ENGINEERING AND PROCESS DEVELOPMENT response and does not allow a n optimum solution for a particular kind of disturbance of either set point (input) or load. The experimental and analytical work of this paper shows that the frequency response method of analysis is entirely practical for fluid processes of the usual kind encountered in chemical processing. Nonlinear dynamic behavior of such processes is not a n important factor in analysis, so long a s the solution is used in a reasonable operating range.
= output/input transfer function = process transfer function = valve transfer function
i KP
KO m
MP Steady State Calculations Agree with Transient Test Data
-r
n
P
The calculation of fluid process capacitance and resistance by employing values determined by steady-state calculation or test seems to be justified by the general agreement obtained in these tests between steady-state test data and transient calculations and test. This conclusion is logical because a discrepancy would not be expected until fluid velocities approached the order of magnitude of momentmuminterchange in the fluid or until gas velocities approach an appreciable Mach number.
5
Wi WO
X
Acknowledgment
X
The authors wish to express their appreciation t o Case Institute of Technology and its Department of Mechanical Engineering and to the University of Minnesota for aponsoring this research while the first author was visiting professor at the former institution during the spring term, 1952. Thanks are also due George M. Lance, instructor in mechanical engineering a t Case Institute of Technology, for certain portions of the test work. Nomenclature = indicated value of pressure, lb./sq. ft. = controlled variable (pressure = P),Ib./sq. ft.
b c
= Laplace transform of c ( t ) = mass flow capacitance, ft. sec.2
Cf
C, C,
= standard volume flow capacitance, fte6/lb. = cu,
e E
= =
f q
=
=
Ib./sq. ft. error (deviation), Ib./sq. f t . Laplace transform of e ( t ) frequency, cycles/sec. 32.2 ft./sec.2
ft. per
u
Y Y*
P W
= controller transfer function = constant = 4 Z-l
= process gain, Ib./sq. inch per lb./s inch = controller gain, Ib./sq. inch per %.,/sq. inch = Ib./ sq. inch in tank per lb./sq. inch on valve = mass, lb. sec.Z/ft. = maximum ratio [ C / R [ in the frequency range from 0 to = polytropic exponent = vessel pressure ( P = c), lb./sq. ft. = standard volume flow rate, standard cu. ft./sec. = fluid resistance, Ib. sec./ft.6 = Ib./sq. ft. per cu. ft./sec. = gas constant = Laplace transform operator = time sec. = absofute temperature, O R. or O K. = process time constant, sec. = valve time constant, sec. = vessel volume, cu. ft. = mass inflow rate, lb. sec./ft. = mass outflow rate, Ib. sec./ft. = valve top pressure, lb./sq. ft. = Laplace transform of x ( t ) = controller output pressure = Laplace transform of y ( t ) = error transfer function = density, lb./cu. ft. = frequency, radians/sec.
literature Cited
(1) Ahrendt, W. R., and Taplin, J. F., “Automatic Feedback Control,” New York, McGraw-Hill Book Co., 1951. (2) Brown, G. S., and Campbell, D. P., “Principles of Servomechanisms,” New York, John Wiley & Sons, 1948.
(3) Chestnut, H.,and Mayer, R. W., “Servomechanisms and Regulating System Design,” New York, John Wiley & Sons, 1951. (4) Eckman, D. P.,and Moise, J. C., presented a t the National Conference, Instrument Society of America, Cleveland, Ohio, September 1952. (5) Jamea, H.M., Nichols, N. B., and Phillips, R. S., “Theory of Servomechanisms,” New York, McGraw-Hill Book Co., 1947. (6) Ziegler, J. G . , and Nichols, N. B., Trans.Am. Soc. Mech. Engrs., 64, 8, 769 (1942). RECEIVED for review July 10, 1952.
A C C E P ~ EJune D 8, 1953.
Solute Transfer from Single Drops in Liquid-Liquid Extraction WILLIAM LICHT, JR., AND WILLIAM F. PANSING1 University of Cincinnati, Cincinnati 2 I , Ohio
T
0 DEVELOP further a n understanding of the fundamental mechanism b y which solute is transferred during liquidliquid extraction i n spray towers, an investigation was undertaken into the process of extraction from single drops passing through a stationary column of solvent. The basic premise, as proposed by Licht and Conway (7), was that in the life of each drop there must be three distinct stages and the mechanism of solute transfer must be studied separately in each stage. The stages are: 1
Present address, Standard Oil C o . (Indiana), Whiting, Ind.
September 1953
I. Drop formation-at
the nozzle or spray tip 11. Drop movement-through the column of continuous stationary solvent phase 111. Drop coalescence-at the interface a t terminal end of the column The motivation for this study came from a n observation of Licht and Conway ( 7 ) regarding stage I, which seemed t o be at variance with other results. Previous investigators working with single drops (8, 7 , 9, 11) have determined t h e amount of extraction during drop formation by making a plot of the logarithm fraction unextracted, or an equivalent variable, versus
INDUSTRIAL AND ENGINEERING CHEMISTRY
1885
ENGINEERING AND PROCESS DEVELOPMENT column height. Since the plot usually gave a relatively straight line, this line wa.s extrapolated back to zeIo column height. The ordinate intercept was conqidereti to he the fraction of the bolute unextracted during drop forniat~ori. During drop formation, the amount of extraction olitained in this way varied from 10 to 50% solute extracted. 1,icht and Conway, however, observed t h a t the tbtal amount of extraction in stages I and I1 occurring in a cdumn 3 inches high was practically independent of drop formation time. I t is difficult t o see how the apparent amount of extraction during drop formation could be so large without being a function of diop formation time. This paper develops some possible theoretical mechanisms of solute transfer in the various stages under this type of experimental condition and describes methods for testing experimental data t o determine the applicability of these mechanisms. The various mechanisms devised were applied t o a study of the extraction of acetic acid from single drops in three different ternary systems. Diffusion into Surrounding Phase Controls Extraction during Drop Formation
The conditionsunder whichsolutetransfer is occurringduringthe formation of a single drop from a submerged nozzle in a stationary solvent phase (in an apparatus such as shown in Figures 1 and 2 ) are probably &s follow-s: While the drop is in the procrss of being formed, the interior of it must be in motion and of relatively uniform concentration. The surrounding continuous phase is undergoing a small amount of viscous displacement due t o the expansion of the drop. Hence, there is no film in the continuous phase surrounding the drop, or rather, a “film” of infinite extent surrounds the drop. To analyze this mechanism mathematically, a bulk of material of relatively uniform and constant concentration in rontact with a n extracting medium of infinite extent is pictured. The rate of extraction of soluk is controllcd by the rate of diffusion of qolute into this infinite medium. For convenience, the unstcady-state diffusion equation in rectangular coordinates is used.
where the area, A , is a fuiiction of time, a5 follows:
A
1/8
= (AT)
=
b2C’ D,G
To solve this problem exactly, the diffusion equation in spherical Coordinates should be used, but in this case the outer radius of the sphere is considered at infinity, a condition which makes it impossible t o obtain a definite solution to the equation. Following the notion used by Highie (S) in the study of gas ahsorption from bubbles, the problem has been simplified t o one of a plane of contact having an area equivalent t o that of asphere. The dispersed phase is assumed l o lie uniformly mixed and of constant composition, C d 0 . Thii latter assumption neglects the decrease in the concentration of the drop during drop formation and hence will be valid o n l ~if ~the total extraction during formation is small. At the interface hetween the dispersed and Continuous phase, equilibrium is a>sunied to be maintained. Taking the initial concentration oC the surrounding m d i u m as zero, the solution of Equation 1 iq: X
The instantaneous rate of solute transfer at the I)oundrZr\’ 0) is given by
(5 =
I886
*,,dWI dt
tx/a
(1)
dV
Recognizing the fact that --, the volumetric feed ratr, is rydt perimentally held constant, aswniing that the drop hcriiis to form from zero volume and is sphrrical throughout t h r formation, and integrating Equation 3 over the timr of drop, Corrnatio.1, f J , the following expression is ol~tairictl Fraction solute extracted during foriii rt ion = I
-
Eliminating the drop diameter, d, in terms of the constant volumetric feed rate, V I ,an alternate form is obtained: (cia)
Whether all assumptions undeilj mg Equation 5 are mct tiepends partly on the deaign of the riozile tip. In this experimental work, the tips were so desigrictl that each drop btai tc t l to form from as nearly zero volume as po--it)le. Mechanisms of Solute Transfer Are Proposed for Drop Movement through Column
Many mechanisms or combiuations of mechanism3 of solutr transfer may be postulated for stage 11. Some of the mole dcfinite mechanisms, permittingmathematical analysis, are considci cd helow. 1. Two-Film Theory. The rate of solute transftv thiough two films in series may be evpressed by AT = K d A ( C d b - c&) (6) .issumine that the concentration of the continuous phasc. is zeio and Kd and A are constant, Equation 6 can be integrateti 01 c r the duration of the falling priiod t o give
log (1 - E ’ )
dbtc‘
3
=
-2.61
t
2 . Diffusion within Drop. It may be assumed that p u ~ ~ molecular diffusion of solute occurs within the drop and that tlic surrounding contiuuous phase offers negligible resistancc to solute transfer. Assuming that the initial conccntration of til(. drop is uniform and that the concentration of the contiriuoiii phase is negligible, the following solution t o the diffusional equ:ition in spherical coordiriatcs is oijtained ( 1 ) :
(8)
3. D s u s i o n within Drop plus Film Resistance. I\ t h i i r l mechanism permitting m a t h ~ m a t i r a lanalysis is that of pul r diffusion within the drop plus an atlclitional film resistance ai ouiltl the drop. Assuming that the initial roncentration of the t l r o ~i)y uniform, that the concrntra~ionof the bulk of the continuour phase is zero, and that the transfer coefficient at the surface i > constant with time, the following solution to the diffusional eqnation in spherical coordinates is o b t a i n 4 ( 1 ) :
n - 1
where h
=
kL
HDd’
and the condition
INDUSTRIAL AND ENGINEERING CHEMISTRY
oil
aliis that
it must be a root of
Vol. 45, No. 9
ENGINEERING AND PROCESS DEVELOPMENT The first four roots of Equation 10 have been presented graphically ( 1 ) . 4. Streamline Diffusion within Drop. Kronig and Brink (6) recently proposed and analyzed mathematically a mechanism in which the interior of the drop is assumed t o contain streamline currents resulting from the drag of the continuous phase against the drop. Solute will be transferred from one streamline t o another by molecular diffusion. The solution was found to be m
1 -E
=
BZe "
8
Table
7
(11)
Slope Log (1 E) us. t
-
Remarks Slope proportional t o I/d if Kd constant with drop size Slope proportional t o I/dl a
Kd
**
-2.61 d
2.
Pure diffusion wlthin drop (no external resistance)
3.
Pure diffusion within drop plus film resistance Streamline convection within drop (no external re sistance)
i
n = l
where B , is a coefficient and u,, is a n eigenvalue. The more important assumptions made in the solution are t h a t the drop is spherical, the resistance in the continuous phase is negligible, and the drop radius is greater than 0.10 cm. The first two eigenvalues and cofficients have been evaluated as:
Mechanisms of Solute Transfer in Stage II
Mechanism
l6Ddt n
.
5.
Dd a
-l7.I5
&Dd
-g
-46.8
a
j
Dd
dZ
-
Transient films
Slope proportional t o
a
l/d2
a
fc
(p+ .)
-2Q5iF
-
Case A
Dd
Slope proportional to
-2.96
d v p
v
UI ~1
BI = 1.32 Bz = 0.73
= 1.678 = 9.83
Case B
5. Transient Films. Another type of solute transfer mechanism was originally proposed by Higbie (3) in connection with gas absorption. H e attempted to show t h a t even though a liquid film may exist around a bubble of gas, the actual time of contact (penetration period) of the gas with the liquid at any point is so short that the film acts eauivalent t o an infinite medium with respect to solute transfer. The problem again involved a sohtion of Equation 1. West et al. (11) have indicated how this theory might also be applied t o liquid-liquid extraction. They have assumed that, in addition t o the transient film in the continuous phase proposed by Higbie, a transient film is constantly being formed in certain areas on the surface of the drop and t h a t this film moves over the surface and is depleted or enriched with solute and then disappears into the bulk of the drop. So long as the concentration at the interface remains constant over the life of the film, Higbie's results should apply t o both phases combined as film resistances in series. The values of the transfer coefficients are given by West (IO) as:
k, = 2
r and k,
-
=
2
ds
wherefc 6 1. It is assumed that,f, is the same for both films. This mechanism is' basically the same as the two-film theory, but it is a more detailed picture of how the two films functipn. West (IO), therefore, applied these concepts t o Equation 7, where 1 Kd is taken as l,kd + H,k, and obtained: -
fat
= -2.95
(14)
There are two special cases of this concept: A. The film on the dispersed-phase side offers negligible D k resistance-i.e., k d = m or ---' W
The explanation of this increased rate of extraction can be found in the difference in behavior of the drop, as noted above. The oscillations of large amplitude have set up stronger eddy currents within the drop as well as in the surrounding medium. The velocity (19.53 cm. per second) is not out of line with thoee found for the smaller range of drop sizes, hence the change in the rate of extraction cannot be credited to this. Further study of this reversal in the rate of extraction should be fruitful. Commercially, this may be important since increased drop size might produce increased throughput without sacrificing extraction efficiency. The reason for the breaks in the straight-line portions of t h e plots for the two smallest drop sizes is not certain. The eddy currents within the drop would be expected t o decrease with time
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 9
ENGINEERING AND PROCESS DEVELOPMENT ~~
~~~~
~~
and the effective resistance t o mass transfer would be expected to increase. The true plot may actually be curved and it is merely a fortuitous result t h a t the breaks appear t o occur a t the intersection of two etraight lines. Pure Diffusion M a y Control Extraction Rate for Carbon Tetrachloride-Oil System
d
e
I n the first two systems investigated, the pure diffusion mechanisms (2, 3, and 4) were not encountered. To determine what effect increased viscosity of the drop might have on the presence of a diffusion mechanism, the carbon tetrachloride-oil system was investigated. The results are plotted in Figure 6 and were obtained in column 11. The viscosity of the carbon tetrachloride-oil-acetic acid mixture was 3.93 centipoises. Mechanisms 1, 5, and 6 were immediately eliminated be’cause the plots are curved. An attempt was made t o fit the pure diffusion mechanism 2 t o the data. Regardless of the value used for the true diffusivity, D,, the curves could not be reproduced with the Diffusion Equation 8. It must be concluded that the curvature of the plots is due t o a continuously changing mechanism. A possible explanation is that the initial rate of extraction is due to the fact t h a t the convection or eddv currents remain in the droD after formation. Because of the viscosity of the drop, dampening of these currents takes place as the drop falls. Eventually the interior of the drop becomes stagnant and pure diffusion within the drop controls the rate of extraction. If this mechanism is the correct one, then the slope of the curves should gradually approach the slope for pure diffusion. The slope of each of the curves was measured at a series of time intervals. A fictitious diffusivity was calculated for each slope and designated as effective diffusivity. These effective diffusivities were then plotted against the corresponding fall times in Figure 13. The effective diffusivity very definitely approaches the value of t h e true molecular diffusivity, 3.30 X 10-8 square cm. per second, estimated by the method of Wilke (18). The results may also be interpreted by the transient film mechanism if i t is assumed t h a t the factor, fc, varies and the diffusivity remains constant. The value of H is not available for this system but again it is presumed that it is small and t h a t Equation 14b applies. This interpretation requires t h a t fc decrease as the drop falls-that is, that the film requires a relatively longer time t o traverse the surface of the drop. This would seem t o be consistent with the dampening of oscillations and eddy currents and the approach t o stagnant conditions in the interior of the droa. On the basis of the data a t hand. there seems to be no w a i to choose between these two interpret&tibns. The drops in this system fell very calmly with no visible distortions, a condition no doubt necessary for pure diffusion eventually t o control the mechanism. I n addition, a s seen in Figure 9, the actual values of the drop velocities are close to the values predicted for rigid spheres. This is also a n indication t h a t oscillations in t h e drops were very small and, therefore, t h a t pure diffusion is likely t o control the rate of extraction. Extraction during Drop Coalescence I s Proportional to Drop Concentration and Diameter
D a t a pertaining t o stage I11 were found only for the methyl isobutyl ketone system. The vertical distance between the two curves for columns I and 11, for approximately the same drop size, represents the amount of extraction in stage 111. Since these two curves are parallel, the amount of extraction in stage I11 must be proportional to the concentration of the drop before entering this stage. I n addition, this proportionality constant for various drop diameters was found t o be proportional t o drop diameter. Therefore, the amount of extraction in stage I11 is proportional t o the concentration of the drop and the drop diameter. September 1953
Conclusions
The difference in behavior between the methyl isobutyl ketone system and the other two systems can be attributed to a difference i n drop behavior. I n the ketone system, the drops were very distorted or flattened out and vibrating. This evidently caused a laminar portion of the continuous phase t o be dragged along with the drop. This was partly verified by the fact t h a t the larger drops fell slower than the smaller drops in this system. Because of these vibrations, the interior of the drop tended t o be mixed. A major portion of resistance to solute transfer was found to be in the continuous phase. However, for the perchloroethylene and carbon tetrachlorideoil systems, the drops were not distorted and did not vibrate. These drops tended t o glide through the continuous phase. Consequently, the continuous phase offered negligible resistance to solute transfer while the disaersed Dhase offered aracticallv all the resistance. In the perchloroethylene system, the investigation of one large size drop which did oscillate and was distorted showed an abnormal increase in the rate of extraction. In this case, the resistance in the drop was probably diminished more than the resistance in the continuous phase was increased. The most important factors in the mass transfer mechanism are, therefore, those properties relating t o the existence and magnitude of oscillations in the falling drop. Acknowledgment
The authors wish to thank Frank B. West for his careful review of the manuscript and many constructive suggestions. Nomenclature
a A
B C d
D E fc,
radius of drop interfacial area series coefficient concentration drop diameter = molecular diffusivity = fraction solute extracted = square root of factor relating life of transient film t o time required for drop t o travel a distance equal to its diam= = = = =
der = Kd/Dd = distribution coefficient (C,/C,) = individual film coefficient = over-all film coefficient lll_
h
H k
K
= degree of association; index t o infinite series N = rate of solute transfer R = resistance Re = Reynoldsnumber S = slope of log per cent solute unextracted versus fall time t = time u = variable of integration; eigenvalue v = drop velocity V = drop volume V I = volumetric feed rate z = linear distance from interface a = constant in series solution
n
Subscripts b
= bulk of phase (uniform)
c
=
d e
f L 0 1 2 3
= = = = = = = =
continuous phase dispersed phase equilibrium concentration formation effective contact time in life of transient film initial concentration st,age I stage I1 stage I11
Literature Cited (1) Carslaw, H. S., and Jaeger, J. C., “Conduction of Heat in Sol-
ids,’’ England, Oxford University Press, 1947. (2) Farmer, W. S., U. S. Atomic Energy Comm., Tech. Inform. Div., Oak Ridge, Tenn., Rept. ORNL-635 (1949).
INDUSTRIAL AND ENGINEERING CHEMISTRY
1895
E N G I N E E R I N G AND PROCESS DEVELOPMENT (3) Higbie, R., Trans. Am. Inst. Engrs., 31,365 (1935). International Critical Tables, Vol. 5, Kew York, 31cGraw-Hil1, 1929.
(4)
( 5 ) Kats, H. &I., 94.S~.thesis in Chemical Engineering, Univ. of
Cincinnati, 1950. (6) Kronig, R., and Brink, J. C., AppZ. Sci. Research, A2, 1942 (1950). (7) Licht, w., and Conway, J. B., IKD. ENG.CHExr., 42, 1151 (1950). (8) Perry, J. H., ed., “Chemical Engineers Handbook,” Sew York, McGraw-Hill, 1950.
(9) Shermood, T. K., Evans, J. E., and Longcor, J. 5’. A., Trans Am. Inst. Chem. Engrs., 35,597 (1939). (10) West, F. B., personal communication, Jan. 1, 1952. (11) West, F.B., Robinson, P. A., iUorganthaler, A. C., Beck, T. R , and MeGregor, D. K.. IND. ESG.CHmf., 4 3 , 234 (1981). (12) Wilke, C. R., Chem. Eng. Progr., 45, 218 (1949).
RECEIVED for review Xovember 19, 1951. ACCEPTED June 6, 1 9 3 . An abstract of the dissertation submitted by William F. Pansing t o the Graduate School of Arts and Sciences of the Unirersity of Cincinnati in partial fulfillment of the requirements for the degree of Ph.D., June 1931.
Cleaning Emulsion Polymerization Equipment Fouled by Synthetic Rubber latex J, S. NETTLETON, M. J. G. DAVIDSON,
AND
H. LEVERNE WILLIAMS
Process Engineering Deparfmenf and Resewch and Developmenf Division. Polymer Corp., ltd., Sarnia, Ont., Canada
R
EDUCED heat transfer coefficients can limit cold rubber pro-
surface. Another method tried Lvith limited success v a s to fill the vessel with hot water and 200 gallons of styrene, agitate for 16 duction rates because there is a lower limit for jacket temhours, drain, wash down the wall with a water jet from the manperature below which the latex freezes. It is therefore most imway, and finally scrape off the softened film before it dried. A pdrtant to keepinternal reactor surfaces clean and free of deposits. refinement of the manual scraping method involved working talc Deposits of rubber on the walls of reactors and equipment used dust into the coagulum which made the job easier when a soft, in the production and testing of synthetic rubbers are undesirable sticky coating was encountered. Refluxing benzene in the reacin other ways. The deposit not only represents loss of rubber but tor was tried without success, These methods were so laborious also becomes detached and contaminates rubber being produced. and inefficient that the more efficient cleaning method described Test equipment and sample lines are readily fouled by such in this paper lyas developed. material. Asymetrical ketones have been used in the laboratory as solPetroleum Naphtha-Cumene Hydroperoxide vents for cleaning laboratory polymerization and test equipment Solvent Dissolves Butadiene-Styrene Film ( 1 ) . The mechanism of the action was suggested b j Kinkler t o Two types of apparatus were used to determine the efficienry be degradation of the polymer to low molecular weight, soluble of solvents. One A as a percolator, illustrated in Figure 1, which fragments by the oxidizing action of peroxy compounds forme i could be immersed in a thermostatically controlled constant by exposure of these ketones to the atmosphere. Such peroy‘ temperature bath when temperacompounds form readily, and it ture higher than room temperawas further shown ( 2 ) t h a t the I / L. tures were desired. h slow stream dissolution of the polymer depends of air caused small portions of upon the presence of oxygen. Reflux, water-cooled solvent t o rise up the tube and The standard reactors in the condenser splash doim over a sample of polyPolymer Corp. plant are glass-lined, mer film removeJ froin the reactors. jacketed, pressure vessels of 3750 The sample of polymer film x a s U. S. gallons’ capacity, manuweighed before placing in the perfactured by the Pfaudler Co. Durcolator (usually 2 grams). h piece ing 8 years’ production of polymers of fine stainless steel or copper over a wide range of temperatures Polymer screen held the polymer suspended and pH levels, the glass linings above the liquid level. After the have become roughened or etched. Screen apparatus was started it ivasallored I n this condition rapid fouling t o operate for 24 or 48 hours. At occurs and frequent cleaning is Solvent the ~ i i of d this time the total volume necessary to maintain productivity. of solvent in the apparatus was Manual cleaning with plastic or measured and any solid material hardwood scrapers has been used Air 4 contained in it was allowed to settle. t o clean t h e r e a c t o r s . T h i s A 10-ml. aliquot of the clean liquid method took 8 t o 12 hours and as drawn off with B calibrated invariably left a thin, tightly adpipet, evaporated to dryness on a hering layer of rubber on most steam hot plate, and the concenof the reactor i d 1 which, besides reFigure 1. Air Lift Pump Circulator or Percolator tration of dissolved solids calcuducing heat trankfer, resultedin more for Dissolving Polymer Film lated. rapid fouling than the clean glass
I
1896
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 45, No. 9
