as proposed by Ng and Rippin (1965) can be equivalent to twice the micromixing parameter of the IEM (Interaction by Exchange with the Mean) micromixing model as developed by Villermaux and coworkers (1981, 1983,1986). Mehta and Tarbell (1983) in their recent elegant work provided a link between the systems approach of an environment model based on population balance principles and a fluid mechanics approach in the framework of turbulence theory. They established that the transfer parameter R is generally equal to the reciprocal of the turbulent mixing time which may be estimated from the Corrin-Rosensweig turbulent mixing theory. The analysis presented here lays no claim to providing a detailed physical insight into the interplay of mixing, reaction and subsequent crystallization but provides rather an eflicient means to evaluate and subsequently predict performance for a continuous crystallizer configuration. The model parameters required should be characterized for the experimental response. The general technique of
935
using a known reactive crystallization system to evaluate the parameters of the model that matches the performance obtained experimentally for the crystallizer should be used. As the micromixing effects are much more important when the reactants enter separately than when they are initially premixed just before entry (see e.g. Treleaven and Tobgy, 1971; Ritchie and Tobgy, 1978, 1979; Ritchie, 1980) the modelling of such reactive precipitators with separate feed streams is presented in Part II of this paper. Acknowledgements-The author wishes to acknowledge with thanks Professors R. Pohorecki and D. W. T. Rippin for their comments on the original manuscript. NOMENCLATURE
Total crystal surface area, m2 (kg solvent)-’ Nucleation order Nucleation rate, No. s-I (kg solvent)-’ Concentration, km01 (kg solvent)-’ Saturation concentration, kmol (kg solvent)-’ AC= Concentration driving force (c - c l), kmol (kg solvent)-’ CV = Coefficient of variation based on weight distribution, % E(8) = Dimensionless residence time distribution (RTD), dimensionless exit age distribution F(8) = Dimensionless cumulative residence time distribution g = Growth rate order G = Linear growth rate, m s-r i = Relative kinetic order (= b/g) j = Index variable k = Reaction rate constant, kg (kmol)-‘s-l ka = Area shape factor k,= Nucleation rate constant, No. s-r kg-’ (kmol kg-‘)-’ k, = Growth rate constant, m s-i (kmol kg-‘)-’ k, = Volume shape factor KR = Relative rate constant, No. s’ -‘m-‘kg-’ L = Crystal size, m, gm mJ= jth weighted moment [equation (3611, No. mi kg-’ MC = Molecular weight of C n = Population density, No. m-’ kg-’ no = Nuclei nonulation density, No. m-’ ka-’ NT = Specific number of cryst&, No. kg-‘rc = Reaction rate, kmol s-’ kg-’ R = Transfer rate coefficient, s-’ t = Residence time, s t’=Age, s T(B) = Dimensionless transfer function x* = Dimensionless concentration of A (EJE,,) xc = Dimensionless concentration of C (cc/Z& W= Instantaneous crystal mass, kmol kg-’ II-VIII = Specific integrals as defined in equations (22-25) (28) A, = b= B= c= c* =
Greek symbols OL= Dimensionless solid deposition rate concentration fi = Dimensionless inlet of B (=&&0) y = Damkahler number (=hTAo7) 8 = Dimensionless residence time (= t/r) u = Dimensionless micromixing parameter (= Rr) A = Dimensionless residual lifetime ( = t/r) pl = jth moment of population density with respect to sire, No. mikg-’
N. S. TAVARE
936
&= Residual lifetime, s p = Crystal density, kg me3 r=Meanmsidencetime,s * =Dimensionless age (=r’/r) Subscripts A, B, C = Components 0 = Inlet W = Weight basis Superscripts E = L= _= *=
Exit of the entering environment Leaving environment Mean, outlet Dummy variable REFERENCES
Baldyga J. and R. Pohorecki, The effect of micromixing on the precipitation process in a premixed feed continuous stirred tank crystallizer (CSTC). A reprint of a paper presented at Conference on Mix&I Colloque Agitation Mecanique, ENSIGC, Toulouse (1986) Becker G. W. Jr and M. A. Larson, Mixing effects in continuous crystallixation. Chem. Engng. Prog. Symp. Ser. 96, 14-23 (1969). Costa P. and C. Trevissoi, Reactions with non-linear kinetics in partially segregated fluids. C/tern. Engng Sci. 27, 204-2054 (1972). Danckwerts P. V., The effect of incomplete mixing on homogeneous reactions. C/tern. Engng Sci. 8, 93-102 (1958). Garside J. and N. S. Tavarc, Mixing, reaction and prccipitation in an MSMPR crystallizer: effects of reaction kinetics on the limits of micromixing. In Industrial Crystallization 84 (Edited by S. J. Jan& and E. J. de Jong), pp. 131-136. Elsevier, Amsterdam (1984). Garside J. and N. S. Tavare, Mixing, reaction and precipitation: limits of micromixing in an MSMPR crystallizer. Chew. Engng Sci. 40, 1485-1493 (1985). Goto S. and M. A. Matsubara, A generalized two-environment model for micromixing in a continuous fiow reactor. I: construction of model. Chem. Engng Sci. 30, 61-77 (1975). Mehta R. V. and J. M. Tarbell, Four-environment model of mixina and chemical reaction I: model develooment. ArChE Jl29, 320-329 (1983). Methot J. C. and P. H. Roy, Segregation effects on homogeneous second order chemical reactions. Chem. Engng. Sci. 26, 569-576 (1971). Ng D. Y. C. and D. W. T. Rippin, The effect of incomplete mixing on conversion in homogeneous reactions. Proc. 3rd European Symp. Chem. Reaction &gng, Amsterdam, pp. 161-165. Pergamon Press, Oxford (1965). Nishimura Y. and M. Matsubara, Micromixing theory via the two-environment model. Chem. Engng Sci. 25, 1785-1797 (1970).
Pohorecki R. and J. Baldyga. The use of a new model of micromixing for determination of crystal sire in precipitation. C/tern. Engng Sci. 38, 79-83 (1983). Rao D. P. and L. L. Edwards, Mixing effects in stirred tank reactors: a comparison of models. Chem. Engng Sci. Zs, 1179-1192 (1973) Rippin D. W. T., Segregation in a two-environment model of a partially mixed chemical reactor. Chem. hgng Sci. 22, 247-251 (1967). Ritchie B. W., Simulating the effect of mixing on the performance of unprcmixed flow chemical reactors. Con. J. Chem. Engng 58, 626-633 (1980). Ritcbie B. W. and A. H. Tobgy, General population balance modelling of unpremixed feed stream chemical reactors: a review. Chcm. Eng. Common. 2, 249-264 (1978). Ritchie B. W. and A. H. Tobgy, A three-enviromnent micromixlnn model for chemical reactors with arbitrarv separate feed streams. Chem. fingng J. 17,173-182 (1979). Takao M. and Y. Murakami, Evaluation on intensitv of segregation of two-environment model for micromi&g. J. Chem. Engng Japan 9, 336-338 (1976). Tavare N. S., Mixing in continuous crystallizers. AK/& JI 32, 705-732 (1986). Tavare N. S., Micromixing limits in an MSMPR crystalliir. C/rem. Engng Technol. 12, l-10 (1989). Tavare N. S., Mixing, reaction and precipitation: IEM micromixing models in continuous crystalliirs. Forwarded for publication (1991). Treleaven C. R. and A. HI To&y, Conversion in reactors having separate reactant feed streams. The state of maximum~mixedness. Chem. Engng. Sci. 26, 1259-1269 (1971). Villermaux J., Drop break-up and coalescence.. Micromixing Eflects in Liquid-Liquid Reactors, in Mdtiphase Chemical Reactors, I (Edited by Rcdrigues, Carol and Sweed) NATO Adv. Study Inst. Series E, Appl. Sci. 52. SijthoR Noon&off (1981). Villermaux J., Mixing in chemical reactors. ACS Symp. Ser. 226, 135-186 (1983). Villermaux J., Macro and micromixing phenomena in chemical reactors in chemical reactor design and technology. NATO ASI Series E: Applied Sciences No. 110 (Edited by Hugo I. de Lasa), pp. 191-243. Martinus Nijhoff, The Hague (1986). Villermaux J. and R. David, Recent advances in the undcrstanding of micromixing phenomena in stirred reactors. Chem. Enrm~ Commun. 21. 1055122 119831. Villermaux i and A. Zoulalian, Etat de‘ melange du Auide dans un r&acteur continu. A propos dun mod&e de Weinstein et Alder. Chem. Engng Sci. 24, 1513-I 517 (1969). Weinstein H. J. and R. J. Alder, Micromixing effects in continuous chemical reactors. Chem. Engng Sci. 22,65-75 (1967). Wcn C.’ Y. and L. T. Fan, Models for Flow Systems and Chemical Reactors. Dekker, New York (1975). Zwietering Th. N., The degree of mixing in continuous flow systems. Chem. Engng Sci. 11, l-15 (1959).
009th1354/92f5.00 +O.OO
Copyright 0 1992Pwgamon
Pms Ltd
MIXING, REACTION AND PRECIPITATION: ENVIRONMENT MICROMIXING MODELS IN CONTINUOUS CRYSTALLIZERS-I. PREMIXED FEEDS N. S. Department
TAVARE
of Chemical Engineering, University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88, Sackville Street, Manchester, U.K. (Received 22 February 1992;ji~l revision received 23 March 1992; received for publication 2 April 1992)
Abstract-The two-environment micromixing model of Ng and Rippin is extended to a process involving an elementary chemical reaction between two reactant species and subsequent crystallization of a product
in a continuous crystallizer. The model formulation in Part I of this paper deals with the premixed feeds case and assumes that the premixed reactants first enter a completely segregated entering environment and subsequently transfer to a maximum-mixedness leaving environment at a specified rate as defined by an environment transfer function. The sensitivity of this model to several process parameters like the DamkBhler number, micromixing parameter and dimensionless inlet concentration of excess reactant is explored. This computationally efficient two-environment model appears to characterize satisfactorily the micromixing effects in a continuous reactive precipitator with premixed feeds.
system. The scope of Part I of this paper is restricted to analysis of premixed feeds only. A variety of environment models have been proposed by many authors in the chemical reaction engineering literature because of their simplicity, versatility and computational economy (see e.g. Table 7; Tavare, 1986). Formulation of models for complete segregation and maximum mixedness for a continuous precipitator with any RTD (residence time distribution) function, as depicted by plug flow vessels with side exits and entrances respeztively, represents the intrinsic micromixing level in an environment (Danckwerts, 1958; Zwietering, 1959). In general, environment models for chemical reactors assume that the vessel volume consists of two or more environments having intrinsic, extreme states of micromixing. The reactors with premixed feeds may be modelled by either parallel (e.g. Villermaux and Zoulalian, 1969) or series (e.g. Ng and Rippin, 1965) environment models. In the parallel environment model the feed may be split into segregated and maximum-mixedness environments and the product is obtained by combining them at the reactor outlet with no interaction among the parallel environments. For the series’ two-environment model the entire feed stream may be fed to a segregated entering environment and subsequently transferred to a maximum-mixedness leaving environment at a prescribed rate, the effluents of these series environments h&g combi&. at the vesse1 exit. Several variations of these environment
INTRODUCTION
Characterization of intermediate micromixing or partial segregation for a reactive precipitation (or crystallization) system is important (Danckwerts, 1958; Becker and Larson, 1969; Garside and Tavare, 1984, 1985; Tavare, 1986, 1989). Pohorecki and Baldyga (1983) attempted the development of a model characterizing a micromixing level. Their single-parameter model was based on continuous mass transfer between a point and its environment to describe the molecular dissipation zone as originally proposed in IEM (Interaction by Exchange with the Mean) model by Costa and Trevissoi (1972). The model parameter was determined from the spectral interpretation of mixing in an isotropic homogeneous turbulent field. The influence of mixing intensity of the product precipitate and its crystal size distribution for a fast reactive system was evaluated. Tavare (1991) also extended an analysis of IEM micromixing model to a process of reactive precipitation. Two specific cases of feed conditions, viz. premixed and unpremixed feds were found to influence signiticantly both reaction and crystallization performance characteristics. The sensitivity of these two cases to several process and model parameters was explored and the model description for the premixed feeds case appeared better to represent a nearly segregated configuration. The purpose of this article is to extend the study of environment models to a reactive precipitation 923
N. S. TAVARB
924
Product E(B) %A EC
niLI
y=.
1
EE m-e.
,_,::I:
T
-
T
.
_I
T TA
B E(B)
+-I A=0
Fig. 1. Schematic representation of two-environment model for the premixed feeds case. models have been proposed differing either in structurei of environment interaction or in terms of transfer rates (see e.g. Weinstein and Alder, 1967; Rippin, 1967; Nishimura and Mats&am, 1970; Methot and Roy, 1971; Rao and Edwards, 1973; Goto and Matsubara, 1975; Takao and Murakami, 1976; Ritcbie, 1980). Environment models are essentially system specific and empirical in nature. Recently some mechanistic implications have been pointed out by establishing an analogy with the theory of isotropic turbulent mixing and its equivalence of other models. Baldyga and Pohorecki (1986) investigated the influence of mixing conditions on a reactive precipitation process for a premixed feeds case by employing atwo-en vironment mode1 under certain simpli5ed assumptions based on experimental results. They assumed that nucleation was a dominant process in the completely segregated entering enviromnent
whereas only crystal growth occurred in the maximally-mixed leaving environment. Effects of residence time, inlet concentration of reactants and micromixing parameter on the product precipitate were evaluated and compared with the experimental results obtained for precipitation of barium sulphate crystals in order to assess the adequacy of the model.
MODELDESCItIt’ilON For the present purpose the mode1 development for a continuous crystallizer will follow similar lines to that used for chemical reactors by Ng and Rippin (1965). A schematic representation of a continuous crystalhxer with premixed feed and its two-environment mode1 is shown in Fig. 1. The archetype overall reaction considered is: A+B+C,
rc=kc,,cs,
(1)
Mixing, reaction and ptipitation-I where the two species, A and B, react together homogeneously with first-order reaction kinetics with respect to each of the reactants; the component A being assumed limiting. Precipitation of the solid product C resulting from this liquid phase reaction occurs simultaneously when the fluid phase becomes supersaturated with respect to component C. Conventional power law expressions of the form: G = k,Ac8,
(2)
B=k,Acb=KxGi,
(3)
and
are used to represent the growth and nucleation kinetics of the precipitation process, respectively. All the product in both solid and liquid phases, together with tuneacted material, leaves the crystallizer through a single exit. In general the feed stream can be ascribed by its RTD, flowrate and compositions. In the model formulation the vessel is divided into two environments, the entering environment in which the fluid elements are completely segregated and the leaving environment which is always in the state of maximum mixedness. Material transfer from the entering environment is assumed to take place at a rate which is proportional to the amount of material remaining in the entering environment and with a specific rate of transfer parameter R which can be used as a micromixing parameter of the model. When R is zero, there is no transfer from the entering to the leaving environment and the whole vessel is completely segregated. When R is infinitely large the entire entering material is immediately transferred into the leaving environment so that the vessel is in the state of maximum mixedness. Intermediate values of R may thus provide degrees of micromixing. In order to provide a unified mathematical formulation of the model and evaluate the performance characteristics of a continuous reactive crystallizer the following dimensionless variables may be defined:
925
where g is the constant of proportionality and also the dimensionless micromixing parameter of the model. Reactants A and B are premixed at the entry of the entering environment where they reside reacting in a completely segregated state for their age + and subsequently transfer to the leaving environment where they spend their residual lifetime 1 reacting in a state of maximum mixedness. Of the material leaving the crystallixer the fraction that enters and remains in the crystallixer for ages between 0 and I,+ is P(+) and is related to the RTD function B(e) as:
E(@)=fg% Of the material of age + the fraction remaining entering environment from equation (4) is: F@(S ) = exp( - rlJI ) and the fraction
(6)
in the leaving environment
FL($) = 1 - exp(-q*).
is: (7)
Entering environment As the entering environment of the vessel is completely segregated each clump of reactive mixture can be considered as a small batch vessel. The concentration profiles of species A and C for this reactive clump can be given as: -&(8-1+x*),
where
(10) The pertinent
1 dW Q=~~,
in the
boundary
x,=1
and
conditions xc=0
are:
at @=O.
(11)
q=Rr.
The transfer of material between the environments is assumed to be at a rate proportional to the amount of material remaining in the entering enviromnent and thus first-order with respect to mass in the entering environment. This can be described by a transfer function which depends on the age rj as: (4)
The population balance equation for such a batch conIiguration with negligible breakage and agglomeration effects is:
d”++o, w with the boundary
conditions
n(L, 0) = 0;
(12) as:
n(0, #) = no.
(13)
N. S. TAVARB
926 The moment transformation respect to size yields:
%
w
=&
of equation
(12) with
The population balance equation for the solid precipitate C in the leaving environment is: &IL
&IL
-+yL;vI*]=o, 2T'"ar.
(14)
’
(27)
where (15) VII =
(16)
(17) with boundary
mn(L,#)exp(-rl$)Hti s0
The appropriate
boundary *=O drl
at * = 0, j = 0, 1,2,3.
(18)
The variation of crystal size in a batch clump may be represented as: dL -= dJI
conditions as
I-r00
are:
9
(29) (30)
The pertinent moment equations through moment transformation with respect to size as:
may be obtained of equation (27)
(31)
(19)
GT’
(28
n(0, II) = no.
conditions:
pj = 0
+l)d$.
with L = 0 at tj = 0 as the initial condition. (32)
Leaving environment Following Ng and Rippin (1965) [and see also Wen and Fan (1975)] the dimensionless concentration profiles for the limiting reactant A and product C in the leaving environment may be written as follows:
-t&xL,(B-l+x:)+q[““t;,
(33)
(34) IV],
(20) with the boundary
+-rx:(B-I+&)
conditions
dp; ==O
as
as: 1-rco.
(21)
+rl[Ilh;;VI]+B,
The weighted moment m, used equations may be evaluated as:
where II =
m [l
50
III = s0 IV= s VI=
- exti-tlJr)lW
(35)
+ A)W.
(22)
in the
moment
m VIILjdL.
mj=
(36)
s0
w exp(-?$)E($
+ A) ddr,
(23) Crystallizer outlet
om+(#)exp(-q$)E($
+ 1) d+,
(24)
x&)exp(-96)E($
+ A) d&
(25)
The appropriate boundary conditions are the same as that taken by Zwietering (1959) for maximum mixedness, i.e.
d”z=O and cu
&‘=O YE
atA+oO
(26)
The mean concentrations at the vessel outlet are obtained by mixing together, as specified by the RTD, the material in the leaving environment with a zero life expectation and the material still in the entering environment with a zero life expectation. The fractional quantity of reactant A and product C still in the entering environment is obtained from IV and VI, respectively, at 1= 0. Thus the mean exit dimensionless concentrations are: .% = [II& +Ivl1,-0.
(38)
&=P&+VIlI,_,.
(39)
Mixing, reaction and ptipitation-I Siilarly the mean population density of the product precipitate C can be obtained as:
A(L) = [IInL(I;) +vII)l*,o,
927
It is convenient to convert the integrals in equations (48) and (49) into diEcrential equations as:
WI
where VII and nL(L) are the population density functions evaluated at A= 0 for the entering and leaving environment from the population balance equations [equations (12, 2711, respectively.
(51) 2
= xcexp[-(q
+ I)*],
with and
&=O
.~?cc=O at $=O,
Application of RTDs Two specific cases of exit age distributions (i.e. RTDs) viz. an exponentially decaying RTD correspending to a backmixed or an MSMPR (mixedsuspension mixed-product removal) crystallizer as: E(@)=exp(-6’)
E(8) = 48 exp( -29).
JI1=l+?j III =
&
m,=exp(--r2)rii:,
Pm (55) For the other RTD of two stirred tanks-in-series [equation (4211the integrals in equations (22) and (23) yield:
exp(-l),
exe(--11,
&(n (44)
(46)
VII = @(L)exp(
-A),
z=A =
SE=
-xA s0
exp[-(rl + WI W,
0
+lMlW
1
exp(-U).
r1+2 >
(57)
+ RL],
(58)
VI =4exp(-21)[fi&
+ a&,],
(59)
where % 1ZE, and Z& are defined, respectively, as:
(48)
-E
(49)
and
1 +-
IV = 4exp(--U)[.Gt,
-
X.%2-
5axcew[-(~
(56)
The integrals in equations (24) and (25) require the numerical evaluations in two parts as:
(47)
where Rz, 2: and PI@(L)are the average dimensionless concentrations of A and C and the average population density function at the crystallizer outlet from the entering environment, respectively. These quantities are detined as:
4 r1+2 (
(45)
VI=feexp(-A),
+&)]exp(-2A)
and III =-
IV =Ziexp(-A),
(54)
where P$ is the average weighted moment at the crystallizer outlet from the entering environment defined as:
(42)
are considered for the present study. For a perfectly backmixed flow system with an exponentially decaying RTD [equation (4111 the evaluations of integrals in equations (25-28) at any residual lifetime 1 yield:
(53)
as the initial conditions. The values of 6 and Zc determined by the integration of equations (51) and (52) for $ + cc will equal Xi and a& respectively. The weighted moment m, in equation (36) for the single tank RTD results in:
(41)
and a function depicting the behaviour of two wellstirred vessels of equal volume in series as:
(52)
sow
XA+ewl-(2 + tlM1dtk
-6
(61)
(62)
xc1 =
and Ha(L) =
m x(L
#)exp[--(rl + l)$l dti.
(50)
_fg=
I
~xc~cxpI-_(2+sWlW.
0
(‘53)
N. S.
928
These integrals [equations @O-63)], as in the case of an exponentially decaying RTD, can be evaluated by transforming them into a set of differential equations and then integrating them numerically from # = 0 until $ + co. The integral in equation (28) can also be evaluated by splitting it into two parts as: VII = 4 exp(-21)[12nF(l)
+ A:(L)],
TAVAM segregation throughout the remainder of its stay within the entering environment. Each of these clumps thus acts as a small batch reactor and transfers the material from the entering environment to the leaving environment. This transfer rate between environments is assumed to be at a rate proportional to the amount of material remaining in the entering environment.
(64) Entering environment
where
A:(L) =
s
an(L, Ilr)exp[-(2
+sMl dr(r (65)
0
The weighted yields:
moments
in equation
(36) for this case
mi=4exp(-21)[Lr$,+fi@,
(67)
where m A$=
$(L)L’dL
(68)
s0 and
s m
lit;=
ii,e(L)LjdL.
(69)
0
MODEL
Computational
EVALUATION
details
In order to evaluate the performance characteristics of a reactive precipitation system described by the proposed model the specific set of parameters is given in Table 1. These physicochemical parameters are similar to those used in the previous studies (Garside and Tavare, 1984; Tavare, 1991). In the foregoing model description it is assumed that the two streams mix completely just at entry of the entering environment after which each clump of premixed feeds is assumed notionally to preserve its Table 1. Parameter vahes used in model calculations Molecular weigh of C, kg kmol~’ Solubility of C, c*=, km01 kg-’ Fad conoeatration of A, based on total flow, c&o. km01 kg-’ B (=%lJG) Y (=&r) rl (=R*) Crystal density of C, pc kg me3 Area shape factor Volume shape factor Mean rraidena time, 5, s Nuckation order, b Nuckation rate cxmstant kb, No./[s~kg(kmol Growth rate order, g Growth rate constant. k.. m/ls~kmol kc’Pl
100 0.001 0.002 I.5 10.0 0.1, 1, 10 2660 3.68 0.525 1000 kg-‘)‘]
The behaviour of a clump in the entering environment is described by the set of 11 equations [equations (8-18)]. Initially only reaction occurs in each of these clumps until the concentration of product C reaches saturation point. As soon as the clump gets supersaturated with respect to C, nucleation and subsequent particle growth begin and result in evolution of the crystal sire distribution. Consequently only those equations depicting the concentration profiles and evaluating average exit concentrations of the entering environment are integrated initially, after a having been set equal to zero, until the concentration of C in the liquid phase reaches the saturation value. Thus a set of four equations [equations (8,9), (51,52)] for the RTD of a single tank and a set of six equations [equations (8,9) and an additional four equations obtained from equations (60-63)] for the RTD of two stirred tanksin-series need to be integrated with the appropriate boundary conditions. Once the clump of the reaction mixture becomes supersaturated with respect to C all the differential equations describing concentrations and moments along with the partial differential equation [equation (12)] and the integrals evaluating the average population density [equation (SO)] or equations (65-66) and its moments [equation (55) or equations (68-69)] at the vessel exit from the entering environment need to be integrated simultaneously with the appropriate boundary conditions. A set of nine differential equations [equations (8-9) (14-l 7), (19), (5 l-52)] for the case of a single stirred-tank RTD and a set of 11 differential equations [equations (t&9), (14-17), (19), additional four equations corresponding to equations (60-63)] for the case of two stirred tanks-in-series RTD need to be considered. All the differential equations involved were integrated by the fourth-order Rung&Lutta method with an integration step length of A$ = 0.0001 starting from the appropriate initial conditions until $ = 10. The partial differential equation [equation (12)] was solved by the modified method of numerical integration along the characteristics with a specified grid length of sixe while the integrals in equations (50) or (65-66) and equation (55) or equations (68-69)
Mixing, reaction and were evaluated numerically using Euler’s formula. In the algorithms the set of differential equations depicting concentrations and moments was initially integrated with a step length of A@ = 0.001 until the increment in size was equal to the grid length of 2 pm used in the solution of the partial differential equation. The growth rate and hence nuclei population density no is delined at the end of the grid and the solution of the partial differential equation moved forward by the time required to increase the size by one grid length (2 pm). The step length for time in evaluating the integrals in equation (SO)or equations (65-66) was also the same, i.e. the time required to increase the size by one grid length (2 pm). The step length for size in evaluating the integrals in equation (55) or equations (68-69) was equal to one size grid length (2 pm). Leaving environment The concentration profiles in the leaving environment are described by equations (20-21), required integrals in equations (22-25) being evaluated analytically in terms of residual lifetime, 1 i.e. equations (43-46) for the RTD of a single stirred tank and equations (56-59) for the case of two stirred tanks-inseries. It is necessary to point out that an opposite sign to that used by Ng and Rippin (1965) [their equation (811 is used to describe the concentration profiles for A and C as in equations (20-21). As the life expectation increases in the opposite direction to teal time their sign convention in their equation (8) is correct for the analytical solutions for the leaving environment. In the present study the numerical integration process starts at A--, ODtherefore the concentrations profiles in equations (20-21) are described in the same sense as the age or real time and have the opposite sign to that of their equation (8). Numerical integration of all the equations for the leaving environment commences from a suitable high value of I, say L = IO,with the initial conditions derived from the concentration prosles [equations (20-21)] to satisfy equation (26) at the entry and finishes at the outlet where 1 = 0 to yield the exit values from the leaving environment. Initially equations depicting the concentration profiles in the leaving environment [equations (20-21)] are integrated with o = 0 until the concentration of C in the liquid phase reaches the saturation value. Once the reaction mixture becomes supersaturated with respect to C the set of seven differential equations [equations (20-21), (31-34), (3711 is integrated along with the partial dilYerential equation [equation (2711 depicting the population balance equation in the leaving environment. The other features used in numerical calculations were similar to those used for the entering environment.
precipitation-1
929
RESULTS AND DJSCUSSION
Using the physicochemical parameters listed in Table I calculations were performed to evaluate the crystallizer performance characteristics as predicted by the proposed model for the premixed feeds case. The resulting product size distributions from equation (40) as illustrated by the conventional population density plots for typical cases of this model are shown in Fig. 2. Also reported in Fig. 2 are the population density plots under otherwise similar conditions for the extremes of micromixing, viz. maximum mixedness (Mode1 I), MM(I) and complete segregation (Model II), CS(Il), from the previous studies (Garside and Tavare, 1985). Corresponding details of all these cases regarding dimensionless concentrations and product size distribution statistics are included in Table 2. According to the mode1 description when 11approaches zero the transfer from the entering to leaving environment reaches zero and the whole vessel therefore tends to be in the completely segregated entering environment. For a large value of n most of the material from the entering environment is transferred to the leaving environment occupying most of the vessel and then the whole reactor approaches the state of maximum mixedness. Thus with an increase in micromixing parameter, u there should be a gradual movement of the performance characteristics from the completely segregated to the maximum mixedness case. Both the reaction and crystallization performance characteristics reported in Fig. 2 and Table 2 appear to show this trend generally and approach these extreme limiting cases of micromixing over the range of micromixing parameter 9. Eflect of Damkiihler number y The dimensionless reaction group y, i.e. the DamkBhler number, can characterize the dimensionless concentration of A at the vessel exit and hence determines the reaction performance keeping all other parameters in Table 1 constant. The sensitivity of both the reaction and crystallization performance characteristics to the DamkGhler number was explored by varying y over the range 0.1-1000, covering a l@-fold range. The results of these calculations are reported in Figs 3 and 4. The variations of dimensionless concentrations of A and C at exit, LZ~and &, with the Damkiihler number y at a typical value of tbe micromixing parameter (u = IO)are shown in Fig. 3. Also included in Fig. 3 are t&se variations for extremes of micromixing levels and RTD of two stirred tanks-in-series. Although the difference between the dimensionless
N. S. TAV~
930
IC
c 10
0
2000
1000
3000
L (pm1
Fig. 2. Product population density plots (data as in Table 1). concentrations at two extreme micromixing levels is small, the calculated dimensionless concentration at any y lies within the small range and follows the correct trend, i.e. with an increase in q movement of concentration value from the completely segregated to the maximum mixedness case. Because of the small range of concentration between two extreme micromixing levels it appears that the dimensionless concentration changes are less sensitive to the micromixing parameter, 9 of this model. Low values of y result in lower reaction rates and yield lower conversion and hence higher _g*. With an increase in y, Z,, decreases and .Q increases rapidly at a lower y and then remains almost constant as a consequence of production of solid C. Similar variation of & has been reported by Ng and Rippin (1965) over the range of y from 0 to 100 in their original development.
Tabk n 0
0.1 1.0 10.0 co
2. Model
pd-nce
characterimks I;, (urn)
CVW (t/s)
(y = lo; Fig. 2) NT x 10-e (No. kc’)
5.
%
0.096
0.540
353
22.5
I.2
Ez 0.111
0.543 0.538 0.545
356 412 958
25.3 32.7 57.9
0.8 1.0 0.4
0.136
0.567
930
50
0.2
Figure 4 depicts the variation of crystallization performance characteristics at the vessel exit, viz. weight mean size, coefficient of variation and specific total number of crystals per unit of solvent mass. Also reproduced in Fig. 4 are these variations for extreme micromixing levels. The critical Damk%ler number, ytit at which the product solid C starts appearing depends on the micromixing parameter q. For a completely segregated system (q = 0) ysrit is lower (~0.1) and for a maximum-mixedness case (q + co) it is about 1. For an intermediate case the critical y lies between these two limits. At low values of y( < y&J only reaction occurs, the reaction rate is slow and controls the overall process. Increasing values of y (> yd) up to a value of about 5 produces significant changes in the sire distributions. The weight mean size increases rapidly and passes through a maximum. The variation of the weight mean size follows a correct trend. With an increase in q the peak in weight mean size increases and the curve moves towards that for the maximum mixedneas case. The coeSicient of variation passes through a slight maximum while the specific number of crystals per unit mass of solvent increases rapidly just after ypit and then mote slowly. Both these performance character&&s show a correct trend with the micromixing parameter q. Further increases in y then
931
Mixing, reaction and pmcipitation-I
0.6
0.6
04
Ix”
0.2
0 0.1
1
10
1000
100
Y
Fig. 3. Effect of DamkiihJernumber y on outlet concentrations of A and C.
produce comparatively small changes in the crystallization performance characteristics. Both weight mean size and coefficient of variation decrease with y, the decrease being more pronounced at higher micromixing parameter q. However, the specific numJxr of crystals per unit mass of solvent shows a slight increase. At higJrer y the reaction rate is fast resulting in local, sharper and earlier supersaturation peaks and thus a CSD having a higJrer number of crystals with smaller mean size and narrower distribution. At high y the results show the the performance characteristics of the CSD produced are closer to those of
a completely segregated system and less sensitive to the micromixing parameter q. EfJt
of micromixing parameter
q
Keeping aJ1 other parameters in Table 1 constant, q was varied over a range of 0. J-1000, thus covering a W-fold range of q. The resuJts of these calculations are reported in Fig. 5, the most sensitive range of q being l-1000. The variations of both zZ* and J& between the extreme micromixing limits are small (Fig. 3 and Tabk 2) and are not sensitive to q over the range. CrystaJJization performance
N. S. TAVARE
932
1200
800
400
0
1
10
100
Y
Fig. 4. Effect of DamkBhler number y on solid product characteristics.
characteristics, however, show substantial changes. The variation of the weight mean size with q shows a sigmoid curve, the point of in&ction being at about q = 10. At higher q the weight mean size exceeds the range predicted by extreme micromixing levels and it may perhaps be due to feeding of slurry from the entering environment to the leaving environment, operating at a higher level of supersaturation. The variation of the coeHicient of variation with q also shows a similar sigmoid curve with the point of inflection at about q =4 and exceeds the range of extreme micrommmg levels. The variation of the specific number of crystals per unit mass of solvent, however, shows an opposite sigmoid curve
showing a rapid decrease over the most sensitive range of q. Eflect of dimensionless
inlet concentration
of B, /I
The effect of the feed concentration for the case of premixed feeds was explored by changing dimensionless inlet concentrations of B, 4 over the range l-100, the rest of the parameters in Table 1 being kept constant. The results of these calculations are represented in Fig. 6. For this case Z* dccrroses because ofanincreascdreactionrateduetohigher~and~ remains practically constant as a consequence of two competitive procemes. Both these concentration profiles appear to lie within the range predicted in the
Mixing, reaction and precipitation-1
933
08
1200
06
600
04
400
0.2
E a 5 1-J
0
I
_I
0.1
10
1
100
’
0.1
looa
9
Fig. 5. Effect of micromixing
parameter q on performance characteristics
at crystallizer outlet.
ber of crystals per unit of mass increases thus producing many smaller particles with narrower distribution at higher /TX
extremes of micromixing (results not shown). With an increase in /3 both the weight mean size and the coefficient of variation decrease and the specific num-
25
1.2 20
Z 0
1.5
$
0.8
0 t e
%
‘0
I
I
I-?
10
z’
19 IX
;
04
-__0.5
/
C-...-..........._.,,(~.
_,.....,,...,....,.....
1
2
5
0
I
0 10
50
loo
B
Fig.
CACE
16/10-11-B
6. Effect of dimensionless
&let concentration crystalker
of B, /I, on performance exit.
characteristics
at
934
-__-
Single
tanks
v=lO
leq.41J
s + 2. 60
2.5
0.1
10
1
100
1000
Y
Fig. 7. Effect of Damkiihler numbery on solid product characteristicsfor the RTD of two stirred tanks-in-se&s.
RTD of two stirred tanks-in-series The performance characteristics of a crystallizer having RTD of two stirred tanks-in-series were evaluated over the range of y from 0.1 to 1000. The variations of & and Xc with y at a typical value of q = 10 are included in Fig. 3 and provide comparison with those for an RTD of a single stirred tank. The results of crystallization performance characteristics are reported in Fig. 7. Also reported in Fig. 7 for comparison are those characteristics of the two extremes of micromixing and micromixing parameter, q = 10 for an exponentially decaying single-tank
RTD. The product size distribution appears more sensitive to the micromixing parameter at higher values of y than that for the single-tank RTD case. For q = 10 both the weight mean size and the coefficient of variation pass through large peaks, thus producing a lesser number of large product crystals with wider distribution than those for the single-tank RTD case at higher values of y. MODEL In
eral
ASSESSMENT
foregoing analysis the effects of sevparameters (y. q. /I,RTD) on reaction and the
Miring, reaction and precipitation-I
crystallization performance characteristics of a reactive precipitation process in a continuous crystallizer are reported. Several other parameters like crystallization kinetics and other operating conditions may have a significant influence4 It is important to note that the present analysis is concerned with a global characterization of micromixing effects on the performance characteristics of a reactive precipitation system in a continuous crystallizer. Dimensionless concentration profiles of A are consistent with those reported in the literature (Ng and Rippin, 1965; Methot and Roy, 1971; Rao and Edwards, 1973). Studies concerning reactive precipitation systems are scarce. Only a study for the case of premixed feeds appears to be reported in the literature by Baldyga and Pohorecki (1986). They have simulated a fast reactive precipitation of barium sulphate crystals with its experimental verification. Although direct comparison with their study is difficult due to different physicochemical parameters used, the trends in the variations of mean size with micromixing parameters and residence time appear consistent with the present study. For their fast reactive precipitation system, the Sauter mean size increases with the micromixing parameter and decreases only slightly with the residence time (or the Damklihler number). Several authors (e.g. Rao and Edwards, 1973; Villermaux, 198 1, 1983, 1986; Villermaux and David, 1983) attempted to establish an equivalence relationship between all one-parameter micromixing models either on the basis of the turbulent mixing theory or by comparing the reaction performance. The micromixing parameter r) of the environment model as proposed by Ng and Rippin (1965) can be equivalent to twice the micromixing parameter of the IEM (Interaction by Exchange with the Mean) micromixing model as developed by Villermaux and coworkers (1981, 1983,1986). Mehta and Tarbell (1983) in their recent elegant work provided a link between the systems approach of an environment model based on population balance principles and a fluid mechanics approach in the framework of turbulence theory. They established that the transfer parameter R is generally equal to the reciprocal of the turbulent mixing time which may be estimated from the Corrin-Rosensweig turbulent mixing theory. The analysis presented here lays no claim to providing a detailed physical insight into the interplay of mixing, reaction and subsequent crystallization but provides rather an eflicient means to evaluate and subsequently predict performance for a continuous crystallizer configuration. The model parameters required should be characterized for the experimental response. The general technique of
935
using a known reactive crystallization system to evaluate the parameters of the model that matches the performance obtained experimentally for the crystallizer should be used. As the micromixing effects are much more important when the reactants enter separately than when they are initially premixed just before entry (see e.g. Treleaven and Tobgy, 1971; Ritchie and Tobgy, 1978, 1979; Ritchie, 1980) the modelling of such reactive precipitators with separate feed streams is presented in Part II of this paper. Acknowledgements-The author wishes to acknowledge with thanks Professors R. Pohorecki and D. W. T. Rippin for their comments on the original manuscript. NOMENCLATURE
Total crystal surface area, m2 (kg solvent)-’ Nucleation order Nucleation rate, No. s-I (kg solvent)-’ Concentration, km01 (kg solvent)-’ Saturation concentration, kmol (kg solvent)-’ AC= Concentration driving force (c - c l), kmol (kg solvent)-’ CV = Coefficient of variation based on weight distribution, % E(8) = Dimensionless residence time distribution (RTD), dimensionless exit age distribution F(8) = Dimensionless cumulative residence time distribution g = Growth rate order G = Linear growth rate, m s-r i = Relative kinetic order (= b/g) j = Index variable k = Reaction rate constant, kg (kmol)-‘s-l ka = Area shape factor k,= Nucleation rate constant, No. s-r kg-’ (kmol kg-‘)-’ k, = Growth rate constant, m s-i (kmol kg-‘)-’ k, = Volume shape factor KR = Relative rate constant, No. s’ -‘m-‘kg-’ L = Crystal size, m, gm mJ= jth weighted moment [equation (3611, No. mi kg-’ MC = Molecular weight of C n = Population density, No. m-’ kg-’ no = Nuclei nonulation density, No. m-’ ka-’ NT = Specific number of cryst&, No. kg-‘rc = Reaction rate, kmol s-’ kg-’ R = Transfer rate coefficient, s-’ t = Residence time, s t’=Age, s T(B) = Dimensionless transfer function x* = Dimensionless concentration of A (EJE,,) xc = Dimensionless concentration of C (cc/Z& W= Instantaneous crystal mass, kmol kg-’ II-VIII = Specific integrals as defined in equations (22-25) (28) A, = b= B= c= c* =
Greek symbols OL= Dimensionless solid deposition rate concentration fi = Dimensionless inlet of B (=&&0) y = Damkahler number (=hTAo7) 8 = Dimensionless residence time (= t/r) u = Dimensionless micromixing parameter (= Rr) A = Dimensionless residual lifetime ( = t/r) pl = jth moment of population density with respect to sire, No. mikg-’
N. S. TAVARE
936
&= Residual lifetime, s p = Crystal density, kg me3 r=Meanmsidencetime,s * =Dimensionless age (=r’/r) Subscripts A, B, C = Components 0 = Inlet W = Weight basis Superscripts E = L= _= *=
Exit of the entering environment Leaving environment Mean, outlet Dummy variable REFERENCES
Baldyga J. and R. Pohorecki, The effect of micromixing on the precipitation process in a premixed feed continuous stirred tank crystallizer (CSTC). A reprint of a paper presented at Conference on Mix&I Colloque Agitation Mecanique, ENSIGC, Toulouse (1986) Becker G. W. Jr and M. A. Larson, Mixing effects in continuous crystallixation. Chem. Engng. Prog. Symp. Ser. 96, 14-23 (1969). Costa P. and C. Trevissoi, Reactions with non-linear kinetics in partially segregated fluids. C/tern. Engng Sci. 27, 204-2054 (1972). Danckwerts P. V., The effect of incomplete mixing on homogeneous reactions. C/tern. Engng Sci. 8, 93-102 (1958). Garside J. and N. S. Tavarc, Mixing, reaction and prccipitation in an MSMPR crystallizer: effects of reaction kinetics on the limits of micromixing. In Industrial Crystallization 84 (Edited by S. J. Jan& and E. J. de Jong), pp. 131-136. Elsevier, Amsterdam (1984). Garside J. and N. S. Tavare, Mixing, reaction and precipitation: limits of micromixing in an MSMPR crystallizer. Chew. Engng Sci. 40, 1485-1493 (1985). Goto S. and M. A. Matsubara, A generalized two-environment model for micromixing in a continuous fiow reactor. I: construction of model. Chem. Engng Sci. 30, 61-77 (1975). Mehta R. V. and J. M. Tarbell, Four-environment model of mixina and chemical reaction I: model develooment. ArChE Jl29, 320-329 (1983). Methot J. C. and P. H. Roy, Segregation effects on homogeneous second order chemical reactions. Chem. Engng. Sci. 26, 569-576 (1971). Ng D. Y. C. and D. W. T. Rippin, The effect of incomplete mixing on conversion in homogeneous reactions. Proc. 3rd European Symp. Chem. Reaction &gng, Amsterdam, pp. 161-165. Pergamon Press, Oxford (1965). Nishimura Y. and M. Matsubara, Micromixing theory via the two-environment model. Chem. Engng Sci. 25, 1785-1797 (1970).
Pohorecki R. and J. Baldyga. The use of a new model of micromixing for determination of crystal sire in precipitation. C/tern. Engng Sci. 38, 79-83 (1983). Rao D. P. and L. L. Edwards, Mixing effects in stirred tank reactors: a comparison of models. Chem. Engng Sci. Zs, 1179-1192 (1973) Rippin D. W. T., Segregation in a two-environment model of a partially mixed chemical reactor. Chem. hgng Sci. 22, 247-251 (1967). Ritchie B. W., Simulating the effect of mixing on the performance of unprcmixed flow chemical reactors. Con. J. Chem. Engng 58, 626-633 (1980). Ritcbie B. W. and A. H. Tobgy, General population balance modelling of unpremixed feed stream chemical reactors: a review. Chcm. Eng. Common. 2, 249-264 (1978). Ritchie B. W. and A. H. Tobgy, A three-enviromnent micromixlnn model for chemical reactors with arbitrarv separate feed streams. Chem. fingng J. 17,173-182 (1979). Takao M. and Y. Murakami, Evaluation on intensitv of segregation of two-environment model for micromi&g. J. Chem. Engng Japan 9, 336-338 (1976). Tavare N. S., Mixing in continuous crystallizers. AK/& JI 32, 705-732 (1986). Tavare N. S., Micromixing limits in an MSMPR crystalliir. C/rem. Engng Technol. 12, l-10 (1989). Tavare N. S., Mixing, reaction and precipitation: IEM micromixing models in continuous crystalliirs. Forwarded for publication (1991). Treleaven C. R. and A. HI To&y, Conversion in reactors having separate reactant feed streams. The state of maximum~mixedness. Chem. Engng. Sci. 26, 1259-1269 (1971). Villermaux J., Drop break-up and coalescence.. Micromixing Eflects in Liquid-Liquid Reactors, in Mdtiphase Chemical Reactors, I (Edited by Rcdrigues, Carol and Sweed) NATO Adv. Study Inst. Series E, Appl. Sci. 52. SijthoR Noon&off (1981). Villermaux J., Mixing in chemical reactors. ACS Symp. Ser. 226, 135-186 (1983). Villermaux J., Macro and micromixing phenomena in chemical reactors in chemical reactor design and technology. NATO ASI Series E: Applied Sciences No. 110 (Edited by Hugo I. de Lasa), pp. 191-243. Martinus Nijhoff, The Hague (1986). Villermaux J. and R. David, Recent advances in the undcrstanding of micromixing phenomena in stirred reactors. Chem. Enrm~ Commun. 21. 1055122 119831. Villermaux i and A. Zoulalian, Etat de‘ melange du Auide dans un r&acteur continu. A propos dun mod&e de Weinstein et Alder. Chem. Engng Sci. 24, 1513-I 517 (1969). Weinstein H. J. and R. J. Alder, Micromixing effects in continuous chemical reactors. Chem. Engng Sci. 22,65-75 (1967). Wcn C.’ Y. and L. T. Fan, Models for Flow Systems and Chemical Reactors. Dekker, New York (1975). Zwietering Th. N., The degree of mixing in continuous flow systems. Chem. Engng Sci. 11, l-15 (1959).