I n d . E n g . C h e m . Res. 1989,28, 1735-1740
P = property such as Tb, M ,I , and d S = specific gravity Tb = normal boiling point, "R X = regression parameter defined in eq 2 x = cumulative volume, weight, or mole fraction Y = regression parameter defined in eq 2 Superscript * = dimensionless parameter defined for property P as P * = (P-P0)IPO Subscripts av = average property for the mixture
0 = initial value for any parameter at x = 0 i = hydrocarbon group in the mixture I = refractive index parameter M = molecular weight parameter T = boiling point parameter S = specific gravity parameter
Literature Cited American Petroleum Institute (API). Technical Data BookPetroleum Refining,4th ed.; Daubert, T. E., Danner, R. P., Eds.; American Petroleum Institute: Washington, DC, 1983;Chapters 2 and 4. Austad, T.; Hvidsten, J.; Norvik, H.; Whitson, C. H. Practical Aspects of Characterizing Petroleum Fluids. Presented at the conference on North Sea Condensate Reservoirs and Their Development, London, May 24-25, 1983. Berge, 0. M.Sc. Thesis, Norwegian Institute of Technology, University of Trondheim, Norway, 1981. Brule, M. R.; Kumar, K. H.; Watanasiri, S. Oil Gas J. 1985,Feb 11, 87-93. Haaland, S. M.Sc. Thesis, Norwegian Institute of Technology, University of Trondheim, Norway, 1981.
1735
Hariu, 0. H.; Sage, R. C. Hydrocarbon Process. 1969,143-148. Hoffman, A. E.; Crump, J. S.; Hocott, C. R. Trans. AIME 1953,198, 1-10, Jacoby, R. H.;Koeller, R. C.; Berry, V. J. Trans. AIME 1959,216, 406-41 1. Kesler, M. G.; Lee, B. I. Hydrocarbon Process. 1976,53(3),153-158. Lee, S. T.; Jacoby, R. H.; Chen, W. H.; Culham, W. E. Presented at the SPE 54th Annual Fall Technical Conference and Exhibition, Las Vegas, Sept 23-26, 1979;paper SPE 8398. Pedersen, K. S.; Thomassen, P.; Fredenslund, A. Ind. Eng. Chem. Process Des. Dev. 1984a,23, 163-170. Pedersen, K. S.;Thomassen, P.; Fredenslund, A. Ind. Eng. Chem. Process Des. Dev. 1984b,23,566-573. Pedenen, K. S.;Fredenslund, A.; Christensen, P. L.; Thomassen, P. Chem. Eng. Sci. 1984c,39(6), 1011-1016. Riazi, M. R.; Daubert, T. E. Hydrocarbon Process. 19808, 59(3), 115-116. Riazi, M. R.; Daubert, T. E. Ind. Eng. Chem. Process Des. Deu. 1980b,19, 289-294. Riazi, M. R.; Daubert, T. E. Ind. Eng. Chem. Process Des. Dev. 1986, 25(4), 1009-1015. Riazi, M. R.; Daubert, T. E. Ind. Eng. Chem. Res. 1987,26,755-759. Rodgers, P. A,; Creagh, A. L.; Prauge, M. M.; Prausnitz, J. M. Ind. Eng. Chem. Res. 1987,26,2312-2321. Smith, H. M. Qualitative and Quantitative Aspects of Crude Oil Composition. Bulletin 642,Bureau of Mines, 1968. Standing, M. B.; Katz, D.L. Trans. AIME 1942,146,159-165. Whitson, C. H. J. Pet. Tech. Trans. AIME 1983a, 275 (Aug), 683-694. Whitson, C. H. Dr. Techn. Thesis, Norwegian Institute of Technology, University of Trondheim, Norway, 1983b. Whitson, C. H. SOC.Pet. Eng. J. 1984,277,685-696. Whitson, C. H.; Andersen, T. F.; Soreide, I. Paper presented at the AIChE National Meeting, Miami, FL, Nov 24-27, 1986.
Received f o r review December 5, 1988 Revised manuscript received June 22, 1989 Accepted August 15, 1989
COMMUNICATIONS The Flow Field of Annular Jets at Moderate Reynolds Numbers T h e flow field of single annular jets was studied at Reynolds numbers ranging from 3400 to 21 500. The three annular nozzles used in the study had a square-edged entrance configuration and length-to-diameter ratios of 1, 2, and 4. For axial locations beyond 10 hydraulic diameters, the Tollmien-Reichardt analysis for circular jets provides a good estimate of annular jet behavior. Furthermore, the pattern of an annular jet is independent of either the Reynolds number or the length-to-diameter ratio. As has been recently pointed out by other investigators (Obot et al., 1984, 1986), much of the past work on free jets does not provide an adequate description of practical nozzles used in industry. This present paper concerns itself with the isothermal velocity fields issuing from annuli with sharp-edged inlets. An example of the industrial use of such a jet is the melt blowing process for producing microfibers (Shambaugh, 1988). Many researchers have studied the behavior of free air jets. Taylor et al. (1951), Donald and Singer (1959), Wygnanski and Fiedler (1969),Donaldson et al. (1970, and Beltaos and Rajaratnam (1974) are only a few examples of the numerous publications on the experimental investigation of free jet characteristics. Most experimental work was done with short, convergent discharge nozzles. Only
Boguslawski and Popiel (1979) and Wall et al. (1980) studied the development of free jets issuing from long tubes. Obot et al. (1984,1986),on the other hand, analyzed the flow fields of air jets issuing from straight, sharp-edged entrance orifice plates. In the melt blowing process, the air jets exit from the die head through very small, annular orifices with a sharp-edged entrance configuration. (For some common types of melt blowing dies, an annular configuration is an approximation of the actual die head; see Shambaugh (1988).) No data, however, were found on the characteristics of free jets issuing from this type of aperture. Consequently, an experimental program was started to study the flow fields of free air jets exiting from small, annular apertures with a sharp-edged entrance configuration. Three different orifice plates were used with
0888-5885/89/2628-~735~01.50/0 0 1989 American Chemical Society
1736 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 Metering Valve Dressure Gauge
~
200
.Air ienum Rotameter
Air
n
s- 1
Tube
Pressure Regulotor Traversing Unit
Figure 1. Schematic diagram of experimental apparatus.
0
Radial Posityon r (rnm)
Figure 3. Development of the axial velocity profile for axial positions near the annular orifice; LID = 4 ; Re = 13600. AIR
AIR
1 2
t L/D=1
a
L/D=2
p
I L/D = 4 vo = 202 m/s Re = 13,600
,
m
L/D=4
Figure 2. Plenum chamber and orifice plates.
d 22
length-to-diameter ratios ranging from 1 to 4. For each discharge plate, the Reynolds number was varied from 3400 to 21 500. This Reynolds number range covers operating conditions typical of commercial melt blowing (Shambaugh, 1988). Above a Reynolds number of 1000, a circular jet may be treated as turbulent (Boguslawskiand Popiel, 1979).
Experimental Equipment and Procedures The experimental setup is shown schematically in Figure 1. The equipment consisted of a flow control and metering system, a plenum chamber with a capillary insert, a square-edged orifice plate, and an impact tube mounted on a traversing unit. The rotameter was calibrated against a mass flow meter with *l.O% accuracy. Figure 2 shows how the air was distributed into the cylindrically shaped plenum chamber with an inner diameter of 50.8 mm and a length of 50.8 mm. The top part of the plenum held a capillary tube with an outer diameter of 1.3 mm. This capillary served as the inner wall of the annular air discharge orifice. The interior of the capillary was sealed off (i.e., the capillary was a solid rod). Experiments were done with three different orifice plates (see Figure 2). All three plates had an overall diameter of 50.8 mm and an air distribution section with a diameter of 29.0 mm. The diameter of the bore, which served as the outer wall of the annular air discharge orifice, was 2.37 mm for each plate. The length-to-diameter ratios of the bores were 1.14,2.12, and 3.95. When the capillary was placed in the bore, the corresponding length to hydraulic diameter ratios were 2.51, 4.66, and 8.75, respectively. All three plates had a square-edged entrance configuration. Velocities were measured with a cylindrical impact (pitot) tube. The impact tube had an outer diameter of 0.71 mm, an inner diameter of 0.45 mm, and a conical nose shape with a cone angle of 25 deg. The impact tube was 22.9 mm long and was connected with 1.19 mm inner diameter hypodermic tubing to four Magnehelic pressure
2 4
Figure 4. Development of axial velocity profile at intermediate axial positions; L I D = 4;Re = 13600.
gauges with full-scale readings of 0.001-0.069 MPa gauge pressure. The impact tube was mounted on a traversing unit built from a microscope sliding stage. Vertical steps of 2.54 mm were used during data collection. At each level, measurements were made in four horizontal directions (OO, 90°, BOo, and 270') at horizontal steps of 0.1 mm. The data from the four directions were averaged into a single data set. Measured pressures were converted to velocities by means of the following formula (Chue, 1975):
In this formula, the velocity is expressed in m/s, the pressure in lbf/ft2,and the density in lbm/ft3. Equation 1 is accurate to within 1% up to a Mach number of 0.85.
Experimental Results Profile Development. The axial velocity profiles in the free jet were measured as a function of both axial position ( 2 ) and radial position (r). Over 5000 velocity measurements were made; data were collected for orifice plate L/ D ratios of 1and 4, for Reynolds numbers of 6800 and 13600, and for z / D h values ranging from 0 to 118. Figures 3-5 are representative of the data that were collected. Similar to the work of previous investigators (Taylor et al., 1951; Beltaos and Rajaratnam, 1974; Obot et al., 1984), rl12, the velocity half-width, is defined (for a given axial position) as the radial distance from the jet axis to the position where u = 0 . 5 ~ The ~ . u, is defined as the maximum velocity in the jet for a given axial position. The uo, on the other hand, is defined as the average velocity of the jet in the plane of discharge (z = 0). Finally,
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1737
3
1.2
14'0
= 4 10
80
40
20
100
120
r1/2 = 0.088(
Figure 12. Decay of maximum velocity: comparison of experiment and theory (LID= 1, 2, and 4). 14
' 2 1
I
A
i".-""; o
Re = 13,600
/
c 4 1
/ORe=l A
Re = 13.600
I
I
L/D = 4 0
40
20
60
80
100
120
140
Z/Dh
Figure 13. Growth of the jet half-width comparison of experiment and theory.
study. Again, no effect of either the Reynolds number or the L I D ratio can be observed. This result does not parallel the data of Obot et al. (1984)for circular jets. They found that the longer orifice plate shows a slower decay of the maximum velocity than the shorter orifice plate. No such trend was observed in this study of annular jets, though part of this difference may be due to the fact that Obot used a wider range of L I D values ( L I D = 1and 12) than were used in our study. Figure 12 shows a predicted curve based on eq 7 from the Tollmien-Reichardt analysis. This equation predicts the maximum velocity decay very well in the region of fully developed flow. Also shown on Figure 12 is the following equation for the least-squares best fit to the data: UO _ - 0.09175
um
- - 2.749
(;h
)
for
z > 10
(10)
Dh
If the outer diameter of the annulus is substituted for the hydraulic diameter, Dh, in eq 10, the slope of the maximum velocity decay will change from 0.092 to 0.20. Obot et al. (1984) observed slopes of maximum velocity decay of 0.19 and 0.17 for circular jets issuing from orifice plates with a length to diameter ratio of 1 and 12, respectively. Figure 13 illustrates the growth of the jet velocity half-width with increasing axial position. Figure 13 contains the dimensionless representation of the data of Figure 9 along with two other curves; as stated previously for Figure 9, no effect of Reynolds number or L I D ratio is observed. Again, this conclusion does not parallel the data of Obot et al. (1984) for circular jets. (As was stated previously, part of this difference could be due to the fact that they used a wider range of L I D values.) Their results showed that a jet issuing from an orifice plate with small
Dh
&+
1.758)
for
z > 10
(11)
Dh
Obot et al. (1984) observed the same slope of the growth rate (0.088) for the jet velocity half-width of a circular jet issuing from an orifice plate with a length-to-diameter ratio of 1. For the growth of the entrainment coefficient (see Figure lo), the best fit line through the experimental data in the region of fully developed flow reveals a constant growth rate slope of 0.12. However, if the outer diameter of the annulus is substituted for the hydraulic diameter, Dh, in the best fit equation, the growth rate slope of the entrainment coefficient will change from 0.12 to 0.26. For a circular jet issuing from an orifice plate with an L I D ratio of 1, Obot et al. (1984) observed growth rate slopes of 0.24 and 0.27 for Re = 13000 and 22000, respectively. In conclusion, over the range of experimental conditions covered in this study, the pattern of an annular jet was observed to be independent of both Reynolds number and the length-to-diameter ratio of the orifice plate. The results can be correlated very easily €or computational purposes. The Tollmien-Reichardt analysis for circular jets predicts the behavior of an annular jet very well in the region of fully developed flow.
Acknowledgment The authors thank John Kayser, Mark Porter, and Brian Milum, who assisted in the collection of the extensive data in this paper.
Nomenclature A = discharge area, m2 C, = isobaric heat capacity, J/(kg K) C, = isochoric heat capacity, J/(kg K) D = outer diameter of annular orifice, mm Dh = hydraulic diameter of annular orifice, mm Di = inner diameter of annular orifice, mm g = gravitational acceleration constant, m/sz k = dimensionless gas constant, k = C,/C, K = kinematic momentum, m4/s2 L = length of the annular orifice, mm p o = static pressure, Pa pi = dynamic pressure, Pa Q = volumetric flow rate, m3/s Qo = discharge volumetric flow rate, m3/s Q, = entrained air volumetric flow rate, m3/s r = radial position, mm r1,2 = jet velocity half-width, mm Re = discharge Reynolds number, Re = P'U&h/PO u = velocity, m/s uo = discharge velocity, m/s u, = maximum velocity of the jet at a specified axial position, m/s z = axial position, mm
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 - A ( 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
