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Determination of actual drop velocities in agitated extraction columns
Pergamon
Chemical En#ineerin# Science, Vol. 50, No, 2, pp. 255 -261, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00
0009-2509(94)00237-1
D E T E R M I N A T I O N O F A C T U A L D R O P V E L O C I T I E S IN AGITATED EXTRACTION COLUMNS J. WEISS, L. STEINER and S. HARTLAND* Department of Industrial and Engineering Chemistry, Swiss Federal Institute of Technology, 8092 Zurich, Switzerland
(Received 5 May 1993; accepted in revisedform 27 July 1994) Abstract--Finding that the axial velocities of individual drops in polydispersed turbulent dispersions with medium hold-up (as found in liquid-liquid extraction columns) cannot be predicted by known equations from the literature, methods were derived for their direct experimental determination. Assuming powerfunction relations between the drop diameter and the drop velocity (for a given system, hold-up and intensity of agitation), the velocities are evaluated from simultaneous measurements of local hold-ups and drop size distributions. Sets of existing data were treated in this way to correlate the actual drop velocity against drop diameter, hold-up and agitation intensity in different types of agitated extraction columns. To check the reliabilityof the proposed method of evaluation, new data were measured with a pilot-plant-sized Kfihni column. Together with the above-mentioned drop size distributions and local hold-ups, response curves to a tracer injection into the dispersed phase were recorded in several positions along the column axis. A method for evaluating the individual drop velocities from the changing shape of these curves and known drop size distribution was derived. Both methods delivered practically identical results so the evaluation of earlier data was justified. The procedures may be used in solutions of the population balance models to reduce the number of unknown parameters. INTRODUCTION
In the simulation of the hydrodynamic performance of countercurrent extraction columns, drop population balance models are frequently used to predict changes in hold-up and drop size distribution along the column height, which in turn determine the flooding and the working load of the given column. Numerous algorithms for solutions of such models ranging from the Monte-Carlo simulation of Rod and Misek (1982) to the simplified model of Laso et al. (1987) have been published. In all cases the variations of the breakage rates, the coalescence rates and the drop velocity with the drop diameter are assumed to be known and are frequently expressed by generalised functions from the literature, the drop interaction rates (breakage and coalescence) usually from an equation of Coulaloglou and Tavlarides (1977). No direct equations exist for the individual drop velocities in denser dispersions but the mean slip velocity (obtained from the specific loadings and the hold-up) has been correlated by different means. Godfrey and Slater (1991) compiled the available equations and proposed their own for different column constructions. However, although their characteristic velocity is a function of the drop diameter, it is not the actual velocity of the drops, and its application in the population balance models is not justified. Also the alternative method of Fan et al. (1987), who observed single drops in an extraction column and scaled the observed velocities up to actual working conditions with
a realistic hold-up, proved as unreliable at high intensities of agitation. Evaluating the breakage and coalescence rates from the measured variations of the hold-up and the drop size distribution along the column height, only a limited number of parameters may be optimised from the experimental data. Especially the drop velocities should be known in advance and not optimised together with the interaction rates. Depending on physical properties of the liquid system, its purity, mass transfer direction and the operating regime, the drop velocities should be determined experimentally for each individual situation, under the actual operating conditions and in the actual column. As the direct measurement (e.g. with a laser-doppler anemometer) still encounters considerable difficulties, we are describing alternative methods for evaluation of the individual drop velocities in turbulent dispersions, starting either from local values of the hold-up and the drop size distribution or from the shape of the response curves after a tracer is injected into the dispersed phase. In this way empirical correlations for a given column arrangement and the liquid system used are obtained which realistically describe the variation of the drop velocity with the drop diameter and the intensity of the energy input. THEORY
Relations between hold-up and relative velocity of drops The mean velocity of drops relative to the continuous phase (slip velocity) is defined as bd
v"
*Author to whom correspondence should be addressed. 255
b,
= -e+ - -1. - e
(1)
256
J. WEISSet al.
If differently sized drops with a discrete volumetric distribution function q~v(d) are present in the column, this velocity is related to relative velocities of the respective drop classes by
~, = Y, v,~ q~oj.
(2)
It was assumed that the influences of the drop diameter, hold-up and the agitation intensity on the actual slip velocity may be expressed by a power function of the following shape: v,(d,e) = 6(1 + e)'(1 - e)#d ~.
(3)
The variation of the slip velocity with the hold-up is expressed by the coefficients c~and ft. The exponent on drop diameter y is expected to be positive. All the free parameters ~ to 6 may be functions of the agitation intensity:
/~ =/~o + n p~
(4)
Y = 7o exp (vln)
1.2. Relative velocities of the drops are calculated from ¢q. (3). 1.3. Mean velocity is calculated from eq. (2) using the measured drop size distributions. 1.4. New estimates of the hold-ups en+l,~ and en+l,2 are obtained from eq. (1). 1.5. Ife~+l,1 differs from en. 1 or e~+1.2 differs from en,2 the procedure returns to step 1.2 for a next iteration changing the iteration number n to n + 1. 1.6. A proper value of hold-up is selected from en, l and e~, 2 as the one with smaller deviation from the measurement. 2. Weighted sum of squares is calculated as s 2 = ~ (e - e . . . . )2w-1
(7)
where the weight w was defined as w = eme~ + 0.01 to account for the fact that very low hold-ups are measured with lower accuracy than the higher ones. 3. Current guess of the parameters is changed according to Marquardt's strategy. 4. The calculation returns to step 1 until a minimum value of s 2 is found.
6 = 6oexp(61n). Multiplying eq. (1) by e ( 1 - e) and rearranging, a quadratic equation for e is obtained, the condition for having two real solutions being (e, + ba - b~)2 > 4bdg,.
(5)
Since the hold-up e must be between 0 and 1, the following condition must be fulfilled: ~, t> ( x / ~ + x/~a) 2-
(6)
The right-hand side of eq. (6) defines a critical velocity which is the mean relative velocity under which no stable operation is possible and the column floods. If the velocity is greater than the critical velocity, eq. (2) has always two solutions with two corresponding hold-up values. This suggests that columns can work in two different hydrodynamic regimes for given phase throughputs. However, it may be shown that only the lower value of the hold-up is stable and the other (corresponding to densely packed drops) may be maintained only under special circumstances. It is especially the drop coalescence which stabilises the column performance by increasing the drop sizes and velocities. Drop velocities from measured hold-ups and drop size distributions The free parameters in eq. (3) may be estimated by the following procedure from simultaneous measurements of local drop size distributions and hold-ups. 1. For an estimate of the free parameters in eq. (3) the corresponding values of the hold-ups are calculated for each experimental run at each of the measuring positions: 1.1. The measured hold-up is taken as the initial estimate of both possible hold-up values e~ and e2.
Drop velocities from response curves The procedure described above may be independently confirmed by evaluating the response curves to tracer injection into the dispersed phase. A short impulse of a detectable material is injected at the inlet of the dispersed phase before the distributor and its concentration is recorded in approximately the same positions where the measurements of local hold-up and drop size distributions were made. Working with a non-coalescing system, relatively low hold-ups, and a distributor at the bottom of the column producing practically uniform drops, the shapes of the response curves within the column can be realistically modelled and the velocities evaluated from their Laplace transform. The procedure is described in the Appendix. EXPERIMENTAL
A stirred extraction column of the Kiihni type with a diameter of 150 mm and an active height of 1.25 m was used. It was equiped with a distributor at the inlet of the dispersed phase consisting of 110 syringe needles, 25 mm long, 0.9 mm ID. The column contained 18 agitated stages with turbine agitators; the free area of the stators was 23%. The same column was used and described in detail by Kumar et al. (1986). Technical grade toluene dispersed in deionised water, without mass transfer, was used as the liquid-liquid system. The column was built into a standard supporting system with four 400 1 storage tanks, pumps and flow controls. The hold-ups were measured by rapid withdrawal of 100 ml of dispersion through sampling ports, reading the volumes of the collected water and toluene after settling. Given the intensive agitation by the turbine stirrers and the large volume of the stages, this is a reliable and reproducible method. Drop size distributions were obtained by continuously withdrawing a sample of the dispersion through a broad capil-
Actual drop velocities lary tube (2 mm), photographing in flashlight and evaluating the distributions manually from projected negatives on a digitaliser connected to a personal computer. The method has been thoroughly tested earlier and is shown to deliver realistic results if the coalescence at the inlet and inside the capillary tube is prevented by working with a system without mass transfer and a reasonably clean environment. To obtain the response curves in the dispersed phase a novel technique was developed. The tracer consisted of a 1% solution of the scintillator BBQ in toluene, and 10 ml of the solution was injected. The photocells were fixed from outside to the glass wall of the column, UV lamps being positioned behind them. The photocells and the lamps were screened from the daylight so the photocells responded to visible light coming from inside the column only. The tracer reaching the stage with a UV lamp emitted blue light with an intensity proportional to its concentration. The light was converted into a voltage signal by the photocells and recorded against time by a data acquisition card in a personal computer and stored on a diskette. After finishing the experiment the results were evaluated off-line. In comparison to trace experiments with a conventional dye the signal noise is decreased by an order of magnitude. Having confirmed the reliability of the evaluation techniques by successful comparison of both methods, the following sets of earlier hold-up/drop size distribution data were recalculated in the same way: • Kumar (1985) worked with the same column and the system of water, o-xylene and acetone, both with and without mass transfer.
257
• yon Fischer (1983) investigated a novel construction of a stirred column with coalescence enhancing plates. His column had a diameter of 72 mm and consisted of 32 stages with total heights of 60 ram. The EC (enhanced coalescence) plates with teflonised lattices were 20 mm high, the hydraulic diameter of the openings being 9 mm. The system was water, toluene and acetone; all experiments were performed under mass transfer conditions. • Bensalem (1985) investigated the behaviour of a reciprocating-plate column of the Karr type. It was 2 m long and its diameter was 76 mm. The plate-stack contained 54 perforated plates with a hole diameter of 16 mm and a free area of 58%. He worked with a constant amplitude of 11 mm, the system was water, toluene and acetone, and both transfer directions were investigated. The main parameters of the equipment used are given in Table 1.
RESULTS
Evaluating our own experimental data and leaving out the parameters without statistical significance the following equation was obtained: v, = 0.0412 exp ( -0.322n)d °'161.
(8)
Table 2 demonstrates good statistical significance of the mixing intensity term, and Table 3 confirms that the statistical significance of the hold-up terms is low so that leaving them out of eq. (3) was justified. Figure 1 compares the hold-ups calculated from eq. (8) and the measured drop size distributions to
Table 1. List of data sources Reference
Type
Height (m)
Diameter (m)
No. of stages
Disp. phase
Mass transfer
This work Kumar (1985) yon Fischer (1983) Bensalem (1985)
K~ihni Kiihni EC Karr
1.2 1.2 2.0 2.0
0.15 0.15 0.075 0.075
18 18 32 54
Toluene o-Xylene Toluene Toluene
-x~, ~/
Table 2. Significanceof the agitation intensity term Sum of squares
Degreesof freedom
Mean square
Variance ratio F
Fc,t
Variable 1 Residual
0.494 0.406
1 30
0.494 0.0135
36.6 --
4.17 --
Variance ratio F
Fcr~t
Table 3. Significanceof the hold-up terms Variable
Sum of squares
Degreesof freedom
Mean square
ct, fl Residual
0.006 0.40
2 28
0.003 0.0143
0.21 --
4.34 --
258
J. WEISSet al. PeedicLioa
0.I
0.0 0.0 0.0
//
//
0.6Predictedhold-rip
0.45
~q+
0.15F
0.03 0.06 0.09 Experlmenl
Fig. 1. Comparison of hold-ups calculated from eq. (8) to experimental data.
0.12 Rmpoam
0
0.12
0
Z .i. +
0.3
r 0.15 0.3 0.45 Mecsumdbold-up
0.6
Fig. 3. Comparison of hold-ups recalculated from eq. (9) to experimental measurements. Kfihni column, o-xylene in water. Kfihni column 1 (Kumar)
method
v,s
=
15.73exp(-1.18n)(1
-
~)
1• 53d i . 1 1 4 , x p ( - 0 2 2 5 n ) .
(9) 0.09
/
0.06 0.03
/
/
/
o0
~
0.03 0.06 0.09 l~D method
0.12
Fig. 2. Comparison of the evaluation methods for drop velocity.
experimental values. In Fig. 2 the two methods are compared against each other by plotting the hold-ups calculated from velocities obtained from the drop size distributions (X-axis) against the hold-ups calculated from drop velocities resulting from the tracer experiments. The bad statistical determination of the hold-up exponents (and also of the exponent on d) was caused by using the distributor in the column to give a very narrow initial drop size distribution and selecting the operation parameters so that the hold-up did not change significantly along the column height. Such simple conditions were necessary to guarantee the accuracy of the response method of drop velocity evaluation, which was the main purpose of these experiments. Having proved that both methods deliver similar results, the large sets of available data listed in Table 1 have been treated. They contain hundreds of drop-size distribution measurements, without and with mass transfer in both directions and in a broad range of hold-ups, involving some points in the dense-dispersion regime. The following equations were obtained.
The exponent on 1 + e was under 0.01 so the term was left out. The exponent on 1 - e was to some degree a function of the agitation speed but the correlation was too uncertain and did not improve the agreement of predicted to measured hold-ups significantly. Equation (9) is valid for data both without mass transfer and with mass transfer in both directions. The comparison of the hold-ups evaluated from the relative velocities calculated from this equation to those measured experimentally is in Fig. 3. The set contained some data with hold-up values changing by several hundred per cent along the column height. The comparison of recalculated to measured hold-ups for other data sources was similar to Fig. 3. EC column (yon Fischer) v,j = 0.454 exp ( - 0.315n) x(l+e)°'4s"(1-e)
- 0 131n
•
0 297 d s" .
(10)
This set contains some data in the dense region; in few cases the character of the dispersion changed along the column height. The coalescence-enhancing lattice plates caused practically perfect coalescence between the stages, so the calculated velocities are higher than in a conventional column. Karr column (Bensalem) v,s = 9.06 exp ( -- 0.163n)d~ 's56.
(11)
The velocities are higher than in columns with a rotating stirrer, probably due to the absence of tangential forces. No significant influence of the hold-up has been found. Spray columns The following equation of Kumar et al. (1980) for the relative velocity of drops in undisturbed denser
Actual drop velocities
259
dispersions has been used for comparison with the recent results: v--~2= 2.725 Ap ( 1 - ~
"~1.s3,
dg
"
(12)
60Mean ml. velocity [mm/l} 40
The above equations are compared with each other for a hold-up of 10% and a low and a high intensity of agitation in Figs 4 and 5, respectively. The physical properties in eq. (12) correspond to toluene and water. With different stirrer dimensions and different principles of agitation an exact comparison would be difficult, but using the lowest and the highest energy inputs in the sets of data in Figs 4 and 5, respectively, the comparison is reasonable - - the columns were tested in a full range of available speeds or frequencies.
Comparison of individual and mean relative velocities Having a large set of experimental data, the mean relative velocities may be calculated from eq. (1) and correlated against the mean Sauter diameter by eq. (3). However, such an equation does not represent properly the individual drop velocities as the exponent on d is overestimated. This is demonstrated in Fig. 6 for Kumar's data. The relative velocities are evaluated from eq. (9) for the mean Sauter diameters and compared to mean relative velocities obtained from eq. (1).
O.10 Relative velocit 0.08
~
0.06
I-j
7 /
0.00 ~ 0
1
2
v¢
3
4
5
Drop dtame~r [mini Fig. 4. Comparison of eqs (8)-(11) for a low intensity of agitation. Kuhni 1: Kumar; Kuhni 2: this work.
0.10 Relative veloett 0,08
~
0.06
/ ''~
/ /
004
/
f~
o.o2
-
0.00 ' ~ 0
-
l
2
a
4
5
Fig. 5. Comparison of eqs (8) (11) for a high intensity of agitation.
I
+;
30
*++~+ + +
I,uG+~* ~"+ 0
+~ ~
6
+ :
~
•
10 15 20 Equation (9) [mm/s]
25
Fig. 6. Comparison of individual relative velocities to those calculated from mean Sauter diameters. Kumar's data (Kiihni 1).
DISCUSSION
The relative velocities of drops in the agitated extraction columns are smaller than the velocities of drops of the same dimensions in undisturbed denser dispersions described by eq. (12). Their magnitude depends not only on the column construction but also on the system properties and its purity. The Kiihni column, being a typical representative of stirred columns with intensive energy input, displayed the lowest drop velocities, and the two data sources provided different results although the physical properties of o-xylene are very similar to those of toluene. Equation (9) has been obtained from data measured both with and without mass transfer. Obviously the mass transfer affects the drop size distribution but did not influence the individual drop velocity. In the EC column with turbine agitators separated by lattices wetted by the dispersed phase the velocities were higher because all drops were coalesced between each two respective stirrers. They emerged from the lattices in the form of continuous streams which were broken into drops only after reaching the next stirrer, so the path the dispersed phase moved in the shape of drops was shorter. In the Karr column the radial component of the drop movement is missing so the velocities are still higher. In all cases the velocities decreased strongly with the intensity of the energy input; in the stirred columns the exponent on d also decreased with the agitation making the drop velocities more uniform. This work should not be seen as an attempt to produce equations for general application but as a description of novel methods for determination of the variation of the drop velocity with the drop diameter for each given special case from generally available data (hold-up and drop size distribution). The assumption that, for one set of local conditions, the drop velocity may be expressed in the form of a power function of its diameter (v, = Cld c2) has been confirmed for all data treated here. Recalculating the
260
J. WEISSet al.
hold-up for each pair of the constants C1 and C2 the result was in all cases identical with the experimental value. The problems arose only when these constants were generalised to be applicable for more than one measurement which produced the scatter observed in Fig. 3. In the simulation of breakage and coalescence processes the drop velocities can now be evaluated independently from the determination of the interaction rates, so the reliability of the calculation is better and the n u m b e r of free parameters to be optimised is significantly reduced.
Indices
c d i j meas n r t v
continuous phase dispersed phase related to column stage i related to drop class j measured value iteration number relative time domain related to volume REFERENCES
CONCLUSIONS
1. In turbulent dispersions the variation of drop velocity with drop diameter may be obtained either from measured local hold-ups and drop size distributions or from tracer experiments in the dispersed phase. Both methods produce very similar results. 2. The drop velocity is approximated by a power function of drop diameter, hold-up and mixing intensity. It decreases with increasing energy input. 3. The mass transfer and its direction affected the drop size distributions but had no influence on the drop velocity so all data from the same source could be correlated together. 4. The correlation of the mean relative velocity evaluated from eq. (1) against the mean Sauter drop diameter (for a large set of data) is not identical with the correlation of the individual relative velocities against the drop diameter. 5. In stirred columns the differences in drop velocity caused by different drop diameters decrease with increasing energy input. In the Karr column the exponent on d was constant.
NOTATION
a~k(s) A(s)
b
c'(t) d g
.~i(s) n s t ~r W
element of A(s) transition matrix specific throughout, m3/m2 s tracer concentration in the ith stage at time t drop diameter, m gravitational acceleration, m/s 2 Laplace transform of concentration ci(t) stirrer speed, pulsation frequency, smean deviation mean time of the response curve at a stage velocity, m/s mean relative velocity of drops, m/s weight
Greek letters
~,/~,~, L~,v P Ap
O-2
free parameters hold-up of dispersed phase density, kg/m 3 density difference variation of tracer curve residence time drop size distribution function
Bensalem, A.-K., 1985, Hydrodynamics and mass transfer in a reciprocating plate extraction column. Diss. ETH No. 7721, Zfirich. Coulaloglou, C. A. and Tavlarides, L. L., 1977, Description of interaction processes in agitated liquid-liquid dispersions. Chem. Engng Sci. 32, 1286. Fan, Z., Oloidi, J. O. and Slater, M. J., t987, Liquid-liquid extraction column design data acquisition from short columns. Chem. Engng Res. Des. 65, 243. Godfrey, J. C. and Slater, M. J., 1991, Slip velocity relationship for liquid-liquid extraction columns. Trans. Instn chem. Engrs 69, 130. Kumar, A., Steiner, L. and Hartland, S., 1986, Capacity and hydrodynamics of an agitated extraction column. Ind. Engng Chem. Process Des. Dev. 25, 728. Kumar, A., 1985, Hydrodynamics and mass transfer in Kiihni extractor. Diss. ETH No. 7806, Zfirich. Kumar, A., Vohra, D. K. and Hartland, S., 1980, Sedimentation of droplet dispersions in counter-current spray columns. Can. J. chem. Engng 58, 154. Laso, M., Steiner, L. and Hartland, S., 1987, Dynamic simulation of agitated liquid-liquid dispersions. Chem. Engng Sci. 42, 2429. Rod, V. and Misek, T., 1982, Stochastic modelling of dispersion formation in agitated liquid-liquid systems, Trans. Instn chem. Engrs 60, 48. von Fischer, E., 1983, Fl/issig-flfissig Extraktionskolonne mit koaleszenzf6rdernden Einbauten. Diss ETH No. 7220, Zfirich. APPENDIX: EVALUATION OF DROP VELOCITIES FROM TRACER INJECTION EXPERIMENTS
Short impulses of a tracer were injected into the feed line for the dispersed phase just before the distributor. The responses were monitored in stages 2, 8 and 17, the total number of stages being 18. The mean residence time of the tracer (first moment of the response curves) is determined by the throughput and the hold-up of the dispersed phase. The variance and the higher moments of the response curves are affected by the mixing mechanism. If the drops of different sizes move with the same velocity or if the rates of coalescence and redispersion are infinitely large, then (with constant hold-up profile) the variance of the residence time distribution would be a21 = const i.
(A1)
If, in the other extreme case, all mixing effects were caused by different drop velocities the variance would rise with the square of the stage number: a~i = const i 2.
(A2)
The evaluation of the response curves thus enables conclusions about the mixing mechanism even in cases when the hydrodynamic conditions in the stages must be described by models with free parameters. The response curve in stage i may be expressed by means of the vector of the concentration functions (c~ (t)..... c~k(t)), where c~ (t) is the tracer concentration in drops of/-class in stage i. With drops of different sizes these functions are not
261
Actual drop velocities directly observable, the measured signal being the weighted average of the elements of this vector:
ci(t) = ~ C~(t)flpvj.
(A3)
1=1
The physical meaning of the terms on the right-hand side corresponds to a sequence of parallel ports with perfect mixing and a sequence of ports with plug flow and partial mixing between them respectively. The shape of the signal at the bottom of the column was assumed as
The Laplace transform is
= 0
20'(s) = ~ 20j(s)qS~j.
(A4)
for t < to - 3v
c](t)=exp(-
t-to v
1=1
If there is no significant flow of the dispersed phase in the direction of the continuous phase then the transition matrix A may be defined by
=0
~__ 1 ,/2(1 - e -3)
(All)
fort>t0+3v.
In Laplace transform this converts to
[20'? l(s) ..... 20~+I(s)] = [20~ (s) ..... 20~(s)]A(s). (A5) The elements of the matrix A express the material transfer between the stages and between the drop classes (breakage and coalescence). The Laplace transform of the response in stage i is calculated from the transition matrix and the signal response in the first stage is [20~ (s) ..... 20~(s)] = [ 2 ° l(s) ..... 20~ (s)] A t- ~.
(h6)
Working without mass transfer and with hold-ups under 0.1 the coalescence rate was low. Breakage occurred mostly in the lowest stage and the mean drop diameter did not change dramatically along the column height. The matrix A was therefore a diagonal one with transition functions for the respective drop classes at the main diagonal. The same model can also be used for cases where either coalescence or breakage are significant. The weights in eq. (A4) must then be changed to reflect the mean drop size distribution for the part of the column between stages 1 and i: =
j=l
20i (s) ~bv1
(A7)
where ~bv,] = il k~l= i q~i.
(A8)
Drop size measurements in stages 2, 8 and 17 being available, the following equations were used instead of eq. (A8): ~b*~ = 4~j
(a9a)
~b,j8 =1~(34~v;2+ 2q~j + ~b~7)
(A9b)
~v*jl 7 = ~(~bvi 1 2 +
(A9c)
and 17 ). ~b~j+ ~b~i
s~lr(dk,e) + 1 I- (1 -- ~l)exp[--sr(dk,e)]
e -st°
[.1 -- e -3(1 -sv)
2(1 - e-a)l_
1 -sv
+
1 - e-3{l+sv) 1 i~sv j . (A12)
There was a distributor at the inlet of the dispersed phase into the column so the incoming drops were practically of the same size and moved with the same velocity into the first stage. The assumption of equal concentration responses in the first stage was therefore justified. The mean time of the response curve for the first stage was estimated from the response time in the second stage and from the mean residence time in a stage: tl = t 2 -
i
z(d;,s)4'vi.
(A13)
j=l
Two alternative equations were tried instead of eqs (A10) and (A12) but the results did not change significantly. Computational procedure The evaluation of drop velocity from the measured response curves required the following steps:
• Conversion of measured response curves into Laplace transforms. • Evaluation of drop velocities from eq. (3) for guessed values of the free parameters. • Estimation of to from eq. (A13). • Evaluation of 20](s), j = 1. . . . . k, from eq. (A12). • Evaluation of transition functions from eq. (A10). • 20~(s), i = 2,8, 17, were calculated from eq. (A7) using eqs (A6) and (A9). • Parameter modification in eq. (12) to improve the fit of Laplace transforms 20~(s) and return to the second step.
.
The parameters in eq. (12) defining the velocity function and the ones in eq. (A 10) defining the character of the mixing in the stages were common for each experimental run. The parameter v in eq. (All) was different for each measured curve. The optimisation criterion for evaluation of the parameters in eq. (12) was the sum of weighted squares of the residuals:
(Al0)
S 2 = ~ [v,(dj,e) - v*(dj, e)]2~,q.
Modelling of stage dynamics The transition function for the flow through the stages was assumed as akk(S) =
20] (s)
(A14)
Chemical En#ineerin# Science, Vol. 50, No, 2, pp. 255 -261, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00
0009-2509(94)00237-1
D E T E R M I N A T I O N O F A C T U A L D R O P V E L O C I T I E S IN AGITATED EXTRACTION COLUMNS J. WEISS, L. STEINER and S. HARTLAND* Department of Industrial and Engineering Chemistry, Swiss Federal Institute of Technology, 8092 Zurich, Switzerland
(Received 5 May 1993; accepted in revisedform 27 July 1994) Abstract--Finding that the axial velocities of individual drops in polydispersed turbulent dispersions with medium hold-up (as found in liquid-liquid extraction columns) cannot be predicted by known equations from the literature, methods were derived for their direct experimental determination. Assuming powerfunction relations between the drop diameter and the drop velocity (for a given system, hold-up and intensity of agitation), the velocities are evaluated from simultaneous measurements of local hold-ups and drop size distributions. Sets of existing data were treated in this way to correlate the actual drop velocity against drop diameter, hold-up and agitation intensity in different types of agitated extraction columns. To check the reliabilityof the proposed method of evaluation, new data were measured with a pilot-plant-sized Kfihni column. Together with the above-mentioned drop size distributions and local hold-ups, response curves to a tracer injection into the dispersed phase were recorded in several positions along the column axis. A method for evaluating the individual drop velocities from the changing shape of these curves and known drop size distribution was derived. Both methods delivered practically identical results so the evaluation of earlier data was justified. The procedures may be used in solutions of the population balance models to reduce the number of unknown parameters. INTRODUCTION
In the simulation of the hydrodynamic performance of countercurrent extraction columns, drop population balance models are frequently used to predict changes in hold-up and drop size distribution along the column height, which in turn determine the flooding and the working load of the given column. Numerous algorithms for solutions of such models ranging from the Monte-Carlo simulation of Rod and Misek (1982) to the simplified model of Laso et al. (1987) have been published. In all cases the variations of the breakage rates, the coalescence rates and the drop velocity with the drop diameter are assumed to be known and are frequently expressed by generalised functions from the literature, the drop interaction rates (breakage and coalescence) usually from an equation of Coulaloglou and Tavlarides (1977). No direct equations exist for the individual drop velocities in denser dispersions but the mean slip velocity (obtained from the specific loadings and the hold-up) has been correlated by different means. Godfrey and Slater (1991) compiled the available equations and proposed their own for different column constructions. However, although their characteristic velocity is a function of the drop diameter, it is not the actual velocity of the drops, and its application in the population balance models is not justified. Also the alternative method of Fan et al. (1987), who observed single drops in an extraction column and scaled the observed velocities up to actual working conditions with
a realistic hold-up, proved as unreliable at high intensities of agitation. Evaluating the breakage and coalescence rates from the measured variations of the hold-up and the drop size distribution along the column height, only a limited number of parameters may be optimised from the experimental data. Especially the drop velocities should be known in advance and not optimised together with the interaction rates. Depending on physical properties of the liquid system, its purity, mass transfer direction and the operating regime, the drop velocities should be determined experimentally for each individual situation, under the actual operating conditions and in the actual column. As the direct measurement (e.g. with a laser-doppler anemometer) still encounters considerable difficulties, we are describing alternative methods for evaluation of the individual drop velocities in turbulent dispersions, starting either from local values of the hold-up and the drop size distribution or from the shape of the response curves after a tracer is injected into the dispersed phase. In this way empirical correlations for a given column arrangement and the liquid system used are obtained which realistically describe the variation of the drop velocity with the drop diameter and the intensity of the energy input. THEORY
Relations between hold-up and relative velocity of drops The mean velocity of drops relative to the continuous phase (slip velocity) is defined as bd
v"
*Author to whom correspondence should be addressed. 255
b,
= -e+ - -1. - e
(1)
256
J. WEISSet al.
If differently sized drops with a discrete volumetric distribution function q~v(d) are present in the column, this velocity is related to relative velocities of the respective drop classes by
~, = Y, v,~ q~oj.
(2)
It was assumed that the influences of the drop diameter, hold-up and the agitation intensity on the actual slip velocity may be expressed by a power function of the following shape: v,(d,e) = 6(1 + e)'(1 - e)#d ~.
(3)
The variation of the slip velocity with the hold-up is expressed by the coefficients c~and ft. The exponent on drop diameter y is expected to be positive. All the free parameters ~ to 6 may be functions of the agitation intensity:
/~ =/~o + n p~
(4)
Y = 7o exp (vln)
1.2. Relative velocities of the drops are calculated from ¢q. (3). 1.3. Mean velocity is calculated from eq. (2) using the measured drop size distributions. 1.4. New estimates of the hold-ups en+l,~ and en+l,2 are obtained from eq. (1). 1.5. Ife~+l,1 differs from en. 1 or e~+1.2 differs from en,2 the procedure returns to step 1.2 for a next iteration changing the iteration number n to n + 1. 1.6. A proper value of hold-up is selected from en, l and e~, 2 as the one with smaller deviation from the measurement. 2. Weighted sum of squares is calculated as s 2 = ~ (e - e . . . . )2w-1
(7)
where the weight w was defined as w = eme~ + 0.01 to account for the fact that very low hold-ups are measured with lower accuracy than the higher ones. 3. Current guess of the parameters is changed according to Marquardt's strategy. 4. The calculation returns to step 1 until a minimum value of s 2 is found.
6 = 6oexp(61n). Multiplying eq. (1) by e ( 1 - e) and rearranging, a quadratic equation for e is obtained, the condition for having two real solutions being (e, + ba - b~)2 > 4bdg,.
(5)
Since the hold-up e must be between 0 and 1, the following condition must be fulfilled: ~, t> ( x / ~ + x/~a) 2-
(6)
The right-hand side of eq. (6) defines a critical velocity which is the mean relative velocity under which no stable operation is possible and the column floods. If the velocity is greater than the critical velocity, eq. (2) has always two solutions with two corresponding hold-up values. This suggests that columns can work in two different hydrodynamic regimes for given phase throughputs. However, it may be shown that only the lower value of the hold-up is stable and the other (corresponding to densely packed drops) may be maintained only under special circumstances. It is especially the drop coalescence which stabilises the column performance by increasing the drop sizes and velocities. Drop velocities from measured hold-ups and drop size distributions The free parameters in eq. (3) may be estimated by the following procedure from simultaneous measurements of local drop size distributions and hold-ups. 1. For an estimate of the free parameters in eq. (3) the corresponding values of the hold-ups are calculated for each experimental run at each of the measuring positions: 1.1. The measured hold-up is taken as the initial estimate of both possible hold-up values e~ and e2.
Drop velocities from response curves The procedure described above may be independently confirmed by evaluating the response curves to tracer injection into the dispersed phase. A short impulse of a detectable material is injected at the inlet of the dispersed phase before the distributor and its concentration is recorded in approximately the same positions where the measurements of local hold-up and drop size distributions were made. Working with a non-coalescing system, relatively low hold-ups, and a distributor at the bottom of the column producing practically uniform drops, the shapes of the response curves within the column can be realistically modelled and the velocities evaluated from their Laplace transform. The procedure is described in the Appendix. EXPERIMENTAL
A stirred extraction column of the Kiihni type with a diameter of 150 mm and an active height of 1.25 m was used. It was equiped with a distributor at the inlet of the dispersed phase consisting of 110 syringe needles, 25 mm long, 0.9 mm ID. The column contained 18 agitated stages with turbine agitators; the free area of the stators was 23%. The same column was used and described in detail by Kumar et al. (1986). Technical grade toluene dispersed in deionised water, without mass transfer, was used as the liquid-liquid system. The column was built into a standard supporting system with four 400 1 storage tanks, pumps and flow controls. The hold-ups were measured by rapid withdrawal of 100 ml of dispersion through sampling ports, reading the volumes of the collected water and toluene after settling. Given the intensive agitation by the turbine stirrers and the large volume of the stages, this is a reliable and reproducible method. Drop size distributions were obtained by continuously withdrawing a sample of the dispersion through a broad capil-
Actual drop velocities lary tube (2 mm), photographing in flashlight and evaluating the distributions manually from projected negatives on a digitaliser connected to a personal computer. The method has been thoroughly tested earlier and is shown to deliver realistic results if the coalescence at the inlet and inside the capillary tube is prevented by working with a system without mass transfer and a reasonably clean environment. To obtain the response curves in the dispersed phase a novel technique was developed. The tracer consisted of a 1% solution of the scintillator BBQ in toluene, and 10 ml of the solution was injected. The photocells were fixed from outside to the glass wall of the column, UV lamps being positioned behind them. The photocells and the lamps were screened from the daylight so the photocells responded to visible light coming from inside the column only. The tracer reaching the stage with a UV lamp emitted blue light with an intensity proportional to its concentration. The light was converted into a voltage signal by the photocells and recorded against time by a data acquisition card in a personal computer and stored on a diskette. After finishing the experiment the results were evaluated off-line. In comparison to trace experiments with a conventional dye the signal noise is decreased by an order of magnitude. Having confirmed the reliability of the evaluation techniques by successful comparison of both methods, the following sets of earlier hold-up/drop size distribution data were recalculated in the same way: • Kumar (1985) worked with the same column and the system of water, o-xylene and acetone, both with and without mass transfer.
257
• yon Fischer (1983) investigated a novel construction of a stirred column with coalescence enhancing plates. His column had a diameter of 72 mm and consisted of 32 stages with total heights of 60 ram. The EC (enhanced coalescence) plates with teflonised lattices were 20 mm high, the hydraulic diameter of the openings being 9 mm. The system was water, toluene and acetone; all experiments were performed under mass transfer conditions. • Bensalem (1985) investigated the behaviour of a reciprocating-plate column of the Karr type. It was 2 m long and its diameter was 76 mm. The plate-stack contained 54 perforated plates with a hole diameter of 16 mm and a free area of 58%. He worked with a constant amplitude of 11 mm, the system was water, toluene and acetone, and both transfer directions were investigated. The main parameters of the equipment used are given in Table 1.
RESULTS
Evaluating our own experimental data and leaving out the parameters without statistical significance the following equation was obtained: v, = 0.0412 exp ( -0.322n)d °'161.
(8)
Table 2 demonstrates good statistical significance of the mixing intensity term, and Table 3 confirms that the statistical significance of the hold-up terms is low so that leaving them out of eq. (3) was justified. Figure 1 compares the hold-ups calculated from eq. (8) and the measured drop size distributions to
Table 1. List of data sources Reference
Type
Height (m)
Diameter (m)
No. of stages
Disp. phase
Mass transfer
This work Kumar (1985) yon Fischer (1983) Bensalem (1985)
K~ihni Kiihni EC Karr
1.2 1.2 2.0 2.0
0.15 0.15 0.075 0.075
18 18 32 54
Toluene o-Xylene Toluene Toluene
-x~, ~/
Table 2. Significanceof the agitation intensity term Sum of squares
Degreesof freedom
Mean square
Variance ratio F
Fc,t
Variable 1 Residual
0.494 0.406
1 30
0.494 0.0135
36.6 --
4.17 --
Variance ratio F
Fcr~t
Table 3. Significanceof the hold-up terms Variable
Sum of squares
Degreesof freedom
Mean square
ct, fl Residual
0.006 0.40
2 28
0.003 0.0143
0.21 --
4.34 --
258
J. WEISSet al. PeedicLioa
0.I
0.0 0.0 0.0
//
//
0.6Predictedhold-rip
0.45
~q+
0.15F
0.03 0.06 0.09 Experlmenl
Fig. 1. Comparison of hold-ups calculated from eq. (8) to experimental data.
0.12 Rmpoam
0
0.12
0
Z .i. +
0.3
r 0.15 0.3 0.45 Mecsumdbold-up
0.6
Fig. 3. Comparison of hold-ups recalculated from eq. (9) to experimental measurements. Kfihni column, o-xylene in water. Kfihni column 1 (Kumar)
method
v,s
=
15.73exp(-1.18n)(1
-
~)
1• 53d i . 1 1 4 , x p ( - 0 2 2 5 n ) .
(9) 0.09
/
0.06 0.03
/
/
/
o0
~
0.03 0.06 0.09 l~D method
0.12
Fig. 2. Comparison of the evaluation methods for drop velocity.
experimental values. In Fig. 2 the two methods are compared against each other by plotting the hold-ups calculated from velocities obtained from the drop size distributions (X-axis) against the hold-ups calculated from drop velocities resulting from the tracer experiments. The bad statistical determination of the hold-up exponents (and also of the exponent on d) was caused by using the distributor in the column to give a very narrow initial drop size distribution and selecting the operation parameters so that the hold-up did not change significantly along the column height. Such simple conditions were necessary to guarantee the accuracy of the response method of drop velocity evaluation, which was the main purpose of these experiments. Having proved that both methods deliver similar results, the large sets of available data listed in Table 1 have been treated. They contain hundreds of drop-size distribution measurements, without and with mass transfer in both directions and in a broad range of hold-ups, involving some points in the dense-dispersion regime. The following equations were obtained.
The exponent on 1 + e was under 0.01 so the term was left out. The exponent on 1 - e was to some degree a function of the agitation speed but the correlation was too uncertain and did not improve the agreement of predicted to measured hold-ups significantly. Equation (9) is valid for data both without mass transfer and with mass transfer in both directions. The comparison of the hold-ups evaluated from the relative velocities calculated from this equation to those measured experimentally is in Fig. 3. The set contained some data with hold-up values changing by several hundred per cent along the column height. The comparison of recalculated to measured hold-ups for other data sources was similar to Fig. 3. EC column (yon Fischer) v,j = 0.454 exp ( - 0.315n) x(l+e)°'4s"(1-e)
- 0 131n
•
0 297 d s" .
(10)
This set contains some data in the dense region; in few cases the character of the dispersion changed along the column height. The coalescence-enhancing lattice plates caused practically perfect coalescence between the stages, so the calculated velocities are higher than in a conventional column. Karr column (Bensalem) v,s = 9.06 exp ( -- 0.163n)d~ 's56.
(11)
The velocities are higher than in columns with a rotating stirrer, probably due to the absence of tangential forces. No significant influence of the hold-up has been found. Spray columns The following equation of Kumar et al. (1980) for the relative velocity of drops in undisturbed denser
Actual drop velocities
259
dispersions has been used for comparison with the recent results: v--~2= 2.725 Ap ( 1 - ~
"~1.s3,
dg
"
(12)
60Mean ml. velocity [mm/l} 40
The above equations are compared with each other for a hold-up of 10% and a low and a high intensity of agitation in Figs 4 and 5, respectively. The physical properties in eq. (12) correspond to toluene and water. With different stirrer dimensions and different principles of agitation an exact comparison would be difficult, but using the lowest and the highest energy inputs in the sets of data in Figs 4 and 5, respectively, the comparison is reasonable - - the columns were tested in a full range of available speeds or frequencies.
Comparison of individual and mean relative velocities Having a large set of experimental data, the mean relative velocities may be calculated from eq. (1) and correlated against the mean Sauter diameter by eq. (3). However, such an equation does not represent properly the individual drop velocities as the exponent on d is overestimated. This is demonstrated in Fig. 6 for Kumar's data. The relative velocities are evaluated from eq. (9) for the mean Sauter diameters and compared to mean relative velocities obtained from eq. (1).
O.10 Relative velocit 0.08
~
0.06
I-j
7 /
0.00 ~ 0
1
2
v¢
3
4
5
Drop dtame~r [mini Fig. 4. Comparison of eqs (8)-(11) for a low intensity of agitation. Kuhni 1: Kumar; Kuhni 2: this work.
0.10 Relative veloett 0,08
~
0.06
/ ''~
/ /
004
/
f~
o.o2
-
0.00 ' ~ 0
-
l
2
a
4
5
Fig. 5. Comparison of eqs (8) (11) for a high intensity of agitation.
I
+;
30
*++~+ + +
I,uG+~* ~"+ 0
+~ ~
6
+ :
~
•
10 15 20 Equation (9) [mm/s]
25
Fig. 6. Comparison of individual relative velocities to those calculated from mean Sauter diameters. Kumar's data (Kiihni 1).
DISCUSSION
The relative velocities of drops in the agitated extraction columns are smaller than the velocities of drops of the same dimensions in undisturbed denser dispersions described by eq. (12). Their magnitude depends not only on the column construction but also on the system properties and its purity. The Kiihni column, being a typical representative of stirred columns with intensive energy input, displayed the lowest drop velocities, and the two data sources provided different results although the physical properties of o-xylene are very similar to those of toluene. Equation (9) has been obtained from data measured both with and without mass transfer. Obviously the mass transfer affects the drop size distribution but did not influence the individual drop velocity. In the EC column with turbine agitators separated by lattices wetted by the dispersed phase the velocities were higher because all drops were coalesced between each two respective stirrers. They emerged from the lattices in the form of continuous streams which were broken into drops only after reaching the next stirrer, so the path the dispersed phase moved in the shape of drops was shorter. In the Karr column the radial component of the drop movement is missing so the velocities are still higher. In all cases the velocities decreased strongly with the intensity of the energy input; in the stirred columns the exponent on d also decreased with the agitation making the drop velocities more uniform. This work should not be seen as an attempt to produce equations for general application but as a description of novel methods for determination of the variation of the drop velocity with the drop diameter for each given special case from generally available data (hold-up and drop size distribution). The assumption that, for one set of local conditions, the drop velocity may be expressed in the form of a power function of its diameter (v, = Cld c2) has been confirmed for all data treated here. Recalculating the
260
J. WEISSet al.
hold-up for each pair of the constants C1 and C2 the result was in all cases identical with the experimental value. The problems arose only when these constants were generalised to be applicable for more than one measurement which produced the scatter observed in Fig. 3. In the simulation of breakage and coalescence processes the drop velocities can now be evaluated independently from the determination of the interaction rates, so the reliability of the calculation is better and the n u m b e r of free parameters to be optimised is significantly reduced.
Indices
c d i j meas n r t v
continuous phase dispersed phase related to column stage i related to drop class j measured value iteration number relative time domain related to volume REFERENCES
CONCLUSIONS
1. In turbulent dispersions the variation of drop velocity with drop diameter may be obtained either from measured local hold-ups and drop size distributions or from tracer experiments in the dispersed phase. Both methods produce very similar results. 2. The drop velocity is approximated by a power function of drop diameter, hold-up and mixing intensity. It decreases with increasing energy input. 3. The mass transfer and its direction affected the drop size distributions but had no influence on the drop velocity so all data from the same source could be correlated together. 4. The correlation of the mean relative velocity evaluated from eq. (1) against the mean Sauter drop diameter (for a large set of data) is not identical with the correlation of the individual relative velocities against the drop diameter. 5. In stirred columns the differences in drop velocity caused by different drop diameters decrease with increasing energy input. In the Karr column the exponent on d was constant.
NOTATION
a~k(s) A(s)
b
c'(t) d g
.~i(s) n s t ~r W
element of A(s) transition matrix specific throughout, m3/m2 s tracer concentration in the ith stage at time t drop diameter, m gravitational acceleration, m/s 2 Laplace transform of concentration ci(t) stirrer speed, pulsation frequency, smean deviation mean time of the response curve at a stage velocity, m/s mean relative velocity of drops, m/s weight
Greek letters
~,/~,~, L~,v P Ap
O-2
free parameters hold-up of dispersed phase density, kg/m 3 density difference variation of tracer curve residence time drop size distribution function
Bensalem, A.-K., 1985, Hydrodynamics and mass transfer in a reciprocating plate extraction column. Diss. ETH No. 7721, Zfirich. Coulaloglou, C. A. and Tavlarides, L. L., 1977, Description of interaction processes in agitated liquid-liquid dispersions. Chem. Engng Sci. 32, 1286. Fan, Z., Oloidi, J. O. and Slater, M. J., t987, Liquid-liquid extraction column design data acquisition from short columns. Chem. Engng Res. Des. 65, 243. Godfrey, J. C. and Slater, M. J., 1991, Slip velocity relationship for liquid-liquid extraction columns. Trans. Instn chem. Engrs 69, 130. Kumar, A., Steiner, L. and Hartland, S., 1986, Capacity and hydrodynamics of an agitated extraction column. Ind. Engng Chem. Process Des. Dev. 25, 728. Kumar, A., 1985, Hydrodynamics and mass transfer in Kiihni extractor. Diss. ETH No. 7806, Zfirich. Kumar, A., Vohra, D. K. and Hartland, S., 1980, Sedimentation of droplet dispersions in counter-current spray columns. Can. J. chem. Engng 58, 154. Laso, M., Steiner, L. and Hartland, S., 1987, Dynamic simulation of agitated liquid-liquid dispersions. Chem. Engng Sci. 42, 2429. Rod, V. and Misek, T., 1982, Stochastic modelling of dispersion formation in agitated liquid-liquid systems, Trans. Instn chem. Engrs 60, 48. von Fischer, E., 1983, Fl/issig-flfissig Extraktionskolonne mit koaleszenzf6rdernden Einbauten. Diss ETH No. 7220, Zfirich. APPENDIX: EVALUATION OF DROP VELOCITIES FROM TRACER INJECTION EXPERIMENTS
Short impulses of a tracer were injected into the feed line for the dispersed phase just before the distributor. The responses were monitored in stages 2, 8 and 17, the total number of stages being 18. The mean residence time of the tracer (first moment of the response curves) is determined by the throughput and the hold-up of the dispersed phase. The variance and the higher moments of the response curves are affected by the mixing mechanism. If the drops of different sizes move with the same velocity or if the rates of coalescence and redispersion are infinitely large, then (with constant hold-up profile) the variance of the residence time distribution would be a21 = const i.
(A1)
If, in the other extreme case, all mixing effects were caused by different drop velocities the variance would rise with the square of the stage number: a~i = const i 2.
(A2)
The evaluation of the response curves thus enables conclusions about the mixing mechanism even in cases when the hydrodynamic conditions in the stages must be described by models with free parameters. The response curve in stage i may be expressed by means of the vector of the concentration functions (c~ (t)..... c~k(t)), where c~ (t) is the tracer concentration in drops of/-class in stage i. With drops of different sizes these functions are not
261
Actual drop velocities directly observable, the measured signal being the weighted average of the elements of this vector:
ci(t) = ~ C~(t)flpvj.
(A3)
1=1
The physical meaning of the terms on the right-hand side corresponds to a sequence of parallel ports with perfect mixing and a sequence of ports with plug flow and partial mixing between them respectively. The shape of the signal at the bottom of the column was assumed as
The Laplace transform is
= 0
20'(s) = ~ 20j(s)qS~j.
(A4)
for t < to - 3v
c](t)=exp(-
t-to v
1=1
If there is no significant flow of the dispersed phase in the direction of the continuous phase then the transition matrix A may be defined by
=0
~__ 1 ,/2(1 - e -3)
(All)
fort>t0+3v.
In Laplace transform this converts to
[20'? l(s) ..... 20~+I(s)] = [20~ (s) ..... 20~(s)]A(s). (A5) The elements of the matrix A express the material transfer between the stages and between the drop classes (breakage and coalescence). The Laplace transform of the response in stage i is calculated from the transition matrix and the signal response in the first stage is [20~ (s) ..... 20~(s)] = [ 2 ° l(s) ..... 20~ (s)] A t- ~.
(h6)
Working without mass transfer and with hold-ups under 0.1 the coalescence rate was low. Breakage occurred mostly in the lowest stage and the mean drop diameter did not change dramatically along the column height. The matrix A was therefore a diagonal one with transition functions for the respective drop classes at the main diagonal. The same model can also be used for cases where either coalescence or breakage are significant. The weights in eq. (A4) must then be changed to reflect the mean drop size distribution for the part of the column between stages 1 and i: =
j=l
20i (s) ~bv1
(A7)
where ~bv,] = il k~l= i q~i.
(A8)
Drop size measurements in stages 2, 8 and 17 being available, the following equations were used instead of eq. (A8): ~b*~ = 4~j
(a9a)
~b,j8 =1~(34~v;2+ 2q~j + ~b~7)
(A9b)
~v*jl 7 = ~(~bvi 1 2 +
(A9c)
and 17 ). ~b~j+ ~b~i
s~lr(dk,e) + 1 I- (1 -- ~l)exp[--sr(dk,e)]
e -st°
[.1 -- e -3(1 -sv)
2(1 - e-a)l_
1 -sv
+
1 - e-3{l+sv) 1 i~sv j . (A12)
There was a distributor at the inlet of the dispersed phase into the column so the incoming drops were practically of the same size and moved with the same velocity into the first stage. The assumption of equal concentration responses in the first stage was therefore justified. The mean time of the response curve for the first stage was estimated from the response time in the second stage and from the mean residence time in a stage: tl = t 2 -
i
z(d;,s)4'vi.
(A13)
j=l
Two alternative equations were tried instead of eqs (A10) and (A12) but the results did not change significantly. Computational procedure The evaluation of drop velocity from the measured response curves required the following steps:
• Conversion of measured response curves into Laplace transforms. • Evaluation of drop velocities from eq. (3) for guessed values of the free parameters. • Estimation of to from eq. (A13). • Evaluation of 20](s), j = 1. . . . . k, from eq. (A12). • Evaluation of transition functions from eq. (A10). • 20~(s), i = 2,8, 17, were calculated from eq. (A7) using eqs (A6) and (A9). • Parameter modification in eq. (12) to improve the fit of Laplace transforms 20~(s) and return to the second step.
.
The parameters in eq. (12) defining the velocity function and the ones in eq. (A 10) defining the character of the mixing in the stages were common for each experimental run. The parameter v in eq. (All) was different for each measured curve. The optimisation criterion for evaluation of the parameters in eq. (12) was the sum of weighted squares of the residuals:
(Al0)
S 2 = ~ [v,(dj,e) - v*(dj, e)]2~,q.
Modelling of stage dynamics The transition function for the flow through the stages was assumed as akk(S) =
20] (s)
(A14)