Comment on studies on axial disperation in fixed beds

Chemical Engineering

and Processing,

107

33 (1994) 107- 111

Letter to the Editor

Comment on studies on axial dispersion in fixed beds (D. J. Gunn, Trans. Inst. Chem. Eng., 47(1969) T351; D. J. Gunn, Chem. Eng. Process., 32 (1993) 333)

PE,,

is correlated

Re, = u,d/v

or (better) E. Tsotsas Dow Deutschland Inc., Solids Processing Engineering Werk Stade, Postfach 1120, 2 1677 Stade (Germany)

Science Laboratory,

Verfahrenstechnik,

In 1969 Gunn published a paper [ 11. trying to explain the behaviour of axial dispersion in packed beds as observed in the course of transient tracer experiments. The same subject was considered by us in 1988 [2]. We criticized several aspects of the derivation by Gunn, and introduced a new, different approach. Recently, another paper [ 31 by Gunn was published in Chemical Engineering and Processing. While this later paper does not contribute any essentially new aspects to the problem, it tries to avert our criticisms in ref. 2 and, on its part, vehemently attacks our model. We gratefully take the opportunity to discuss axial dispersion in packed beds once more, since there is evidently still a considerable lack of understanding of the phenomenon. In this letter we: briefly restate the problem for readers not familiar with it; discuss the model of Gunn and show why we consider it to be erroneous or, at best, inconsistent; refer to our own model and to its comparison with experimental findings; address some problems of theoretical and practical importance which are, in our opinion, still unsolved.

The problem

The results of transient tracer experiments in packed beds are usually presented as effective P&let numbers PK., = @ID,,

(1)

where D,, is the coefficient of axial dispersion, u, the average superficial flow velocity, and d the particle diameter.

02%2701/94/$7.00 SSDI 0255-2701(93)00504-W

number (2)

the molecular

P&let number (3)

Pe, = uod/6

E.-U. Schhinder Institut fiir Thermische Universit%t Karlsruhe, Postfach 6980, 76 128 Karlsruhe (Germ~y)

with either the Reynolds

where v is the kinematic viscosity of the fluid and 6 is the fluid-tracer diffusion coefficient. The respective phenomenology is as follows. (i) As Pe, -+O, D,, is directly proportional to 6 and PE,, to Pe,. Axial dispersion is obviously governed by molecular diffusion in the axial direction. (ii) As Pe, + 00, D,, is directly proportional to u,,, so that PE,, attains a constant value of about 2. This was explained very clearly by Aris and Amundson [4] in 1957 by the equivalence between the packed bed and a cascade of ideal mixing vessels at large flow velocities. The discussion of Gunn on page T352 of ref. 1 is essentially a repetition, without quotation, of ref. 4. The interested reader may note that a similar derivation for radial dispersion, based on probabilistic arguments, was given by Baron [5] in 1952. (iii) At medium Pe, values and for gases (low Schmidt number media) a weak maximum of the PE,,(Pe,) curve is observed. (iv) At medium Pe,, values and for liquids (large Schmidt number media) the weak maximum mentioned above is sometimes also present. After the maximum, the effective P&let number PE,, does not immediately tend to 2 (as with gases). Instead, an extended plateau smaller of nearly constant PE,, values, considerably than 2, is observed. Effect (iii) is not very important. Our main concern in the present letter, as well as in ref. 2 is effect (iv) in close combination with effect (ii). Mathematics,

physics and the model of Gunn

The model of Gunn is described by eqns. (28) and (29) of his original paper [I], along with eqns. (23) and (26), which explain the abbrevations, and the respective initial conditions. This description should be physically and mathematically complete, self-sufficient, and invariant throughout the derivation. It is therefore advisable to forget everything in the paper (ref. 1) given before it; especially the discussion on page T352. On this basis we see that Gunn unequivocally defines a system of capillary flow with maldistribution (a fast,

(I, 1994 ~~ Elsevier Sequoia. All rights reserved

108

uniform stream surrounded by static fluid) and with diffusion as the only mechanism for the relaxation of concentration profiles. Dispersion in flow systems of this kind had been thoroughly investigated long before 1969 by Aris ([6], 1956) and Taylor ([7], 1953) though neither of these references are cited in Gunn’s paper. The final result of refs. 6 and 7 may be put in the compressed form l/P&

= (l/&J

+ @o/K)

(4)

Here the effective as well as the molecular P&let number are built with the average flow velocity and the average diameter of the capillary. Both the shape of the capillary and the flow distribution within it may be arbitrary. Their influence is condensed into K, which is a constant. To give an example, for the parabolic profile in a circular tube with diffusion of the tracer (the classical Taylor solution) a value of K = 192 is obtained. As can easily be seen, the limiting case Pe,+cc

n PE,,+O

(5)

is yielded from eqn. (4) (the dispersion coefficient depends on the square of the velocity at large velocities). On the contrary, Gunn claims to have derived the limiting case Peo+ainPEa,=2

(6)

Since Gunn presupposes the very same physical system as Taylor and Aris, there are good reasons for a careful control of his derivation. A critical point is the transition from eqn. (40) to eqn. (41). Two possibilities exist: (i) Stick to the system as defined by Gunn himself. We must then assume that the system retains its memory up to the observed cross-section, in the same way that memory is retained in the empty tube of Taylor, and in the same way that a manipulated die will never forget its previous results. Consequently, we must substitute the time t by the product nT everywhere in eqn. (40) where n is the number of repetitions of the basic stochastic process and T is the time constant of a single process. In this manner the exponential function of eqn. (40) turns out to be exp( --anT)

- 1

(7)

and not n exp( -0)

- 1

(8)

as stated in eqn. (41) of Gunn. Here Q is an abbreviation for all parameters except time of the argument of the exponential. It can easily be seen that eqns. (7) and (8) are identical only if the quadratic term in the expansions of

the exponential functions is neglected. this term the limiting case PEo+co~PEa,-+~

Da,=0

By neglecting (9)

is derived. Axial dispersion is not appearing, and an injected ideal pulse would leave the bed with the fast stream as an ideal pulse. Alternatively, consequent development of eqn. (7) up to the quadratic term leads to our eqn. (26) (ref. 2) 1/P&x = (( 1 -19) /@In

(10)

Since n has to be a very large number, this is equivalent to eqn. (5). In other words, by consequent expansion to the quadratic term the same limiting relationship is obtained as for the entire family of Taylor-Aris solutions (eqn. (4)). In the original paper of Gunn [ 1] the transition from eqn. (40) to eqn. (41) has been outlined very briefly. On this basis we anticipated the elementary mathematical mistake of simply forgetting to put n in the argument of the exponential function. From his new, more detailed explanations [3] we understand that this has not been the case. We therefore proceed with the second possibility for the transition. (ii) Apply eqn. (40) to only one basic stochastic process, for time T. Repeat this n times, and apply the central limit theorem. In this case eqn. (41) is, indeed, obtained without injuring mathematics. However, the use of the central limit theorem presupposes the repetition of independent single steps. Since one step is indentifled with each particle layer, it presupposes the destruction of information, i.e. the loss of the system memory, at the end of every particle layer. This can be done only by mixing and is inconsistent with the model definition by Gunn since there mixing is not accounted for, and diffusion is the only mechanism able to relax the concentration profiles. To use probabilistic language Gunn is, on the one hand, manipulating his die by the introduction of diffusion in his basic equations while, on the other hand, he is expecting random results from this manipulated die. The only way to (partly) overcome the dilemma would be to define Gunn’s model on a penetration or surface renewal basis by introducing the use of the central limit theorem as an additional major assumption, not as a mathematical triviality. Even by this generous interpretation, two further discrepancies cannot be resolved. The first one, as already pointed out in that part of our 1988 paper repeated by Gunn in ref. 3, concerns the probability parameter p. This parameter is supposed to describe at the same time both Taylor dispersion, i.e. the combination of flow maldistribution with diffusion, as well as the series-of-mixers effect. This is quite re-

109

markable since the two mechanisms are, in physical terms, entirely different. We still consider p as a chameleon parameter, changing its meaning on need and demand. The second descrepancy concerns the derivation of Gunn leading to eqn. (40), an asymptotic, long-time solution as evidenced by the transition from eqn. (21) to eqn. (22) or (25), which is done by using only the first term of the series expansion, as well as in the definitions of /3 and y (eqns. (26) and (23)). However, a long-time solution presupposes the penetration of mass up to the geometrical boundary of the considered body. We do not have any difficulty in accepting that this is fulfilled in a Taylor-Aris solution, i.e. when the system is allowed to develop over a very large number of particle layers (actually, the Taylor-Aris solutions are, by clear definition, valid only in this limit). However, we have serious difficulties in accepting without control that the same should also be true for only one particle layer, as inherently anticipated in the derivation of Gunn. Therefore, we proceed with the following check. The short-time and long-time solutions for mass transfer to a cylinder after a step change of concentration at its surface are, respectively, Sh = 2/(,,&

4%)

(11)

and Sh = 5.78

Here, the Fourier

(12) number

Fo = h/s2

(13)

(14)

Now take d = 1 mm as a typical particle diameter, and take 40% of it, i.e. 0.4 mm, as the approximate diameter of the voids in the bed. Further assume, for the sake of simplicity, an equal subdivision between the fast stream and the stagnant fluid, thus obtaining s = 0.2 mm for each of them. With the typical value of 6 = 10e9 m* SK’ for difiusion in liquids, the criterion t > 1.524 s

t = d/u = 0.4 s

of 0.40) and (16)

is obtained, which is smaller than 1.5 s and does not fulfill the criterion for the applicability of the long-time solution. The discrepancy becomes even larger if we consider the following. (i) The height of one particle layer in a random bed is smaller than the particle diameter d. (ii) The velocity in the fast stream will be greater than u (in the case of equipartition between the flow regions by a factor of two). (iii) Every deviation from equipartition will inevitably increase the geometrical dimension s of one stream. The ‘penetration time’ of eqn. (15) will increase by s2 for this stream. By switching to larger Reynolds numbers, say Re, = 10, the deviation between the residence time available in one particle layer and the residence time necessary for the solution of Gunn to be valid becomes overwhelming. Clearly, Gunn is adopting the Taylor-Aris approach for one particle layer, by working with long-time solutions, while simultaneously denying it, by restarting the process after every particle layer. The result is, in our opinion, inconsistent.

Fo is defined as

with s the diameter of the cylinder. The respective Sherwood number Sh is built also with s. By simply regarding the intersection of the asymptotes (eqns. ( 11) and ( 12)) the range of validity of the long-time solution can, generously, be obtained as Fo > 4/(~5.78~) = 0.03811

of u = 0.0025 m s-i (for a bed porosity finally, a residence time of

(15)

is obtained from eqn. (14) for the validity of Gunn’s derivation. Now compare this number with the actual residence time for one particle layer at the typical Reynolds number of Re, = 1, with a kinematic viscosity of typically v = 10m6 m2 s-i and d = 1 mm. A superficial flow velocity of u0 = 0.001 m s-‘, an interstitial flow velocity

Our model and experiments

Contrary to Gunn’s model, our model from 1988 [2] is characterized by a consequent application of the Taylor-Aris long-time analysis. The evolution of profiles is not interrupted when proceeding from one particle layer to the next. Two instruments are used in order to, in spite of this, avoid the limiting relationship eqn. (5) and obtain, instead, the correct expression eqn. (6). (i) Axial as well as radial dispersion due to and sustained by hydrodynamic mixing (turbulence), incorporated a priori, empirical and per definition into the fast stream. (ii) The cross-sectional area occupied by stagnant fluid assumed to decrease with increasing Bow velocity and vanishing entirely at Re, -) CO. In this way’we separate from each other things which do not belong together and interpret: (i) The dispersion at medium Pe, values in large Schmidt number fluids, i.e. effect (iv) from the introduction as Taylor dispersion, initiated by the combination of flow maldistribution and lateral mass transfer. (ii) The limiting case Pe,, -+ 00 n PE,, = 2, as the dispersion in a cascade of ideal mixers. This asymptotic

110

law emerges from the dispersion features of the fast stream per se. Since this approach appears to be conceptually clear and consistent, and since we are not aware of any mathematical error, elementary or not, we have every reason to favour it over Gunn’s model. Furthermore, we have little reason to accept Gunn’s criticism concerning the comparison of our predictions with experimental results. In ref. 3 Gunn says that our model does not properly describe changes of the Schmidt number in the range 500-2000, and tries to support this with Fig. 2 of his new paper. Frankly, we never expected that it would. In our opinion a variation of the Schmidt number by a factor of 4 to 5 is simply too small, so that its impact is overlapped by experimental inaccuracy. The latter is not due to bad experimentors, we know the papers listed by Gunn well and have a high opinion of the authors, it is due to the randomness of packed beds and to the inherent difficulty of measuring within them. Inaccuracy is great not only in axial dispersion measurements but also in all other packed bed investigations, e.g. radial dispersion, wall-heating experiments etc. (see ref. 8 for several examples). Most people having serious contact with packed beds and other particulate systems are well aware of this, and a researcher with the experience and merits of Gunn should know it best. Anyway, to state the point clearly, we think thdt reliable trends can be obtained only by a really large change of the Schmidt number, for example the change by a factor of about 1000 when going from gases to liquids. Therefore, our ambition in ref. 2 was only to provide an explanation for the respective observations, and this explanation appears to work very well. Our plot of calculated PE, numbers (Fig. 3 in Gunn’s paper) has a finer subdivision rather in order to illustrate the behaviour of the model than with the purpose of a more detailed comparison with measurements.

Open questions

Finally, we would like to address some aspects which, though not mentioned in the paper of Gunn, possess considerable theoretical and practical importance and are not yet well understood. We put a number of questions and add some opinions and comments without trying to find out any final answers. What about macroscopicalflow

maldistribution?

The macroscopical flow maldistribution (wall channeling) and the radial velocity profiles which are generally assumed to describe it should induce a large, clearly detectable Taylor dispersion. Unexpectedly, they do not. We feel that the macroscopical Taylor dispersion is

suppressed by the microscopical Taylor dispersion, since relaxation time for radial mass transfer in the microscale is much shorter than in the macroscale. Some arguments for this explanation are given in ref. 2, but are far from being a proof. What part of D,,, as measured by transient experiments, is real backmixing?

tracer

In our opinion, genuine backmixing occurs only at the limits Pe, --t 0 (molecular diffusion) and PeO + CC (turbulence). Other investigators, see for example Hiby [9], go even further and deprive the Pe,, + co, PE,, = 2 asymptote of the character of real backmixing. Taylor dispersion, as well as any other form of dispersion due directly or indirectly to velocity profiles, can hardly be considered as backmixing. It is rather an artifact of the measuring method, changing its appearance from one experimental situation to another. In this context, we should not forget that the intention of Taylor [7] has not been to measure axial dispersion coefficients in the circular tube with laminar flow and then use it for other purposes, but to apply the tracer method to a situation with an exactly defined velocity profile and derive binary diffusion coefficients (tracerfluid) from the long time behaviour of the system. For our taste, we often tend to expect too much from the Taylor method, as well as from the transient tracer method itself. May axial dispersion coeficients D,,, measured by transient tracer experiments, be used in steady state situations? In our opinion, only those parts of D, corresponding

to real backmixing may be used in steady state situations. More generally, only these parts are invariant upon things like the existence or non-existence of accumulation terms, the existence or non-existence of distributed sources or sinks (i.e. of chemical reaction), the permeability or impermeability of the side wall etc. Therefore, we suggest simply superposing the PeO -0 and Pe, + 00 asymptotes in order to obtain a formula for general practical use. In this context, readers with a special interest in packed bed reactors may find the recent discussion between Gunn, Vortmeyer, Stewart, Shapiro and Brenner [lo] interesting.

References 1 D. J. Gun, beds, Trans. 2 E. Tsotsas packed beds 15-31. 3 D. J. Gun, Process., 32

Theory of axial and radial dispersion in packed Inst. Chem. Eng., 47 (1969) T351-T359. and E.-U. Schliinder, On axial dispersion in with fluid flow, Chem. Eng. Process., 24 (1988) On axial dispersion (1993), 333.

in fixed beds,

Chem. Eng.

111 4 R. Aris and N. R. Amundson. Some remarks on longitudinai mixing or diffusion in fixed beds, AZChE .Z., 3 (1957) 280282. 5 T. Baron, Generalized graphical method for the design of fixed bed catalytic reactors, Chem. Eng. Progr., 48 (1952) 118-124. 6 R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. R. Sot. London, Ser. A, 235 (1956) 67-77. 7 Sir G. Taylor, Dispersion of soluble matter in solvent flowing

slowly through a tube, Proc. R. Sot. London, Ser. A, 219 (1953) 186-203. 8 E. Tsotsas, tiber die W&me- und Stofftibertragung in durchstrcmten Festbetten, VDZ-Fortschritt-Berichte, Reihe 3, Nr. 223, VDI-Verlag, Dusseldorf, 1990. 9 J. W. Hiby, Longitudinal and transverse mixing during singlephase flow through granular beds, in P. A. Rottenburg (ed.), Interaction between Fluids and Particles, Inst. Chem. Eng., London, 1962, pp. 312-325. 10 Letters to the editor, AZChE J., 38 (1992) 1675-1680.