Chemical Reaction EngineeringâBoston - ACS Publications

6 A Novel Method for Determining the Multiplicity Features of Multi-Reaction Systems VEMURI B A L A K O T A I A H

and D A N LUSS

Downloaded by CORNELL UNIV on June 18, 2017 | http://pubs.acs.org Publication Date: September 16, 1982 | doi: 10.1021/bk-1982-0196.ch006

University of Houston, Department of Chemical Engineering, Houston, TX 77004

The q u a l i t a t i v e multiplicity f e a t u r e s o f a lumped -parameter system i n which s e v e r a l r e a c t i o n s occur sim u l t a n e o u s l y can be determined i n a systematic f a s h i o n by f i n d i n g the o r g a n i z i n g s i n g u l a r i t i e s o f the steady - s t a t e equation and its u n i v e r s a l u n f o l d i n g . To illus trate the technique we determine the maximal number o f s o l u t i o n s of a CSTR in which Ν parallel, first-order r e a c t i o n s with equal and h i g h a c t i v a t i o n energies occur as w e l l as the i n f l u e n c e of changes in the resi dence time on the number and type o f s o l u t i o n s . We d e s c r i b e here a new technique based on the s i n g u l a r i t y and b i f u r c a t i o n t h e o r i e s f o r p r e d i c t i n g the m u l t i p l i c i t y f e a tures of lumped-parameter systems i n which s e v e r a l r e a c t i o n s occur simultaneously. Our purpose i s mainly t o i l l u s t r a t e the power of the technique and present some n o v e l r e s u l t s . A more d e t a i l e d a n a l y s i s i s presented elsewhere [1 2 J . We use the technique to answer the f o l l o w i n g questions: 9

(a)

What i s the maximum number of s t e a d y - s t a t e s o l u t i o n s f o r a lumped-parameter system i n which s e v e r a l chemical r e a c t i o n s occur simultaneously, and f o r what values of the parameters w i l l t h i s occur?

(b)

What are a l l the q u a l i t a t i v e l y d i f f e r e n t types o f b i f u r c a t i o n diagrams which d e s c r i b e the dependence o f a s t a t e v a r i a b l e (such as the temperature) on a design o r operat ing v a r i a b l e (such as the feed temperature o r flow r a t e ) and f o r what parameter values w i l l a t r a n s i t i o n from one type t o the other occur?

0097-6156/82/0196-0065$06.00/0 © 1982 American Chemical Society Wei and Georgakis; Chemical Reaction Engineering—Boston ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

CHEMICAL REACTION

66

ENGINEERING

H e u r i s t i c D e s c r i p t i o n of the Theory Consider a n o n l i n e a r s t e a d y - s t a t e equation o f the form F(x,X, ) - 0

(1)


2

where χ i s a s t a t e v a r i a b l e , λ i s a b i f u r c a t i o n v a r i a b l e and £ i s a v e c t o r of parameters. F i s assumed to be smooth with respect to a l l the v a r i a b l e s . The graph of χ versus λ which s a t i s f i e s Eq. (1) f o r a f i x e d £ i s d e f i n e d as a b i f u r c a t i o n d i a gram. A l o c a l b i f u r c a t i o n diagram d e s c r i b e s t h i s dependence i n a s m a l l neighborhood of some p o i n t , w h i l e a g l o b a l b i f u r c a t i o n diagram d e s c r i b e s i t f o r a l l χ and λ w i t h i n the domain of i n t e r est. The parameter space £ c o n s i s t s o f regions with d i f f e r e n t types of b i f u r c a t i o n diagrams. There e x i s t some h i g h l y degen e r a t e p o i n t s ( s i n g u l a r p o i n t s ) at which boundaries of v a r i o u s regions coalesce so that i n t h e i r neighborhood s e v e r a l d i f f e r e n t types of l o c a l b i f u r c a t i o n diagrams e x i s t . These p o i n t s a r e c h a r a c t e r i z e d by the v a n i s h i n g of a f i n i t e number o f d e r i v a t i v e s of F with respect to χ and λ. I t i s u s u a l l y p o s s i b l e t o f i n d a smooth, n o n l i n e a r and i n v e r t i b l e change of coordinates (χ,λ,£) •> (y,y,a) that t r a n s forms the s t e a d y - s t a t e equation (1) i n t o a polynomial f u n c t i o n G(y,y,a) = 0, having a l l the q u a l i t a t i v e features o f equation (1) i n the neighborhood of these s i n g u l a r p o i n t s . A polynomial G, which can represent a l l the l o c a l b i f u r c a t i o n diagrams e x i s t ing next to a s i n g u l a r p o i n t of Eq. (1) and which contains the minimal number of parameters a . i s c a l l e d the u n i v e r s a l u n f o l d i n g of the s i n g u l a r i t y . Our a n a l y s i s i s based on the f o l l o w i n g theorem [3]. Suppose that the steady-state equation (1) has a s i n g u l a r p o i n t at which 1

0

F(x°,X , °) = 0 £

•^J 9X

0

2)

(2)

1

then i n the neighborhood o f (χ°,λ°,£°), the u n i v e r s a l u n f o l d i n g of F ( x , X , ) i s : £

(i)

1

G(y,p,a) â y**

- a^y*

1

1

Γ

2

- ct^^ " -

£

G(y,y a) = y*"*" - α ^

provided

1

(χ°,λ°, °) |f

provided

(ii)

-

....-c^y - μ - 0 (3)

(χ°,λ°, ) < 0. £

- ....-a y 2

2

- α

χ

+ \iy = 0

-55- (χ°,λ°, °) = 0 £

Wei and Georgakis; Chemical Reaction Engineering—Boston ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

(4)

(5)

(6)

6.

BALAKOTAIAH


and

AND LUSS

— ^ 9x

Multiplicity of Multi-Reaction Systems

(χ ,λ ,£ )

(χ ,λ ,£ ) > 0.

67

(7)

The maximum number o f s o l u t i o n s o f equation (1) i s r+1 next t o such a s i n g u l a r p o i n t . Moreover, a l l the l o c a l b i f u r c a t i o n d i a grams of the f u n c t i o n F can be determined by the a n a l y s i s o f the simpler polynomial f u n c t i o n G. The values o f λ (within the domain o f i n t e r e s t ) at which the number of s o l u t i o n s o f Eq. (1) changes are c a l l e d b i f u r c a t i o n points. At these p o i n t s F = 3F/8x = 0. Using b i f u r c a t i o n theory i t can be shown that the nature o f a b i f u r c a t i o n diagram can change only i f the parameter values cross one o f three hypers u r f a c e s [3]. The f i r s t c a l l e d the H y s t e r e s i s v a r i e t y (H) i s the s e t o f a l l p o i n t s i n the parameter space £ s a t i s f y i n g

F(x,A,£) = g

(Χ,λ,£) = ^ | (Χ,λ,£) = 0. 3x

(8)

E l i m i n a t i o n of χ and λ from these three equations gives a s i n g l e a l g e b r a i c equation i n £ , d e f i n i n g a hypersurface. When £ values cross the Η v a r i e t y two b i f u r c a t i o n p o i n t s appear o r disappear and the nature o f the b i f u r c a t i o n diagram changes as shown i n Figure l . a . The I s o l a v a r i e t y ( I ) i s the s e t o f a l l p o i n t s £ s a t i s f y i n g F

( χ

λ

(x>*,£) - f " > > £ >

=

χ

λ

| f < > >£>

9

- °·

When £ crosses t h i s v a r i e t y two b i f u r c a t i o n p o i n t s appear o r d i s appear so that e i t h e r the b i f u r c a t i o n diagram i s separated l o c a l l y i n t o two i s o l a t e d graphs (Figure l . c ) or one i s o l a t e d curve appears o r disappears (Figure l . b ) . The Double L i m i t v a r i e t y (DL) i s the s e t o f £ values s a t i s fying F(

X;L

, X , £ ) - F(x ,X,£) - 0 2

3F 3F -g^ (χ ,λ,£) - -gj (χ ,λ,£) = 0 x φ x . (10) The number o f b i f u r c a t i o n p o i n t s does not change as £ crosses t h i s hypersurface, but the r e l a t i v e p o s i t i o n o f the b i f u r c a t i o n p o i n t s changes as i l l u s t r a t e d by F i g u r e s l . d and I.e. These three hypersurfaces d i v i d e the g l o b a l parameter space £ i n t o d i f f e r e n t regions i n each o f which a d i f f e r e n t type o f b i f u r c a t i o n diagram e x i s t s . 1

2

±

2

Ν P a r a l l e l Reactions i n a CSTR The

s i n g u l a r i t y and b i f u r c a t i o n t h e o r i e s

can be used t o



68

CHEMICAL REACTION ENGINEERING

s.

/ r \

s

/

b X

Χ

:3d ι

2> m

z>

c

i Ρ λ

Figure 1. Possible forms of transformation of an unstable bifurcation diagram (middle column) into either one of two possible stable forms (left or right column) at the Hysteresis (a), Isola (b, c) and Double Limit varieties (d, e).


6.

BALAKOTAIAH AND LUSS


p r e d i c t the q u a l i t a t i v e m u l t i p l i c i t y features o f lumped para meter chemically r e a c t i n g systems. We consider as an example a non-adiabatic CSTR i n which Ν p a r a l l e l , f i r s t - o r d e r r e a c t i o n s k. A ^ ~ ^ i i = 1,2, ,N p

±

occur. To s i m p l i f y the a l g e b r a i c manipulations we assume that the a c t i v a t i o n energies of a l l the r e a c t i o n s are equal. The species and energy balances can be combined to give a s i n g l e equation f o r the dimensionless steady-state temperature Θ:


Λ

F(0,Da,£) = (l-KxDa)0 - aDa0 where

Ε Ύ

=

-

c

D

e"x p i1 - +θ/γ ρ ν

ο τ-τ ο

B

Ύ (

ο

ο

Δ Η

/pC

(11)

/

ο

Da = Vk.. (Τ )/q I ο T

V

i • - Λ, p o io

X

' Τ -Τ

_

α = Ua/Vk-(T )pc l o p

a

, θ

Χ -

ρRT τ"

Ν Β V DaX Σ ^ - 0 i=l i

=

k

(T

)/k

(

i i o lV

v

The dimensionless v a r i a b l e s a r e defined so that changes i n the flow r a t e (residence time) a f f e c t only Da which i s s e l e c t e d to be the b i f u r c a t i o n parameter. We s h a l l determine the maximum number o f steady-state so l u t i o n s and a l l the b i f u r c a t i o n diagrams (Θ vs. Da) o f Eq. (11). We consider s e p a r a t e l y two cases; A d i a b a t i c case (a = 0) and γ » Here Eq.

θ

(11) s i m p l i f i e s to

fl

Ν B.V.Dae V ) = θ - Σ g- - 0 i = l 1+VjDae Λ

F(0,Da,B

(13)

I t can be shown [1] that the set o f equations

* « U 3Θ has a s o l u t i o n Λ θ = θ° - 2

= -^i 2Ν

=

0

(

1

4

)

9 Θ

N

Σ 1/i i=l


(15)

70


v

i

D

a

v

D a

• ±°

=

°

(

1

z

)

" i

β χ

θ

ρ(- °)/

ζ

16

±

where z^ are the zeros of the Legendre polynomial of order Ν d e f i n e d over the u n i t i n t e r v a l (0,1) and w.^ are the correspond i n g Gauss-Legendre quadrature weights. Moreover, at any s i n g u l a r point defined by Eqs. (14) 9

2N+1


3 Θ

. — 9Da

2Ν+1

< 0 ' U

u

(17} ' /

The q u a l i t a t i v e features of the l o c a l b i f u r c a t i o n diagrams (Θ vs. Da) of Eq. (13) i n the neighborhoood of any s i n g u l a r point d e f i n e d by (14) are same as those of i t s u n i v e r s a l u n f o l d i n g G(x,X

a) = x

2 N + 1

- a

2 0 X T

l

ZN-1

W

- . . . .-OLX - λ = 0 1

X

(18)

Assume that (6°,B?,v?,Da°) i s a s o l u t i o n of (14). Because of the symmetry of the^problem any permutation of the (B°,V^,Da ) i s also a solution. Therefore, there e x i s t N! separate para meter regions i n each of which the steady-state Eq. (13) has (2N+1) s o l u t i o n s . Eq. (18) can have f o r any N, e i t h e r zero, two, four, . . . or 2N b i f u r c a t i o n p o i n t s . A l l the p o s s i b l e l o c a l b i f u r c a t i o n diagrams can be constructed by a method described i n [1]. Moreover, i t can be proven [1] that any g l o b a l b i f u r c a t i o n d i a gram of Eq. (13) must be s i m i l a r to one of the l o c a l b i f u r c a t i o n diagrams of Eq. (18). For N=l, Eq. (18) describes the cusp s i n g u l a r i t y Q

Οίχ,λ,αρ = χ

3

- α χ χ

- λ - 0.

(19)

The I s o l a and Double L i m i t v a r i e t i e s do not e x i s t i n t h i s case. The H y s t e r e s i s v a r i e t y (a =0) d i v i d e s the space i n t o two regions (a^ > 0 and < 0) corresponding to the two b i f u r c a t i o n diagrams shown i n F i g u r e s 2.a and 2.b. These two are a l s o the only p o s s i b l e g l o b a l b i f u r c a t i o n diagrams (Θ vs. Da) f o r Eq. (13) as the H y s t e r e s i s v a r i e t y (B =4) d i v i d e s the B^ space i n t o two r e g i o n s . For N=2, F(x,X

Eq.

α) = χ

(18) 5

defines the b u t t e r f l y s i n g u l a r i t y

- a x 3

3

- a x 2

2

- αχ χ

- λ = 0.

(20)

The H y s t e r e s i s and the Double L i m i t V a r i t i e s d i v i d e i n t h i s case the (α.,ο^,οΟ space i n t o seven regions corresponding to the seven b i f u r c a t i o n diagrams shown i n Figures 2.a-g.




Figure 2.


Classification of the bifurcation diagrams of Equation 18 for Ν (a,b)forN = 2(a-g).


72


S i m i l a r l y , the H y s t e r e s i s and Double L i m i t v a r i e t i e s d i v i d e the g l o b a l parameter space of ( i > 2 » 9 ^ regions having the b i f u r c a t i o n diagrams shown i n F i g u r e s 2.a-g. Because of the e x i s t e n c e of two s i n g u l a r p o i n t s there e x i s t two i s o l a t e d parameter regions corresponding to each of the f i v e b i f u r c a t i o n diagrams shown i n Figures 2.c-g. B

B

V

i

Non-Adiabatic Case (α φ 0) and γ »

n

t

0

s

e

v

e

n

θ


We consider f i r s t the s p e c i a l case o f equal coolant and feed temperature (Τ = Τ ). I t i s proven i n [2] that Eq. (11) has N! s i n g u l a r p o i n t s c R a r a c t e r i z e d by

_Λ-9Σ_-ο

= i l - i?I -

F

9 6

3Θ

2

" 3Θ

2 Ν

"

9

D

«i)

a

and 3 9 Θ

2 N + 1

2

F 2Ν+1

9F > 363Da

where θ° and ν °

Da° are d e f i n e d by Eq.

±

a° D a

0

0.

(16)

and

= θ°-1

B.° = ι

1

,. z^l-zj

,

θ

— ( θ

(22)

ο_

1 }

The q u a l i t a t i v e features of the steady-state Eq. (11) i n the neighborhood of these s i n g u l a r p o i n t s are the same as those of the u n i v e r s a l u n f o l d i n g x

G(x,A,a) = χ -α

χ

- O&2N

"

2n-l

X

" · · ·

a

" 2

x

+ λ χ = 0

(23)

Eq. (23> has at most 2Ν+1 s o l u t i o n s and up to (2Ν+1) b i f u r c a t i o n p o i n t s . An I s o l a v a r i e t y e x i s t s i n t h i s case so that the b i f u r c a t i o n diagrams are more i n t r i c a t e and c o n t a i n i s o l a s ( i s o l a t e d branches) i n a d d i t i o n to the h y s t e r e s i s loops. In the case of a s i n g l e r e a c t i o n Eq. (23) d e s c r i b e s the pitchfork singularity δ(χ,λ,α) = χ

3

- α χ 2

2

+ λ χ - α

= 0

(24)

3 The H y s t e r e s i s v a r i e t y of Eq. (24) i s ou = 27α- while the Isola variety is = 0. The two v a r i e t i e s d i v i d e the (α^,α ) 2



6.

BALAKOTAIAH

AND LUSS

73


plane i n t o four regions w i t h d i f f e r e n t b i f u r c a t i o n diagrams. The H y s t e r e s i s and I s o l a v a r i e t i e s o f the steady s t a t e Eq. (11) were constructed i n [4] and a r e shown i n F i g u r e 3. Four d i f f e r e n t types of b i f u r c a t i o n diagrams denoted as b,c,d and f i n F i g u r e 4, e x i s t next to the p i t c h f o r k s i n g u l a r i t y , which i s l o c a t e d a t α = exp(2) and = 8. An a d d i t i o n a l b i f u r c a t i o n diagram, shown as case a i n F i g u r e 4, e x i s t s i n the g l o b a l (α,Β^) plane. Z e l d o v i c h and Z y s i n p r e d i c t e d a l r e a d y i n 1941 [5] that f i v e d i f f e r e n t types o f b i f u r c a t i o n diagrams e x i s t i n t h i s case. When 9 ^ 0 and γ i s f i n i t e , Eq. (11) has h i g h e r order singularities. The coordinates o f the s i n g u l a r p o i n t s are cumbersome expressions reported i n [2]. For Ν-1 Eq. (11) has a unique s i n g u l a r p o i n t , the u n i v e r s a l u n f o l d i n g o f which i s the winged cusp s i n g u l a r i t y (25) I t was shown i n [6] that the H y s t e r e s i s and I s o l a v a r i e t i e s d i v i d e the (α-,ο^,οΟ space i n t o seven regions w i t h d i f f e r e n t b i f u r c a t i o n diagrams. A c o n s t r u c t i o n o f the H y s t e r e s i s and I s o l a v a r i e t i e s of the s t e a d y - s t a t e Eq. (11) has shown that the seven b i f u r c a t i o n diagrams shown i n F i g u r e 4 are the only ones that e x i s t i n the g l o b a l parameter space (α,Β,θ ,γ) [ 4 J . I t i s important to note that w h i l e the s e l e c t i o n of the b i f u r c a t i o n v a r i a b l e does not a f f e c t the maximal number o f s t e a d y - s t a t e s o l u t i o n s , i t a f f e c t s the number and type o f b i f u r c a t i o n diagrams. For example, i f we s e l e c t e d the coolant or feed temperature as the b i f u r c a t i o n v a r i a b l e then Eq. (18) would be the u n i v e r s a l u n f o l d i n g f o r both the a d i a b a t i c and the cooled case and no i s o l a s would e x i s t [ 1^,2]. Concluding Remarks The s t e a d y - s t a t e equations d e s c r i b i n g lumped parameter systems i n which s e v e r a l r e a c t i o n s occur simultaneously c o n t a i n a very l a r g e number o f parameters. Thus, i t i s i m p r a c t i c a l t o conduct an exhaustive parametric study t o determine t h e i r features. The new technique presented here p r e d i c t s q u a l i t a t i v e features of these systems such as the maximum number o f s o l u t i o n s , parameter values f o r which these s o l u t i o n s e x i s t and a l l the l o c a l b i f u r c a t i o n diagrams. C o n s t r u c t i o n o f the three v a r i e t i e s enables the d i v i s i o n o f the g l o b a l parameter space i n t o regions with d i f f e r e n t b i f u r c a t i o n diagrams. We have used t h i s technique t o determine the q u a l i t a t i v e f e a t u r e s of s e v e r a l m u l t i - r e a c t i o n systems and the r e s u l t s w i l l be reported elsewhere [1,2]. I t i s expected that t h i s method w i l l become the standard t o o l f o r p r e d i c t i n g the q u a l i t a t i v e m u l t i p l i c i t y features o f these systems.



74


Figure 3.

A schematic of the Hysteresis and Isola varieties of Equation 11 for Ν = 1, θ = 0, and γ -> oo. ΰ

D = H(T)V/q Q

Figure 4.

0

Bifurcation diagrams describing the dependence of the dimensionless temperature θ on the flow rate (D ) for the single reaction case. a


6.



Acknowledgement We are t h a n k f u l to the N a t i o n a l Science Foundation f o r support o f t h i s research. Legend o f Symbols


a Β c DI Ε ΔΗ k £ q Τ V x,y α γ θ λ, y ρ

heat t r a n s f e r area dimensionless heat of r e a c t i o n concentration heat c a p a c i t y Damkohler number a c t i v a t i o n energy heat of r e a c t i o n r a t e constant v e c t o r of parameters flow r a t e temperature volume state variables parameters v e c t o r dimensionless a c t i v a t i o n energy dimensionless temperature bifurcation variables density r a t i o o f r a t e constants defined by Eq.

(12)

Subscripts ο i c

i n l e t conditions i - t h r e a c t i o n o r i - t h element coolant

Superscripts ο

singular point

coordinate

Literature Cited [1] [2] [3] [4] [5] [6]

Balakotaiah, V.; Luss, D. Chem. Eng. S c i . accepted f o r publication. Balakotaiah, V.; Luss, D. Chem. Eng. Sci. submitted for publication. G o l u b i t s k y , M.; Schaeffer, D. Comm. on Pure and Appl. Math. 1979, 32, 21-98. Balakotaiah, V.; Luss, D. Chem. Eng. Comm. accepted f o r publication. Z e l d o v i c h Ya. B.; Z y s i n , Y. A. J. T e c h n i c a l P h y s i c s . 1941, 11, 502. G o l u b i t s k y , M.; K e y f i t z , B. L. SIAM J . Math. A n a l . 1980, 11, 316-339.

RECEIVED April 27, 1982.


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