Liquid distribution in packed columns

Chemical

Engineering Science, Vol. 47. No.

8. pp. 2095-2097,

1992.

oolx~25cq92 ss.00 + 0.00 Q 1992 Persamon Press Ltd

Printed in GreatBritain.

Liquid distribution in packed columns (First received 21 March 1991;

accepted in revised form 30 August 1991)

INTRODUCTION

Liquid redistribution in packed columns takes the form of a developing liquid flow in the wall region as liquid is transferred between the wall and the bulk regions of packing. The transport of liquid within the packing has been studied particularly for older packings such as raschig rings and berl saddles. For newer, high-porosity plastic and metallic packings, there is much less information; however, preliminary experiments suggest that the phenomenon is at least as important as for older packings (Ibrahim, 1986). The preliminary experiments suggest moreover that the manner in which the liquid flow approaches the equilibrium value is a little different from the pattern of older packings as shown, for example, in the experiments of Gunn and Farid (1978). The mathematical description of redistribution was attributed at first to the equation

potential flow can describe, for example, liquid redistribution in gas-saturated packings without gas flow that are not welldescribed by the isotropic model. Is the basic assumption of isotropy valid? The 1978 analysis, in which the potential field was supposed isotropic, gave the equation

(2) so that, if the spread factor K is constant. it cancels out and the equation for a steady potential distribution is obtained. If the field is not isotropic, different axial and radial spread factors may be introduced. If those spread factors K, and K, are constant, the equation describing the potential field is K

au _=-a.7

~a rar

au r-.( aZ >

~ZP *ss+--

K,, r

(1)

where u is the axial component of liquid velocity, z and r are the axial and radial coordinates and K is known as the liquid spread factor. This equation was used to describe the spread of liquid from a point source into an unconfined packing. The same equation has been used to describe redistributiorrwhen liquid is distributed uniformly. It is now accepted that, for a confined liquid flow, the enhancement of flow in the neighbourhood of the wall and the corresponding depletion in the bulk of the packing, are due to the effect of the walls of the vessel. It might be expected that the influence of the surrounding walls would be expressed as boundary conditions to eq. (l), and different sets of boundary conditions have been suggested by Cihla and Schmidt (1957) Porter and Jones (1963), and Dutkai and Ruckenstein (1968); but the descriptions are phenomenological in nature and not related to the physics of the wall effect. A mechanistic description of the wall effect was given by Gunn (1978) who pointed out that the layer of packing next to the wall included a region of higher porosity in which individual packing elements are aligned by contact with the wall. The resistance to flow offered by the aligned elements constituting the wall region is therefore less than the bulk region of packing, so increasing flow in the wall region. The permeability to liquid flow in the wall region is greater than the bulk permeability. In this paper the basic idea of a potential-driven redistribution is developed further, an extended equation for redistribution is derived and an analytical solution is given for the extended equation. The solution is compared with some preliminary measurements of liquid redistribution in high-porosity plastic packings. THE POTENTIAL FOR LIQUID REDISTRIBUTION In a detailed discussion of possible mechanisms for redistribution within packings, Gunn (1978) outlined the possible influences of pressure, gravity, axial and radial velocities, and interfacial tension acting in isolation and in cooperation. While the basic equation of continuity must be satisfied, it does not, by itself, allow the velocity field to be solved. However, the early experimental observations were interpreted in terms of a potential-driven flow, and measurements of liquid redistribution in raschig rings under conditions of two-phase flow are well-described by a potential-driven model that includes the effect of axial and radial liquid velocities in an isotropic potential field. It is therefore worthwhile to ask whether an extension of the basic concepts of

a

rap -

ar ( ar >

=o

(3)

where the axial and radial velocities associated potential field are u=

-KK,,--

ap

u=

az’

If a coordinate transformation

ap ar

is now introduced,

x7

K.4.5 x = ex,

z=

-?&---.

with the

-

K RS

g=

Jr-7

2 RS

(5)

then on substitution into eq. (3) the potential field equation is obtained in isotropic form. If a stream function + is now defined so that 1 ati U=2nrdr

(6)

1 a* -2rrrax

“=

(7)

the partial differential equation (3) may then be cast in the form

aq

a++

~+~-;~=O.

1w

(8)

This differential equation describes the distribution of the stream function within the bulk of the packing, that is to say beyond one particle diameter from the wall. The boundary conditions are determined by the continuity of potential and flow within the wall region, and at the boundary of the bulk region. This has been discussed fully elsewhere (Gunn, 1978) and that discussion applies here with the sole change of replacing the permeability K for the bulk region by the permeability or axial spread factor K,,. The solution for the isotropic case is given by Gunn (1978), and the solution for the anisotropic case is obtained by using the transformation inverse to eq. (5). For an axisymmetric distribution, F(R), of liquid at the top of the packing the stream function is

2095

2a: J,(hR)R=

* G

= mz~ C(2 + 24)* + a:] CJ&.)1* X X

exp[a#(ZL 1

- Z)]

+ exp (a,@Z)

1 + exp (2cQL) RF(R)J,(qR)dR

+ BR*

> (9)

Shorter

2096 where the following Z=“,

dimensionless

variables

Communications

are used,

Q,=80l/min -Theoretical Experimental 1.0 c x

ri

8 = 0.086 _~___-------x-x

and the a, are the roots of (2 + 2&J,(a) When the distribution packing,

+ c&(a)

= 0.

b

(5

(11)

of liquid is uniform over the top of the

0.4

c

x

X-Y

0.2 t

-X-X 2

ti IL, = yRZ

at

Z=O,OtR<

1.0

x

.r.-.-----ye 6 8

4

112)

and the stream function for the anisotropic case is related to the variables R and Z when the initial distribution is uniform by * --=4(1 tiT

A = 0.8 4 = 0.662

12

10

Dimensionless

Fig. 1. Comparison of stream function

X-X

14

height

of theoretical and experimental values for 1” plastic pall rings. (-) 8 = 0.086; (- - -) e = 1.

++-A) Total liquid flowrate

n=1

x

cc2 + WY + dlJ,(n.)

exp [a,9(2L

-I

- Z)]

A PRELIMINARY

+-f-R’ 1++

of liquid distributed

COMPARISON

kg m I m2 S

l.O-

+ exp (a,@Z)

1 + exp (2a,BL)

with (1 - y) as the fraction region.

= 10.876 y = 0.8 +=5.09

o.e*.x_x (13)

$

in the wall

3

R

x

0.929 Y x x x x

=

x-~-x-x-xTrx

0.6 . 0.4

.

R=0.560

Y. ~~xxx~xxxxux

WITH

0.2

EXPERIMENT

Because of developments in newer materials and methods of fabrication, the use of ceramic raschig rings has been greatly reduced, and replaced by high-porosity plastic and metallic packings. The newer packings can support much higher gas and vapour flows because the dry porosity of such packings is greater than 0.9. In a series of preliminary experiments, Ibrahim (1986) has supplemented the experimental measurements of Farid and Gunn (1978) by carrying out a series of experiments on 1” plastic pall rings supplied by the Norton Company. He used equipment of Farid (1977) for the initial distribution of water at the top of the packing and in the wall region so that the division of liquid between the two regions could be controlled. However, the liquid distributor was modified by the insertion of several additional tubes to allow operation of the column at gas rates up to twice the rates that could be sustained by ceramic raschig rings. The arrangement for the measurement of liquid distribution at several pack heights was the same as that described by Farid (1977). The experimental measurements of stream function at three fixed positions and for several different bed lengths were fitted to eq..(13) by non-linear least squares. The two parameters 4, the equilibrium flow ratio, and 0, the fraction of radial to axial spread factor, were found by minimising the sum of squares of the experimental data about eq. (13) by changing 4 and 8. A set of experimental data compared to eq. (13) is shown in Fig. 1 for plastic pall rings at an initial distribution given by y = 0.8. The agreement between the experimental data and eq. (13) is very close. The value of the ratio 8 required for this experiment in which the gas flow rate was zero, was close to 0.10. Therefore, in this particular case the radial spread factor was abour 1% of the axial spread factor. The dotted line on this figure corresponds to equal axial and radial spread factors as assessed in the 1978 analysis. The dotted line shows substantial disagreement between experiment and the 1978 analysis, but agreement between experiment and the present theory is close. In the 1978 experiments of Farid and Gunn, there was also found a significant difference between experiment and theory

,x&x-X

R =

0

0.179

&x~xyx-x-x-x’.x’x’xxx-x-x-x-x-~ 2 4 6 Dimensionless

height

. 8

of pocking,

10

I/q

Fig. 2. Comparison of experimental and theoretical values of stream function for $’ raschig rings without gas flow, 0 = 1 (Farid and Gunn, 1978).

Total

liquid flowrete Ges flowrate

l.O-

0.8 5, -x-x-x,x 5

OX

+

0.4

= 10.876 = 0.250 y = 0.8 4 = 4.14

kg m / m* s kg m I m* s

R = 0.829 _X_X_X_X-X-X-X-~X-X

-

R = 0.550 L X _ ‘k7(7c

?u-+Lx_&Lw

x-x-x

0.2 R - 0.179 x.x,x,w-x-x-x-~-~~~-x~~~~~~-~ 0

2 Dimensionless

4

6 height

of pocking,

8

10

I / ri

Fig. 3. Comparison of experimental and theoretical values of stream function for $” raschig rings with gas flow, 6 = 1 (Farid and Gunn, 1978). for raschig rings in the absence of gas flow. A typical experimental result for this case is shown in Fig. 2 with the theoretical line corresponding to 8 = 1, for equal axial and radial spread factors; a value of 0 much less than one in this particular case would also have greatly improved agreement between experiment and theory. For experiments in which there was two-phase flow over the raschig rings the best value of 8 was indeed close to one, as shown in Fig. 3 for raschig rings under conditions of twophase flow. However, for high-porosity modem packings the

Shorter Communications value of 8 or the ratio K&K,, may be seen subsequently to be an important index of the hydrodynamic characteristics of liquid flow over packings.

Greek a, Y

D. J. GUNN

:

Department of Chemical Engineering University College Swansea Singleton Park, Swansea SA2 8PP, U.K.

2097

letters roots of eq. (11) fraction of liquid distributed in the bulk region defined by eq. (5) defined by eq. (10) stream function total liquid flow to column

NOTATION

=w JO

K K RS K RS Km.

L P PW %u r 2 U ”

x z

REFERENCES

cross-sectional area of wall region Bessel function, zero order, first kind packing spread factor axial permeability and spread factor radial permeability and spread factor permeability (axial spread factor) in the wall region total length of column potential for redistribution potential for redistribution in the wall region liquid volumetric flow rate in the wall region radial coordinate radius of bulk packing dimensionless radial coordinate ( = r/ri) axial velocity radial velo&y transformed axial coordinate according to eq. (5) axial coordinate

Cihla, Z. and Schmidt, O., 1957, Coil. Czechoslovak. Chem. Commun. 22, 896. Dutkai, E. and Ruckenstein, E., 1968, Liquid distribution in packed columns. Chem. Engng Sci. 23, 1365. Farid. M. M., 1977, Studies in chemical reactor engineering: liquid dispersion and distribution in two phase flow in packed columns. Ph.D. thesis. University of Wales. Farid, M. M. and Gunn, D. J., 1978, Liquid distribution and redistribution in packed columns. II: experimental. Chem. Engng Sci. 33. 1221. Gunn. D. J.. 1978, Liquid distribution and redistribution in packed columns. I: theoretical. Chem. Engng Sci. 33, 1211. Ibrahim, A. A., 1986, Liquid distribution in packed columns. M.Sc. thesis. Universitv of Wales. Porter, K. E. and Jones, G. C., 1963, A theoretical prediction of liquid distribution in a packed column with wall effects. Trans. Instn Chem. Engrs 41, 240.

ChcmfcalEq~tnrerin~Science,Vol. 47, No. 8, pp. 2097-2t0O. 1992. Printedin Great Britain.

oo(R2509/92 ss.00 + 0.00 8 1992 Perpmon Press Ltd

Pressure drop for laminar flow of viscoelastic fluids in static mixers (Received 2 July

1991; accepted

INTRODUCTION

Static mixers are widely used in processing of highly viscous Newtonian and non-Newtonian fluids. Recently. Shah and Kale (1991) have presented an empirical correlation between friction factor&,, and generalised Reynolds number, Re;,,, for Newtonian and inelastic fluids. They have defined fsnr in terms of porosity of a given static mixer and ReX, assembly. Their correlation was developed for Kenics and Sulzer type static mixers only. Although they have demonstrated that viscoelastic fluids exhibit higher pressure drop, no correlation has been rep&ted in the literature for viswelastic fluids. EXPERIMENTAL Figure 1 represents a schematic view of the experimental set-up. Two different pipe sizes, 27 mm and 54 mm in diameter, were fitted with 24 elements of static mixers of corresponding size. Three different types of static mixers, Kenics,

for

publication 15 November 1991)

Sulzer and Komax were used which were fabricated by M/s. MAMKO Ltd. Aqueous solutions of sodium salt of carboxy methyl cellulose (CMC. MV grade, M/s.. Cellulose Products of India Ltd.), polyacrylamide (PAA, Rishfloc-440, MV grade, M/s. Rishab Chemicals) and glycerol (Hindustan Lever Ltd.) were used as teat fluids. Flow rates were varied from 1.1 x 10e6 to 1.1 x 10-*m3s-l. Pressure drop was measured using a mercury manometer. The flow properties were measured using Weissenberg Rheogoniometer (model R19) and HAAKE rotoviscometer (model RV3) over the shear rate range 4-1OOOs-‘. The viscous behaviour was described by a power law, ‘T -

KW”

and the primary normal stress different similar relation, Tll

-

722

=

pair.

(1) was described by a (2)