Hydrodynamics and heat transfer in packed beds with liquid upflow

Hydrodynamics and heat transfer in packed beds with liquid upflow

Chemical Engineering and Processing, Hydrodynamics A. S. Lamine*, 385 and heat transfer in packed beds with liquid upflow M. T. Colli Serrano an...

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Chemical Engineering and Processing,

Hydrodynamics A. S. Lamine*,


and heat transfer

in packed

beds with liquid upflow

M. T. Colli Serrano and G. Wild

Lahoratoire dey Srienre.~ du G&e F-54001 Nancy Cedex (France) (Received

31 (1992) 385-394





?Vanc,v, B.P. 451, I, rue Grandzrille,

June 22, 1992)

Abstract The authors investigate the radial heat transfer in packed columns with liquid flow and compare their results with those of studies carried out with gas flow. The heat transfer experiments are performed in a packed bed heated electrically through the wall with upward flow of water and aqueous solutions of ethylene glycol. The particle diameter has been varied from 1 to 6 mm, and the reactor inner diameter is either 50 or 100 mm. Radial and axial temperature profiles are measured inside the packed bed. The heat transfer parameters of the packed bed are obtained by fitting the parameters of a two-dimensional, homogeneous model to the measured profiles: the radial effective thermal conductivity of the packed bed (A,), and the wall heat transfer coefficient. The conductivity of the packed bed incresaes linearly with the fluid velocity hut when the ratio of A, to the thermal conductivity of the fluid is plotted against the fluid Peclet number (Pe, = Re, Pr), the slope is found to he lower for liquid than predicted for gas. This different hehaviour is explained by differences in the flow regimes. The cxpcriments in gas flow are usually carried out in turbulent flow, whereas the experiments in liquid flow are in an intermediate regime (between laminar and turbulent). A new correlation is proposed for this regime, depending on both Prandtl and Reynolds number of the fluid. Results on heat transfer coefficient at the wall and on wall channelling are also presented.

Synopsis L’ttude Porte sur le transfert de chaleur duns des colonnes ci garnissage, avec kcoulement de liquide; les rPsultats obtenus sont comparb ci ceux des etudes portant sur I’tcoulement de gaz. Le rgacteur (Fig. 1) est chau#k en paroi par des rksistunces klectriques, son diam&re est 50 ou 100 mm, le liquide utilisk est de l’euu ou des solutions aqueuses d’&hyl&e glycol, le garnissuge consiste en des billes de verre de 1 ci 6mm de diam;tre. Des sondes conductimktriques concentriques, placdes en sortie de rkacteur, permettent @ar injection de traceur) de mettre en evidence un kcoulement prkfkrentiel ci la paroi du reacteur dans le cas de t&s fuibles d&bits de liquide. Cet efSet croit avec la taille des particubs et avec la densitk de jlux imposke d la puroi (Fig. 4). Les rtkultats obtenus sont comparb avec des modtYes d’Ccoulement de la littkruture. Les mesures des paramktres thermiques sont eflectukes dans des conditions telles que cet koulement prifkrentiel est nkgligeable. On mesure les profils de temperature 6 l’int.krieur du lit (en [email protected] positions radiales et ci la paroi) et on en dkduit les paramktres thermiques (conductivitd radiale Lquivalente du lit, A, et coeficient de transfert de chuleur


to whom




be addressed.

ri la paroi, clw) par [email protected] des proJils mesurks avec des profls calrulks par modtYe homog&ze (eqns. (I)- (4)). La conductivitk thermique du lit croit linkairement avec le d&bit liquide (Fig. 5), en accord avec les etudes antkrieures me&es aver du gaz, mais elle est plus faihle que ne le pkdisent les correlations proposkes dans les dites Ptudes, par exemple le modele de Schliinder (eqn. (6)) qui fait l’h_vpotht?e d’un mtlange parfait du fruide dans les pores. L’explication avancke est l’influence de l’hydrodynamique sur le transfert de chaleur. La plupart des essais me&s avec du gaz sont en rkgime turbulent, alors qu’avec le liquide, le regime est intermtdiaire (entre laminaire et turbulent). Les modkles prPdisant que la conductivitt! thermique du lit croft liniairement avec le nombre de P&let (Pe, = Re, Pr) ne conviennent plus si le rkgime hydrodynamique dtipend du nombre de Reynolds. Des comparuisons avec des rksultats obtenus par d’autres uuteurs en transfert de mat&e (Fig. 9) permettent d’tlurgir lu discussion. Une nouvelle corrklation est proposPe pour l’estimation de la conductivitk thermique (eqn. (7)), qui dkpend ri la fois du nombre de Reynolds et du nombre de Prandtl du liquide, vuluble en tkoulement laminaire inertiel, et vPr$Pe pour 5 < Re, < 120 et 4 < Pr < 22.5 (Fig. 8). La rksistunce au transfert d la paroi (1 la,) est environ igale au rupport du diambtre de particule cj la conductivit& thermique du lit (Fig. 11).


1992 ~ Elsevier Sequoia. All rights reserved



Tubular packed bed reactors are widely used in the chemical and petrochemical industry, especially for solid-catalysed heterogeneous reactions. If the reaction is exothermic the heat of reaction has to be removed by cooling the walls, in order to avoid as much as possible hot spots, production of undesired species or catalyst deactivation. Thus the prediction of heat transfer parameters such as thermal conductivity of the bed and wall heat transfer resistance by accurate correlations is necessary in order to be able to predict temperature profiles in the reactor. Heat transfer in packed beds has been extensively analysed in the literature (for recent reviews on this subject, see Lemcoff et al. [ 1] or Tsotsas, [2]), but so far most of the research work in this field has been carried out with gas flow. Therefore, it is interesting to investigate whether the correlations established from gas flow are still valid in liquid flow and to discuss the possible differences of behaviour between gas and liquid Bow. Another important aspect should be mentioned: liquid-solid biocatalytic reactions feature in some cases of packed bed reactors with delicate temperature control problems. More frequent are the gas-liquid-solid packed bed reactors. An accurate description of heat transfer in such reactors requires the knowledge of heat transfer in liquid-solid packed beds. We present here results obtained with liquid flow, the heat parameters are deduced by fitting the computed temperature profiles to experimental ones, the former ones being calculated with a two-parameter homogeneous model which assumes plug flow. Previous investigations on air flow show that the radial effective thermal conductivity (A,) in a packed bed only depends on the Peclet number of the fluid Pe, (linearly) (see e.g. Bauer [3]) and on the tube-to-particle diameter ratio, while the wall Biot number (defined as ~l,d//i,, CL, being the heat transfer coefficient at the wall of the reactor) only depends on the particulate Reynolds number. Only experiments carried out with different fluids can allow discussion of whether more parameters (such as fluid properties or hydrodynamic regimes. .) are necessary to describe the heat transfer in a packed bed.


Figure 1 shows the experimental equipment. It mainly consists of a 100 mm i.d. packed column of 2.7 m height. The column is made of glass, except for the heating section. The liquid is supplied at the bottom of the column. In most cases the liquid used is water; in some recent experiments we use ethylene glycol/water mixtures (40

thermocouples 3

I!! ,

Fig. 1. Experiment



al equipment.

and 60% ETG). The liquid first flows through a calming section of one meter length which allows a uniform fluid temperature profile in the bed at the entrance of the heating section; the latter consists of a brass tube electrically heated with a constant heat flux; it is one meter long and is made of four thermally isolated pieces (separated by insulating joints) in order to limit axial heat conduction in the wall. The superficial velocity of the liquid is in the range of 0.02-1.5 cm SC’. The packing is made of glass spheres (diameter 1, 2, 4 or 6 mm). Some results have also been obtained in a smaller column (50 mm id.) with similar packings and liquid velocities. In both columns temperature profiles in the bed and wall temperatures were measured at different levels in the bed. In the 100 mm i.d. column 20 thermocouples (of type K and protected by a stainless steel tube of 1 mm o.d.) are placed at different axial and radial positions in the bed and at the wall (two axial and ten radial positions). In the 50 mm i.d. column, 18 thermocouples (of the same kind) are placed at different axial and radial positions in the bed and at the wall (eight axial and four radial positions). The wall thermocouples are soldered into holes (drilled in the tube wall) and the inner surface of the tube has been polished thereafter. The thermocouples in the bed are set in a comb, the support tube is of 6 mm o.d. The measurements are made in stationary regime and each temperature profile is measured ten consecutive times and averaged.


The hydrodynamics are investigated by using a salt tracer technique. A small amount of a 1M KC1 solution is injected at the bottom of the column. Conductimetric probes measure the transient mean salt concentration at two cross-sections of the column (inlet and outlet of the heating section). At the outlet the conductimetric probe consists of five concentric probes allowing eventually for the determination of average velocities at different values of the radius. Hydrodynamics

and heat transfer models

Fig. 2. Temperature profiles (calculated and measured).

Hydrodynamics : plug Jlow model with axial dispersion The response curve (mean outlet concentration) is well represented by the plug flow model with axial dispersion. The parameters of the model, the mean residence time of the liquid and the axial dispersion Peclet number, are obtained by minimization of the difference between the calculated and the experimental response curves (using the classical imperfect pulse technique, see e.g. Villermaux [4]).

- cq lindrical

Heat transfer: two-parameter homogeneous model Heat transfer in a packed bed reactor can be described by homogeneous or hcterogencous models. The heterogeneous model considers separately the temperature of each phase as well as the heat transfer between the different phases but, for practical purposes, leads to a far too complicated description of the phenomena. Tn the case of slow, not highly exothermic reactions the difference between these temperatures is low enough to allow the use of a homogeneous model, which yields much simpler equations. The results are therefore represented by using a twoparameter homogeneous model, assuming plug flow of fluid and considering the packed bed as an equivalent homogeneous medium; the parameters are the bed effective radial conductivity A, and the wall heat transfer coefficient a,. This parameter takes into account a supplementary heat transfer resistance at the wall, due to a lower radial dispersion of the fluid in the region near the wall. For a cylindrical packed bed, in steady state condition, the heat balance gives:


The axial second derivative is assumed to be very small compared to the other terms. This assumption has been checked by several authors (e.g. Dixon 1.51). The boundary conditions are: _ constant heat flux at the wall: for r = R

3T := 0 ar


for r = 0.

The solution of this equation, given by Carslaw and Jaeger [6] is then used to fit the computed temperature profiles to the experimental ones, and to determine the heat transfer parameters A, and c(,. The heat transfer coefficient verifies: (4)

T, being measured and T,. = R being extrapolated from the computed profiles. A detailed description of this classical fitting technique may be found in Gutsche [7]. As shown by Fig. 2, this procedure leads to a satisfactory fitting of the temperature profiles.

Hydrodynamics Plug Jo w assumption The flow in the reactor is represented by the plug flow model with axial dispersion, but the heat transfer is described by a homogeneous model assuming plug flow. The axial dispersion coefficient, D, has thus to be small enough, which is considered to be verified when the axial dispersion Peclet number PE, = u,,H/D, is higher than 100 (H being the height of the measuring section). Figure 3 shows PE, versus liquid flow rate. PE, > 100 and the plug flow For L > 1 kg mm2 s-l, assumption is justified, but for L < 1 kg mm2 s - ‘, axial dispersion can be important. Case of Iow flow rates: wall channelling Due to higher porosity close to the wall, the local velocity is increased. This effect is furthermore increased by a lower viscosity near the heated wall. Local measurements of residence time distribution (at different radial positions in the bed) have been carried out in the 100 mm i.d. packed bed by means of tracer experiments and five concentric conductimetric probes. Such measurements implicitly neglect the radial mixing of the



I 2

_ -1 I

.d.?rnrn q *-4mm . cl-mm





Fig. 3. Axial dispersion





to obtain an analytical velocity profile (for isothermal flow). Table 1 shows a comparison of measured wall and core velocities with calculated velocities by the models of Martin and Vortmeyer. The wall superficial velocity (u,) is measured in a 1 cm large zone, close to the wall, and the core superficial velocity (u,) in a 1 cm diameter zone at the axis of the reactor. These local velocities are made dimensionless by dividing them by the superficial velocity (uz = u,/u,, and u: = u,/u,,). The local velocities deduced from both models have been averaged for these two intervals and a simple modification has been brought to the model of Martin by using also a twostep viscosity profile in order to take into account the effect of heating. The experimental results are in good agreement with these studies: the channelling is more important at low velocities (high flow rates tend to render the velocity profiles more uniform), at larger temperature difference and bigger packing. Calculations by the model of Martin are closer to experimental results because they take into account the wall-to-core temperature difference, which has a great influence on channelling. But even these values are systematically higher than the measured ones, this is not surprising since the measuring technique used here inevitably neglects radial mixing of the tracer between the inlet and the outlet probes; this radial mixing flattens the measured velocity profiles. Systematic measurements have been carried out with small wall-to-core temperature differences in order to limit this channelling phenomenon and thus to justify the plug flow assumption.


L(kglm 2s,


versus liquid flow rate.

liquid and thus underestimate the real non-uniformity of the velocity. At low liquid velocities (L < 1 kg me2 s-‘, i.e. u,, d 1 mm s- ‘), these measurements confirm the existence of a velocity profile. No non-uniformity could be detected for higher liquid flow rates. This channelling, which has been detected in isothermal flow, for the biggest particles (4 and 6 mm) is strongly enhanced by wall heat flux (Fig. 4a). The channelling also increases with the particle size (Fig. 4b). Several theoretical studies on flow profiles in packed beds allow prediction of these velocity profiles. Martin [8] applied the Ergun equation to a two-zone packed bed, assuming a higher porosity in the wall region (its thickness was assumed to be one particle radius) and deduced from it a two-step velocity profile. Vortmeyer and Schuster [9] solved the Brinkman equation in order

L-l kglm% Tw-Tc=OK





























IL-O,5 kg/m%]

Fig. 4. Detection

of channelling

by tracer


a veraged

local liquid velocities.


389 TABLE 1. Comparison 191

of measured

L(kgm-’ 2 4 4 4

wall and core velocities


0.5 0.5 1


with calculated


by the models

AT( “C)



25 25 0 35

1.2 1.4 1.1 1.3

1.47 I .69 1.43 1.67

Heat transfer As shown earlier, the radial thermal conductivity of the bed and the wall heat transfer coefficient are obtained by fitting the computed temperature profiles (deduced from the two-parameter homogeneous model, assuming plug flow) to the experimental profiles. The temperature difference between the wall and the core of the bed is kept around 10 “C, in order to avoid preferential flow near the wall.

Thermal conductivity of the bed

0.‘7 0.4 0.9 0.6

0.73 0.60 0.75 0.61

of Martin


[8] and of Vortmeyer

u:u,*(Vort. 159 1.89 1.80 1.80


0.70 0.58 0.62 0.62

0.5mms-‘, corresponding to Re, < 5) were not precise and probably too far from plug flow to consider the discrepancy as significant (see section ‘Case of low flow rates: wall channelling’ on channelling effect). Former measurements obtained in the smaller column (50 mm i.d., see Gutsche [7]) show similar trends but the slopes (A,/& versus Re,) are always somewhat higher than in the larger column. This difference can be explained by a lower number of thermocouples in the smaller column (three thermocouples in the core of the bed instead of nine) inducing a less precise measurement. These results are nevertheless in satisfactory agreement with later results in the larger column.

Qualitative description The radial thermal conductivity of the packed bed with fluid flow is usually described in the following form [3]: n, = i,, + 1,


where I,, is the effective bed conductivity at zero flow rate and 1, the contribution of convective mixing. In case of liquids, i,, is usually small compared to /2, (which would no longer hold true for gases). In Fig. 5, the fitted values of the radial thermal conductivity are plotted against the Reynolds number (for different packings, in the 100 mm i.d. column). For single phase flow, A, appears to be a linear function of fluid velocity. The slope of A,/,$ versus Re, is slightly increasing with column to particle diameter ratio. The measurements at low velocity (u,, = 0.15 and





Fig. 5. Radial thermal ings (D = 100 mm).

60 conductivity


Comparison with dzflerent correlations and models Table 2 presents different correlations of the literature for bed conductivity with either gas or liquid flow. Since these models do not depend on the same non-dimensional numbers, for a general comparison of these correlations, one has to distinguish gas and liquid flow. Figure 6(a) shows a rather good agreement of the different correlations for gas flow (even with the correlations obtained from liquid flow studies). On the other hand, most of the correlations issued from gas flow studies are not valid for liquid flow as shown on Fig. 6(b), because they are not dimensionless. A comparison of our experimental results with these correlations gives the following: the results are about 30% lower than predicted by Yagi [ IO]; the other works concern a different range of Reynolds number. However, the experimental thermal conductivities have been compared with the model of Bauer [3], which is derived from a mixing cell model (Schlundcr [ 161) and should not depend on the fluid. According to this model, neighbouring parts of the flow around the particles meet in the void, mix totally there and then separate again. This model leads to a radial Peclet number of dispersion equal to eight in an infinite bed (PE,, =8). Bauer and Schliinder [14] proposed a correction factor taking into account the ratio tube-to-particle diameter (k = 2 - (1 - 2d/D)*) and by a mixing length depending on the shape of the particles (X = 1.15d for spheres).

100 Red120

of the bed for different


A, _ J”,, ,& -. 1,. + K Pe,

with !$

= 8{2 - (1 - 2d/D)2)}



2. Correlations:



of packed






Red Pr


Yagi [IO]


glass, steel sph.

water downflow

7-125 3-5


cyl. catalysts


50-900 zz 0.7




0.18-200 z4.5



= +

+ 0.09 Pe, ‘f

De Wash, Froment [ 1 l]

A, = i”,, + ~~


Re,( 1 par.)

1 + 120(d/D)l

0.0025 n, = 1,” + 1 + 46(d,D)2


Bauer, Schliinder

[ 141

Chen et al. [ 151


2n = 0.6296 & &

Re,” 79


&0 ~




’ 8(2 ~ ( 1~ 2d/D)‘) Ped(sph’)



A, = 0.199 + O.OlD/d + fA = L E., + 8(2A,

This work


1 Pc, 15.39 + D/d > 1 + 20/Pe,

[ 121

Li et al. [13]





1 + 14.5(d/D)2

1.15 ( 1 - 2d/D)‘)

Pr0.6 Re













I n

0 + l

Yagi-L DeWash-G Lerkzik-L m I.i-G





-x- This work-l,




Fig. 6. Comparison of different correlations; (a): gas, (b): liquid (L from liquid flow study and G from gas flow).

downflow spheres


500-3600 % 0.7



200 1800 z 0.7

spher. catalyst


100-500 4 0.7


glass spheres

water aq. sol. of glyc. upflow

I-120 4-22.5


10 60

In the case of gas flow, this model has been successfully tested with different experimental studies, in mass transfer as well as in heat transfer. Results of different authors have been compared by Bauer and Schliinder [ 161. Rcccnt results of heat transfer in packed bed with gas flow are presented by Nilles and Martin [17]. Qualitatively this model agrees well with our experimental data obtained with liquid flow: linear increase and slope increasing with D/d ratio, but the slope is found to be two times lower in liquid flow than in gas flow. In Bauer and Schliinder’s model, the conductivity of the bed only depends on two parameters: Pe, and d/D. This simple description of heat transfer can only be successful when the hydrodynamic regime is independent of Re,, which is true first at very low Reynolds number (in Darcy regime, or creeping flow, but in this case, molecular conduction is important) and secondly, in highly turbulent flow, which yields complete mixing. This second condition is usually satisfied in gas flow measurements where the Reynolds number ranges usually from 100 to 10 000 (whereas in liquid flow, it is from 1 to 120). Therefore it seems necessary to take the hydrodynamic regime into account, in order to model heat transfer in packed beds with liquid flow.


Influence of the flow regime In the case of water flow, at a given flow rate, an increase of temperature from 20 “C to 60 “C would slightly decrease the Peclet number (by 8%), whereas the Reynolds number will increase by a factor higher than two. At medium Reynolds number (10-100) a strong change in viscosity induces a strong change in mixing and a subsequent change in bed conductivity, while Peclet number remains nearly constant. Experiments also prove that, at a given flow rate and at a given wall-to-axis temperature difference, the conductivity increases with liquid temperature. We therefore suggest, in the same manner as other heat transfer correlations (such as forced convection in tubes where the dependence of heat transfer coefficient on Reynolds and Prandtl number depends on flow regime) to distinguish diffcrcnt flow regimes. The model of Schliinder is an upper limit to dispersion, as it supposes perfect mixing, it is thus valid for high Reynolds number, often reached in gas flow, but not in liquid flow. For low and intermediate Reynolds number other models or correlations have to be proposed. In porous media, the transition from laminar to turbulent regime is quite progressive. According to Dybbs and Edwards [ 181 four different regimes can be considered, depending on the Reynolds number: - in the region of low Reynolds numbers, the flow through the packed bed is first laminar (with no mixing: Darcy regime or creeping flow, up to Re, x l-7), then becomes inertial laminar (with increasing mixing but not complete mixing, up to Red z 100) and unsteady laminar (up to Re, z 200); _ in the region of large Reynolds numbers (over regime becomes turbulent Re ,, z 200) the hydrodynamic and the mixing is complete. According to this description the present results in liquid flow fall in the range of inertial laminar flow. More results in the intermediate range of Reynolds numbers would be necessary in order to verify whether

these hydrodynamic transitions are relevant in the description of heat transfer. Since the results plotted in a A,/& versus Pe, form present a lower slope than that expected at high Reynolds (according to the mixing cell model and gas flow results), this suggests that in an intermediate region where the regime is getting more and more turbulent, the slope is increasing and, in completely turbulent flow, is becoming the same as in gas flow. We cannot check whether, with increasing turbulence one reaches with liquids the complete mixing limit, but by plotting PE,( =p, Cp,u,,d/n,) versus Re, (Fig. 7) we observe maxima which correspond to this increase of the slope in the intermediate regime. Correlation of our own data Our experimental investigation also suggests a linear. increase of bed conductivity with fluid flow rate, we therefore propose a correlation in the form (valid in the range of 5 < Re, < 120 and 4 < Pr < 22.5), the Prandtl number exponant being chosen by fitting: A, &0 7 = 7 + K(d/D)


Pr”-” Re,


with l/K(d/D) = 8( 2 - ( 1 - 2d/D)‘), according to Bauer and Schliinder. This correlation represents our experimental data + 10% in the 100 mm column (as shown on Fig. 8) and f 30% in the smaller column. Some recent results in the 100 mm column with aqueous solution of glycol, allowing to vary the Prandtl number by a factor of five (Pr = 15 or 22.5 instead of 4-6 for water), are in quite good agreement with this correlation ( + 10%) and confirm the necessity of taking fluid properties into account (by means of the Prandtl number) when modelling heat transfer in liquid flow. Comparison to mass transfer results These results have also been compared to mass transfer results. Several experimental studies have shown higher radial dispersion coefficient for gases than for



10 PO









120 0 0

Fig. 7. Radial heat transfer Peclet number versus Reynolds (with some fitting curves showing the trend: decrease of PE, after a maximum value).






Fig. 8. Parity plot of correlation (7): bed radial conductivity liquid flow, valid for 5 < Rc, < 120 and 4 < Pr < 22.5.



number, one can discuss the shape of PE, versus Pe, (or Re,) curves (see Table 3): - If Recrit_hydroPr or Recrit_b,,droSc is larger than Pe,,lt.t,annfer the convective transport becomes dominating before the mixing is complete, k is expected to increase (towards its maximum value in turbulent regime) and PE, versus Pe, to have a maximum. This is the case for heat transfer with liquid flow and for mass transfer in both cases. The experimental results confirm this trend in heat transfer (Fig. 7) and in mass transfer (Fig. 9 and Tsotsas [2]). _ If Recr,t_hydroPr or Recrit_hydroScis smaller or equal to PeC,,t-t,ansre,the mixing is already complete when the convective transport becomes dominating, therefore, k is expected to be constant, and PE, versus Pea to have no maximum (this is the case for heat transfer in gas flow, results of Nilles [ 171).


10 1 Heat

transfer (present work)



0 001



Fig. 9. Comparison of present heat transfer data, collected by Tsotsas [2].



data to mass transfer

liquids (de Ligny, [20] and Dixon, [ 191). Tsotsas [2] collected and discussed a large number of data literature; the range of our experimental heat transfer data has been added to these mass transfer data and the agreement is satisfactory (Fig. 9). When describing the heat transfer mechanism in packed beds, two regions have to be distinguished as shown by Tsotsas [2,21] depending on the Peclet number: _ in the region of low Pcclct numbers A, z A,,,, the molecular conduction is dominating; _ in the region of high Peclet numbers A, z A,, the heat transfer is mainly due to the mixing of the fluid in the voids of the bed; the convective transport is dominating. The critical value of Peclet number for transition is defined for /I,,//z, = 2,-/d, = K Pe, (bearing in mind that this transition is gradual in a porous medium). According to Dybbs and Edwards [IS], the critical Reynolds number for incipient turbulent flow is of the order of 200. One can remark that for Reynolds numbers above 200 the PE, number characteristic for mass transfer becomes constant (Tsotsas [2]). Now, by comparing the values of the critical Peclet number (determining heat transfer mechanism) and the critical Reynolds number (determining hydrodynamics), multiplied by either the Prandtl or the Schmidt TABLE

3. Heat and mass transfer


I,, li,


air water

z 15 N 1.5

zo.12 N 0.06

Mass transfer

a,, /a f



5 0.25 (Tsotsas,




Fig. 10. Schematic



of the different




Peclet numbers

Heat transfer



As shown in Fig. 10, we propose to distinguish also different flow regimes for heat transfer description. We propose a correlation for the inertial laminar flow regime in the form A,/,$ =f(Re,, Pr, d/D); the correlation of Bauer, in the form A,/& =f(Pe,, d/D) (and others) is valid for the turbulent flow regime. More data in the intermediate regime would be necessary to be able to predict the bed conductivity in this case, but it must he kept in mind that, for practical purposes, in liquid flow the usual range of flow rates corresponds to the inertial laminar flow.

-1-4 (Tsotsas,

Pe,,,, z 125 z 25 SC


-1 z 1000

Pr 5 0.7 =5

Pe,,,JSc Zl 5 0.001

Pe,,,, IPr % 180 -5

393 to-



peiiiF to

Fig. 1 I. Biot numbers

PB = Fts P, 100

versus Reynolds

1 000


Wall heat transfer coefJicient The wall heat transfer coefficient CI, is deduced from a temperature difference which is varying from negligible values to a few degrees, by q,/(T,., - TrF J = a,.,( T, being measured, and T, = R being extrapolated from the computer profiles). The determination of this parameter in our working conditions is thus not precise at all, therefore, despite its interest, only broad results will be presented here. With liquid flow the wall heat transfer coefficient CI, is found to be of the order of magnitude of A,/d which means that the Biot number (Bi = a,d/A,) is about equal to one; this is in good agreement with a simple physical reasoning taking the bed conductivity in a wall region of one particle diameter width to be half of the conductivity in the bulk of the bed. The relative importance of wall heat resistance is decreasing with decreasing packing diameter. The Biot number also decreases slightly with Reynolds number (Re, = p,u,,d/p,). The results are in satisfactory agreement with the correlation proposed by Dixon and Cresswell [22] as shown in Fig. 11: Bi = 3 Red-0.25


In the case of the smallest particles (1 mm) the wall heat transfer resistance was negligible, the heat transfer coefficient could not be measured, since the temperature gap at the wall was too small to be determined.


The heat transfer in tubular packed bed reactors in liquid flow has been studied and heat transfer parameters have been compared to gas flow results, pointing out similarities and differences. The homogeneous model represents quite well the heat transfer in a packed bed, except for very low velocities. In case of low liquid velocities preferential flow at the wall has been measured by a tracer technique and results are in rather good agreement with models of preferential flow.

The convective part of the bed conductivity is proportional to the liquid velocity and to the particle diameter, as in gas flow. Hydrodynamic effects on heat transfer phenomena have been shown. The discrepancy between liquid flow and gas flow results is explained by the differences in the flow regime stressing the need of taking these effects of the hydrodynamics (and therefore of the viscosity) into account. The comparison to mass transfer results allow a wider discussion and confirms this point. A new correlation is proposed for bed conductivity, valid in the range of intermediate Reynolds numbers (5 120) and tested for Prandtl number from 4 to 22.5. The wall heat transfer resistance is found to be nearly equal to particle diameter to bed conductivity ratio.


The authors wish to thank the Institut Pttrole for their financial support.





D D,> D, d H k K L Ped PC PC

Pr 4w R Red r SC T,

heat capacity of liquid, J kg-’ Km’ reactor diameter, m radial and axial dispersion coefficient, m2 s L ’ packing particle diameter, m height of measuring zone, m constants liquid mass flow rate, kg m - * s ’ = Red Pr, Peclet number of the fluid = u,, H/D,, axial dispersion Peclet number = p,u,,C,,d/A, or u,,d/D,, radial heat or mass transfer Peclet number = ,+C,r/&, Prandtl number of the fluid wall heat flux density, W m-’ reactor radius, m = Ld/p,, Reynolds number of the liquid radial coordinate, m Schmidt number (impulse to mass molecular diffusivity ratio) liquid temperautre in the centre of the reactor, K entrance (of heating section) liquid temperature, K outlet (of heating section) liquid temperature, K wall temperature, K liquid superficial velocity, m s - ’ local liquid superficial velocity, m s- ’ core and wall zone liquid superficial velocity, ms-’

core and wall zone dimensionless liquid perficial velocity (u: = u,/u,, , u$ = u,/u,,) axial coordinate, m


wall heat transfer coefficient, W m - ’ K- ’ bed dispersion coefficient without liquid flow, m’s_’ difTusivity coefficient in fluid without packing, m2s-l

liquid thermal conductivity, W m - ’K - ’ fluid thermal conductivity, W m- ’ K- ’ convective part of A,, W m-’ K-’ bed conductivity without liquid flow, W m-’ K-’ = A,, + A,, bed radial conductivity, W m-’ K- ’ bed axial conductivity, W m _’ K - ’ liquid viscosity, Pa s liquid density, kg m 3

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