Ind. Eng. Chem. Res . .1989,28, 1740-1741
1740 Greek Letters
= virtual kinematic viscosity of turbulent flow, m2/s po = air dynamic viscosity at discharge conditions, Pa s to
po p'
4
= air density at static conditions, kg/m3 = air density at discharge conditions, kg/m3 = entrainment coefficient
Literature Cited Beltaos, S.; Rajaratnam, N. Impinging Circular Turbulent Jets. Proc. Am. SOC.Civ. Eng. J . Hydraulics 1974, 100 (No. HYlO), 1313. Boguslawski, L.; Popiel, Cz. 0. Flow Structure of the Free Round Turbulent Jet in the Initial Region. J . Fluid Mech. 1979,N (Part 31, 531. Chue, S. H. Pressure Probes for Fluid Measurement. Prog. Aerospace Sci. 1975, 16 (No. 2), 147. Donald, M. B.; Singer. H. Entrainment in Turbulent Fluid Jets. Trans. Inst. Chem. Eng. 1959, 37, 255. Donaldson, C. Dup.; Snedeker, R. S. A Study of Free Jet Impingement. Part 1. Mean Properties of Free and Impinging Jets. J . Fluid Mech. 1971, 45, 281. Obot, N. T.; Graska, M. L.; Trabold, T. A. The Near Field Behavior of Round Jets at Moderate Reynolds Numbers. Can. J . Chem. Eng. 1984, 62, 587. Obot, N. T.; Trabold, T. A.; Graska, M. L.; Gandhi, F. Velocity and Temperature Fields in Turbulent Jets Issuing from Sharp-Edged
Inlet Round Nozzles. Ind. Eng. Chem. Fundam. 1986,25, 425. Reichardt, H. Gesetzmassigkeiten der freien Turbulenz. VDI-Forschungsh. 1942, 414, 1961. Shambaueh. R. L. A MacroscoDic View of the Melt-Blowing Process for Prlducing Microfibers. i n d . Eng. Chem. Res. 1988,27,2363. Taylor, J. F.; Grimmett, H. L.; Comings, E. W. Isothermal Free Jets of Air Mixing with Air. Chem. Eng. Prog. 1951, 47 (No. 41, 175. Tollmien, W. Berechnung Turbulenter Ausbreitungsvorgiinge 2. Angew. Math. Mech. 1926, 6,468. Wall, T. F.; Nguyen, H.; Subramanian, V.; Mai-Viet, T.; Howley, P. Direct Measurements of the Entrainment by Single and Double Concentric Jets in the Regions of Transition and Flow Establishment. Trans. Inst. Chem. Eng. 1980,58, 237. Wygnanski, I.; Fiedler, H. Some Measurements in the Self-Preserving Jet. J . Fluid Mech. 1969, 38 (Part 3), 577.
* Author to whom all correspondence should be addressed. Marc A. J. Uyttenclaele, Robert L. Shambaugh* Department of Chemical Engineering and Materials Science, The University o f Oklahoma Norman, Oklahoma 73019 Received for review March 13, 1989 Accepted July 31, 1989
Use of Residence Time Distribution Information and the Batch Pelletization Equation To Describe an Open-circuit Continuous Pelletizing Device A method to obtain the response of a continuous pelletizing device from its batch response is presented. It is assumed that the pellet growth kinetics is linear and the residence time distribution (RTD) is size independent. Under these assumptions, results obtained by the above method agree with those from a more detailed mathematical model. Pelletization, also known as balling or granulation, is an important unit operation employed in several industries for the size enlargement of fine particulate systems, including minerals, chemicals, and pharmaceuticals (Knepper, 1962). Most commonly, this operation is carried out in revolving drums, cones, disks, or pans. A thorough analysis of the process requires detailed knowledge of the pellet growth mechanisms and the transport characteristics of the pelletizing charge through the pelletizing device. Recently, Kapur et al. (1981) developed a model for an open-circuit pelletization drum by considering the balling drum as consisting of m perfect mixers in a series and by writing a generalized population balance equation for the jth mixer. After solving a complex set of equations, they obtained the expression for the mean pellet mass in the product under steady-state conditions as wont
=
d o ) ( 1 + $)m
t 1)
where g(0) is the mean granule mass of the feed at the inlet of the drum, h is the rate parameter, T is the mean residence time, and m is the number of perfect mixers in series. Although the model developed by Kapur et al. is quite adequate and suitable for simulation, the purpose of this communication is to show that, under certain conditions and assumptions (which are usually valid in a normal pelletizing operation), relationships between input and output variables of a continuous pelletizing circuit can be obtained without the requirement of a detailed model for pellet transport through the device. If the equations describing the batch process are linear and the pellets of 0888-5885/89/2628-1740$01.50/0
all sizes are characterized by a single residence time distribution (RTD), then the steady-state response of a continuous pelletizing drum can be obtained by summing the batch response of the system to an appropriately weighted train of impulses of input. Such an approach has already been used for developing simple lumped parameter models for open- and closed-circuit grinding mills (Herbst et al., 1978). It has been shown by Kapur and Fuerstenau (1964) and by Sastry and Fuerstenau (1973) that, in the batch pelletization of comminuted fine particulate materials, the predominant mechanism of pellet growth is the direct coalescence between agglomerates. It has also been established by Fuerstenau (1980) that pellet transport through industrial-scale pelletizing drums is essentially of plug flow type and by Bhrany et al. (1962) that a pelletizing disk behaves like a perfect mixer. A theoretical model of the batch pelletization process has been proposed by several workers (Kapur and Fuerstenau, 1969; Kapur, 1972) to describe the evolution of the pellet size distribution. According to this model, which uses a population balance approach, a general equation for pellet growth by the coalescence mechanism can be written as
n ( X _ f J - X ( y , r - y , t )n(y,t) n(r-y,t) dy N ( t ) '(disappearance by coalescence)
(2)
where n(x,t) dx = the number of pellets in the mass in0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1741
+
terval x to x dx at time t , N ( t )= J t n ( x , t ) dx = the total number of pellets at time t , and A(x,y,t) = the binary collision-coalescence rate function for the pelletizing pair of masses x and y. Again, it has been shown by Kapur and Fuerstenau (1969) that the pelletization of a wide range of particulate systems can be represented by a random coalescence process in that the rate of coalescence between two colliding pellets or granules is independent of their masses. Mathematically this implies that A(x,y,t) = A ( t )
It has also been shown (Kapur and Fuerstenau, 1969) that, unless the feed particles are excessively fine, the coalescence rate function is independent of time. An important implication of these two facts is that the model equation becomes linear and has a solution of the form (Kapur, 1972) (3)
where p ( t ) is the average pellet mass at time t and p ( 0 ) is the average particle mass of the feed at the drum inlet. A description of a continuous pelletizing device can now be written by invoking the additive property of linear systems (Himmelblau and Bischoff, 1968). The additive property implies that the output response of a linear system to a series of inputs is equal to the summation of output responses of the system to individual input signals. Recognizing this important property and assuming that pellets of all sizes are characterized by a single residence time distribution, the average mass of product pellets in an open-circuit pelletizing device at steady state can be written as PCont
=
JmpBatch(t)
E(t)dt
(4)
which is an average of the response of a batch pelletizing device weighted with respect to the distribution of residence times of pellets in the device. E ( t )dt in eq 4 is the exit age distribution of pellet mass, and it represents the mass fraction of pellets in the output that have resided in the device for the time interval t to t + dt. If the transport of the material through the pelletizing device is plug flow type, then it can be represented by a series of perfect mixers, and the residence time distribution E ( t ) can be represented as a y distribution function:
( m - l)!e)x:-(.
dt
(5)
where m is the number of perfect mixers in a series and T is the mean residence time. Substituting eq 3 and 5 into eq 4 yields
pcont
=
(
e-"W1 d t
which for a plug flow type residence time distribution, that is when m is large, becomes
which is a convenient equation for digital simulation of the open-circuit pelletizing device. Incidentally, eq 6, as expected, is identical with eq 1 obtained by Kapur et ai. (1981) by considering the drum as consisting of m mixers in a series and writing a population balance equation for pellets in the jth mixer. Although the method described above gives less information that can be obtained from the Kapur model (Kapur et al., 1981), it provides a simple and elegant method of obtaining the response of a continuous pelletizing device from the batch solution and RTD information. However, it must be remembered that the above described procedure is valid only when the batch kinetic equation is linear and the residence time distribution is not size dependent. Acknowledgment This research was supported by the California Mining and Mineral Resources Research Institute through Grant G1124106 from the Bureau of Mines. Literature Cited Bhrany, V. N.; Johnson, R. T.; Myron, T. L.; Pelczarski, E. A. Dynamics of Pelletization. In Agglomeration; Knepper, W. A., Ed.; Interscience: New York, 1962; pp 229-247. Fuerstenau, D. W. Mechanisms and Kinetics of Green Pellet Growth in Laboratory and Industrial Scale Pelletizing Systems. Preprints, European Symposium, Particle Technology, Amsterdam, June 1980; DECHEMA Frankfurt, 1980; Vol. B, pp 1200-1222. Herbst, J. A,; Grandy, G. A.; Mika, T. S. On the Development and Use of Lumped Parameter Models for Continuous Open- and Closed-Circuit Grinding Systems. Trans. ZMM,Sect. C 1978,20, C193-198. Himmelblau, D. M.; Bischoff, K. B. Process Analysis and Simulation; Wiley; New York, 1968; p 324. Kapur, P. C. Kinetics of Granulation by Non-random Coalescence Mechanism. Chem. Eng. Sci. 1972,27, 1863-1869. Kapur, P. C.; Fuerstenau, D. W. Kinetics of Green Pelletization. Trans. AIME 1964, 229, 348-355. Kapur, P. C.; Fuerstenau, D. W. A Coalescence Model for Granulation. Ind. Eng. Chem. Process Des. Dev. 1969, 8, 56-62. Kapur, P. C.; Sastry, K. V. S.; Fuerstenau, D. W. Mathematical Models of Open-circuit Balling or Granulating Devices. Znd. Eng. Chem. Process Des. Dev. 1981,20, 519-524. Knepper, W. A. Agglomeration; Interscience: New York, 1962. Sastry, K. V. S.; Fuerstenau, D. W. Mechanisms of Agglomerate Growth in Green Pelletization. Powder Technol. 1973, 7,97-105.
* To whom all correspondence should be addressed. 'Current address: Pitman-Moore, Inc., P.O. Box 207, Terre Haute, IN 47802. Let
then
Vikram P. Mehrotra: Douglas W. Fuerstenau* Department of Materials Science and Mineral Engineering, University of California Berkeley, California 94720 Received for review October 4 , 1988 Revised manuscript received February 13, 1989 Accepted July 17, 1989
