Investigation of the Multiplicity of Steady States with the Generalized

40 Investigation of the Multiplicity of Steady States with the Generalized Recycle Reactor Model

Downloaded by CORNELL UNIV on May 18, 2017 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch040

TAI-CHENG YANG and HERBERT WEINSTEIN Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616 BARRY BERNSTEIN Department of Mathematics, Illinois Institute of Technology, Chicago, Ill. 60616

The effect of incomplete mixing on the multiplicity of steady states and reactor stability is studied with the generalized recycle reactor model. This model represents an infinite number of states of micromixing at any given level of macromixing. The concept of segregation of mass is extended to include that of temperature. Two extreme cases in energy segregation are studied. The high thermal conductivity model behaves as an isothermal reactor. The low thermal conductivity model represents complete segregation of energy. The method of invariant imbedding is used to solve the equations. Conversion curves show that for a given residence time distribution, multiple solutions can exist at some states of micromixing whereas only a single solution exists at other states. Stable optimal conditions can be found for each mixing pattern by varying the adjustable inlet conditions.

he multiplicity of steady-state operating points of chemical reactors has been studied for both varying reactor models and varying reaction models [see for example, Bilous and Amundson ( I ) , Aris ( 2 ) , and an excellent review of this work by Perlmutter ( 3 ) ] . Almost all the analyses reported i n the literature used the " i d e a l " p l u g flow tubular ( P F T R ) and backmix (or C S T R ) reactors, the P F T R w i t h recycle, and the P F T R w i t h axial a n d / o r radial dispersion. T h e limitation of all these models is that when using them to cal culate chemical conversion, one usually can only couple a single description of the small-scale mixing, called micromixing, w i t h each description of the gross mixing patterns w h i c h determine the distribution of holding or residence times of the reacting fluid. This latter type is called macromixing and is essentially a linear description of a nonlinear mixing process. T h e operation of a reactor is not, however, limited to the discrete state of micromixing usually attached by a particular model to the state of macro-

T

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Hulburt; Chemical Reaction Engineering—II Advances in Chemistry; American Chemical Society: Washington, DC, 1974.


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mixing measured for that reactor. There are, i n fact, an infinite number of micromixing states possible for any reactor w h i c h is described only b y its residence time distribution ( R T D ) , except for the P F T R . Because the state of micromixing is not necessarily that described by a single parameter model, instabilities may not be discovered i n reactor design w i t h these models. It is important, therefore, to be able to search through a spectrum of possible micromixing states for a given R T D to determine if multiple steady states can exist. In this paper a model is formulated for the behavior of an adiabatic reactor system i n w h i c h macro- and micromixing states can be varied inde pendently. This model is then used to investigate the effects of variable micromixing on the number of steady-state operating points of a reactor system defined b y its macromixing level or residence time distribution. The kinetic rate expressions used are limited to the nth-order type, and heat transfer across the boundary of the reactor system is not considered. Method Generalized Recycle Reactor M o d e l for M i c r o m i x i n g . The R T D for η equal-sized C S T R s i n series w i t h recycle has been developed by F u et al. (4). The R T D ΕΤΛ,Ν E i t )

1 V* /

R

\

m

(R +

= R h { r + l )

^

_

l)mn mn mn-l n

t

(ran -IV.

i

n(R+

«

/ix

)t

'

has first moment t = V/q and dimensionless variance 1 + n(l

nR

+

(2)

R)

where R is the recycle ratio. Both E(t) and σ have, as parameters, R and n. The dimensionless variance, σ , or the first and second moments are usually taken to be enough information to describe the R T D because higher moments of experimental R T D curves are greatly affected by measurement error. D o h a n and Weinstein (5) imposed two assumptions to this η stirred tanks i n series w i t h recycle model—viz., (a) allow for perfect mixing at the junction between the recycle stream and fresh feed stream, (b) operate the reactor internal to the recycle loop under a completely segregated flow condition. Thus, they separated the effects of micromixing and macromixing w i t h a fixed ôr . T h e y showed that a negligible error is introduced because moments of the R T D higher than the second are not held constant. H o l d i n g σ constant, one can vary η and R simultaneously to describe different levels of micromixing for a fixed state of macromixing. F o r the special case of σ = 1, a single-stirred tank w i t h variable recycle can be used to describe any level of micromixing. A d d i t i o n of the E n e r g y Balance E q u a t i o n . T o study the multiplicity of steady states of a nonisothermal reacting system, we must add an energy b a l ance equation to the generalized recycle reactor model. T o do this we must model the heat transfer within the reactor i n a manner analogous to the model ing of the mass transfer. The state of micromixing corresponds to this mass transfer modeling. W h e n the thermal conductivity is very high, heat transfers swiftly through the reacting fluid. I n the limiting case of infinite thermal conductivity, tem perature is uniform throughout the reacting fluid down to the molecular scale. 2

2

2

2

2



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This statement defines our h i g h thermal conductivity model. I n this case there is no segregation of energy. O n the other hand, when the thermal conductivity is very low, heat transfer through the reacting fluid is very slow. . I n the limiting case of zero thermal conductivity, energy is completely segregated. This is our low thermal conductivity model. F o r an exothermic reaction, a "point" of fluid [defined b y Danckwerts ( 6 ) ] is, according to the low conductivity model, exposed to low temperature w h e n the concentration of reactant is h i g h and to high temperature when the concentration of reactant is low. According to the high conductivity model, however, a point of fluid sees only a single tempera ture during its passage through the reactor. W e limit ourselves here to these two limiting cases to describe the mode of heat transfer within the reactor. The model based on the generalized recycle reactor model w i t h the temperatures is shown i n Figure 1. A t point A , we allow perfect mixing. Inside the reactor, reacting fluid is completely segregated. Temperature is either completely segregated or uniform. CR 13

Dx

n

m

where η is reaction order and D = k C ~ V/q. W e also note that x = x in the limit. L o w T h e r m a l Conductivity M o d e l for One-Stirred T a n k w i t h Recycle ( σ = 1). In the low thermal conductivity model, batch reactor information on concentration and temperature are coupled. Mass and energy balances around point A give 0

n

1

Ao

m

t

2

1

1 +

^

-

Γ

Γ

Λ

R

+t4-b*< ' 1 + R'

+

Γ Γ Λ *

(14)

(

1

5

)

A t the exit of the system, Xf

= f

œ

Xbatch £(*) 0.5. F o r R < 0.5, there is only a single operating point for each inlet temperature. I n both h i g h and low con ductivity cases, multiple steady-state operating points can be avoided by oper ating at inlet temperatures above about 181°C while the lower limit on inlet temperature is not the same for both cases. One might be tempted to explore further the possibility of operating at low inlet temperatures, — 120°-140°C, and almost complete conversion indicated b y this figure. T h e results for the h i g h and low conductivity cases for the single-stirred tank and second-order kinetics are shown i n Figure 3. These results differ 4

R

Ao

g

0

11

A o

p

t


40.

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539


ι

ι

Γ

REGION OF MULTIPLE STEADY STATES

sfe —zfe— —sou —sfe —"Sfe"

180

1

-1

-1

-1

EXIT TEMPERATURE , ° C

HIGH T H E R M A L CONDUCTIVITY M O D E L i.o

J — - - i —

\

i ^ U -

\

\

ξ .6

ι

χ.

1 R

/ / A / / / / / / / / / / / / / / /

/ //

1 120

1

ι 140

ι ι - T ^ 1 1 1 160 180 200 FEED TEMPERATURE, ° C

R

=

0

R=5 R = R

/ / /

110

=

min R=.l R=.25 R=-5 R=l R=2

1

1 220

max

.1

= O D

.....i 240

1

260

LOW T H E R M A L CONDUCTIVITY M O D E L Figure 2. Conversion as a function of temperature for RTD variance, σ — 1, V2-order reaction (physical constants given in text) 2

from the previous cases i n only two respects: (a) the segregated flow, R = 0, yields the maximum conversions for the h i g h conductivity model; (b) the degree of the effects is different. T h e m i n i m u m inlet for w h i c h h i g h conversion can be obtained without a multiplicity of steady states is still about the same. The lower bound on the inlet temperature for the multiple steady-state region is, however, higher than for the y2-order case, about 160°C. It is interesting to extend the criterion for stability to the model of onestirred tank w i t h recycle, l o w conductivity case. O n e can show from the asymptotic solution, e.g., for the second-order reaction ( 7 ) ,


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REGION OF MULTIPLE STEADY STATES

200 240 280 EXIT TEMPERATURE, ° C

320

360

400

HIGH THERMAL CONDUCTIVITY M O D E L 1.0 j — .9.8-

R - R

max R=2 R=l

=

.2.1 -

120

140

160 FEED

180 200 TEMPERATURE, ° C

220

240

260

LOW THERMAL CONDUCTIVITY M O D E L Figure 3. Conversion as a function of temperature for RTD variance, σ = 1, second-order reaction (physical constants given in text) 2

dT dX

(32)

0

m

F o r endothermic reactions, Ρ < 0, and we always have dT /dX < 0, w h i c h means the system is stable. F o r moderate exothermic reactions, Ρ > 0, and we still have dT /dX < 0 and a stable system. O n l y for highly exothermic reac tions, Ρ > > 0, is dT /dX > 0, indicating that the system has a region of instability. 0

0

m

m

0

m


40.

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Recycle

Reactor

541

Model

η-Stirred Tanks i n Series w i t h Recycle, σ < 1. Calculations were made on the model of η stirred tanks i n series w i t h recycle, w i t h both high and low conductivity cases and w i t h second-order kinetics. The results for σ = 0.5 are shown i n Figure 4, and those for σ = 0.2 are shown i n Figure 5. Both figures indicate there is essentially no micromixing effect for the high conduc tivity model. The band of conversions caused by variation i n micromixing, R = 0 to R > is essentially a line. In these two figures the effects of micromixing are more apparent i n the results for the low conductivity model. The trends are the same as i n Figure 3. A s σ becomes smaller, the permissible 2

2

2

max


2

LOW THERMAL CONDUCTIVITY

MODEL

Figure 4. Conversion as a function of temperature for RTD variance, σ* = 0.5, second-order reaction (physical constants given in text)


542

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CHEMICAL

FEED

TEMPERATURE,

°C

LOW T H E R M A L CONDUCTIVITY

MODEL

Figure 5. Conversion as a function of temperature for RTD variance, σ* == 0.2, second-order reaction (physical constants given in text) amount of micromixing becomes smaller, and the effects of varying the micromixing level must diminish (cf. the low conductivity case of Figure 4 w i t h those of Figure 5 ) . The band of inlet temperatures w h i c h covers the range of large conversion change is m u c h narrower for σ = 0.2 than for σ = 0.5. These latter two figures also indicate that calculations made for a reactor model w h i c h allows for a single value for the degree of segregation can give misleading results on the existence of multiple steady states. However, at least i n these figures if the inlet temperature is chosen h i g h enough either to avoid multiple states where the simple reactor model indicates their presence, or high 2

2


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543

enough to obtain close to complete conversion i n the case where the simple model does not indicate the presence of multiple steady states, then h i g h con version and a single steady-state operating point are ensured. Intermediate conversion cases cannot be discussed i n detail here because the single set of physical constants chosen provides for a very large gradient i n conversion over a short temperature range. Nomenclature A Β C C D Ε Ε (t) G(t) G (Θ) ΔΗ / k k Κ η ρ Ρ q A


p

Κ

0

R R t t Τ V χ y

g

E/R T , dimensionless AH C /p C T , dimensionless Concentration of A , gram-mole/liter Specific heat, cal/gram-mole, °K k C ~ V/q, dimensionless Activation energy, cal/gram-mole Residence time distribution of a system First passage time distribution of internal reactors Dimensionless first passage time distribution of internal reactors Heat of reaction, cal/gram-mole of reactant Degree of segregation Reaction rate constant, ( m o l e / l i t e r ) " / h r Frequency factor k C ~ τ/η, dimensionless Order of reaction or number of tanks Group of variables given by Equation 27 AH C / C ,°K Volumetric flow rate, liters/hr, or group of variables given by Equation 27 Recycle ratio Ideal gas law constant, 1.987 cal/gram-mole, °K T i m e , hr V/q, hr Temperature, °K Total volume of reactor ( s ), liters Dimensionless concentration, C / C Dimensionless temperature, T/T g

R

0

Q

Ao

p

n

Ao

0

l

1

a

n

Ao

w

ι

n

Ao

P

p

A

A o

0

Greek Symbols Θ I p σ τ φ

2

ί / τ or nt/τ, dimensionless time A dummy variable M o l a r density, gram-mole/liter Dimensionless variance of residence time distribution V/(1 + R)q,hr A parameter

Subscripts f m ο

L e a v i n g or final conditions Inlet condition to internal reactor ( s ) Inlet or fresh feed conditions to a system

Literature Cited

1. Bilous, O., Amundson, N. R., A.I.Ch.E.J. (1955) 1, 513. 2. Aris, R., Chem. Eng. Sci. (1969) 24, 149.


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CHEMICAL REACTION ENGINEERING-II

3. Perlmutter, D. D., "Stability of Chemical Reactors," Prentice-Hall, Englew Cliffs, N. J., 1972. 4. Fu, B. J., Weinstein, H., Bernstein, B., Shaffer, A. B., Ind. Eng. Chem. Proc. Design Dev. (1971) 10, 501. 5. Dohan, L. Α., Weinstein, H., Ind. Eng. Chem., Fundamentals (1973) 12, 64. 6. Danckwerts, P. V., Chem. Eng. Sci.(1958) 8, 93. 7. Yang, T. C., Ph.D. Thesis, Illinois Institute of Technology, Chicago (1974) 8. Kalaba,R.,Ruspini, Ε. H.,J.Computat. Phys. (1971) 8, 489. January 2, 1974.


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Investigation of the Multiplicity of Steady States with the Generalized

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