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Mass transfer across liquid—liquid interfaces
Analytica Chimica Acta, 167 (1985) 171-181 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
MASS TRANSFER ACROSS LIQUID-LIQUID Part 2. Calculation of Mass-transfer Coefficients Automated Falling-drop Apparatus
ULF HILLGREN*
INTERFACES from Experiments
with an
and INGER NAHRINGBAUER
Institute of Inorganic and Physical Chemistry, Faculty of Pharmacy, Center, Box 574, S-751 23 Uppsala (Sweden)
Uppsala Biomedical
(Received 25th June 1984)
SUMMARY The mass transfer of 4-methoxy-N,N-dimethylbenzylamine from aqueous drops to cyclohexane was studied by the falling-drop method with computer-controlled equipment. Different contact times were achieved by letting the drop-forming device ascend or descend to previously defined levels in the column containing the continuous phase. The overall mass-transfer coefficient was evaluated from the relationship between contact time and solute concentration in the donating phase. Whether or not the continuous phase must be replaced between the measurements needed to calculate the overall masstransfer coefficient is discussed in detail. Corrections for the gradually increasing concentration of the transported solute in the receiving phase are proposed and tested. The order of contact times (decreasing or increasing) is shown to be of great importance for these corrections.
Rate constants for the transport of a drug between immiscible solvents have been investigated by means of a falling-drop (or rising-drop) method [l]. The falling (or rising) time of the droplet was varied by using three columns of different lengths. The contact time between the dispersed and continuous phases for the free fall (rise) of the droplet was measured by a stop-watch. After the fall (rise) through the continuous phase, the dispersed phase was collected in 5-ml samples, and pH and the absorbance were measured. In order to accelerate the collection of data as well as to improve the accuracy of the determinations, computer-controlled apparatus was developed [ 21. The contact times between dispersed and continuous phases, the free-fall rates of the droplets, and their mean radius were measured by utilizing the capacity of the computer to measure time. Further, the dispersed phase of the required pH and solute concentration was prepared automatically. Preliminary tests of the apparatus [2] indicated high precision in the determination of the overall mass-transfer coefficient. The present paper deals with the effect on overall and individual mass-transfer coefficients when different experimental procedures are used.
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o 1985 Elsevier Science Publishers B.V.
172 THEORY
A molecule passing between two immiscible solvents encounters a total resistance to mass transfer consisting of the two liquid-phase resistances and the resistance of crossing the interface [3, 41. In considerations of mass transfer across a liquid/liquid interface, hydrodynamic and physicochemical effects must be carefully separated. The liquid-phase resistances are governed by the hydrodynamics of the system. As in most biological and technical systems, the hydrodynamics is very complex for a drop moving in a continuous medium because the transport to and from the interface takes place to a varying degree by convective diffusion [5, 61. The interfacial resistance depends on physicochemical effects, such as interactions between solute and solvent; acidity constants, K,, and distribution constants, &, are thus of great importance. The flux of mass transfer from a droplet to the continuous phase can be described [4] by
J = Kapp (cl - cdm)
(1)
where the apparent mass-transfer coefficient K app= [l/k,,
+ l/kal + (i/12,, +
The apparent partition coefficient m = (1 + an+/&)-’
k,l&l(k,&u)
i/k,W1l
is
-’
(2)
[2] is = (1 + an+/K,)-‘Kd
(3)
provided that the solute is a base and only neutral particles are distributed from the aqueous to the organic phase. Mass-transfer coefficients for the and rate constants for the energy barrier near the bulk are km1 and &, interface are F2.r, FZdl and kg, kdz, respectively. The dispersed phase is represented by i and the continuous phase by 2. In the dispersed aqueous phase, the overall rate of mass transfer will be dcJdt
= -(3/r)
K,,(cl
- c2/m)
(4)
where r denotes the radius of the drop. When sink conditions and zero concentration are applied to the continuous phase, integration of Eqn. 4 gives In cl = -(3/r)K_t
+ In cl,o
(5)
The concentration of the dispersed phase on detachment of the droplet from the nozzle, c1,o in Eqn. 5, is difficult to measure experimentally. It is more convenient to evaluate the apparent mass-transfer coefficient from the linear dependence of In cl on the contact time, t K aPP= -(r/3)[d(ln
c,)/dt]
(6)
From Eqn. 6, the mass transfer of the solute has to be determined at different contact times, i.e., at different heights of fall. Theoretically, the sink conditions are valid only for the very first drop
173
formed. Deviation from the sink conditions starts when a drop is influenced by the mass transfer of the preceding drops. Accumulation of the solute during fall of the drop might be significant after the passage of a number of drops and the effect will be more pronounced at a high rate of partition and a slow molecular diffusion in the continuous phase. Without a thorough knowledge of the hydrodynamics, it is difficult to correct for the gradual increase of the deviation from the sink conditions during a single run. EXPERIMENTAL
Chemicals Cyclohexane (pro analysi quality) was used as continuous phase and the dispersed phase was an aqueous buffered solution of the solute. Both phases were mutually saturated before use. The buffers used were sodium phosphate or borate systems (pro analysi quality). All water used was redistilled from glass. The solute was 4-methoxy-NJ!-dimethylbenzylamine hydrochloride (MDBA; Hassle Lakemedel AB, Gothenburg) which was used as received. All measurements were made at 25°C and an ionic strength of 0.1 mol l-‘, achieved with appropriate concentrations of the buffer components or by the addition of sodium chloride. The acidity constant used was a mixed constant, calculated from Ki = [B] aH+/[HB] +, where [B] and [HB]+ are the concentrations of the basic and acidic forms, respectively, of the solute; it was determined by potentiometric titration. The distribution constant, K,, of each substance used was determined by shaking, for 1 h in a thermostatted bath, equal volumes of cyclohexane and buffer solution containing the solute. Subsequently, the phases were separated and centrifuged for 30 min at ca. 2000 rpm. The concentrations of the solute in both phases were determined spectrophotometrically at 226 nm. The value of Kd was calculated from K, = D(1 + un+/K:), where D is the quotient of the total concentrations of the solute in the organic and aqueous phases, respectively. The procedure was applied at different pH values, but chosen to give D= 1. The application of concentrations instead of activities was considered to be justified by the use of very dilute solutions and constant ionic strength. Equipment A detailed description of the equipment and processing technique has already been given [ 21. The principal aim of each experiment was to evaluate the mass transfer of solute from drops of buffer solution during their free fall through the continuous phase. The time of contact between dispersed and continuous phases was recorded only during the free-fall period of a droplet. The contact time was varied by changing the height of free fall. Generally five different heights were used (0.60, 0.90, 1.20, 1.65, 2.00 m). The absorbance of the dispersed phase was measured at the end of the run when about 135 drops or 9.9 ml of the dispersed phase had passed
174
through the continuous phase. This quantity was necessary to avoid dilution effects in the aqueous phase originating from the preceding run. The upper level of the continuous phase was kept at least 0.2 m above the coil connected to the drop-forming nozzle. Between different sets of runs, the spectrophotometer was calibrated and a fresh coalesced layer of aqueous phase was formed at the bottom of the column. In the supporting tubes and burettes, any aqueous phase remaining from the preceding set of runs was replaced or highly diluted. About 40 ml of the buffer solution to be used was utilized for calibration, rinsing, etc. The concentration of the continuous phase was assumed to be constant throughout a single run, although this is not strictly true (see above). RESULTS
AND DISCUSSION
For convenience, the continuous phase was replaced with fresh solvent only when the mass transfer at a given pH value was to be measured for all the contact times utilized to evaluate KaPP. In order to justify this procedure, corrections for the effect of the solute transported into the continuous phase during the preceding run were examined. If the value of cz is not zero but is independent of cl, integration of Eqn. 4 gives
ln hi
- cz.i/~)l(c1.0
-c&Q1
= -(3/r) &G
(7)
where i denotes the number of runs within one set. One set is defined here as the number of runs used to obtain Kapp. Mass transfer during drop formation is likely to differ from mass transfer during the free fall of the drop [8, 91, and the concentration in the drop at t = 0 (i.e., the start of free fall) is difficult to estimate. The value of c1,o was therefore calculated by a stepwise procedure. The c~,~, obtained from the intercept when ln cl*1 was plotted versus t, was inserted in Eqn. 7. Another c1,o was calculated by applying the resulting Kapp and the concentrations cl,, and c2,1 to Eqn. 7. The presented cl,o was obtained by repeating these two steps once, starting with the new c~,~. The following procedures were tested for the collection of the data used to calculate KaPP: replacement of the continuous phase by a fresh solvent between runs; no replacement of the continuous phase until one complete set of runs had been completed and the order of contact times was increased or decreased. Depending on how the runs were done, c2,i was calculated in different ways. If the continuous phase is replaced by fresh solvent between the runs, i.e., after each change of the height of fall, a realistic assumption will be C 2,i
=
0m5
(vl,ilv2,i)
(Cl.0
-
cl,i)
(8)
where Vl,i (m”) is the total volume of the dispersed phase used in run i. With regard to the unprocessed volume of the dispersed phase, limited by the coalescence interface and the spectrophotometer cuvette, the concentration
175
Cl%i,which is measured at the end of the run, corresponds to the concentration of the dispersed phase when only about half the number of drops have passed the continuous phase. This assumption justifies the insertion of 0.5 in Eqn. 8. As 2.2 m of the column had a volume of 4.5 X low4 m3, the volume of the continuous phase involved in the mass-transfer process was calculated from V2.i = h, 4.5 X 10F4/2.20 where hi, in metres, is the height of fall in run i. When the continuous phase was not replaced with a fresh solvent until one set of runs had been completed and when the runs were done at decreasing heights of drop fall, the following formula was used =
c2,i
c2,i-l
+
0m5
tvl,i-1
/VZ,,-1)
tcl,O
-cl,i-l)
+
0.5(v1,
i/V2,i)(cl,0
-Cl,r) (9)
A similar equation was used when the runs were done with increasing heights of drop fall c2,i
=
C2,r-l
+
0e5
(v2,i-l (vl,i/V2,i)
Iv2,i) (Cl.0
+
0.5
tvl,i-I
lV2.i)
(Cl.0
-Cl,*--1)
-Cl,i)
(10)
The different sets of runs can be compared by inserting Eqns. 8-10 in Eqn. 7 for the appropriate cases. For large values of m (i.e., for pH > pKA when & > lo), the effect of the solute transferred to the continuous phase can be neglected (cf. Eqns. 3 and 4). However, a decrease in pH implies a decrease of m, the effect of the transferred solute increases and a correction is needed. This is demonstrated in Table 1, which shows&_, calculated from Eqns. 6 and 7, respectively. A TABLE 1 Results at various pH values with the corresponding K,, PH
6.233 6.471 6.482 6.693 6.710 6.950 7.186 7.322 7.634 7.926 8.071 8.427 8.655 8.870 8.987
calculated from Eqns. 6 and 7.
ms*)
m (Eqn. 3)
Kapp (10’ (Bqn. 6)
Wqn. 7)
0.03 0.06 0.06 0.09 0.10 0.17 0.29 0.39 0.80 1.54 2.12 4.52 7.10 10.50 12.75
0.42 0.72 0.73 1.19 1.09 1.92 2.60 3.03 4.60 5.40 8.65 12.16 11.41 11.56 12.01
0.53 0.88 0.86 1.42 1.30 2.29 3.01 3.42 4.99 5.74 9.31 12.66 11.79 11.82 12.25
26.2 22.2 17.8 19.3 19.3 19.3 15.8 12.9 8.48 6.30 7.63 4.11 3.33 2.25 2.00
0.6 0.8 0.8 1.3 1.3 2.4 3.5 4.1 7.4 10.1 18.8 32.3 34.0 35.1 38.2
is the percentage increase of Kapp when Eqn. 7 is used instead of Eqn. 6. bMaximum value of c,/c,.
aAKap*
176
series of runs was done at pH 6.2-9.0, with MDBA as transported solute. The values of K, and pKi at 25°C were found to be 41.5 + 1.6 and 9.34 + 0.01, respectively, calculated from a van’t Hoff plot (In K vs. l/T) [9] of data obtained at 5-7 different temperatures from 15-45°C (the errors indicate the 95% confidence limits [lo]). As can be seen from column 5, the correction increases from 2% at pH 9.0 to 26.2% at pH 6.2. It may be noted that to some degree the effect of a decrease of m is counteracted by the accompanying decrease of c2 (cf. Eqn. 7 and column 6, Table 1). This variation of cZ is due to the slower mass transfer at decreasing values of m (cf. Eqn. 2). Compounds with values of K, < 1 can be discussed in the same way except that the correction will also have to be done at pH > pK: because m < 1 for all pH values. The correction for the transferred solute is expected to have a different influence on the sizes of the calculated individual mass-transfer coefficients. Because it was impossible to separate the different types of coefficients in Eqn. 2, the equation was simplified by introducing k1 and kz defined as + l/k,)-’ and k2 = (l/k,, f l/k,)-‘: k1 = (l/k,, K app = [l/k1 +
WwW’
(11)
According to Eqn. 11, k1 = Ifa,_, for m 9 1. Because the correction of K app is small in this case, the refined value of kI is expected to be influenced only slightly by the effect of the transferred solute. For m << 1 (i.e., for K(, Q 1 or/and at low pH), the l/k1 term in Eqn. 11 is negligible and kz = Kappm-l. According to the above discussion, the correction of K,,, could be significant for m < 1, which underlines the importance of the correction leading to a proper value of k2. This was demonstrated by calculation of the combined mass-transfer coefficients, kI and k2. Nonlinear regression treatment of the data presented in Table 1, using the logarithmic form of Eqn. 11, was done by means of the MINUIT-D-506 program [ll] on a NORD-10 computer. The results obtained were kI = 1.24 (kO.24) X lo4 m s-l and k2 = 1.21 (kO.18) X lo4 m s-l, using the K app obtained from Eqn. 6. When corrections were made, and Kapp calculated from Eqn. 7 was used, the value of k2 increased to 1.54 (kO.27) X 10m4 m s-l, whereas the value of kI was not significantly changed at 1.22 (kO.25) X 10m4 m s-l. The errors give a parameter change resulting in an increase of the residual sum of squares corresponding to an approximate 95% confidence level. An F-test [lo] was used for the calculation according to the formula S - So = So{ b/(n -p)] F(p, II - p, 0.95) - 11, where S is the residual sum of squares when the parameter is changed by one error, S,, is the minimum residual sum of squares obtained from the regression, p is the number of parameters and n is the number of observed data points. In order to test the proposed corrections for the deviation from the sink conditions in the continuous phase, additional experiments were done. In all runs, the pH was 7.10-7.20, and MDBA was the transported solute. At pH 7, the value of m was 0.2 and the need for a significant correction was
177
expected. A lower value of pH was avoided to ensure that KaPP could be determined precisely. The slight pH difference between the sets of runs was corrected for by normalizing I&.,,, to the same pH value. The following formula, based on Eqn. 11, was used: K aPP,n
=
[l/k,
Kapp
+
Wzm)l
[l/k, + Wm,W-’
(12)
where m, is the apparent partition coefficient (Eqn. 3) at pH 7.1, and K is the normalized Kapp at the same pH. %‘\he first sets, the continuous phase was replaced with a fresh solvent after each change of the height of fall. The volume of the continuous phase, exposed to mass transfer, depended on the position of the nozzle and varied for each run. Between runs, a fresh coalesced layer of pure buffer solution was formed after the column had been filled with a fresh continuous phase. In order to evaluate Kapp, equivalent runs were done consecutively at five different heights. Another 13 sets of runs were done in the same way. The values of Kapp obtained by using Eqns. 6 or 7 and 8 and normalization (Eqn. 12) are shown in Table 2. The correction produced by using Eqns. 7 and 8 increases the value of Kapp insignificantly from 3.26 to 3.33 with an average deviation of kO.16 in both cases. One set of runs used for one evaluation of Kapp is presented in Table 3. An inspection of the third column shows a decrease of c2 with increasing contact times, which is probably a combined effect of the stepwise increase of the volume of the continuous phase and the decreasing rate of mass transfer. Despite this variation of c2, the correction of Kapp according to Eqns. 7 and 8 will be slightly positive. This can be explained by the lower percentage effect of the correction at a high value of c1 (i.e., at a short contact time) compared with the percentage effect at a lower value of cl present at a longer contact time (cf. Eqn. 7). A comparable series of experiments was done in which the continuous phase was not replaced with a fresh solvent until a complete set of runs used to TABLE 2 Results obtained when the continuous radius 2.58 X lO+ rd.) PH
7.023 7.142 7.143 7.152 7.176 7.173 7.188
*a
(10” m s-1
Cl (mol l-‘)
K app,n Eqn. 6
Eqns. 7, 8
5.0x 5.0 x 5.0x 5.0 x 1.0 x 1.0 x 1.0 x
3.16 3.05 3.65 3.16 3.02 3.29 3.02
3.22 3.10 3.73 3.23 3.07 3.35 3.08
10’ 10” lo6 10-5 lo4 lOA lo4
sThe concentration phase.
phase was replaced after each run. (Mean drop
PH
7.132 7.099 7.103 7.106 7.095 7.088 7.063
Cfa
(mol l*) 3.8 3.8 3.8 4.8 4.8 4.8 4.8
x x x x x x x
lo-’ lolo+ lo4 1O-4 lolOA
K app,n (IV
ms-’ 1
Eqn. 6
Eqns. 7, 8
3.20 3.30 3.12 3.55 3.28 3.58 3.29
3.26 3.36 3.18 3.64 3.34 3.66 3.35
of MDBA in the aqueous solution before its contact with the organic
178 TABLE 3 The set of runs at pH 7.095 (cf. Table 2)* Run no.
(Cf ,i In (cl ,0 -
Cl (10-M)
1 2 3 4 5
4.02 3.77 3.52 3.20 2.96
3.55 2.34 2.26 2.11 2.03
-0.1428 -0.2070 -0.2756 -0.3709 -0.4489
Contact
w/m) C,,i/mT
time
-0.1480 -0.2121 -0.2824 -0.3798 -0.4597
(S)
3.85 5.55 7.26 9.90 11.85
*The value of cl,0 was calculated as 4.637 x lo4 mol 1-l; cI was evaluated from the absorbance at 272 nm and c, was derived from Eqn. 8. TABLE 4 Results obtained when the continuous phase was only replaced after each set of runs. Otherwise the experiments were done under the same conditions as the analyses presented in Table 2. K app,n UO-Sms”)
PH
CT - 5 7.177 7.173 7.173 7.171 7.171
X
K app,n Wm5 m s-l 1
PH
Eqn. 6
Eqns. 7, 10
1O-5 mol 1-l 2.83 2.70 3.00 3.01 2.91
3.43 3.21 3.54 3.59 3.45
cf = 5 7.088 7.100 7.081 7.081 7.076 7.092
X
Eqn. 6
Eqns. 7,10
lo4 mol 1-l 2.63 2.75 2.54 2.75 2.80 2.78
3.15 3.33 2.98 3.34 3.30 3.38
calculate Kapp had been obtained. The results are given in Table 4. These runs were done with increasing contact times within each set. The corrections were based on C2,ig calculated from Eqn. 10, and they resulted in a sharp increase of Kapp (about 20%) from 2.80 (kO.ll) to 3.34 (rt0.13). The details of the last set of these runs are given in Table 5. As shown in the third column, the solute concentration of the organic phase, c2, increases considerably at increasing contact times, because of the solute remaining from the preceding run. As previously discussed, the percentage effect of the correction is greater at longer contact times than at shorter times. Both these effects result in a considerable underestimate of Kapp if no corrections are made. A comparison of the above results indicates that when Eqn. 7 is used for the correction, it is justified to complete a set of five runs without replacement of the continuous phase. The corrected mean values of Kapp are 3.33 X lo-’ kO.16 and 3.34 X l(rs kO.13 m s?, respectively (Tables 2 and 4). No significant effect on the results was observed for a change in concentration
179 TABLE 5 The last set of runs given in Table 4 (pH 7.092)a Run no.
;;a4
1 2 3 4 5
4.09 3.86 3.66 3.34 3.15
M)
Cl (lo+
ln(C*” -c&n) (C, .O -c&n)
M)
3.54 7.02 9.11 10.18 11.95
-0.1420 -0.1999 -0.2531 -0.3446 -0.4031
aThe value of c, +, was calculated as 4.714
x
Contact time (5) 3.80 5.53 7.33 9.97 12.05
-0.1471 -0.2151 -0.2795 -0.3875 -0.4656
lOA mol 1-l; c, was derived from Eqn. 10.
from 5.0 X 1W5 to 5.0 X lo4 mol 1-l (cf. Tables 2 and 4). A more extensive study of the dependence on the concentration will be published later. The order of the heights of fall could be essential in obtaining the most accurate results. If the set is started with the longest contact time, i.e., with the nozzle at the top of the column, the whole column of the continuous phase will be available to the solute in the first run; but only the lower part of the continuous phase will contact the solute if the set is started at the lowest height. In both cases, the subsequent runs in each set will result in a non-uniform concentration of the solute in the continuous phase along the column, but the effect on mass transfer will be different depending on the order of the runs. If the runs are done with increasing height of fall, each drop will experience a stepwise increase of c2 as it falls in the column. For the opposite order, there is no such effect as each separate experiment involves part of the continuous phase which is assumed to have no discontinuities of concentration along the column. Table 6 shows the results of TABLE 6 Results obtained with increasing and decreasing orders of contact timesa PH
K app,n UO”m Eqn. 6
PH
s* 1 Eqns. 7, 10
Increasing contact times 7.142 2.26 7.127 2.27 7.197 2.08 7.184 2.29 7.165 2.42 7.145 2.34
2.59 2.60 2.31 2.57 2.75 2.70
Mean value Mean deviation
2.59 0.10
2.28 0.07
Decreasing 7.146 7.136 7.129 7.128 7.128
K app,n (lo-’
m s-l)
Eqn. 6
Eqns. 7, 9
contact times 2.59 2.74 2.85 2.84 2.83 2.77 0.08
2.51 2.65 2.75 2.74 2.68 2.67 0.07
mol l*) was transported from the aqueous aMDBA (initial concentration 5 X lo* drops to the continuous phase of cyclohexane. The mean drop radius was 2.62 X 10” m. Kapp,” was obtained as in Table 2.
180 TABLE 7 The first set of runs with decreasing contact time (Table 6)” Run no.
:iO-
1 2 3 4 5
3.65 3.92 4.25 4.53 4.78
M)
”(lo-’ 2.77 7.51 11.55 15.59 19.54
M)
ln(c,,Jc,,,) -0.3669 -0.2956 -0.2147 -0.1509 -0.0972
In
Contact time (6)
(c I*,.-c2,i/m) (c,,, -c,,,lm)
-0.3760 -0.3154 -0.2367 -0.1717 -0.1141
aThe drop radius was 2.611 X 10” m (pH 7.146); c,,~ was calculated as 5.268 mol l* ; c, was calculated from the absorbance at 226 nm and c1 from Eqn. 9.
11.87 9.67 7.60 5.21 3.49 x
10”
six sets with decreasing and increasing contact times. With increasing contact times, underestimated values of Kapp are expected from Eqn. 6 (see above), compared to Eqn. 7, and this was confirmed in practice (Table 6). Table 7 gives details of the first set with decreasing contact times in Table 6. There is a sharp increase of cz with decreasing contact times, which implies a significant overestimate of Kapp when derived from Eqn. 6. However, as discussed above, the percentage effect of the correction of cz depends on cl, which in this case has a reduced effect on the value of Kapp. This is experimentally verified in Table 6 (right side). The mean value of K app based on Eqn. 6 is insignificantly higher than the corresponding value derived from Eqn. 7 with c2 corrected according to Eqn. 9. The corrected mean values, 2.67 X lo-’ and 2.59 X 10B5 m C’ which are similar, and the corresponding agreement of Kapp, presented in Tables 2 and 4, suggest that equivalent results can be obtained irrespective of the order of the contact times, provided that the appropriate corrections are made. However, the procedure with decreasing contact times seems more acceptable because the drops then experience a relatively uniform concentration throughout the contact time. During large series of experiments such as described above, the main problem is to keep the system unchanged. It is well known that the slightest contamination in either phase can markedly influence the mass transfer [12], probably because of a change in the hydrodynamics of the system. For instance, the results presented in Tables 4 and 6 were based on experiments done at two times three years apart. A comparison of the mean values of the corrected Kapp,” shows a significant discrepancy which is probably due to contamination-induced differences. We express our sincere thanks to Mr. Bo Larsson for his skilful contribution concerning computer handling. We are also indebted to Prof. Allan Agren and Prof. Lars-Olof Sundelijf for all facilities placed at our disposal. Grants received from the IF Foundation for Pharmaceutical Research are gratefully acknowledged.
181 REFERENCES 1 A. Brodin and A. &ren, Acta Pharm. Suet., 8 (1971) 609. 2 I. Nahringbauer and B. Larsson, Anal. Chim. Acta, 161 (1983) 153. 3 H. Brenner and L. G. LeaI, A.I.Ch.E.J., 24 (1978) 246. 4 J. A. Shaelwitz and K. T. Raterman, Ind. Eng. Chem. Fundam., 21 (1982) 154. 5 C. V. Sternling and L. E. Striven, AIChE.J., 6 (1959) 514. 6 H. Sawistowski, in C. Hanson (Ed.), Recent Advances in Liquid-Liquid Extraction, Pergamon Press, Oxford, 1971, Ch. 9. 7 V. Zimmermann, W. Halwachs and K. Schiigerl, Chem. Eng. Commun., 7 (1980) 95. 8 J. S. Vrentas, H. T. Liu and J. L. Duda, Chem. Eng. J., 21 (1981) 155. 9 A. N. Martin, J. Swarbrick and A. Cammarata, Physical Pharmacy, Lea & Febiger, Philadelphia, PA, 1969. 10 N. R. Draper and H. Smith, Applied Regression Analysis, Wiley-Interscience, New York, 1966. 11 F. James and M. Roos, CERN/DD Internal Report 75/20 (1976). 12 L. Mekasut, J. Molinier and H. Angelino, Chem. Eng. Sci., 34 (1979) 217.
MASS TRANSFER ACROSS LIQUID-LIQUID Part 2. Calculation of Mass-transfer Coefficients Automated Falling-drop Apparatus
ULF HILLGREN*
INTERFACES from Experiments
with an
and INGER NAHRINGBAUER
Institute of Inorganic and Physical Chemistry, Faculty of Pharmacy, Center, Box 574, S-751 23 Uppsala (Sweden)
Uppsala Biomedical
(Received 25th June 1984)
SUMMARY The mass transfer of 4-methoxy-N,N-dimethylbenzylamine from aqueous drops to cyclohexane was studied by the falling-drop method with computer-controlled equipment. Different contact times were achieved by letting the drop-forming device ascend or descend to previously defined levels in the column containing the continuous phase. The overall mass-transfer coefficient was evaluated from the relationship between contact time and solute concentration in the donating phase. Whether or not the continuous phase must be replaced between the measurements needed to calculate the overall masstransfer coefficient is discussed in detail. Corrections for the gradually increasing concentration of the transported solute in the receiving phase are proposed and tested. The order of contact times (decreasing or increasing) is shown to be of great importance for these corrections.
Rate constants for the transport of a drug between immiscible solvents have been investigated by means of a falling-drop (or rising-drop) method [l]. The falling (or rising) time of the droplet was varied by using three columns of different lengths. The contact time between the dispersed and continuous phases for the free fall (rise) of the droplet was measured by a stop-watch. After the fall (rise) through the continuous phase, the dispersed phase was collected in 5-ml samples, and pH and the absorbance were measured. In order to accelerate the collection of data as well as to improve the accuracy of the determinations, computer-controlled apparatus was developed [ 21. The contact times between dispersed and continuous phases, the free-fall rates of the droplets, and their mean radius were measured by utilizing the capacity of the computer to measure time. Further, the dispersed phase of the required pH and solute concentration was prepared automatically. Preliminary tests of the apparatus [2] indicated high precision in the determination of the overall mass-transfer coefficient. The present paper deals with the effect on overall and individual mass-transfer coefficients when different experimental procedures are used.
QOO3-2670/65/$03.30
o 1985 Elsevier Science Publishers B.V.
172 THEORY
A molecule passing between two immiscible solvents encounters a total resistance to mass transfer consisting of the two liquid-phase resistances and the resistance of crossing the interface [3, 41. In considerations of mass transfer across a liquid/liquid interface, hydrodynamic and physicochemical effects must be carefully separated. The liquid-phase resistances are governed by the hydrodynamics of the system. As in most biological and technical systems, the hydrodynamics is very complex for a drop moving in a continuous medium because the transport to and from the interface takes place to a varying degree by convective diffusion [5, 61. The interfacial resistance depends on physicochemical effects, such as interactions between solute and solvent; acidity constants, K,, and distribution constants, &, are thus of great importance. The flux of mass transfer from a droplet to the continuous phase can be described [4] by
J = Kapp (cl - cdm)
(1)
where the apparent mass-transfer coefficient K app= [l/k,,
+ l/kal + (i/12,, +
The apparent partition coefficient m = (1 + an+/&)-’
k,l&l(k,&u)
i/k,W1l
is
-’
(2)
[2] is = (1 + an+/K,)-‘Kd
(3)
provided that the solute is a base and only neutral particles are distributed from the aqueous to the organic phase. Mass-transfer coefficients for the and rate constants for the energy barrier near the bulk are km1 and &, interface are F2.r, FZdl and kg, kdz, respectively. The dispersed phase is represented by i and the continuous phase by 2. In the dispersed aqueous phase, the overall rate of mass transfer will be dcJdt
= -(3/r)
K,,(cl
- c2/m)
(4)
where r denotes the radius of the drop. When sink conditions and zero concentration are applied to the continuous phase, integration of Eqn. 4 gives In cl = -(3/r)K_t
+ In cl,o
(5)
The concentration of the dispersed phase on detachment of the droplet from the nozzle, c1,o in Eqn. 5, is difficult to measure experimentally. It is more convenient to evaluate the apparent mass-transfer coefficient from the linear dependence of In cl on the contact time, t K aPP= -(r/3)[d(ln
c,)/dt]
(6)
From Eqn. 6, the mass transfer of the solute has to be determined at different contact times, i.e., at different heights of fall. Theoretically, the sink conditions are valid only for the very first drop
173
formed. Deviation from the sink conditions starts when a drop is influenced by the mass transfer of the preceding drops. Accumulation of the solute during fall of the drop might be significant after the passage of a number of drops and the effect will be more pronounced at a high rate of partition and a slow molecular diffusion in the continuous phase. Without a thorough knowledge of the hydrodynamics, it is difficult to correct for the gradual increase of the deviation from the sink conditions during a single run. EXPERIMENTAL
Chemicals Cyclohexane (pro analysi quality) was used as continuous phase and the dispersed phase was an aqueous buffered solution of the solute. Both phases were mutually saturated before use. The buffers used were sodium phosphate or borate systems (pro analysi quality). All water used was redistilled from glass. The solute was 4-methoxy-NJ!-dimethylbenzylamine hydrochloride (MDBA; Hassle Lakemedel AB, Gothenburg) which was used as received. All measurements were made at 25°C and an ionic strength of 0.1 mol l-‘, achieved with appropriate concentrations of the buffer components or by the addition of sodium chloride. The acidity constant used was a mixed constant, calculated from Ki = [B] aH+/[HB] +, where [B] and [HB]+ are the concentrations of the basic and acidic forms, respectively, of the solute; it was determined by potentiometric titration. The distribution constant, K,, of each substance used was determined by shaking, for 1 h in a thermostatted bath, equal volumes of cyclohexane and buffer solution containing the solute. Subsequently, the phases were separated and centrifuged for 30 min at ca. 2000 rpm. The concentrations of the solute in both phases were determined spectrophotometrically at 226 nm. The value of Kd was calculated from K, = D(1 + un+/K:), where D is the quotient of the total concentrations of the solute in the organic and aqueous phases, respectively. The procedure was applied at different pH values, but chosen to give D= 1. The application of concentrations instead of activities was considered to be justified by the use of very dilute solutions and constant ionic strength. Equipment A detailed description of the equipment and processing technique has already been given [ 21. The principal aim of each experiment was to evaluate the mass transfer of solute from drops of buffer solution during their free fall through the continuous phase. The time of contact between dispersed and continuous phases was recorded only during the free-fall period of a droplet. The contact time was varied by changing the height of free fall. Generally five different heights were used (0.60, 0.90, 1.20, 1.65, 2.00 m). The absorbance of the dispersed phase was measured at the end of the run when about 135 drops or 9.9 ml of the dispersed phase had passed
174
through the continuous phase. This quantity was necessary to avoid dilution effects in the aqueous phase originating from the preceding run. The upper level of the continuous phase was kept at least 0.2 m above the coil connected to the drop-forming nozzle. Between different sets of runs, the spectrophotometer was calibrated and a fresh coalesced layer of aqueous phase was formed at the bottom of the column. In the supporting tubes and burettes, any aqueous phase remaining from the preceding set of runs was replaced or highly diluted. About 40 ml of the buffer solution to be used was utilized for calibration, rinsing, etc. The concentration of the continuous phase was assumed to be constant throughout a single run, although this is not strictly true (see above). RESULTS
AND DISCUSSION
For convenience, the continuous phase was replaced with fresh solvent only when the mass transfer at a given pH value was to be measured for all the contact times utilized to evaluate KaPP. In order to justify this procedure, corrections for the effect of the solute transported into the continuous phase during the preceding run were examined. If the value of cz is not zero but is independent of cl, integration of Eqn. 4 gives
ln hi
- cz.i/~)l(c1.0
-c&Q1
= -(3/r) &G
(7)
where i denotes the number of runs within one set. One set is defined here as the number of runs used to obtain Kapp. Mass transfer during drop formation is likely to differ from mass transfer during the free fall of the drop [8, 91, and the concentration in the drop at t = 0 (i.e., the start of free fall) is difficult to estimate. The value of c1,o was therefore calculated by a stepwise procedure. The c~,~, obtained from the intercept when ln cl*1 was plotted versus t, was inserted in Eqn. 7. Another c1,o was calculated by applying the resulting Kapp and the concentrations cl,, and c2,1 to Eqn. 7. The presented cl,o was obtained by repeating these two steps once, starting with the new c~,~. The following procedures were tested for the collection of the data used to calculate KaPP: replacement of the continuous phase by a fresh solvent between runs; no replacement of the continuous phase until one complete set of runs had been completed and the order of contact times was increased or decreased. Depending on how the runs were done, c2,i was calculated in different ways. If the continuous phase is replaced by fresh solvent between the runs, i.e., after each change of the height of fall, a realistic assumption will be C 2,i
=
0m5
(vl,ilv2,i)
(Cl.0
-
cl,i)
(8)
where Vl,i (m”) is the total volume of the dispersed phase used in run i. With regard to the unprocessed volume of the dispersed phase, limited by the coalescence interface and the spectrophotometer cuvette, the concentration
175
Cl%i,which is measured at the end of the run, corresponds to the concentration of the dispersed phase when only about half the number of drops have passed the continuous phase. This assumption justifies the insertion of 0.5 in Eqn. 8. As 2.2 m of the column had a volume of 4.5 X low4 m3, the volume of the continuous phase involved in the mass-transfer process was calculated from V2.i = h, 4.5 X 10F4/2.20 where hi, in metres, is the height of fall in run i. When the continuous phase was not replaced with a fresh solvent until one set of runs had been completed and when the runs were done at decreasing heights of drop fall, the following formula was used =
c2,i
c2,i-l
+
0m5
tvl,i-1
/VZ,,-1)
tcl,O
-cl,i-l)
+
0.5(v1,
i/V2,i)(cl,0
-Cl,r) (9)
A similar equation was used when the runs were done with increasing heights of drop fall c2,i
=
C2,r-l
+
0e5
(v2,i-l (vl,i/V2,i)
Iv2,i) (Cl.0
+
0.5
tvl,i-I
lV2.i)
(Cl.0
-Cl,*--1)
-Cl,i)
(10)
The different sets of runs can be compared by inserting Eqns. 8-10 in Eqn. 7 for the appropriate cases. For large values of m (i.e., for pH > pKA when & > lo), the effect of the solute transferred to the continuous phase can be neglected (cf. Eqns. 3 and 4). However, a decrease in pH implies a decrease of m, the effect of the transferred solute increases and a correction is needed. This is demonstrated in Table 1, which shows&_, calculated from Eqns. 6 and 7, respectively. A TABLE 1 Results at various pH values with the corresponding K,, PH
6.233 6.471 6.482 6.693 6.710 6.950 7.186 7.322 7.634 7.926 8.071 8.427 8.655 8.870 8.987
calculated from Eqns. 6 and 7.
ms*)
m (Eqn. 3)
Kapp (10’ (Bqn. 6)
Wqn. 7)
0.03 0.06 0.06 0.09 0.10 0.17 0.29 0.39 0.80 1.54 2.12 4.52 7.10 10.50 12.75
0.42 0.72 0.73 1.19 1.09 1.92 2.60 3.03 4.60 5.40 8.65 12.16 11.41 11.56 12.01
0.53 0.88 0.86 1.42 1.30 2.29 3.01 3.42 4.99 5.74 9.31 12.66 11.79 11.82 12.25
26.2 22.2 17.8 19.3 19.3 19.3 15.8 12.9 8.48 6.30 7.63 4.11 3.33 2.25 2.00
0.6 0.8 0.8 1.3 1.3 2.4 3.5 4.1 7.4 10.1 18.8 32.3 34.0 35.1 38.2
is the percentage increase of Kapp when Eqn. 7 is used instead of Eqn. 6. bMaximum value of c,/c,.
aAKap*
176
series of runs was done at pH 6.2-9.0, with MDBA as transported solute. The values of K, and pKi at 25°C were found to be 41.5 + 1.6 and 9.34 + 0.01, respectively, calculated from a van’t Hoff plot (In K vs. l/T) [9] of data obtained at 5-7 different temperatures from 15-45°C (the errors indicate the 95% confidence limits [lo]). As can be seen from column 5, the correction increases from 2% at pH 9.0 to 26.2% at pH 6.2. It may be noted that to some degree the effect of a decrease of m is counteracted by the accompanying decrease of c2 (cf. Eqn. 7 and column 6, Table 1). This variation of cZ is due to the slower mass transfer at decreasing values of m (cf. Eqn. 2). Compounds with values of K, < 1 can be discussed in the same way except that the correction will also have to be done at pH > pK: because m < 1 for all pH values. The correction for the transferred solute is expected to have a different influence on the sizes of the calculated individual mass-transfer coefficients. Because it was impossible to separate the different types of coefficients in Eqn. 2, the equation was simplified by introducing k1 and kz defined as + l/k,)-’ and k2 = (l/k,, f l/k,)-‘: k1 = (l/k,, K app = [l/k1 +
WwW’
(11)
According to Eqn. 11, k1 = Ifa,_, for m 9 1. Because the correction of K app is small in this case, the refined value of kI is expected to be influenced only slightly by the effect of the transferred solute. For m << 1 (i.e., for K(, Q 1 or/and at low pH), the l/k1 term in Eqn. 11 is negligible and kz = Kappm-l. According to the above discussion, the correction of K,,, could be significant for m < 1, which underlines the importance of the correction leading to a proper value of k2. This was demonstrated by calculation of the combined mass-transfer coefficients, kI and k2. Nonlinear regression treatment of the data presented in Table 1, using the logarithmic form of Eqn. 11, was done by means of the MINUIT-D-506 program [ll] on a NORD-10 computer. The results obtained were kI = 1.24 (kO.24) X lo4 m s-l and k2 = 1.21 (kO.18) X lo4 m s-l, using the K app obtained from Eqn. 6. When corrections were made, and Kapp calculated from Eqn. 7 was used, the value of k2 increased to 1.54 (kO.27) X 10m4 m s-l, whereas the value of kI was not significantly changed at 1.22 (kO.25) X 10m4 m s-l. The errors give a parameter change resulting in an increase of the residual sum of squares corresponding to an approximate 95% confidence level. An F-test [lo] was used for the calculation according to the formula S - So = So{ b/(n -p)] F(p, II - p, 0.95) - 11, where S is the residual sum of squares when the parameter is changed by one error, S,, is the minimum residual sum of squares obtained from the regression, p is the number of parameters and n is the number of observed data points. In order to test the proposed corrections for the deviation from the sink conditions in the continuous phase, additional experiments were done. In all runs, the pH was 7.10-7.20, and MDBA was the transported solute. At pH 7, the value of m was 0.2 and the need for a significant correction was
177
expected. A lower value of pH was avoided to ensure that KaPP could be determined precisely. The slight pH difference between the sets of runs was corrected for by normalizing I&.,,, to the same pH value. The following formula, based on Eqn. 11, was used: K aPP,n
=
[l/k,
Kapp
+
Wzm)l
[l/k, + Wm,W-’
(12)
where m, is the apparent partition coefficient (Eqn. 3) at pH 7.1, and K is the normalized Kapp at the same pH. %‘\he first sets, the continuous phase was replaced with a fresh solvent after each change of the height of fall. The volume of the continuous phase, exposed to mass transfer, depended on the position of the nozzle and varied for each run. Between runs, a fresh coalesced layer of pure buffer solution was formed after the column had been filled with a fresh continuous phase. In order to evaluate Kapp, equivalent runs were done consecutively at five different heights. Another 13 sets of runs were done in the same way. The values of Kapp obtained by using Eqns. 6 or 7 and 8 and normalization (Eqn. 12) are shown in Table 2. The correction produced by using Eqns. 7 and 8 increases the value of Kapp insignificantly from 3.26 to 3.33 with an average deviation of kO.16 in both cases. One set of runs used for one evaluation of Kapp is presented in Table 3. An inspection of the third column shows a decrease of c2 with increasing contact times, which is probably a combined effect of the stepwise increase of the volume of the continuous phase and the decreasing rate of mass transfer. Despite this variation of c2, the correction of Kapp according to Eqns. 7 and 8 will be slightly positive. This can be explained by the lower percentage effect of the correction at a high value of c1 (i.e., at a short contact time) compared with the percentage effect at a lower value of cl present at a longer contact time (cf. Eqn. 7). A comparable series of experiments was done in which the continuous phase was not replaced with a fresh solvent until a complete set of runs used to TABLE 2 Results obtained when the continuous radius 2.58 X lO+ rd.) PH
7.023 7.142 7.143 7.152 7.176 7.173 7.188
*a
(10” m s-1
Cl (mol l-‘)
K app,n Eqn. 6
Eqns. 7, 8
5.0x 5.0 x 5.0x 5.0 x 1.0 x 1.0 x 1.0 x
3.16 3.05 3.65 3.16 3.02 3.29 3.02
3.22 3.10 3.73 3.23 3.07 3.35 3.08
10’ 10” lo6 10-5 lo4 lOA lo4
sThe concentration phase.
phase was replaced after each run. (Mean drop
PH
7.132 7.099 7.103 7.106 7.095 7.088 7.063
Cfa
(mol l*) 3.8 3.8 3.8 4.8 4.8 4.8 4.8
x x x x x x x
lo-’ lolo+ lo4 1O-4 lolOA
K app,n (IV
ms-’ 1
Eqn. 6
Eqns. 7, 8
3.20 3.30 3.12 3.55 3.28 3.58 3.29
3.26 3.36 3.18 3.64 3.34 3.66 3.35
of MDBA in the aqueous solution before its contact with the organic
178 TABLE 3 The set of runs at pH 7.095 (cf. Table 2)* Run no.
(Cf ,i In (cl ,0 -
Cl (10-M)
1 2 3 4 5
4.02 3.77 3.52 3.20 2.96
3.55 2.34 2.26 2.11 2.03
-0.1428 -0.2070 -0.2756 -0.3709 -0.4489
Contact
w/m) C,,i/mT
time
-0.1480 -0.2121 -0.2824 -0.3798 -0.4597
(S)
3.85 5.55 7.26 9.90 11.85
*The value of cl,0 was calculated as 4.637 x lo4 mol 1-l; cI was evaluated from the absorbance at 272 nm and c, was derived from Eqn. 8. TABLE 4 Results obtained when the continuous phase was only replaced after each set of runs. Otherwise the experiments were done under the same conditions as the analyses presented in Table 2. K app,n UO-Sms”)
PH
CT - 5 7.177 7.173 7.173 7.171 7.171
X
K app,n Wm5 m s-l 1
PH
Eqn. 6
Eqns. 7, 10
1O-5 mol 1-l 2.83 2.70 3.00 3.01 2.91
3.43 3.21 3.54 3.59 3.45
cf = 5 7.088 7.100 7.081 7.081 7.076 7.092
X
Eqn. 6
Eqns. 7,10
lo4 mol 1-l 2.63 2.75 2.54 2.75 2.80 2.78
3.15 3.33 2.98 3.34 3.30 3.38
calculate Kapp had been obtained. The results are given in Table 4. These runs were done with increasing contact times within each set. The corrections were based on C2,ig calculated from Eqn. 10, and they resulted in a sharp increase of Kapp (about 20%) from 2.80 (kO.ll) to 3.34 (rt0.13). The details of the last set of these runs are given in Table 5. As shown in the third column, the solute concentration of the organic phase, c2, increases considerably at increasing contact times, because of the solute remaining from the preceding run. As previously discussed, the percentage effect of the correction is greater at longer contact times than at shorter times. Both these effects result in a considerable underestimate of Kapp if no corrections are made. A comparison of the above results indicates that when Eqn. 7 is used for the correction, it is justified to complete a set of five runs without replacement of the continuous phase. The corrected mean values of Kapp are 3.33 X lo-’ kO.16 and 3.34 X l(rs kO.13 m s?, respectively (Tables 2 and 4). No significant effect on the results was observed for a change in concentration
179 TABLE 5 The last set of runs given in Table 4 (pH 7.092)a Run no.
;;a4
1 2 3 4 5
4.09 3.86 3.66 3.34 3.15
M)
Cl (lo+
ln(C*” -c&n) (C, .O -c&n)
M)
3.54 7.02 9.11 10.18 11.95
-0.1420 -0.1999 -0.2531 -0.3446 -0.4031
aThe value of c, +, was calculated as 4.714
x
Contact time (5) 3.80 5.53 7.33 9.97 12.05
-0.1471 -0.2151 -0.2795 -0.3875 -0.4656
lOA mol 1-l; c, was derived from Eqn. 10.
from 5.0 X 1W5 to 5.0 X lo4 mol 1-l (cf. Tables 2 and 4). A more extensive study of the dependence on the concentration will be published later. The order of the heights of fall could be essential in obtaining the most accurate results. If the set is started with the longest contact time, i.e., with the nozzle at the top of the column, the whole column of the continuous phase will be available to the solute in the first run; but only the lower part of the continuous phase will contact the solute if the set is started at the lowest height. In both cases, the subsequent runs in each set will result in a non-uniform concentration of the solute in the continuous phase along the column, but the effect on mass transfer will be different depending on the order of the runs. If the runs are done with increasing height of fall, each drop will experience a stepwise increase of c2 as it falls in the column. For the opposite order, there is no such effect as each separate experiment involves part of the continuous phase which is assumed to have no discontinuities of concentration along the column. Table 6 shows the results of TABLE 6 Results obtained with increasing and decreasing orders of contact timesa PH
K app,n UO”m Eqn. 6
PH
s* 1 Eqns. 7, 10
Increasing contact times 7.142 2.26 7.127 2.27 7.197 2.08 7.184 2.29 7.165 2.42 7.145 2.34
2.59 2.60 2.31 2.57 2.75 2.70
Mean value Mean deviation
2.59 0.10
2.28 0.07
Decreasing 7.146 7.136 7.129 7.128 7.128
K app,n (lo-’
m s-l)
Eqn. 6
Eqns. 7, 9
contact times 2.59 2.74 2.85 2.84 2.83 2.77 0.08
2.51 2.65 2.75 2.74 2.68 2.67 0.07
mol l*) was transported from the aqueous aMDBA (initial concentration 5 X lo* drops to the continuous phase of cyclohexane. The mean drop radius was 2.62 X 10” m. Kapp,” was obtained as in Table 2.
180 TABLE 7 The first set of runs with decreasing contact time (Table 6)” Run no.
:iO-
1 2 3 4 5
3.65 3.92 4.25 4.53 4.78
M)
”(lo-’ 2.77 7.51 11.55 15.59 19.54
M)
ln(c,,Jc,,,) -0.3669 -0.2956 -0.2147 -0.1509 -0.0972
In
Contact time (6)
(c I*,.-c2,i/m) (c,,, -c,,,lm)
-0.3760 -0.3154 -0.2367 -0.1717 -0.1141
aThe drop radius was 2.611 X 10” m (pH 7.146); c,,~ was calculated as 5.268 mol l* ; c, was calculated from the absorbance at 226 nm and c1 from Eqn. 9.
11.87 9.67 7.60 5.21 3.49 x
10”
six sets with decreasing and increasing contact times. With increasing contact times, underestimated values of Kapp are expected from Eqn. 6 (see above), compared to Eqn. 7, and this was confirmed in practice (Table 6). Table 7 gives details of the first set with decreasing contact times in Table 6. There is a sharp increase of cz with decreasing contact times, which implies a significant overestimate of Kapp when derived from Eqn. 6. However, as discussed above, the percentage effect of the correction of cz depends on cl, which in this case has a reduced effect on the value of Kapp. This is experimentally verified in Table 6 (right side). The mean value of K app based on Eqn. 6 is insignificantly higher than the corresponding value derived from Eqn. 7 with c2 corrected according to Eqn. 9. The corrected mean values, 2.67 X lo-’ and 2.59 X 10B5 m C’ which are similar, and the corresponding agreement of Kapp, presented in Tables 2 and 4, suggest that equivalent results can be obtained irrespective of the order of the contact times, provided that the appropriate corrections are made. However, the procedure with decreasing contact times seems more acceptable because the drops then experience a relatively uniform concentration throughout the contact time. During large series of experiments such as described above, the main problem is to keep the system unchanged. It is well known that the slightest contamination in either phase can markedly influence the mass transfer [12], probably because of a change in the hydrodynamics of the system. For instance, the results presented in Tables 4 and 6 were based on experiments done at two times three years apart. A comparison of the mean values of the corrected Kapp,” shows a significant discrepancy which is probably due to contamination-induced differences. We express our sincere thanks to Mr. Bo Larsson for his skilful contribution concerning computer handling. We are also indebted to Prof. Allan Agren and Prof. Lars-Olof Sundelijf for all facilities placed at our disposal. Grants received from the IF Foundation for Pharmaceutical Research are gratefully acknowledged.
181 REFERENCES 1 A. Brodin and A. &ren, Acta Pharm. Suet., 8 (1971) 609. 2 I. Nahringbauer and B. Larsson, Anal. Chim. Acta, 161 (1983) 153. 3 H. Brenner and L. G. LeaI, A.I.Ch.E.J., 24 (1978) 246. 4 J. A. Shaelwitz and K. T. Raterman, Ind. Eng. Chem. Fundam., 21 (1982) 154. 5 C. V. Sternling and L. E. Striven, AIChE.J., 6 (1959) 514. 6 H. Sawistowski, in C. Hanson (Ed.), Recent Advances in Liquid-Liquid Extraction, Pergamon Press, Oxford, 1971, Ch. 9. 7 V. Zimmermann, W. Halwachs and K. Schiigerl, Chem. Eng. Commun., 7 (1980) 95. 8 J. S. Vrentas, H. T. Liu and J. L. Duda, Chem. Eng. J., 21 (1981) 155. 9 A. N. Martin, J. Swarbrick and A. Cammarata, Physical Pharmacy, Lea & Febiger, Philadelphia, PA, 1969. 10 N. R. Draper and H. Smith, Applied Regression Analysis, Wiley-Interscience, New York, 1966. 11 F. James and M. Roos, CERN/DD Internal Report 75/20 (1976). 12 L. Mekasut, J. Molinier and H. Angelino, Chem. Eng. Sci., 34 (1979) 217.