Liquid distribution in packed columns

Liquid distribution in packed columns

Chemical Engineering Science, 1968, Vol.23, pp. 1365-1373. Pergamon Press. Printed in Great Britain. Liquid distribution in packed columns E. DUTKA...

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Chemical Engineering Science, 1968, Vol.23, pp. 1365-1373.

Pergamon Press.

Printed in Great Britain.

Liquid distribution in packed columns E. DUTKAIt and E. RUCKENSTEINS t Research Institute “Chimigaz”, Rignov, Romania SPolytechnic Institute, Bucharest, Romania (Received 10 March 1968) Abstract- An equation for the liquid distribution in a packed column is established by using a diffisiorr al model for the spreading of the liquid in the packing and by assuming that the variation of the flowrate in the wall region is determined by the difference between two flow-rates, one towards the wall and another one returning from the wall. The predictions of the theory are compared with experimental flow profile for four packings. Though the constants appearing in the theoretical equations were determined independently, by means of experiments achieved in some limiting conditions, the agreement between the theoretical and expehmental distributions is very good. INTRODUCTION

THE USUAL design of a packed column is based

on the assumption that the liquid and the vapour are uniformly distributed over its cross-section. Numerous experiments [ I-51 have shown however that there exists a tendency of the liquid to collect at the wall and that an equilibrium distribution is achieved over the cross-section after the liquid has covered a sufficiently long distance in the packing. The usual explanation for the radial motion of the liquid towards the wall of the column is based on the larger void fraction at the wal1[4,5]. It is however very probable that the &se is more complex and that the differences of the curvature radii of the wall and of the packing and the capillary and wetting forces also play an important part [6]. The problem was treated theoretically by means of a diffusional model [7- 151. This implies the assumption that the radial motion of an element of liquid takes place with equal probabilities towards the wall or the center of the column. The first clear formulation of the model is due to Cihla and Schmidt [ Ill, who have used the following diffusional type equation for the distribution of the liquid in a cylindrical bed:

ensemble of discrete particles and not as a continuum as above. The corresponding finite difference equation was used by him instead of Eq. (1). Cihla and Schmidt [ 111 have solved Eq. (1) by using at the wall the boundary condition $=O

for

r=a.

(2)

This boundary condition was suggested to them by the fact that the liquid does not penetrate into the wall. In reality, the diffusional model is valid only up to the distance a- 6 from the centre, 6 being the thickness of the wall region, since in the last region the equal probability assumption (see above) is no longer valid. The correct boundary condition must express the fact that the diffusion flux at the distance a - 6 equals the variation of the liquid flow-rate in the wall region -D27r(a-8)

(5 > =s. r=a_6

(3)

Porter and Jones [ 131 assuming that w,,,=

v

(4)

(1) In a recent paper[ 151, Jameson has treated the problem by considering the packing as an

have obtained a boundary condition which, though more realistic than that of Cihla and Schmidt, leads however to an equation for V

1365

E. DUTKAI

and E. RUCKENSTEIN

which gives values too large as compared to the experimental ones. The departing point of the present work was a new boundary condition suggested to us by an analogy with the process of accumulation of a surface active agent at a fluid interface. The transfer of the surface active agent towards the interface obeys the convective-diffusion equation. At the interface the diffusion flux equals the variation of the superficial flow-rate of the surface active agent. The penetration into the interface takes place via an adsorption-desorption mechanism. In a packed column the distribution of the liquid also satisfies a diffusional type equation and Eq. (3), valid for the wall region, is similar to that used for the surface active agent. We shall assume that the penetration of the liquid into the wall region takes place also via an “adsorption-desorption” mechanism, the ‘adsorption’ rate being proportional to WrEa-* and the ‘desorption’ rate to V. In this manner one may write for the wall region = kW,,,_6-

k’V = g.

(5) If diffusion is the rate determining step, then one may consider that equilibrium is achieved between ‘adsorption’ and ‘desorption’ and consequently that k Wr=a-8 - k’ V = 0.

basis of the new boundary condition equations for the distribution of the liquid, but also to determine for a number of packings the three phenomenological constants appearing in the equations. It may be noted that though these constants were determined independently, by means of experiments achieved in some limiting conditions, the agreement between the theoretical and experimental distributions is very good.

THEORETICAL

It was mentioned in the Introduction that the theoretical problem consists in the solution of Eq. (1) for the boundary conditions =kW

= -k’V=g +a

dz’

(7)

Equation (7) differs slightly from Eq. (5). Owing to the fact that 6 Q a, a - 6 was replaced in those equations by u. Considering the case in which for Z = 0 the liquid is uniformly distributed over the crosssection, we have also that W= W,=const.

for

Z=O

and

0 < r
The solution of Eq. (1) for the boundary conditions (7) and (8) is (see Appendix 1)

(6)

In this particular case Eq. (5) leads to the boundary condition suggested by Porter and Jones which, as mentioned above, does not agree in a completely satisfactory manner with experiment. It will be shown below that Eqs. (5) lead to a good agreement with experiment. By using the language inspired from the above analogy one may say that as compared to Porter and Jones’ contribution, in the present paper one considers that both the diffusion and the “adsorption-desorption” determine the rate of the accumulation process. The aim of the present paper is to obtain on the

x

Jdq,r/a) J,(q,)

exp[-

qn2F]}

(9)

and

1366

x em [-423] (10)

Liquiddistributionin packedcolumns

k' k

where

a-

k Tra2k,,

~rD f l = "k

1-(Vd~ra2Wo) 1 -- Vp _

(VJWo)

1raZVp"

(11) This equation is a consequence of Eq. (13) and of the material balance

and qn are the roots of the equation

2J, ( qn) [ flqn -- -~n] -- Jo ( qn ) =0.

Wo

~ra2Wo = ~ra2W®+ V®.

(12)

For large values o f Z (Z --> oo) W® -- ~ l+a

(13)

V® = - - ~ kWo k' l + a "

(14)

(3) The returning coefficient k' by feeding the liquid only at the wall so as to form a film and by determining the wall flow-rate for a sufficiently short column. The column being sufficiently short it is possible to neglect in Eq. (7) kWr=a as compared to k' V and one obtains dV

dZ = - k ' v For the flow-rate of the liquid inside a radius equal to d, there results a

2~r

l+4(l+a)~

(18)

which by integration leads to In V = In Vo--k'Z,

o0

rWdr=~d

(17)

(18')

= V0 being the value of V for Z = 0.

0

1

x J • ( qexp[-q.~-~jj~. n d[ / aoDZ]) )

(15)

The three coefficients D, k and k' appearing in the above equations can be determined in the most simple manner from experiments carried out in some limiting conditions as follows: (1) The 'diffusion' coefficients in a column with no wall effect by feeding the liquid from a point source located axially at the top of the column. The height of the packing is such that the liquid does not reach the wall of the column. I f Q is the flow-rate of the source, one may write / 9 W = ~ e -m4°z. 4,rDZ

(16)

(2) The ratio k/k' by determining the equilibrium distribution and by using Eq. (13) or (14). It is convenient to use the following equation

EXPERIMENTAL Two columns were used. One of them of 150 mm dia. and the other of 250 mm dia. Each column is equipped at the bottom with a collecting device consisting of a set of four concentric metal cylinders, assembled to the bottom of the column. Together with the wall of the column the four cylinders delimite five annular areas, from which liquid can be collected to a set of measuring vessels. The outer cylinder is placed at a distance of 3 mm from the wall thus enabling the determination of the flow-rate at the wall. The other cylinders are equally spaced. For the determination of the "diffusion coefficient" D the liquid was fed through a pipe 4 mm i.d. placed axially at the top of the packing. In the experiments carried out for determining the liquid radial distribution, an initial uniform distribution over the cross-section was achieved by means of a distributor having a flat bottom into which 100 respectively 200 tread-mounted brass nozzles of 2 mm dia. are uniformly disposed. The uniform distribution of

1367

E. DUTKAI

and E. RUCKENSTEIN

Table 1. Values obtained for the various phenomenotogical coefficients 150 mm Column Packing 10 mm 15 mm 25 mm 15 mm

Raachig rings Raschig rings Raschig rings Intalox saddles

D (cm)

(cm-‘)

(cm)

0.138 0.1% 0.274 0.169

o*oso 0JMo 0.036 0.060

540 568 6.61 7.40

k’

k

the liquid was verified in each experiment by collecting the liquid which leaves the distributor directly in the collecting device. The verification was made by bringing the movable distributor near to the collecting device. The uniformity is achieved by adjusting the heights with which the bottom plate of the distributor is exceeded by the nozzles. For the determination of the coefficient k’ one uses a short column equipped with an over-flow which enables the feeding of a thin liquid film on the wall. Before each experiment the verticality of the column was thoroughly verified. The experiments were performed with water and no gas phase was circulated through the column. The flow-rate of water in the column of 150 mm dia. was comprised between 250 and 750 l/hr and in the column of 250 mm dia. between 800 and 1500 l/hr. In Table 1 are synthetised the values obtained for D, k and k’ by means of the methods mentioned above. We notice that these constants are practically independent of W. (Tables 4 and 5). The diffusion coefficient was determined by plotting log ( WZ/Q) vs. rMz/Z. The intercept and the slope of the straight line (see Eq. (16))

log?=-

log4&-O-4342402

bf2

k’lk

250 mm

Column a

Q

(cm+?) 0*00925 Q.612 0*00705 O-805 0.00544 1.040 OJ_M)810 0.700

(c!L)

(cknll_k

6.25 666 8.04 8.45

O+Nl800 0+)0600 OW448 O-00710

0.250 O-334 O-447 0.282

The computation of the ratio k’/k was made by means of Eq. (17). In Table 3 the pertaining experimental data are given for the four packings. The determination of the returning coefficient k’ was made by means of Eq. (18’) valid, as mentioned above, for short columns. The same value of k’ was obtained for various values of Z if Z < 15 cm. For larger values it is no longer possible to neglect W,=, in Eq (7). The

(16’)

allow one to compute D. It may be noted that the same value for D is obtained from the slope or the intercept. In Table 2 are given the experimental data used to obtain the diffusion coefficient and in Fig. 2 log ( WZ/Q ) is plotted vs. t-,$/Z for one of the packings (Rashig rings of 15 mm). 1368

Table 2. Experimental datafor the determination of D

Z (cm)

r2 (cm*)

W.S/Q

rM21Z W.Z/Q (cm)

(cm-l)

0.293 0.885 1.665 0.631

0.335 0.107 0.026 0.178

0.1% O-984 2.581 0.631

0.320 O-126 0.020 0.200

0.293 1.475 0.252 1.260

0.224 0.077 0.237 0.094

l-475 0.885 2,325 0.121

o-053 o-119 0.016 O-398

10 mm Raschig rings 15 25 35 35

4.46 22.12 58.22 22-12

0.589 0.366 o-109 0.435

15 mm Raschig rings 22.50 22.50 22.50 35.00

440 22.12 58.22 22.12

0,375 0.478 0.128 O-487

25 mm Raschig rings 15@0 15@0 17.50 17.50

4.443 22.12 4.40 22.12

0.394 0442 0.357 0460

15 mm Intalox saddles 15.00 25*00 25.00 36.30

22.12 22.12 58.22 440

0.302 0.406 0.091 0.288

(S = area of the segment)

Liquid distribution in packed columns 7

ISmm Rarhig

rhg8

I.0 040 0.60 050 E

/

2

Fig. 2. log ( WZ/Q ) vs. r,g/Z.

E

-----l

L i



WOkl

2 -liquid Fig. 1. Experimental apparatus. 1 -Column; device; 5 -pipe of distributor; 3 -nozzle; 4 -collecting 10 mm dia.; 6-flow-meter.

In order to verify the equations obtained for the liquid distribution, the flow-rates in each of the five annular areas were determined for various heights of the packings. The height of the packing was increased until the equilibrium distribution was achieved. In our experiments the equilibrium was attained for heights comprised between 1 and 1.5 m. The data, expressed as fractions from the total flow-rate, are plotted together with the theoretical curves, in Figs. 3-6. Also, in Fig. 7 the present equation is compared with that obtained by Porter and Jones. For the coefficients D, k and k’ were used the values from Table 1, determined as mentioned above. CONCLUSION

experimental data used for the computation k’ are given in Table 5.

of

The experimental results confirm the present theory. The main assumption being expressed

Table 3. Experimental data obtained for V, Packing

150 mm column

250 mm column

IV, = 2.83 l/cm2 hr W, = 240 l/cm2 hr 10 mm 15 mm 25 mm 15 mm

Raschig rings Raschig rings Rawchig rings Intalox saddles

0.380 0445 o-510 0.410

1369

0.200 0.250 0.310 0.220

E. DUTKAI

and E. RUCKENSTEIN

Table 4. Experimental data obtained for V, and k’lk150 mm column Wdl/cm%r)

1.41

VP k’lk

Packing: 15 mm Raschig rings 250 mm column

2.83

4.24

1.60

2.40

3.00

0.460

0.445

0.435

O-263

O-250

0.232

040664

040705

040740

04IO561

040600

040662

Table 5. Experimental data for the determination of k 150 mm

Column

V,

V

250 mm Column k’

Z

Packing

k:

VLl

V

k’

av.

(UW

(Vhr)

(cm-r)

(cm-r) 0.050

(cm)

Whr)

(I/hr)

(cm-‘)

15 mm Raschig rings

5-o 5.0 5-o

113.40 195.00 113.40

88.33 150.65 93.10

0~0500 195Go o-0515 336.70 o-0395 195.00

152.40 263.30 160.20

o-0493 0.0492 o-0393

195-00 113.40

15940 95.17

oG403 0.035 1

336.70 195.00

274.45 163.45

04409

25 mm Raschig ring

5-o 5-o

15 mm Intalox saddles

5.0 5.0

195.00 113.40

162.65 84.22

0.0363 0.0596

336.70 195.00

279.70 145.00

o’0354 O-0372 0.0593

5.0

195*00 14460

0.0600

336.70

247.90

0.0611

10 mm Raschig rings

o 040 0.036 o 060

IOmm Raschig rings 15mm

15omm column

i! (cm)

250 cobnn

15Omm eckmn

Fig. 3. Flow in each circular segment vs. Z for 10 mm Rashig theoretical curve, 0 experimental points; rings: W, = 2.83 I/cm*hr for 150 mm column; WI,= 2-40 I/cm*hr for 250 mm column.

Roschig rings

Z (cm)

250mm

column

Fig. 4. Flow in each circular segment vs. Z for 15 mm Rashig rings:----theoretical curve, 0 experimental points; W, = 2-83 l/cm% for 150 mm column; IV,,= 240 l/cm*hr for 250 mm column.

1370

Liquid distribution in packed columns 25mm

I530

IO0

60

250mm c&ma

Roschigrhgs

I30 I50

15 30

60

IO0

Ztcm)

15Omm

130 150 c0hn

Fig. 5. Flow in each circular segment vs. Z for 25 mm Rashig rings:BheoreticaJ curve, 0 experimental points; W, = 2.83 l/cm*hr for 150 mm column; W, = 240 I/cm*hr for 250 mm column.

Lkm)

uthors theoretical curve, Fig. 7. Flow at the wall:----------Porter and Jones theoretical curve, 0 experimental points; 150 mm column W, = 2.83 l/cmzhr.

15mm lntdox aoddbs

Jo,JI Bessel functions of the first kind coefficient of flow towards the wall returning coefficient from the wall 472 Eigenvalues given by Eq. ( 12) Q flow-rate of the point source r radial variable in polar coordinates average radius of a circular segment F-M k k'

Ii++

00

II

I

b?m

a0

l50mm

I I l-?-l ID0 130!300630 cohmw

am)

1 60 250

I

II

IO0

130!50

r-i and r,

cohllrwl

Fig. 6. Flow in each circular segment vs. Z for 15 mm lntalox theoretical curve, 0 experimental points; saddles: W, = 2.83 l/cm% for 150 mm column; W, = 2.40 I/cm*hr for 250 mm column.

by Eq. (5) one may conclude that the rate of penetration of the liquid in the wall region depends on the rate of diffusion through the packing towards the wall region and on the rate of liquid accumulation (equal to the difference kW,,, k’V). NOTATION

: D

radius of the column radius of an inner zone of liquid diffusion coefficient 1371

S V VCI V,

the two radii of a circular segment area of a circular segment flow-rate at the wall value of V for 2 = 0 value of V for 2 + co

mV VP ma2W, W flow-rate per unit area WO value of W for 2 = 0 W, value of W for 2 * (x Yl Bessel function of the second kind z distance to the top of the packing (cm) ff,P quantities defined by Eqs. (11) 6 thickness of the wall region constants in Eq. (A. 15) from the &&, appendix

E. DUTKAI and E. RUCKENSTEIN REFERENCES [l] KIRSCHBAUM E.,Z. Ver. dt. Ing. 1931 75 1212. [2] WEIMANN M., Chem. Fabr. 1933 6411. [3] BAKER T., CHILTON T. H. and VERNON H. C., Trans. Am. Inst. them. Engrs 1935 31296. [4] KIRSCHBAUM E., Destillier und Rektifiziertechnik. Springer 1960. [5] KIRSCHBAUM E., Chemie-Zng-Tech. 1956 28 639. [6] SHERIDAN M. B. and DONALD M. B., Ind. Chemist 1959 35 439,487. 171 THERMANN K., Destillieren und Rekt.$zieren. Spamer 1928. [8] TOUR R. S. and LERMAN T., Trans. Am. Inst. them. Engrs 1939 35 719. [9] TOUR R. S. and LERMAN T., Trans. Am. Inst. them. Engrs 1944 46 79. [lo] MULLIN J. W., Trans. Inst. them. Engrs 1959 37 89; 1960 38 308. [l l] CIHLA Z. and SCHMIDT O., C&t Czech. Chem. Commun. 1957 22 896. [12] CIHLA Z. and SCHMIDT O., CoNn Czech. Chem. Commun. 1958 23 23. [13] PORTER K. E. andJONES M. C., Trans. Inst. Chem. Engrs. 1963 41240. [14] KOLAR V. and STANEK V., Colln. Czech. Chem. Commun. 1965 30 1045. [15] JAMESON G. J., Trans. Inst. them. Engrs 196644 198.

APPENDIX

+ q&. = 0.

Let us solve the equation

Equation (A. 10) has a solution of the form

(A.11

(A.13)

for the following boundary conditions: W= W,=const.

forz=O

(A.3

=E=kW,=,-k’V. &

where F(r) satisfies a Bessel type equation

(A.3)

Introducing the function y=2?r

(A.12)

(A.14) One obtains

I’r(W(I

F=A.J,(~)+B,y,~)

W,,)dr

(A.15)

where

and eliminating V by means of the equation (A.5) 0

+ -X for r = 0.

deduced from Eq. (A.3), there results The boundary condition (A. 12) leads to the equation (A.6) #= y=O

for

z=O

2[Pq,-$1

(A.16)

(A.7) for the eigenvalues qn, where

The new substitution JIcy+k

The function $ has therefore the form

wf12 k’ a2+&

(A.9)

leads to (A.lO) for

z=o

JI= i

1372

(~~18)

The boundary condition (A. 11)leads to akW+x --= k’ a*+_ k mk’

(A.ll)

A.rJl(~)exp-[~z].

I: AJ,(q,x) Ii=*

= ‘P(X) x=;.

(A.19)

Liquid distribution in packed columns Since (q.r-41np)

~~xJI(q”x)J*(qlnx)dx =4~~;(q,)J*(qA)--4”J:(4n)J1(4n).

there results, by means of Eq. (A. 16), that (4”%3[

J; xJI(qnx)Jl(q&)

=A.[~~XJIP(q,x)dx+2~~11(q,)].

(A.20)

(A.22)

For the coefficients A, one obtains

dx+2BJ*(qfn)J1(q.)]

=o. 1+: (A.21)

A. = !$&_ $+&

[2sq”*_3+q”‘++)~

(A.23)

Using Eqs. (A.19) and (A.21), one may write &rJ1(qmMx)

dx+zMl)J,(q,)

=

The quantities W and V are given by Eqs. (9) and (10) from the paper.

RCume-On etablit une equation pour la distribution de liquide dans une colonne gamie, en utihsant un modtle a diffusion pour la repartition du liquide darts la garniture et en supposant que la variation du debit prbs de la paroi est determinte par la difference entre deux debits, un vers la paroi et I’autre venant de la paroi. Les previsions apporttes par la theotie sont comparees a la courbe experimentale de debit pour quatre colonnes gamies. Bien que les constantes qui apparaissent dans les equations theoriques aient CtC dCterminCes independamment au tours d’exptriences realisees darts des conditions tres limit&es, on remarque que l&cord est satisfaisant entre les distributions theotiques et exp&imentales. Zusammenfassnng-Es wird eine Gleichung fur die Fltissigkeitsverteilung in einer Fhllkorpersaule aufgestellt, wobei ein Diffusionsmodell fiir die. Ausbreitung der Fiiissigkeit im Bereich der Fiillkiirperverwendet wird; dabei wird angenommen, dass die Schwankung der Striimungsgeschwindigkeit im Wandgebiet vom Unterschied zwischen den beiden Striimungsgeschwindigkeiten. d.h. dejenigen in Richtung auf die Wand hin und detjenigen von der Wand zurhck, abh>. Die theoretischen Voraussagen werden mit experimentellen Striimungsproflen von vier FiillkSrpermaterialien verglichen. Obwohl die in den theoretischen Gleichungen auftretenden Konstanten unabhiingig an Hand van Experimenten unter gewissen Grenzbedingungen bestimmt wurden, konnte doch eine sehr gute Ubereinstimmung zwischen theoretischen und experimentellen Verteilungen festgestellt werden.

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Vol. 23 No. I I-F