The limits of applicability of pressure drop correlations

2994

Shorter Communications EDGAR N. RUDISILL M. DOUGLAS LEVAN’ Engineering

Superscript dimensional

variable

Department of Chemical University of Virginia Charlottesville, VA 22903-2442, U.S.A.

REFERENCES

NOTATION

numerical

B c

constant

F/C;

ii> i k: K 1+ L M Pe r* Shw u* “Z W x x* Y z z+ Greek A

concentration, mol/m” mixing cup c&ceniration diffusion coefficient of solute in fluid, m2/s shape integer: 0 for plane, 1 for cylinder ieqs (1) and 112)l . __ membrane mass transfer coefficient based on inside area, m/s numerical constant characteristic length (inside radius or half-width), m eigenvector Kummer’s function Peclet number (U * 1*/D*) radial coordinate, m wail Sherwood number (ktl*/D*) average velocity, m/s axial velocity [eqs (13) and (21)] transformed variable [eq. (24)] X*/l* coordinate normal to flow direction, m transformed variable [eq. (24)] wPe)(z*ll*J axial coordinate, m letter eigenvalue

Subscripts m index W wall value ‘Author

to whom correspondence

should be addressed.

Abramowitz, M. and Stegun, I. A., 1970, Handbook ofMathematical Functions, --_ DD. 504-505. National Bureau of Standards, U.S. Department of Commerce, Washington, DC. Cooney, D. O., Davis, E. J. and Kim, S.-S., 1974a, Mass transfer in parallel-plate dialyzers-a conjugated boundary value uroblem. Chem. Emma J. 8. 213-222. Coohey, D. b., Kim, S.-S. and Davis, El J., 1974b, Analyses of mass transfer in hemodialyzers for laminar blood flow and homogeneous dialysate. Chem. Engng Sci. 29, 173&1731. Davis, E. J., 1973, Exact solutions for a class of heat and mass transfer problems. Can. J. them. Engng 51, 562-572. Drew, T. B., 1931, Mathematical attacks of forced convection problems: a review. Trans. Am. Inst. them. Engng 26,2&80. Jakob, M., 1949, Heat Transfer, Vol. 1, pp. 451-459. Wiley, New York. Kim, J.-I. and Stroeve, P., 1988, Mass transfer in separation devices with reactive hollow fibers. Chem. Engng Sci. 43, 247-257. Kim, J.-I. and Stroeve, P., 1989, Selective and enhanced mass transfer in hollow fiber membranes with facilitated ionSci. 45, 99-114. pair transport. J. Membrane Newman, J., 1969, Z’he Graetz Problem, pp. 2-15. Report UCRL-18649, Lawrence Radiation Laboratory, University of California, Berkeley, CA. Stroeve, P. and Kim, J.-I., 1987, Separation in mass-exchange devices with reactive membranes, in Liquid Membranes: Theory and Applications (Edited by R. D. Noble and 3. D. Way), ACS Symposium Series No. 347, pp. 39-55. Tereck, C. D., Kovack, D. S. and LeVan, M. D., 1987, Constant-pattern behavior for adsorption on the wall of a cylindrical channel. Znd. Engng Chem. Res. 26, 1222-1227.

Chemical Engineering Science,Vol. 45, No. 9, pp. 2994-2998, 1990. printi in Great Britain.

The limits ( First

received

ooos-zso9~

of applicability

17 November

$3.00 + 0.00

Q1990FergamonPmspk

of pressure

1989; accepted

INTRODUCTION

Two different approaches can be distinguished among the correlations for pressure drop across a fixed or expanding bed of solid particles. In the first approach a general form of the Ergun equation is used with a combination of two terms contributing to the total pressure drop:

drop correlations

in revised form

16 February

1990)

where the functionsf,(u) and fs(u) are independent of the bed voidage. The second approach is based on the common voidage dependence for all flow conditions and the pressure drop correlation can be generally written as (Foscolo et al., 1983)

=f*(U)&-y1 Lap\ \ /n.e

E)

L

wheref*(u) depends only on the flow regime, and exponent n has values between 4.6 and 4.8 [e.g. Foscolo et al. (1983), Kmiee (1976) and Richardson and Jeronimo (1979)]. Both of the above approaches are widely used for the prediction of pressure drop, bed expansion and minimum

Shorter Communications fluidization or bubbling velocities. A natural question arises concerning the limits of validity of eqs (1) and (2). An attempt to solve this problem is given below. THEORETICAL Starting with a general equation for the drag force on the particles in a bed:

2995

there is a minimum on the Q vs E curve. That would meanwith respect to the given interpretation of the criterion Qnbthat the particIes with higher values of sti would have an increasing tendency to a freely bubbling bed. However, experience with non-expanding powders shows that the highest values of .sti exhibit large particles of irregular shape which in turn have a high tendency to local channelling during fluidization. This contradiction suggests that the validity of eq. (4) should be limited exclusively to the decreasing part of the function. The minimum can be found from the relation

where u d LQ and E s Ed, it was shown in our previous paper (PunEochZ et al., 1990) that the following expression can be derived from eq. (3) taking into account eq. (1):

3 --_=o E’

1 (1 - E + Re*)’

(9)

and for the limiting value of E it holds that Q=(-$2)M,z=,_s>RP,+; where Re* is a modified

E = 0.634(Re*

(4)

Reynolds

Thus, the range of validity relation

number defined as

E < 0.634(Rc*

Re=-.

By combining

cd PPI

+ 1).

of eq. (4) is limited + I).

(12) = G, eq. (4) can be 1

+L

nb =l-sti+R&

.smb’

(7)

It has to be noted that, for the bed with negligible expansion, the left-hand side of eq. (7) is equivalent to the criterion of the quality of fluid&&ion, Q,, suggested in PunEochBi’ et al. (1989, 1990). i.e.

(8) Higher values of Q,,,/ imply easier formation of bubbles and consequently an increasing tendency to a freely bubbiing regime. Vice versa, lower values of Qrnl correspond to the evolution of spouting, local channelling, etc. As criterion (8) is based on the idea of the perturbation of force equilibrium at the onset of bubbling, it is obvious that eq. (7) represents the generalized form of Q,,,./ which is also applicable for expanding powders. A graphical representation of eq. (4) is given in Fig. 1. It can be seen that for the given value of Re* -z 0.577

:35

(11)

eqs (4) and (10) we have

P

For the onset of bubbling, where ( F.& written in the form

(10) by the

.ss

.4s

Fig.

1. Unlimited

i.e. the analytical expression for the curve limiting the validity of eq. (4) (Cc. Fig. 2). When using eq. (2) to evaluate Q we obtain

where 4.6 d n Q 4.8. As eq. (2) was successfully tested for the latest stages of the expansion of fluidized systems (Foscolo et al, 1983). it can be expected that eq. (13) can also be used for expanding beds. Considering the practical identity of eqs (12) and (13), and taking into account the above conclusions, the following requirements can be stated for the physical validity of eq. (l), particularly with respect to the form of functions fi(u) and

f2W (9 Relation

(11) must be valid for all corresponding values (Re*, 6). (ii) With increasing E relation (11) has to converge to eq. (10) for expanding systems, i.e. 0.634(Re*

.65

.75

EPS Q vs E function.

+ 1) -P E for sufficiently large E.

.a5

-95

Shorter Communications

2996

I

4’ .3S

.4S

.SS

.65

.75

.a5

.9S

EPS Fig. 2. Q vs E function with limit curve.

Both of the above statements arc the necessary conditions for the application of eq. (1) in the wide range of experimental conditions (from packed to expanded bed). A problem which arises is the tack of knowledge of the functions fi(u) and f&) and hence of Re’. There are many empirical correlations in which both the functions are approximated by constants. Relation (11) gives the possibility of determining the range of validity for these correlations in the following way: the value of Re* is estimated from the given correlation and the validity of relation (11) is tested. In the positive case the correlation is accepted; otherwise eq. (2) is recommended.

at the minimum bubbling point using several sets of experimental data. From eq. (5) it follows that Re* = (1.75/150)+ Re.

RESULTS To confirm theoretical considerations given above, the applicability of the Ergun equation (Ergun, 1952) was tested

Table

Material

(k$&‘)

t = 20°C Type of equation*

#J

2.4 5.4 4.5 6.7 11.1 3.1 7.6

1.00 1.00 0.69 0.65 0.61 0.38 0.37

0.422 0.407 0.511 0.513 0.526 0.688 0.700

0.652 0.675 0.658 0.667 0.685 0.643 0.655

1 1 1 1 1 2 2

1.3

0.91

0.425

0.643

1

46.1 12.6 21.0 49.7 54.5

0.98 0.80 0.74 0.56 0.41

0.457 0.443 0.456 0.517 0.584

0.728 0.915 0.752 0.856 0.804

: 1 1 1

0.359 0.505 0.505 0.63 0.715 0.505 0.715

Sand

2650

0.27

Sand Corundum Ash Keramzit

2650 3330 1680 1570

0.55 1.05 0.715 1.325 1.425

1180

powders:

Re,,

2500 2500 I330 1330 1330 950 950

Diakon

1. Non-expanding

6, (mm)

Glass bead Glass bead Coal Coal Coal Polyethylene Polyethylene

smb

LIMt

0.078 0.118 0.141 0.155

0.04 0.09 0.12 0.16

1 1 1 1

0.542 0.496 0.48 1 0.465

0.634 0.635 0.635 0.635

1 1 1 1

0.190 0.220

0.22 0.36

1

0.452 0.442

0.636 0.637

:

tLlM = 0.634(Re& + 1) (Ret, = 0.0124 Re,,). ‘1 = Ergun equation, 2 = eq. (2).

(14)

The classification relation (11) was applied for the minimum bubbling conditions. Missing shape factors in Geldart (1973) and Rietema (1973) were deliberately substituted by the value $ = 1. The results for non-expanding materials given in Table 1 show that the application of the Ergun equation is very acceptable. Particles of polyethylene represent the extremely irregular, low-density material, which is not classifiable in Geldart’s system. The data for expanding powders arc summarized in Table 2. The values of hb for fresh and spent catalysts are close to the limiting curve, which gives the possibility of the

Reference

Ho et al. (1983)

PunEoch6i

Geldart

et al. (1989)

(1973)

Shorter Communications Table 2. Expanding

Mate#al

Cracking

catalyst

Fresh catalyst

Spent catalyst

750

loo0

1500

24.5 61 62 65 73 99 128

t = 20°C

Re,,

%Zb

LIM’

Cl

0.673 0.608 0.535 0.56 1 0.576 0.482 0.459

0.634 0.634 0.634 0.634 0.634 0.634 0.634

Type of equation’ 2 1 1 1

0.01 0.02 0.03 0.04 0.06 0.07

0.645 0.629 0.645 0.645 0.638 0.627 0.630

0.634 0.634 0.634 0.634 0.634 0.634 0.634

4s 62 75 87 95 115

0.02 0.02 0.03 0.04 0.05 0.06

0.675 0.630 0.635 0.600 0.610 0.610

0.634 0.634 0.634 0.634 0.634 0.634

2 l(2) 2(l)

41

0.776 0.765 0.742 0.730 0.701

0.634 0.634 0.634 0.634 0.634

2 2

0.00

Reference

Rietema (1973)

: 1

25 39 55 68 75 100 108

39.7 55 57 78 103

Polypropylene

powders:

2997

Geidart

(1973)

ii2j

Geldart (1973)

&) l(2)

Rietema (1973) ; 2

+LIM = 0.634(Re$, + 1) (Re:, = 0.0124 Re,,). ‘1 = Ergun equation, 2 = eq. (2).

NOTATION

application of both types of correlations. In agreement with the results of Gibilaro et al. (1988), the decreasing particle diameter and/or particle density restrict the use of the Ergun equation. The data for a cracking catalyst show surprisingly low values of sti when compared with the other catalysts. However, the missing values of the shape factor do not allow more detailed analysis of this fact. The cohesive forces play the decisive role in the behaviour of polypropylene, which is the typical material for application of eq. (2).

cross-sectional area of bed, m3 mean particle diameter, m drag force on the particles in the bed, N function [see eq. (I)] function [see eq. (I)] function [see eq. (2)], kg m-* se2 effective weight of the particles in the bed, N height of the bed, m coefficient [see eq. (2)] pressure difference across the be& Pa function [see eq. (411 bed temperature, “C linear velocity of gas, m/s Reynolds number [see eq. (6)] modified Reynolds number [see eq. (S)]

CONCLUSIONS

The necessary conditions for the applicability of eq. (1) in the wide range of operational conditions were formulated. It was shown that eq. (2) is a limiting case of eq. (1) with respect to the voidage function. The criterion for classiEcation of pressure drop correlations was suggested and tested on experimental data. The problem arising due to uncertainty of the values of the shape factor indicate that the proposed deterministic approach should be substituted by fuzzy reasoning in the future. M. PUNCOCHAfi+ J. DRAHOS J. C:ERMAK Institute of Chemical Process Fundamentals Czechoslovak Academy of Sciences Rorvojov6 135 165 02 Prague 6-Suchdol Czechostovakia

‘Author

to whom correspondence

should be addressed.

Greek

letters

4

porosity of bed viscosity of gas, Pa s density of gas, kg/m3 density of particle, kg/m3 shape factor of particles

Subscripts mb mf %s

minimum bubbling conditions minimum fluidization conditions at linear velocity E( and porosity e

& c1 Pf

PP

REFERENCES Ergun, S., 1952, Fluid Row through packed columns. Chem Engng Prog. 48,89-94. Foscolo, P. U., Gibilaro, L. G. and Waldram, S. P., 1981 A

Shorter Communications

2998

unified model for particulate expansion of fluid&d beds and flow in fixed porous media. Chem. Engng Sci. 38, 1251-1260. Geldart, D., 1973, Types of gas fluidization. Powder Technof. 7, 285-292. Gibilaro, L. G., Di Felice, R. and Foscolo, P. U., 1988, On the minimum bubbling voidage and the Geldart classification for gas-fluidized beds. Powder Technol. 56, 21-29. Ho, T. C., Yutani, N., Fan, L. T. and Walawender, W. P., 1983, The onset of slugging in gas-solid fluidized beds with large particles. Powder Technol. 35, 249-275. Kmiec, A., 1976, Some remarks on the Richardson-Zaki equation. Chem. Engng J. 11, 237-238.

Chemical Printed

Enginewing

Science,

in GruntBritain.

Vol. 45. No.

9, tq~. 299&3001.

An elongational

(Received

PunEochBi, M.. DrahoH, J., Bradka, F. and &-ma, J., 1989, Characterization of the quality of fluid&&on in gas-solid fluid&d beds. International Conference on Mechanics of Two-phase Flow, Taipei, Taiwan, June, pp. 481-485. PunEochBi, M., DrahoB, J. and CermBk, J., 1990, The dependence of fluidization regime upon the bed structure at the onset of fluidization. Chem. Engng Sci. 45, 764-766. Richardson, J. F. and Jeronimo, M. A. de S., 1979, Velocity-voidage relations for sedimentation and fluidization. Chem. Engng Sci. 34, 1419-1422. Rietema, K., 1973, The effect of interparticle forces on the expansion of a homogeneous gas-fluid&d bed. Chem. Engng Sci. 28, 1493-1497.

1990.

mm-2509/90 s3.00 + 0.w Q 1990 F’cqamon Press pk

flow model for drop breakage in stirred turbulent dispersions

15 January

1990; accepted

INTRODUC?TION The value of the average drop size in a stirred liquid-liquid dispersion is required for the analysis of rate processes occurring in it. This is normally achieved through the determination of the maximum diameter of stable drops, d,,,. A number of workers have reported expressions for the prediction of d,,,. Most of them use the basic expression first derived by Hinze (1955) by comparing the inertial stress due to turbulent velocity fluctuations with the restoring stress due to interfacial tension. His final expression is d 11111 = c’(We)_O.6. D Sprow (1967) found the value ofthe constant to vary between 0.126 and 0.15, whereas Lagisetty et al. (1986) found it to be 0.125. Coulaloglou and Tavlarides (1976) have discussed the various correlations available in the literature based on eq. (1). It has been reported by many workers (Arai et al., 1977; Konno et al., 1982; Lagisetty et al., 1986; Davies, 1987; Calabrese et al., 1986) that the maximum drop size increases with an increase in the d.ispersed-phase viscosity. Equation (1) cannot be used to predict the effect of dispersed-phase viscosity as it neglects the viscous forces generated in a drop prior to its breakage. Arai et al. (1977) were the first to propose a model for predicting d,,, which incorporated the effect of dispersed-phase viscosity. They described the breakage phenomenon through a Voigt element which simultaneously takes both the restoring stress due to interfacial tension and the viscous stress due to flow inside the drop, into account. They assumed that the turbulent pressure fluctuation is periodic and the drop breaks when the deformation strain reaches a critical value. Lagisetty et af. (1986) modified the model by considering the interfacial stress value to pass through a maximum and the breakage process to be completed within the lifetime of an eddy. They tested the model for Newtonian, non-Newtonian and Bingham plastic dispersed phases. Koshy et al. (1988a) successfully predicted the effect of surfactants on d,,, by intro-

for publication 21 March

1990)

ducing in the model of Lagisetty et al. (1986) another stress, generated due to the difference in static and dynamic interfacial tensions. The same basic ideas of the model of Lagisetty et al. (1986) have also been successfully employed for predicting d,. values when the dispersed phase is mildly viscoelastic, having time constants of the order of 0.1 s or less (Koshy et al., 1988b). However for hiily viscoelastic dispersed phases, no drop breakage was observed experimentally. Instead, the dispersed phase was found to be present in the form of big blobs which elongated into threadlike structures when they passed through the impeller zone. The model was unable to predict such behaviour. All the models used to predict the effect of viscosity of the dispersed phase on d,, implicitly assume shear flow in the drop during its deformation prior to breakage. They also assume a somewhat arbitrary condition for breakage. It is known that there is elongational flow inside an eddy. During eddy-drop interaction, therefore, the drop is likely to undergo elongational rather than shear flow. As the drop will take the shape of an elongated cylinder, the condition for breakage would naturally arise out of jet instability theory, unlike the contrived one now used. Further, the hypothesis of elongational flow inside the drop has the potential of explaining the failure to break highly vi-elastic fluids into drops (Koshy et al., 1988b). Thus, an attempt has been made in this work to propose an alternate framework for analysing drop breakage based on elongational flow during its deformation prior to breakage. This model is different from the existing shear flow models both in the nature of flow assumed and the condition necessary for breakage. THE MODEL Voigt model When a drop experiences a stress due to turbulent velocity fluctuations, it tends to deform. As its shape undergoes a change, the interfacial tension tends to restore it to its original shape.. Further, the deformation is associated with an extensional flow inside the drop which generates a viscous