Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
useless to consume such an energy over the whole volume and it is more logical to localize it near the transfering wall. Such a conclusion seems to recommend special cells such as the "Swiss Roll Cell" (Robertson, 1975) in which the interelectrode distance is very small and fixed by a porous plastic cloth, which also improves mass transfer rates. Nomenclature C, = capital cost per unit time of operation, per unit current-carrying area of channel, F/m2 s C, = cost of electrical energy delivered to electrodes, F/J C, = cost of pumping energy delivered to fluid, F/J CT = product cost per coulomb passed through the system, F/C CT,opt = optimum product cost, F/C Cs = ferricyanide concentration, mol/m3 D = ferricyanide diffusion coefficient, m2/s d = channel equivalent diameter, m d, = particle diameter, m e = intercylinder distance, m F = Faraday's constant, (9.649 X lo4 C/equiv) g = acceleration due to gravity, m2/s h = channel thickness, m i = average current density over current-carrying area, A/m2 iLim = limiting value of i, A/m2 k = overall mass transfer coefficient L = length of current-carrying section, m Re = (u-h)/v,Reynolds number Sc = v / D , Schmidt number S = area of current-carrying region, m2 u = superficial liquid velocity, m/s uopt = optimum liquid velocity, m/s
103
Greek Letters a = factor, defined in eq 2 , taken to be 0.8 in numerical examples y = apparent electrical conductivity of working fluid, mho/m yo = electrical conductivity of working fluid, mho/m A = conventional friction factor, defined by eq 7 p~ = fluid density, kg/m3 ps = particle density, kg/m3 t = packed or fluidized bed porosity v- = fluid kinematic viscosity, m2/s 7 = average fluid shear stresses, N/m2 P = pressure drop, in flow direction, along current-carrying section, N/m2 A V = potential drop between electrodes, V Literature Cited Colburn, A. P., Am. Inst. Chem. Eng. Trans., 29, 174 (1933). Davies, S. J., White, C. M., Proc. Roy. SOC.London, Ser. A, 119, 92 (1928). Ergun, S.,Chem. Eng. Prog., 48 (2), 89 (1952). Isaacson, M. S., Sonin, A. A , , Ind. Eng. Chem. Process Des. Dev., 15 (2), 313 (1976). Neale, G. H., Nader, W. K., A.I.Ch.E. d., 19 ( I ) , 112(1973). Norris, R. H., Streid, D. D., Trans. Am. SOC.Mech. Eng., 62, 525 (1949). Robertson, P. M., Schwager, F., Ibl, N., d. Electroanal. Chem., 65, 883 (1975). Schlichting, H., "Boundary-layer Theory", McGraw-Hill, New York, N.Y., 1968. Sonin, A . A . , Isaacson, M. S., Ind. Eng. Chem. Process Des. Dev., 13, 241 (1974). Sonin, A . A . , paper presented at the 143rd Meeting.of the Electrochemical Society, Chicago, Ill., 1973. Storck, A., Vergnes, F., Le Goff, P,, Powder Techno/., 12, 215 (1975) Storck, A.. Thesis, Nancy, 1976. Storck, A., Coeuret, F., Electrochim. Acta, in press, 1977
Recaved /or reuiew December 8,1976 Accepted September 12,1977
COMMU NlCATlONS
Analysis of Continuous Spouted-Bed Granulation
A generalized analysis of continuous spouted-bed granulators is provided. The analysis is based on Uemaki and Mathur's model (1976) which encompasses the mechanisms of particle growth by solute deposition, particle breakage and abrasion, and the elutriation of undeposited droplets. Expressions for the variation of the mean particle diameter with time are derived for the cases of total bed weight and total number of particles in the bed being constant. It is shown that for long operating times, the mean particle diameter converges to a finite value. This steady-state mean diameter depends on the solid properties, the system design, and the operating conditions but not on the initial particle size.
In a recent article, Uemaki and Mathur (1976) considered the operation of a continuous spouted-bed granulator producing ammonium sulfate particles. In their study, Uemaki and Mathur examined the complicated mechanisms of particle growth by solute deposition, particle breakage, and dust formation by particle abrasion and undeposited droplets. They formulated a simple, yet physically sound, model which encompassed all these mechanisms and expressed them in measurable quantities. However, the final result of their analysis is physically questionable as it indicates that the mean particle diameter will increase (or decrease) indefinitely with time. Since they assume the total mass of the bed is
constant, their eq 10 implies that eventually the granulator will consist of either a single large particle or very fine particles that at some point will be elutriated out of the bed. This is a rather unrealistic result because usually when two competing mechanisms exist in a continuous system they tend to balance each other after a certain operating time. For a spouted-bed granulator it is expected that for some finite mean particle diameter the rate of particle growth is balanced by particle breakage and abrasion and that a steady-state mean diameter is reached. Uemaki and Mathur derived their expression under two assumptions: (a) that the bed weight is constant, and (b) that
0019-7882/78/1117-OlO3$01.00/0 0 1978 American Chemical Society
104
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
the ratio between the formation rate of fresh nuclei and the number of particles in the bed is constant. The physical basis of the second assumption is questionable. I t implies that the formation rate of fresh nuclei is proportional to the number of particles in the bed and with the constant weight assumption, it means that the particle breakage decreases as the particles grow. In practice, however, large particles tend to break more readily than small particles. This communication provides a revised, generalized analysis of spouted-bed granulation. The analysis is based on the mechanisms of particle growth and breakage proposed by Uemaki and Mathur incorporated into an unsteady-state material balance over the solids in the granulator. The incorporation of a transition term is essential since shortly after start-up it is difficult to control the product withdrawal rate such that the rate of solute addition to the bed is identical with the sum of the elutriation and the product withdrawal rates. The transition term also allows us to generalize the analysis to batch granulators. A general expression for the change in the mean particle diameter with time is derived under the assumption that the formation rate of fresh nuclei is independent of the particle size. This is a reasonable approximation of the actual physical behavior in the granulator since the mean particle diameter varies very slowly with time and usually changes over a relatively limited range. The general expression for the change in the mean particle diameter with time is then integrated for two special cases: (a) the total number of particles in the bed being constant, and (b) the total weight of the bed being constant. A procedure to analyze and interpret data of of spouted-bed granulators is described. Consider the continuous spouted bed granulator described schematically in Figure 1. In accordance with Uemaki and Mathur's notation, the material balance for the solids in the granulator is expressed by
LXL = P
+ E f dWldt
(1)
where d Wldt is the accumulation rate of the solute in the bed. Equation 1 can be expressed in terms of the particles withdrawn and demolished per unit time, the undeposited spray droplets, and the dust generated from surface abrasion by
L X L = n,E,
+ newe + S t D f dWldt
(2)
where
ELUTRIANTS
1 E * newe+ s
+D
SOLUTION
Figure 1. Schematic description of a spouted-bedgranulator.
where yj and dpj are the weight fraction and the mean diameter of the j t h fraction. Due to the excessive mixing in the spouted-bed granulator, it is reasonable to assumcthat no particle segregation occurs inside the bed; thus 2, = db.(This assumption can be readily examined experimentally.) Also, as indicated above, it is assumed that the breakage and abrasion of particles are independent of the particle size, Le., 2, = a b . I t fOllOWS that
and
z,= Loe = -w b
(4b)
An overall population balance on the particles in the granulator gives dNb -- n - np - ne - n d
dt
where n is the rate of generation of fresh nuclei, and n d is the rate of particles breakage to form these nuclei. Thus ( n - n d ) is the net addition rate of particles to the bed. In most cases n >> n d and n d can be neglected. The derivative of eq 3 with respect to time is
Substitution of eq 5 into eq 6 and substitution of the latter into eq 2 give 3NJp2d (Zp)
-
(n
Xab3p
(3) =Nb6 where N b i is the number of particles of weight u b i inside the granulator and N b = Z i N b i . In the expressions above, Lo,, De, and Lob and a,, and z b are the mean particle weights and diameters of the product particles, the bed particles demolished to elutriated dust, and the particles inside the bed, respectively. The mean particle diameter is readily obtained from sieve analysis by i
nbiWb, = N b E b
ap,
z,
=
TP
= dt
(7)
s -D)
Equation 7 is the general expression for the variation of the particle diameter, with time. It can be integrated whenever the manner in which N b changes with time is known. (This can be obtained experimentally.) It is interesting to consider two special cases which approximate many practical operations: (a) the number of particles in the bed, N b , being constant, and (b) the weight of solids in the bed, W, being constant. In the first - case, integration of eq 7 with the initial condition that 2, = d,o a t t = 0 gives
a,,
The weight of the solids in the bed is given by
W=
- nd)2p3
6 - - ( L X -~
6 (LXL-S-D) r z-3 - XP
(n
- nd)
1
1
For the second case, eq 3 and 7 are combined and rearranged
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
105
to give EXPERIMEUAL DATA FROM UEMPXl AND MATHUR'S RUN 5
-
Integration of eq 9 with the initial condition that t = 0 gives
111
6
( n - n d ) J p 0 3 - - ( L X L- S - D ) =P
a,= z p o a t
W' 6 5 4 SSD' 0 0 5 LX,. 0 97 E: 0 I 1
KG KGIHR KG/HR KG/HR
/
1
1
LXL-S -D =t W
(10)
1
I
Equations 8 and 10 express the variation of the mean particle diameter, with time under the specified conditions. In both cases, as the operating time becomes very large, the mean particle diameter converges to a finite limit (steady-state diameter) given by
zp,
2
4
6
I
1
1
I
1
8 1 0
1
40
20
l
/
I
I
I
w m o o
TIME, HOUR
Figure 2. Comparison between theory and experimental data.
parameters. The following procedure is suggested. First, the mean particle diameter a t steady state, should be determined for the specific system and operating conditions. The can be readily obtained by operating the system value of for a long duration and measuring the mean diameter of the is determined product particles as a function of time. Once the left side of either eq 8 or 10 can be plotted as a function of time. From the slope of the curve ( n - nd)/Nb or ( L X L - s D ) / W can be calculated. Substitution of these quantities into eq 11 provides the values of L X L - S - D and n - n d independently and allows the calculation of other parameters. For example, the rate of particle demolition to dust can be obtained from ne = ( E - S - D)/iijpwhere E is the total elutriation rate which can be measured directly. Once the key parameters are calculated, their dependency on the system design and the operating conditions can be investigated.
z,,,
zpm
By combining eq 11and 10 or &_onecan express the variation of the mean particle diameter, d,, with time in terms of the For the case of constant bed steady-state diameter, weight we obtain
zpm.
=-
LXL-S-D t W
(12)
Equations 8 and 10 confirm the hypothesis that the mean particle diameter reaches a finite steady-state value a t long operating times. Note that the mean particle diameter a t steady-state is independent of the initial particle size; it is a function of bed geometry, the operating conditions, and the solution injection rate. The rate of dust generation from undeposited droplets, S, is determined by the contact efficiency between the spray droplets and the particles. This in turn depends on the geometry of the bed, the location and orientation of the atomizers, and the operating conditions. The rates of dust generation, D , and formation of fresh nuclei, ( n - nd), are determined by the physical properties of the particles and the flow conditions. The mean particle diameter at steady state may be either larger or smaller than the initial average particle diameter in the bed. Uemaki and Mathur measured a reduction in the mean particle diameter in some of their granulation tests. The validity of the theoretical analysis can be readily checked against Uemaki and Mathur's experimental data. Unfortunately, these workers did not attain steady-state conditions in their experiments. Nevertheless, eq 12 can be compared with experimental data by fitting different values of Figure 2 shows such a check for Uemaki and Mathur's run no. 5 . Although the data are given over a range where the specific value of 2,- is not critical, the agreement between the theory and experiments is quite good and it is consistent with Uemaki and Mathur's observation that the mean diameter continues to increase in 9.5 h runs. Equations 8 and 10 express the variation of the mean particle diameter with time in terms of parameters which are not measured directly. Consequently, it is necessary to manipulate the experimental data t o determine the desired operating
apm.
zpm
Acknowledgment
I would like to thank Professor E. J. Crosby of the University of Wisconsin, Madison, and Dr. G. L. Brown of Union Carbide Corporation for their comments. Professor Mathur provided additional data which made the comparison between theory and experiment possible. I gratefully appreciate his assistance. Nomenclature D = rate of dust generation by surface abrasion of particles in bed, kg/h & = mean particle diameter of the solids in bed, cm d e = mean particle diameter of particles demolished to form elutriated dust, cm ZP = mean particle diameter of withdrawn product, cm d,, = mean particle diameter of solids in bed at steady state, cm E = rate of dust elutriated out of bed, kg/h i = summationindex L = mass flow rate of feed solution, kg/h Nb = number of particles in the bed n = rate of generation of fresh nuclei, particles/h nd = rate of particle demolition to generate fresh nuclei, particledh ne = rate of demolition of bed particles to elutriated dust, particles/h n p = rate of withdrawing product particles, particles/h P = product withdrawal rate, kg/h
106
Ind. Eng. Chem. Process Des. Dev.,Vol. 17, No. 1, 1978
S = rate of elutriation of undeposited spray droplets, kg/h t = time, h W = weight of solids in the bed, kg z b = mean weight of individual particles in bed, kg iZe = mean weight of individual particles demolished to elutriated dust, kg iZP = mean weight of individual particles in product stream, kg X L = mass fraction of solute in the solution p = density of the particles, kg/cm3
Literature Cited Uemaki, 0..Mathur, K. B.. Ind. Eng. Chem. Process Des. Dev., 15, 504 (1976).
Department of Chemical Engineering Texas Tech University Lubbock, Texas 79409
Uzi Mann
Received for review January 5,1977 Accepted August 15, 1977
