Chemical Reaction Engineering—Plenary Lectures - ACS Publications

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

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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.

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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.

VILLERMAUX

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

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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

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

C H E M I C A L REACTION ENGINEERING

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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

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

6.

viLLERMAUX

Mixing in Chemical Reactors

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

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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

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

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

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-

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

6.

VILLERMAUX

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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

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ENGINEERING

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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

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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

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

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

Mixing in Chemical Reactors

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.

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

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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. =

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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

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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

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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

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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

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= 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

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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

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

6.

VILLERMAUX

Mixing in Chemical Reactors

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 .

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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

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

178

C H E M I C A L REACTION ENGINEERING

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).

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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 -

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

6.

VILLERMAUX

179

Mixing in Chemical Reactors

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

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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

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

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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

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1

1

1

t p

t

1

1

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DI S

2-5

'~

s'

_

/

2

. /

/

/

%

^ ' /

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

~

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

/ ^

^---ΖΓ--*

V

.

M

«

—L -

τ HOURS

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

1

Q

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

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6.

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181

Mixing in Chemical Reactors

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

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m

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c

1

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m

2

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1

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D

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

2

182

CHEMICAL

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

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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.

REACTION

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