Mixing in Chemical Reactors - ACS Publications - American Chemical

6 Mixing in Chemical Reactors JACQUES V I L L E R M A U X

Downloaded by UNIV OF SYDNEY on May 3, 2015 | http://pubs.acs.org Publication Date: July 28, 1983 | doi: 10.1021/bk-1983-0226.ch006

Institut National Polytechnique de Lorraine, Laboratoire des Sciences du Génie Chimique, CNRS-ENSIC, Nancy, France

Recent progress in the understanding and modelling of mixing phenomena in chemical reactors is reviewed. The following items are discussed: characterization of the degree of mixing, Eulerian approach to turbulence theory with application to modelling combustion, macromixing and residence time distributions, phenomenological models for mixing earliness i n the age space, mechanism of micromixing i n physical space and segregation phenomena at the microscopic l e v e l , mixing in stirred tanks and static mixers. An example is presented of the application to continuous free radical polymerization. It is concluded that a comprehensive and predictive theory for micromixing should not be sought through turbulence theory alone, but rather in phenomenological interaction models, whose parameters could be interpreted on a fundamental basis by this theory. Contacting reactants and subsequent mixing of r e a c t i n g species i s one of the major features c o n t r o l l i n g the behaviour of chemical r e a c t o r s . In s p i t e of recent advances, a u n i f i e d theory p r o v i d i n g the engineer with general r u l e s a p p l i c a b l e to any case of competit i o n between mixing and chemical r e a c t i o n i s s t i l l l a c k i n g . However, d i r e c t i o n s i n which such a theory should be sought are now i n view. The aim o f t h i s chapter i s t o review recent and s i g n i f i cant c o n t r i b u t i o n s which may l e a d t o such a general treatment. Of course, the f i e l d of mixing i n chemical engineering i s immense. Therefore, t h i s review w i l l be r e s t r i c t e d , with perhaps a few exceptions, to those phenomena where mixing and chemical reactions are c l o s e l y l i n k e d . The purely p h y s i c a l aspects of mixing won't be considered, although t h i s important operation also poses unsolved problems to i n d u s t r y . That mixing i n chemical r e a c t o r s i s a subject of very a c t i v e research i s a t t e s t e d by the recent p u b l i c a t i o n of s e v e r a l review 0097-615 6/ 8 3/0226-013 5 $ 13,80/ 0 © 1983 American Chemical Society In Chemical Reaction Engineering—Plenary Lectures; Wei, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.


136

C H E M I C A L REACTION

ENGINEERING

papers, which appeared a few years ago or while the present was i n p r e p a r a t i o n (J_-6) . Instead o f d u p l i c a t i n g these reviews and r a t h e r than present i n g an exhaustive l i t e r a t u r e survey, I have chosen t o emphasize the points which seem the most c r i t i c a l and c o n t r o v e r s i a l , or those where s i g n i f i c a n t progress has been made i n recent years, and to give personal views on these s u b j e c t s . In doing so, I ac cept the r i s k that t h i s review may appear p a r t i a l or incomplete to those authors whose (sometimes e x c e l l e n t ) papers w i l l be l e f t out. That mixing i n chemical reactors i s also a problem of v i t a l importance to i n d u s t r y appears i n some, too r a r e , papers d e s c r i b i n g the e f f e c t of a g i t a t i o n on y i e l d i n p i l o t p l a n t reactors (7.) or the d i f f i c u l t i e s of conserving a good s e l e c t i v i t y when s c a l i n g up a mixing device to the commercial s c a l e (8). I t i s a p i t y that p r o p r i e t a r y requirements r e s t r a i n the p u b l i c a t i o n of such indus t r i a l case s t u d i e s , whose a n a l y s i s would be o f the g r e a t e s t i n t e r est t o t h e o r i c i a n s o f mixing. But t h i s i s a general problem i n Chemical Reaction Engineering. Nevertheless, i t i s encouraging to see i n d u s t r i a l p a r t i c i p a n t s attending meetings and g i v i n g t h e i r o p i n i o n about the relevance of research topics and about what should be done to improve our knowledge i n the f i e l d (9,10). This should prevent "too many academics pursuing too many non-problems"

(Jl). The l a s t point that should be o u t l i n e d i n t h i s i n t r o d u c t i o n i s that mixing i n chemical reactors r e a l l y c o n s t i t u t e s an o r i g i n a l chapter of Chemical Reaction Engineering. Since the p i o n e e r i n g work of Danckwerts ( 12) , Zwietering (13) and others, i t i s c l e a r that chemical r e a c t i o n engineering c o n s t i t u t e s a s c i e n t i f i c d i s c i p l i n e with i t s own methods. The RTD and the segregation concepts f o r i n s t a n c e , have completely renewed the d e s c r i p t i o n of the beha v i o u r of r e a c t i n g mixtures i n r e a c t o r s . I am convinced that f u t u r e progress can be expected i n pursuing novel concepts rather than i n more s o p h i s t i c a t e d combinations of chemical k i n e t i c s and f l u i d me chanics. Of course t h i s doesn't mean that we must neglect the con t r i b u t i o n of the l a t t e r d i s c i p l i n e , as w i l l be seen below. D e f i n i t i o n and c h a r a c t e r i z a t i o n of the degree of mixing A d e f i n i t i o n of mixing i s proposed i n many papers and t e x t books (14, 15, 16, 2), and t h i s point w i l l not be discussed here. Several c r i t e r i a are used f o r a q u a n t i t a t i v e c h a r a c t e r i z a t i o n of the " q u a l i t y of mixing". These have been r e c a l l e d by Hiby (17). In a non-uniform mixture, l e t p(C) be the l o c a l c o n c e n t r a t i o n d i s t r i b u t i o n of a species. p(C) may be c h a r a c t e r i z e d by i t s mean C, its_ variance and the average value AC of the d e v i a t i o n AC = |C - C| from the mean. From these q u a n t i t i e s , s e v e r a l c r i t e r i a may be de f i n e d (17), namely 6 = AC/C, 6 = A C / C and ό = σ/C. Another critérium i s Δ = AC/AC , where AC i s the value o f Sc before mix i n g (or at the reactor°inlet). Smarting from two_streams o f ( r e duced) c o n c e n t r a t i o n 0 and 1, then AC = 2 C ( l - C). The m a x

max

σ

In Chemical Reaction Engineering—Plenary Lectures; Wei, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

6.

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137

Mixing in Chemical Reactors

I n t e n s i t y of Segregation d e f i n e d by Danckwerts is I = σ^/σ = o / C ( l - C) , but /ΐζ = σ/σ bas a l s o been used. I f two species A and Β are mixed, the d e f i n i t i o n of I becomes 02) *s "~ A B^Ao*^Bo* R e l a t i o n s h i p s between a l l these i n d i c e s are obvious. Their common property i s that the s m a l l e r they are, the b e t t e r the q u a l i t y of the mixture. The corresponding degrees of homogeneity may a l s o be defined as 1 - δ, 1 - 6 , 1 - / Ï J , e t c . . . Mixing times are the times r e q u i r e d f o r 6, or Δ, or I to f a l l from t h e i r i n i t i a l value (before m i x i n g ) , down to some pres c r i b e d small value ( f o r instance 0.05 or 0 . 0 1 ) . We s h a l l see that i n Lagrangian coordinates, - σ /(do /dt) i s a l s o a mixing time. The e s t i m a t i o n of mixing times i n s t i r r e d tanks w i l l be discussed i n a further Section. The s c a l e of segregation gives i n f o r m a t i o n on the s i z e of se gregated regions. Let c(x) and c(x+r) be the f l u c t u a t i n g concen t r a t i o n s at p o i n t s χ and x+r (c = C - C), then the a u t o c o r r e l a t i o n f u n c t i o n of c o n c e n t r a t i o n i s (2):g (r) = c(x) c ( x + r ) / o . The s c a l e of segregation i s then defined as : 2

s

0

0

g

=

c

c

σ

g


2

2

L

s

=

8 s ( r ) dr ο

In turbulence theory, t h i s i s a l s o known as the c o n c e n t r a t i o n mac r o s c a l e , which plays an important r o l e i n the i n t e r p r e t a t i o n of micromixing phenomena. A f i r s t remark about these mixing i n d i c e s i s that a c e r t a i n confusion p r e v a i l s i n the l i t e r a t u r e owing to the m u l t i p l i c i t y of d e f i n i t i o n s . I t would be d e s i r a b l e to adopt one s i n g l e measure f o r the degree of mixing (and/or homogeneity), based f o r instance on the reduced variance of the c o n c e n t r a t i o n d i s t r i b u t i o n . A second remark i s that an experimentally measured degree of mixing depends on the s p a t i a l r e s o l u t i o n of the probe used f o r mea s u r i n g " p o i n t " values of the concentration. In l i q u i d s , the most commonly used methods are e l e c t r i c a l c o n d u c t i v i t y (18, 19), l i g h t a b s o r p t i o n , fluorescence (30) and che m i c a l methods based on the c o l o r change of an i n d i c a t o r under the i n f l u e n c e of an instantaneous r e a c t i o n (21, 22). The s p a t i a l r e s o l u t i o n of p h y s i c a l methods ( o p t i c a l , e l e c t r i c a l microprobes) i s about 100 ym (19) so that these are w e l l s u i t e d to macromixing s t u d i e s but cannot compete w i t h chemical methods f o r the study of mixing at the molecular s c a l e . An o r i g i n a l method based on the con tinuous i n j e c t i o n of r a d i o a c t i v e t r a c e r s i n an i n d u s t r i a l mixer has a l s o been proposed (23). In gases, c o n c e n t r a t i o n f l u c t u a t i o n s have been measured using a c a t a l y t i c wire (24). The d e f i n i t i o n s of the degree of mixing presented above aim at a l o c a l c h a r a c t e r i z a t i o n of the mixture homogeneity i n the phy s i c a l space. There a l s o e x i s t more i n d i r e c t mixing i n d i c e s . The segregation index J of Danckwerts (12) i s one of the most famous ones. I t a p p l i e s to continuous r e a c t o r s and r e l i e s upon the v a r i ance of age : J = Var a /Var a,where a i s the age of a molecule, p


C H E M I C A L REACTION ENGINEERING


138

dp the mean age at " p o i n t " Ρ and Var denotes the variance of the d i s t r i b u t i o n s over a l l " p o i n t s " or a l l molecules i n the r e a c t o r . J = 1 i n a t o t a l l y segregated mixture ( a l l the ages are the same w i t h i n a " p o i n t " α = αρ) and J takes a minimum value depending on the residence time d i s t r i b u t i o n when mixing e a r l i n e s s i s at a mi nimum. J i s s t i l l a very popular q u a n t i t y i n the l i t e r a t u r e where i t i s used by many authors to compare t h e i r micromixing models. However I t h i n k that w h i l s t i t c o n s t i t u t e s some u s e f u l reference for mixing e a r l i n e s s , i t i s of l i t t l e i n t e r e s t f o r the design of chemical r e a c t o r s as i t cannot be measured d i r e c t l y nor be used for a s t r a i g h t f o r w a r d c a l c u l a t i o n of chemical conversion. Another i n d i r e c t mixing index was proposed by Ogawa et a l . (25, 26), based on i n f o r m a t i o n theory and on the e s t i m a t i o n of the "entropy" of a mixture. This "entropy" i s defined from the t r a c e r d i s t r i b u t i o n among η zones i n the r e a c t o r as Η = - £ V? In p^, where V* i s the reduced volume of zone i and p i t h e i = l " p r o b a b i l i t y " of occurence of the t r a c e r i n zone i (see reference (25) f o r more d e t a i l s ) . As mixing proceeds, the degree of homoge n e i t y of the mixture i s defined as Μ = H/H^, where H^ i s the f i n a l entropy of the homogeneous mixture. This method was used to study the mixing r a t e i n s t i r r e d v e s s e l s . M was p l o t t e d as a f u n c t i o n of time from a record of the concentrations at v a r i o u s places i n the v e s s e l . This allowed a comparison of e f f i c i e n c y of d i f f e r e n t a g i t a t o r s (26). As an i n t e r e s t i n g g e n e r a l i z a t i o n , the macromixing ho mogeneity i n a continuous r e a c t o r having a RTD E ( t ) may be defined as (25) : M = - J " E ( t ) In E ( t ) dt,from which i t i s seen that M= 0 i n a plug flow r e a c t o r and M = 1 i n a continuous s t i r r e d r e a c t o r . E u l e r i a n approach: M i x i n g and

Turbulence

Turbulence theory provides a c l a s s i c a l approach to mixing phenomena. This i s a n a t u r a l way f o r mechanical engineers and spe c i a l i s t s of combustion, who are very f a m i l i a r w i t h the methods of f l u i d mechanics. However, when complex chemical r e a c t i o n s are i n volved, the use of the formalism of turbulence alone seems to lead to a deadlock, as has been pointed out by s e v e r a l authors. An ex c e l l e n t p r e s e n t a t i o n of the s t a t e of the a r t can be found i n the recent l i t e r a t u r e e s p e c i a l l y by Brodkey (16, _2, 27) and P a t t e r s o n (3). These reviews r e v e a l no major breakthrough, and only slow progress on a d i f f i c u l t road. As the r e s u l t s of turbulence theory are sometimes used i n c o r r e c t l y i n the l i t e r a t u r e , i t seems h e l p f u l to r e c a l l (Table I) the p r i n c i p a l q u a n t i t i e s c h a r a c t e r i z i n g v e l o c i t y and c o n c e n t r a t i o n f l u c t u a t i o n s , based on the assumption of homogeneous i s o t r o p i c turbulence. Most of them can be deduced from s p e c t r a l measurements. I f the techniques f o r determining v e l o c i t y f l u c t u a t i o n s p e c t r a are w e l l e s t a b l i s h e d (hot wire anemometer, l a s e r - d o p p l e r anemometer...) r e l i a b l e methods f o r monitoring c o n c e n t r a t i o n f l u c t u a t i o n s are l e s s known. I n t e r e s t i n g data were r e c e n t l y obtained by using l o c a l conductometry microprobes i n t e r f a c e d w i t h a high g a i n , f a s t response


6.

viLLERMAUX


conductimeter (28, 29, 30), but there are s t i l l problems of space and time r e s o l u t i o n (19) and new methods should be developed f o r the determination of concentration f l u c t u a t i o n s s p e c t r a i n chemi c a l r e a c t o r s . I t i s c l e a r from Table I that there are two p a r a l l e l f a m i l i e s of c h a r a c t e r i s t i c s p e r t a i n i n g to v e l o c i t y f l u c t u a t i o n s on one hand and to c o n c e n t r a t i o n f l u c t u a t i o n s on the other hand. This d i s t i n c t i o n i s not always made i n the l i t e r a t u r e , some authors using f o r instance v e l o c i t y macro or m i c r o s c a l e s , (which are b e t t e r known) i n place of c o n c e n t r a t i o n s c a l e s . I t must a l s o be r e c a l l e d that d i f f e r e n t length and time s c a l e s have very p r e c i s e meanings and should not be used at random on the s i n g l e b a s i s o f dimension a l a n a l y s i s . The macroscales L and L c h a r a c t e r i z e l a r g e i n i t i a l eddies, whose s i z e i s g e n e r a l l y comparable to that of the i m p e l l e r i n s t i r r e d tanks, or that of the i n l e t tubes i n tubular r e a c t o r s . Taylor and C o r r s i n microscales Xf and A are those of the maximum of d i s s i p a t i o n e i t h e r of the turbulent k i n e t i c energy, or of the segregation. In t h i s respect, the C o r r s i n microscale X and the a s s o c i a t e d time constant T play an important r o l e . The c l a s s i c a l treatment of C o r r s i n , widely used i n the l i t e r a t u r e on mixing, y i e l d s the f o l l o w i n g expressions : f


139

g

g

s

s

1/3

2/3

(3-1)

f o r gases

τ Se

s

1/3

2/3

il

3

1/2 In Sc

for liquids

(3-2)

In l i q u i d s , the second term of (3-2) i s o f t e n n e g l i g i b l e and T ifc 2 ( L 7 ) / . The problem i s to estimate the turbulent energy d i s s i p a t i o n per u n i t mass ε from macroscopic data. For i n s t a n c e , i n s t i r r e d tanks, i t i s not obvious that a l l the power Ρ d i s s i p a t e d at the shaft contributes to ε and s e v e r a l authors (2_, 3J_, 32) have been led to introduce an unknown " e f f i c i e n c y " η such that ε = ηΡ/pV. An a d d i t i o n a l d i f f i c u l t y i s that ε i s not uniform w i t h i n the tank volume and may vary by a f a c t o r 10 from one place to the other. (See S e c t i o n on s t i r r e d tanks). In order to estimate T , Patterson (3) proposed a s s i m i l a t i n g :e Tc, L to L and found ( Ι ^ / ε ) / * = (Lf/ε)' (Ι^/ε) = q/ε A t y p i c a l value f o r i n an i n d u s t r i a l s t i r r e d tank i s a few m i l l i m e t e r s (19). The smallest s i z e f o r turbulent eddies i s given by the Kolmogorov microscale λ^. Energy loss below t h i s s i z e only occurs v i a viscous d i s s i p a t i o n . Here a l s o , s e v e r a l s c a l e s have been i n t r o duced i n the framework of turbulence theory, depending whether ve l o c i t y or concentration f l u c t u a t i o n s are considered, namely λκ, λβ, and Xç (see Table I ) . In l i q u i d s , λκ i s t y p i c a l l y between 10 and 100 urn. The Kolmogorov microscale λκ i s f r e q u e n t l y used i n the i n 1

s

s

3

e

c

1

s

1 / 3

f



constant

dissipation

time

Kolmogorov m i c r o s c a l e

Viscous

Taylor

.2

0

A l l directions

Taylor microscale

Spectra

energy d i s s i p a t i o n

2

= 10

λ

rt

dt

= (ν

3 2

...2

~,2

f

/ε)

J

ο

Γ

10

J

3/2

dt

dk,

f lOv

,1/2

B a t c h e l o r m i c r o s c a l e λ_

)dk

E(k.)dk. 1 1

E(k,)

k, E ( k

Γ ο

2

1

k

dk,

1

E(k,)

E(k.)

direction

- -»D : η n

-> D : η η

(£ = D)

(ε /2ν)

1 / 2

ν

the l a t t e r being v a l i d f o r newtonian f l u i d s o n l y . The average v a l ue o f e over the whole r e a c t o r volume leads t o the o v e r a l l e f f i ciency " e f f " of the mixing process t e l l i n g us how the viscous d i s s i p a t i o n i s u t i l i z e d to promote the t h i n n i n g of the laminae (107, 102, 103, 105). Let us give three examples (the reader may t r y to f i n d these r e s u l t s by a p p l i c a t i o n o f ( 7 - 2 ) ) . 1) Shear flow ( U = Gy, U = 0, U = 0, laminae i n i t i a l l y nor mal to Ox) (99, 104) : x

δ/δ

y

2

2

2

2

δ = δ

x

= αχ, Uy = - ay, U

δ/δ

1 / 2

(7-5)

= 0, laminae i n i

z

exp(- at)

(7-6)

3) S t r e t c h i n g at constant v e l o c i t y (υ = 0)

z

2

= - G t / ( 1 + G t ) and δ = 6 (1 + G t ) "

2) Stagnation flow ( U t i a l l y normal to Oy) (104)

U

z

= G / ( l + Gt), U

χ

= - G/(l + Gt) and δ = δ (1 + G t ) "

y

= 0,

x

1

(7-7)

This k i n d of s t r e t c h i n g occurs f o r instance i n " t a f f y p u l l " . I f the sheet i s f o l d e d up a f t e r a given s t r e t c h i n g d u r a t i o n t and the r e s u l t i n g sheet i s streched up again, then the number of f o l d s i s 2 ' o and δ = δ 2"" ' o. In a l l these processes, e f f i c i e n t mixing i s achieved when the laminae are p e r i o d i c a l l y r e o r i e n t e d w i t h r e spect t o the d i r e c t i o n o f s t r e t c h i n g (101). This remark i s a l s o im portant i n the design of s t a t i c mixers. Q

t

t

t

t

0

1

E r o s i v e (or d i s p e r s i v e ) Mixing. O t t i n o s treatment assumes continuous motion, namely connectedness of m a t e r i a l s u r f a c e s , and hence c o n s e r v a t i o n of t o p o l o g i c a l features (105). Conversely, one may t h i n k of a mixing process t h a t would g r a d u a l l y p u l l o f f s m a l l er fragments from the segregated clumps by t u r b u l e n t f r i c t i o n at t h e i r e x t e r n a l surface. This i s the b a s i s f o r the " S h r i n k i n g Aggre-


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167


gate" (SA) Model of P l a s a r i et a l . (71). The p e e l i n g - o f f process i s c h a r a c t e r i z e d by a mass t r a n s f e r c o e f f i c i e n t h which i s assumed to be expressed by the Calderbank-Moo Young c o r r e l a t i o n , a p p l i c a ble to small p a r t i c l e s immersed i n t u r b u l e n t media : hl/ S i n s t i r r e d r e a c t o r s (119, 120). The p a r t i c l e s are assumed to be s p h e r i c a l (radius R) and the equations f o r r e a c t i o n / d i f f u s i o n are s o l v e d i n v a r i o u s cases: V e r s i o n I : P a r t i c l e s i n i t i a l l y c o n t a i n i n g A immersed i n pure Β (A i s not allowed t o d i f f u s e w i t h i n the p a r t i c l e , other species a r e ) . V e r s i o n I I : symmetrical case (par t i c l e of n o n - d i f f u s i n g Β immersed i n A). The parameters are two Thiele c r i t e r i a = k B R / D * t / t , Ε = A / B and α = V /V . The model p r e d i c t s the y i e l d o f S : X = 2S/(2 S + R) a t the end of the r e a c t i o n (when a l l Β i s consumed). I t was developed f o r batch and semi-batch r e a c t o r s (119, 120), and l a t e r extended to continuous s t i r r e d r e a c t o r s v i a a somewhat complicated procedure (121-112). Some c r i t i c i s m may be adressed, to the MIRE-model, i n s p i t e o f i t s great i n t e r e s t : a r b i t r a r y choice o f s p h e r i c a l shape, a s s i m i l a t i o n of R to h a l f the Kolmogorov m i c r o s c a l e (which i s not obvious as we have seen above) and above a l l , assumption that the i n i t i a l r e a c t a n t i n the p a r t i c l e cannot d i f f u s e o u t s i d e , which creates an unwanted dissymmetry between A and Β when V^ = V-Q. The r e a c t i o n / d i f f u s i o n c o m p e t i t i o n can a l s o be simulated by the IEM-Model. I t s u f f i c e s to set t = t = yL /J«. The equivalence w i t h the MIRE-model w i l l be discussed below. More s o p h i s t i c a t e d mechanisms may be considered, where chemi c a l r e a c t i o n takes place during v a r i o u s stages of Beek and M i l l e r , or a combination of these : - E r o s i v e mixing f o l l o w e d by r e a c t i o n : Two unmixed r e a c t a n t s come i n t o contact i n a CSTR by e r o s i o n of f r e s h aggregates. The e r o s i o n product i s e i t h e r a m i c r o f l u i d (71) or small segregated p a r t i c l e s of mixed reactants undergoing f u r t h e r i n t e r a c t i o n by mo l e c u l a r d i f f u s i o n (108). - S t r e t c h i n g of p a r t i c l e s and simultaneous r e a c t i o n . For r e p r e s e n t i n g r e a c t i o n and d i f f u s i o n i n s t r e t c h i n g l a m e l l a r s t r u c t u r e s , O t t i n o and Ranz (101) introduced a "warped time" t defined by : 11

1

t n

R

m


=

m

2

2

i

Q

D

R i

Q

0

A

B

s

2

D

w


170



t

w

= /

c

Jo

JL^ H t < )

2

d

f

or Ιίϋ =S>

a

dt

ENGINEERING

2

(7-10)

where the r a t e of change of the s t r i a t i o n thickness i s given by ( 7 - 2 ) . The equations f o r r e a c t i o n / d i f f u s i o n are then e a s i e r to solve i n Lagrangian coordinates. In the same way Bourne, Angst et a l . ( 1 2 2 , 1 1 4 ) extended t h e i r MIRE-Model by assuming that the s i z e of the p a r t i c l e s decreases according to ( 7 - 5 ) . This would mean that a r e d u c t i o n of the s t r i a t i o n thickness below the Kolmogorov m i c r o s c a l e i s conceivable. Reaction and d i f f u s i o n i n d i s t o r t i n g s t r u c t u r e s were a l s o s t u d i e d by Palepu et a l . ( 3 1 ) and Spalding ( 1 2 3 ) . The I.E.M. Model accounts f o r such deformations i n assuming that the micromixing time i s a s p e c i f i e d f u n c t i o n of the p a r t i c l e age (32). We may now complete the l i s t of equivalences c i t e d i n ( 6 - 9 ) by a new one between the d r o p l e t - d i f f u s i o n model of Nauman ( 8 1 ) or the MIRE-model of Rys, Bourne et a l . ( 1 1 9 - 1 2 0 ) , and the IEM Model. The c o n d i t i o n f o r equivalence i s t = t ^ , where tp i s given by ( 7 - 9 ) . The IEM-Model i s a lumped v e r s i o n of the d i s t r i b u t e d para meter r e a c t i o n / d i f f u s i o n model. Numerical s i m u l a t i o n s prove that r e p l a c i n g c o n c e n t r a t i o n p r o f i l e s by average values does not change the o v e r a l l conversion and y i e l d i n the p a r t i c l e very much, even f o r " s t i f f systems" of f a s t r e a c t i o n s . Figures 11 and 12 are exam p l e s of such equivalences. Therefore, my o p i n i o n i s that i n most a p p l i c a t i o n s , a simple lumped parameter model may be used i n place of s o p h i s t i c a t e d d i s t r i b u t e d models. This saves much computer time and the d i f f e r e n c e i n the r e s u l t s i s not greater than f o r instance that induced by a change i n the a r b i t r a r y assumptions concerning the p a r t i c l e shape. The s i m u l a t i o n s v i a the IEM model revealed an i n t e r e s t i n g property. A p a r t i a l l y segregated f l u i d may be considered as a mix ture of m a c r o f l u i d ( f r a c t i o n 3) and m i c r o f l u i d ( f r a c t i o n 1 - 3 ) . I t comes out that the r a t i o (1 - β)/(3 i s always c l o s e to that of two c h a r a c t e r i s t i c times ( 3 2 ) . In the case of e r o s i v e mixing of two reactants i n a CSTR (1 - ft/S % x / t % 4 τ/t . In the case of r e a c t i o n and d i f f u s i o n or premixed r e a c t a n t s i n a CSTR ( f i g u r e 1 3 ) : ( 1 - 3 ) / 3 D % ^ ^ Β * interesting r u l e of thumb f o r r a p i d e s t i m a t i o n of the extent of segregation. m

e

m

T h i s

i s

a n

d

I d e n t i f i c a t i o n of segregation by chemical methods. P a r t i a l segregation can be s t u d i e d through i t s i n f l u e n c e on the conversion and y i e l d of chemical r e a c t i o n s . For i n s t a n c e , l e t us denote by X and Xmicro the l i m i t i n g extents of r e a c t i o n one would ob serve i n a w e l l macromixed r e a c t o r . I f the r e a c t o r i s p a r t i a l l y segregated : X = ΒΧ^,-ο + (1 - 3) XmicroThis a l s o holds f o r the y i e l d of an intermediate product and i s the b a s i s f o r the determination of 3 or conversely, f o r the pre d i c t i o n of X ( 3 2 ) . Fast consecutive-competing r e a c t i o n s A + B ÎLL>R, R+B _JL> S (kj >> k ) are e s p e c i a l l y i n t e r e s t i n g : i f a small amount of Β i s mixed i n t o an excess o f A, R i s formed and immedim a c r o

2



6.

VILLERMAUX

Figure 11. and the IEM reaction i n for perfect second-order


111

Equivalence between the d r o p l e t d i f f u s i o n model (81) model f o r a zero-order r e a c t i o n and a second-order a CSTR. The Damkohler numbers are such t h a t f = 0.5 micromixing. The agreement i s e x c e l l e n t f o r the r e a c t i o n , more approximate f o r the zero-order one.



172

CHEMICAL REACTION ENGINEERING

Figure 12. Equivalence between the r e a c t i o n / d i f f u s i o n model and the IEM model f o r second-order consecutive competing r e a c t i o n s k

l &2 A + Β R, R + Β S. The curves represent t h e average con c e n t r a t i o n s v s . (reduced) time i n a s p h e r i c a l p a r t i c l e immersed i n a bath o f constant composition (C^ 0.105, C = 0 ) . I n i t i a l concentrations i n the p a r t i c l e : C = 0, CBO = 1> R 0· k C g R / ^ = 2 f o r A and R. Β cannot d i f f u s e w i t h i n the p a r t i c l e (S> = 0 ) . k-^/k^ = 10. F-β represents the t o t a l production o f R (equivalent c o n c e n t r a t i o n i n the p a r t i c l e ) . B a s i s f o r the e q u i valence t . - t . = 0 L / j ; β = 3/5, L = R/3 (sphere). The agreement i s s a t i s f a c t o r y . I n much " s t i f f e r " c o n d i t i o n s (k^> > k ) , the agreement i s always good f o r the o v e r a l l produc t i o n o f R, even i f the i n d i v i d u a l c o n c e n t r a t i o n p r o f i l e s become different. =

R

A

C

Q

Q

=

2

2

Q

2

D

2

/// f

;

t /t R

0.1

1

D

10

Figure 13. M i c r o f l u i d / m a c r o f l u i d volume r a t i o v s . r e a c t i o n / d i f f u s i o n time r a t i o . Key t o curves: 1 t o If, s i m u l a t i o n w i t h the IEM model, t = t p ; 1 t o 3, second-order r e a c t i o n k CAo = 2 ( l ) , 5 ( 2 ) , 10(3); U, second-order consecutive competing r e a c t i o n s = 0-β , k-,/kp = 2, t ^ = l / k ^ C ^ ; 5, r e a c t i o n and d i f f u s i o n i n a s l a b (See Rer. 3 2 . ) . m

2

ο


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173


a t e l y converted to S at the contact of Β i f the f l u i d i s segregated. Therefore, X i s some kind of segregation index : X i < X < 1, and β = ( X - ° X , ) / ( l " * ,micro)· The method was e x t e n s i v e l y e x p l o i t e d by Bourne and coworkers, who enumerated the q u a l i t i e s of a good r e a c t i o n f o r i n d u s t r i a l tests ( 124) (t-R % t , d i s t r i b u t i o n of products depending on segre gation, i r r e v e r s i b l e r e a c t i o n , known mechanism, easy a n a l y s i s and regeneration, safe and inexpensive chemicals). Unfortunately, the r e a c t i o n s proposed to-date do not f u l f i l l a l l these requirements : Azo-coupling (119,137), n i t r a t i o n of aromatic hydrocarbons (125), bromation of r e s o r c i n (124) , bromation of 1-3-5 trimethoxybenzene ( 126). Some of them e x h i b i t complex pH e f f e c t s (124, 127, 111), but a l s o a good s e n s i t i v i t y t o s e g r e g a t i o n . For i n s t a n c e , the amount o f 2-U dibromoresorcin i n the di-isomer may vary from 30% to 60 % when the a g i t a t i o n speed passes from 0 to 360 r.p.m. (124). A new r e a c t i o n i s proposed i n t h i s Symposium by Barthole et a l . (128) : the p r e c i p i t a t i o n of baryum s u l f a t e complexed by EDTA i n a l k a l i n e medium under the i n f l u e n c e of an a c i d . I t has many of the advantages c i t e d above, but the r e a c t i o n i s not s t r i c t l y i n s t a n t a neous and thus dependent on macromixing. Fluorescence methods may also be employed (20). In a recent s e r i e s of papers (129, 112, 113, 114), Bourne and coworkers presented a thorough i n v e s t i g a t i o n of segregation i n s t i r r e d reactors (2.5 and 63 dm ) with various i n l e t p o s i t i o n s . As a t e s t r e a c t i o n they used the coupling of 1-naphtol (A) with d i a z o t i s e d s u l p h a n i l i c a c i d (B). They succeeded i n apply ing the MIRE-Model provided that the p a r t i c l e s i z e 2R be 2 to 9 times smaller than the Kolmogorov m i c r o s c a l e . The agreement was improved by assuming s t r e t c h i n g of p a r t i c l e s : 6 = δ ( 1 + t / t ) / where t = 2 ( v / ε ) ' (122, 114). They showed that the best mixing conditions were achieved with i n l e t tubes placed j u s t beneath the turbine. They concluded from t h e i r i n t e r p r e t a t i o n , r e l y i n g on the Kolmogorov microscale, that the scale up of the state of segrega t i o n requires keeping ε and thus N d^ constant. The i n t e r p r e t a t i o n of these b e a u t i f u l experiments should be discussed with much care, taking i n t o account p o s s i b l e macromixing e f f e c t s (comparison of r e a c t i o n time and c i r c u l a t i o n time t ) , i n t e r v e n t i o n of d i f f e r e n t mixing processes (of stage 2) preceding the d i f f u s i o n a l one,and the s p a t i a l d i s t r i b u t i o n of ε i n the tank. Beside consecutive-competing r e a c t i o n s , instantaneous (gener a l l y acid-base) r e a c t i o n s are a l s o used as an i n d i c a t o r of segre gation, e s p e c i a l l y i n m u l t i j e t tubular r e a c t o r s . O t t i n o (102) de duced the r e l a t i o n s h i p t ^ ( t ) between "warped" and r e a l time from the comparison between experimental conversion X(t) along the axis of a tube and the t h e o r e t i c a l expression X ( t ) . a was then c a l c u lated by (7-10) and the e f f i c i e n c y e f f ( t ) by : (l/a )da /dt = eff(t)(D : D ) . e f f ( t ) was found to decrease as a f u n c t i o n of t with 6 = 1/a of the order of 10 ym. In the same paper, the average e f f i c i e n c y " e f f " was estimated i n a s t i r r e d reactor, " e f f " was found to decrease from 20-30 % f o r t < 0.25 s to 1-10 % l a t e r , but here a l s o , macromixing e f f e c t s ( c i r c u l a t i o n 0

8

g

8

m i c r o

m

c

r

o

s

s


Q

3

2

0

1

2

ô

3

c

w

v

1 / 2

v

v

v


2

- 1

2

174


time ?) cannot be excluded. Bourne et a l . (116) a l s o performed ex periments w i t h NaOH/HCl mixing behind a Sulzer s t a t i c mixer i n a tube, ε was estimated by U(Ap/£)/p and the i n f l u e n c e of the v i s c o s i t y was e s p e c i a l l y s t u d i e d , and found to be n o t i c e a b l e . Assuming a d i f f u s i o n a l mechanism f o r mixing, these authors conclude that C o r r s i n s micromixing time T cannot e x p l a i n t h e i r observations and that the i n f l u e n c e of v i s c o s i t y suggests a d i f f u s i o n time ba sed on Kolmogorov m i c r o s c a l e . The same c o n c l u s i o n was drawn from experiments i n s t i r r e d tanks, where the IEM-model seemed unable to account f o r a l l experimental data. This i s i n c o n t r a d i c t i o n w i t h the r e s u l t s of Pohorecki and Baldyga (117) who a l s o studied the r e a c t i o n of NaOH and HCl c o n t r o l l e d by mixing i n a tube. They found that the IEM-model was a p p l i c a b l e w i t h t = 2 τ ^ ( L ^ / e ) ' where L was equal to the diameter of the i n j e c t i o n tube, ε was deduced from the assumption of i s o t r o p i c homogeneous turbulence. This problem of mixing w i t h chemical r e a c t i o n has drawn the a t t e n t i o n of many authors i n the l a s t few years. Takao et a l . ( 130) studied the a l k a l i n e h y d r o l y s i s of c h l o r o a c e t a t e i n a batch s t i r red r e a c t o r ( t ^ % t ) ; t h e i r r e s u l t s , obtained on the b a s i s of the IEM model, can probably be explained by macromixing ( t ^ t ^ 1/N). Miyawaki et a l . (118) s t u d i e d the r e a c t i o n of C O 2 + N H 3 and C O 2 + OH" i n m u l t i j e t tubes and i n s t i r r e d r e a c t o r s . In tubes, the data are compatible w i t h the IEM model (X = 1 - e x p ( - t / t ) ) w i t h t ^ C o r r s i n s T , whereas i n s t i r r e d tanks conversion i s probably c o n t r o l l e d by r e c i r c u l a t i o n ( t ^ 1/N) as i n reference (130). Murakami et a l . (131) developed a model e q u i v a l e n t to the IEM model for i n t e r p r e t i n g mixing i n batch s t i r r e d r e a c t o r s (1 and 50 t) both w i t h a non r e a c t i v e t r a c e r and i n the presence of three r e a c t i o n s of d i f f e r e n t r a p i d i t y . They found that N t could be c o r r e l a t e d as a f u n c t i o n of the a g i t a t i o n Reynolds number Nd /v and a Damkohler number. Costa and L o d i (136) a s s i m i l a t e d the IEM mixing time t to (v/ε) ' w i t h a c o r r e c t i o n depending on the Schmidt number, but without any experimental support. Hanley and C a l l (132) suggested e x p l o i t i n g c o n c e n t r a t i o n f l u c t u a t i o n s at the o u t l e t of a CSTR to c a l c u l a t e micromixing parameters. Ghodsizadeh and A d l e r (133) pro posed an i n t e r e s t i n g method based on d i l a t o m e t r y to f o l l o w the course of an acid-base r e a c t i o n i n a batch r e a c t o r . Bhatt and Z i e g l e r ( 134) determined the m a c r o f l u i d f r a c t i o n i n a CSTR by t a k i n g i n t o account the n o n - i d e a l i t y of the segregated phase RTD and by assuming i n t e r a c t i o n by r e a c t i o n and d i f f u s i o n . Bryant (135) considered the case of zero order r e a c t i o n i n fermenters by assu ming d i f f u s i o n a l l i m i t a t i o n s i n p a r t i c l e s having the Kolmogorov s i z e λ£· Palepu et a l . (31) used the "warped -time method d e s c r i bed i n (102) to estimate the s t r i a t i o n thickness 6 as a f u n c t i o n of time i n a m u l t i j e t tube and i n a s t i r r e d r e a c t o r . A f t e r an i n i t i a l decrease, i t seems that 6 tends to X but there i s s t i l l some strange behaviour (climb of 6 before s t a b i l i z a t i o n ) and the pro blem of the e f f i c i e n c y i n the c a l c u l a t i o n of ε i s a l s o posed. From t h e i r own experiments (109, 71, 108), V i l l e r m a u x and coworkers suggest (32) that depending on experimental c o n d i t i o n s , and chemi1

s

1

m

3

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c a l r e a c t i o n s , there are s e v e r a l mixing mechanisms f o r stages 2 and 3 of Beek and M i l l e r o c c u r i n g simultaneously and i n t e r a c t i n g w i t h chemical processes. This might e x p l a i n the d i s c r e p a n c i e s ob served between authors w i l l i n g to i n t e r p r e t t h e i r experiments by one s i n g l e mechanism. Conclusion. There are s t i l l u n c e r t a i n t i e s i n the f i n a l i n t e r p r e t a t i o n of mixing and chemical r e a c t i o n a t the molecular l e v e l . The IEM model seems to provide a simple b a s i s f o r r e p r e s e n t i n g i n t e r a c t i o n between p a r t i c l e s , even by molecular d i f f u s i o n . The pro blem i s to decide what i s hidden behind the micromixing time t ? C o r r s i n s time constant T (32) ? A d i f f u s i o n constant based on Kolmogorov microscale (113, 114) ? F u r t h e r research should be de veloped i n the f o l l o w i n g d i r e c t i o n s : - Search f o r r e l i a b l e t e s t r e a c t i o n s obeying the c r i t e r i a s t a t e d i n reference ( 124). These r e a c t i o n s should be usable i n i n d u s t r i a l r e a c t o r s and perhaps be l e s s " s t i f f than those proposed by Bourne and coworkers. - Design of experiments where hydrodynamic c o n d i t i o n s are per f e c t l y c o n t r o l l e d : small s t i r r e d r e a c t o r s w i t h high power input (no macromixing e f f e c t s , c i r c u l a t i o n time t 0.5 m ). They showed that c i r c u l a t i o n times t were log-normally d i s t r i b u t e d : 3

Q

f(t ) c

=

1

exp

/2-ïïot

the mean value t

In t - μ c σ/2

c and n a t u r a l variance s being given by z


176


2

2

2

ENGINEERING

2

= exp(y + σ / 2 ) and s = ( t ^ ) ( e x p σ - 1). I f Η i s the height of l i q u i d i n the tank, they found t ^ H/(Nd ) and s ^ H ' / ( N d ) . They also studied the terminal mixing time 0 , required f o r redu cing c o n c e n t r a t i o n gradients down to a s p e c i f i e d l e v e l by m u l t i p l e r e c i r c u l a t i o n s . They found 6 /"tf^ = A + B( s / t ) , s / t > 0.8, so that 0 ^ H ^_ /(Nd ) when A i s small. The power input i s thus Ρ % (Nd) (V/t ). Many c o r r e l a t i o n s f o r mixing time (see above) have been proposed i n the l i t e r a t u r e (142). One of the most comprehensive treatments of t h i s problem was published by Khang and L e v e n s p i e l (143), on the b a s i s of a r e c y c l e model. 6 i s defined as the time constant f o r the exponential decrease of pseudo-periodic o s c i l l a tions a f t e r a pulse i n j e c t i o n of t r a c e r i n a batch s t i r r e d reactor. When Re > 10 , they obtain f o r turbines : 3

7

3

3

m

2

m

1

3

c

3

m

2

c

m


4

2

5

(8-2)

= 0.9 % 1.5 P/(pN d )

(8-3)

Ν e (d/d ) * m

3

T

3

= 0.5 % 0.1 p/(pN d )

and f o r p r o p e l l e r s N0

(d/d )

m

T

2

3

5

C o r r e l a t i o n s f o r jet-mixing can be found i n (157). Experimental data on mixing times may be used to estimate the o v e r a l l e f f i c i e n c y f o r batch mixing of viscous f l u i d s , according to the method proposed by Ottino et a l . (107). In a d d i t i o n to these macromixing c h a r a c t e r i s t i c s , many au thors have determined turbulence parameters and t h e i r s p a t i a l d i s t r i b u t i o n w i t h i n the tank volume by measuring v e l o c i t y and concen t r a t i o n fluctuations(144-147, 19, 158) . In a t y p i c a l i n v e s t i g a t i o n (19) concerning a s e m i - i n d u s t r i a l tank (0.15 m ) and aqueous me dium, the f o l l o w i n g s p a t i a l v a r i a t i o n s were found : u = 5 to 30 % of TrNd, L f = 4 to 150 mm, Xf = 1 to 5 mm, ε/ε = 0.2 to 2.5, c'/C = 2 to 10 χ 10"" ( f o r eddies > 100 ym). This shows that a s t i r r e d tank i s f a r from being the homogeneous and uniform system assumed i n many academic papers. 3

f

4

C e l l models. In order to p r e d i c t chemical conversion i n s t i r red tanks, P a t t e r s o n and coworkers (3, _39, 40) d i v i d e d the tank volume i n t o 30 mixing segments connected by s p e c i f i e d flowrates Q^j % Nd . The turbulence l e v e l i n each segment i s c h a r a c t e r i z e d by L (^ d ) and ε(^ d ) (HDM model). Mann and coworkers (148,149) also studied a model where c e l l s (or segments) are connected accor ding to the average flow p a t t e r n . Commutation according to a spe c i f i e d p r o b a b i l i t y at each c e l l ' s o u t l e t allows a s t o c h a s t i c path to be simulatedj f o r instance f o r a flow f o l l o w e r . They thus ob tained c i r c u l a t i o n time d i s t r i b u t i o n s very s i m i l a r to experimental ones (135, 140, 141). 3

2

s

T

Multiphase s t i r r e d tanks. This item w i l l be reviewed only ve ry b r i e f l y as the subject was r e c e n t l y covered i n e x c e l l e n t and


6.

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177

quasi-exhaustive surveys by J o s h i e t a l . f o r g a s - l i q u i d r e a c t o r s (150) and T a v l a r i d e s and Stamatoudis f o r l i q u i d - l i q u i d r e a c t o r s 05, 40 . J o s h i e t a l . gave a thorough comparison of c o r r e l a t i o n s for N0 . This product seems to increase w i t h the gas-flowrate. L i t t l e i s yet known about the s t a t e of mixing of the dispersed gas and on the i n f l u e n c e of s o l i d i n suspension. T a v l a r i d e s presents a s o p h i s t i c a t e d model f o r r e p r e s e n t i n g coalescence and breakage of d r o p l e t s i n l i q u i d - l i q u i d d i s p e r s i o n s . The model r e l i e s on the p o p u l a t i o n balance equation and s t i l l r e q u i r e s the adjustment of 6 parameters. The s o l u t i o n of such equa t i o n s i s d i f f i c u l t and r e q u i r e s the use of Monte-Carlo methods (151) . The e f f e c t of coalescence and break-up of d r o p l e t s on the y i e l d of chemical r e a c t i o n s was s t u d i e d by V i l l e r m a u x (33). M i c r o mixing e f f e c t s may occur even i n batch r e a c t o r s i f there i s a drop s i z e d i s t r i b u t i o n and mass-transfer c o n t r o l . Although p r a c t i c a l r u l e s f o r the design and scale-up of l i q u i d - l i q u i d r e a c t o r s are a v a i l a b l e as Oldshue showed i n the case of a l k y l a t i o n ( 152), many problems remain unsolved (.5) : mass t r a n s f e r e f f e c t s , high hold-up f r a c t i o n s (> 20 % ) , l a r g e d e n s i t y d i f f e r e n c e s , high v i s c o s i t i e s , i n f l u e n c e of s u r f a c t a n t s .


m

Conclusion : areas f o r f u t u r e research. Mixing i n s t i r r e d reactors i s no longer the e m p i r i c a l o p e r a t i o n i t used to be, ("mostly a r t and very l i t t l e s c i e n c e " (153). For i n s t a n c e , Oldshue summarized u s e f u l r u l e s f o r the scale-up of fermenters (153). How ever, s e v e r a l current problems are s t i l l w a i t i n g s o l u t i o n . These were reviewed i n an e x c e l l e n t paper by Kipke (154). Future r e search should be d i r e c t e d towards - Turbulent phenomena - Large volumes ( s p a t i a l unhomogeneities) - Multiphase systems ( g a s - l i q u i d , l i q u i d - s o l i d , l i q u i d - l i q u i d , gas-liquid-solid) - Non newtonien media, rheology problems (155, 156) - Search f o r s i m p l i f i e d models and new concepts - Less dimensional a n a l y s i s Less c l a s s i c a l devices : s t a t i c mixers S t a t i c mixers have been e s s e n t i a l l y developed s i n c e 1970. About 30 types of these devices are known (159). Their e f f e c t i v e ness can be c h a r a c t e r i z e d i n two ways : by__the r e d u c t i o n of σ/C (See above) along the mixer a x i s (159), σ/C = a exp(-mz/d ) or by the increase of s t r i a t i o n number produced by passing through η m i xing elements (160) : δ /δ = b . For i n s t a n c e , w i t h the Hi-Mixer (161), 6 /6 = 4 . A f a c t o r of 5 i s easy to o b t a i n f o r m w i t h respect to an empty tube, at the ex pense of a corresponding increase i n pressure-drop. A comparison between the e x i s t i n g types of s t a t i c mixers (162) shows that most of them have an e q u i v a l e n t e f f e c t i v e n e s s . The case of Sulzer-Mixers has been e s p e c i a l l y s t u d i e d (163), i n c l u d i n g use w i t h gases (164). t

n

0

Q

n


178


Mixing i n s t a t i c mixers considered as chemical r e a c t o r s was e s s e n t i a l l y s t u d i e d by Nauman ( 165, 166). This author proposed a model which c o n s i s t s of a t u b u l a r r e a c t o r comprising Ν zones i n laminar flow ( p a r a b o l i c v e l o c i t y p r o f i l e ) . Mixing between each zone i s achieved accross a plane by a permutation of the r a d i a l po s i t i o n of f l u i d p a r t i c l e s ( r j ^ » τ^), i n t h i s way the f l o w r a t e i s kept unchanged . Several cases are considered : complete mixing (permutation at random), complete flow i n v e r s i o n ( r 2 = 1 ~ \> W2 = 1 " Wj), p a r t i a l i n v e r s i o n . I n the f i r s t case, Ν = 0 c o r r e s ponds to a CSTR and Ν °° to a plug-flow r e a c t o r . I t i s shown that the best chemical conversion i s obtained w i t h complete flow i n v e r s i o n . The RTD i n a Kenics mixer comprising 8 elements could be r e presented by t h i s model w i t h Ν = 3 and complete mixing. S t a t i c mi xers could be used as chemical r e a c t o r s f o r s p e c i f i c a p p l i c a t i o n s (reactants having large v i s c o s i t y d i f f e r e n c e s , p o l y m e r i z a t i o n s ) but the published data are s t i l l very scarce and a d d i t i o n a l informa t i o n i s r e q u i r e d f o r assessing these p o s s i b i l i t i e s . Beside s t a t i c mixers, there are p r a c t i c a l l y no a l t e r n a t i v e s to the " u b i q u i t o u s " s t i r r e d tank, i f one excepts loop r e a c t o r s (167) and the somewhat s p e c i a l back-flow mixer ( 168). Imagining en t i r e l y new p r i n c i p l e s f o r mixing r e a c t a n t s i s a challenge f o r f u ture researchers. F i r s t estimations show that an "informed" mixing system, working as a Maxwell demon would be much more e f f e c t i v e than our present devices (169).


T

Mixing of S o l i d s . This p o i n t i s a c t u a l l y very important but deserves a s p e c i a l review, and w i l l not be t r e a t e d i n t h i s paper. The reader w i l l f i n d the s t a t e of the a r t and a l i t e r a t u r e survey i n three e x c e l l e n t papers (170, 171, 172). An example of the importance of mixing e f f e c t s i n chemical reactors : continuous f r e e r a d i c a l p o l y m e r i z a t i o n . One might now ask the question : are segregation e f f e c t s r e a l l y important i n p r a c t i c e or i s micromixing "a s o l u t i o n i n search of a problem" (173), a mere i n t e l l e c t u a l e x e r c i s e f o r academics who are short of o r i g i n a l PhD subjects ? I t i s true that micromixing e f f e c t s can g e n e r a l l y be neglec ted i n the design of r e a c t o r s f o r simple and slow r e a c t i o n s . How ever, as has been pointed out i n the preceding Sections i n the case of f a s t r e a c t i o n s w i t h unmixed r e a c t a n t s , chemical conversion could be e n t i r e l y c o n t r o l l e d by mixing, and induce dramatic v a r i a t i o n s i n the d i s t r i b u t i o n of products. The p r a c t i c a l examples of combustion and r e a c t i o n s i n l i q u i d suspension are e s p e c i a l l y i l l u minating i n t h i s respect. Another area where micromixing plays a c a r d i n a l r o l e i s con tinuous p o l y m e r i z a t i o n . The subject i s t r e a t e d elsewhere i n t h i s Symposium, and was reviewed by Nauman ( 173) and Gerrens ( 174) a few years ago. There fore a thorough d i s c u s s i o n of mixing e f f e c t s i n polymer r e a c t o r s would go beyond the scope of t h i s paper. I t i s l i k e l y that s i g n i -


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f i c a n t progress has been made s i n c e these reviews, but they are d i f f i c u l t to assess because of a lack of published data. Two examples w i l l show the importance o f micromixing e f f e c t s on the s t r u c t u r a l c h a r a c t e r i s t i c s of polymers ( 175). The f i r s t example i s a simula t i o n o f f r e e r a d i c a l p o l y m e r i z a t i o n i n a CSTR. The r e t a i n e d k i n e t i c scheme i s kd,f A

> 2R k

R + M

*—> R k

Initiation

P

Propagation

t P

R + Ρ

>Ρ + R

Transfer to Polymer


k R + R

-—> Ρ + Ρ

Termination ( D i s p o r p o r t i o n a t i o n )

F i g u r e 14 shows a p l o t of the D i s p e r s i o n Index DI = M /M wersus the conversion X of the monomer f o r segregated flow (S; and w e l l micromixed flow (M). The dramatic i n f l u e n c e of segregation can be n o t i c e d a t high conversion, e s p e c i a l l y w i t h t r a n s f e r to po lymer. Moreover, an i n t e r e s t i n g e f f e c t i s observed w i t h d i l u t e d and slow i n i t i a t o r s , namely an i n v e r s i o n of the r e l a t i v e p o s i t i o n of S and M curves when the t r a n s f e r constant k p i s increased. This doesn't happen w i t h concentrated and f a s t i n i t i a t o r . The se cond example i s an experimental one (176). Continuous polymeriza t i o n of styrene was c a r r i e d out i n a CSTR and i n cyclohexane s o l u t i o n i n order to keep the v i s c o s i t y low and constant. The D i s p e r s i o n Index was measured as a f u n c t i o n of space time and a g i t a t i o n speed. L i m i t i n g curves f o r segregated flow (S) and w e l l micromixed flow (M) were c a l c u l a t e d from batch experiments. C l e a r evidence for segregation e f f e c t s can be seen on f i g u r e 15 which shows that p e r f e c t micromixing may be very d i f f i c u l t to achieve, even w i t h strong a g i t a t i o n and low v i s c o s i t y . Besides these l a b o r a t o r y experiments, the a n a l y s i s of indus t r i a l r e a c t o r s may a l s o r e v e a l segregation e f f e c t s , as f o r instance i n r e a c t o r s f o r f r e e r a d i c a l p o l y m e r i z a t i o n of ethylene where the i n i t i a t o r feedstream i s l i k e l y to be mixed by an e r o s i v e process ( 175) . P o l y m e r i z a t i o n and polycondensation r e a c t o r s o f f e r s an es p e c i a l l y i n t e r e s t i n g f i e l d f o r f u t u r e a p p l i c a t i o n s of micromixing. w

t

General c o n c l u s i o n The end of t h i s survey leaves us w i t h the f e e l i n g that r e search on mixing i n chemical r e a c t o r s i s a very l i v e l y area, where problems have been attacked from s e v e r a l d i r e c t i o n s (turbulence theory, RTD and mixing e a r l i n e s s , segregation and micromixing . . . ) . I f the major concepts have been i d e n t i f i e d , there i s s t i l l a need f o r a u n i f i e d theory a l l o w i n g a - p r i o r i p r e d i c t i o n s from the s o l e knowledge of physicochemical p r o p e r t i e s and operating parameters, even i f encouraging progress has been made i n t h i s d i r e c t i o n . Without r e p e a t i n g the conclusions drawn at the end of each



180

CHEMICAL REACTION ENGINEERING

Q2

, Q6 , 0.8

04

t

,X

Figure ih. Free r a d i c a l p o l y m e r i z a t i o n i n a CSTR. D i s p e r s i o n Index DI v s . conversion X. S - segregated flow, M = w e l l - m i c r O mixed flow, I = l i n e a r p o l y m e r i z a t i o n k = 5 x ιοί L mol"- * h k = 1,5 x 10 L mol" h " , k = 0.033 1Γ' , f = 0.5, A = 3 x 10" mol L , S =7.12 mol L"" ( s o l v e n t ) , M = 3.56 mol L " . Curves 2 t o U: T r a n s f e r t o polymer. 2 : k p = 3-5 x 1 0 mol L " h , 3 : k = 1.05 x I0h I T m o l " h " , k : k p = 1.05 χ 10 L mol" h"* , A = 3 x 10 mol L " , k = 0.33 h " (other parameters un changed) . 1

1

1

1

Ρ

1

1

J

1

Q

3

1

t

1

1

1

1

t p

t

1

1

1

d

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2-5

'~

s'

_

/

2

. /

/

/

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__/ N-100 RPM (o)N-300 x)

~

" " J ^ N-1800 3 N.2700 N-3600

/ ^

^---ΖΓ--*

V

.

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—L -

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Figure 15. E f f e c t of segregation on p o l y m e r i z a t i o n of styrene i n cyclohexane s o l u t i o n . Standard CSTR with h b a f f l e s and a 6-blade t u r b i n e , V = 670 cm , Τ = 75 °C. D i s p e r s i o n Index DI vs. space time. Influence o f a g i t a t i o n speed. Curves S (segre gated flow) and M (we11-micromixed flow) c a l c u l a t e d from batch experiments^ I n i t i a t o r : PERKADOX l 6 , A = 0.033 mol I T , k = 5 x 10" s " , f = 0.85; M = 6.65 mol L " , S = 2.22 mol I T . 3

1

5

d

1

1

Q

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Q



6.

VILLERMAUX

181


S e c t i o n , we may t r y to summarize the s i t u a t i o n as f o l l o w s : R e s i dence time d i s t r i b u t i o n s are now a w e l l e s t a b l i s h e d t o o l but pro gress i s s t i l l d e s i r a b l e f o r m u l t i p l e i n l e t / o u t l e t systems, un steady s t a t e operations, f l u i d s w i t h t i m e - v a r i a b l e p r o p e r t i e s and inhomogeneous r e a c t i n g media. More a t t e n t i o n should be paid to i n t e r n a l age d i s t r i b u t i o n s and r e l a t e d q u a n t i t i e s such as c i r c u l a t i o n time d i s t r i b u t i o n s i n s t i r r e d tanks. I n continuous r e a c t o r s , models f o r mixing e a r l i n e s s d e s c r i b i n g the t r a n s f e r between Ente r i n g and Leaving Environments are superabundant. Further r e d u c t i o n of segregation by i n t e r a c t i o n between f l u i d p a r t i c l e s can be conve n i e n t l y represented by simple models (exchange w i t h the mean, c o a l e s c e n c e - d i s p e r s i o n ) , but s e v e r a l stages f o r mixing, each w i t h t h e i r own time constants should be considered, p o s s i b l y i n s e r i e s or i n p a r a l l e l . There i s s t i l l a problem as to the u l t i m a t e stage of mixing by molecular d i f f u s i o n where i t i s not c l e a r whether the mixing time i s t = T ^ (Lg/ε) ' or t = ϋ ^ λΖ/Jb. C a r e f u l l y designed experiments (no macromixing e f f e c t s , p e r f e c t l y defined hydrodynamic p a t t e r n s ) and new chemical t e s t r e a c t i o n s would be welcome i n t h i s respect. The method of " c h a r a c t e r i s t i c times" i s e s p e c i a l l y h e l p f u l f o r determining which processes are c o n t r o l l i n g . These are f o r instance the space time τ f o r a continuous r e a c t o r ; a c h a r a c t e r i s t i c time f o r i n t e r n a l macromixing p a t t e r n , e.g. the c i r c u l a t i o n time t i n a s t i r r e d tank ; one or s e v e r a l r e a c t i o n times, e.g. t = 1/kC^" ; and one or s e v e r a l micromixing times t , e.g. t (erosion) or t ^ £ / ^ , or t _ = l/ω or ^ (v/ε) ' , or T ^ (Lg/ε)^' . Comparison between a l l these times allows the determination of the mixing regime, sometimes q u a n t i t a t i v e l y . I t was thus e s t a b l i s h e d that the m i c r o f l u i d / m a c r o f l u i d volume r a t i o was n e a r l y equal to t ^ / t p . Another c h a r a c t e r i s t i c of current r e search i s a gradual and f o r t u n a t e merging between the E u l e r i a n approach of F l u i d Mechanics and the Lagrangian approach of Chemi cal Engineering. Measurement of c o n c e n t r a t i o n f l u c t u a t i o n s should be developed, both i n presence and i n absence of chemical reac tions i n order to o b t a i n r e l i a b l e s p e c t r a l data. However, the f i nal s o l u t i o n to micromixing problems should not be sought i n t u r bulence theory alone, but r a t h e r i n phenomenological i n t e r a c t i o n models, whose parameters could have a fundamental i n t e r p r e t a t i o n by t h i s theory. This i s the wish of most i n d u s t r i a l s : "In g e n e r a l , i n d u s t r y would plead f o r l e s s s o p h i s t i c a t e d mathematical models and more phenomenological models g i v i n g us more understanding of what's going on i n the tank" (10). Among other recommendations (9, _K), 27, 154), there i s a general agreement f o r encouraging research on large volume r e a c t o r s , g a s - l i q u i d - s o l i d systems and mixing of non newtonian f l u i d s . New ideas on e n t i r e l y novel mixing p r i n c i p l e s and equipments would a l s o be welcome. But above a l l , theory w i l l progress i n a d i r e c t i o n u s e f u l to p r a c t i t i o n e r s i f more experimen t a l data on r e a l i s t i c i n d u s t r i a l s i t u a t i o n s are a v a i l a b l e to r e searchers . 1

m

3

s

m

β

c

1

R

m

2

e

D

1

c

D

G


2

182

CHEMICAL

Literature 1. 2. 3. 4. 5. 6. 7. 8.


9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

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ENGINEERING

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