a simplified hydrodynamic model for a pulsed sieve-plate extraction column
TRANSCRIPT
ELSEVIER
Chemical Engine ring
8 Gessing
Chemical Engineering and Processing 36 (1997) 385-395
A simplified hydrodynamic model for a pulsed sieve-plate extraction column
Swati Mohanty *l Alfons Vogelpohl Imtitut fiir Therm&he ?/erfflhiensteclzrzi, TU Claustlznl, 0-3367s Clmstlml, Gerr!xq
Received 10 January 1997; received in revised form 5 March 1997; accepted 6 March 1997
Abstract
A simplified stage-wise hydrodynamic model for simulation of a pulsed sieve-plate extraction column has been presented. The model is based on a drop population balance which takes into account drop breakage and coalescence in each stage. Each stage is represented by the actual column compartment between two sieve plates. Although experimental break-up parameters have been used in the model, the coalescence coefficients have been taken as the fitting parameters and the values which gave the best fit with the experimental data have been used. The predicted values have been compared with data from two different diameter pilot plant columns and the agreement is found to be quite satisfactory for the system butylacetate-water, The model can be used for coalescence parameter estimation of different drop sizes, which will be taken up as part of the extension of the present work. 0 1997 Elsevier Science S.A.
Keywords: Liquid-liquid extraction; Modeling; Pulsed sieve-plate
1. Introduction
Although liquid-liquid extraction is the second most important unit operation in the chemical industry, next only to distillation, the literature is not so exhaustive as in the case of distillation. In order to design an opti- mum extractor, the hydrodynamic and the mass trans- fer behavior of the system should be well understood. The dispersed phase in the case of liquid-liquid extrac- tion undergoes changes and loses its identity continu- ously as the drops break and coalesce. This makes the study of hydrodynamics quite complex.
The pulsed sieve-plate extractor (PSE), which has been found to have high throughput, high separation efficiency and insensitivity towards contamination of the interface, has led to its wide applicability, particu- larly in the extraction of radioactive materials. An empirical model for predicting the hydrodynamics in a PSE has been proposed by Kumar and Hartland [l] and several population balance models have been pro-
* Corresponding author. Current address: Regional Research Lab- oratory [CSIR), BhubanesLvar 751013, India.
0255-2701/97~$17.00 Q 1997 Elsevier Science S.A. All rights reserved. PUSO255-2701(97)00010-X
posed by various authors. A stagewise model for the transient behavior of a sieve-plate extraction column, taking into account the back flow and assuming con- stant hold-up, has been developed by Blass and Zim- merman [2].
Garg and Pratt [3] have developed a population balance-model for a pulsed sieve-plate extraction taking into account experimentally determined values for drop breakage and coalescence. For drop breakage, the as- sumption made by them is that a drop breaks into two equal size daughter drops, and for drop coalescence, the assumption is that the drop coalesces with drops of the same and adjacent size intervals. Major work on the population balance model has been carried out by Casamatta and co-workers who applied the model to various types of extraction columns, a good review of which is presented by Gourdon et al. [4]. Dimitrova Al Khani et al. [5] have applied this model for d.ynamic and steady-state simulations of a pulsed sieve-plate extraction column with the assumption that the drop breaks into three equal sized drops and taking the coefficients accounting for drop coalescence as the fitting parameters. Milot et al. [6] have used the same
386 S. hfoolmnry, A. Vogelpohl;‘Ci~erwicai Engineering and Processing 36 (1997) 385-395
basic model for the dynamic simulation of a pneumati- cally pulsed liquid-liquid extraction column. assuming that the drop breaks into two drops of identical size and the coalescence frequency between two drops fol- lows the equation proposed by Delichatsios and Prob- stein 1’71. Haverland et al. [8] have applied the model for steady-state simulation of a pulsed sieve-plate extrac- tion column. Although they have taken into account drop breakage, drop coalescence has been assumed to be negligible in the operation range considered. The daughter drop size distribution has been represented by a Beta function. Later studies [9>10] also show that the daughter drop size distribution of a drop when it undergoes breakage fits well with the beta function. Haverland [ll] has also used a simplified stagewise model to predict the hold-up and Sauter mean diameter for a pulsed sieve-plate column and has found it to fit well with experimental data. He has, however, neglected drop coalescence.
In this study, a simplified stagewise model has been proposed for simulation of a pulsed sieve-plate extrac-
Column Material Glass Diameter 80 and 225 mm Active height 4.3 m
Plates Hole diameter 4mm Plate free area 39.0 % Plate spacing 100 mm
-TL +------Butylacetate
1 +-Water
ii- Pulsator
Fig. 1. Schematic diagram of a pulsed sieve-plate extraction column.
Table 1 Break-up parameters for the butylacetate-water system and 4 = 4 mm
Apf’ (m s-‘)
0.015 0.0175 0.020
Break-up parameters
4isiab km WlZ (mm) (mm)
2.63 6.62 1.00 2.28 6.35 0.94 2.32 6.04 0.86
wlx k
0.342 0.150 0.363 0.154 0.383 0.163
tion column (Fig. I), taking into consideration drop breakage and coalescence. The system chosen is buty- lacetate-water without any mass transfer. The model output has been validated against experimental data obtained from two pilot plant columns having diame- ters of 80 and 225 mm and an effective height of 4.3 m.
2. Model development
The basic unsteady state population balance model as given by Casamatta and Vogelpohl [12] and used by several others is as follows:
= + Rfhd) t0
where P(h,d) is the volumetric drop size distribution function of size d at height h, and when integrated over all drop sizes at height h, gives the hold-up 50 at that height, i.e.
I
4mx VV) = P(h,d)bd
0 zr(h,d) is the drop velocity of diameter d at height h, D(h) is the dispersion coefficient at height h, R(U) is the net generation of drops of size n at height i2.
The first term on the right-hand side of Eq. (1) is due to convection and the second due to the axial disper- sion. The simplified steady-state model is based on the following assumptions:
(1) Each compartment between two sieve plates is considered as a stage which represents a perfectly mixed stirred tank with a drop population homogeneously scattered and the hold-up representative of that mea- sured in the stage considered.
(2) The coalescence takes place just below the sieve plate and drop break-up while the drops pass through the sieve plate.
(3) The effect of axial dispersion has been neglected. The axial dispersion increases with increasing column diameter. The extent to which it increases depends on
Tabl
e 2
Oper
ating
co
nditio
ns,
calcu
late
d an
d ex
perim
enta
l ho
ld-up
an
d Sa
uter
m
ean
diam
eter
for
se
vera
l ru
ns
Para
met
er
Run
no.
1: 4
036
KCD
(m3)
1
x 10
-10
9x
lo-”
2X
IO-“’
8x
lo-”
2.5
x IO
-‘” KC
E (s)
1
x10-
5 Ix
lo-5
I x
1o-5
1
X IO
-5
I x
10-5
m
2.
7 2.
54
2.65
2.
52
2.6
A,J(
m
s-‘)
0.01
5 0.
015
0.01
75
0.01
75
0.02
0 6.
(m
s-
1)
0.06
0 0.
061
0.06
1 0.
061
0.06
I
t&
(111
s-
‘)
0.06
1 0.
061
0.06
L 0.
061
0.06
0 Co
l. di
a.
(mm
) SO
22
5 80
22
5 80
4 (m
m)
4 4
4 4
4
2:
4006
3:
40
33
4:
4003
5:
40
35
Expt
. Cd
. Ex
pt.
Cd
Expt
. Cd
. Ex
pt.
Cal.
EXpl.
Ca
l.
Sarilc
r m
d.
(mm
) 1 2 3 4
Hold-
up,
q~ (“
A)
1 2 3 4
2.54
2.
58
2.43
2.
57
2.41
2.
30
2.29
2.
43
2.35
2.
55
2.42
2.
37
11.4
12
.68
13.1
6 13
.9
13.3
8 14
.58
14.8
13
.55
16.0
9 15
.1
13.5
2 16
.02
2.5
2.28
2.
4s
2.22
2.
38
2.15
2.
34
2.04
2.
17
2.37
2.
48
2.0’0
2.
41
2.37
2.
40
2.13
2.
29
2.04
2.
28
I .96
2.
1 I
2.0
1 2.
34
1.99
2.
39
2.26
2.
39
2.19
2.
27
2.25
2.
29
2.17
2.
1 I
2.17
2.
29
2.10
2.
39
2.46
2.
40
2.30
2.
27
2.37
2.
29
2.23
2.
13
2.33
2.
27
2.25
13.1
4 12
.8
13.0
0 15
.11
14.0
4 15
.0
14.0
3 17
.47
17.0
6 8.
20
9.10
9.
20
13.6
0 17
.3
13.4
9 16
.09
14.8
8 19
.8
14.3
7 19
.09
17.4
6 10
.30
9.97
Il.5
0 13
.60
16.1
13
.40
19.2
4 14
.59
19.9
14
.10
21.8
0 16
.63
Il.30
10.3
4 12
130
13.4
5 17
.7
13.2
5 17
.94
14.6
9 20
.6
13.9
1 20
.55
IS.9
0 11
.30
10.4
8 13
.10
6:
40’05
7:
40
27
8:
4029
9:
40
30
‘: B
9x
10-I’
5x
lo-
” 9x
lo-
” 4x
IO
-” -3
0 1
x 10
-S
I x1
0-5
1 x
1o-5
I
x 10
-S
-‘ \ 2.
1 3.
3 0.
020
0.01
75
0.06
1 0.
047
0.06
1 0.
047
225
SO
4 4
3.2
0.02
0 0.
047
0.04
7 80
4
3.3
9 0.
015
2 0.
047
s 0.
047
s.
80
h 5.
4 $ 2 OS
Expt
. Ca
l. Ex
pt.
Cal.
Expt
. Ca
l. Ex
p1.
Cal.
9 c La
$
2.32
2.
50
2.63
2
2.20
2.
39
2.48
2‘
09
2.
17
2.38
2.
42
k 2.
17
2.55
2.
39
3 %
3 IO
.16
7.8
8.74
Il.2
6 8.
90
9.36
5
11.5
1 9.
30
9.68
11
.49
9.80
9.
83
5
388 S. Mohanty, A. Vogelpohl/Chemical Engineering and Processing 36 (1997) 385-395
the system chosen. In the present study, the model has been validated against data for the system buty- lacetate-water, for which the influence is less impor- tant. It has been shown that the influence of the axial dispersion coefficient (D(h)) on the result is not sig- nificant [4].
The discrete steady-state model is based on making a volume balance of every drop size d, before and after the passage through a sieve plate. The volumet- ric distribution qn then depends on the volumetric dis- tribution q, _ 1 and can be written as:
dd)=q,-l(4+M4
where q,(d) is defined as
(21
f’,, (4 q,(d) = - v??
(3)
Here, qn(d) is the volume fraction of drop of diame- ter d at the 17th stage, q,?- l(d) is the volume fraction of the drop of diameter d at the (II - 1)th stage, R,(d) is the net generation of the drops of diameter d at the n th stage ( = RB + RC), RB is due to break-up (= RB- + RB’), and RC is due to coalescence (= RC+ + RC-).
The term RB- as given by Haverland is as follows:
m - = 2(d), - lsln - ,(d) s
d max pJ* =
zW)q, - dd*)q-rAd”,d) Jd” d
(4)
(5)
where Z(d) is the probability of a drop of diameter d breaking into smaller drops, q&d*& is the fraction of drops of diameter P breaking into drops of di- ameter d, and L-I,, is the diameter of the largest drop in the population.
The discretized form of the terms RCf and RC- can be explained as given below. Coalescence of
2- x
x 6
l- -3
0; 0 05 1 1.5 2 2.5 3 3.5 4 4.50
Column height, m
Fig. 2. Comparison of calculated and measured hold-up and Sauter mean diameter for run no. 3 when KCD = 2 x IO- lo m3 and KCE = 1 x 10-S s.
CoIm diamkr = 80 mn 1.0 - Hole ctian&er=4nxn 7 -I 0.8 -iJ~.=;~=o.ml m’s
0.6 - A$=“.0175 lxis
;;- I_E”1’“‘““[ I : -I-
r- - Height: 4.26m
1.0 1 =_ &ght : 3.21 m
0.8 - 1 ,
.,
Height: 2.15 m
1.0 t ,- -8 B$jS: 1.09 m
0.0 1.0 20 3.0 4.0 Drop diamta cd), mn
Fig. 3. Comparison of calculated and measured drop size distribution for run no. 3 when KCD =2 x lO-‘O m3 and RCE- 1 x 10e5 s.
drops may take place due to two phenomena: (a) when two drops collide with each other; (b) due to difference in rising velocities, where a smaller drop may be absorbed by a larger one.
If the coalescence coefficient in any stage between drops of diameter di and diameter 4 is given by KCD(cl,&), then the number of collisions which result in coalescence is -Ad q(4) q(4) v(di)
- Acr, KCD(d,,rlj) l z;(dj) (6)
Then the decrease in the volumetric fraction of the drops of diameter d due to collision with other drops in the population is given by
s @&us -Q?“~ q _ ,(d.) q _ ,(d) n n KCD (di,d) Sd, u(d)
0 v(diJi) Ml
KCD(d,d)tl(d) 6d (7)
The first term applies for all drops whereas the sec- ond term applies only when ri is not greater than
S. Afohanry, A. Vogelpohl/ Chemical Engineering and Processing 36 (1997) 385-395 389
- calculated SxAer mea diameter
oo- I 2 2.5 3 3:s 4 4.9
Column height, m
Fig. 4. Comparison of calculated and measured hold-up and Sauter mean diameter for run no. 4 when KCD = 8 x 10 - ” m3 and KCE = 1 x low5 s.
d max /2’13. This is due to the fact that when two drops of equal size coalesce, their contribution to the num- ber of drops lost is twice as high as when they coa-
0.6 - 0.4 - 0.2 - 0.0 I-_/ I
Height: 3.21 m
7 E 0.8
5 0.6
i 0.4 2 0.2 ‘2 0.0 3 1.0 ‘2 0.8 ti 2
0.6 g 0.4
0.2 0.0
Height : 1.09 m 0.8 0.6 0.4 0.2 0.0
2.0 3.0 4.0 Drop diameter (d), mm
5.0
Fig. 5. Comparison of calculated and measured drop size distribution for run no. 4 when KCD= 8 x lo-” m’ and KCE= 1 x 10V5 s.
lesce with drops of unequal size. Since the maximum size of the drops in the population is &,,: only drops smaller than c1,,,/21’3 can combine with an equal size drop so that the resultant diameter is not larger than d Ill&X.
If KC’E(di,cl,) is the probability of coalescence due to absorption of drops of diameter (i, and IJ, during the time the two drops spend in the stage, then the number of drops formed due to coalescence between drops L& and (r, is given by [13,14]:
(8)
and the volume fraction of the drops of diameter d lost due to their coalescence with other drops in the population is given by
x KCE(d,,d) Sdi v(d)
Therefore,
RC- =
4
(d&, - &)‘I3
+ qn - l(4) qn - l(d) ___ ~ z(d; + d)2
0 44) v(d)
x (ju(d,) - u(d)l)KCE(di,d) 6d, v(d) (10)
Similarly, an increase in the volume fraction of drops of diameter d due to coalescence by the above two phenomena can be written as
RC+ = qn - l(4) q,z - ,(cl,) 44)
~ KCD(dJJ 6d, u(d) v@>
+ c
‘LZ”~ qn - t(4) qu - ,(cl,) n(di + dJ 0 l-44) v(rlj)
Since only drops smaller than or equal to d/2l” can combine with a drop larger than or equal to itself to give a drop of diameter cl, the upper limit of the integral in Eq. (11) has been taken as d/21’3. Drops larger than d/2113 will combine with drops smaller than d/2 1/3 to give drops of diameter d which have already been considered when integrating from 0 to d/2113.
Tabl
e 3
Hold-
up
and
Saut
er
mea
n dia
met
er
assu
min
g no
co
ales
cenc
e
Run
no.
Calc.
(%
) an
d d,
, (m
m):
q
nz
Sam
e as
whe
n co
ales
cenc
e is
con
sider
ed:
I 2 3 4
wz F
or
best
fit
with
ex
pt.
p:
I 2 3 4
1 2
3 4
5 6
7 a
9
V d
13
(P
d 13
v
d 13
v
d 13
v
d 13
v
d 13
V
d 13
v
43
V d
13
992 =
2.
25
m=
1.8
13.0
9 2.
55
13.7
5 2.
46
14.1
2 2.
41
14.8
6 2.
34
15.0
4 2.
23
17.6
3 2.
11
9.22
2.
46
10.5
3 2.
28
8.80
2.
61
14.4
7 2.
4 15
.06
2.34
16
.45
2.26
17
.09
2.21
18
.15
2.09
21
.02
2.01
IO
.31
2.3
12.3
8 2.
12
9.54
2.
44
15.3
9 2.
33
15.9
3 2.
28
18.1
6 2.
21
18.7
2.
16
20.1
8 2.
05
23.3
8 1.
98
10.9
3 2.
23
13.3
2 2.
07
9.99
2.
31
16.0
0 2.
29
16.5
1 2.
25
19.4
8 2.
18
19.9
1 2.
14
21.3
9 2.
03
25.1
2 1.
97
11.2
7 2.
20
13.7
3 2.
05
10.2
7 2.
33
2.4
2.4
2.45
2.
45
2.25
1.
6 3.
3 2.
9 2.
8 12
.56
2.54
13
.44
2.46
13
.58
2.40
14
.62
2.33
15
.04
2.23
16
.59
2.10
9.
22
2.46
10
.13
2.28
8.
50
2.61
13
.70
2.39
14
.61
2.34
15
.46
2.26
16
.65
2.21
18
.15
2.09
18
.99
1.99
10
.31
2.3
11.6
7 2.
12
9.15
2.
44
14.4
2 2.
32
15.2
7 2.
28
16.6
4 2.
20
18.0
0 2.
16
20.1
8 2.
05
20.2
5 1.
96
10.9
3 2.
23
12.4
0 2.
07
9.54
2.
31
14.8
7 2.
29
15.8
3 2.
25
17.3
7 2.
17
18.9
0 2.
14
21.3
9 2.
03
20.7
0 1.
95
11.2
7 2.
20
12.7
0 2.
05
9.77
2.
33
Saute
r me
an
diam
eter,
mm
Hold-
up,
%
S. Mohunty, A. Vogelpold / Chemical Engineering and Processing 3ZTT9??7j7)38395 391
1.0 1 Height: 3.21 m
-g 0.8 3 0.6 g 0.4 ‘S 0.2 E 0.0 -I
3 1.0 ,g Height: 2.15 m 0.8 ,
E g
0.6 0.4 0.2 0.0
Height: 1.09 m 0.8 0.6
0.4 0.2 0.0
1.0 2.0 3.0 4.0 5.0 Drop diarreter (d), m-n
Fig. 7. Comparison of calculated and measured drop size distribution for run no. 4 when KCD = KCE = 0.
(14)
where S is the cross-sectional area, and r’, and r’d are the volumetric flow rates of the continuous and dis- persed phase, respectively.
The slip velocity is related to the characteristic ve- locity by the following relation:
z =f(cp) = (1 - cp)‘” ( .5)
where u is the characteristic velocity and nl is a vari- able which is a function of the local hydrodynamic condition characterized by the Reynolds number rela- tive to the drop [15]. The factor (1 - ~7) takes into account the effect of the swarm. The simplest relation given by Gayler et al. [16] assumes VI to be equal to unity. Other empirical relations between slip velocity and characteristic velocity are presented by Maude and Whitmore [17], Mersmann [I 81 and Godfrey and Slater [19].
The relationship between the characteristic velocity and the terminal velocity for a pulsed sieve plate has been given by Hussain et al. [20] as
Lf -= 1 - E(d) (16) w,
where
d m(d) = -J-- - APf (A,f j2 1 + q’Hdr + 1 + 0.95H- 1 + 28.3H (17)
and
47’ = 0.0268 + 0.0365~ (18)
LI being the single drop rising velocity or the charac- teristic velocity under pulsation, and ~2, the terminal velocity.
In the present work, H/, has been evaluated using the equation given by Kaskas [21]. The Reynolds number for a spherical drop moving under the force of gravity in a continuous medium is given by
4 4 Re,= 3c J d
(19)
where
For a rigid particle with R, < 2 x lo’, the drag co- efficient is given by
3.2. Correction for hold-up
During pulsation, the drops accumulate beneath the sieve plate before passing through it. This type of regime is obtained for not very intense mixing or with a small throughput. In such cases, a correction has to be made to the calculated hold-up. The com- partment is assumed to be of two parts. one with a volume equal to A(H - AJ and the other A *A,, where A, is the pulsation amplitude, H is the distance between the sieve plates and A is the cross-sectional area of the sieve plate. In the second area, qP is the hold-up just beneath the plate. The value qDF can be calculated by a simple evaluation in the steady-state regime. During a pulsation cycle, the mass balance gives
Qd = Ud,pf (20)
392 S. Mohunry, A. V’ogeipohl j Chrmical Engineering and Processing 36 (1997) 385-395
where ud,p is the volume of the drops accumulated under the plate:
li d.p = A,~, (21)
From Eqs. (20) and (21), qP can be calculated. The corrected hold-up, qP+, in a compartment is
thus calculated from the following equation [8], which is the average of the hold-up, (p,,, and that calculated by the simulation model, 9, so that
(22)
3.3. Break-up pnrnnleters
To describe the break-up process it is necessary to define a probability of break-up: 2. This breakage probability is defined as the fraction of those drops breaking while passing through the sieve plate.
The break-up probability is given by the simple equation
(23)
The &,,,: d,,, and ~1-7 values for the system butylac- etate-water for various plate geometries and pulsa- tion intensities are available [l I].
3.4. Dnughter size distributiolz funcrion
The daughter drop size distribution is described by a classical statistical distribution, the beta function [91?
&( d ) -= d IMT NP, q”) (24)
where the normalizing factor, B(p,q”) is defined by the gamma function as follows:
(25)
The two shape factors p and 4” are related to the experimental first and second moments (xl3 (mean) and s,, (standard deviation)) by
P x -- 13-P+q’r
and
(26)
(27)
The experimental values of -y13 and s13 for the system butylacetate-water for various plate geometries and
pulsation intensities are given by Haverland [l l] and have been correlated as follows:
,3= 1++ [ 1 - ,il.X x L
0.5 for+%
L
s 13=ks for *> 1
(29)
The constants nzx and k, will depend on the pulsation intensity and hole diameter.
The dimensionless daughter drop size distribution function, qTT, is to be converted into a dimensional daughter size distribution function since in Eq. (5), qTT has a dimension of (length) - ‘. Therefore,
(31)
3.5. Sawer wei7n dianleter
The characteristic drop diameter in the population is generally represented by the Sauter mean diameter defined as:
(32)
4. Results and discussion
A computer program written in Turbo Pascal 6.0 was executed on an IBM 486 personal computer to solve the above model equations. Eqs. (12)-(32) was solved to estimate the drop size distribution, hold-up and Sauter mean diameter in each compartment of the extraction column.
The drop size interval chosen was 0.2 mm and the number of classes was varied depending upon the smallest and the largest drop size in the column. The drop size distribution at the distributor obtained from pilot plant data served as the input data. The system chosen was butylacetate-water as recommended by the EFCE (European Federation of Chemical Engi- neering). Data from two= columns, 80 and 225 mm, were used for comparison with the predicted values. The distance between two plates was 100 mm. The break-up parameters experimentally determined by Haverland [I l] were used and are shown in Table 1.
The exponent, r~, in Eq. (15) was found to lie be- tween 2.1 and 3.6 depending upon the flow rate and column diameter. As pointed out by Godfrey and Slater [19]$ m = 1 is not valid for low hold-up and the correlations available in the literature for estimating m do not cover a wide range of physical properties or
S. dfohan(y, A. Vogelpohl/ Clzeinicnl Engineering mzd Processing 36 (1997) 385-395 393
column geometry. The ~12 values calculated using these correlations led to ambiguous results and, hence, were not used. In the present study, the values of LIZ which were found to give the best fit with the experimental data were used. These are lower than those calculated from correlations and are presented in Table 2. As was to be expected, the m value increases with de- creasing agitation intensity.
In the absence of experimentally determined values for the parameters KCD and KCE, these were taken as the fitting parameters to get the best fit with the experimental data. While studying the effect of these two parameters, it was seen that thl: drop size distri- bution could be well represented by KCD rather than KCE. This shows that coalescence is mainly due to collision rather than absorption in the operating re- gion considered. Hence, a very low value of KCE was used which had little effect on the drop size distribu- tion. The KCE and KCD used in this work, therefore, represent a mean value for coalescence between all drop sizes. However, attempts are still being made to determine these parameters experimentally at the In- stitut fur Thermische Verfahrenstechnik, Clausthal (Germany), which would increase the validity of the model.
The operating conditions and the predicted and cal- culated Sauter mean diameter and hold-up values for several runs are given in Table 2. Figs. 2 and 4 show the measured and calculated hold-up and Sauter mean diameter along the length of the column for runs 3 and 4, respectively. Figs. 3 and 5 show the measured and calculated drop size distributions at dif- ferent heights of the column for the same runs. The agreement between the calculated and predicted val- ues is quite satisfactory. There was an appreciable increase in the value of KCD for the smaller diameter column and it was almost constant for the larger di- ameter column with increase in pulsation intensity. The parameter was also higher in the case of the smaller diameter column, indicating that coalescence in the smaller diameter column is higher than in the larger diameter column. From the drop size distribu- tions, we see that the the model overpredicts coales- cence at lower heights. This may be due to the assumption of equal probability of coalescence be- tween all drop sizes, In reality, the probability of coa- lescence between larger drops is smaller than between smaller drops. So at heights not far off from the distributor, the coalescence would be low. Coales- cence is appreciable when the drops have undergone breakage after passing through a number of sieve plates.
Computer runs were also made for KCE and KCD set equal to zero to compare the results with the experimental data, assuming no coalescence between
the drops. The results are shown in Table 3. It can be seen that the predicted Sauter mean diameters do not compare well with the experimental values. The error involved in prediction of the hold-up is of the same order of magnitude. The 1~ values used in these runs were same as those used for previous runs where coalescence was considered, except in the case of two runs, as given in Table 3, which led to ambiguous results. Computer runs were also made by changing the IIZ values to get the best fit with experimental data as given in Table 3. Although the hold-up values im- proved, there was no effect on the Sauter mean di- ameter. Fig. 6 shows the comparison between calculated and experimental Sauter mean diameter and hold-up for run no. 4, and Fig. 7 shows the comparison between calculated and experimental drop size distribution at different heights of the column for the same run. A comparison of the drop size distribu- tion shows that it deviates from the experimental drop size distribution and that the distribution when coalescence is considered is closer to the experimental values. However, the present simplified model can be used to predict the hydrodynamics in a pulsed sieve- plate extraction column quite satisfactorily and the deviation is well within the experimental error. The model validity would be increased if experimentally determined values of KCD and KCE as a function of drop size, column geometry, hold-up and pulsation intensity could be used. With carefully designed ex- periments, the model can be used for estimation of the parameter KCD for different drop sizes. Since the break-up parameters are already known, the only un- known parameter is KCD. The parameter for all drop sizes can be optimized to get the best fit with the experimental values. This will be part of an extension of the present work and a better prediction is envis- aged.
5. Nomenclature
A, pulsation amplitude (m) Ar ~~ Archimedes number Km) normalizing factor in Eqs. (24) and (25) Cd drag coefficient d drop diameter (m or mm) 4 sieve-plate hole diameter (m or mm) d 13 Sauter mean diameter (m or mm)
P dispersion coefficient (m’ s-r) pulsation frequency (s-l)
g acceleration due to gravity (m sP2) h height (m) H distance between two sieve plates (m) KCD coalescence coefficient due to collision
Cm’>
394
KCE
P
P 4”
4
R RBf RB-’ RF RC-’ Re s13
s
t 11
u
ti
3
VE N’r ‘$ 30 We x13
z
S. hfokznty, A. Vogelpoid,’ Chemical Engineering ad Processing 36 (1997) 38S-395
coalescence coefficient due to absorption (4 constant in Eqs. (28) and (23), respec- tively parameter of daughter drop size distribu- tion volumetric drop size distribution (m- ‘) parameter of daughter drop size distribu- tion volumetric drop distribution as defined in Eq. (3) (m-r) dimensionless and dimensional size distri- bution (m- ‘) generation term (m-i) generation due to breakage (m-r) death term due to breakage (m-l) generation due to coalescence (m-r) death term due to coalescence (m-l) Reynolds number standard deviation of the dimensionless daughter drop distribution cross-sectional area of the column (m’) time (s) characteristic drop velocity (m s-l) drop volume (m’) linear velocity (m s - ‘) volumetric flow rate (m3 s- ‘) as defined in Eq. (17) slip velocity (m s-l) terminal velocity (m s-l) Weber number mean diameter of the dimensionless daughter drop distribution break-up probability
Greek letters J%J> gamma function 17 viscosity (MPa s or kg m-l v kinematic viscosity (m2 s-‘) P density (kg mm3) rJ surface tension (mN m-i) Y hold-up 47l as defined in Eq. (18)
Subscripts C continuous phase d dispersed phase max maximum MT mother drop n stage number P plate P-P corrected stab stable 100 probability of breakage is 1
S-
Acknowledgements
One of the authors, Swati Mohanty, gratefully ac- knowledges the Alexander von Humboldt Foundation, Bonn, for financial assistance, and the Director, Re- gional Research Laboratory (CSIR), Bhubaneswar, for permission to publish this paper.
References
[l] A. Kumar, S. Hartland, Prediction of dispersed phase hold-up in pulsed perforated plate extraction columns, Chem. Eng. Process. 23 (I) (1988) 41-59.
[Z] E. Blass, H. Zimmerman, Mathematische Simulation und Ex- perimentelle Bestimmung des Instationaren Verhaltens einer Fliissigkeitspulsierten Siebbodenkolonne zur Fliissig-fliissig- Extraction, Verfahrenstechnik 16 (9) (1982) 652-690.
[3] M.O. Garg, H.R.C. Pratt, Measurement and modelling of droplet coalescence and breakage in a pulsed-plate extraction column, AIChE J. 30 (3) (1984) 432-441.
[4] C. Gourdon, G. Casamatta, G. Muratet, in: J.C. Godfrey.
Pl
I61
[71
I81
191
ilO1
I111
1121
u31
u41
I151
I161
1171
M.J. Slater iEds.), Liquid-Liquid Extraction Equipment, Wi: ley, Chichester, 1994, p. 137. S. Dimitrova Al Khani, C~. Gourdon, G. Casamatta, Dynamic and steady state simulation of hydrodynamics and mass trans- fer in liquid-liquid extraction column, Chem. Eng. Sci. 44 (6) (1989) 1295-1305. J.F. Milot, J. Duhamet, C. Gourdon, G. Casamatta, Simula- tion of pneumatically pulsed liquid-liquid extraction column, Chem. Eng. J. 45 (2) (1990) 11 I - 122. M.A. Delichatsios, R.F. Probstein, The effect of coalescence on the average drop size in liquid-liquid dispersion, Ind. Eng. Chem. Fundam. 15 (2) (1976) 134-137. H. Haverland, A. Vogelpohl, C. Gourdon, G. Casamatta, Sim- ulation of hydrodynamics in a pulsed sieve-plate column, Chem. Eng. Technol. 3 (1987) 84-91. M. Cabassud, C. Gourdon, G. Casamatta, Single drop break- up in a Kfihni column, Chem. Eng, J. 44 (1990) 27-41. K. Eid, C. Gourdon, G. Casamatta, Drop breakage in a pulsed sieve-plate column, Chem. Eng. Sci. 46 (7) (1991) 159.5-1608. H. Haverland, Untersuchungen zur Tropfendispergierung in Fliissigkeitspulsierten Siebboden-Extraktionskolonnen, Disserta- tion, Technical University of Clausthal, Germany, 1988. G. Casamatta, A. Vogelpohl, Modelling of fluid dynamics and mass transfer in extraction columns, Ger. Chem. Eng. 8 (1985) 96-103. X. Zhang, R.H. Davis, The rate of collisions due to Brownian or gravitational motion of small drops, J, Fluid Mech. 230 (1991) 479-504. X. Zhang, O.A. Basaran, R.M. Whan, Electric field-enhanced coalescence of liquid drops, Sep. Sci. Tech. 30 (3-9) (1995) 1169-I 187. P.J. Bailes, J.C. Gledhill, J.C. Godfrey, M.J. Slater, Hydrody- namic behavior of packed, rotating disk and Kfihni liquid-liq- uid extraction column, Chem. Eng. Res. Des. 64 (1) (1986) 43-55. R. Gayler, N.W. Roberts, H.R.C. Pratt, Liquid-liquid extrac- tion. IV. A further study of hold-up in packed columns, Trans. IChemE 31 (1953) 57-68. A.D. Maude, R.L. Whitmore, Generalised theory of sedimen- tation, Br. J. Appl. Phys. 9 (1958) 477-480.
1181 A. Mersmann, Zum Flutpunkt in Fliissig/Fliissig Gegen- stromkoionnen, Chem.-Ing.-Tech. 52 (12) (1980) 933-942.
S. Mohnnty, A. Vogelpohl / Chemical Engiuewing cd Processing 36 (1997) 385-395 395
[19] J.C. Godfrey, M.J. Slater, Slip velocity relationships for liquid- liquid extraction columns, Trans. IChemE Part A 69 (1991) 130-141.
[20] A.A. Hussain, T.B. Liang, M.J. Slater, Characteristic velocity of drops in liquid-liquid extraction pulsed sieve-plate column,
Chem. Eng. Res. Des. 66 (6) (1988) 541-559. [21] A. Kaskas, Berechnung der Stationaren und Instationaren Bewe-
gung von Kugeln in Ruhenden und Striimenden Medien, Diplo- marbeit, Lehrstuhl fur Thermodynamik und Verfahrenstechnik der Technische Universitat Berlin, Gemrany, 1964.