Flow Control Operation of a Plug-Flow Tubular ... - ACS Publications

46 Flow Control Operation of a Plug-Flow Tubular

Downloaded by UNIV OF MASSACHUSETTS AMHERST on May 31, 2018 | https://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch046

Reactor with High Heat Diffusivity Y U . P. G U P A L O , V . A . N O V I K O V , and Y U . S. R Y A Z A N T S E V The Institute for Problems in Mechanics, USSR Academy of Sciences, Moscow, USSR

1. Formulation of the problem I t i s known that c o n t r o l l e d operation of chemical r e actors at n a t u r a l l y unstable conditions should be of i n t e r e s t i n the design of some commercial reactors because the unstable or s o - c a l l e d intermediate s t a t e s may o f f e r a d e s i r a b l e compromise between a s t a t e of very low a c t i v i t y or conversion on the one hand and a s t a t e of poor s e l e c t i v i t y on the other [1]. In the present paper the model of a t u b u l a r r e a c t o r with n e g l i g i b l e mass diffusivity and high diffusivity of heat i s considered. The dimensionless equations with the boundary and initial conditions governing the unsteady mass and heat t r a n s f e r i n the one-dimensional plug flow t u b u l a r r e a c t o r with high heat d i f f u s i v i t y can be w r i t t e n i n the form

Weekman and Luss; Chemical Reaction Engineering—Houston ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

46.

GUPALO ET AL.

Flow Control of Plug Flow Tubular Reactor

563

where X i s a s p a t i a l coordinate ( 0 ≤ X ≤ L), L i s a reactor length, t i s a time, c i s a concentra t i o n of r e a c t i v e species i n r e a c t o r volume, c is a feed concentration of r e a c t i v e s p e c i e s , ξ i s an ex tent, u i s flow v e l o c i t y , ε i s the r a t i o of a bulk fluid volume to a t o t a l one, Τ i s a temperatu re; V , S are a r e a c t o r volume and a surface of r e a c tor w a l l s , * g density and a heat capacity


o

c

of f l u i d ;

j* , c s

Q

a

r

e

a

are a density and a heat capacity

of c a t a l y s t , T i s a feed flow temperature, h i s a heat of r e a c t i o n , k i s a pre-exponential f a c t o r f o r r e a c t i o n r a t e , Ε i s an a c t i v a t i o n energy, R i s a gas constant, u* i s c h a r a c t e r i s t i c reactant flow v e l o c i t y * In obtaining Eqs (1.1) and (1.2) mass d i f f u s i v i t y has been neglected; high d i f f u s i v i t y of heat and f i r s t order Arrhenius k i n e t i c s f o r one step exother mic chemical r e a c t i o n has been assumed. These assump t i o n s can serve as a f a i r approximation f o r some kinds of f l u i d i z e d - b e d r e a c t o r [g] . The Eq. (1.2) can be derived f o r m a l l y by i n t e g r a t i o n over the t o t a l length of the r e a c t o r . The s o l u t i o n s of the steady-state forms of Eqs (1.1) and (1.2) can be w r i t t e n as Q

0

The dependence of the steady-state temperature Θ on parameter ν f o r f i x e d values of θ£ , and g obtained from Eq. (1.6) are presented i n F i g . 1, which shows that the m u l t i p l i c i t y of the steady s t a tes i n p o s s i b l e . F o r example three steady-state tem peratures correspond to the value ν = v » I t i s known that the upper and lower steady s t a t e s 0 , 0"" are s t a b l e , meanwhile the intermediate steady s t a t e i s unstable [ 3 ] · 0

e

+

2. The method of c o n t r o l Consider the p o s s i b i l i t y of s t a b i l i z a t i o n of the un^ s t a b l e intermediate steady s t a t e by the method of p r o p o r t i o n a l c o n t r o l (see, f o r example [ 4 ] ). Up t o now the theory of chemical r e a c t o r c o n t r o l was f o c u -


CHEMICAL REACTION ENGINEERING—HOUSTON

564

sed primerely upon the c o n t r o l of a s t i r r e d r e a c t o r 1^2 · The f i r s t example of the a n a l y s i s of the un stable steady s t a t e s t a b i l i z a t i o n f o r d i s t r i b u t e d pa rameter r e a c t o r was given r e c e n t l y by Oh and Schmitz


[1]· In the case under consideration the c o n t r o l l e d v a r i a b l e i s the temperature θ and the manipulated v a r i a b l e i s the flow v e l o c i t y ν . The feed back r e l a t i o n s h i p i s as f o l l o w s v ( t ) = v { l + ά [ θ ( ΐ - ΐ ) - θ|]} 0

(2.1)

4

where θ| i s intermediate steady-state temperature, f j i s a time l a g , d i s s t a b i l i z a t i o n parameter. In view of (2.1) the Eq. (1.6) becomes, respec tively, -g exp (- β/θ°) - θ° + 1 - exp — = : a.= 0 (2.2) v [ l + d(9°- 9|)J The r e l a t i o n s h i p (2.2) shows that i n the pre sence of c o n t r o l the upper and lower steady-state temperatures depend on parameter d and can be de termined as the i n t e r s e c t i o n points of curve 1, with l i n e s θ° = θ| + d ( v - v ) . corresponding to d i f f e r e n t values of d ( l i n e s 2-7) i n F i g . 1; The dependence of θ° on d r e s u l t i n g from Eq. (2.2) are pointed out i n F i g . 2. I t can be seen that f o r d > d the i n termediate steady s t a t e turns to be the lower one and f o r d > d± i t becomes the s i n g l e steady s t a t e . The value of d can be obtained from the c o n d i t i o n that 0

c

c

the l i n e θ° = 9Î + d ( v - v ) i s tangent to the curve 1. 0

β d

= —

?

v

0

Γβ g exp(- β/θ|) exp L E . +

I f the i n e q u a l i t y dy s t a t e condition considered as d>d_.

d >d

Ί (

2

>

3

)

takes place the s t e a -

θ° = θ| should s a t i s f y the s o - c a l l e d slope f o r s t a b i l i t y . Therefore one can expect the system of r e a c t o r c o n t r o l to be e f f e c t i v e This q u a l i t a t i v e conclusion requires r i g o -

rous approaches. In order to analyse the s t a b i l i t y of steady state under c o n t r o l , ire use the small ^pertur bation method. By s u b s t i t u t i n g Θ ( Ό = 9 ° + e ' W and ξ(χ,-ε) = ?/(x) + ξ ' ( χ , Ό i n t o Eqs (1.1)-(1.4) and (2.1) one can obtain by the Laplace transform the so l u t i o n f o r θ'(C) and ξ ' ί χ , Ό . I t can be found that a l l the s i n g u l a r i t i e s of t h i s s o l u t i o n s are poles and


46. GUPALO ET AL.

Flow Control of Plug Flow Tubular Reactor

565

are determined by the roots of c h a r a c t e r i s t i c equation. Y(s)

= s

2

0

0

+ a^s exp (- sv £

«—h-

-

-

b

exp(- | ) ] 0

C

1 + exp (- | ) ]

θ " -

0

, bχ exp ( - -o)

( p - i s the Laplace transform v a r i a b l e ) . The neces sary and s u f f i c i e n t c o n d i t i o n f o r s t a b i l i t y of the c o n t r o l l e d r e a c t o r to small perturbations i s that the r e a l parts of the roots of Eq. (2.4) are negative. Therefore the c o n t r o l l e d r e a c t o r are s t a b l e i f a l l the roots of the Eq. (2.4) l i e i n the r i g h t h a l f - p l a ne of the complex plane s = χ + i y . We chose the countor Γ » Γ + Q where fj i s the r i g h t h a l f - c i r c l e of large radius R with the centre l o c a t e d i n Zero point and Γ i s the part of the imaginary axis l y U R . I t can be shown that the increament of argument V(s) on Π| at R oo i s equal t o 2^r f o r any value of a · Λ

ζ

3· The i d e a l c o n t r o l i n Eq.

In the s p e c i a l case of i d e a l c o n t r o l by putting (2.4) f j , = 0 one can obtain

Y ( s ) = s 2 + fljs -

+ £ 2 ( 1 - e~ )/s s

(3.1) £,=

- a3 " 4» a

&z=

a

5

- a 6f

^3= i a

+

a

2

The f u n c t i o n ( 3 · Ό depends on s as w e l l as on three parameters & , Ω,χ . To analyse the s t a b i l i t y of c o n t r o l l e d r e a c t o r we have to f i n d thedomain i n three-dimensional space ( Çl^ Ω > ^3 ) ^ k& k η

9

2

n

w


c

CHEMICAL REACTION ENGINEERING—HOUSTON

566

there are no roots of V ( s ) with p o s i t i v e r e a l p a r t s . To do t h i s we consider the behaviour of the f u n c t i o n Y ( s ) on Γ , I t can be seen that the r e a l p a r t s of roots of Y ( s ) vanish f o r those values of , Ώ-ζ* Ώ which belong to the surface defined by the f o l l o wing equations 2


3

= -y

ο

2

sm y

+ a y

Ί

,

o

=

Q

3

1 - cos y

—z—

(3.2)

1 - cos y

04 y < 0 0 The a n a l y s i s of Eq. ( 3 · 2 ) reveals that f o r the domain bounded by the surface (3.2) at 0 £y < 2JT

and plane Sl - - f l ^ the increament of argument V (s) on i s equal to -2X . Therefore the t o t a l i n c r e a ment on Γ -+ i s nought and i s the s t a b i l i t y 1

ή

domain because i t contains no roots of V (s) with po s i t i v e r e a l parts. The r e s u l t s obtained permit to analyse the i n fluence of the c o n t r o l on the intermediate steady s t a t e s t a b i l i t y . The s e c t i o n of the s t a b i l i t y domains and the point A corresponding to intermediate steady temperature θ° = θ| f o r d i f f e r e n t values of parame ter d are shown i n F i g . 3 . The s e c t i o n of s t a b i l i t y domain corresponding to d = 0 i s dashed i n F i g . 3 · I t i s seen that the intermediate steady s t a t e becomes s t a b l e when d > d . I f d = d the point A achieves the boundary of the s t a b i l i t y domain. The value d can be obtained from the equation Sl^—fï^, which i s i d e n t i c a l to Eq. (2.4). Thus the intermediate steady s t a t e becomes s t a b l e when i t turns to be the lower steady-state. The numerical a n a l y s i s shows that i n the n o n l i n e a r case the value of d depends on the pert u r b a t i o n amplitude. F o r example i f d = d the i n t e r mediate steady s t a t e i s s t a b l e as the temperature perturbations are l e s s than Δ θ (see F i g . 2). 2

4. The influence of time l a g I t has been shown that i f d > d and f = 0 the c o n t r o l l e d steady s t a t e i s s t a b l e . To study the e f f e c t of time l a g on s t a b i l i t y we consider the i n crease of argument of f u n c t i o n (2.4) on f f o r d i f f e r e n t values of t

Flow Control Operation of a Plug-Flow Tubular ... - ACS Publications

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