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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
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
183
fl, = C,/Co. The dimensionless moments of the population,
f , are 0 P
I\
(4) The dimensionless plug flow moment equations become
where f j = dfj/dz. The initial conditions are fj(0) = 0 (j = 0, 1, 2 , . * *)* The total solids concentration per unit volume of solids-free liquid is
MT = pKvm3(x) = [CO- C b ) I Weight Porcrnt Wator
where p is the crystal density of PETN, 1.77 g/cm3, and K , is the volume shape factor based on Stokes' law. In terms of the dimensionless quantities, eq 6 becomes
Figure 1. Solubility of PETN in aqueous acetone solutions.
where n is the number population density of the system and G is the linear growth rate, dL/dt. The internal coordinate is taken as the Stokes diameter, L , as determined by sedimentation techniques. The external coordinate system is represented by the single crystallizer length dimension, x . For small particles and high velocities, V,, the external linear velocity, may be taken as the plug flow velocity, u,. Assuming that growth rate G remains independent of size, an empirical observation known as McCabe's AL law, which often holds true, then eq 1 reduces to an
uX-
ax
an + G-=aL
0
dm , dx
+ (0)jBO - jGmj-1
At the exit of the reactor, z = 1. If the reaction is complete at z = 1,then MT/CO p, = 1 and
+
P, = P(1)
(8)
This assumption of fast precipitation, complete within the physical length of the reactor, is verified experimentally (in the subsequent Experimental Section) by weighing the product for a given volume of solids-free liquid. Substituting eq 8 into eq 7
dividing eq 7 by eq 9 and rearranging
with side conditions, n(0,x)= BO(x)/G(x) (nuclei density) and n(L,O) = 0 (unseeded). Equation 2 is the plug flow equation that includes the following assumptions: steady state is achieved; linear velocity = plug flow velocity = constant; negligible breakage is achieved; growth rate is independent of size. The population balance can be transformed to an equation in terms of the moments of the distribution by multiplying eq 2 by LjdL, and integrating from zero to infinity as shown by Randolph and Larson (1971). Thus U, 2
(6)
=0
(3)
where j = 0, 1 , 2 , . . . . The quantity (0)j is zero when j # 0 and one when j = 0. The equations given by eq 3 are the moment equations for the plug flow model. We will now introduce the retention time, 7, defined by 7 = T/uxrwhere F is the effective length of the plug flow reactor in cm and u, is the plug flow velocity in cm3/s. The quantities CO,the initial concentration ( x = 0), and C,, the concentration of the saturated solution, will also be used. It is noted that the parameter T is an effective length which, with the assumed empirical growth and nucleation kinetics, gives full reaction within the actual physical length of the reactor. Thus, the model is, as stated, applicable only to fast precipitation reactions. It will be convenient to use dimensionless quantities normalized in terms of the initial conditions where x = 0. The dimensionless external coordinate becomes z = x/T, the dimensionless internal coordinate becomes t = L/GoT, and the dimensionless population density becomes y = n/n$ where noo = Boo/Go. The dimensionless concentration becomes P = C/Co and the dimensionless saturated concentration becomes
(10)
This is the relationship for the concentration as a function of the normalized external coordinate. The normalized external coordinate, z , will be referred to as the reaction coordinate. For the case in which P, = 0
B. Dispersed Plug Flow Model. Consider the plug flow of a fluid, on top of which is superimposed some degree of intermixing, the magnitude of which is independent of position within the vessel. This is called the dispersed plug flow model (Levenspiel, 1972). For eddy diffusion in the x direction modeled with an effective eddy diffusivity, the governing differential equation is given as an an a2n U, - G - = D (12) ax aL ax2 where the parameter D , called the axial eddy diffusivity coefficient, uniquely characterizes the degree of backmixing between adjacent reactant volume elements during flow. The dimensionless group (D/u,T), called the vessel dispersion number, is the parameter that measures the extent of axial dispersions. Thus
+
u h
-
- 0 negligible dispersion, hence, plug flow U,,
D U,,
m
large dispersion, hence, mixed flow
This model represents satisfactorily flow that deviates not too greatly from plug flow (Levenspiel, 1972).
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Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
Equation 1 2 may be rewritten as
-DT
a2n ax
an dn + U, + G -= 0 ax aL
Again, utilizing the moment transformation, the reactor equations can be written as
-D 2 dm d 2 m ' + u, 2- (0)jBO - jGmj-1 = 0 dx2 dx Using the definitions in section A, we obtain ,
(14)
plug flow:
D d2fj dfj . Bo G ( 0 )~ j -fj-l =0 (15) u,X dz2 dz Boo Go Defining the Peclet number as the reciprocal of the vessel dispersion number, Pe = uxT/D, the resulting dispersed plug flow moment equations become (Randolph and White, 1977)
+
+
with the boundary conditions fj(0) - 1/Pe fj(0) = 0, and fj(1) = 0. The addition of the second derivative term to model flow dispersion requires an additional boundary condition to complete the set. The conventional reactor and condition of zero gradients is used. Equation 10 from section A holds for the dispersed plug flow model, namely
and for p, = 0
C. Kinetics. Equations 5 and 16 contain the quantities BO/BoOand GIGO, which represent the normalized nucleation and growth rate functions, respectively. 1. Nucleation. The initial nucleation rate, Boo, is the rate of formation of nuclei at z = 0, the entrance of the reactor. The assumption is made that the initial nucleation rate is a function of the ratio, R . Equation 9 may be written in the form r
,
giving growth ratio GIGO as G -= (1- z ) b (23) GO In summary, the moment equations, including the assumed kinetic models, are given as
1
The total nucleation, BO, is assumed to be the product of the initial nucleation and a position decay function of the form (1- z ) G . The expression for Bo is thus assumed to be
f, - (0),(1 - z ) a - j(1 - ~ ) ~ f , - =1 0
1 .. dispersed plug flow: -f, - f, Pe
+ (O)l(l - z + j(l -
(24)
) ~
~ ) ~ f , - 1=
0
(25)
Equations 24 and 25, together with the appropriate boundary conditions, can be solved numerically to obtain the j t h moments of the distribution. Equations 24 were numerically solved with a Runge-Kutta numerical technique, and eq 25 were solved by means of a finite difference technique. The resulting package programs are available at LASL. D. Inversion of the Moments, The sol-ition of either the plug flow or dispersed plug flow moment equations gives a set of (f,1 dimensionless moments. The number of moments used in this study is 10. The set of moments can be approximated by the equation
from which the dimensionless population distribution function, Yk, may be calculated for a given $k and using the matrix inversion techniques described by Randolph and Larson (1971). The cumulative weight fraction distribution is
where n ( p ) is the population density of particles having size less than L. Or, in terms of the dimensionless variables, the weight distribution is given as
(28) whers y ( p )is the dimensionless population density of particles having dimensionless size less than E . Equation 28 may be approximated by the sum
with the nucleation ratio Bo/Booas
BO BO0 2. Growth. The initial growth rate at the entrance is GO. The initial growth rate is Elated to the mass-weighted mean particle size, E, as follows: L, the mass-weighted mean particle size, is defined by the ratio of the 4th and 3rd moments.
-= (1 - z ) G
(20) thus, by eq 4,
Go is assumed to be a function of the experimental parameter, R. The total growth rate, G, is assumed to be the product of the initial growth rate and position decay function of the form (1 - z ) b , namely
E. Comparison of Theory with Experiment. The different,ialequations representing the dispersed plug flow model are solved for a given set of parameters a, b, and Pe. The resulting dimensionless moments are inverted to obtain the population distribution functions and finally, the cumulative weight fraction distribution is calculated by means of eq 29. This cumulative weight fraction distribution can be directly compared with experimentally determined distributions. The coefficient of variation, or relative dispersion, is the measure of dispersion stated as a function of its average, that is C.V. = 2 (30) L where CJ is the standard deviation from the mean for the distribution.
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
185
Table I. Experiment Holding Times and Flow Rates
Acetone solution flow rate. cm3/s
Water flow rate. cm3/s
R
Holding Time, 0.5 s Total Flow Rate, 18.0 cm3/s 3.6 6.0 9.0
RESERVOIR
ROTAMETER
GLASS Y-TUBE O-RING
BALL VALVE POLYPROPLYENE
CONNECTOR
STATIC M I X E R
14.4
'14
12.0 9.0
lh
Holding Time, 2.0 s Total Flow Rate, 4.5 cm3/s 1.13 3.38 2.25 2.25 3.0 1.5 3.38 1.13
1
%?
1 2
3
the static mixer by means of an O-ring connector. Figure 3 is a schematic of the static mixer and Y-tube connector. Because of the continuous flow division reversal and flow inversion at every element of the mixer, conditions approaching plug flow are approached (Bor, 1971). The holding time of the static mixer is calculated by the equation volume of mixer (cm3) holding time(s) = total volume flow rate (cm3/s)
Figure 2. PETN ciystallization apparatus.
Stainless Steel Elements
Glass Y- t u b e
Figure 3. Static mixer. The coefficient of variation may be calculated from the moments by
and thus
(Note that rno(wt) = 1.) The coefficient of variation and mean from the theoretical model were fit to the experimental values in an initial scan of parameter values. Final parameter estimation was accomplished by fitting the entire weight-size distribution using eq 29. Experimental Section A. Apparatus. The crystallization apparatus consists of three sections: (1)the constant flow feed system, (2) the static mixer, and (3) the filtering system. A diagram of the apparatus is shown in Figure 2. 1. Constant Flow Feed System. There are two identical feed systems. The liquid level is controlled in the reservoir by an airtight, polyethylene tube (feedback loop) that connects the top of the reservoir to the top of the bottle. The solution is fed to a rotameter ahead of the static mixer. The rotameters were calibrated using either pure acetone or water. 2. Static Mixer. The effluent from the rotameters mix together at the first element of the static mixer by means of a glass Y-tube with a glass divider. The Y-tube is connected to
Table I gives the scheme for holding times and flow rates used in the experiments. 3. Filtering System. Small samples (about 2 g) were collected on a 350-cm" tared Buchner funnel with a fritted glass disk (medium porosity) mounted on a 2-L suction flask. The volume of the solids-free liquid collected in the flask was measured. The PETN sample was washed with distilled water and dried under vacuum for 18 h at 65 "C. The dried sample was weighed on a single-pan analytical balance. The data were used to calculate the M T value for the run. Large samples (30 to 50 g) were collected on a 29-cm diameter stainless steel screen nominally rated at, 5 pm. B. Procedure. The acetone-PETN solution (about 10 L) was adjusted to 20 & 3 "C and placed into the polyethylene bottle. The P E T N concentration was 0.5 wt % for R 5 1,and 5.0 wt YOfor R > 1. The distilled water (about 20 L) was adjusted to 20 & 3 "C and placed in the second polyethylene bottle. The solutions were allowed to flow into their respective reservoirs and the bottles and feedback loops were checked for air leaks. The control valves were adjusted for the proper R value to be used. The total flow rate was checked with a stopwatch and graduate. After allowing a t least 30 s for equilibration, the metal screen was placed into position for collection of the sample. C. Particle Measurement. 1. Photelomet -r Technique. The estimation of particle size distribution fo; subsieve particles below the 45-pm sieve size was made possible by a particle sedimentation-in-liquid method developed by King and Panowski (1948). In this method, a representative P E T N sample is dispersed homogeneously in a 50 vol % mixture of n-octane and 1,1,2,2-tetrachloroethanein a sedimentation tube. The intensity of a collimated beam of light passed through the suspension is measured after varying sedimentation times, and at varying levels. The particle size distribution is calculated by means of Stokes' law. The samples were wet sieved using PETN-saturated ethanol as the transfer fluid. The following sieve sizes were used: 250, 177, 125, 88, 62, and 45 pm. Photelometer results were used for sub 45-pm particles. The correlation of photelometer results with sieve analyses was previously tested for HMX (octahydro-1,3,5-7tetranitro-s-tetrazocine). The two sizing techniques overlapped exactly in the 20-50-pm size range and
186
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2. 1978
Table 11. Particle Size Characteristics
R value
Av
z,wn
12.2
Av coefficient of variation 0.34 0.35
12.7
0.40
3oox
looox
Figure 5. Scanning electron micrographs, PETN-3856 (R = '/z).
size L (rrnl
Figure 4. PETN experimental particle Size distribution can he treated as equivalent methods of crystal size measurement. 2. Scanning Electron Microscopy. Each sample was photographed at 100,300, and 100OX. The samples were gold shadowed to avoid any thermal effects on the heat-sensitive particles. Results A. Experimental Particle Size Distributions. Two typical cumulative weight distribution curves for all R values are shown in Figures 4a and b. The error bars represent standard deviations. Photelometer runs were sufficient for R < 1;for R 2 1,the photelometer method was supplemented by sieves. Table 11summarizes the particle size characteristics of the products. Representative product scanning electron micrographs for low R ratios are shown in Figure 5. Crystals grown with higher R ratios of 1 to 3 had a hlockier hahit, approaching tabular form, as shown in Figure 6. E. Theoretical Model Predictions. 1. Particle Size Distribution. The plug flow model calculations predict distributions that are significantly narrower than the experimentally determined ones. The coefficients of variation predicted by the plug flow model range from 0.11 to 0.21. The inclusion of the Peclet number causes the predicted distributions to broaden as Pe is lowered. A Peclet value of 250 gives results very nearly approaching plug flow values. A series of about 500 computer runs was made with combinations of the nucleation parameters, a, ranging from 1.0 to 4.0, the growth parameter, b, ranging from 0.5 to 1.7, and the diffusion parameter, Pe, ranging from 25 t o 250. Additional runs were made to refine the calculations.
300%
looox
Figure 6. Scanning electron micrographs, PETN-3883 (R = 3). 2. Effect of Varying Parameters. The effect of decreasing a (longer nucleation period) is primarily to increase the
coefficient of variation. I t also tends to decrease the mean particle size, predicting a wider distribution of slightly smaller particles. The primary effect of increasing b (shorter growth period) is to decrease the mean particle size. It also increases the coefficient of variation, predicting smaller particles having a wider distribution. The primary effect of decreasing Pe (increasing diffusion) is to increase the coefficient of variation
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
187
6ot /
" 40
Concentration
SIZE L ( pm)
Figure 7. Comparison of theoretical values with experimental particle size distributions.
.-c
g
tem is equal to or less than 0.5 s. Thus, the resulting model parameters are effective kinetic quantities. These effective kinetics are valid for this system, but may not be valid if significantly higher flow rates are used wherein the actual retention time becomes less than the equivalent reaction time. The effective initial growth rate in pm/s is
-Concentration ----Growth
---Nucleation
0.6
\
LL
Figure 9. Growth rate kinetics along reaction coordinate.
\
Go = 33.68 + 90.58R
(36)
or
The effective growth rate kinetics are
G = (33.68 0
0.0
0.4
0.6
0.8
I,O
Figure 8. Theoretical kinetics along reaction coordinate. with a small increase of particle size. The resulting best fit parameters, considering all sets of experimental data, were a = 3.75, b = 1.56, and Pe = 51. The resulting predicted particle size distributions using these parameters compared with experimental results are shown in Figures 7a through f. Note that the poorest data fits were at the larger R ratios where the blocky crystal habit was obtained. The poorer fit was undoubtedly due to the change in crystal habit. It was not considered feasible to modify the model to accommodate changes in particle habit. Certain parameter combinations outside the range of values chosen gave oscillatory CSDs when the moments were inverted. This was considered an artifact of the numerical technique used. 3. Kinetics. A least-squares linear fit for vs. R gives the equation, for E in ym
+ 18.56R)
(33) with a linear correlation factor of 0.99. From eq 2 1 we have (34) For a = 3.75, b = 1.56, and Pe = 51, the ratio v3(l)/f4(l)] is 2.44. Thus, for GOTin ym GOT= 16.84
+ 45.29R
(38)
with G in pm/s. The nucleation parameter as given by eq 17 was given as
Reaction Coordinate, 2
= (6.90
+ 90.58R)(1 - 2)1.56
(35)
Four experimental runs were made with R = 1 and 0 5 s holding time. Four more runs were made with R = 1and a 2-s holding time. The resulting particle size distributions, using the two holding times, were essentially the same. This information suggests that the equivalent reaction time of the sys-
Bo0 1 (17) Co(1 - P J - ( P K J G o T ) ' T ~ s ( ~ ) where p = 1.77 g/cm3, K , = ~ / 6f3(l) , = 0.006745 (using best fit parameters), and T = 0.5 s. Thus K - p,) - (16.84 + 45.29R)3
Bo0 Co(l
(39)
where K = [pK,~f'(l)]-l = 320 cm3/g s. The total effective nulceation rate kinetics are
Bo = 320(16.84 + 45.29R)-' Co(1 - P,)(l
-z
) ~ (40) , ~ ~
with Bo in nuclei/cm3 s. Figure 8 is a plot of the normalized concentration of PETN in the acetone-water solvent, P ( z ) , the normalized growth rate, G ( z ) , and the normalized nucleation rate, B o ( z ) ,as functions of the reaction coordinate, z . The early part of the reaction appears to be due to the combination of rapidly decaying nucleation with a linearly decaying growth. In the later part of the reaction, there is essentially no nucleation, the growth rate being the primary mechanism at this stage. The change of growth rate as a function of concentration is shown in Figure 9. The reaction proceeds from right to left along the concentration axis. The relationship is nearly linear, except for the initial and final stages of the reaction. The variation of the first four moments along the reaction coordinate is shown in Figure 10. The zeroth moment may be thought of as the total number of crystals in the volume under consideration. The rather large slope a t the early part of the reaction is due to the initially high nucleation rate. The first moment represents the sum of the lengths of all the crystals in the distribution, the second moment is related
188
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 2, 1978
Table 111. Summary of Theoretical Equations
Quantity
Equation
Av deviation from experimental results, %
+
Mass-weighted mean particle size, pm Effective growth rate, Fm/s Effective nucleation rate, nuclei/cm3 s
L = 6.90 18.56R G(R,.z)= (33.68 t 90.58R)(1 - z ) ~ , ~ ~ B0(R,z)= 320(16.84 + 45.29R)-3Co(l - p)(1 - z ) ~ . ~ ~
Concentration a
8.9 8.9 28.7 a
Not determined.
I
I
0.20
I
0.16
/-
/'
--7 ---------broth
/'
First
Sec,ond
I
-I
LIL~ :---I
0.04
e--
0.0 0
-------
0.2 0.4 0.6 0.8 Reaction Coordinate, Z
100
Figure 10. Variation of moments along reaction coordinate,
to the specific surface area, and the third moment is related to the specific mass of crystals in the distribution. Conclusions Crystal and kinetic properties of interest for PETN produced in a plug flow crystallizer can be calculated for a given acetone/water ratio, R . These values include nucleation and growth rates, mass-weighted mean particle size, and concentration profiles. The crystals obtained from the continuous plug flow crystallizer have narrow distributions and are fairly uniform in crystal habit. The dispersed plug flow model calculations provide a reasonably good prediction of these properties. This study demonstrates a continuous reproducible recrystallization technique and a predictive model for the preparation of PETN. No fundamental importance is attached to the kinetic relationships used in the model; however, they do accurately represent the magnitude of growth and nucleation and permit an empirical description of the system, which should be useful in describing other precipitation processes. Acknowledgments The authors wish to acknowledge the support of several people at LASL who assisted in this project. In particular, we wish to acknowledge Dr. Peter G. Salgado, Dr. William A. Cook, Dr. Michael Steuerwalt, Mr. Robert E. Smith, and Dr. Robert H. Dinegar. This work was performed at the Los Alamos Scientific Laboratory of the University of California under the auspices of the Energy Research and Development Administration Contract W-7405-ENG-36.
Nomenclature a = nucleation rate parameter, constant b = growth rate parameter, constant BO = nucleation rate, nuclei/cm3 s BOO = initial nucleation rate, nuclei/cm3 s Co = initial concentration, g/cm3 C, = concentration of saturated solution, g/cm3 C.V. = coefficient of variation D = axial eddy diffusivity coefficient f j = j t h dimensionless moment of the population G = growthrate,@m/s Go = initial growth rate, pm/s K , = volume shape factor, constant L_ = internal coordinate, pm or cm L = mass-weighted mean particle size, pm or cm m . = j t h moment of the population = solids concentration, g/cm3, clear liquid basis n = population density, [number/(unit length)(unit slurry volume)] no = nuclei population density, [number/(unit length) (unit slurry volume)] N = number of crystals Pe = Peclet number or diffusion parameter, constant R = ratio of PETN-acetone solution flow rate to flow rate of water t = time, s u, = plug flow velocity, cm3/s W = cumulative weight fraction distribution x = external coordinate, cm lc = effective reactor length, cm y = dimensionless population density z = dimensionless external coordinate (reaction coordinate)
dl-
Greek Letters /3 = dimensionless concentration /3, = dimensionless saturated concentration A[ = dimensionless width of crystal size range p = particle density, g/cm3 u = standard deviation 'T = retention time, s
Literature Cited Bor,T., Br. Chern.€ng., 16(7),610(1971). King, D. R., Panowski. J. B., "Determination of Particle Size Distribution and Surface Area by Photometry," Los Alamos Scientific Laboratory Report LA-1267-MS, 1948. Levenspiel, D., "Chemical Reactor Engineering," p 272, 2nd ed, Wiiey, New York, N.Y., 1972. Randolph, A. D., Larson, M. A., "Theory of Particulate Processes," pp 19-61, Academic Press, New York, N.Y., 1971. Randolph, A. D., White, E. T., Chern. €ng. Sci., 32, 1067 (1977). Roberts, R. N., Dinegar, R. H.,J. Phys. Chern., 62, 1009 (1958). Scott, C. L., paper presented at 5th Symposium on Detonation, Pasadena, Calif., Aug 18-21, 1970 (Proceedings issued on ONR, DR-163, 148) (1970).
Receiued for review March 29, 1977 Accepted December 20, 1977
Presented at the Atlanta AIChE Meeting, Mar 10, 1978.
