968 Ind. Eng. Chem. Res. 1990,29,968-977

Generalized Description of Fluid Flow, Void Fraction, and Pressure Drop in Fixed Beds with Embedded Tubes

Felix A. Schneidert and David W. T. Rippin* Technisch-Chemisches Laboratorium, Eidgenossische Technische Hochschule, CH-8092 Zurich, Switzerland

Esmond Newsod Swiss Aluminium R & D, CH-8212 Neuhausen am Rheinfall, Switzerland

Generalized definitions of voidage, superficial mass flow, and hydraulic diameter were combined to compare axial and radial flow fixed beds with respect to their mean void fraction and pressure drop. Experimental measurements were made with two separate radial flow configurations in which substantial changes in the configurations could be made. For relatively open beds (volume occupied by cooling tubes is 25%), the pressure drop could be represented by a general correlation for beds with particles of widely dispersed sizes. For more closely packed tubes, the pressure drop correlation had to be modified to incorporate factors for extension of the length of the flow path and for the variance of the mass flux due to flow restrictions between the tubes. The use of values of these factors derived directly from the geometry of the radial flow configuration gave satisfactory predictions of the pressure drop for all configurations studied.

The increasing pressure drop limitation of fixed-bed axial flow reactors with decreasing catalyst particle size has been successfully circumvented by using radial flow reactors (RFRs). More active catalysts can therefore be fully exploited with higher effectiveness factors and in- creased production rates per unit volume of reactor, as in ammonia synthesis (Dybkjaer and Gam, 1984), methanol synthesis (Linde, 1983), catalytic reforming, and auto ex- haust converters. Design criteria for RFRs have been proposed (Chang et al., 1983), the highest conversion being achieved when the flow is uniformly distributed in the axial direction, the direction of the radial flow, inward or out- ward, being of secondary importance.

A conventional packed-bed reactor consists of a bundle of tubes filled with catalyst through which the reactant flows in an axial direction. The auxiliary cooling fluid flows around the tubes and heat is transferred to it through the heat-exchange surface of the tube walls (Figure 1A).

An adiabatic radial flow reactor (Figure 1B) was de- veloped for ammonia synthesis (Hansen, 1964). In recent years, alternative flow configurations have been developed, among them being the radial flow reactor in which the bed of catalyst particles surrounds the bundle of tubes through which the cooling fluid flows (Figure IC), as described by Ohsaki et al. (1980, 1984) and Dobson (1982). The di- rection of flow of reactants through the catalyst bed is then orthogonal to the axes of the cooling tubes.

Many previous axial and radial flow designs have op- erated adiabatically, with heating or cooling usually sep- arated from the reaction. In nonadiabatic axial flow re- actors, such as those used for selective oxidation, the performance is limited by the pressure drop and heat transfer. The nonadiabatic radial flow configuration in which the particle bed surrounds the cooling tubes allows for a more flexible reactor design. The severity of tem- perature and pressure gradients in the bed can be further reduced by adjustment of the cooling/ heating capacity as the reaction progresses, and the optimal reaction path can be followed more closely than in quench systems (Ohsaki

* To whom enquires may be addressed. 'Present address: Swiss Aluminium R & D, CH-8212 Neu-

1 Present address: Paul Scherrer Institute, Department 4FB, hausen am Rheinfall, Switzerland.

CH-5303 Wurenlingen, Switzerland.

et al., 1985). The nonadiabatic radial flow reactor thus has considerable advantages, not only over adiabatic fiied-bed reactors but also over nonadiabatic axial flow reactors.

However, the RFR configuration with the particles surrounding the heating/cooling tubes complicates the description of fluid flow. The overall characteristics of the bed of particles within a reactor depend upon the local characteristics of the particles, the arrangement of the cooling surface within the reactor, and the influence ex- ercised by the cooling surface on the structural arrange- ment of the particles in the neighborhood of the surface.

Three types of bed characterization are of interest. (a) Mean Void Fraction. This depends only on the

bed structure with no directional consideration. (b) Pressure Drop. This depends upon the bed

structure and the direction of flow or the path followed by the fluid through the bed.

(c) Heat Transferred from the Fluid Flowing through the Bed to the Cooling Surface. This depends upon the flow path and the structure between that path and the cooling surface through which the heat is trans- ferred.

The present paper is concerned with the comparison of the mean void fraction and the pressure drop in axial and radial flow reactors. The comparison of heat-transfer performance will be taken up in a later paper.

First the characterizing the parameters will be defined, which are common to the particle bed independently of whether it is located in an axial or a radial configuration. The predictions of the correlations obtained by other workers for the void fraction and the pressure drop in axial flow configurations (i.e., for beds of particles inside circular tubes) will be compared with the experimental results obtained in the present work for a range of radial flow configurations. Proposals are made for modifying the correlations for the axial flow configuration that retain the same general form but incorporate additional factors to account for special features of the radial flow configuration. With these modifications, satisfactory performance pre- dictions are obtained, and a common basis is arrived at for characterizing both the axial and the radial configu- rations or, equivalently, beds of particles both within and surrounding tubes.

Generalized Description of Fixed-Bed Reactors. Since there is a variety of usage in the characterization of

0888-5885/90/2629-0968$02.50/0 Q 1990 American Chemical Society

Ind. Eng. Chen 1. Res., Vol. 29, No. 6, 1990 969

Figure 1. Qpes of fixed-bed reactors. (A) Axial flow reactor (AFR). (B) Adiabatic radial flow reactor. (C) Nonadiabatic radial flow re- actor (RFR).

Embedded tubes - Radial Flow Reactor RFR

Fi l led tubes - Axial Flow Reactor AFR

v v b

Cooling system: tubes and liquid.

Fixed bed wi th packing

+ General flow direction of reactants

Figure 2. Fixed-bed and cooling system volumes in radial and axial flow reactors.

fixed beds, a set of consistent definitions has to be made that are sufficiently general to cover axial, radial, or other flow configurations.

(1) Volumes and Heat-Transfer Surfaces. Within the total reactor volume, V,, the cooling system occupies a volume Vc and provides a cooling surface of area Fc for exchange between the fluid flowing through the bed and the auxiliary cooling fluid. The remainder of the reactor is occupied by the fixed bed of particles of volume v b (Figure 2). The particle bed can be further divided into V,, the volume of the solid particles themselves, and the interstitial volume, Ve, which is available for fluid flow ( v b

In the simplified description used in the subsequent analysis, auxiliary installations such as distributors, con-

= vp + ve = vr - V,).

'b V C

V Cooling system: tubes and liquid.

v b Fixed bed wi th packing

(3 8 8 m)

0 V e Effective free reactor volume

u v ..::I::. Packing (solid phase)

Figure 3. Definition of void fractions.

nectors, etc., are not incorporated into the reactor volume. (2) Void Fractions. Two void fractions are required

to describe the relations between these three volumes. They can be the conventional void fraction, q,, of the bed, together with either E,, the fraction of volume remaining after installation of the cooling system in the reactor shell, or the fraction of free volume in the whole reactor shell, where er =

(3) Surfaces and Hydraulic Diameters. Three types of surface are defined Fr, the area of contact between the particle bed and the reactor shell; F,, the heat-exchanger surface between the particle bed and the cooling fluid; and F,, the total external surface area of the particles making up the bed.

Hydraulic diameters are defined with reference to sur- face-to-volume ratios within cylindrical tubes and have no directional orientation. The classical definition for an infinite fixed bed of spheres (Dp = particle diameter), as used in pressure drop correlations (VDI Wheat las , 1984), is

If additional surfaces are in contact with the fixed bed, such as, in our case, the outer shell r and the cooling system c, the free volume and the wall area must be correspond- ingly modified:

v b - v~ Dh*P = 4Fr + Fc + Fp (3)

For structured fixed-bed reactors, three definitions of the hydraulic diameter may be considered D, for the empty reactor shell, Dh,b for the fiied-bed geometry, and Dbp for the interparticular free volume:

whole reactor volume Vr = 4- (4) Dhs = 4fixed-bed-shell contact area F,

= 4- vb (5) fixed-bed volume Dh*b = 4fixed-bed-shell + cooling area F, + Fr

970 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990

(6) free interstitial volume - v e -

Dh*p = 4packing + cooling + shell 4Fp + F, + F r

The interconnections between these values are

Description of Representative Configurations A complete industrial-scale reactor may have cooling or

other geometrical arrangements that vary with the location in the reactor. For purposes of characterization, the reactor should, if necessary, be divided into a number of homo- geneous regions.

(1) Standard Axial Flow Reactor. The standard axial flow reactor used for solid-catalyzed gas reactions with large heat loads contains within the reactor shell a bundle of tubes filled internally with catalytic packing. Reactants flow through the tubes in parallel and are distributed and collected at the bottom and top of the shell. The heat- exchange surface is defined by the internal tube diameter,

he arrangement of the bank of tubes is characterized (as in the radial flow reactor) by two parameters: S,, the distance between the centers of tubes in a row; and S,,.the distance between the center lines of successive rows. (Smce the coordinate z is used for the direction of flow of reac- tants, the inter-row distance for the radial flow reactor is designated as Sz.)

The characterizing parameters for the pressure drop and heat transfer of the reactant fluid in the axial flow reactor are

For this configuration, there is essentially no direct contact area between the particle bed and the reactor shell:

D t p = Dt,i*

Fr,AFR = 0, Dhj,AFR = (9)

Fc,AFR = v b / ( v b / F c ) = Vb/(Dt/4)

%,AFR = (7/4)D?/(s,s2) (10)

Dh,b,AFFt = Dt

(11) D4p,AFR = l /Dt + (1 - q,)(1.5/Dp)

(2) Radial Flow Reactor Hydraulic Diameters. In the flow arrangement used in radial flow reactors recently described in the literature (Figure IC), the reactor com- prises a cylindrical shell containing the particle bed within which the cooling tubes are arranged in concentric rings. The reactants enter at the center of the shell, flow over cooling tubes orthogonal to their axes, and are collected at the outer surface of the reactor shell. The external cooling tube diameter is representative for heat transfer:

by the inter-tube distance (S,) and the inter-row distance (SJ as before. At this stage, the mean flow velocity of the reactants across the tube bank in the radial direction is assumed to be constant. Changes in velocity and possible changes in tube arrangement between the center and the circumference of the reactor shell are not considered. Thus, with reference to a complete industrial radial flow reactor, local behavior is being characterized.

For the particle bed and the cooling surface of the radial flow reactor,

cb

= Dt,a (Figure 4A). DtK e local geometry of the tube configuration is described

Fr,RFR = 0, DhjW = O3

A

C

B

D

Figure 4. Characterization of bed and tube arrangements. (A) Radial flow over cooling tube D,. (B) Tube spacing S,, S,. (C) Flow constrictions B,, Bd. (D) Inscribed circle Dbj.

When local conditions are considered in the interior of the bed, there is in this region no direct contact area between the particle bed and the reactor shell:

F~,RFR = Vc/(Vc/Fc) = Vc/(Dt/4)

C~,RFR = 1 - (7/4)Dt2/(S,S2) (12)

Constructional Flexibility. In contrast to the axial reactor, the structure of the particle bed in the radial flow reactor depends on the tube arrangement. This can be characterized by two additional dimensionless parameters:

relative distance between tubes in a row xz = S,/(2S2)

DZ = Dt/(2Sz) relative tube diameter

Any possible triangular arrangement of the RFR tubes must lie within the region of the DZ-XZ space bounded by the constraints

DZ < 1

DZ < XZ separation on the z axis separation on the x axis

(XZ2 + 1)1/2 D Z < diagonal separation

This region is shown in Figure 5 together with locations of the radial flow configurations for which experimental results are reported later. Values of the hydraulic bed diameter in the feasible region are shown in Figure 6:

Ind. Eng. Chem. Res., Vol. 29, No. 6,1990 971

Table I. Dimensions of Experimental Configurations Dp, 4, s,, s,, B,, Bd, Dbj, Dh,n D4b, D4p9

code mm mm mm mm mm mm XZ DZ q, cc cr mm mm mm mm DP/D4, Preliminary Experiment

0 3.44 20 37.7 30.0 17.7 15.4 0.63 0.33 0.39 0.72 P 3.44 20 42.8 29.0 22.8 16.0 0.74 0.34 0.38 0.75 Q 3.44 20 47.6 26.0 27.6 15.2 0.92 0.38 0.38 0.75 R 3.44 20 52.1 25.0 32.1 16.1 1.04 0.40 0.38 0.76 S 3.44 20 56.2 22.0 36.2 15.7 1.28 0.45 0.38 0.75 T 3.44 20 59.9 20.0 39.9 16.0 1.50 0.50 0.38 0.74

Main Experiment A 2.28 27 36.4 27.0 9.4 5.6 0.67 0.50 B 2.28 27 36.4 32.0 9.4 9.8 0.57 0.42 C 2.28 27 36.4 37.0 9.4 14.2 0.49 0.36 E 3.48 27 36.4 27.0 9.4 5.6 0.67 0.50 F 3.48 27 36.4 32.0 9.4 9.8 0.57 0.42 G 3.48 27 36.4 37.0 9.4 14.2 0.49 0.36 J 5.07 27 36.4 32.0 9.4 9.8 0.57 0.42 K 5.07 27 36.4 37.0 9.4 14.2 0.49 0.36 L 5.07 27 36.4 42.0 9.4 18.8 0.43 0.32

1.05

DZ

0.70

T

. . . . . . . 0.35

0.00

0.00 0.40 0.80 1.20 1.60 2.00 xz

Figure 5. Feasible region for tube arrangements.

Dh,b 12 sz 10.00

3.39

0.09 0 .88

0 .

2.00

0.02

Figure 6. Hydraulic bed diameter for feasible configurations.

Experimental Apparatus. An experimental apparatus was constructed to represent a segment of an RFR reactor in which heat transfer but no reaction occurred. Heated air was passed through a fixed bed of particles in which were embedded six rows of water-cooled tubes. The total volume of particle bed and cooling tubes for the range of configurations used varied from ca. 15 to 25 L.

The external diameter of the cooling tubes was D, = 27 mm, and the distance between centers of tubes in a row was Sx = 36.4 mm. Sz, the distance between the rows, could be varied. Glass spheres for packing were available in three uniform sizes. In order to ensure uniformity of packing between the cooling tubes, which were horizontal, quite vigorous vibrations were used. Thus, it was not practical to study low-density packing configurations. Figure 7 and Table I show the configurations studied and the letters used to identify the results from each configu- ration in subsequent presentation of results.

0.39 0.42 0.39 0.51 0.38 0.57 0.42 0.42 0.40 0.51 0.39 0.57 0.42 0.51 0.41 0.57 0.40 0.63

0.28 21.8 146 34.6 1.31 0.099 0.29 24.8 164 39.8 1.31 0.086 0.29 27.8 195 41.8 1.31 0.082 0.29 32.1 214 45.3 1.31 0.076 0.29 37.9 260 45.1 1.31 0.076 0.28 44.9 308 45.1 1.31 0.076

0.16 12.2 597 17.9 0.86 0.20 15.3 338 24.0 0.87 0.22 18.9 310 30.3 0.87 0.17 12.2 597 17.9 1.35 0.20 15.3 338 24.0 1.32 0.22 18.9 310 30.3 1.32 0.21 15.3 338 24.0 1.99 0.23 18.9 310 30.3 1.94 0.25 22.9 299 36.3 1.93

Code descriDtion i n GraDhs; . . .

Row to row distance SZ 27 32 37 42 [ m m l

a I

Figure 7. Identification of tube arrangements studied.

0.127 0.095 0.075 0.194 0.145 0.115 0.211 0.168 0.140

Table I also shows various characterizing parameters of the configurations used. When the wall effect due to the reactor shell is entirely absent, Dhs - 03. The experiments were carried out in a reactor segment where the sur- rounding insulated walls exercised a certain influence on the geometrical arrangement of the particles. However, from Table I it can be seen that the hydraulic diameter (Dh,r) associated with the surrounding walls in the main experiment is at least an order of magnitude greater than the hydraulic diameter (&b) associated with the exchange between the bed and the cooling area. Thus, wall effects are not expected to be large. Hydraulic bed diameters (Dh,b) and space velocities were close to values used in industrial reactors for selective oxidation. Hydraulic particle diameters (Dh,p) were selected to be equal to or less than those used in practice (Dp - 3-7 mm in industrial applications).

Additional results from a prototype apparatus with tube diameter D = 20 mm were also included in the evaluations.

The ranges of local Reynolds numbers (Re,,) and path corrected Reynolds numbers (Reb,") are shown on the axes of Figures 13, 14, and 16 in brackets.

Measurements were made of the void fraction, pressure drop, and heat transfer in the bed. A full description of the apparatus, the experimental procedures, and the re- liability of the results is given by Schneider (1989).

Mean Void Fraction: Comparison between Axial and Radial Flow Reactors. The interstitial velocity and, in consequence, the pressure drop are very sensitive to the mean void fraction in a particle bed. For packed tubes, as found in the axial flow reactor, the ratio of tube to

972 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990

Table 11. Correlations for Mean Void Fraction in Particle Beds (Brauer (1971) and Achenbach (1982)) and for Embedded Tubes (Aesse and Ringer (1982))"

kd SE(kd) k,, SE(k,J R2, % DF From Brauer (1971): tb = 0.375 + 0.34(0,/0,)

expt 0.362 0.004 0.254 0.034 74.8 17 lit. 0.375 0.34

From Achenbach (1982): t b = 0.375 + 0.78(0,/DJ2 expt 0.377 0.002 1.006 0.112 81.4 17 lit. 0.375 0.78

From Hesse and Ringer (1982): 'b = 0.345 -t 0.646(0,/0b,i) expt 0.359 0.006 0.157 0.027 65.0 17 lit. 0.345 0.646

"Experiment" gives results of fitting experimental data to an expression of the same form for comparison with coefficient values from the cited references.

1 0 420

C P

0 400

0 .380

0 000 0 050 0 100 0 150 0 200 p,o hb

Figure 8. Correlations for mean void fraction, 6 . Literature cor- relations for packed tubes [(l) Brauer (1971); (2) blchenbach (1982)] compared with experimental resulb [(letters and 3) quadratic fit; (4) linear fit].

particle diameter influences the void fraction significantly. A number of different sets of experimental results and

forms of the correlation have been reported in the litera- ture as shown in Table 11.

For comparison with the radial flow results, two current correlations for packed tubes of linear and quadratic form in the particle to tube diameter ratio were used.

Hesse and Ringer (1982) reported experimental results for the voidage of a packed bed surrounding rectangular bundles of tubes. As a correlating parameter, they defined Db,i, the diameter of the largest circle that could be in- scribed in the space available to the packed bed between the tubes (see Figure 4D):

Db,i = [(XZ2 + 1) /2 - DZ]2S, (16)

Experimental values of the voidage for several RFR pitches filled with spheres of different sizes were compared with the two published correlations for packed tubes. For the RFR, the tube diameter was replaced as a correlating parameter by the hydraulic mean diameter of the bed, Dhb. (For flow within tubes, Dh,b is identical with Dt.)

Both the linear and quadratic forms of the correlation for packed tubes gave acceptable fits to the RFR data, the fit of the quadratic form being slightly better (Figure 8). The coefficients in the literature correlations of the same forms for packed tubes were in the neighborhood of gen- erous confidence bounds (-99%) for the estimates of these coefficients from the fit to the RFR experimental data, the quadratic form again being notably better.

The relative success of the quadratic form of correlation suggests that the ordering effects, directed from the walls into the fixed bed, are quite strongly sensitive to the presence of more neighboring walls due to a kind of bridging effect between adjacent patterns. (See pictures

of local voidage distribution in AFR and RFR pitches in Schneider and Rippin (1988).)

Hesse and Ringer's parameter, Db,i, was not successful in correlating the results, their literature value for the coefficient of linear dependence being 4 times greater than that obtained by fitting the experimental results (with rather poor accuracy). It appears that this correlation is not transferable from the rectangular tube configuration used to obtain it.

Directed Effects in Fixed Beds. Different Levels of Structure. Most phenomena of interest occurring in fixed beds have a directional character. In the simplest case, a single potential field exists along the bed, resulting, for example, in steady heat flow by conduction or the establishment of a velocity profile for fluid flow through the bed.

For most fixed-bed configurations, the microstructure, comprising the local inter-relationships of the particles and the inter-particulate network of channels, is independent of the wider environment and does not require special consideration when comparisons between configurations are made. The properties of the microstructure will also be considered to be isotropic. Changes in the packing structure in the neighborhood of boundaries of the bed will be considered later.

In the intermediate size range of the structure corre- sponding to boundaries of the bed or cooling facilities, items occurring as obstacles or creating dead zones in the flow must be considered individually.

Completely dead zones can be removed from the active bed volume, whereas obstacles affect the length and cross section of flow paths through the bed.

The macrostructure describes the arrangements of the fixed bed and any cooling facilities along the axis of the reacting fluid flowing through the bed. The possibility of wide variations in this structure along the flow path in a single reactor is an important advantage of the RFR that, is not available in the AFR.

It is in the intermediate size range that differences in configuration exercise their effect upon directional be- havior such as flow patterns. Thus, in comparing the performance of AFR and RFR configurations, attention will be concentrated at this intermediate scale. In the axial flow reactor, the fluid traverses a bed of uniform structure and dimension in the direction of the mean flow. In the radial flow reactor, the flow direction and the cross section available for flow change continuously as the fluid traverses the banks of cooling tubes. If a common basis is to be found for interpreting flow and heat-transfer character- istics, allowance must be made for these additional effects in radial flow.

After the general definition of mass fluxes, correction factors are derived for path length and variation of the flow cross section.

Mass Fluxes. An average mass flux (mass/(area.time)) is derived from the mass per unit time divided by a cor- responding cross-sectional area. Depending on the defi- nition of this area, different mass fluxes are obtained: G, in the empty reactor shell; G, (or Go), the commonly used superficial mass flux in the fixed bed volume; and Gb, the mean local mass flux in the interstitial volume between the particles.

If the path through the bed does not follow the axial direction ( z ) of the mean flow, a path extension factor, f,, may be defined. This extended path length implies a reduction in the mean cross-sectional area available for flow and an increase in mass flow rate per unit area. G,, and Gb," take into account the extension of the flow path.

Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990 973

M id1 ine is z

A I I I I I I I I I I I I 1

s z ' fv .S2

Flow a r c Figure 9. Construction of geometrical flow paths through the RFR tube configuration for the definition of geometrical correction fac- tors.

For a given mass per unit time, W, flowing in a reactor of volume V, over an axial distance L,,

G, = WL,/Vr (17)

Flow Path Stretching Factors (f,). Two methods are used to estimate the length of the flow path of a fluid in a bed containing circular tubes arranged on a triangular pitch.

Circular Arc Length. An arc of a circle is constructed connecting the midpoints of the two narrowest cross sec- tions of the flow as seen in Figure 9. The relative ex- tension of this flow path compared with flow along the z axis is

f v = ?/sin Y (22)

where y = a-2 arctan (l/DZ). I t is of interest to note that this extension factor is

independent of the relative separation, XZ, of tubes in the row. This flow path avoids the front and rear of the tubes and might be particularly appropriate if stagnant zones build up there.

Midline. The midline path is constructed to pass through the midpoint of each section of the bed traversed by the flow (Figure 9). The bed dimension is measured in the radial direction relative to the tube around which the flow is passing. This path penetrates more deeply into the region immediately in front of and behind each tube.

For most practical tube arrangements, the midline es- timate of the path extension factor gives values ca. &lo% higher than the arc length estimate, which has a value of ca. 1.10 (h0.05) (Figure 10). In practice, other factors will also disturb the path length so that an accuracy of better than 5-10% should certainly not be expected.

Mass Flux Variance Factor ( f e ) . In pressure drop correlations, the kinetic energy term contains the square of the velocity. When the flow cross section varies, use of the mean velocity will result in systematic errors.

The correlation factor (fe) is introduced to relate the mean value of the square of the mass flux, needed to ap- proximate the kinetic energy, to the square of the mean value of the mass flux. This factor can be derived directly from the geometry of the fixed bed (Figure l l ) , since the

-~ 0.00 0.15 0.30 0.45 0.60 DZ

Figure 10. Flow path stretching factors. Midline estimate for a range of values of XZ of 0.1-2:(1) integral, (2) linear three-point approximation, (3) arc length estimate (independent of XZ), (A, B) predictions for experimental configurations.

A

I Z j ...... Fixed bed

Cool i ng System

4 Standard

I f low oath I ( = ' % I

4 Stretched f low path I ( E $ * sz,

B C Figure 11. Geometrical construction of variance factors in addition to flow path stretching. The resulting fixed-bed model is shown in B for stretching only and in C for both factors combined.

mass flux is inversely proportional to the width of the flow cross section (B,):

V(l/Bv) = E(l/B,2) - E(1/Bv)2

with

E(l/Bv) = L,Il/B,(u) du/L ,

E(l/B,2) = Lvj1/B,2(u) du/L ,

where the integration is taken over the variable u along the path L,.

974 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990

Table 111. Correlations for Pressure DroD in Fixed Beds' range parameter (incl path factor)

source Reynolds no. e or p Dh fh: laminar f& turbulent

VDI Atlas (1984) -0.2 < Reb < 20000 -0.7 < Reb < 15000 0.2 < Reb < 30000 300 < < 20000

Reichelt (1972)

Metha and Hawley (1969) 0.2 < Reb < 9 Paterson et al. (1986) 42 < Reb < 1500

Reichelt (1972) 20 < Reb < 15000 200 < Reb < 1500

1 , 1 2 0 1

I

Spheres 0.37 < c < 0.42 0.20 < e < 0.64 i3 < 0.1 0.07 < p < 0.6 0.02 < p < 0.13 0.045 < p < 0.28

Cylinders p < 0.1 0.04 < p < 0.48

General

Dh.p,= 142 4.13/Reb0.'

Dh,p,= 133 1.73 'h,P 133 1/(1.7p + 2.0@* + 0.58) Dh,p 133 2.33

Dh,p,.. 142(0.4/0°.78 4.13/Re$.1(0.4/e)0.78

&,,.. 133(1 1.228) 2.33 exp{l.GC((l - p)* - I)]

Dh,p,- 178 2.08 Dh,p 178 l/(3.0@ + 2.402 + 0.48)

Dh,p,- 133 2.33

0.320 0.360 0.400 0.440 0.480 DZ

Figure 12. Variance factor estimates for conditions of main ex- periment. (A) Flow arc, (C) midline, (B, D) simplified approxima- tions.

The arc length or midline approximation considered earlier can be used for the path. The average of the linear and two-point linear approximations for the shape of the bed boundary (Figure 12) agrees well with the exact in- tegration over the path.

For the tube configuration considered, values of the mass flux variance factor fall in the range ca. 1.05-1.2. Extreme variations resulting from the use of different models fall well within the range *lo%.

Pressure Drop. General Fixed-Bed Correlations. The general correlation for pressure drop in fixed beds originally due to Ergun (1952) is cited by Bird et al. (1960) in the form

This form can be simplified by use of relevant local var- iables as in VDI Warmeatlas (1984) (see Table 111):

G d h p

77 Reb = -

Pressure Drop in Radial Flow Reactors. Direct Analogy. For a packed bed containing randomly dis- tributed particles of widely varying sizes, the VDI Atlas recommends a voidage fraction correction to the above equation valid in the t range 0.195-0.64:

1 . 0 0 4 L

1Re.b

Figure 13. Prediction of VDI analogy (eq 26) (calculated pressure drop over measured) vs log local Reynolds number (Reynolds num- ber in brackets) for main experiments as coded in Table I.

1.05 1.40 1 . 7 5 2.10 2 . 4 5 ( 1 1 ) ( 2 5 ) ( 8 6 ) (126) ( 2 8 2 )

Use of this equation for the radial flow reactor configu- ration with embedded tubes implies that the tubes are treated as particularly large particles.

This equation has been used with the mass flux and hydraulic diameter definitions for the RFR (eqs 19 and 11). The void fraction, t, (eq 1 and Figure 3), is here the fraction of interstitial volume in relation to the total bed volume including cooling tubes.

Quite good agreement (*-lo%) was found between this model and experimental measurements in preliminary equipment with cc - 0.75 (er - 0.29).

The main experiments were carried out with a more closely spaced arrangement of tubes, t, - 0.4-0.6. The model increasingly overestimates the pressure drop by up to 60% as cc is reduced; i.e., the tubes are more densely packed (Figure 13). In fact, for the main experiments the values of er (0.16-0.25) are partly outside the region for which the void fraction correction was established. Fur- thermore, the regular arrangement of the close packed cooling tubes is quite different from the random mixture of particles of various sizes used to derive the voidage fraction correction. The agreement found for the prelim- inary experiments is quite surprising considering the differing circumstances.

Even if the value of t, falls within the range for which the correction was derived, the model predictions would be very questionable for a tube arrangement that was highly asymmetrical, with tubes closely spaced within a row but the rows widely separated from one another ( B d >> in B, in Figure 4C). Modified Description of Pressure Drop To Account

for Flow Paths in RFR. A modified correlation is pro- posed in which the effects of the cooling tubes are ac-

1.40 -

E

0

I L

J

- 20 z .......................................................................................... ....................... 0.80 . I

1 .05 1.40 1.75 2.10 2.45 2.80 IRe.bv (11) (25) (56) (126) (282) (631)

Figure 14. Prediction of model equation (27) including stretching and velocity variance for arc length model vs log path-corrected Reynolds number.

counted for by use of the two geometric correction factors, f v and fez

The Reynolds number is expressed in terms of the mod- ified mass flux, Gb,v (eq 21), which includes the path stretching correction. In the voidage correction q, is used instead of cr since the effect of the tubes is now accounted for by the other factors, fv and fe.

The discrepancies between correlation and experiment are substantially reduced, and the systemic error, which in Figure 13 increases as tubes are packed more closely (cases A and E), is much less evident in Figure 14. Pre- dictions using the midline path correction are about 20% higher than those using the arc length shown in Figure 14, indicating the sensitivity of pressure drop predictions to these geometric factors.

Use of the arc length path stretching factor (fv) alone considerably improves the average level of the model predictions compared with Figure 13. However, the pre- diction errors still vary systematically with tube packing density (or cc) for different configurations. These system- atic errors are substantially reduced by further incorpo- ration of the mass flux variance factor (fe) as shown in Figure 14.

The model still predicts pressure drops that on the av- erage are about 10% higher than observed experimentally.

In the AFR, when the particle size is significant com- pared with the diameter of the packed tube, the structure of the particle bed allows channeling in paths of lower resistance along the walls, resulting in reduced pressure drop. These effects are qualitatively well-known for packed tubes, but no general model or description is widely accepted (Metha and Hawley, 1969; Reichelt, 1972; Achenbach, 1982; Vortmeyer and Schuster, 1983; Griffiths, 1986).

A similar effect of favorable flow paths along the tube walls as suggested in Figure 15 can be postulated on the basis of the observations of the particle structure in radial flow beds reported by Schneider and Rippin (1988).

Precise estimation of the effects of this complex flow pattern would be very difficult. In the AFR, the wall flow and the core flow are parallel through the length of the tube. However, in the RFR, the three types of flow shown in Figure 15 follow one another sequentially for any stream element. The influence of wall effects on pressure drop is expected to be less marked than in the AFR. (See Schneider (1989) for further discussion.)

Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990

9 Average Forced f low Reduced f low through along the f low across centra l core tube wa l l s the packing

1 ayer s

975

Figure 15. Postulated flow patterns resulting from nonrandom arrangement of particles around cooling tubes in RFR (wall effects).

1 . 20 - + 2o ..................................................................................................................

&I. R/&. H

0

J J J

1.05 1 .40 1.75 2.10 2.45 2.80 IRe.bv (11) (25) ( 5 6 ) (126) (282 ) (631)

Figure 16. Prediction of model equation (27) including additional correction for wall effects from Paterson’s AFR correlation.

Paterson et al. (1986) proposed a wall effect correction to the pressure drop equation for flow in packed tubes as a function of the particle/tube diameter ratio. To assess the order of magnitude of this effect for the RFR, the tube diameter is replaced by the hydraulic mean diameter, Dh,p,RFR (eq 14), and predictions using the resulting cor- rected correlation are shown in Figure 16.

The systematic discrepancies shown in Figure 14 for closely packed tubes (bases A and E) are more than com- pensated for by this wall effect correction. Since this is admittedly an overcorrection, the model fit is as good as can be expected without a more detailed study of wall effects or other details of the flow and bed structure. The accuracy of the pressure drop predictions for the RFR compare favorably with those obtained by other workers for the well-studied system of axial flow through packed circular tubes.

In the experimental equipment used in this study, the cooling tube arrangement did not vary along the flow path, and the physical properties of the fluid were effectively

976 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990

constant. This will often not be the case for a complete radial flow reactor. The pressure drop equation (27 ) can then be used in differential form:

feGb.v2 f v f 0.4 f i ,

At radius r of the radial flow reactor, Gb,v and the cor- rection factors f, and fv can be determined from the specification of the tube geometry at that radius. Local values of fluid density and viscosity (needed for the Rey- nolds number) can then be used to calculate the pressure drop over one or more tube pitches. This procedure can be repeated for different reactor radii from the inlet to the outlet to determine the total pressure drop. Decoupling of the determination of the geometric correction factors, f, and f,, from variations in fluid properties is justified provided the variation of these properties over one tube pitch is small compared with the variations in mass flux due to changing flow cross section over a single tube pitch. This assumption is usually valid.

Conclusions A consistent nomenclature is defined that facilitates the

characterization of fixed beds of widely differing geome- tries. Nonoriented properties can be described in terms of this characterization for fixed beds of any form. Thus, known correlations for mean bed voidage in packed tubes are extended to the RFR configuration.

Similar extensions can be made for the effects oriented in the flow direction if corrections are made for extension of the flow path and the variance of the velocity. The pressure drop in the RFR is successfully characterized by extension of the packed tube correlations.

Heat-transfer effects call for orientation in at least two directions, the flow direction of the fluid and the direction of heat transfer orthogonal to the flow and the cooling surface. Geometrical considerations are even more im- portant. The description of these rather complex effects in the radial flow geometry will be taken up in a later publication.

Acknowledgment

Swiss Aluminum (Neuhausen) is acknowledged for partial funding for F.A.S. and substantial workshop sup- port. General guidelines on commercial reactor design from Alusuisse Italia are appreciated.

Nomenclature f = factor p = pressure, N m-2 u = axis for an average flow path through fured bed of the RFR x = axis across the flow direction, across the cooling tubes in

y = axis along the cooling tubes in the RFR z = axis along the general flow direction in the RFR B = width, m D = diameter, m F = surface area, m2 G = mass flux, kg m-2 s-l L = length in flow direction z , m S = spacing distance, m V = volume, m3 W = mass flow rate, kg t = voidage fraction 7 = dynamic viscosity kg m-l s-l p = density kg rn--3

the RFR

Subscripts a = outer b = fixed bed c = cooling installation (or heating) e = empty h = hydraulic i = inner p = packing r = reactor t = tube u , x , y, z = along corresponding axes w = wall M = measured value R = calculated value 0 = free bed superficial AFR = for axial flow reactor RFR = for radial flow reactor

Dimensionless Numbers (See also Figure 3)

Reb = DhpGb/V = Dh Go/(CbB) Reo = DpGo/II

Reb,v = D’h,pGb,v/o = B h , p G d v / v

Literature Cited

Achenbach, E. Druckverlust in durchstromten Kugelschuttungen bei hohen Reynolds-Zahlen. Chem.-Ing.-Tech. 1982,54,66; Chem.- Ing.-Tech. 1982, MS No. 966182.

Bird, B. R.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960.

Brauer , H. Grundlagen der Einphasen-und Mehrphasenstromungen; Sauerlander: Aarau and Frankfurt, 1971.

Chang, H. C.; Saucier, M.; Calo, J. M. Design Criteria for Radial Flow Fixed-Bed Reactors. AIChE J. 1983,29, 1039-1041.

Dobson, B.; Whyman, P. J. M.; Doy, R. J.; Engel, M. 0.; Davies, R. Chemical Reactor and Process. European Patent Application EPO 082 609 Al, 1982.

Dybkjaer, I.; Gam, E. A. Energy Saving in Ammonia Synthesis- Design of Converters and High Activity Catalysts. Chem. Econ. Eng. Reu. 1984, 16 (9), 29-35.

Ergun, S. Fluid Flow through Packed Columns. Chem. Eng. Prog.

Griffiths, N. B. The Flow in and Structure of Narrow Packed Beds. Ph.D. Dissertation, Cambridge University, England, 1986.

Hansen, H. J. Process and Apparatus for Performing Reactions in the Gaseous Phase. U S . Patent 3 372 988, 1964.

Hesse, P.; Ringer, D. Heat Transfer and Pressure Drop of Fixed Bed Reactors with Submerged Tube Bundles. Heat Transfer Proc., 71st Int. Heat Transfer Conf. 1982, 6, 19-23.

Linde. The Linde Variabar Process for the Production of Methanol with the New Isothermal Reactor. Chem. Econ. Eng. Reu. 1983,

Metha, D.; Hawley, M. C. Wall Effect in Packed Columns. Ind. Eng. Chem. Process Des. Dev. 1969, 8, 28C-282.

Ohaski, K.; Nishimura, Y. Development of a New Methanol Reactor. Presented at the AIChE Spring National Meeting, Anaheim, CA, 1984.

Ohsaki, K.; Zamma, J.; Kobayashi, Y.; Watanabe, H. Process and Apparatus for Reacting Gaseous Raw Materials in Contact with a Solid Catalyst Layer. UK Patent Application GB 2 046 618 A, 1980.

Ohasaki, K.; Shogi, K.; Okuda, 0.; Kobayashi, Y.; Koshimizu, H. A Large Scale Methanol Plant Based on IC1 Process. Chem. Econ.

1952,48, 89-94.

15 (4), 14-16.

E n i Rev. 1985, I7 (5), 31-38. Paterson. W. R.: Burns. J. R. M.: Griffiths. N. B.: Kesterton. K. R.:

Paverley, A. J. Experimental ’Studies of Transport Processes in Packed Beds of Low Tube-to-Particle Diameter Ratio. Proceed- ings, World Congress 111 of Chemical Engineering, Tokyo, 1986; Society of Chemical Engineers: Tokyo, Japan, 1986; Vol. 2 pp

Reichelt, W. Zur Berechnung des Druckverlustes einphasig durchstromter Kugel und Zylinderschuttungen. Chem.-Ing.-Tech.

Schneider, F. A. Wiirmetransport, Druckabfall und Leervolumen- verteilung in radial durchstromten Festbettreaktor. Doctoral Dissertation 8799, ETH Ziirich, 1989.

Schneider, F. A.; Rippin, D. W. T. Determination of the Local Voidage Distribution in Random Packed Beds of Complex Geom-

304-307.

1972,44, 1068-1071.

Ind. Eng. Chem. Res. 1990,29,977-988 977

etry. Znd. Eng. Chem. Res. 1988,27, 1936-1941. VDI Wheat laa . Bereehnungsbhtter ftir den WheBbergang, 4th

ed.; VDI-Verlag: Diisseldorf, 1984. Vortmeyer, D.; Schuster, J. Evaluation of Steady Flow Profiles in

Rectangular and Circular Packed Beds by a Variational Method.

Chem. Eng. Sei. 1983,38,1691-1699.

Received for review September 13, 1989 Revised manuscript received February 12, 1990

Accepted February 22, 1990

Structure of Dilute Supercritical Solutions: Clustering of Solvent and Solute Molecules and the Thermodynamic Effects

RongSong Wu,? Lloyd L. Lee,**+ and Henry D. Cochrant School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019, and Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

The objective of this paper is to exhibit, on the molecular level, the relationships between the microscopic structure and thermodynamic properties of dilute supercritical solutions by application of the integral equation theories for molecular distribution functions. To solve the integral equations, we use Baxter’s Wiener-Hopf factorization of the Ornstein-Zernike equations and then apply this method to binary Lennard-Jones mixtures. A number of closure relations have been used: such as the Percus-Yevick (PY), the reference hypernetted chain (RHNC), the hybrid mean spherical approximation (HMSA), and the reference interaction-site (RISM) methods. We examine the microstructures of several important classes of supercritical mixtures, including the usual “attractive”-type and the less known “repulsive”-type solutions. The clustering of solvent molecules for solvent-solute structures in the attractive mixtures and, correspondingly, the solvent cavitation in the repulsive mixtures are clearly demonstrated. These are shown to be responsible for the large negative growth of the solute partial molar volumes in the attractive case and the positive growth in the repulsive case. Integral equations also afford us a unique opportunity to study the micros- tructures of solute-solute interactions. There is strong evidence of solute-solute aggregation in extremely dilute supercritical mixtures, a picture consistent with experimental fluorescence spec- troscopy evidence.

1. Introduction It has been known for over a century (Hannay and

Hogarth, 1879, 1880) that gases at slightly supercritical temperatures can exhibit, for a solute of low volatility, solubility that increases upon compression from that predicted for an ideal gas mixture with solvent power orders of magnitude higher than the ideal gas prediction. Other properties of supercritical solutions (Eckert et al., 1983, 1986; Johnston et al., 1988; Subramanian and McHugh, 1986; Collins et al., 1988) can also vary dra- matically with changes of temperature or pressure near the critical point (CP) of the pure solvent. Recent results (Kim and Johnston, 1987a,b; Debenedetti, 1987; Cochran et al., l987,1988a,b; Pfund et al., 1988) have interpreted these observations in terms of solvent structure about a dissolved solute molecule: properties change dramatically near the CP as the correlation in density fluctuations becomes long-ranged and the size of the cluster of solvent molecules about each solute increases.

The purpose of the present paper is to use the integral equation calculations to explore the relationship between structure and properties of dilute supercritical solutions. We briefly review the pertinent experimental observations and the previous theoretical results relating to the so- lubolvent cluster in typical supercritical solutions. Then we extend this examination in two areas: first, we consider the relationship between structure and properties when the characteristics of the solvent and the solute are re- versed (i.e., when solvent T, > solute T, and solvent V, > solute VJ. Second, we propose to consider the distribution

University of Oklahoma. *Oak Ridge National Laboratory.

of solute molecules about a central solute molecule to understand the nature and effects of solute-solute aggre- gation for states near the solvent CP.

To begin, we review the experimental observations that have suggested clustering of solvent molecules around a solute molecule in dilute supercritical solutions (solute partial molar volume (PMV) measurements, solubility measurements, solute absorption and fluorescence spectra, and others) as well as some observations that have sug- gested a high degree of solute-solute aggregation in dilute supercritical solutions. We shall also review briefly the prior theoretical interpretations of these observations.

The experimental observations of the large negative growth of the solute PMV by Eckert and co-workers (Eckert et al., 1983, 1986) of dilute solutions of large or- ganic molecules dissolved in supercritical C02 and CzH4 were seminal in the developing understanding of the re- lationship between properties and structure of supercritical solutions. The partial molar volume was determined from density measurements as a function of solute concentration for dilute solutions; the density was measured by the vi- brating tube technique. These workers observed that the partial molar volume of solutes (naphthalene, tetra- bromomethane, camphor) became negative and very large in magnitude (ca. 100 times the solvent’s bulk molar volume) near the solvent CP. They concluded “...that the solution process represents a disappearance of more than 100 mol of solvent/mole of solute. This suggests some sort of clustering process, perhaps akin to electrostriction about an ion in a protic solvent.”

Kim and Johnston (1987a,b) and Debenedetti (1987) both presented molecular interpretations of the partial molar volume results to suggest that each solute is sur- rounded by a cluster of solvent molecules which grows to

0 1990 American Chemical Society