Chem. Eng. Technol. 13 (1990) 273-277

Transport Effects in Catalytic Fluidized-bed Reactors*

273

Abdulla A. Shaikh and James J. Carberry**

The majority of the fluidized-bed reactor models are rooted in the tacit assumption that transport between the bubble, cloud, and emulsion phases occurs in series with chemical reaction. A more realistic model that anticipates simultaneous transport and reaction is presented in terms of a fluid-bed effectiveness factor which embraces the limits between chemical and mass transport control. Analysis of the predictive capability of this model vis-a-vis the Kunii-Levenspiel model reveals signal differences in chemical conversion.

1 Introduction

The importance of the catalytic fluidized-bed reactor (FZBR) in the chemical and petroleum industries has stimulated the development of various empirical and deterministic models to describe the performance of this important class of heterogeneous reactors. The diverse models heretofore set forth have been admirably summarized in recent monographs [ 1, 21 and authoritatively reviewed by Wen [3] and Grace [4].

A close examination of the FZBR modelling literature reveals that the so-called bubble (or bubbling bed) models have become the most popular. Among the bubble models, the two basic models are those of Davidson and Harrison [5] and Kunii and Levenspiel 16, 71. These have been modified over the years, e.g., the Toor-Calderbank [8] modification of the Davidson [9] model and the Fryer-Potter [lo] modification of the Kunii- Levenspiel model. Central to these models is the specification of an effective bubble diameter and the tacit assumption that transport between bubble (dilute), cloud and emulsion phases occurs in series with catalytic reaction.

Industrial FZBRs generally host rather rapid rates of reaction. Witness the contact times - of the order of seconds. Witness employed values of gas velocity relative to that of minimum fluidization - of the order of 25 to 50 (e.g., in catalytic crack- ing). Real FZBRs are also usually devoid of bubbles but mask- ed by “snakes” of solids-lean gas passing tortuously through a solids-rich emulsion phase. In fact, industrial fluid beds are often so well baffled (with horizontal elements) that bubble growth is discouraged and solids backmixing severely minimized.

Yet, whatever the nature of the solids-lean phase (bubbles or “snakes”), we assert that in real FZBRs the existence of bub- bles is questionable and, more importantly, the typical reaction vigour is incompatible with the series reaction-transport description employed in the majority of the existing models. In this work, we elaborate on an earlier concept [ 11, 121 and

* Paper presented at the Third National Meeting of Chemists, Dhahran, Saudi Arabia, March 12 - 14, 1989.

** Dr. A.A. Shaikh, Assoc. Professor of Chemical Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia, and Dr. J.J. Carberry, Professor of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, U S A.

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fashion a model rooted in a more realistic notion of fluid bed behaviour. The model involves mass transport from a solids- lean phase to a solids-rich, cloud-emulsion phase during which transport and reaction occur simultaneously. The predictive capability of the model is subsequently contrasted with that of the celebrated Kunii-Levenspiel model 16, 71.

2 Reaction-Diffusion Model

It is assumed that a reactant gas of concentration A, (in the bub- ble) is transported to the bubble-cloud interface and, thereafter, it is transported and undergoes reaction simultaneously in the cloud phase. That portion of the gas which escapes unreacted in the cloud and into the emulsion phase, reacts simultaneously while it is transported into that emulsion phase.

The terminology of Kunii-Levenspiel is employed as shown in Fig. 1. For linear isothermal kinetics, A + products, a species balance based on plug flow in the bubble (dilute) phase yields ’) :

Eq. (1) can be integrated to provide an expression for the con- version of species A:

where e b is contact time and qD - is the fluid-bed effectiveness factor based on simultaneous diffusion-reaction. We define this factor as:

(4)

The interfacial flux in Eq. (4) can be evaluated by solving the one-dimensional local continuity equation:

1) List of symbols at the end of the paper.

0930-7516/90/0408-0273 $3.50+ .25/0

274 Chem. Eng. Technol. 13 (1990) 273-277

(8) k Y C k y e Vec 42 = -, and 11,' = ~ . kP Bi = ~

m c ' K b c Kbc

Note that the bubble-cloud mass transfer coefficient is defined as K b c = D,/Zf..

The limiting behaviour of the effectiveness factor can be established by a detailed analysis of Eq. (7) under asymptotic conditions. Some distinct cases have been considered and are summarized in Table 1. This compilation of limiting forms of qD - allows one to determine the global rate of reaction (i.e., qD - k A b ) and the controlling effects.

EMULSION PHASE

' . . :. . .

Fig. 1. Details of the Kunii-Levenspiel model. Table 1. Summary of asymptotic behaviour of effectiveness factor.

Phase Bubble Cloud Emulsion

transfer transfer -+ Event reaction reaction reaction

Volume Vb "c Ve

Volume of solids

Volume of bubbles Yb Yc Ye

Kbc Kce Transfer coefficient -

with the boundary conditions:

dA z = O , -aD,- = a k , ( A b - A ) ,

dz

Condition (6a) anticipates interphase transport resistance be- tween the dilute and cloud phases, while condition (6b) merely states that reactant escaping consumption in the cloud, undergoes reaction in the dense emulsion phase.

The local boundary-value problem, Eqs (5) and (6), can be easily solved to yield an expression for the effectiveness factor. In dimensionless form, we obtain:

I . [ 11 + Y e c v e c l 4 I I, + [ 4 + $2 4 1 1 1

V D - R =

11,' + 4 tanh 9 (7)

where Bi is a Biot number which reflects the gas-side resistance, and 4 and 11, are Thiele moduli. These groups are defined as:

Limiting conditions

(a) Bi + 03

4J << 1 $ << 1

1 1

(b) Finite Bi 1 1

1 + ($'/Bi) 4J << 1 $ << 1 1 + 11 + yet V,,1 ( C 2 W

1 4J >> 1 << 1

[1 + Yec V,,l 4J

(c) Bi -+ 03

>>

(d) Finite Bi 4J >> 1 $ >> 1

1 Bi

[1 + Yec v,,1 7 BI -

*2

3 Comparison with Kunii-Levenspiel Model

Few studies are available in the literature in which experiments were carried out to discriminate among the rival bubble models. Chevarie and Grace [13, 141 have compared the distribution of selected component concentrations in a FZBR found experimentally with those calculated theoretically. The experimental and calculated data were most compatible in the case of the Kunii-Levenspiel (K - L) model.

The K - L model also has an important advantage from the computational viewpoint. Namely, given physicochemical pro- perties of the reacting system and a single adjustable parameter (i.e., an effective bubble diameter), it is possible to secure the values of all other parameters in the model. These characteristics of the K - L model have certainly decided its continued popularity and application (see, e.g., [15 - 171).

When applied to linear isothermal kinetics, the K - L model leads to a relationship for the conversion of species A [6, 71 which can be expressed in terms of an effectiveness factor as well:

where the effectiveness factor for this model is given by:

Chem. Eng. Technol. 13 (1990) 273-277

0

275

I I 1

Note that, in general, yb = 0.01, so that it can be ignored for all practical purposes.

We are now in a position to compare our model, i.e. Eqs (3) and (7), with the K - L model, i.e. Eqs (9) and (10). Both models have been applied to the ozone-decomposition reaction system of Kobayashi and Arai [18]. The relevant physicochemical parameters can be found in [7] and [l 11. The empirical correlations, necessary for evaluating the transfer coefficients, Kbc and K,,, and the solids-to-bubbles ratios, yc and ye, can also be found in the same references.

Fig. 2 shows how the effectiveness factors vary with reaction vigour and gas-side resistance. Note that, in contrast to the D - R model, there is no allowance for gas-side resistance in the K - L model. Note also that qD-R approaches asymptotic values (at low and high k values) which are consistent with those given in Table 1 . Fig. 3 dramatizes the differences be- tween the effectiveness factors. It is clear that the simultaneous reaction-transport concept, upon which the D - R model is bas- ed, is of little consequence in the case of slow reactions. However, the opposite is true in the case of fast reactions.

Recalling that conversion is exponential in q for a simple first- order reaction, the departure of the ratio of effectiveness factors from unity with increasing k surely leads to drastic differences in predicted conversions, as shown in Fig. 4.

Fig. 2. Effect of reaction rate constant on effectiveness factors.

I Z I a , = I o

RATE CONSTANT [ l / s ] 0'

Fig. 3. Effect of reaction rate constant on the ratio of effectiveness factors.

d

- 0

I .

5 8 - E d

. 12) B i = 1.0 m 0 I31 B i = I00 0

As stated earlier, the usefulness of the K - L model is rooted in specifying a bubble diameter and it is therefore of interest to consider the effect of this parameter. Fig. 5 shows how the ratio of effectiveness factors changes with bubble size and gas-side resistance. Clearly, the observed differences must be reflected in model predictions of conversion as shown in Fig. 6 .

4 Concluding Remarks

In essence, the model set forth here is similar to that evoked to describe the reaction between an absorbed gas and a co-reactant

276 Chem. Eng. Technol. 13 (1990) 273-277

1 0 0 , * 0 1

12) 81.1 0

13) 01-100

0

0

00?20 100 200 300 4 0 0 5 0 0 600 700 B O O 9 0 0 1

BUBBLE DIAMETER [ Cm]

Fig. 5. Effect of bubble diameter on the ratio of effectiveness factors.

1 1 ) B i -0.1

12) B i *1.0

1 3 ) B I -100

0 ,- 0000 100 2 0 0 300 400 5 0 0 600 TOO BOO 900 1 0

BUBBLE DIAMETER C cml

Fig. 6. Effect of bubble diameter on the ratio of conversions of species A predicted by the models.

in the liquid phase. In this instance, the emulsion phase is equivalent to the liquid phase. The solids-lean phase is equivalent to the well identified gas phase in gas-liquid systems - though, undoubtedly, the gas-liquid interface is, while acknowledged, not well defined in the presence of dispersion and coalescence phenomena. Even less certain are our bases for definition of the fluid bed. We can assert that the fluid bed is multiphase in nature. But, in contrast to bubble models, it is

one is obliged to search for qualitative, not quantitative, phenomena. Which is to state that an intuitively more realistic (but hardly perfect) model must be developed if for no other purpose than to demonstrate some of the not-too-subtle inade- quacies of previous bubble models.

The FZBR model, which we develop and apply in this work is based on the postulate that mass transport from the solids-lean phase (not necessarily a bubble) to the solids-rich, cloud-emul- sion phase is accompanied by simultaneous reaction. The model leads to a fluid-bed effectiveness relationship which per- mits specification of limiting forms of the global rate, i.e. chemical to mass transport control.

While we readily admit that quantitative predictions escape us - primarily due to imprecision of the diverse transport coefficients - our model reveals signal differences with respect to previous oversimplified series reaction-transport theses. In short, this work demonstrates the inadequacies of transport coefficient correlations and the potential deception implicit in the series reaction-transport FZBR models.

Acknowledgement

The authors are grateful to the King Fahd University of Petroleum & Minerals for support of this and related work. We also thank Mr. S.M. Al-Subaii for his help in performing some of the computations.

Received: September 6, 1989 [CET 2451

Symbols used

a [ c ~ * I A , [mol/cm3]

Interfacial area concentration of species A in solis-. -lean (bub- ble) gas phase concentration of species A at reactor inlet Biot number, defined by Eq. (8) diffusion coefficient of species A reaction rate constant gas-side mass transfer coefficient mass transfer coefficient between solids-lean (bubble) and cloud phases mass transfer coefficient between cloud and emulsion phases velocity of a bubble rising through a bed volume of cloud phase volume of emulsion phase ratio defined as V,/ V, conversion of species A, defined as (1 -Ab/A,) distance along reactor height distance into cloud-emulsion phase thickness of cloud film

Greek symbols

Yb

Yc

ratio of solids dispersed in bubbles to volume of bubbles in the bed ratio of solids in cloud-wake region to volume of bubbles in the bed

doubtful that industrial units exhibit such neat behaviour. Thus, Ye ratio of solids in the emulsion to volume of bubbles in the bed

Chem. Eng. Technol. 13 (1990) 277-288 277

Yec ratio defined as +ye/yc

qo - qK Ob

@ ic

effectiveness factor based on diffusion-reaction model effectiveness factor based on Kunii-Levenspiel model contact time defined as height of reactorlu, Thiele modulus defined by Eq. (8) Thiele modulus defined by Eq. (8)

References

[ l ] Yates, J.G., Fundamentals of Fluidized-Bed Chemical Processes, Butterworth, London 1983.

[2] Doraiswamy, L.K., Sharma, M.M., Heterogeneous Reactions: An- alysis, Examples and Reactor Design. Volume I : Gas-Solid and Solid-Solid Reactions, Wiley, New York 1984.

131 Wen, C.Y., Flow Regimes and Flow Models for Fluidized Bed Reac- tors, in: Recent Advances in the Engineering Analysis of Chemically Reacting Systems, Doraiswamy, L.K., (ed.), Wiley Eastern, New Delhi, 1984.

[4] Grace, J.R., Fluid Beds as Chemical Reactors, in: Gas Fluidization Technology, Geldart, D. (ed.), Wiley, New York 1986.

[5] Davidson, J.F., Harrison, D., Fluidized Panicles, Cambridge University Press, New York 1963.

[6] Kunii, D., Levenspiel, O., Ind. Eng Chem. Fundam. 7(1968) p. 466. [7] Kunii, D., Levenspiel, O., Fluidization Engineering, Krieger, New

York 1977. [8] Toor, F.D., Calderbank, P.U., in: Proc. Int. Symp. on Fluidization,

Netherlands University Press, Amsterdam 1967. [9] Davidson, J.F., Trans. Inst. Chem. Eng. 39 (1961) p. 230.

[lo] Fryer, C., Potter, O.E., Ind. Eng Chem. Fundam. 11 (1972) p. 338. [ l l ] Carberry, J.J., Chemical and Catalytic Reaction Engineering,

McGraw-Hill, New York 1976. [12] Carberry, J.J., Trans. Inst. Chem. Eng. 59 (1981) p. 15. 1131 Chevarie, C . , Grace, J.R., Ind. Eng Chem. Fundam. 14 (1975) p. 79. [14] Chevarie, C., Grace, J.R., ibid. 14 (1975) p. 86. [15] Irani, R.K., Kulkami, B.D., Doraiswamy, L.K., Ind. Eng Chem.

Process Des. Dev. 19 (1980) p. 24. [16] Guigon, P., Large, J.F., Bergougnou, M.A., Chem. Eng J.

(Lausanne) 28 (1984) p. 131. [17] Tabis, B., Chem. Eng Sci. 43 (1988) p. 1615. 1181 Kobayashi, H., Arai, F., Kagaku Kogaku (Chem. Eng Japan) 29

(1965) p. 885.

Sensitivity Analysis of Catalytic Conversion of n-Cs

Andrzej Wolff

The present paper is a case study of an application of sensitivity analysis in chemical kinetics. Emphasis is laid upon chemical interpretation of sensitivity information and on identification of the most important model parameters. The kinetic model for reforming of c6 hydrocarbons pro- posed by Mobil [14] is extended to the analysis of the behaviour of n-hexane conversion in an adiabatic reactor. The importance of six initial conditions (feed composition and initial temperature) is analyzed by the computation of normalized first order sensitivity gradients (yp/y) (6yi/6yp). The relative importance of 21 model parameters aj is estimated by the computa- tion of normalized sensitivity gradients of the type (aj/yi) (6yi/6aj). The influence of the decisive model parameters AH: and AH; (activation enthalpies of benzene hydrogenation and methyl cyclopentane isomerization, respectively) as well as operating parameters is presented. The pro- blem of uncertainty in the value of AH: and its influence on the model solution is also shown. Finally, some advantages of the application of normalized gradients to the explanation of process behaviour are discussed.

1 Introduction

Mathematical modelling and consecutive sensitivity analysis of a process model is an important tool for a better understanding and explanation of the behaviour of complex chemical systems. The typical approach to mathematical modelling is well known but the necessity for and advantages of sensitivity analysis in verifying and predicting the model behaviour are not quite SO

well appreciated. The sensitivity notion concerns the influence of changes in the values of parameters of a mathematical model on the steady state solution of the system. The validity of this concept has its origin in the existence of uncertainties in the model parameters as well as in the experimental data used for the model development. This approach was already reflected in several papers dealing with mathematical modelling and sen- sitivity analysis of chemical reactors, reaction mechanism and kinetics [ 1 - 101.

* Dr. habil. A. Wolff, Institut fur Technische Chemie I , Universitit Erlangen-Nurnberg, Egerlandstr. 3, D-8520 Erlangen, Permanent ad- dress: Instytut Inzynierii Chemicznei i Chemii Fizycznej, Politechnika Krakowska, ul. Warszawska 24, 31-155 Krakow, Poland.

A mathematical model is referred to as sensitive when a small change in its parameters leads to Significant changes in the model solution. Appreciable sensitivity is undesirable and the

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