Boundary conditions of liquid phase reactors with axial dispersion

The Chemical Engineering Journal, 11 (1916) 19-Z @Elsevier Sequoia S.A., Lausanne. Printed in the Netherlands

Boundary Conditions of Liquid Phase Reactors with Axial Dispersion WOLF-DIETER

DECKWER and ENNO A. MAHLMANN

Institut fiir Technische Chemie, Technische Universittit Berlin, I Berlin 12 (F.R.G.) (Received 12 December 1974; in final form 13 August 1975)

Abstract

the molecular diffusion coefficient D, turbulent diffusion coefficient Dt.

Numerical calculations of concentration profiles in axially dispersed flow reactors with different properties from section to section show that the conversion may depend considerably on the kind of boundary conditions applied at the boundary surfaces between sections. The jump condition of Danckwerts and the non-jump condition of Wehner and Wilhelm yield different concentration profiles. An experimental study of an isothermal liquid phase reactor divided into sections with different properties reveals that no back effect can be found as would be predicted by the application of Wehner and Wilhelm3 boundary conditions. This result and earlier experimental findings in the literature lead to the conclusion that the mechanism of effective turbulent diffusivity in liquids does not provide any back transport of mass. Therefore the application of the boundary conditions of Wehner and Wilhelmfor flow reactors is not justified if mass dispersion is caused predominantly by effective turbulent diffusivity. From a macroscopic point of view, the jump condition of Danckwerts appears to be more appropriate as it predicts conversion profiles which agree well with experimental results.

D,, =D,

+Dt

and an effective (1)

Turbulent diffusivity is caused by convective flow and depends on Reynold’s number in various ways according to the kind of equipment used. It is usually ‘assumed to be a random type transport mechanism whose single steps are large compared with molecular dimensions but small compared with a characteristic length of the flow such as tute diameter or particle size in packed beds (dv > 100 p). Relations for Dt and its physical interpretation for many different flow patterns are available in the literature lm7. In most applications of industrial importance turbulent diffusivity prevails over molecular diffusion; hence D, = Dt . When considering dispersion in reactors, the balance equations become second order differential equations which are subject to boundary conditions. Danckwertsa originally proposed these boundary conditions and they were apparently generalized by Wehner and Wilhelm9~10. However these authors did not discriminate between molecular and effective turbulent diffusivity. Results of a theoretical and experimental study of the boundary conditions for a liquid phase reactor with dispersion are reported in this paper.

INTRODUCTION

BOUNDARY CONDITIONS

When calculating the performance of chemical reactors and separation equipment, models are used which take into account the flow pattern and different mixing phenomena which may occur when a fluid phase flows through any kind of flow duct. Among the variety of models developed, the axial dispersion model seems to be of particular significance if one looks over the literature of the past 20 years. The influence of dispersion or effective diffusion is usually described by analogy to Fick’s law for molecular diffusion. Generally, the axial dispersion coefficient D, is composed of two parts:

From considerations of axial dispersion effects in steady state tubular chemical reactors, Danckwerts* solved the governing differential equation by applying boundary conditions which he derived from the condition of continuity of mass flow across the surfaces at reactor entry and exit:

19

(2) (3)

W.-D. DECKWER, E. A. M#HLMANN

20 These boundary conditions have caused much discussionr1-*6 because Danckwerts obtained the condition given by eqn. (3) partly from intuitive arguments and because an abrupt concentration jump occurs at reactor entry. The problem was solved by Wehner and Wilhelm9~ro who included an infinite fore and aft section in their treatment of reactors with effective turbulent diffusion. The same general topic had already been treated by Damkoehlerl’ . Wehner and Wilhelm solved the mass balance equations simultaneously with a first order reaction taking place in ihe finite central zone. In addition to the condition of continuity of mass flow at boundary surfaces, they assumed that the concentration is continuous near any point. Therefore at the entry boundary of the reaction zone the conditions

C(O_) =

c(o+)

(5)

were used, and similar conditions at the exit. By introducing these conditions it was possible to avoid concentration jumps at the entrance surface. Wehner and Wilhelm also showed that the boundary conditions (4) and (5) predict a marked back effect from the reaction zone to the fore section. On the basis of the paper of Wehner and Wilhelm9 and the fundamental work of Taylor 1 , the dispersion model has been applied to many different kinds of flow vessels including those with axial and radial dispersion2y3. The Wehner-Wilhelm boundary conditions were apparently proved to be true in general for reactions with arbitrary kinetic laws13 and the Danckwerts boundary conditions were shown to be limiting cases of the Wehner-Wilhelm conditions”. Both sets of boundary conditions yield the same conversion profile in the reaction zone. However, most of these results were obtained from mathematical studies. REACTORS

WITH DIFFERENT

PROPERTIES

FROM

SECTION TO SECTION

With regard to the boundary conditions, an interesting situation occurs if an axially dispersed flow reactor is +

reactor with dispersed

flow dwded

divided into sections with different properties as shown in Fig. 1. In the chemical industry, reactors are frequently divided into sections in order to minimize instalment costs and reactor volume and to provide optimal reactor performance. Zones with different flow velocities, dispersion coefficients and reaction rates may result from heating or cooling equipment. In catalytic reactors, the kind and amount of catalyst, its support material and the type of packing can be varied, yielding sections with different porosity and catalytic activity. When modelling reactors with sections of different properties, the question arises which boundary condition is to be applied at the surfaces between sections-the jump condition of Danckwerts or the non-jump condition of Wehner and Wilhelm. A theoretical study reveals that the two kinds of boundary conditions do not yield equal conversions for such a reactor. For convenience, the calculations were carried out for a reactor which is divided into three sections only. Dispersion in the entrance and exit sections is assumed to be negligible. Hence, for an isothermal first order reaction, the reactor is described by the following differential equations: x1 d2C

dC

Dar

Per dx2

du

-?=O

(6) OGxQxr

x2

-x,d2C

dC (7)

1 -x2

dZC

Pe, ti2 -~---lxc=o

dC dx

Da3

(8) 2 x2


1

The concentrations are taken relative to the entrance concentration C = c/co. Peclet and Damkoehler groups relate to section length. x1 and x2 denote the axial coordinate at the end of sections 1 and 2. Applying the condition of continuity of mass flow and of concentration at the boundary surfaces between the sections,

m N sections ---+

Fig. 1. Axially dispersed flow reactor divided in sections of length xj+l - xi.

BOUNDARY CONDITIONS OF LIQUID PHASE REACTORS

the boundary

ax;) -

conditions

are

dC(x;) ---=C(X:)-y-T

XI

pe

21

x2

-

x dC(x;) 1

2

lk

(9) C(x ;) = C(xT) x2

C(xZ)--g--=

-

(10) x1

1 - x2 dC(x;) C(x:) - pe __

dC(x;)

2

dx

3

dx

(11) C(xi) = C(xZ)

a=0

(13)

CLX =

qx;)

-

2

Danckwerts wehncr-

5

(12)

An analytical solution of eqns. (6)-(12) using the boundary conditions of Danckwerts (eqn. (2) at reactor entrance and eqn. (3) at the exit) was given by Langemann and Kolbel”. In this study, it is assumed that each section of the reactor can be treated independently when using the Danckwerts boundary conditions at section entries. At section ends the concentration gradient is assumed to be zero. Thus at the boundary between sections 1 and 2

c(x;)

1

0.2----

_

!s!g

x*

(14)

2

and similar equations are valid at the other boundary surface at x2.

NUMERICAL RESULTS

Calculated concentration profiles for a reactor divided into three sections of equal length with various properties and a first order reaction are presented in Fig. 2. Contrary to the Wehner-Wilhelm arrangement with only one reaction zone, it can be seen that the conversions do not agree at reactor exit for the different boundary conditions. In Fig. 2 the Damkoehler group in the last section was varied, whereas all other properties were kept constant. With increasing sink in the last section, the back effect becomes more pronounced as the concentration in the preceding sections decreases. The influence of the reaction rate in the last section ranges up to the first section. Both sets of boundary conditions lead to differences in conversion at reactor exit. However, theoretical predictions of both sets of boundary conditions differ considerably from each other at the end of the second section. This has to be

0

0.2

0.4

0.6

0.8

1.0

p

x

Fig. 2. Influence of the Da in the last section on the concentration profiles for a first order reaction.

attributed to back transport of mass which is caused by Wehner-Wilhelm’s condition of continuous concentration. Of course, the back effect depends strongly on the dispersion as can be seen from Fig. 3, where the influence of the Peclet number on the location of the concentration profiles is shown. In Fig. 3, the rate group is zero in the middle section and the dispersion varies in this section. #en using Danckwerts’ condition at section boundaries, the concentration remains constant in the second section and its value is the same as that at the end of section 1. With the non-jump condition, the concentration decreases in sections 1 and 2 depending on the dispersion in the second section. The concentration profiles in the last section differ slightly. Extreme cases are only shown in Fig. 3. Similar results were obtained for other orders of reactions at isothermal steady state conditions. Since

0

02

0.4

0.6

0.6

t.0 -

x

Fig. 3. Dependence of the concentration on the Pe of the middle section for a first order reaction.

W.-D. DECKWER,

22

E. A. MXHLMANN

calculations indicated that the back effect may be considerable and, particularly, that the concentration profiles differ markedly at the end of the second section. This could be verified experimentally. The reactor used was a Plexiglass tube of 9 cm internal diameter and with a length of 8 1 or 113 cm. It was divided into three isothermal sections with different dispersions. In addition, the temperature in the last section could be varied giving a different reaction rate there. Special care was taken to verify the model assumption at the boundary surfaces between the sections. Therefore any rigid separation of the different sections was avoided. For instance, the boundary surface between the second and third zones was a schlieren layer which resulted from a slight density difference in the liquid. This density difference was caused by the increased temperature in the last section. The reaction studied was the formation of a thiazolium chloride from chloropropanone and thiourea in aqueous solutions. The formation of the chloride could easily be followed by conductivity measurements. The activation energy was found to be 16.95 kcal gmol-l. Each series of runs was carried out by increasing the temperature in the last section in a stepwise manner, thus increasing the reaction rate. All other properties of each section were maintained approximately constant. The dispersion coefficients in the different sections were determined from measurements of the residence time distribution. Values of D,, between 0.2 and 1.2 cm2 s-l were found. This indicates that molecular diffusivity can be neglected. Details of the experimental arrangement and measurements as well as the determination of the reaction kinetics and dispersion coefficients are described in ref. 20. Since the kinetic data and the dispersion coefficients were obtained by independent means, fitting of theoretical predictions to experimental findings could be avoided. The experiments may thus be used to discriminate between the rival boundary conditions at the surfaces between the reaction zones. cal

0.6 -

0.5

pe=2 Do,= l.2 0

0.2

Da2=0 0.L

Da3= 1.2 0.6

0.6

1.0

p---tx

Fig. 4. Influence of Pe on the concentration catalytic reaction (A + B = 2B).

for an auto-

the differential equations are then non-linear, a quasilinearization technique and the implicit difference method were applied to obtain a solution19Jo. It was found that the Danckwerts boundary conditions always yield a higher conversion at reactor exit for reactions in which the rate decreases with the progress of the reaction. On the other hand, if the reaction rate runs through a maximum, as is the case with autocatalytic reactions, the boundary conditions of WehnerWilhelm yield higher conversions since they allow the back transport of reaction product which increases the rate. This can be seen from Fig. 4 where, for low Peclet numbers in the middle section, a very large difference of the conversion can be observed. EXPERIMENTAL PHASE

REACTOR

INVESTIGATION DIVIDED

INTO

OF A LIQUID SECTIONS

With reference to the treatment of Wehner and Wilhelm, it should be emphasized that, to date, no experimental work has appeared in the literature which definitely proves the influence of the parameters of the reaction zone on the profile in the fore section when only mass dispersion occurs. This may be due to the fact that the jump condition of Danckwerts and the continuity condition of Wehner-Wilhelm predict identical profiles for the reaction zone. However, the numerical simulation showed that the situation is changed when several reaction zones, each with different properties, are taken into account. Experiments were therefore carried out in a liquid phase reactor divided into sections in order to determine which boundary conditions are relevant to axially dispersed flow reactors. It was thought that such a study would easily vield the nredicted feedback if it existed at all. Numeri-

EXPERIMENTAL

RESULTS

For the particular reaction and the experimental arrangement used in this study, both sets of boundary conditions yield theoretical conversions at reactor exit which do not differ by more than the experimental error. This can be seen from Fig. 5, where calculated conversion profiles are compared with measured values

BOUNDARY CONDITIONS OF LIQUID PHASE REACTORS

r -I---

5

10

r

5 6

23

06~

Pe,:136

Pe,= 73

Da, =0022

CkyaOQl

RfO11 0% 0830 OL96 0260

----ii

0

OL

02

es

06

10 I

Fig. 5. Influence of Da3 on the conversion profiles: calculated from Wehner and Wilhelm; A measured; ____--__calculated from Danckwerts.

Fig. 6. Comparison of calculated and measured conversions at the end of the second section (cf: Table 1).

for a typical series of runs. However, it has already been pointed out that the different boundary conditions may predict very different concentrations at the end of an intermediate reaction zone. Solving the differential equations of the dispersion model with the conditions of Wehner and Wilhelm yields the full lines given in Fig. 5. It can be seen that the calculated conversions deviate considerably from the experimental findings. The difference increases with increasing Daa. Figure 6 confirms this observation for some other runs. The data for these runs are given in Table 1. Figure 6 presents computed and measured conversions at the end of the second section as functions of the logarithm of the Damkoehler number of the last zone. Contrary to the predictions using the boundary conditions of Wehner and Wilhelm, the application of Danckwerts’ boundary conditions leads to theoretical conversions which agree very well with the measured values. The use of Danckwerts’ conditions actually means that each reactor section is treated completely independently and there is no mutual interference between sections. Since the conversions measured at the end of the

second section demonstrate that the rate properties of the last section do not influence the concentration in the preceding zone, the boundary conditions of Wehner and Wilhelm are not appropriate to reactors where axial dispersion is predominantly caused by effective turbulent diffusion. DISCUSSION The boundary

conditions of Wehner and Wilhelm, in particular the continuity condition for the concentration, appear to be most logica13>13 from a physical viewpoint when the underlying mechanism of dispersion or effective diffusion as described by Fick’s law for molecular diffusion is accepted. However, there is experimental evidence that this law cannot always be applied to dispersive processes. Wehner and Wilhelm and also Danckwerts did not distinguish between molecular and turbulent diffusivity. Hiby carried out very careful experiments on dispersion in liquid flow systems and did not find any back transport at all. He injected a dye tracer in a bed of glass spheres and observed that the dispersion model describes the measured concentration distribution only

TABLE 1 Experimental conditions for the runs shown in Fig. 6 RIM 1 3 4 9

Q (1 h-l)

:;;ol

8.48 12.60 11.82 8.32

0.0363 0.0434 0.0364 0.0357

a, chloropropanone;

b, thiourea.

cbO/caO

Pel

Dal

Pe2

Da2

Pe3

1.55 1.53 3.13 4.01

16.9 9.6 10.3 9.7

0.011 0.018 0.014 0.035

8.0 8.5 8.2 6.6

0.046 0.075 0.062 0.142

0.52 0.97 0.90 0.65

I-‘)

W.-D. DECKWER, E. A. MXHLMANN

24

at distances from the point of infusion which are large compared with the particle diameter, because many independent repetitions of velocity variations are needed to obtain a probability function. Near the point of injection the dispersion model fails to predict the tracer distribution. Obviously, Fick’s law only correctly describes the widening and spreading of an input signal far away from the source point and a relative back transport is observed only if one moves with the mean flow velocity. This was also the original result of Taylor’. This feature of dispersion is taken into account in mathematical models of chemical reactors which do not explicitly apply Fick’s law to simulate dispersive flow. In the Deans-Lapidus2r procedure, the void volume of a bed of catalytic particles is substituted by a two-dimensional array of ideal mixers with two outlets to the subsequent row of cells which are offset by half a cell. StewartZ2 proposed a wavelike propagation of dispersive flow. We may conclude from these considerations that no feedback can be found in experiments with a reactor divided into sections of different properties. In this study the boundary surface between two zones was a boundary between two regions with different dispersion and reaction rate. Because the dispersion provides no back transport, the properties of the following section cannot influence the conversion in the preceding section as would be predicted from the conditions of continuous concentration and continuous mass flow at boundary surfaces. Correspondingly, the boundary surface between the fore and the reaction zone of the original Wehner-Wilhelm treatment is a surface behind which product is generated. The validity of the non-jump condition would require back transport of reaction product in the fore section which disagrees with the experimental findings of this study. Quite recently Wicke23 discussed th.: boundary conditions of catalytic fixed bed reactors and concluded that, in gas phase systems where effective turbulent diffusivity predominates, the boundary conditions of Wehner and Wilhelm are not correct. Wicke pointed out that these conditions refer only to that part of the overall dispersion which represents molecular diffusion. This part can be neglected in almost all cases of practical significance as the back transport by diffusion will not extend to more than a fraction of the particle diameter. Since there is no back transport of mass by turbulent diffusion in all cases of single phase flow through any kind of flow duct, the WehnerWilhelm boundary conditions cannot be applied to reactors with dispersion or effective diffusion of mass. Of course, the conditions are correct for molecular dif-

fusion processes and for effective diffusion of heat in fixed beds, when different coefficients are applied to transport downwards or upwards relative to the flow direction. Furthermore, an effective back transport of mass is observed in several kinds of two-phase flow24-29 where the Wehner-Wilhelm conditions may be appropriate. Since the experimental concentration profiles agree with numerical computations using the jump conditions of Danckwerts, this condition appears to be more suitable for isothermal one-phase reactors with effective turbulent diffusion. Although a concentration jump appears to be very unreasonable, Standart14 pointed out that there are no a priori reasons why properties must be continuous at boundary surfaces. Standart applied the principles of irreversible thermodynamics and thus his results are general. Moreover, they are not connected to any particular mixing model and are unspecific. The main results of Standart’s analysis are: first, dispersion fluxes may vanish at boundaries in certain cases, and second, the jump condition of Danckwerts is a natural consequence of the second law of thermodynamics if the dispersion is markedly increased in the flow direction at boundary surfaces between sections. Therefore, from a macroscopic point of view, discontinuities may be possible which justify the application of the jump condition of Danckwerts.

NOMENCLATURE

C

D ax & Dt Da L N Pe U X

c/co, dimensionless concentration axial dispersion coefficient molecular diffusivity effective turbulent diffusivity Damkoehler number, related to section length length of entire reactor total number of sections Peclet number for mass, related to section length linear flow velocity dimensionless axial coordinate

Subscripts

1, 2,3

refer to section

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25 18 H. Langemann and H. Kolbel, Verfahrenstechnik (Mainz), I (1967) 5. 19 W.-D. Deckwer, H. Kblbel and H. Langemann, Chem. Eng. Sci.. 27 (1972) 1643. 20 W.-D. Deckwer and E. A. M%hlmann, Third Int. Symp. on Chemical Reactor Engineering, Evanston/Ill.. I9 74. in H. M. Hulburt (ed.),Adv. Chem. Ser., 133 (1974) 334. 11 H. A. Deans and L. Lauidus, A.I.Ch.E. J., 9 (1963) 129. 22 W. E. Stewart,-Chem. Eng. hog. Symp. Ser., 61 (i965) 61. 23 E. Wicke, Chem.-Ing.-Tech., 47 (1975) 547. 24 A. Klinkenberg, Chem. Eng. Sci., 23 (1968) 92. 25 S. Hartland and J. C. Mecklenburgh, Chem. Eng. Sci., 21 (1966) 1209. 26 S. Hartland and J. C. Mecklenburgh, Chem. Eng. Sci., 23 (1968) 286. 27 W.-D. Deckwer, U. Graeser, H. Langemann and Y. Serpemen, Chem. Eng. SC;, 28 (1973) 1223. 28 W.-D. Deckwer. R. Burckhart and G. Zoll, Chem. Ens. ScL. 29 (1974) 212;. 29 K. Schuegerl, Chem. Eng. Sci., 22 (1967) 793.