Chemical Engineering Science, 1974. Vol. 29, pp. 747-751. Pergamon Press. Printed in Great Britain

CATALYTIC REACTIONS IN TRANSPORT REACTORS

K. C. PRATT Department of Chemical Engineering and Chemical Technology, Imperial College, London SW7 2BY,

England

(Received 21 August 1973)

Abstract-By assuming plug flow of both gas and solids phases, and simple kinetics, a model of the transport reactor has been prepared which accounts for the unsteady state behaviour of the catalyst particles. Computational results are presented to show the influence of the major variables. As in a steady state analysis, the Thiele modulus serves to indicate the catalyst effectiveness.

1. INTRODUCTION 2. THE MODEL

In a transport reactor, the catalyst or solid reactant is carried through a pipeline by the reacting gases. In principle the system presents many advantages to the designer. For consecutive reactions, where an intermediate product is desired, the close control of residence time facilitates the maintenance of the desired conversion level. Further, in contrast to fixed and fluid bed reactors, the catalyst loading can be varied independently of the gas flow. Thus, in the catalytic cracking of petroleum feedstocks, where the conventional fluid bed system is not suited to the demands of fluctuating feedstock rate and composition, up to 90 per cent of the cracking may be done in a riser before entering the conven- tional dense phase system [ l-31.

We shall consider a system in which both particu- late and gas phases are in plug flow. The catalyst particles are spherical and of uniform size. For simplicity, we assume that only a single reaction is occurring, and that this reaction is first order, ir- reversible, and isothermal.

Considering an element of the reactor, if there is no volume change during the reaction, a mass bal- ance for the reacting component in the gas phase leads to

In future applications, advantage could be taken of the increased heat transfer rates in riser systems 141 to impose a temperature gradient on the reaction, enabling an optimum path to be followed. Also, since the gas flow is fully turbulent, ensuring that the solids are well distributed, and reaction conditions uniform at each point along the reactor, the possibility of hotspots developing is remote.

Under the usual conditions of operation, experi- mental evidence suggests that axial diffusion can be neglected[5], thus Eq. (1) becomes

u*_ 30(1-a) ac dl Ry- arlrzR. (2)

Despite these advantages, little or no use has been made of the transport reactor outside the pet- roleum industry, though the system should be suited to other fast reaction or adsorption proces- ses. The main reason for this neglect appears to be a lack of operating and design experience. A few experimental studies of non-catalytic reactions are available[5-71, but there is a need for systematic experimental and theoretical study of catalytic reactions in transport reactors. This work is an at- tempt to produce a model of the system, and must account for the non-equilibrium behaviour of the catalyst particles.

It should be noted that while the above results con- sider steady state operation in the gas phase, the particles themselves do not attain steady state.

The mass balance equation for the particles is

E aC #C 2 ac kc --=2----- D at ar rar D

Equations (2) and (3) must be solved simultane- ously subject to the following boundary conditions. (a) At the entrance to the reactor the catalyst parti- cles are filled with an inert diluent

C(O,r)=O OSrSR. (4)

747

748 K. C. PRAIT

(b) The reacting component has a concentration cu at the entrance to the reactor

c(0) = co = C(0, R). (5)

(c) Spherical symmetry in the particles

$+,O)=O.

(d) There is no external mass transfer resistance for the particles

c(t, R) = c(l). (7)

The solution can be simplified by expressing Eq. (2) in terms of a time derivative with respect to particle motion, thus

Udc 30(1-o) ac --=---- u dt R a arh (8)

Equations (3)-(S) are solved using Laplace Transforms[8], the solution is

P =1+&&(1-o) URZp,2 -- -- 2u (Y &+&lJxo) I( m (10)

The kms and p,s are the roots of

ff&$pn = I-(k,R)cot(kR), (11)

and epPn + k k;=-T (12)

In order to express the solution in dimensionless form we put

x. = k,R, (13) and

l Z2 y. = 7

Thus the concentration profile in a catalyst particle is given by

and

F =1+?p-y.+~y.2 2 2x. 6P xn 05)

The x.s and yns are the roots of

xn = - y. - &. (17)

and

&Y = l-ficotfi. (18)

M, P, and 4 are dimensionless groups given by

DL M=~Q (19)

p_U(1-4 u a

m=R& (21)

The group C#J will be recognised as the Thiele Mod- ulus of the catalyst particles.

Finally, by making use of Eq. (7), we obtain the concentration of the reacting component in the gas phase as a function of the dimensionless length l/L,

(22)

F, is given by Eq. (16).

3. COMPUTATIONAL RESULTS

Using an iterative technique, the first 10 eigen- values of the transcendental Eqs. (17) and (18) were found for 5 values of rk2, namely, 1000, 100, 10, 1, O-1, and 5 values of P; 0~1,0~05,0~025,0~01,0@05. In practice, the series in Eqs. (15) and (22) con- verged rapidly, and only the first 4 eigenvalues were used. A representative set of results is contained in the figures accompanying this paper.

4. DISCUSSION

In Figs. 1 and 2, conversion is plotted as a func- tion of the parameter M, for various values of c$ and P. As the catalyst particles are not at steady state, the quantity (1 -C/C&~ does not, strictly speaking, represent the conversion. However, as the volume of the reactor occupied by catalyst is

I I

0 20 40 60 80 loo M

Fig. 1. Conversion as a function of the parameter M, for various values of P. 4 = 10.

Catalytic reactions in transport reactors 749

Con ,o_ P.005 0025 am 0005 I

08

06

04

02

lr,,:

0 200 400 600 800 1000 M

Fig. 2. Conversion as a function of the parameter M, for various values of P. 4 = 1.

generally only of the order of 2 per cent, then (l- C/G) gives a good approximation to the reactant conversion.

Since the quantity R21D represents the time con- stant for diffusion, then the parameter M may be thought of as being a measure of the number of diffusion time constants which the particle spends in the reactor. Thus, as expected, the effect of in-

In Fig. 4, a set of typical axial conversion profiles of the reactant in the gas phase is presented. It can be seen that the profile is almost linear at low catal- yst circulations.

Of perhaps the greatest interest is the behaviour of the catalyst particles themselves. Figures 5-8

COnV

1 o- P-0 05 0 025.

0 02 04 06 OS 1O"L

Fig. 4. Axial conversion profiles for various values of P. M = 5,d = 100.

creasing M is to increase the conversion. The parameter P also has an obvious signifi-

CIC,

0 5 - cance, indeed, it is easy to show that IlL~Ol

I

P=&& (23) 0 6 t

where p is the catalyst circulation ratio (weight of 02 catalyst circulated/weight of gas). Figure 3 isolates 0 L -

03

the influence of catalyst circulation ratio, and de- monstrates the diminishing return from increasing catalyst circulation at conversions greater than 80 per cent.

0 2

From the designers point of view, a family of curves similar to Figs. 1 and 2 would be the most 0 02 04 06 06 10 r/R useful. Having selected the catalyst (hence d), and the desired conversion, the designer can then deter-

Fig. 5. Concentration profiles within the catalyst particle

mine corresponding values of reactor length and at various values of 1/L. 4 = 100, P = 0.025, M = 5.

catalyst circulation. The optimum combination would be determined by external factors, such as power requirements, regeneration, and space limi- tations.

o a

0 0.01 a02 0.03 0.04 a05 p

Fig. 3. Conversion as a function of the parameter P, for Fig. 6. Concentration profiles within the catalyst particle various values of hf. I$ = 0.1, at various values of l/L. I$ = 10, P = 0.025, M = 20.

750 K. C. PRATT

TIC I I -0 -i . IiL.01

02 0.6 -

0.3

0.4 - 04

05

06

02- 08

10

I

0 02 04 06 06 IO r/R

Fig. 7. Concentration profiles within the catalyst particle at various values of l/L. 4 = 1, P = 0.025, M = 100.

OS- liL.0 1

06: 02

03

04. 04

0 02 04 06 08 10 r/R

Fig. 8. Concentration profiles within the catalyst particle at various values of l/L. 4 = 0.1, P = 0.025, M = 1000.

display radial concentration profiles within the par- ticles at various stages of their passage through the reactor. Values of 4 range from 100 to 0.1. Consid- eration of steady state effectiveness factors [93, in- dicates that for values of 4 greater than about unity, the rate is diffusion controlled. Examination of Figs. 5-g shows that similar criteria will apply to the unsteady state case treated here. The transition from diffusion to reaction control over the range of values of 4 is striking. For 4 = 100, it is clear that only a fraction of the available catalyst is being util- ised, while for 4= 0.1, the almost uniform con- centration profiles indicate virtually full use of av- ailable active surface.

Some difficulty was experienced in obtaining convergence of the series in Eq. (15) for very small values of l/L, even when the number of eigen- values used was increased. Since xn and y. are not

given explicitly, it is difficult to test Eq. (15) for convergence. However, as the values of l/L above which convergence is obtained are so low, this is not a practical problem in the use of Eq. (15). For 4 = 0.1, convergence was obtained above l/L = 040005, and the initial development of the profile is shown in Fig. 9. It can be seen that C/c, rises to a maximum at the centre of the particle, before de- creasing as the particle traverses the reactor.

Fig. 9. Development of the concentration lrofile within the catalyst particle in the inlet region of the reactor. 4 =

0.1, P = 0.025, M = 1000.

Finally, some mention should be made of the fluid mechanics involved in the assumption of plug flow in both the particulate and gas phases. As yet, no basic theory has been successful in an a priori prediction of two phase gas solids flow. Available experimental evidence [ 10, 1 l] suggests that the effect of added solids is to promote plug flow of the gas phase. Solids concentration tends to be uniform across the tube[l2], but a parabolic type velocity profile results in a non-uniform mass velocity[ll]. At very high solids loadings, there is some evidence of solids backmixing at the walls [13]. There is also an implicit assumption that the particles are in- stantly accelerated to the velocity U, at t = 0. For the size of particles likely to be encountered in these systems, this is reasonable. The specification of plug flow in the solids phase is perhaps the least justified of the assumptions involved in this model. However, no theory exists at the present time which will give an adequate representation of the solids velocity profile.

The existence of a non-uniform solids mass flux, backmixing of solids, and the behaviour of the catalyst particles in the mixing zone are problems requiring further study.

CONCLUSIONS

This work has shown that a model of the trans- port reactor operating under idealized conditions can predict the influence of the major variables.

Catalytic reactions in transport reactors 751

C concentration of reactant in the reactor, g mole/cm3

cu concentration of reactant at reactor entr- ance, g mole/cm3

c concentration of reactant in catalyst particle, g mole/cm

D effective diffusion coefficient of reactant in the catalyst particles cm2/sec

E axial dispersion coefficient cm*/sec k, P roots of Eqs. (11) and (12)

k first order rate constant per unit volume of catalyst pellet, set-

1 length along reactor, cm L length of reactor, cm M dimensionless group defined by Eq. (19) P dimensionless group defined by Eq. (20) r radial co-ordinate, cm r radius of catalyst particles, cm t time, set

t: particle velocity, cmlsec interstitial gas velocity, cm/set

x.9 Y roots of Eqs. (17) and (18)

Greek symbols a voidage in reactor

Consideration of the results shows that the relation- ship between the Thiele modulus and the catalyst effectiveness is similar to that obtained from a steady state analysis. Further work is needed to extend the model to cover situations which are more kinetically realistic.

NOTATION

p catalyst circulation, g catalyst/g gas c voidage of catalyst particle

pp gas density, g/cm3 ps bulk density of a single catalyst particle,

g/cm & Thiele modulus for catalyst (Eq. 21)

[II

121

r31

141

151

t61

171

181

191

r101

[III

1121

II31

REFERENCIZS

BRYSON M. C., HULING G. P. and GLAUSSER W. E., Hydroc. Proc. 1972 85. STROTHER C. W., VERMILLION W. L. and CONNER A, J., Ibid. 89. PIERCE W. L., SOUTHER R. P., KAUFMAN T. G. and RYAN D. F., Ibid. 92. SADEK S. E., Znd. Engng Chem. Proc. Des. Deu. 1972 11 133. JEPSON G., POLL A. and SMITH W., A.I.Ch.E.-I. Chem. E. Joint meeting, London 13-17 June 1965. GAUVIN W. H. and GRAVEL J. J. O., Symposium on the Interaction Between Fluids and Particles, p. 250. Instn Chem. Engrs London 1%2. YANNAPOULOS JT C., THEMELIS N. J. and GAUVIN W. H.. Can. _I. &em. Ennnn 1%6 44 23 1. CRANK J., 2% Mathematics of-D&ion. Oxford University Press, London 1970. SAITERFIELD C. N., Mass Transfer in Hetero- genous Catalysis. M.I.T. Press, Clinton 1970. DOIG I. D. and ROPER G. H., Znd. Engng Chem. Fundls 1%7 6 247. PRATT K. C., Ph.D. Thesis, University of Mel- bourne 1%9. SO0 S. L., TREZEK G. J., DIMICK R. C. and HOHNSTREITER G. F., Znd. Engng Chem. Fundls 1%4 3 98. VAN ZOONEN D., Symposium on the Interaction Between Fluids and Particles p. 64. Instn Chem. Engrs London 1%2.