ChemicalEngineering
Science,
Mixing
models D. P. RAO and LOUIS L. EDWARDS University of Idaho, Moscow, Idaho 83843, U.S.A. 22 April 1971; accepted6
developed for studying the effect of time distribution and intermediate degrees of segregation on chemical reactor performance were shown to be equivalent comparison of conversions obtained for a second order chemical reaction with mixed feed. Conversions with mixed feed were compared for a single CSTR and a reactor with a 2CSTR time distribution usine values of the micromixing for both of the models. The Monte Carlo Coalescence Model of Spielman and Levenspiel, Macromixed feed model of Manning et al and the TwoEnvironment Model of Ng and Rippin were found to give nearly results for (1) a single second order (2) consecutive second order reactions, stepwise addition polymerization without termination taking place in a single CSTR with mixed feed. values of the micromixing for the respective Villermaux and were extended to reactors with unmixed feed with special to a second order chemical Results calculated Levenspiel and Kattan and Adler usina values of the micromixing parameter.
engineering with mixing proposed continuous These reactors mainly divided into two i.e. and tank reactors (CSTR). Ideal for above two plug flow with complete radial and complete deviations from mixing in actual chemical reactors modeled in several several reviews are available on this [ 1,6, lo]. For plugflow reactors, the deviations from ideality occur due to axial mixing caused by various factors (e.g. velocity profile, eddy and molecular diffusion, presence of packing, etc.). For stirred tank reactors deviations from ideality occur due to low stirring rates or viscous fluids. Mixing models are generally based on the concepts of macromixing, characterized by the residence time distribution, and micromixing, characterized by the degree of segregation intermediate between the state of complete segregation and maximum mixedness. These concepts
were introduced originally by Danckwerts [4] and Zwietering [ 171. This paper is concerned with the following singleparameter models for intermediate degrees of segregation: Modell, the Monte Carlo coalescence model of Spielman and Levenspiel [13] for a CSTR; Model2, Macromixed feed model of Manning et al. [71 for a CSTR; Model3, Twoenvironmental model of Ng and Rippin [8] for arbitrary residence time distributions; Model4, the Monte Carlo residual life time model of Kattan and Adler[6] for arbitrary residence time distributions; Model5, General parallel feed reactor model of Villermaux and Zoulalian [ 151 for arbitrary residence time distributions. The mixing parameter for each of these models is listed in Table 1. The models are compared by calculating the reactor exit concentrations for a particular residence time distribution and feed condition and at the same value of the mixing parameter. This is possible because the mixing parameters for all five models under consideration have the same upper and lower limits, i.e. infinity and
1179
D. P. RAO and LOUIS
to the Monte Carlo coalescence model of Spielman and Levenspiel for a single CSTR with an ideal residence time distribution and mixed feed.
Table 1. Identification of mixing models Model number
Authors
1
Spielman and Levenspiel [ 131
112
2
Manning et a[[71
Q/F
3
Ng and Rippin [8]
RT
4
Rattan and Adler 161
112
5
Villermaux and Zoulalian [ 151
K
L. EDWARDS
Mixing parameter
DISCUSSION
zero. The scope of this study and the overall conclusions are shown in Table 2. It should be noted in the bottom row of Table 2 that models 2, 3 and 4 have been extended to unmixed feed conditions. A stepwise addition polymerization reaction without termination has been studied for intermediate degrees of segregation by applying models 1, 2 and 3 (row 3 of Table 2). As a general observation, it can be stated that the model of Ng and Rippin is the easiest to use from a computational standpoint. Also, when the physical situation simulated by these models is considered as discussed below, it appears that the model of Ng and Rippin is equivalent to the Monte Carlo residual life time model of Rattan and Adler for any arbitrary residence time distribution and mixed feed, and is also equivalent
OF MODELS
Ng and
[email protected]] simulated the intermediate degrees of segregation for an arbitrary residence time distribution by considering the reactor to consist of two environments which correspond to the extreme cases of mixing, i.e. complete segregation and maximum mixedness. As depicted in Fig. 1, the reactor feed enters the segregated environment. Material from the segregated environment either leaves the reactor or is transferred to the maximum mixedness environment. The output from the reactor is a combination of the flows from the two environments. A specific model which Ng and Rippin worked out in detail assumes that the rate of transfer from the entering to the leaving environment is proportional to the amount of material remaining in the entering environment. Thus, if m represents the mass of a particular quantity of material in the entering environment at any time r, the equation for the rate of transfer from the entering to the leaving environment can be written as
r!!!EdmmmR dt
Table 2. Scope of model comparisons and overall conclusions Chemical reaction
Models studied
Residence time distribution (RTD)
Feed condition
Ideal single CSTR
Mixed
Single second order
1,2,3,4and5
Ideal single CSTR
Mixed
Consecutive second order
1,2and3
Ideal single CSTR
Mixed
Ideal 2CSTR
Mixed
Stepwise poly1,2and3 merization reaction without termination Single second 3and4 order
Ideal single CSTR
Unmixed
Single second order
1180
1,2,3,4andS
Conclusions Results nearly identical except Model 5 All models give nearly identical results All models give nearly identical results Both models give nearly identical results All give different results
Mixing effects in stirred tank reactors: a comparison of models Age
a=0
a=co
The Monte Carlo residual life time model of Kattan and Adler[6] simulates the intermediate degrees of segregation by considering finite coalescence rates between fluid elements all having the same residual life time, A, as shown in Fig. 2. If the radial mixing is infinite, the reactor corresponds to a maximum mixedness reactor and if the radial mixing is zero, the reactor corresponds to a completely segregated reactor. As the coalescence between fluid elements having a given value of residual life time, A, is regarded as a random process, this situation again appears to correspond to the transfer from the entering to the leaving environment in the model of Ng and Rippin. This is made clearer when one compares Figs. 1 and 2.
_
111111111111 Segregated environment
Product
t t t t t t t t t t t Mowmum
I,
mixedness envircnment I
x =o
X=CO Life
expectahon
Fig. 1. Twoenvironment model of Ng and Rippin[l].
where R is a transfer coefficient that is assumed constant. Integrating the above equation, m=mgRt
(2)
where m. is the value of m at t = 0. A physical meaning of the above equation, as given by Ng and Rippin, is that every point or fluid element in the entering environment has an equal probability of being transferred to the leaving environment. In the case of a CSTR, Ng and Rippin’s model is equivalent to saying that the feed port is situated away from the impeller and hence the feed stream is completely macromixed before material from it reaches the impeller. The impeller region is taken to be the leaving environment which is in a state of maximum mixedness. The exit stream will be a mixture of the flow from the impeller region and the segregated region. In comparison, consider the Monte Carlo coalescence model of Spielman and Levenspiel [ 131. Here, the reactor is assumed to consist of a finite number of fluid elements of equal size which coalesce and immediately redisperse. All fluid elements have equal probability of coalescence. Thus it is obvious that for mixed feed, as long as a fluid element does not coalesce with another fluid element, it stays in the entering environment and because every fluid element has an equal probability of coalescence, this appears to correspond to the assumption made by Ng and Rippin about the transfer from the entering to the leaving environment.
Finite radial x=0
Ltfe expectotlon 
mixing
=.ab?dpr x=co
Fig. 2. Monte Carlo residual lifetime model of Kattan and Adler[6].
Figure 3 shows the macromixed feed model of Manning ef al. Here, the feed and the recirculating stream remain segregated in the reactor until they reach the impeller where they get completely micromixed. The ratio of the flow rate of the recirculating stream, Q, to that of the feed stream F, i.e. Q/F, is taken to be the mixing parameter. At a first glance, one finds some similarity between this model and the model of Ng and Rippin with respect to the nature of mixing. O&put
Micromixer
1181
4
Macromixer
I
i
I I I
i P Feed
co _
‘F
i __
L
_J j I
I
Fig. 3. Micromixed feed model of Manning et al. [7].
D. P. RAO and LOUIS L. EDWARDS
an error in a model developed by Weinstein and Adler [16] for intermediate mixing states which consists of a maximum mixedness and a completely segregated reactor in parallel. They presented the results with correction, and also proposed a more general model than that of Weinstein and Adler. This model consists of two reactors in parallel, one being a completely segregated reactor and the other a maximum mixedness reactor as shown in Fig. 4. It is assumed that feed divides into two streams, one of them entering the segregated reactor and the other entering the maximum mixedness reactor. The fraction of the total feed having residence times between x and x+ dx, which enters the completely segregated reactor is written as s(x)&
(3)
= g(x)f(x)dx
where g(x) is an arbitrary function and f(x) is the residence time distribution function, The remaining fraction enters the maximum mixedness reactor, m(x)dx = (1 g(x))f(x)dx.
(4)
One particular form for g(x), as suggested by Villermaux and Zoualalian is eew where K is a X=CU
Lifeexpectotmn
Moxnnummlxe&ws
”
x=0
reactor
v+(x)dx
/
constant. This is analogous to the model of Ng and Rippin. The difference between the model of Villermaux and Zoulalian and that of Ng and Rippin can however be clearly seen when one compares Fig. 1 with Fig. 4. In the model of Ng and Rippin, part of the feed is transferred from the segregated environment to the maximum mixedness environment, whereas in the model of Villermaux and Zoulalian, the feed divides into two streams before entering the parallel reactors and no transfer between reactors occurs. However, the fraction of the volume in complete segregation is the same in both models if K is equated to RT of Ng and Rippin. When one uses the procedure of Rippin [ 111 for calculating the degree of segregation, J, the model of Villermaux and Zoulalian gives the same expression for J as that derived for the model of Ng and Rippin, i.e.
(5) This leads one to conclude again that J is nonunique. This nonuniqueness of the degree of segregation, J, has also been shown by Rippin [l l] and recently by Nishimura and Matsubara [91. APPLICATION
A B
Products
by using the various models shown in Table 1. The application of the model of Ng and Rippin [8] for the above reaction gives rise to an equation of the form
Ao
Feed
 1 [ 1 + 12k&,m
+ vl
l1
 4krCA,,r)eqEi( 7)
Segregatedreactor 0=Q)
a=0
Fig. 4. Parallel reactor model of Villermaux and Zoulalian
I151. 1182
l
e Ei(q) k7Cm (6)
where
Age
FOR MIXED
Table 3 shows the results obtained for a second order chemical reaction of the type
c A=R7 c Rr
Product
OF MODELS FEED
Rr+l ‘I = k&A,, ’
3 0.614 0.490 0.337 0,240 0.164 0*092 0.057 0.034
2
0.615 0.491 0.337 0.240 0.163 0.091 0.057 0.034
10
0.49 0.168 0.032
4 0.617 0.498 0.353 0.26 O186 0.114 0.076 0.050
5
I
0.62 0.49 0.36 0.27 0.18 0.109 0.068 0.042
1 0.618 0.50 0.357 0.262 0.186 0.110 0.072 0.046
2

0.617 0.497 0.351 0.258 0.183 0.110 0.071 0.045 0.50 0.041
4
3
50
Mixing parameter
1 0.63 0.50 0.36 0.275 0.1% 0.130 0.082 0.054
5 0.618 0.50 0.358 0.269 0.1% 0.128 0.090 0.062
0.618 0.50 0.358 O270 0.20 0.13 0.089 0.059
2

0.618 0.499 0.356 0.266 0.194 0.122 0.083 o055
0.19 0.054
4 3
200
3. Fraction of feed unconverted for a second order chemical reaction with mixed feed in a single CSTR
0.618 0.50 0.358 0.27 0.20 0.131 0.094 0.067
5
D. P. RAO and LOUIS
The results for the model of Villermaux and Zoulahan are obtained by solving the differential equation for the maximum mixedness section on a digital computer using a fourthorder RungeKuttaGill scheme. The results for the model of Spielman and Levenspiel, and Kattan and Adler are obtained from the literature. The necessary equations for Manning’s model are solved on a digital computer. A comparison of the results shows that the models of Spielman and Levenspiel, Manning et al., and Ng and Rippin agree very well whereas the model of Villermaux and Zoulalian differs. This again, substantiates the conclusion of Rippin [ 111 that though the degree of segregation, J, is the same for two types of models, conversions need not be the same. Also, because the computation time for the models of Ng and Rippin, and Manning et al. is much less compared to the Monte Carlo coalescence model of Spielman and Levenspiel or the equivalent population balance model of Curl[3] the former models may be preferred for design purposes. Table 4 shows some results obtained for a consecutive second order reaction of the type A+Bk’R A+B Table 4. Dimensionless
’
,S
exit concentrations
where R is the desired product and k, and k2 are the rate constants for the two reactions. The models studied with this reaction scheme were those of Spielman and Levenspiel, Manning et al. and Ng and Rippin. The later model requires computation
application of the model of Manning et al requires trial and error solution, each trial involving computation of segregated reactor concentrations. The model of Spielman and Levenspiel requires numerical integration of the batch reactor differential equations over a short interval of time. This was accomplished by using the modified Euler’s method. Dashed lines in Table 4 indicate that solution could not be obtained. The three models give nearly identical results. As the models of Ng and Rippin, and Kattan and Adler have been intuitively shown to be identical, and as they are applicable for any arbitrary RTD, a 2CSTR RTD was selected for
for a consecutive second order chemical reaction in a single CSTR with mixed feed (k, = 10.0, C,,/C,, = 1.0, r = 1O)
Model 1 Spiehnan and Levenspiel [ 131 Mixing parameter
L. EDWARDS
Model 2 Manning et al. [7]
Model 3 Ng and Rippin [8]
C*’
CLf’
C.¶’
CA’
C,’
CR’
C”’
CB’
CIZ’
1 10 100
0.25750 0.28347 0.32010
0.17400 0.18648 0.20820
0.6590 O61955 0.56798
0.2478 1 
0.16461 
0.65973 
0.25148 0.28159 0.30336
0.16602 0.18359 0.19856
0.65942 0.60629 0.54344
0.1
1 10 100
0.38516 0.40763 0.43661
0.11016 0.11%6 0.12812
o33983 0.30438 0.25490
0.37663 
0.10131 
0.33839 
0.38040 0.39864 0.41979
[email protected] 0.10973 0.12213
0.33392 O29973 0.25516
1.0
1 10 100
0.51092 0.51329 0.52130
0.07644 0*08044 0.09137
0.05460 0.05384 0.04878
0.50225 
0.06834 
0.05407 
o50553 0.50447 0.50461
0.068519 0.074305 0.085949
0.054272 0.05265 0.049336
10.0
1 10 100
0.53323 0.53493 0.53990
0.07176 0.07516 0.08515
0.00529 o+IO53 oXtO
0.52437 
0.0638 1 
0.00529 
0.52774 0.52556 0.52350
0+64015 0.069139 0%%52
0.0053198 0.0052985 100.0 O+XX2708
1184
k/k,
Mixing effects in stirred tank reactors: a comparison of models
comparison purposes. Kattan and Adler report results for this case[6]. Table 5 shows the results obtained by the two models. They are nearly identical. The Monte Carlo coalescence model of Spielman and Levenspiel [ 131 was also used to study the effect of mixing on stepwise addition polymerization reaction without termination for the case of mixed feed. The extreme cases of complete segregation and ideal mixing have already been treated by Tadmor and Biesenberger [ 141. The polymerication reaction is M,+M
k > M,,,.
dM, =kMIM dt dM, _ kM(M,_, M,) dt
A, =
5 PM,.
2
I
%=
(11)
2
= kMh,
(12)
kM[2&+hol.
(13)
The initial conditions for A1 and A2are (Ml)o as initiator is included in the polymer chain Tadmor and Biesenberger[l4] showed that the expres
200
Model 3 Ng and Rippin [8]
Model 4 Rattan and Adler [6] 0.40 0.429 0.436 0.437
0.39 0.437
0.39 0.429 0.434 0.437
048
048
ca
0.131
0.107 0.127 0.131
0.083 0.109 0.130 0.131
0
0.01
0.01 0.0166 0.022 0.0264 0.0297 0.032
0.011 0.0154 0.025 0.028 0.030 0.032
0.002 0.0056 0.0073 0.0092 0.0113
0.002 1 oW6 0.007 O*OlOl 0.0113
10 200
0
) 1000
Calculated [6]
0
10 50 200 1000 co
50 200 1000 a,
(10)
Ao= (M,)o
10 50 cc 0
20
weight dis
Using Eqs. (7)( 10) and assuming that moments of the molecular weight distribution can be treated as continuous variables, it can be shown that
Table 5. Fraction of feed unconverted for a second order chemical reaction with mixed feed in a reactor with a 2CSTR residence time distribution Mixing parameter
(9)
5=1
(7)
krcm
for x > 1.
The nth moment of the molecular tribution is defined as
In this scheme Ml represents initiator, M, is the polymer having chain length x (including the initiator molecule), and M is monomer. The batch reactor equations are:
where (Ml)o is the initiator concentration at r = 0 as well as the total polymer concentration throughout the reaction period.
(8)
0.032 o+lO2
0*0113
1185
D. P. RAO and LOUIS
sions for M and A1remain the same for extreme cases of complete segregation and complete micromixing, as the rate expressions for M and A1 correspond to first order reactions. They also found that the expressions for A2 differ for the two mixing extremes. The equations describing the intermediate states of mixing for the polymerization reactions shown above obtained by applying the TwoEnvironment model of Ng and Rippin are as follows. Solution of the batch reactor expressions for M, Al and A2 yields the following equations M = M, epst (14)
L. EDWARDS
along with the expressions for III and II given in the Appendix. The resulting differential equation can be solved for the monomer concentration in the leaving environment, ML, to give
ML= (sT+Y$zz+ 1) *
(20)
Applying Eq. (AS) the expression for the mean concentration of monomer leaving the reactor, A, can be shown to be
ML%_ 1+sr’
A, = (Ml),+ [A,Az= (M1&
M,( 1 eest)
(15)
W,h,12+3b2(M~h
(16)
where s = k(M,),. The above expressions for a batch reactor can be substituted into Eq. (A3) of the Appendix and it can be shown that IVMO=M,eh17S7+~7+1 7 Iv
(Ml),
=
(17)
1 (Md~e+7 7
sM, eh/7 1) (ST+R7+
(23) Again, the value of the first moment is independent of the mixing parameter, RT. The expressions for (A2)Land A2can be shown to be
RT+l
+ (R7+
(22)
1)
(18)
_
IV(M,),
3M0 +& >
oC2)~ = ($$5+
(for the first moment, A,)
(3W+2M021(M,M CRT+1) (ST+ RT
= ePA17 (Ey;O+3M0 K + (Ml),
1
(19)
r=
~ML[L,+~(~)LI
and h,
[email protected],h.
(17) can be substituted
(24)
where
(for the second moment, A,). Equation
1)
+ (M,)o+r~
Mo21(Ml)o !1R7+1+1+&+2S7
_ (3Mo + 2M02/ (Ml) O) (ST+&+ 1)
(2s,R+‘;T1+ 1)
into Eq. (A4)
and
1186
(25)
Mixing effects in stirred tank reactors: a comparison of models
As was shown in the case of a single and consecutive second order reaction schemes that the model of Ng and Rippin agreed closely with that of Manning et al., it would be interesting to examine if the same agreement would be obtained in the case of polymerization reaction. First, it is easy to show that the expressions for a and x1 are independent of the mixing parameter, R’ = Q/F of Manning et al. and are similar in form to Eq. (21) and (23) respectively. The expression for X2 in the case of mixed feed can be shown to be
1
Mo2/(MI) o (3Mo + 2M02/ (MI) O) . c26j +2s7+z?r+ 1(ST+&+ 1) Equation (26) can be simplified further by substituting the expression for (h2)L from Eq. (24). The expression forx, is
(Rr+ 1) (2~7+Rr+ 1)
+ (::;.
3M,,+2 (z)O
1 (l+R~+m)

X2 =R’ (I+m:f?R’+m)
(RT+ 1) + (Ml)o (1 tt$s;;);
(3Mo) (RT) (87) + (ST+ 1) (sT+RT+ 1)
+ (Ml),, (l+s~)~
2M02 (~7)~ (RT) + (MI),, (ST+ 1)2 (sT+RT+
1)
2M,,2 (RT) (8~)~ + (MI)o (ST+ 1)2 (sT+RT+
1)2’
+ $$+3M,++ [ (27)
The expressions for x2, for the limiting cases of complete segregation, i.e. Rr + 0, and maximum mixedness, i.e. Rr + CQcan be derived from Eq. (26) and they are found to be
(X2)
RnO
(M,)o
=
[
1+32+x (2Z)
1
(28)
and 62)
&rm
=
(M,), [1+32+2Z2]
(29)
+&I
krMo
and 
1
112= l+kr(M,)o’ Equations (28) and (29) are of course, identical to those obtained by Tadmor and Biesenberger [141.
(l+R’+2m)
I
(MA,{3Mo
(,:,+L) (30)
Table 6 shows some preliminary results for the case of mixed feed polymerization calculated using the Monte Carlo Coalescence model of Spielman and Levenspiel, the TwoEnvironment Model of Ng and Rippin, and the macromixed feed model of Manning et al. The results for all three models are nearly the same. EXTENSIONS
1 +kdMdo
(:,“,“!%)
*
where
z=
R’ +ST)
AND
UNMIXED
FEED
As the models of Spielman and Levenspiel and Rattan and Adler were also applied to unmixed feed situations, it was felt that it would be interesting to extend the other models to the unmixed feed case and compare all the models. The extension of the TwoEnvironment model to unmixed stoichiometric feed for a second order chemical reaction gives rise to the following equation (see Appendix for derivation) 1187
CL CL
1
40 20 5
Mixing parameter
(MI),
0.0125 0.0125 0.0125 100 0.025
400 200 100
k
0.2 0.2 0.2 0.4
M0
50.0 10o 50O 10.0
7
0.2115 0.2050 0.2095 0.409
Model 1
0.2118 0.2050 0.2095 o41
Model 2
0.2118 O2050 O2095 0.41
Model 2 4.025 4.425 3.8574 7.33
Model 1 4.16 446 3.86 7.30
4.025 4405 3G3540 7.32
Model 3
reaction with mixed feed in a single CSTR
Model 3
Table 6. First three moments of the molecular weight distribution for a stepwise polymerization (kI./mole set; (M,),, M,moles/l.; 7 set)
Mixing effects in stirred tank reactors: a comparison of models
CA= RT
1+s 2 krCAO
C AO R+r+l 1 +RT+
I (31)
1’
It is interesting to note that for the value of reaction rate constant, k, equal to infinity the above expression reduces to
c.Am_ 1 C A,,
RT+~’
(32)
It can also be shown easily that the models of Manning et al., and Villermaux and Zoulalian give the same form as Eq. (32) in the limiting case of k equal to infinity. It is also interesting to note that Eq. (32) is similar in form to the expression for the intensity of segregation derived by Brodkey [2] from the work of Rosenswieg [ 123
l
I,= 1
+E$’
(33)
A.9
Equation (33) was also derived recently by Evangelista et al. [5] for a coalescing system corresponding to the model of Spielman and Levenspiel using the population balance approach of Cur1[3]. But they called I, “Zwietering’s degree of segregation” which is usually denoted by the letter J or I, by Brodkev [2]. In this context, a discussion of the concepts of I, and J seems appropriate. Danckwerts [4] defines the intensity of segregation, I,, for reactors with unmixed feed. The effect of the residence time distribution on the expression for I, is not dealt with anywhere in the literature because of the complexities involved. Danckwerts [4] also defines another intensity of segregation J, for reactors with mixed feed but having an arbitrary residence time distribution. Here also, molecular mixing plays an important role, though not as important as in reactors with unmixed feed. However, it should be recognized that I, and J as defined by Dankwerts are two different parameters not to be confused with one another.
Table 7 shows the results obtained for unstoichiometric feed, second order mixed, chemical reaction by using the models of Ng and Rippin, Spielman and Levenspiel, Manning et al., Villermaux and Zoulalian, and Kattan and Adler. A comparison of the different models shows that at low values of krCAO and high values of Rr, the models agree very closely, and the model of Ng and Rippin and even that of Manning et al gives a good agreement with that of Spielman and Levenspiel. The limiting values for conversion as k7CA0 + ~0, obtained from the models of Ng and Rippin, Manning et al, and Villermaux and Zoulalian are identical as already shown above. However, these values differ significantly from those of Spielman and Levenspiel. The reason for this may be obtained if one recalls the previous analogy drawn between the model of Ng and Rippin and that of Spielman and Levenspiel for mixed feed, i.e. the coalescence between fluid elements is equivalent to a transfer from the entering to the leaving environment. However, in the case of unmixed feed, a coalescence between two fluid elements containing only A does not cause a transfer between the environments. Thus, the effective value of the mixing parameter for the model of Spielman and Levenspiel is lower than Z/2.
CONCLUSIONS
AND
DISCUSSION
From the above discussion of models and results obtained, following conclusions can be stated: 1. The model of Spielman and Levenspiel [ 131 for intermediate states of mixing between complete segregation and maximum mixedness for a single CSTRidealRTD with premixed feed is shown to be identical to the TwoEnvironment model of Ng and Rippin [S] for the same values of the mixing parameters of the two models, i.e. Z/2 and Rr respectively. Also, the model of Manning et a1.[7] is found to agree closely with these two models. 2. The Monte Carlo residual life time model of Kattan and Adler[6] and the TwoEnvironment model of Ng and Rippin[S] are shown to be
1189
D. P. RAO and LOUIS Table 7. Fraction of feed unconverted
Mixing parameter
0.1
L. EDWARDS
for a second order chemical reaction with unmixed feed in a single CSTR
Model 1 Spielman and Levenspiel [ 131
Model 2 Manning el Qi. [71
Model 3 Ng and Rippin [8]
Model 4 Kattan and Adler [6]
Model 5 Villermaux Zoulalian [ 151
0,958 0.924 O921
0.955 O918 0.912 0.91
099 0.96 O.%
0.939 0.917 O916
1 10 100 00
l0
1 10 100 m
0.78 0.63 0.62
0.805 0.653 0.624
0.81 O656 O615 0.62
o91 0.78 o70
0.750 0.624 0.618
10.0
1 10 100 cc
0.62 0.35 0.27
0.617 0.333 0.278
0.635 0.337 0.269 0.27
O62 0.45 0.42
0.579 O295 O271
loo*0
1 10 100 m
060 0.22 0.108
O528 0.170 0.102
0.548 0.178 0.104 0.094
0.62 0.24 0.15
0.52 0.148 0.097
1 10 100 cc
0.60 o19 0.046
0.505 0.111 0.041
0.516 0,119 0.041 0.031
0.62 0.20 0~044
0.504 o105 0.035
1 10 100 m
0.63 0.20 O038
0.50 o095 0.019
0.505 o0991 0.0199 0.01
0.62 0.20 0.04
0.50 0.094 O016
lOOO*O
10,000*0
identical for premixed feed and same values of 112and RT respectively. 3. The models of Ng and Rippin[8], Manning et al. [7], Villermaux and Zoulalian [15] are extended for unmixed feed cases. The model of Ng and Rippin is shown to agree closely with that of Speilman and Levenspiel for relatively low values of /WA,, and high values of RT in the case of a second order reaction. 4. From the above conclusions, it could be stated that the model of Ng and Rippin[8] is the simplest to use and should be applied for reactors with mixed feed and the Monte Carlo model of Kattan and Adler [6], which is a more general model than that of Spielman and Levenspiel, be used for simulating reactors with unmixed feed. The application of either the model of Ng and
Rippin or the model of Kattan and Adler to tubular reactors with axial dispersion is not clear, the primary reason being that the axial dispersion can be caused by several effects, e.g., (1) Taylortype dispersion, (2) turbulent and molecular diffusion, (3) backmixing due to the presence of packing in the case of packed beds, (4) coalescence and redispersion of droplets in the case of a twophase system. Thus, the residencetime distribution obtained by tracer studies cannot be purely attributed to either macro or micromixing. However, it could be visualized that the tracer residence time distribution is a result of the sum of the effects of micro and macromixing. Hence, in order to utilize the models of Ng and Rippin or that of Kattan and Adler, it appears necessary to find physical parameters to separate the contribution of the two processes of
1190
Mixing effects in stirred tank reactors: a comparison of models
maining in the segregated environment fraction of the feed with residence time between x and x + dr feed rate to the reactor (see Fig. 3) intensity of segregation (unmixed feed) degree of segregation (mixed feed) mixing parameter of Villermaux and Zoulalian kz second order reaction rate constants monomer concentration initiator concentration feed concentration of monomer MCI concentration of polymer of chain length x m mass of material in the entering environment in the model of Ng and Rippin m(X)& fraction of the feed with residence time between x and x+ dx staying in the maximum mixedness environment Q flow rate of the impeller stream (see Fig. 3) R mixing transfer coefficient of Ng and Rippin (defined by Eq. 1) t time residence time of a fluid element ; volume of the reactor
mixing to the residence time distribution. The recent results obtained with the Monte Carlo Coalescence model of Rao and Dunn [ 101 throw some light in this direction. The Monte Carlo Coalescence model of Rao and Dunn simulates axialdispersion in a tubular reactor due to coalescence and redispersion of droplets at different axial positions. In this model, every collision process is assumed to result in coalescence and subsequent redispersion. However, in the simulation of the coalescenceredispersion step in the model, if the droplets were allowed to move from one axial position to another without colliding with another droplet, it would cause a spread in the residence time distribution. Thus, it appears possible that by incorporating the effect of droplet motions which do not result in coalescence, but cause only a spread in residence time distribution, one can identify the separate effects of macromixing and micromixing. Acknowledgements The authors are grateful for a National Science Foundation grant (GK10043) which supported this work. NOTATION
reactorexit concentration of reactant, A, dimensionless CA’, CB’, CR’ reactorexit concentration of species A, B and R respectively C AO feed concentration of A to the reactor c‘4m concentration of A in the reactorexit for an infinitely fast second order reaction or tracerinput C AL C,
Greek symbols ff age x life expectation L nth moment of the polymer molecular weight distribution scalar microscale of turbulence h, 7 average residence time
REFERENCES [I] BISCHOFF K. B.,Znd. Engng Chem. 1966 58 18. [2] BRODKEY R. S., The Phenomena of Fluid Motions, p. 345. AddisonWesley, Reading, Mass. 1967. [3] CURL R. L., A.Z.Ch.EJI 1963 9 175. [4] DANCKWERTS P. V., Chem. Engng Sci. 1958 8 93. [5] EVANGELISTA J. J. et al., A.I.Ch.EJll969 15843. 161 KATTAN A. and ADLER R. J., Paper presented at 62nd National A.1.CH.E. M&ing, Salt Lake City, Utah. [7] MANNING F. S. etal., A.I.Ch.EJll965 11723.
1191
D. P. RAO and LOUIS
L. EDWARDS
PI NG D. Y. C. and RIPPIN [91 [lOI Hll [I21 [131 [I41 WI [161 [I71
D. W. T., Third European Symposium on Chemical Reaction Engineering, Amsterdam, p. 161. Pergamon Press, Oxford 1965. NISHIMURA Y. and MATSUBARA M., Chem. Engng Sci 1970 25 1785. RAO D. P. and DUNN I. J., Chem. Engng Sci. 1970 25 1275. RIPPIN D. W. T., Chem. Engng Sci. 1967 22 247. ROSENSWIEG R. E.,A.I.Ch.EJI 1964 1091. SPIELMAN L. A. and LEVENSPIEL O., Chem. Engng Sci. 1965 20 247. TADMORA. and BIESENBERGER A., Znd. Engng Chem. Fundls 1966 5 336. VILLERMAUX J. and ZOULALIAN A., Chem. Engng Sci. 1969 24 1513. WEINSTEIN H. and ADLER R. J.. Chem. Enene Sci. 1967 22 65. ZWIETERING Th. N., Chem. EngAg Sci. 1959 l’i 1.
APPENDIX
The mean concentration
RESTATEMENT OF NG AND RIPPIN’S MODEL AND EXTENSION TO UNMIXED FEED Here, the Eqs. (4)(6), (8) and (9) of Ng and Rippin [8] are rewritten for the purpose of the present derivation. Volume of material in the leaving environment with life expectationh toX + dh
of the leaving fluid is written as
C” = T[ (II c”L)A=o+ (IV),=, C,,].
(As)
The entering environment is assumed to be completely segregated and hence conversion in this environment is nil in the case of umnixed feed. Thus, the expression for IVC,, in the case of unmixed feed becomes m
(1e~)f(a+h)do=IIVdh.
IVC,, = cdo eeR”f(a +A)da. 7 I
(Al)
Rate at which material of life expectation A to A + dh is transferred from the entering to the leaving environment = R (volume of material of life expectation A to A+ dh in the entering environment)
For a singleCSTR with ideal residence t&e distribution defined byf(t) = e%, the expression for CA in the case of unmixed feed can be easily derived as shown below. The expressions for IVCAO, III and II can be shown to be respectively as IVC,,
VRdh = eR”f(a+A)da 7 I 0
= III RVaX.
Where p(C,,) unit volume.
+
SeA/&
646)
7
III
_
e+
1
(A7)
7 Rr+l and II,!!??.
= IV C,,RVdA.
R (III CALIV CA,,) II
is the rate of disappearance
(A8)
(A31
The concentration of reactant in the leaving environment, CAL, is determined as a function of the life expectation by solving the material balance equation
2 = p(C,,)
=
b42)
Rate at which reactant of life expectation A to A +dA is transferred from entering to the leaving environment VRaX = C,(a) e&f(a+A)da 7 I 0
(A3a)
0
(A4)
Substituting the expressions for IV&,, (A4) and solving for C,,, we get c
=
AL

III and II in Eq.
1+ u’l+4k7c”o 2~C*0
and from Eq. (AS) it can be shown that
of reactant per
1192
c
*
=
RT l+~l+4k~Ca0 RT+~ 2krC,,
I
1 R7+1’
(A9)