HUNGARU\NJOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPREM 
Vol. 30. pp. 137- 142 (2002) 

STUDY ON THE TURBULENCE IN PULSED STAGEWISE EXTRACTION 
COLUMNS: TURBULENT MACROSCALE 

G. ANGELOV and C. GOURDON1 

(Institute of Chemical Engineering, Bulgarian Academy of Sciences, Acad. Bonchev St., Bl. 103, 1113 Sofia, 
BULGARIA 

1Ecole Nationale Superieure d'Ingenieurs en Arts Chimiques et Technologiques, Toulouse, FRANCE) 

Received: February 06, 2002 

The authors studied a particular turbulent parameter - turbulent macroscale, which characterizes the size of large energy 
containing eddies, and it is one of the basic parameters with essential role in the concept of turbulent motion. Pulsed 
turbulent liquid flow in stagewise column type apparatuses with internals of disks and rings is considered. The 
distribution of turbulent macroscale L over the stage is determined as depending on stage geometry and Re number. 

· Insignificant changes of L are registered in the course of pulsation cycle. Similarly, the dynamic flow parameters, 
expressed by the value of Re number, pulse amplitude and frequency, demonstrate slight and not pronounced influence 
on the size of turbulent macroscale. Most important effect on L has the stage geometry with pronounced influence of 
interplate distance and lesser dependence on plate free area. It was found that the zones of large L-values correspond to 
zones of intensive drop breakage, located experimentally. 
A correlation is proposed for determination of the mean value of turbulent macroscale as a function of stage geometry 
parameters. Comparison with other correlations is made along with discussion on their advantages and inconveniences. 

Keywords: plate extraction colunms,-flow hydrodynamics, pulsed flow, turbulen~l!lacroscale 

Introduction 

As known from practical experience, the turbulence 
increases the efficiency of transfer processes. Many 
chemical apparatuses operate under turbulent flow 
regimes, although in most cases they are empirically 
tuned. Studies on turbulent flows are helpful for 
understanding and quantification of the influence of 
turbulence on process intensification. This paper 
focuses on a particular turbulent parameter - turbulent 
macroscale. It characterizes the size of energy 
containing eddies and is one of the basic parameters 
with essential role in the concept of turbulent motion. 
As far as the energy containing eddies produce larger 
interphase contact surface by dispersing greater drops to 
smaller droplets, it is worthy to study them through the 
behavior of their characteristic parameter. 

Turbulence model 

The expression for turbulent macroscale is derived 
applying dimensional analysis with assumption that the 
amount of dissipated turbulent energy is determined by 

the energy containing large-scale motion at high Re 
numbers [1,2] 

L=cJs2!3:... 
£ 

(1) 

where L is turbulent macroscale characterizing the size 
of the large energy containing eddies, k is the kinetic 
energy of turbulent motion per unit mass, e is the 
dissipation rate of kinetic energy, CL is an empirical 
constant of order of unity [1]. 

Regarding Eq.( 1 ), larger L-value means higher 
kinetic energy with respect to its dissipation rate. 
Consequently, larger turbulent macroscale means larger 
level of non-dissipated energy that can provoke drop 
breakage, respectively, increase of the interphase 
surface. Thus, the zones of higher values of L should be 
expected to be places of drop breakage and interface 
surface production. 

It is seen from Eq.( 1) that L depends on the ratio of 
two other terms - kinetic energy and its dissipation rate 
(k and e). Their distribution over the flow field can be 
derived from their transport equations, included in the 
standard k- e model of turbulence {2}. 



138 

::. 
1_<:: p:•' 
r "?:: ,D H 

-
'F' 
i..,j~ 

n: 
n. 

a) b) c) 

Fig.] Apparatus scheme: a) internals axonometry; b) general 
column scheme; c) limits of simulation domain 

where U; is the mean velocity vector, v - kinematic 
viscosity, Vt- turbulent viscosity, x - space coordinate, 
c,., C~<> C8, C,r, C82 are constants of the model taking 
standard values, recommended in the relevant literature 
[3]. 

In previous papers [ 4,5] we have reported in details 
the full set of model equations ~or the considered case 
along with the method of resolution and boundary 
conditions for the particular apparatus geometry 
described below. So, adapted k- E model equations are 
resolved in order to determine the distribution of L. 

Apparatus description 

The turbulent macroscale L is studied in case of 
stagewise pulsed columns (Fig.l) with .internals of discs 
and rings (called also discs and doughnuts column [6J). 
This apparatus design has shown great performance in 
various solvent extraction applications [7-10] and is 
produced in large scale for industrial applications [11]. 

Immobile discs and rings are alternately placed at 
equal distance in cylindrical column body forming a · 
series of identical stages. The turbulence is enhanced by 
a piston like mechanical device inducing osciUati 1g 
motion to the fluids. 

As should be expected, the geometrical periodicity 
creates flow periodicity along the column. Previous 
experimental hydrodynamic studies have shown that in 
n~ality the flow picture is repeatedly reproduced on the 
adjacent stages. and flow symmetry with respect to 
column axis exists [12}. So, a typical flow pattern is 
observed in the vertical cross~section of a stage limited 
by the column wail. and two rings {Fig.Jc). 

The stage geometry is defined by two dimensionless 
parameters: 

- the ratio of interplate distance H and column 
diameter De bounds the stage height to column size 

h=HfDc (4) 

the ratio of plate cross-section open to flow SP and 
column cross section Sc defines the plate free area e 

e= Spl Sc =D/ I D/ = (D/- Di) I D/ (5) 

The free area of both plate configurations - disc and 
ring, is taken equal as practiced in the real DRC design. 

Results and Discussion 

The results for turbulent macroscale L are obtained by 
simulation of incompressible Newtonian liquid flowing 
in a typical stage at various flow regimes and variable 
stage geometry. Their effect on turbulent macroscale L 
is registered and discussed below. 

Evolution of Turbulent Macroscale during a Pulsation 
Cycle 

The pulsation created by a piston device corresponds to 
periodic sinusoidal pulsation described by the 
expression 

U(t) = nApf cos(2nft) (6) 

where U(t) is the superficial velocity of pulsed flow at 
the instant t, Ap is pulsation amplitude, f = 1/T is 
pulsation frequency, T is the period of pulsation. 

The results show that turbulent macroscale L is 
slightly changed during the pulse period T. For 
example, the maximal values of L at two moments of 
the pulsation corresponding to rather different instant 
flow velocities (tff = 0 - maximal velocity and tff = 0,2 
- flow velocity approaches zero) are changed from 
3,8.10-2 m to 4,1.10"2 m. It can be conclude'd that the 
value of the turbulent macroscale L does not depend 
significantly on the instant flow velocity during the 
pulsation cycle. 

Influence of Flow Regime 

The above result implies that the flow regime in the 
apparatus should not affect pronouncedly the turbulent 
macroscale. This suggestion is verified below. 

The mean flow regime is defined by Renumber 

UD 
Re=~ 

v 
(7) 

where Um is the mean superficial flow ·velocity; v is 
kinematic viscosity. 

Integrating Eq.(7) over a period of pulsation, the 
mean flow velocity is obtainr.d 



0.05 .---------------

~ 0.04 
E. 
--' 
<ll 

~ 0.03 
§ 
ro 
E -5i 0.02 
:; 
.0 
:; 
f-

0.01 

0.00 -t-----.-----,------1 
4000 8000 12000 16000 

. Renumber 

Fig.2 Influence of Re number on turbulent macroscale L: • 

Lmax (h=0,155; F=0,43); 0 Lm (h=0,155, F=0,43); e Lmax 
(h=0,086; F=0,23); 0 Lm (h=0,086; F=0,23) 

and Eq.(7) becomes 

Re= 
2A fD 

p c 
v 

(8) 

(9) 

In this study, Re number of a water flow has been 
var~ed from 5 000 to 15 000. It corresponds to pulse 
regimes with different intensity of the turbulence - from 
unstable transition regimes at lower Re to established 
turbulent regimes at higher Re xalues [13]. 

Fig.2 represents maximal (Lmax) and mean (L,J 
values of turbulent macroscale on the stage as 
depending on Re number. The results illustrate two 
e~treme s~ge configurations - . smaller interplate 
distance with small plate free area and larger interplate 
distance with large plate free area. It is seen that in all 
~ases the flow regime has slight and not pronounced 
mfluence on the size of turbulent macroscale. Similar 
weak influence was observed by Aoun et al. [6] that 
have reported a correlation for L with small power 
tenns for the quantities A, and f. 

On the contrary, the curves on .Pig.2 indicate 
qualitatively for stronger influence of the stage 
geometry on both mean and maximal L-values. 

Influence of Stage Geometry 

The range of variation of geometry parameters in this 
study is 

- Plate free area e = 0,2 - 0,5 
- Interplate distance to column diameter ratio h = 0,08 

-0,4 

The above values have been selected by practical 
considerations [14]. For example, a direct vertical flow 
through the colunm will reduce apparatus efficiency due 
to axial mixing phenomena. To avoid it, the disc 

139 

; : : ; : : : : ~ ~ :: ~ .! ! ; ~ ; ; t !: ! ~ ~ 1 ~ '""' "'~: 
'<- ~ ~ .-~ .. .., " 

Fig.3a Size and distribution of turbulent macroscale Lon 
stages at same interplate distance and different plate free area e: 

e=0,23; Linax= 3,8.10-2 m; l!llli!l'li!ili2,5.10-2 <L<3,5.10-2 m; 
- 3,5.10.

2 
< L < 3,8.10-2 m 

..• 
·~.., .. 

• ~~. ~ "' \'J ·• " 1) 

", r ,, 

' 
' ' . .. 

Fig.3b Size and distribution of turbulent macroscale Lon 
stages at same interplate distance and different plate free area e: 

e = 0,43; Lmax = 4,1.10-2 m; -2,5.10-2 < L < 3,5.10-2 m; 
-3,5.10'2 < L< 4,1.10-2 m 

diameter Dr should not be smaller than ring aperture Dr-
The above requirement limits the possible upper value 
of e to 0,5, at which Dr= Dd. 

Influence of Plate Free Area 

The distribution of L on the stage is shown for a lower 
and a higher value of the free area at fixed interplate 
distance h (Fig.3). It is seen that in case of small free 
area, the zones of high L-values are smaller in volume 
and are foU1ld in both parts of the stage over and under 
the disk. In case of large free area, the zones of high L-
values become larger. However, the zone under the disk 
bas disappeared. The maximal values of L in these two 
cases are rather similar, i.e. there are no significant 
changes in the turbulent macroscale due to plate free 
area, only the volume and location of the zones are 
affected. · 

Influence of Interplate Distance 

Fig.4 illustrates the case of different interplate distance 
h at fixed plate free area. lt is seen that the interplate 
distance affects strongly the volume and the maximal 
values of the turbulent macroscale. When doubling the 
interplate distance, L increase significantly, the maximal 



140 

FigAa Size and distribution of turbulent macroscale L on 
stages at same plate free area and different interplate distance h: 

h:=;0,086; Lmax = 1,9.10"2 m; -1,5.10-2 <L< 1,9.10-2 m 

L-value is increased as much as twice, and the zones of 
high L-values become much larger in volume. The 
conclusion is that the interplate distance is the most 
important parameter controlling the size of turbulent 
macroscale. 

The results of Figs. 3 and 4 refer to ascending (down 
to up) flow. The picture for the descending phase of the 
pulsation is symmetrical with respect to disk - the 
marked zones take mirror positions on the other side of 
the disk. In all cases, larger values of turbulent 
macroscale are located in the median zones between the 
disk and respective ring. 

The experimental observations [15] in a column of 
same geometry have shown that namely these median 
zones in the stage are places of intensive drop breakage. 
This fairly good correspondence of modeling and 
experience attributes more reliability to the simulation 
results obtained here. 

Correlation Development 

Summarizing the influence of geometry and dynamic 
parameters. the following general tendencies have been 
put in evidence: 

- The dynamic parameters Ap and f (Re) have 
negligible influence on L; 
The turbulent macroscale rises when increasing the 
interplate distance or/and plate free area; 

- The influence of interplate distance on L is stronger 
than that of plate free area. 

Additional considerations are taken into account in 
the development of corresponding correlation: 

~ When plate free area e approaches 1 - case of 
hollow column. no matter the value of h. L should 
tend to a value Lo corresponding to hollow column; 

- The interpiate distance parameter h =HIDe shm·ld 
control L when h S 1 • i.e. until the interplate 
distance is smaller than the column diameter. At 
interplate distance larger than column diameter. the 
latter becomes the limiting geometry dimension and 
the influence of h should diminish. 

An expression translating these tendencies has been 
developed 

Fig.4b Size and distribution of turbulent macroscale Lon 
stages at same plate free area and different interplate distance h: 

h = 0,155; Lmax = 3,8.10-2 m; -1,5.10-2 < L < 2,5.10-2 m; 
llllllllliii2,5.10-2 <L<3,5.10.2 m; -3,5.10-2 <L<3,8.10:

2
m 

The unknown empirical terms have been identified 
by a procedure based on Gauss-Newton algorithm, 
using the mean space and time values (Lm) of computed 
discrete values of turbulent macroscale at various 
geometry parameters: 

DJLo=22,38; a= 1,49; 13 = 0,48 

Comparison with Existing Correlations 

For a hollow column, Brodkey [16] has proposed the 
expression (subscript o designates the terms in a hollow 
column) 

(11) 

From Eqs.(l) and ( 11) it is derived 

k 312 D 
Lo~-0-=_c 

Sa 8,8 

(12) 

This result shows also, that the turbulent macroscale 
in a hollow column depends only on column diameter 
and is independent on flow regime (Re). 

Our correlation (10) can provide the value of Lin a 
hollow column. In this limit case, the free area e 
becomes equal to 1 and Eq.( 1 0) is reduced to 

DJL., = DJLo = 22,38 
from where 

(13) 

The results (12) and (13) are of the same order. 
Their similarity could be considered as relatively good, 
accounting that the empirical coefficient CL in (1) has 
been taken approximately equal to 1. 

More fine coincidence is found applying another 
approach. The friction velocity in the wall zone Ur is 
bound to the kinetic energy and its dissipation rate by 
the equations [1,2] 

(14) 



E=U/!CkL 

From (14) and (15) it is obtained 

(15) 

(16) 

In the above relation Ck = 0,41 is von Karman 
constant [2], C'" = 0,09 is standard constant of k-£ 
turbulence model [3]. 

Applying Eq.( 16) for hollow column with use of our 
result for Lo (13), it follows 

Eo= 8,9 k 312 
D o 

c 

(17) 

Comparing Eq.(l7) to the relation of Brodkey (11), 
encouraging similarity is seen. 

Aoun et al. [6] have proposed the correlation 

(18) 

with the following values of the empirical coefficients: 

a=8,4.10-2; b=0,27; c=0,31; d=-0,05; g=-0,05 

The above values show relatively weak dependence 
of Lm on the pulsation parameters Ap and f. This is close 
to our observations that allowed us to develop a simpler 
correlation excluding the participation of dynamic 
terms. 

Regarding the free area term e in (18), its influence 
is even more important than that of interplate distance. 
In our opinion, the main parameter defining the space 
where the vortices can develop and extend to the 
available geometry limits, is the interplate distance and 
not the plate free area. For this reason, it seems more 
logical to expect stronger influence of the interplate 
distance as we have registered, studying the stage 
distribution of L. This dependence is clearly reflected 
by the correlation (10). 

Unfortunately, correlation (18) cannot be 
unconditionally applied for parameter values outside the 
particular studied range, which is approximately the 
same as our parameter range. For example, the increase 
of h will produce unlimited rise of Lu,. It is known, 
however, that the size of eddies is limited by the column 
diameter when interplate distance is too large. 

In the considered case of hollow column, the 
influence of dynamic parameters still remains, which is 
in conflict with relation (11) and its result {12). 
Nevertheless, if assume for hollow column case h=l and 
e=l, Eq.(IB) produces for Lo values around 0,04Dc, 
which is close to our result (13). In case of various stage 
geometry parameters, correlations (10) and (18) give 
rather similar results although their different structure. 

Conclusions 

The turbulent macroscale L of a pulsed liquid flow in 
stagewise column apparatuses with internals of discs 
and rings was studied by use of k~E model of turbulence. 

141 

The local distribution of L on the stage is thoroughly 
studied as a function of flow regime and stage geometry 
parameters. Small variation of L during the pulsation 
cycle is registered, i.e. the size of large eddies does not 
depend on the instant superficial flow velocity changes 
during the pulsation. Similarly, low dependence ofL on 
Renumber, pulse amplitude and frequency is observed. 
Strong influence of the geometry parameters on L was 
found, not too pronounced for the plate free area, but 
rather significant as regarding the interplate distance. 
The zones of high values of turbulent macroscale are 
located. They correspond in position to the zones where 
intensive break-down of drops has been experimentally 
observed. , This fact attributes reliability to the results 
obtained. A correlation for determination of mean value 
of turbulent macroscale is proposed that reflects 
adequately the influence of main process parameters. 

Acknowledgements 

The study is carried out with partial financial support 
from CNRS - France and Bulgarian Academy of 
Sciences. 

SYMBOLS 

Ap pulsation amplitude, m 
ell, cb c,, cEh c,2 

standard constants of k- £ turbulence model, -
Ck von Karman constant, -
De column diameter, m 
Dd disc diameter, m 
Dr ring diameter, m 
e plate free area,-
£ dissipation rate of turbulent kinetic energy, m

2
s-

3 

f pulsation frequency, s·1 

H interplate distance, m 
h ratio of interplate distance and column diameter, ~ 
k turbulent kinetic energy, m

2
s-

2 

L turbulent macroscale, m 
Lm mean value of turbulent macroscale, m 
Lmax maximal value of turbulent macroscale, m 
Lo turbulent macroscale for hollow column, m 
v kinematic viscosity, m

2
s·

1 

v1 turbulent viscosity, m
2
s-

1 

Re Reynolds number, -
SP plate cross-section open to flow, rrt 
Sc column cross section, m

2 

t time, s 
T period of pulsation, s 
Ur friction velocity, ms-

1 

Ui mean velocity vector, ms-1 

U mean superficial flow velocity, ms"
1 

m ~ 
U(t} flow superficial velocity at the moment t, ms 
x space coordinate, m 



142 

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