ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT fQUR PLbNTS
uininw
COST LOCUS
P R Q O V C T I Q H BUILD-UP
THREE
/A\
m.Anrs
TWO PLANTS
ii
0
CAPACITY
OUTPUT
Figure 17.
IN UNITS
Production Build-up and Cutback Loci
will show that a t OD the amount of dollars saved by shutting down plant A (area A A ) just equals the additional out-ofpocket outlay for running other production units from OB to OD (area BB). -4t any point greater than joint output OD, area BB would be larger than A A thereby encouraging the continued operation of plant A. At any joint output less than OD it is cheaper to operate with A in standby. This procedure can be repeated until the last plant itself is run back to zero after all others have been eliminated. In this way a complete schedule of minimum costs for all outputs can be developed. Build-up Procedure Follows Same Principles Used to Develop the Cutback Method
In the previous sections a method was outlined for cutting back the joint output of a number of plants producing a homo-
geneous product from capacity of the complex to zero output. I t was mentioned earlier that present business conditions (195354) made the cutback method distinctly more timely. However, the same principles can be used to develop a minimum cost locus from zero production to the capacity of all plants operating simultaneously. During a depression when most or all plants of a complex are in standby it is necessary to develop a plan for increasing production (as business conditions permit) a t minimum over-all cost. If all of the plants are in standby then the solution of the problem begins by considering which of the plants in standby has the least net annual cost of start-up (annual start-up minus annual standby charge). However, just as soon as or as long as any plant in the complex is in operation, a balance must be struck b e h e e n increasing production from a going unit and bringing another plant on stream. Output in the plant with the least net start-up cost is increased from zero to a point where the saving in cutting back an increment of production just equals the cost of bringing the second plant on stream. From this point on both plants are operated jointly a t equalized marginal cost until it pays to take the third plant out of standby. This process is repeated until all plants are on stream and operating simultaneously a t capacitv. The principles underlying the build-up method are exactly the same as those outlined in detail for the cutback method. The schedules developed through the uee the catback and buildup methods (see Figure 17) \Till be different because they result from two different situations. The cost X - Y (exaggerated for emphasis) is the “inertia” cost of start-up -the penalty which must be paid for allowing a complex of plants to grind to a halt. However, in either event, if the method selected truly reflects the circumstances of joint output (build-up or cutback), then it will yield the best procedure for adjusting output either up or down and ensure that any desired level of output will be produced a t the minimum over-all cost t q the multiplant complex. Literature Cited (1) Allen, R. G. D., “llathematical Analysis for Economists,” p. 366, London, Macmillan, 1950. ( 2 ) Lfayer, K. &I., Chem. Eng.,60, 214-16 (1953). R E ~ E I V E D for revieiv january 29, 1953. ACCEPTED April 7, 1954
Mechanism of Solute Transfer from Droplets LIQUID-LIQUID EXTRACTION F.
H. GARNER AND A. H. P. SKELLAND
Deparfmenf o f Chemical Engineering, The Universify o f Birmingham, Birmingham 75, England
I
N ORDER to improve the efficiency of the operation of liquid-
liquid extraction for the separation of components of solutions, a more complete knowledge is required of the factors affecting its basic mechanism-solute transfer between two liquid phases, one of which is dispersed in the other. Relatively few papers on transfer t o or from single drops have appeared. In the earlier work of Sherwood, Evans, and Longcor ( I S ) the size and number of drops were known, giving the area available for transfer, and hence the variation in the transfer coefficient, K , could be determined. They extracted acetic acid from methyl isobutyl ketone and benzene droplets with water and found that about 40% of the solute was extracted before the drop
June 1954
left the nozzle. Comparison between experimental and theoretical transfer coefficients during rise led them t o postulate a degree of circulation within the drops which assisted transfer. West, Robinson, Morgenthaler, Beck, and MeGregor ( 1 6 ) repeated some of this work on the benzene system, but found only 14 to 20% extraction during formation and concluded that the drops were internally stagnant. In a later paper, West, Herrman, Chong, and Thomas (16) explained the discrepancy in terms of contamination of the benzene-acetic acid solution due to its passage through Tygon tubing. The rate of transfer in the system benzene-acetic acid-water was also substantially altered by addition of various alcohols to the organic phase.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1255
Licht and ConiTay ( 2 1 ) separated the extraction process in a spray column into three aspects, drop formation, free rise or fall, and coalescence at the end of the column. For the extraction of acetic acid from wtter droplets by isopropyl ether, methyl isobutyl ketone, and ethyl acetat?, they found the extraction during drop formation to be 5 , 8, and 1 i % , respectively, irrespective of drop size and formation time. Coulson and Skinner ( 3 )transferred propionic and benzoic acid from solution in water to benzene drops and measured the transfer coefficient, K , during drop formation by forming a drop then collapsing it. A s suming that the transfer rate was the same during formation and subsequent collapse, they shovied K t o be inversely proportional to formation time, but independent of drop size. K during free rise of the drops increased with drop size but decreased wiLh increasing interfacial tension, and they shomd that it is the time of contact rather than the height of rise B which is important. Comparison between experimental and theoretical amounts transferred Figure 1. Apparatus indicated a degree of circulation A. Drops formed above within the drops. surface The authors of this paper 8. Dispersion nozzle imhave investigated the diffusion mersed of acetic acid from droplets of nitrobenzene falling in water. The distribution ratio for the acetic arid at I o n concentrations is about 16 to 1 in favor of the n-ater.
!i
Water-Acetic Acid-Nitrobenzene System l o Used for Extraction Studies
Materials. Acet'ic acicl contained not less than 99.5% by weight of CE-I1COOII. Birmingham tap n-ater was used; titrations showed it to have negligible aciditmyand a total solids content of 44 p.p.m. Sitrobenzene of maximum acidity, 0.05 nil. I S acid per 100 grams, was used for most of the Ivork. For a few of the runs (curves of Figure 7 labeled "50%" and "20%" and curves 1 and 2 of Figure 4), t'he nitrobenzene supplied vas further purified by distillation through a packed column, giving a product of refractive index nZ$ = 1.33137. Apparatus. To provide a constant head, a 25-cc. buret was fitted with a tightly fitting rubber stopper through s.hich passed a thin-walled glass tube to admit a slon- stream of air to a point within the nitrobenzene-acetic acid solution in the buret. h protractor and pointer m r e attached to the buret tap, and a variation of about 9 " on the protractor covered a range of drop formation times from 0.5 t o 50 seconds. The buret was recalibrated to allow for the air tube and drop sizes found from the volume required t o form a given number of drops. 1-arious size glass nozzles were attached t o the buret by polythene tubing, which is not noticeably attacked by the solution. (a 25-ml. portion of a 6y0 solution of acetic acid in nitrobenzcue shon-ed no detectable change in refractive index, detwmined by an Abbe refractometer, after a 12-day period of contact with 0.0 gram of tubing.) I n the earlier work on variation of drop size, an attempt was made t o eliminate extraction during drop formation. Drops vr-ere formed above the surface of the water and entered it only after 1256
detachment. It was found necessary to surround the nozzle with a tube of diameter somewhat larger than the nozzle, and this surrounding tube entered the water surface to a depth of 0.5 inch. This provided a suitable meniscus for the drops to enter the Tvater without breakup, though the height between meniscus and nozzle tip 1%-as critical for each nozzle (Figure 1, A ) . It x a s later found that this merely substituted another solution effect like that in drop formation, and in all other experiments the nozzles were immersed in the water (Figure 1, B ) . Borosilicate glass extraction columns were used which werc about 5 cm. in diameter with effective heights of i 1 , 60, 40, 20, and 8 em. At the foot of each column was a 60" conical section f i om I+hich di ops were immediately withdraivn through a capillary exit via a tap and collected in a 25-cc. measuring cylinder. The duration of the runs and the droplet fall velocities were measured by stop watches, and from 5 to 15 cc. of disperse phase was passed through the column during most of the runs. The water temperature throughout the uorli was 19" & 2" C. All acid concentrations were determined by titration with aqueous potassium hydroxide. Sufficientethyl alcohol was added t o the nitroben~enesamples to maintain a single phase. A comoarison was made between the total acid initiallv in the disperse phase and the sum of the acid finally in the disperse phase plus that transferred to the water. Most of the runs balanced within 3% and all vtithin 6 t o 7%. Over-all Transfer Coefficient Decreases with Increased Time of Drop Formation
Extraction from Drops during Formation. Experiments were carried out to measure directly the extraction during droplet formation in the Fater-acetic acid-nitrobenzene system. Two borosilicate glass nozzles were used; one thick-walled nozzle had an outside diameter of 0.514 cm. and an inside diameter of 0.049 cm., 11hile the thin-walled nozzle had an outside diameter of 0.516 em. and an inside diameter of 0.364 cm. Each nozzle was ground with fine carborundum flour so that the plane of the tip J? as a t right angles to the nozzle axis. It 7%-asnecessary to grease both the thick-walled and the thin.ivallecl tips with a thin film of while Vaseline in order to get the nomle tip completely wetted by the disperse nitrobenzene phase. It n-as assumed that any modifying effects on extraction due to the Yaseline would be similar for each nozzle, but, in any case, reproducible wetting of the tips vithout the Vaseline seemed almost impossible. The diameter of the droplet is determined mainly by the external diameter of the capillary provided that the face is wetted. Thus, these two capillary tubes gave drops of about the samc di-
Table I.
Experimental and Theoretical Data during Drop Formation
Initial disperse phase C O ~ C I I . = 0.001048 gram-mole solute/cc. Water concn. = 0 H = 0.0625 Dispersing nozzle immersed Dsol, Diani. a t Forma&, CorDetachtion Fracrespondnent, Time, tion Krlexp., ing t o k-dtheor., KdaxD./ Cni. Sec. Extd. Cm./Soc. Am, Cm. Cm./Sec. &Itheor. Thick-Walled Capillary Nozzle 0.576 0,583 0.602 0.608
2.7 6.5 43.4 61.6
0.302 0.347
0.595 0.699 0.599 0.602 0.599
2.7 2.8 3.5 27.3 29.8
0,218 0.19i 0.237 0,521 0.53
0.635 0.616
0.02210 0.01105 0.00406 0.00335
0.43 0.45 0.47 0.47
0,00187 0.0012
11.8 9.2
0.00047 0,00043
8.7 7.8
0.00187 0.00183 0,00165 0.00059 0.00057
8.2 7.2 8.6 7.83 7.8
Thin-Walled Open Nozzle 0.01630 0.01320 0,01425
0.00465
0,00441
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTXY
0.46 0.46 0.46 0.47 0.46
Vol. 46, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Kd =
W
where W is the extraction per drop, t the time of formation, and ACI, the mean concentration difference between the two phases before and after formation. Concentrations are based on the disperse phase; the continuous phase concentrations are converted to the equivalent equilibrium disperse phase concentrations by use of the distribution ratio, H . A , is the equivalent mean drop surface such that the area of surface multiplied by the time of contact is the same as for that of the growing droplet. A , is derived as follows: TIME OF FORMATION, SEC.
Figure 2. Effect of Time of Drop Formation and Nozzle Construction on Fraction Extracted Diameter, Cm. Inside Outside
0 Thick-walled nozzle 0
Thin-walled nozzle
0.049 0.364
0.514 0.516
The simplifying assumption is made that the drop grows from zero volume at time to and is always a true sphere until its detachment a t time t l . If u and d are volume and diameter of any sphere, then d = 2 /3 ( 6 v / ~ ) ~ Surface /~, area A = r d 2 = ):(r , If the drop volume changes is
SO
dV that - = constant, then A a t any time t at
equal to
Equating the product of surface and time, and introducing A,n ameter, but the linear velocity of the liquid in the narrow capillary was much greater for a given formation time, leading to greater turbulence during drop formation. The continuous phase consisted of 25 ml. of distilled water placed in the cone a t the foot of the 7.5-em. column. The drops fell some 4 or 5 mm. before leaving the column by its capillary exit; the conical wall of the column was about a centimeter from the drop surface a t detachment, The proximity of this wall should not have a marked effect because with the small number of drops used-20 to 50-the final aqueous concentrations, when converted t o equilibrium concentrations in the disperse phase, tvere usually less than 1% of the final or extracted disperse phase concentrations. Some extraction may occur during withdrawal of the drops from the column, and it is probable that the figures obtained represent maximum values for extraction during formation. The drops initially contained about 6% by volume of acetic acid in nitrobenzene. Formation Time and Nozzle Size. The results, listed in Table I and plotted in Figure 2, show that there is a marked difference in the rate of extraction from the two nozzles up to times of formation of a t least 6 seconds. At 2.5 seconds, transfer during formation using the narrox capillary exceeds that using the wide capillary by about 50%, evcn though Figure 3 shows that, a t this formation time ( 2 . 5 seconds), the volume of a drop a t detachment is 0.097 cc. for the narroTv capillary and 0.115 cc. for the wide capillary; the corresponding surfacc,s of these drops, assuming them t o be spheres, are 1.02 and 1.14 square em., respectively. The wide capillary, therefore, gives a drop whose surface, a t detachment, is about 11.8% greater than that of the drop produced by the narrow capillary a t a formation time of 2.5 seconds. Clearly then, the agitation inside drops forming on the narrow capillary is substantially greater than in those forming on the wide one, a t least for short formation times, resulting in increased extraction. At formation times exceeding about 6 seconds the extraction is about the same for either nozzle, and the rate of extraction decreases. Figure 2 show that a 25-fold increase in formation time -from 2 to 50 seconds-produces only about a twofold increase in fraction extracted. The drop sizes produced by the two nozzles a t different formation times are compared in Figure 3. Over-all Transfer Coefficient during Formation. During formation the droplet surfare is continually changing with time and, accordingly, an integrated mean drop surface, A,, was used in the expression lune 1954
whence
The calculated values obtained for E d e x p . will be only approximate in view of the simplifying assumptions made in obtaining A,, but should be of interest for comparison purposes. Kdexp.is plotted against formation time in Figure 4,which shows that a rapid fall in the transfer coefficient occurs with increasing time of formation up to about 20 seconds, beyond which the transfer coefficient falls very slightly.
Table II.
Experimental Data for Entrance Effects plus Free Fall with Varying Drop Size Initial water concn. 0 Drops formed above surface Column Time of Fall FracHeight, of Single tion Cm. Drop, See. Extd. E
Log Mean Drop Dlam., Cm.
Kd, Cm./Sec,
50% Solute Mixture Initial Disperse Phase Concn. = 0 00862 Gram-hIole/Cc 0.284 0.339 0.406 0 443
71 71
71
71
8 7 7 7
3 7 4 2
0 0 0 0
989 977 961 956
0.0268 0 0318 0 0335 0 0354
20% Solute hilxture Initial Disperse Phase Concn. = 0 00347 Gram-RIole/Cc. 0.26 0.316 0.383 0.455 0.51
71
71 71 71 71
7.7 7.2 6.5 6.3 6.1
0.973 0.944 0.917 0.89 0,872
0.0205 0.0221 0.0249 0,0274 0.0292
4% Solute Mixture Initial Disperse Phase Concn. = 0.000685 Gram-hlole/Cc. 278 278 338 411 0 411 0.485 0.547 0 278 0.338 0.338 0.411 0.485 0.547
0 0 0 0
71 71 71 71 71 71 71 8.0 8.0 8.0 8.0 8.0 8.0
7.3 7.3 6.5 6.1
6.1 5.8 5.7 0.82 0.73 0.73 0.69 0.65 0.64
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.525 0.535 0.483 0.500 0.505 0.440 0,448 0.261 0.189 0.196 0 . I74 0.161 0.160
0.00503 0.00512 0.00595 0.00808 0.00812 0,0088 0.00945 0.0172 0.0162 0.0168 0.0187 0.0217 0.0241
1257
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT The results are shown in Figure 6 and indicate that, for the 4% solut,ion, thew is not any great variation in fraction estracted wit,h drop size, xhen entrance effects are excluded. Column Height. Experiments n-ere made using effective column heights ranging from 1.5 to 71 em. and with tw-o disperse phase solutions initially containing 6 and 35 volume 470 of acetic acid in nitrobenzene. In these experiments the dispersing nozzle vias immersed in t h e continuous phase. Initial drop volume TIME OF FORMATION, SEC. was 0.117 cc. and formation time 2 seconds for the 6% mixt)ure, and 0.02 cc. and 0.5 Figure 3. Effect of Time of Drop Formation and Nozzle Construction on Drop seconds for t,he 35% mixture. Volume Fraction8 extracted are plotted against 3 Thick-walled nozzle column height in Figure 7 , in which curve 1 7 Thin-walled nozzle refers t o the 6% and curve 2 to the 35% solution. Ext,rapolation of curve 1 to zero 0 0 4 , ,, , , , , , , , , , , , , , , , , , , , , , , , , column height ehom that the fraction extracted during drop t I I I4 formation is about 17%. In t,he experiments during drop iormnt,ion for this mixture, drop diameter (0,608 em.), and formation time ( 2 seconds), t'he fraction extracted during formation was directly measured to be 16%. This appears close enough to juEtify this procedure of extrapolation to zero column height, though Licht and Pansing ( I S ) consider the validity of sur:h extrapolation t o be doubtful. Similar extrapolation of curve 2 is rather difficult, but the curve shorn that about 367, of extractable material is removed during drop formation and the first 1.5 cni. of fall. ~
0
I , 10
,
,
,
,
,
,
,
20
,
,
,
30
Concentration Changes Significantly Alter Over-all Transfer Coefficient during Free Fall
I , , .
40
50
TIME OF FORMATION, SEC.
Figure 4. Effect of Time of Drop Formation on Overall Transfer Coefficient during Formation
0
Over-all transfer coefficients hased on the disperse phase n eie calculated by Equation 1, in which A , and i C l m correspond to the log mean of the drop diameters and concentrations, respectively, before and after pasEage through the column. Drop Size with Entrance Effects Included. These droplets
Thick-walled nozzle
0 Thin-walled nozzle
Solute Concentration and Drop Size Affect Extraction during Free Fall
Drop Size and Inlet Concentration. In these particular experiments all drops were formed outside the cont'inuous phase. Five thick-walled nozzles were used, of outside diameters ranging from 0.128 to 1.545 cm. Fraction extracted is plotted against log mean drop diameter in Figure 5 . (The abscissa gives the log mean of t,he drop diameter before and after passage through the columns.) Curves 1, 2, and 3 are for droplet solutions initially containing 50, 20, and 4%, respectively, of acet'ic acid in nitrobenzene, using the TI-em. column. Curve 4 is for the 4% solution using the 8-cm. column. Entrance effects (due t o the droplets passing through the \vater surface) are included in all four curves, xhich shoTy that, the fraction extracted increases Tvith reduction in drop diameter and that, in the 71-em. column, for a given log mean drop diameter, the fractional ext>raction increases n-ith inlet, solute concentration. Curves 3 and 4 for the 4% solut,ion shon- that beta-een 30 and 40% of the extraction in the 71-em. column is effected within the first 8 em., depending on drop size (Table 11). Drop Size with Entrance Effects Eliminated. In Figure 5 the values of curve 4 were subtracted from the corresponding values of curve 3 to obtain fractions ext,racted for the 4% solution versus drop diameter with entrance efiects excluded.
1258
U
02
8
1
0 3
'
'
I
05
04
0 6
DROP DIAMETER, CM
Figure 5. Effect of Drop Size on Fraction Extracted with End Effects Included
1. 2. 3. 4.
Initial Concn., % 50 20 4
4
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Column Height, Cm. 71 71 71
8
Vol. 46, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT were formed above the surface of the continuous phase, and the experimental coefficients corresponding to passage through the water surface and sub3equent fall through the 71-cm. column are plotted against log mean drop diameter in Figure 8. K d e r p . increases substantially with increase in drop size and, for a given log mean drop diametei, K d increases with increasing inlet acid concentration. Drop Size with Entrance Effects Excluded. Additional experiments a t various column heights showed that there %-as noteworthy extraction while the drop penetrated the water surface. To eliminate this effect, further runs were made on the 4% solution in an 8-em. column, and then values of Kdexp.were calculated for the length of fall between the 8- and 71-cm. points. The resulting values, when plotted against drop diameter, give curve 2 of Figure 9. Since the distribution ratio for the solute greatly favored the water phase (approximately 16 to 1 at low concentrations), most of the transfer resistance must be in the nitrobenzene droplet. Consequently, the increase in Kd with drop diameter must be predominantly due t o reduction in effective thickness of the disperse phase film. This probably corresponds to a greater speed of internal circulation within larger drops, presumably due to the greater skin frictional drag on the drop surface a t the higher fall velocities obtained with such larger drops. These values of Kdexpwith entrance effects excluded give essentially the same form of curve if plotted against the local Reynolds number (in which d is the drop diameter, V its fall velocity, and p and p are density and viscositv of the outer phase). Change in Solute Concentration. Values of Kd were obtained for various lengths of free fall by subtracting results in a short column from those in the longer ones (Table 111).
18% Solution. Experiments were made on a solution initially containing 35% of acetic acid, but three runs a t an effective column height of 8 em. gave an average droplet concentration at the 8-cm. point (free fall datum) of 17.8 volume %. Drops were formed a t a thin-walled glass nozzle immersed in the water, and the average drop volume during free fall was 0.0135 cc. Values for this solution (Figure 10, C) show that the fall in Kd is substantial as the drop progresses down the column and as the solute concentration decreases from 17.8% a t 8 em. t o 2.33% a t 71 em. from the jet.
i’ Figure 7. 1. 2.
DROP DIAMETER, CM.
Figure 6. Effect of Drop Diameter on Fraction Extracted with End Effects ExcIud e d Initial concn.
4%,
column height 71-8 cm.
4% Solution. Drops were formed by a thin-walled glass nozzle immersed in the water, and for the 6% solution the average drop volume during the fall was 0.118 cc. For the solution initially containing 6 volume % of acetic acid in nitrobenzene, the shortest column height used was ‘7.5 em., and at the 7.5-em. point the drop concentration had fallen to 4.3%. Values for this solution are plotted in Figure 10, A . The Concentration change doivn the column is small, 4.3 volume yo solute at 7.5 cm. and 1.’73%at 71 cm. Kd has the value 0.015 em. per second throughout, no variation being detectable. 13% Solution. Additional experiments were made on a solution which contained 13.1 volume % of solute a t the free fall datum 8 em. from the jet; the average drop volume during free fall for this solution was 0.0134 cc. The data in Figure 10, B , show a reduction of almost 50% in Kd, corresponding to a droplet concentration change from 13.1 t o 3.29% a t the point 71 cm. from the jet. June 1954
, ’ 20
I,,
, * 40 ’ 60 COLUMN HEIGHT, CM. I
,
80
Effect of Column Height on Fraction Extracted
Initial droplet concn. 6% acetic acid Initial droplet concn. 35% acetic acid
Factors relevant t o this variation of Kd dovin the column may be change in drop diameter and variation with concentration of distribution coefficient, droplet viscosity, solute diffusivities, and interfacial tension. ENDEFFECTS. It is possible that disturbances caused by detachment from the nozzle may persist for several centimeters into the column. It was felt, however, that any such effects were largely damped out in the first 8-em. of fall, and visual observations seemed t o support this view. Accordingly, a column with an effective height of 8 cm. was used to establish a free fall datum 8 cm. below the nozzle. DROPDIAMETER.Considering Figure 10, C, the drop diameter at 8 em. from the jet was 0.316 em. and at 71 em. was 0.296 em. Kd for the column section 8 t o 20 cm. exceeds that for the section 8 t o ’71 em. by about 0.008 cm. per second, and this is many times more than can be explained by change in drop size (Figures 8 or 9). DISTRIBUTIOR’ CoEFrICIER’T. It iTas found experimentally that increasing solute concentration increased the distribution ratio in favor of the organic phase (Figure 11). Thus, from the relation1 ship, = H/kc increase in concentration should cause a de-
+ 2,
crease in Kd rather than the observed increase ( H =
&robensene/
Cm,,).
DISPERSE PHASE VISCOSITY.From the droplet compositions a t the foot of each column, the disperse phase viscosity and interfacial tension can be obtained for points down the 71-em. column. For Figure 10, 23, the disperse phase viscosity varies only between 1.63 and 1.705 cs. and for Figure 10, C, only between 1.71 and 1.60 cs. Such variation is considered negligible as far as variation in Kd is concerned.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1259
ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT
Table 111. Log Mean Drou Diam., Cm.
Experimental Data a t Varying Column Heights Initial Disperse Phase Concn., Gram-Mole/Cc.
Column Height, Cin.
Time of Fall, See.
Fraction Extd.
Dispersion Nozzle Immersed in Water 0.608 0.608 0.608 0.608 0.608 0.608 0.608 0,608 0.608 0,608 0.608
0 00107 0 00107 0 00107 0 00107 0 00107 0 0 00107 00107 0 00107 0 00107 0 00107 0 00107
0.305 0,304 0,305 0.304 0.305 0.305 0.308 0.305 0.306 0.310 0.307 0.306 0.309 0.310
0 0 0 0
70 70 70 59 59 19 39 19 19 8 0 8 0
6 6 6 5 5 33 2 2 0 0
3 3 3 5 5 99 0 0 74 74
0 685 0 673
0 655 0 643
0 612 00 535 516 0 385 0 389 0 261 0 271
Drops Formed aboi e Surface 00515 00515 00532 00516 0 00515 0 00512 0 00515 0 00515 0 00532 0 00532 0 00615 0 00515 0 00518 0.00515
71 71 71 60 60 60 40 40 40 40 20 20 8 0 8 0
7 7 7 6 6
6
4 4 4 4 2 2 0 0
8 8 8 75 75 75 8 8 8 8 5 5 96 96
0 0 0 0 0 0 0 0 0
0
0 0 0
0
901 924 918 909 915 904 861 878 852 860 800 795 643 628
There is no detectable change in K d vi-ith length of free fall for the 6% solution (Figure 10, A ) for Tvhich the change in interfacial tension down the 71-cm. column is only 2.8 dynes per cm. Suhstantial reductions in Kd .rc.ith increasing fall length are, however, shown by Figure 10, B and C, for TThich the reductions in solute concentration lead to increases in interfacial tension over the greatest fall length of 6 8 and 9.4 dr nes per em. Experimental and Theoretical Transfer Coefficients during Drop Formation Are Compared
The theoretical over-all coefficient, K d t h e o r was calculated on the assumption that transfer was to a constant size, stagnant sphere that is equivalent in that the product of surface X time of contact is the same as for the groving drop. Its diameter is d,, corresponding to A,, derived earlier in this paper. Diffusivity of acetic acid in water was shown by Thorvert ( 1 4 ) to be 0 88 X 10-6 square cm. per second at 20" C., and the diffusivity of acetic acid in nitrobenzene 17-as calculated by the method of Arnold ( 1 ) to be 0.665 X 10-5square cm. per second. The continuous phase film coefficient vias calculated from the expression given by Higbie ( 9 )
k, = 2
Dispersion Nozzle Immersed in Water 0.309 0.309 0.311 0.311 0.31 0.31 0.31 0.31 0.31 0.31 0.314 0.314 0.319 0 319 0.319 0.333 0.333 0.333 0.333 0.333
0.00627 0.00627 0.00604 0 00604 0.00627 0.00627 0.00627 0.00627 0.00627 0.00627 0.00627 0,00627 0.00604 0.00604 0,00604 0 00804
70 70 70 70 60 60 60
40 40 40 20 20 8.0 8.0 8.0 1.5
7 57 7.57 7 57 7 57 6 4 6 4 6 4 4 26 4.26 4.26 2.13 2.13 0.8 0.8 0.8 0 IR
0.933 0.95 0.946 0 952 0.936 0.945 0.94 0.883 0.888 0.893 0.775 0.78 0.585 0.582 0.587
(3)
where D1 = solute diffusivity in continuous phase. This expression is said to apply to unsteady-state diffusion through a surface into a stagnant film of infinite thickness and gives the effective transfer coefficient for a time of diffusion, t,. The disperse phase film coefficient was calculated uiing r-ipressions given by Geddes (8)
ks
-r
i=
- log, B 3t
(4)
\There B is an abbreviation for the expression
SOLUTE DIFFUSIVITIES. Evidence for diffusivity change Kith concentration is meager, but data in the International Critical Tables (10) show that the diffusivity of acetic acid in water actually falls with increasing concentration. Arnold (1) examined the diffusivity data of Dummer for acetone, ethyl benzoate, and ethyl acetate diffusing in nitrobenzene in concentrations of solute to 20%. He concluded that diffusivities did not vary substantially over this concentration range for these systems. It is conBidered unlikely, then, that the observed increase in Kd with increase in concentration in this system can be accounted for by any increase in solute diffusivities with rise in concentration. INTERFACIAL TENSION.It appears, therefore, that this change in Kd with concentration may be connected with the interfacial tension which has been shown in a previous paper ( 7 ) to have a marked effect on the Reynolds number a t which circulation begins ivithin a droplet. Interfacial tension can be considered as one aspect of the surface forces which may give rise to more or less regular orientation of all molecules near the interface. Such orientation and resultant increased viscosity a t the interface may modify solute transfer from droplets in two ways, by directly obstructing the passage of solute moleculee, and by reducing the intensity of circulation within the drop through interference Kith movement of the interface. The possible existence of a Helmholtz electrical double layer a t the interface may introduce further modifications, particularly There polar solutes are concerned. Interfacial tension \\as taken as a measure of the interfacial properties, and from the droplet composition at the foot of each of the shorter columns the droplet interfacial tension mas obtained for points do1v.n the 71-em. column from Figure 12. 1260
45
INDUSTRIAL
where n
=
an integer number.
c, - ci may be regarded as the fractional extraction achieved 13s unsteady-state diffusion through a stagnant sphere of diameter 2r in timet. The solute concentration is assumed uniform throughout the sphere a t a value of C, when diffusion begins, but the conccntration at the interface remains constant a t a lower valur, Ci. 1 B was evaluated by extrapolation from a graph of In - versus
9
B
given by Farmer ( d ) , where d, = 2r, t = time of drop forma-
tion, and D = solute diffusivity in the disperse phase. The individual theoretical film coefficients cvere then used to give the over-all coefficient by use of the distribution coefficient, H ( H = disperse concentration/aqueous concentration a t equilibrium), which, within the concentration range used in these experiments on drop formation, could be taken as constant at 0.0625, whence
The data appear in Table I, which also gives the ratio of corresponding values of I
The expression is stated valid for Reynolds number (Re) dm range 2 to 800. Where - = rate of evaporation (gram-moles dt per second), and u = continuous phase Schmidt group = Dip __ P
For a sphere, 4 % lnlt.concn.
0.01
Y
"
. r
whence DROP DIAMETER, CM.
Figure 8. Effect of Drop Diameter on Over-all Transfer Coefficient with End Effects Included
Thus even a t such long formation times as 50 seconds, the ratio is about 8 or 9, showing that the agitation is considerable, whichever nozzle is used. At shorter times of formation the ratio increases sharply; for formation times of 2.7 seconds the narrow capillary gives a ratio of 11.8, as compared with 8.2 for the wide capillary, Calculations for Free Fall Period Are Related to Experimental Data
The following calculations are for the free fall period only and are compared with the experimental results for the 4% solution in which end effects are excluded. The disperse phase film coefficient was calculated using Equations 4 and 5, where t was the time of free fall, using the graph of D+t/r* versus B given by Geddes (8). These equat,ions assume stagnancy within the drops. The continuous phase film coefficient was obtained from the partly theoretical expression given by Frossling (6) for the rate of evaporation of liquid drops in a moving air stream
The validity of this equation for a liquid phase may be uncertain, since the numerical constants were empirically evaluated for a gas phase and the difference between Schmidt numbers for gases and for liquids is wide. Errors introduced, however, should not be serious in this system since, with the high distribution 1 ratio, the factor, l~d)constituted more than 95% of the right-hand side of Equation 6. The film coefficients were combined by Equation 6 to give &theor. and plotted as curve 1 in Figure 9 for comparison with the experimental values of curve 2. The experimental curve is considerably above the theoretical one, indicating that the interior of the droplets is not stagnant, but that transfer is assisted by substantial agitation within the drops. The ratio Kdexp./Kdtheor. increases from 2.68 a t a drop diameter of 0.278 em. to 4.56 for a drop 0.547 cm. in diameter. This is presumably due to increasing speed of internal circulation caused by greater skin frictional drag as the drop size is increased. Since circulation within a droplet is caused by the viscous drag on its surface, then the maximum circulation speed will correspond to a point on the circumference traveling from leading to rear pole of the drop during the time taken for the drop to fall a distance equal to its own vertical axis. If, however, internal re-
DROP DIAMETER, CM.
Figure 9. Comparison between Theoretical and Experimental Over-all Transfer Coefficients with End Effects Excluded 1. 2. 3.
A. 6,
Theoretical, rigid drops Experimental, no apparent oscillation of drops Theoretical, circulating drops (Higbie equation) Experimental, large oscillating drops C. Experimental, higher concentrations than 2
and A
June 1954
4 %' 20
" " " 40 " " ' 6I 0 ' LENGTH O F FREE FALL, CM.
"
" 80
Figure 10. Over-all Transfer Coefficients with End Effects Excluded for Varying Lengths of Free Fall A. B. C.
4% Solution a t free fall datum 13% Solution a t free f a l l datum 18% Solution a t free fall datum
INDUSTRIAL AND ENGINEERING CHEMISTRY
1261
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT sistance t o circulation is appreciable, then the circulation speed mal- be less than this maximum value. Consequently, times may be too short for establishment of the steady flow conditions within the boundary layers near the interface, as required by the LeivisKhitman theory-time must be taken for penetration of the films by the solute.
E
IA
!
1 1
Comparison of Circulation within Drop during Formation and Free Fall I s Significant
The conclusions reached are t o be regarded as qualitative only, but are considered to give furthrr indication that the circulation speed within these falling drops is xiell below the maximum possible.
Figure 1 1. Equilibrium Distribution of Acetic Acid between Nitrobenzene and Water a t 19' C.
An expression for transfer during penetration of a solvent film by a solute-that is, before steady flow conditions have been established within the film-has been given by Higbie ( O ) , and other workers (3,15, 1 6 ) have applied his expression to the particular case of transfer into films which are moving over the interface of spherical drops. The authors believe that this procedure is inadequate where spherical or near spherical surfaces are concerned, but, assuming its validity for the moment, transfer coefficients for the inner and outer films moving round the interface have been calculated using Higbie's expression, Equation 3. In the calculations, Le, corresponding to the maximum circulation speed vias used and equaled the time for a drop to fall its own diameter. (Spherical drops were assumed.) By applying Equation 6 the over-all coefficient, K d t i l e a r , was obtained and plotted in Figure 9 (curve 3 ) for comparison with Z
