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175
The MIP scheduler was shown to be uniformly superior. The simulation model can readily be used to carry out debottlenecking studies, to evaluate plant operating policies, and to perform parameter sensitivity studies. This application study also pointed out some of the difficulties associated with evaluating results of combined simulation case studies and presented some tentative approaches. It is hoped that this demonstration of the utility of the GASP IV simulation language combined with MIP scheduling models will encourage wider use of these versatile tools in analyzing processes employing batch-semicontinuous operations. Acknowledgment The authors wish to express their thanks to Professors A. A. B. Pritsker and B. A, McCarl of Purdue University for advice and consultation in this work. The research was supported in part under NSF Grant GK42142. Literature Cited
Demand
F i g u r e 4.
L e v e l , Froclion o i
Normal
Effect of decreasing line operating costs.
the ancillary inventory and distribution functions has been presented. A heuristic myopic and a mixed integer linear programming based longer-term scheduling mechanism were formulated and tested within the framework of the simulator.
Duncavage, T. D.. M.S. Thesis, Purdue University, 1974. Fishman, G. S., "Concepts and Methods in Discrete Event Digital Simulation." Wiley, New York, N.Y., 1973. McCarl, B. A., Barton, D., Schrage, L., Station Bulletin, No 24, Department of Agricultural Economics, Purdue University, West Lafayette, Ind., 1973. McCarl, B. A., Santini, J., Department of Agricultural Economics, Purdue University, West Lafavette, Ind., 1976. Overturf, B. W.. Reklaitis, G. V., Woods, J. M., Id,Eng. Chem. Process Des. Dev.. 17, 161 (1978). Pritsker, A. A. B., "The GASP IV Simulation Language," Wiley, New York, N.Y.. 1974. Sargent, R. G., "Statistical Analysis of Simulation Output Data," presented at Symposium on the Simulation of Computer Systems IV, Boulder, Colo., Aug 1976.
Received for review March 8, 1977 Accepted December 19, 1977
Mass Transfer between Two Liquid Phases in a Spray Column at the Unsteady State L. Steiner, M. Horvath, and S. Hartland" Swiss Federal Institute of Technology, Department of lndustrial and Chemical Engineering, Zurich, Switzerland
A model based on the assumption of noncoalescing drops moving uniformly through a cascade of well-mixed hypothetical stages with backflow of the continuous phase is presented to describe mass transfer at the unsteady state in a spray column. Good agreement was obtained when the computed variation of concentration with time at several different points was compared with that obtained experimentally from a pilot plant column. The model may also be used to predict the variation of concentration along the column when the steady state is reached.
Introduction While the spray column cannot compete with the sophisticated and efficient extractors now used in industry, it has the advantages of simplicity, low operational costs, and insensitivity to impurities. Moreover, it provides a suitable standard for checking hypothetical models and theoretical principles, which then may be extended to other extractor types by considering additional influences. Most of the difficulties in scaling up extractors are caused by nonideality of internal flows, rather than by the complexity of the mass transfer phenomena themselves. It is assumed that if the internal flows are properly expressed, the description of the mass transfer rates may be reduced to several standard sit0019-7882/78/1117-0l75$01.00/0
uations such as occur in the formation and movement of drops, films, and jets. It is expected that the corresponding mass transfer coefficients will be applicable to different column sizes and types so that it should be possible to considerably reduce the number of pilot plant experiments required. In the ideal case the mass transfer coefficients would be calculated or measured in small-scale laboratory equipment, leaving the pilot plant experiments for the determination of the hydraulic parameters. Such experiments are much simpler and exclude the costs for solvent recovery and complicated analysis. The behavior of the actual extractor could then be predicted, using the measured values in a suitable model. It has been shown earlier (Steiner and Hartland, 1974) that
0 1978 American Chemical Society
176 Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
0
2
1
4
3 time
5
6
7
(mml
Figure 2. Response of column to rectangular impulse of tracer injected close to inlet of continuous phase. (Shaded area shows shape of impulse in plug flow.) Medium backmixing, f = 0.3.
-
A
-
B
Figure 1. Drops in upper part of column: A, non-coalescing,mass transfer C D, B, coalescing, mass transfer D C; continuous phase, water (C); disperse phase, o-xylene (D); solute, acetone.
axial mixing is one of the most important parameters and that plug flow, assumed in older computing procedures, is a rare exception in practice, so that the axial mixing should be allowed for in all calculations. On the other hand, radial mixing (or rather segregation) is only important for large columns and even then it may he considerably reduced by constructional means, certainly easier than the axial mixing. It is therefore assumed in this paper that the concentration is constant in all horizontal cross sections of the column. In earlier papers (Steiner and Hartland, 1974,1975), the backmixing models were discussed generally and it was shown that the hackflow model (Schleicher, 1960) is the best one in practice as it is versatile and suitable for numerical calculations. The necessity of obtaining generallyapplicable solutions eliminated the possibility of using analytical procedures.
should be introduced in such a way that no large-scale vortices are produced and no large-scale mixing exists. By injecting tracer substances into each of the phases and evaluating the responses a t the other end of the column, it was established that the backmixing in the continuous phase is considerable, the response being of the shape shown in Figure 2, where the shaded area indicates how the impulse would appear if ideal plug flow existed in the column. On the other hand, the dispersed phase flows much more regularly, the residence time distribution being caused by the different axial velocities of drops having slightly different diameter, rather than by their backmixing. The corresponding situation is shown in Figure 3. (These experiments will he described in detail later.) All these observations should he incorporated into the model which, however, should remain applicable in practice; Le., the complexity and the necessary computer time should be kept as low as possible. Keeping this in mind, a modification of the common hackflow model is derived with the continuous phase represented as a cascade of well-mixed hypothetical stages with backflow streams. The drops in the dis-
Theory By observing and photographing the behavior of spray columns up to 10 cm diameter it was established that this type of column is more suitable for systems in which there is little coalescence between the drops. This is because there is no mechanism for the redistribution of the dispersed phase along the column length, so that the drops soon become very large and the interfacial area decreases sharply. In this case other extractor types should he used for the task intended. Examples of coalescing and noncoalescing swarms of drops are shown in Figure 1.Therefore, in a spray column operating under normal conditions, the drops keep their identity until they coalesce with the interface a t the other end of the column. In this way there is no mixing of the contents of individual drops with surrounding drops and the solute accumulates in each of them independently. I t was observed that the drops generally moved in the main direction of the flow with occasional stops and even short returns hackstream, although these hack movements were short and a drop never travelled back along the entire column height. If the spray column is properly designed the phases
0
20
60
40
ttme
60
100
15
Figure 3. Response of column to impulse injected into dispersed phase.
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
177
persed phase maintain their identity and move in plug flow through each stage in the column exchanging solute with the continuous phase at the rate determined by the instantaneous concentration difference. Experiments with drops containing a photochromic dye activated by a laser beam have shown that the drops are mixed internally so that constant concentration may be assumed throughout the drop volume a t any time. The actual shape of a moving noncoalescing drop is an ellipsoid; however, the difference in the lengths of the respective axis is usually small so that approximation by a sphere with the mean Sauter diameter d32 is possible. This diameter is defined as being six times the drop volume divided by its surface area. Using this approximation, the mass transfer rate to the drop is r = - dY = - 6Kd (Y* - Y ) (1) dt Pyd32 in which y is the concentration in the drop expressed as a mass ratio. The drop passes through a series of hypothetical stages with a perfectly mixed continuous phase. In the unsteady state the concentration in each of these stages is a function of time. The number of hypothetical stages is arbitrary, but it should be chosen in such a way that the concentration change in the continuous phase is small. (In our work good results were obtained when the height of the hypothetical stage was between 50 and 100 mm). Both the equilibrium relation and the change of the concentration in the continuous phase may then be approximated by linear functions so that
Carrying out the material balance for the continuous phase around a typical stage with backflows, a result similar to that for the standard backflow model is obtained
y* = mx
where t f = d323 pyj/(6L) is the formation time and j the number of jets in the distributor. The concentration in the dispersed phase a t the exit from the first stage is therefore
(2)
and x = xo
+ at
(3)
both m and a being considered constant during the residence time of the drop in each stage, although otherwise they are functions of concentration and time, respectively. Subscript 0 refers to the time when the drop enters the section. Using these assumptions the mean mass transfer rate during the residence time of the drop in the stage may be calculated as follows. The rate r is differentiated, recognizing that it is a function of the concentrations in the dispersed and continuous phases, both being in turn functions of time. -dr=
dt
(2)dx dx
ydt’
(;;),dt---dy
-
6Kdma Pyd32
6Kdr Pyd32
- L h rn (9) l - f P x
where u = H/{p,A (1- t ) } is the actual velocity of the continuous phase in the axial direction. To obtain the equations for the first stage where the drops are formed, the stage is divided into formation and flow zones, the solute concentration in newly formed drops being denoted by yif. Using this concentration, the transfer rate in the flow zone is similar to the typical stage as given by eq 7, using yif instead of yn-l. The mass transfer during drop formation is expressed by 6Kdf rf = -(Y*I- yi) (10) Pyd32 where Kdf is the mass transfer coefficient related to the final surface area of the drops formed and the initial driving force. As the time of formation is even shorter than the residence time of a drop in a typical section, y*1 may be considered constant during the formation and the concentration in a freshly formed drop is given by Yif
= Yi
y1 = yi
+ rftf
+ rftf + T I -U6
(11)
(12)
and the equivalent of the term 7, in eq 9 is rlf
= (rftf
+ pi)
(13)
Recognizing that the backflow in the continuous phase neither enters nor leaves the column and neglecting the mass transfer during the coalescence of the drops in the upper part of the column, the balance equations for the first and last stages, respectively, are
(4)
or
--dr --6Kd dt a m - r Pyd32
(5)
Denoting by ro the rate at the time to, this equation integrates to
r = am
6Kd - (am - ro) exp (- ( t - to)) Pyd32
(6)
The mean mass transfer rate during the residence time of the drop in stage n with a height 6 is then
where u = L/(Atp,) is the actual velocity of the disperse phase and rn0 = (6Kd/pyd32) (y*, - yn-l) is the mass transfer rate at the time when the drop enters the stage. The concentration in the drop changes during its residence time in the stage to
To obtain the concentration profile, eq 9, 14, and 15 may be solved numerically using a computer library subroutine based on, say, the Runge-Kutta-Merson procedure. If only the steady-state profile is needed, the right-hand sides of these equations are set equal to zero and the solution obtained by using procedures for the solution of sets of nonlinear algebraic equations. This is usually much less laborious and needs only about 20% of the computer time necessary for full integration from zero to the steady-state concentration. The numerical integration is done in short time intervals selected by the computer according to the accuracy demanded. To obtain the coefficients a in eq 6 and 7, the value of the x-derivatives from the previous time interval are stored for use as a’s in the following one. Given the shortness of these intervals the error is negligible and disappears completely as the calculation reaches the steady state. In this original form the equations are written to yield the concentration profiles in both phases from given values of the mass transfer coefficient and all necessary hydrodynamic parameters. It would be possible to
178
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
I
‘1
n
d
d
6
i/ i
0
11
$s 9
1L
9 I
I
1
t t
Figure 4. Flow sheet of experimental arrangement: 1,column; 2, distributor for light phase; 3, inlet of continuous phase; 4,water tank; 5 , oil tanks; 6, rotameters; 7, tracer tank; 8, solute tank; 9, centrifugal pumps; 10, metering pump; 11, heat exchangers; 1 2 , sampling funnels; 13, test tube magazines; 14, conductivity cells; 15, phase separators; 16, tracer distributor for backmixing measurements.
modify them to obtain the mass transfer coefficient from the experimental profile, but since small changes in the profile cause large changes in the coefficient, such evaluation is inaccurate. It is much better to calculate the coefficients indirectly, using an optimization procedure which varies the mass transfer coefficients and calculates the corresponding profiles till the sum of the squares of the deviations from the experimental points is minimized. Experimental Section
To verify the model, unsteady-state concentration profiles at six positions along a spray column were measured together with all necessary hydrodynamic parameters, such as the backmixing coefficient, holdup of the dispersed phase, and the drop diameter. A pilot plant sized column was built for this purpose out of QVF glass parts combined with specially designed pieces made of stainless steel and Teflon. Only these three materials were in contact with the liquids. The diameter of the column and the length of the working section were 100 mm and 2 m, respectively. The unsteady-state work requires measurements of high precision; among others, the following precautions were observed: the column was placed in a thermostated room, fine temperature control being achieved by heat exchangers in both phases so that the operating temperature was maintained to within 0.1 K. The flow rates of the two phases were maintained constant by controlling the position of the rotameter floats with photoelectric cells. The solute was injected by a dosing pump after the corresponding rotameter so that its concentration did not affect the solvent flow. The system used was water, o-xylene, and acetone, transferring acetone from the continuous water phase into the dispersed xylene phase. This system is suitable for test runs as the phases are practically immiscible a t low acetone concentrations and the analysis is therefore possible from density or refractivity measurement correlated against a gas chromatograph. The density changes do not affect the flows of the pure solvents so calculations are easily made using relative mass fractions. Four sampling ports were drilled through the
glass wall of the column, as shown in Figure 4. Both phases were continuously removed from the column using specially constructed sampling devices as described later. Circular magazines with 128 5-mL test tubes were placed under the outlet from the sampling probes and rotated at preset time intervals (usually 30 s for a test tube). The time delay in the sampling probe was very short, (less than 3 s) since a surplus of the coalesced dispersed phase escaped again in newly formed drops from the top of the funnel. The collected samples were analyzed manually at the end of the experiments. In this way the concentration profiles at the ends of the column and at four intermediate points were obtained for both phases. The sampling was started with pure solvents flowing and continued until the steady-state concentrations were reached. The sampling funnels shown in Figure 5 were made of stainless steel so that they were wet by the continuous phase and there was no sticking of drops on the connecting tube and the outer side of the funnel. The funnel was lined with Teflon and there was a thin Teflon rim on its lower side where the drops coalesced at such a rate that the funnel was always completely filled with the disperse phase which was continuously withdrawn for analysis. The surplus escaped at the top of the samplers forming new drops. Since the drops did not stick to the connecting tube, the continuous phase was also withdrawn without contamination. The drop diameter was measured photographically, taking pictures at regular intervals at two positions along the column. The negatives were evaluated on an X-Y reader and the punched tape produced there was fed to a computer to calculate the mean Sauter diameter and its distribution. The drops were usually quite uniform in size, the standard deviation being of the order of 0.1 mm, due to the construction of the distributor with ground stainless steel tubes. The mean drop diameter was between 2 and 5 mm. The holdup of the dispersed phase was measured by a hydrostatic method connecting the top and the bottom of the column with a membrane pressure difference transmitter, the
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
m D
C
Figure 5. Detail of sampling device: C, continuous phase; D, dispersed phase; CW, column wall.
feed lines being filled with the continuous phase. The method was checked against direct measurement, when a part of the column was suddenly sealed off by a pair of specially constructed flat gate valves so that the enclosed volume of the dispersed phase was directly measurable. I t was found that the hydrostatic method gave a systematic deviation, the results being 8%too low, which was allowed for in the calculations. To measure the backmixing in the continuous phase, a steady-state tracer injection was used. A solution of potassium chloride was injected close to the exit of the continuous phase and the liquid collected from the sampling ports was analyzed conductometrically to find the backstream concentrations a t different distances from the point of injection. This method gives more reliable results than upstream injections of short impulse, as mean values are obtained over a longer time interval ( 5 to 15 min). A typical experiment began by running pure solvents through the column at pre-set rates. After a 1-h initial period, provided to wet the needles in the distributor and the sampling devices with the dispersed phase, pictures were taken to measure the drop size and a backmixing experiment was performed; then the acetone flow was started and the sampling begun. After some time the holdup stabilized again, indicating that the new steady state had been reached. The sampling was then stopped and new measurements of drop size and the backmixing were carried out. On completion of these measurements the solvents were re-distilled and the column prepared for a new experiment.
Results As may be seen from the equations given in the theoretical section the following parameters are necessary to simulate the concentration profiles and their time dependence: drop diameter, holdup of the dispersed phase, flow rates of both solvents, feed concentration of the solute in both phases, backmixing coefficient in the continuous phase, and the mass transfer coefficients for forming and freely ascending drops. All these parameters were available from the experimental data, the mass transfer coefficients being evaluated from the concentration profiles. The coefficient for drop formation was obtained from a special set of measurements, carried out with a sampling funnel placed 15 mm above the nozzles of the distributor. All relevant formulas, listed by Skelland and Minhas (1971),were considered in an attempt to correlate the data but the deviations were large. Finally, the following formula in MKS units was used for interpolation, the standard deviation being about 15% (16)
where u j is the velocity in the distributor nozzle. It should be recognized that this equation is based on 16 observations with one liquid system, so it is only applicable for interpolation of parameters for this system. More mass transfer experiments
179
of this type are in progress and a more generally applicable correlation will be presented later. The mass transfer coefficient for the ascending drops was calculated from the measured steady-state concentration profiles by a minimization procedure in which the same profiles were simulated by varying the mass transfer coefficient till the sum of the squares of the deviations between the experimental and simulated values were minimized. The values of the calculated coefficients used in the simulation are given in Table 11. Five typical unsteady-state runs with different parameter settings are shown here to demonstrate the agreement between the the6ry and experiment. Table I gives the measured values of the hydrodynamic parameters, indicating the strong dependence of the drop diameter and the holdup on the presence of the solute. The concentration profiles at sampling points 1and 4 (i.e., 300 mm from the column ends) are given in Figures 6 to 10. These positions were selected for presentation as they represent the change of the concentration inside the column, free of end effects. However, in the calculation all six concentrations were obtained. The profiles were generated using the values in Table I, assuming the change in the drop diameter to be proportional to the concentration of the solute a t the bottom of the column, where the drops were formed. This approximation was sufficiently accurate for the given purpose. The lines in Figures 6 to 10 show the unsteady-state profiles calculated by the computer from the information given in Tables I and 2 using eq 7 to 12. A typical example of a steady-state profile which minimized the sum of squares of deviations from the experimental points is given in Figure 11.
Discussion T o follow the behavior of a column of the size used in the unsteady state is a difficult task which tests the experimental techniques severely. It was found that a small change in the profile causes a large change in the mass transfer coefficient, since the driving force also changes and that the uniformity of the initial distribution of the solute in the continuous phase is of enormous importance. Perfect agreement between the observed and calculated profiles cannot therefore be expected and the agreement found here must be considered as good. The model may be recommended for general use with spray columns as it takes into account the changes in physical properties of the phases along the column and yet minimizes the computational difficulties. The alternatives are the classical backflow model and several statistical models which, however, are not yet sufficiently developed. In comparison with the backflow model, the one presented here cuts the number of necessary equations in the numerically solved set by half and eliminates the backflow coefficient in the dispersed phase, which represents a purely formal parameter for our case of noncoalescing drops in a spray column when the actual backflow of the drops is insignificant. The profiles computed in this work and shown in Figures 6 to 10 may be obtained in practically identical form from the backflow model, the mass transfer coefficient for the moving drops being similar to the ones shown here (within about 15%). However, the backflow coefficients for the dispersed phase which must be evaluated from the concentration profiles, together with the mass transfer coefficient, may reach unrealistic values. For a model with 15 hypothetical stages used in this work, the number of drops moving backwards would sometimes have to be equal to half the number of drops moving forward, which was never confirmed by observation. Furthermore, the backmixing coefficient in the dispersed phase cannot be obtained by steady-state injection of marked
180
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
Table I. Experimental Results
Expt no.
H
L
171 19 222 231 25
12.63 17.15 16.92 12.55 5.24
3.35 1.87 3.57 1.87 4.97
xi
Without mass transfer d32 x io3 t x 102
x 102 3.3
3.91 4.90 3.70 4.85 2.74
11.1
5.8 6.5 8.2
With mass transfer d32 X lo3 t x 102
5.20 1.97 5.70 1.49 11.70
3.39 4.34 2.89 4.40 2.65
7.58 4.31 11.80 2.44 18.04
Table 11. Calculated Parameters
Expt no.
Kd X
171 19
lo3
5.15 15.00 6.84
222
231 25
12.82
5.59
Kdf
0.357 0.169 0.314 0.171 0.479
Without mass transfer E , x 104 a 22.00 27.10 32.20 26.40 10.70
Table 111. Algorithm for Unsteady-Stage Profile Generation Known parameters:
L , H, Xi, Yi, d32, Kd, Kdf, m = m h ) , t, Initial conditions: Xn
p X , pr
= yn = 0 N~ = 0 (in eq 6)
With mass transfer E , x 104 a
80.00 24.10 92.40 18.40 256.20
7
t
6
-
27.80 35.20 26.10 28.30 12.70
134.20 59.60 249.00 33.30 408.50
5 -
Calculate phase velocities: u = H/((1- c)Ap,) u = L/(tp,A)
Formulate the set of equations to be solved by computer using an integration routine in the following order: eq 10 and 11 eq 1 using yif in definition of driving force F1 eq 6 and 7 Y1 eq 12 rlf eq 13 dxlldt eq14 Using a loop formulate the equations for stages 2 to N : Yif
+
rl
rn
Pn Yn
dx,ldt
eq 1 eq 7 where N, equals the dx,/dt obtained for the previous step, for first step N~ = 0 eq 8 eq 9, for n = N use eq 15
Solve for x on computer till steady state is reached.
drops close to the coalescing interface since there is simply no backflow of the drops over large distances. Statistical models may be postulated replacing the uniformly moving drops as used here, by drops moving with different velocities or stopping occasionally for short intervals. A model of this kind was developed by Reynier and Rojey (1972) for packed co-current columns. However, only the response of the column to a rectangular impulse without mass transfer was considered. A model for the computation of concentration profiles would be possible theoretically, but to keep the versatility on the present level a Monte Carlo simulation would be necessary, following the history of a large number of individual drops. Such a procedure would need considerably more computer time than the present model, and, for reasonably uniform drops, would not change the results considerably. However, it may be useful for coalescing drops.
0
1
2
3
4
5
6 time
7
+
8
9
lminl
Figure 6. Changes of concentration in both phases as measured at highest and lowest samplingdevices (Figure 4), after feed of constant composition was introduced into the column. Upper two curves are for continuous phase; lower curves for dispersed phase. Experiment no. 171.
The dependence of the drop size and the holdup on the concentration of the solute represents a considerable difficulty in the evaluation of all unsteady-state experiments. For this reason the method of evaluating mass transfer coefficients by injection of a short impulse of the solute, as described earlier (Steiner and Hartland, 1974), was replaced by injecting the solute for a sufficiently long period to maintain the steady state throughout the column for several minutes. In this way, actual mass transfer coefficients for drops rising in swarms may be obtained with good accuracy for the price of about five times the amount of solvents necessary for the unsteady-state measurements. However, the experimental technique is much simpler and the results are easier to calculate.
Conclusions 1. The unsteady-state concentration profiles along the length of a spray column may be simulated assuming that noncoalescing drops move uniformly through a cascade of mixed hypothetical stages with backflow of the continuous phase. Numerical solution of the equations given in this paper requires knowledge of the flow rates of both phases, inlet so-
Ind. Eng. Chem. Process Des. D :v., Vol. 17, No. 2, 1978
0
1
2
3
4
5
6
181
7
8
9
7
8
9
8
time 0
2
1
3
4
5 time
6
7
8
9
lminl
Figure 9. Same as Figure 6. Experiment no. 231.
lmin,
Figure 7. Same as Figure 6. Experiment no. 19.
a
7 -
7
-L 4
I
l f l ++
/
2
3
+
2
1
1
0
I / 0
//
,
1
2
0 1
0 3
4
5 time
6
7
8
2
3
4
5
3
lminl
time
6 lrnlnl
Figure 10. Same as Figure 6. Experiment no. 25.
Figure 8. Same as Figure 6. Experiment no. 222.
lution concentration in both phases, drop diameter, holdup of the dispersed phase, backmixing coefficient of the continuous phase, equilibrium relationship for the system in question, and the mass transfer coefficients for drop formation and movement. 2. If feeds with constant solute concentrations are introduced, the profiles approach those for steady-state operation as time increases, so the equations may be used to obtain the steady-state solution. The same result may be obtained by setting the time derivatives to zero and solving the set as a system of algebraic equations. 3. The model described here delivers results which are similar to those obtained by solution of the classical backflow model with backmixing in both phase, providing that the backflow coefficient in the dispersed phase is properly selected. However, the number of necessary equations is halved and the backflow coefficient, which is merely a formal parameter in our case of noncoalescing drops, is not needed.
5 1
P
4
1
I
0
0.5
1
I
I
1.5
2
column height
\mi
Figure 11. Steady-state profile obtained by optimization of mass transfer coefficient (experiment no. 171).
182
Ind. Eng. Chern. Process Des. Dev., Vol. 17, No. 2, 1978
Nomenclature A = cross-sectional area of column, m2 a = interfacial area, m-1 d 3 2 = mean Sauter diameter of drops, m E , = dispersion coefficient in continuous phase, mz/s f = backflow coefficient = (E, = u6(f 0.5)) H = mass flux of continuous phase, kg/s j = number of nozzles in distributor Kd = mass transfer coefficient related to driving force in dispersed phase, kg/m2 s Kdf = mass transfer coefficient during drop formation L = mass flux of dispersed phase, kg/s m = equilibrium factor defined by eq 2 r = mass transfer rate, m/s P = mean value of mass transfer rate, m/s t = time, s u,u = actual axial velocities of continuous and dispersed phases, respectively, m/s x,y = concentrations of continuous and dispersed phases, respectively, expressed either in mass fractions or in relative mass fractions uI = velocity in distributor nozzles
+
Greek Symbols = rate of change of concentration in continuous phase 6 = height of hypothetical stage, m p = density, kg/m3
= holdup of dispersed phase
Subscripts i = in f = drop formation n = typicalstage N = last stage measured from drop distributor 0 = time when a drop enters a stage x = continuous phase y = dispersed phase
Literature Cited Mecklenburgh,J. C., Hartland, S.,"The Theory of Backmixing,"Wiley London, 1975.
Reynier, J. P., Rojey, A., Chem. Eng. J., 3, 187 (1972). Schleicher,C.A., AlChE, J., 6 , 529 (1960). Skelland,A. H. P., Minhas,S.S..AlChE, J.. 17,1316 (1971). Steiner. L., Hartland, S..Proc. Int. Solv. Extr. Conf. ISEC, Soc. Chem. lnd. L o m n (1974).
Steiner.L., Hartland,S.,Chem. Rundschau, 3, 28,(1975). Received for review March 25, 1977 Accepted October 24, 1977
CY
We would like to thank the Kommission zur Forderung der wissenschaftlichen Forschung for financial assistance.
A Model for the Precipitation of Pentaerythritol Tetranitrate (PETN) Thomas Rivera. and Alan D. Randolph Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545
A continuous precipitation method for the preparation of crystalline pentaerythritol tetranitrate (PETN) was developed. The process involves the precipitation of PETN from an acetone solution by the addition of water in a Kenics Static Mixer. The principal independent variable is the ratio, R, of the acetone-PETN solution flow rate to the flow rate of water. A mathematical model based on dispersed plug-flow equations adequately represents the physical process. The relationships developed can be used to predict particle size distributions and estimate the effective kinetics involved in the precipitation process. The mass-weighted mean particle size, L, of the precipitated PETN is a linear function of R. The initial nucleation and growth rates are exponentially decaying functions of position z. The nucleation exponent is 3.75; the growth rate exponent is 1.56. The value of the diffusion parameter, Pe, is 5 1.
Introduction An important consideration in the applications of high explosives is the ability to predict the physical properties of the material. In the case of a crystalline explosive, the particle size distribution and crystal habit are important (Scott, 1970). Batch processes commonly used in the preparation of high explosives frequently result in large variations in the products and usually provide little information about the kinetics of the process. A method for the preparation of a reproducible crystalline product having a known particle size distribution and crystalhabit is desirable. The method of recrystallization employed in this work involves the precipitation of P E T N from an acetone solution by the addition of water in a static mixer. The primary independent variable is the ratio, R , of the PETN-acetone solution flow rate to the flow rate of water. PETN is a symmetrical, nonpolar organic compound [C(CH20N0z)4],having a formula weight of 316.15. It forms colorless, nonhygroscopic crystals. Bulk PETN is readily 0019-7882/78/1117-0182$01.00/0
compressible and has a maximum density of 1.77 g/cm3. PETN is insoluble in water and soluble in acetone. Figure 1 gives the solubility of PETN in mixtures of acetone and water (Roberts and Dinegar, 1958). The objectives of this study are to develop a continuous reproducible recrystallization technique for the preparation of PETN, which has definite crystalline properties, and to develop a model that provides some information on the kinetics and predicts the crystalline properties of the product. Of
Crysta11izer
A Kenics Static Mixer was used as the crystallizer in this study. The crystallizer configuration is described with plug flow and plug flow with dispersion models, developed as follows. A. Plug Flow Model. At a steady state, with negligible breakage, the population balance is given by Randolph and Larson (1971) as 0 1978 American Chemical Society
