Fixed bed catalytic reactor modelling

FIXED

BED CATALYTIC THE

HEAT

REACTOR

TRANSFER

W. R. PATERSON*

MODELLING

PROBLEM

and J. J. CARBERRY?

Shell Department of Chemical Engineering, Cambridge University, (Received

Cambridge

1 July 1982)

Abstract-The issue of radial heat transport in fixed beds is discussed and discrepancies between observed and computed hot spots are attributed to neglect of axial dispersion of heat, which neglect also accounts for observed length-dependent radial conductivities.

The Nusselt number for heat transport at the wall is inferred to be JvUW =5.73(Z)

“* Pr(O.llRe, + 20.64) R#“I

which is in accord with recently secured Sherwood numbers for mass transport at a packed wall. The recent success of unconventional reactor models is discussed.

WFRODUCTION In 1953, Danckwerts

[ 11 published his celebrated paper on residence time distribution in continuous contacting vessels, including chemical reactors, and thus provided methods for measuring axial dispersion rates. Earlier, Bernard and WiIhelm[2] had described radial dispersion in packed beds by a Fickian model, which soon found theoretical support [3,4]. An analogous description of heat transport using effective conductivity became available [S, 61. An additional resistance to radial heat transfer near the bed wall was then identified, and quantified as an effective wall heat transfer coefficient [7]. By the early 1960s. the now-conventional model for the packed bed catalytic reactor had emerged. It was then possible for Wilhehn[S] to set forth the status of a priori design of such reactors, with due recognition of the role of transport phenomena in dictating reactor performance. The resulting model is a two-dimensional quasi-continuum model-that is, it involves continuous radial and axial variation of temperature and composition and it is, at least for the depiction of long-range gradients, a pseudo-homogeneous model. Thus it describes rates of material transport across the bed as being due to effective single phase transport rather than separate solid and gas phase events. Moreover, it is a plug-flow model, for Carberry and WendelI had shown axial dispersion to be negligible, except for the shortest of beds. The merit of this model is that it succeeds in capturing many of the important properties of packed reactors in a qualitatively ccrrect way. Moreover, it can often be made to give useful quantitative agreement with steadystate performance by suitable adjustment of parameter *Author to whom correspondence should be addressed. tPermanent address: Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A.

values, such as activation energy, pre-exponential factor, effective radial conductivity or wall heat transfer coefficient. But it is striking that after all these years there is scarcely a published example of a successful a priori prediction of performance of a non-isothermal non-adiabatic packed bed reactor. By “a priori” is meant that the kinetic and transport parameters should have been measured independently and then brought together without adjustment. Though perhaps not generally recognised, it seems clear that for the important and difficult class of exothermic reactions, the discrepancy between prediction and observations is methodical: to wit, the model predicts higher hot spot temperatures than are observed (e.g. [lO-133). Such methodical errors cannot be attributed to faults in the rate equations: they must result from shortcomings in the model. Lest we be accused of presumptive ambition in seeking ~1pn’ori prediction, we readily confess that in practice one must settle for much less, namely a model, adjusted as needs be, which permits useful but prudent interpolation and extrapolation during process development. Currently, even if a suitable catalyst formulation has been devised and approximate operating temperature, pressure and compositions identified, modelling may offer disappointingly little help in choosing catalyst particle size and shape, tube diameter and flowrate. The trade-0ffs-e.g. small particles yield high effectiveness factor and effective wall heat transfer coefficient but low effective radial conductivity and high pressure dropoften cannot be explored thoroughly in limited time and cost, and so even modest help from a model would be welcome. Although we accept that parameter adjustment may be necessary, we urge that as many parameters as possible be independently measured, for otherwise a high parameter cross-correlation may render such adjustment physically meaningless. 175

W. R.

176

PATERSON

It thus behooves us to describe the conventional model, to search for the causes of its failings and to survey the alternatives. We lay some stress on prediction of hot-spot temperature, because it may govern exothermic reactor design for reasons of catatyst damage or the onset of parasitic reactions. For endothermic reactions, parametic sensitivity is probably less severe, but we note that in steam reforming the accurate prediction of tube skin temperature is important since creep of the reactor tube wall is temperature sensitive.

and J. J.

CARBERRY

are deleted from equations (I) and (2), equation (4) is retained, wdile equation (3) is replaced by inlet dimensionless temperature and concentration profiles, for which the simplest forms are just z=o;

f=l,C=l.

In both these models the magnitude of wall Biot number dictates the relative magnitude of the temperature difference over the bulk of the bed to that at the wall, i.e.

1. THE CONVENTIONAL MODEL OF THE PACKED BED REACTOR In dimensionless form the steady-state continuity equations for the cylindrical fixed bed are[ 141, for mass

af

1

---

a’j

az Pe n nX7-GL

(1)

ar ) =+! dzj lr aj 7 ( Y?+--

and heat at

1

a2

3%

Pe,n az2

a “, --(Et+‘ Pe,m arz

=x!L&

(2)

r ar

where z = Z/Z,; I = R/Ro; f = C/Co; t = T/TO; n = ZJd,; m = Rold,,; a = ZJR,; 0 = Z,iu. The 1.h.s. of equations (1) and (2) thus describe the reactor in terms of aspect ratios n, m and a, and the mass and thermal Peclet numbers Pe, E respectively, which dictate dispersion in the axial and radial directions. So

pe =!!&?E. Fe LI

D,’

The customary z=*

a

=!!& Ka,

boundary

4&l. Per=D, I

conditions

Fe, = +.

are

j=1++j) Pe,n

z=o

az 2-0

af a.2

z=l

and

r

Va)

5 = 0. dZ

At bed centerline aj/&- and War = 0, while at the wall, in the absence of a wall catalyzed event, aj/ar = 0. For heat transfer at the wall

=

h,(T - Tw)

or at

(ar._, > =-pg(t-tw) l,

-

h.&

PC&

= Bi, = Sr,Pe,m

where St,.,= h,lpuC, and h,,&lk,, = Bi,. Equations (l)-(4) comprise the axially-dispersed two dimensional model discussed below. The conventional model-the plug flow two dimensional model-is a simplification of this set; the axial second derivative terms

So the key issue of radial heat transport and in exothermic systems the magnitude of the ‘hot spot’, depend critically upon the proper prediction of the wall coefficient h, and the effective radial thermal conductivity of the packed bed. k,, = PC&,. Traditionally the Peclet numbers have been measured in the absence of reaction 5% and Q = 0 in eqns (1) and (2)). Thus radial Peclet numbers for fixed beds are secured by application of eqns. (1) and (2) ignoring the axial dispersion terms. It having been shown that simulation ,of the two-dimensional fixed bed hosting highly exothermic reactions is quite insensitive to the radial mass Peclet number but extremely sensitive to E, and St, (14), we shall therefore focus critical attention on these thermal transport parameters, For extant correlations of k,, and h, reveal distressing disagreement data exhibit an and in the case of k,,, experimental incomprehensible dependency upon bed length. A partial explanation of the failings of the conventional model The systematic over-prediction of hot-spots implies that the conventional model underestimates rates of heat removal from the bed. We offer two explanations. First, the careful application of the conventional model to heat transfer experiments demonstrates its inadequacy because its key parameter-_k.-declines with increase in bed depth[lL181. It has been shown[lS] that introduction of an axial dispersion term into the heat transfer sons reaction model removes the anomaly of lengthdependency and results in a statistically-adequate fit to experimental data. Moreover, since, in the usual heat transfer experiments, the axial temperature gradient declines monotonically with increase in bed length, an analysis of results neglecting the axial dispersion term leads to erroneously low values of k., and h, for use in the conventional model. So the axial dispersion-marked interpretation of Dixon et al. yields higher-than-usual rates of radial heat transfer, helping to explain the systematic error. Of course, inclusion of an axial dispersion term in the model can also directly lower the predicted hot-spot temperature. Secondly, it would seem that the beds of low (D,/d,), used for the conduct of reactions with large heat effects,

Fixed bed catalytic reactor exhibit higher rates of radial transfer than do the wider beds used in many heat transfer studies[t6,17,19]. This phenomenon may be related to the “shoulder” or “hump” observed in measured radial temperature profiles-that is, a zone of low temperature gradient and thus high k,,. We conjecture that this zone has a characteristic size related to packing diameter, and thus occupies a larger fraction of the bed diameter for low values of DJd,, which possibly explains anomalously high values of k.,. Radial heat transfer is then a function of not only Re, but tn. It is now clear why the conventional model can often be made to fit observed data, though it cannot predict them a priori: the fitting process usually involves inthere exist creasing the values of k., or h,. However, cases where attempts to fit the conventional model fail, for example in SO2 oxidation[20], and in cases of multiple steady states (see below). We must then turn to alternative models, though we recognise that in many cases the conventional model captures many of the important phenomena, and possesses such flexibility in its structure, that it is capable of being fitted to observations sufficiently well to allow intelligent yet guarded interpolation. 4. ALTEXINATTVEI MODELS

A. Axially dispersed plug flow Two considerations prompt one to entertain this model. First, Dixon et al. [ 181 reveal that an axial dispersion term is required for proper interpretation of heat Young and transfer experiments. Secondly, Finlayson[20] teach that long-accepted conclusions on the unimportance of axial dispersion do not simply translate from the adiabatic and isothermal cases to the non-adiabatic non-isothermal instances. The real importance of axial dispersion of heat, as opposed tomass, that Pe, < is heightened by the discovery 2 [18,20,22,23]. Further they [201 fitted an axially dispersed two-dimensional model with physically reasonable parameters to an SO* oxidation reactor which could not be successfully modelled by the conventional model. But it was found that although an axial dispersion term was required, the model predictions were rather insensitive to the value of E, used; a result consistent with Dixon el al.[18] whose estimates of %, had wide marginal confidence intervals. One difficulty with the axially dispersed plug flow model is the uncertainty regarding suitable boundary conditions at bed entrance and exit. The Danckwerts conditions[l], which have been plausibly generalised to two dimensions[20], are widely accepted as suitable for the case where dispersion is caused by molecular mechanisms. But their applicability is in some doubt for packed bed reactors at industrial flowrates where fluid mechanical dispersion dominates so that upstream transport of material relative to a fixed plane is negligible [24]. Guun [25] and Tichacek [26] have argued that to avoid the implication of upstream transport across the exit plane, the Danckwerts condition

z=z,;

modelling

177

should be replaced by Z+m, C+ C, for the case where there is an open tube downstream of the packed bed. Presumably similar reasoning may have led to Kalthoff and Vortmeyer’s [27] suggestion $$=O for the exit plane of an inert-packed section downstream of a packed reaction section. This is an approximate way of forbidding dispersion, in this case of heat, across the exit plane, by deleting the dispersion term from the partial differential equation at that position. In their heat transfer study, Dixon et al.[18] found strong experimental support for a Gunn condition Z-m, rather

than a Danckwerts

$$=o.

(6)

has argued, on similar grounds, the Danckwerts condition

Z=O;

uC(-m)-&JO+)= C(O’)

should

(5)

condition

z=zo; Wicker281 bed entrance

T+T,

be replaced Z=O;

that at the

-Da%

= C(O-3

by

uC(-m)-uC(O+)= c(c)+)

-E,@%

= C(O_)

(7)

where B is the molecular diffusivity and x is the inverse of bed tortuosity. He further shows that for Re, > 1, these conditions are well approximated by C(0’)

= C(G)

= C( - 00).

The heat transfer study of Dixon et al. casts little light on the choice of inlet b.c. Although their inlet b.c. is reducible to a Danckwerts set, they used no alternative for comparison and in any event the measured temperatures were remote from the bed entrance and thus, perhaps, insufficiently sensitive to choice of b.c. to permit discrimination. Further, the effective solid phase axial conductivity will presumably be large enough to mean that 9 in eqn (7) cannot be replaced by k,, the fluid thermal conductivity. There is certainly ample evidence that heat, unlike material, can be transported upstream. Perhaps a useful estimate for a value of k,. to be used in a Wicke entrance b.c. upC,T(

- m) - upC,T(O+)

= - k.-,,$

(8)

l+C

==a

is to be had by assuming

the effective

solid phase

con-

178

W.

R.

PATERSON

ductivity to be isotropic and inferring a value of kaO from the various correlations available for the “stagnant” term in the bed effective radial conductivity (e_g_[29]): or by finding a correlation for the “stagnant” term in the bed effective axial conductivity (e.g. [30]). Clearly some issues are not resolved for this model. However the mathematical difficulties of a b.c. at infinity are surmountable[31], and the apparent parametric insensitivity to E, suggests that highly accurate values of it are not required. B. Cell models Models such as that of Deans and Ldpidus[32] allow upstream transport of neither material nor heat. However, a modification known as the backflow cell upstream transport of heat alone, model incorporates and is thus consistent with experimentally observed temperature-related multiple steady state phenomena[33], as is the axially dispersed two dimensional model. Moreover, by its preclusion of upstream material transport, the backflow cell model can also account for isothermal multiple steady state phenomena inexplicable with the axially dispersed plug flow mode1[34], although preclusion of this upstream material transport is not a necessary condition for isothermal multiple steady state phenomena. The necessary extension of the backflow cell model to account for ‘radial dispersion of heat and material would seem to be straightforward. C. Crossflow model Deficiencies in the axially dispersed two-dimensional model have been known for some time[35]. It predicts an infinitely fast signal propagation and incorrectly[24] implies upstream transport of material. It has been shown that no linear continuous second order pde model can satisfy these and other reasonable requirements [36], but the non-Fickian crossflow model which describes both axial and radial dispersion of material[37] does so. Extension of this model to describe heat transfer would presumably require introduction of an effective solid phase conductivity to permit upstream transport of heat and faster radial transport of heat than of material. D. Velocity profile models A number of workers have recently investigated the advantages of incorporating into their models the velocity profiles which are known to exist in packed beds as a consequence of the radial variation of voidage, particularly near the wall. Schlunder [38] and Martin [39] view the bed as consisting of two channels: a cylindrical core of voidage E = 0.4 and an annular zone near the wall with E = 0.5. They have thus explained the anomalously low particle-fluid heat and mass transfer coefficients obserexplved in packed beds at very low Rep, alternatively icable by use of an axially-dispersed one dimensional model[40]. The two channel model has also been used by Carbonell to analyse the apparent axial dispersion of material in residence time distribution studies[41]. Chang [42] has used a voidage, and hence velocity, profile which is a continuous function of the radial co-ordinate

and J. J. CARBERRY to show that this non-Fickian model yields conversion predictions for an isothermal irreversible first order reaction in close accord with those from an axiallydispersed plug flow model. However, these models not only fail to yield a satisfactory description of the rather unimportant radial dispersion of material (which the axially dispersed two-dimensional model does succeed with[43]), but are not written to account for radial and axial dispersion of heat. This omission is rectified by Kalthoff and Vortmeyer [27], who include appropriate terms and thus obtain a satisfactory model of the oxidation of ethane over supported palladium in a wall-cooled bed. Ahmed and Fahien[44] combine radial heat transport with a velocity profile representation to obtain a successful model of the same SO, oxidation data as modelled by Young and Finlayson. This model is based on faulty voidage measurements [45], so that in detail it is erroneous. However, it approximates to a two-channel model and so its predictions may be little affected by such detailed error. It has been claimed[46] that this model yields an adequate description of the heat transfer data of De Wasch and Froment [ 151.

DISCUSSION

The most general model reviewed above is that of Kalthoff and VortmeyerC271 who considered the testing case of multiple steady states in a short bed (72mmx 80 mm) at low Re,( = 7). The SO2 oxidation data considered by Young and Finlayson and by Ahmed and Fahien came from a bed 1.50 mm x 52 mm at Re, = 50. For such conditions, it seems that one must incorporate into the model axial dispersion of heat and/or velocity profiles. But many industrial reactors are much longer and narrower and operate at much higher Re, to obtain high heat transfer rates. What then? The axial dispersion of material will usually be negligible. The importance of axial dispersion of heat should be assessed-perhaps using Mears’ criteria[47], although we must caution that they are derived using generalised Danckwerts b.c.‘s. Alternatively, one might first use a one- or two-dimensional plug flow model and then assess the importance of the omitted axial dispersion term[l6,46]. The importance of velocity profiles under such circumstances needs study. The values of heat transfer parameters in the conventional and the axially dispersed two-dimensional models may be a source of difficulty. We recommend consideration of the equations of Cresswell et al. 119,491 for values of k,.. Values of h, pose greater problems and clearly call for further research. In the meantime, the only correlation available which derives from an analysis accounting for the length dependence of parameters in the conventional model is that of Dixon et al.[18], for air in a bed of ceramic spheres with 5 4 DJd, < 12, and 35 i Rep < 535. Based upon their results and the definition of the wall Biot number (eqn 4), the wall Stanton number functionality is secured: Bi

y

=~=,St k CI

.pe m =5.73(2m)“= I

ReFmZ

.

(9)

Fixed bed catalytic reactor modeiling From a fit to Dixon’s data[l7] &=O.ll+Y. r

P

axial aspect ratio, &/dP Nusselt number at the wall (based on particle diameter) axial Peclet number for mass radial Peclet number for mass axial Peclet number for, heat radial Peclet number for heat Prandtl number heat generation dimensionless radius R/R0 radial position global reaction rate particle Reynolds number Sherwood number at wall Stanton number at wall reduced temperature, T/T,, temperature superficial fluid velocity axial position reactor length

(10)

Therefore St

w

= 5.73d2m(O. v

11+ 20.64/Re )

(11)

or in terms of the Nusselt number Nu, = h,d, = 5.73aPr(O. rc,

11 Re, + 20.64) . Re:%’

(12)

Preliminary analysis of recent data from this laboratory for mass transfer at the wall of a packed bed support eqn (12) insofar as

179

Greek symbols

p c x 0

fluid density void fraction inverse of bed tortuosity contact time, &/u

Subscn’pts SUMMARY

Two recent successful models of packed bed reactors-those of Kalthoff and Vortmeyer and of Ahmed and Fahien-diff er substantially from the conventional (plug flow two-dimensional) model. It may be that at high Re, the conventional model will perform adequately, but if so, fidelity of prediction to reality will rest critically upon sound values of k., and It,. Proper correlations of both parameters emerge only when thermal axial dispersion effects are anticipated in the assessment of these radial transport coefficients; and then perhaps invoked in the modelling of the chemically active fixed bed. The heretofore disappointing performance of the conventional model for a p&n’ prediction has been caused by its essential shortcomings and the imprecision of the parameter values used. Acknowledgement-J. J. C. gratefully acknowledges the support of the Science and Engineering Research Council of the United Kingdom. NOTATION

aspectratio,ZOIRO

Biot number, eqn (4) species concentration species concentration at chemical equilibrium fluid heat capacity particle diameter axial dispersion coefficient, mass radial dispersion coefficient, mass tube diameter molecular diffusivity reduced concentration C/C0 wall heat transfer coefficient axial thermal conductivity axial thermal conductivity, at the bed entrance plane effective radii conductivity fluid phase conductivity axial dispersion coefficient, heat radial dispersion coe5cient, heat wall-fluid mass transfer coefficient radial aspect ratio, Rold,

a

axial r radial 0 inlet w wall

REFERENCES

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180

W. R. PATERSON and J. J. CAKHOKKY

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APPENDIX Since WCplace some weight on the analysis of Dixon et a1.[18], a few words are in order regarding criticisms of it. Dixon et al. observed a length dependence on the heat transfer parameters, h, and k.,, of the conventional model and found that it could be eliminated by adopting the axially dispersed two-dimensional model with appropriate b.c.‘s. The alternative suggestion[50] that the dependence is caused by developing velocity profiles seems implausible, since the bed velocity profile is presumably established in the lengthy packed inlet section of Dixon’s apparatus, excepting only any effect of the temperature field on velocity. Similarly, to attribute the dependence to developing temperature profiles seems to beg the question-an adequate model should represent the axial and radial development of the temperatur~tield. Certainly the analysis of Dixon et al. yields values of Pe, in agreement with those of Votruba er al.[22], Gunn et al.[2, 251 and Wicke et a[.[231 and consistent with the Pe. data of Hsiang and Haynes [52] for beds of low D/d,. As Lerou and Froment remark[53], the physical meaning of the effective axial thermal dispersion coefficient is not established by these results. We speculate that it may originate in part in the existence of (well-developed) radial velocity profiles in packed beds.