1300
Znd. Eng. Chem. Res. 1991,30,1300-1306
Bull. Chem. SOC. Jpn. 198Se,58, 2015-2022. Okamoto, K. I.; Yamamoto, y.; Tanka, H.; Tanaka, M. Kinetics of Heterogeneous Photocatalytic Decompoeiton of Phenol over Anatase TiOz Powder. Bull. Chem. SOC. Jpn. 198Sb, 58, 2023-2028. Pelizzetti, E.; Pramauro, E.; Minero, C.; Serpone, N.; Borgarello, E. Photudegradation of Organic Pollutants in Aquatic Systems Catalyzed by Semiconductors. In Photocatalysis and Environment: Trends and Applications; Schiavello, M., Ed.; NATO AS1 Series C237;Kluwer: Dordrecht, 1987;pp 469-497. Primet, M.; Pichat, P.; Mathieu, M. V. Infrared Study of The Surface of TiOP 11. Acidic and Basic Properties. J. Phys. Chem. 1971,75, 1221-1226. Rabek, J. F. Mechanisms of Photophysical Processes and Photochemical Reactions in Polymers: Theory and Application; Wdey New York, 1987;pp 581-586. Sadana, A.; Katzer, J. R.Catalytic Oxidation of Phenol in Aqueous Solution over Copper Oxide. Ind. Eng. Chem. Fundam. 1984,13, 127-134. Sakata, T.; Kawai, T.; Hashimoto, K. Heterogeneous Photocatalytic Reaction of Organic Acids and Water. New Reaction Paths Besides the Photo-Kolbe Reaction. J. Phys. Chem. 1984, 88,
2344-2350. Sclafani, A.; Palmiearo, L.; Schivaello, M. Influence of the Preparation Methods on TiOz on the Photocatalytic Degradation of Phenol in Aqueous Dispersion. J. Phys. Chem. 1990,94,829-832. Serpone, N.; Au-You, A. K.; Tran, T. P.; Harris, R.AM1 Simulated Sunlight Photoreduction and Elimination of Hg(I1) and CH8Hg(11) Chloride Salts from Aqueous Suspensions of TiOP Solar Energy 1987,39,491-498. Sharifian, H.; Kirk, D. W. Electrochemical Oxidation of Phenol. J. Electrochem. SOC.1986,133,921-924. Sucre, V. S. De; Watkinson, A. P. Anodic Oxidation of Phenol for Waste Water Treatment. Can. J. Chem. Eng. 1981,59,52-59. Sudoh, M.;Nomura, N.; Keide, K. Oxidation Degradation of Aqueous Phenol Effluent by Electrochemical Reaction. Kagaku KOgaku Ronbunshu 1984,IO, 43-48. Takahashi, N.; Kabuki, 0. Oxidative Decomposition of Phenol and Ethylene Glycol by Ozone. Chem. SOC.Jpn. 1976,5,862-868. Yoshido, F.; Minra, Y. Gas Adsorption in Agitated Gas-Liquid Contactors. Ind. Eng. Chem. Des. Dev. 1963,2,263-268. Received for review June 12, 1990 Accepted December 22, 1990
Velocity and Temperature Fields of Annular Jets Biswaroop Majumdar and Robert L. Shambaugh* Department of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
The velocity and temperature fields of single, annular free jets were examined a t Reynolds numbers ranging from 4630 to 10540. The seven different annular nozzles used in this study had squareedged entrance configurations, length-to-diameter ratios of 1-4, annulus inside diameters of 0-1.30 mm, and annulus outside diameters of 1.89-2.46 mm. Nozzle operating temperatures ranged from ambient to 392 OC. For all seven nozzles, the patterns of velocity and temperature distributions are describable by a single set of correlations. The patterns are independent of Reynolds number, length-to-diameter ratio, operating temperature, and annulus inside diameter. Because of the independence from annulus inside diameter, the correlations are very close to correlations that have been developed previously for free jets issuing from simple round holes.
Introduction Heated air jets are widely used in industry to transfer heat, mass, and momentum from the hot jet to another medium. Most of the published literature on heated jets does not address the problem of practical jets with sharp-edged inlets (Obot et al., 1984,1986). A recent paper by Uyttendaele and Shambaugh (1989) first addressed the problem of isothermal discharge of air through sharpedged, annular orifices. This paper expands their work in two respects. First, this paper addresses the problem of nonisothermal discharge through annular orifices. Second, the experiments of this paper cover a much wider range of annular geometries. As a consequence, the results of this paper have very broad applicability. The meltblowing process for producing microfibers (Shambaugh, 1988) is an example of a process that utilizes a heated annular jet. In melt blowing, a molten polymer is injected into a high-velocity gas stream. The force of the gas rapidly attenuates the polymer into fine fibers. Knowledge of the velocity and temperature distributions in a jet field can be used to predict the performance of a melt-blowing process (Uyttendaele and Shambaugh, 1990). For an overall review of past work on free jets, refer to Obot et al. (1984,1986) and Uyttendaele and Shambaugh (1989).
* Author to whom correspondence should be addressed.
Two different die assemblies were used for our experiments. The first die head assembly-die head A-was identical with that investigated at isothermal conditions by Uyttendaele and Shambaugh (1989). This die was also used in melt-blowing tests by Uyttendaele and Shambaugh (1990). For die head A, experiments were run with two orifice plates with length-to-diameter ratios of 1 and 4. The annulus outside diameter (d,,)and inside diameter (di) were constant in these experiments. The second ansembly-die head B-was the same as that described by Kayser and Shambaugh (1990). Die head B was run with five different combinations of annulus do and di values. The actual geometries studied are detailed in the experimental section.
Experimental Equipment and Procedures The first part of our experiments was run on a modification of the apparatus described by Uyttendaele and Shambaugh (1989). Their apparatus (die head A; see Figure 1) was converted to nonisothermal use by the addition of a 4-kW gas heater in the gas feed line, a 500-W band heater on the die head block, and a series of 104-W heat tapes that were wrapped around the gas lines. Fiberglass insulation was placed over the heated lines and the instrument connections on the plenum chamber. For die head A the exit temperature was varied from room temperature to 310 OC, and the exit Reynolds number
0888-588519112630-1300$02.50/0 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1301
t 0
25.4mm
AIR INLET
I
Figure 1. Die head A. The flow field studied was downstream from the small annulus at the center bottom of the figure. See Figure 2 for details on the small annulus.
Figure 3. Die head B.
+
do
4
Figure 2. Details of small annulus on die head A or B.
(based on hydraulic diameter) ranged between 4630 and 10050. Temperature was controlled to within *1 "C of the set point during the experiments. Two square-edged configuration orifice plates with length-bdiameter ratios (L/do)of 1.14 and 3.95 were used with die head A (see Figure 2). The annulus do and di values were kept constant a t 2.37 and 1.30 mm, respectively. Temperatures were measured with a fine ironconstantan thermocouple of the exposed junction type. The junction diameter was 0.7 mm, and the junction end of the therocouple was oriented parallel with the direction of gas discharge from the n o d e (i.e., the thermocouple was pointed up into the nozzle). This arrangement minimized the disturbance of the gas flow. The thermocouple was mounted on a traverse unit built from a microscopic sliding stage. Vertical (z direction) steps of 2.54 mm were used during data collection. At each level, measurements were made in four horizontal directions (O", SOo, 180°, 270") at horizontal (r direction) steps of 0.1 mm. The pressures were measured and converted to velocities in the same way as described in detail in Uyttendaele and Shambaugh (1989). The second die head configuration (die head B; see Figure 3) corresponded to the annular die assembly described by Kayser and Shambaugh (1990). The gas heating and feed apparatus was the same as that used for die head A. Besides the geometrical differences between die head A and die head B, die head B was used with a small 350-W, 50.&mm (inside diameter) band heater that enveloped the bottom section of the die head. For die head B the experiments were run with five different hydraulic diameters ( d h ) ranging from 0.617 to 2.456 mm; the L / d o ratios
Table I. Dimensions of Die Head Annuli (See Figure 2) die head confign do, mm di, mm dh, mm L i d , Lidh A A-1 2.37 1.30 1.07 1.14 2.51 A-2 1.30 2.37 1.07 3.95 8.75 B B-1 0.00 2.46 1.29 2.46 2.46 B-2 2.46 0.83 1.63 1.29 1.95 B-3 2.46 1.27 1.19 1.29 2.67 B-4 1.89 0.83 1.06 1.68 3.00 1.89 B-5 1.27 0.62 1.68 5.12
ranged from 1.3 to 1.7. The exit temperatures ranged from room temperature to 392 "C, and the exit Reynolds number was varied from 5460 to 10 540. The jet temperatures and pressures were measured in the same manner as described above for die head A. A summary of the various geometries studied in our experiments is listed in Table I. The primary objectives of running the experiments on die head B were to examine the effects of variable annulus do and di (and hence dh)on the velocity and temperature profiles and identify the dimensionless groups needed to correlate these parameters. Comparisons of the correlations developed for the two die heads were made. Results and Discussion The axial temperature and velocity profiles of the nonisothermal jets were measured both as a function of the axial position ( 2 ) and the radial position (r) for die heads A and B. Over 8000 individual velocity and temperature measurements were taken for the two die heads. The geometries tested are listed in Table I. An excess temperature Ojo was defined as the difference between the jet temperature at the nozzle and the ambient temperature (23 f 1 "C) in the laboratory. The excess temperatures tested were 80, 151, 155,162,207, 218,227,241, 252, and 287 "C for configurations A-1 and A-2. For configurations B-1 through B-5, the excess temperatures tested were 163, 165, 166, 168, 170, 182, 209, 255, 259, 264, 266, 332, 348, and 369 "C. Axial Profile Development. Some of the typical centerline temperature and velocity decay profiles for die head A are shown in Figures 4 and 5, respectively. No significant differences were noted in temperature and velocity profiles for the two nozzles selected for our study
1302 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
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Z(d3 Figure 6. Centerline temperature decay aa a function of Z(d,) for die head A.
with L / d o ratios of 1 and 4. Figures 6 and 7 show the centerline velocity and temperature data plotted against Z(do),a nondimensional axial distance. By incorporating the density difference between the heated jet and the room temperature air, the Z(do)permits the generalization of previous work on isothermal jets (Uyttendaele and Shambaugh, 1989) to nonbothermal jets. The ambient-to-jet density ratio ( p - / p o ) varied from 1.00 to 2.25 in our tests. The profiles of Figures 6 and 7 clearly indicate that the region of fully developed flow starts a t an axial position of about 4 orifice diameters (approximately 10 hydraulic diameters), which agrees with the observations made by Uyttendaele and Shambaugh (1989) for the case of iso-
Figure 8. Centerline temperature decay for die head B.
thermal annular jets. Uyttendaele and Shambaugh correlated their data in terms of the hydraulic diameter (dh). Nevertheless, their data would correlate as well with do, because in their analysis both do and dhwere held constant. It will be explained below why do is preferable to dh. Figures 6 and 7 illustrate that both the centerline temperature and velocity decay profiles are independent of the outlet temperature, Reynolds number, and the L / d o ratio of the orifice plate. For circular jets, Obot et al. (1984, 1986) found that both maximum temperature and velocity decay more slowly for a longer orifice plate than for a shorter orifice plate. No such trend was observed in our analysis of annularjets, though part of this difference may be attributed to the fact that Obot used a wider range of L / d o values ( L / d o= 1 and 12). For the temperature and velocity decays, the following least-squares best fits to the data are given on Figures 6 and 7 for die head A: ejo/eo = o.239z(d0) + 0.421 (1) u ~ , / u , = 0.2242(d0) + 0.693
(2)
These correlations provide a predictive scheme for the centerline temperature and velocity profiles for annular dies. Figures 8 and 9 show the temperature and velocity data plotted against Z(do)for die head B. As with die head A, the centerline temperature and velocity decay profiles are independent of the exit temperature and Reynolds number. In addition, the experiments with die head B show that the profiles are independent of the hydraulic diameter when do is constant. The generalized correlations devel-
Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 1303 0
Re 7,140 8,090 9,890 9,970
0258 A 165 0348 0 168
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5,620 5.620 5.820 5.820 5,820 5.820 5,820 5.620 5.820 5.820
Figure 11. Development of temperature profile at various axial positions (for die head A).
Figure 9. Centerline velocity decay for die head B.
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.
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CORREUTION FOR DIE HEAO A (1986)
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-
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1986)
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Figure 10. Figure 7 replotted with Z(dh) on the abscissa.
oped in the case of die head B for temperature and velocity decays are given by the following equations:
ejo/eo = 0.2022(d0)+ 0.480
+
ujo/u0 = 0.2322(d0) 0.673
Figure 12. Nondmensional radial temperature profilea for die head A. 1.o
(3) (4)
A comparison of (3) and (4)with (1)and (2)shows that the correlations developed for die head B are in close agreement with the correlations developed for die head A (see Figures 8 and 9). Obot et al. (1986)used the same parameters (djo/Oo, u. /uo, and Z(dJ) to correlate their data, and they observed &pes of 0.260 and 0.204 for temperature and velocity decays, respectively, for circular jets emerging from a nozzle with L / d o = 1. Figures 8 and 9 include plots that compare die head A, die head B, and Obot et al.'s correlations. Although Obot et al.'s temperature decay correlation agrees fairly well with our results, their velocity decay correlation predicts somewhat lower decay than our experimental observations. Part of the difference may be attributed to the widely different geometries used in the two studies. Figure 10 is similar to Figure 8 except that the parameter Z(dh)is used in the abscissa. Figure 10 shows that a steady increase in the data slope occurs with increasing dh. But, as Figure 8 shows, use of Z(do)eliminates this variability. Hence, as was aforementioned, Z(do)was selected as the correlating parameter. For the data of Figure 10,the inside diameter varied from zero to nearly 70% of the outside diameter. An inside diameter of zero corresponds to the limiting case of the annular jet becoming a circular jet.
0.8
'
(Obot et 01..
241 241 241 241
9.570 9,570 9,570 9,570
2.1 3.2 6.4 9.7
1986)
4 r-/tl/2
Figure 13. Similar to Figure 12, except data taken at higher excess temperatures and variable L/d,.
Radial Profile Development. Extensive measurementa of velocity and temperature distributions were made 180°,270O) at each of the in the radial directions ( O O , No, axial positions investigated. Typical profiles for die head A are shown in Figure 11. For the same die head, the data is represented in dimensionless form in Figures 12-14. In Figures 12 and 13 the abscissa is the radial position normalized with tl12!the temperature half-width. In Figure 14 the radial position is normalized with r1/2,the velocity half-width. Such dimensionless representations of the temperature and velocity profiles in the region of fully
1304 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 -
'[
Re
0.6
5.840 5.840 5.840 7,120 7,120 7,120 9,770 9,770 9,770
-
L/d.
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b
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0
2.4
5
1
10
15
20
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25
z/d0 Figure 17. Growth of temperature half-width with increasing axial position for die head A.
12.7 35.6 7.6 20.3 10.2 23.4
0
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8,090 9,050 9,050
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'
'
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'
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'
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Figure 16. Nondimensional radial velocity profiles for die head B.
developed flow are independent of not only the axial position but also the hynolds number, outlet temperature, and L/doratio. Also on Figures 12-14 are the experimental correlations used by Obot et al. (1986) to fit data from circular jets. These correlations also work well for our annular jets. Even near the jet dkcharge, the temperature correlation fits fairly well, through near the discharge the velocity correlation is in error. Obot et al.'s (1986) data also show that, for near-nozzle conditions, the temperature correlationgives better results than the velocity correlation. Figures 15 and 16 show the dimensionless radial temperature and velocity profiles for die head B. Only representative data for the four different geometries studied
5
I
10
15
20
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z/do Figure 18. Growth of temperature half-widthwith increasing axial position for die head B.
have been included in these figures. For die head B the normalized temperature and velocity profiles also follow the Gaussian correlation used by Obot et al. (1986). Besides showing no dependence on any of the parameters mentioned above for die head A, the dimensionless representations of Figures 15 and 16 are independent of both do and dh used in the experiments on die head B. Thus, the numerical constant (0.693 = In 2) cited in the correlating equations does not depend on the geometrical variations employed in our experiments with die head B. Spreading and Entrainment Characteristics. Figure 17 represents the variation of the temperature half-width ( t l 2) with increasing axial position for die head A. The hdf-width data were obtained by interpolation between measured poinb along the radial profiles of temperatures of the axisymmetric jet. No dependence wm found on the exit temperature, Reynolds number or the L / d o ratio. As suggested by the isothermal velocity measurements of Uyttendaele and Shambaugh (19891, tlI2/dowas assumed to vary linearly with %/dofor %/do> 4, the region of fully developed flow. A least-squares best fit to the data yields tlp/dO = 0.113z/d0 + 0.202 (5) A similar representation of the half-width data is given in Figure 18 for die head B for the range of conditions and
geometries used in our experiments. For this die head the following equation gives the least-squares best fit to the data: tlp/dO = 0.109z/do + 0.155 (6)
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1305
252
4.740 8
6,800 6,800 9.570
4
241
I
;/do( Figure 19. Growth of velocity half-width with increasing axial position for die heads A and B.
Also shown on Figure 18 are plots of the correlation for die head A (from Figure 17) and Obot et al.'s (1986) correlation for a circular nozzle with L/do = 1. As can be seen, the three correlations on Figure 18 all fit the data quite well. Figure 19 shows the variation of velocity half-width with increasing axial position. Data from both die heads A and B are included in this figure. The least-squares best fit to the data is rl12/do = 0.112z/d0 + 0.040 (7) Consistent with the earlier observations for tl/z, rljZwas found to be independent of exit temperature, Reynolds number, L/do ratio, and hydraulic diameter. Also shown on Figure 19 is Obot et al.'s (1986) correlation for a circular nozzle with L/do = 1. Obot et al. observed that nearly all investigators have found that tllz is somewhat larger than rl12; this is also true for our data. At any axial location z from the nozzle, the mass flow rate is given by the following integral over the axial component of the mean velocity: M ( z ) = l p ( r , z ) u(r,z)2?rr dr
(8)
The mass flow rate of entrained flow is given by
M,(z) = M ( z ) - M,
(9)
As suggested previously by Obot et al. (1986) and Uyttendaele and Shambaugh (1989), eq 8 was evaluated numerically by use of the trapezoidal rule. As also suggested by these previous researchers, the boundary points r = 0 and r = ro.l were used. These boundary points define the distance from the jet axis to the position where the velocity of the jet is equal to 1/10 of the centerline value in the same axial plane. This boundary definition minimizes the effect of unreliable data near the jet boundaries and provides a consistent basis for comparison of results that are generated with different nozzle configurations. An entrainment coefficient q is defined as 0 = M,(Z)/M, = M ( Z ) / M , - 1 (10)
Figure 20 is a plot of \k versus zld, (pm/p)1/2for a selection of experimental conditions. No significant effect of Reynolds number, Lld,, or B., on the entrainment coefficient was observed. The best ht line through the experimental data gives a growth rate slope of 0.237. This is in good agreement with the slope of 0.262 which was determined by Uyttendaele and Shambaugh (1989) in their study of isothermal jets with nozzles identical with those used in
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d0
)"2
Figure 20. Entrainment coefficient as a function of axial distance for die head A. 2.0
1.5
e P
0
0
51.0
0
0 0
'
b
A
I
0.5 0.0
I I
I
2
4
I
6
8
10
__. 12
14
Z/dO
Figure 21. Variation of jet heat content with axial distance for die head A.
this study. The results of Obot et al. (1986) can also be favorably compared to our data. When their mass entrainment data is normalized by dividing through by the corresponding values of M,,the slope is 0.27 for a circular jet issuing from an orifice plate with L/do = l,e, = 96 "C, and Re = 13000. Heat Content of the Jets. The heat content of the annular jet for any downstream location is given by
H ( z ) = l C , p ( r , z ) u(r,z) B(r,z)2nr dr
(11)
This heat content H ( z ) is normalized by dividing through by H,,the exit heat content of the jet. The H, is defined by H , = Moc,e, (12) Similar to the calculation of q, the integration limits of
H ( z ) were established as r = 0 and r = ro.l. Figure 21 includes results calculated from a range of experimental conditions. In addition, the data of Sforza and Mons (1978) and Obot et al. (1986) are included on Figure 21. Even though these two groups of investigatorsused circular orifices, not annuli, their data falls in the same range as our data. (The data of Obot et al. were produced with a sharp-edged nozzle with L/do = 1 and exit temperature Bio = 96 "C.)The trends observed in Figure 21 are consistent with the criteria for the constancy of heat flow at consecutive cross sections (Abramovich, 1963). In order to incorporate the influence of localized heatand/or mass-transfer characteristics, a fixed area was
1306 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
Nomenclature 1
A
50 c
-40
cn
i
A
1
1 1
4.740 7,120 7.120
252 162 162
1.0 0.5
1.0
0.422 0.211 0.422
A
tI
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= I
A
A
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A
A
m
A
D
0
. I
I
4
Figure 22. Jet heat content for r 5 1.0-mm region for die head A with 0, = 162 and 252 "C.
considered in the evaluation of the heat content in the jet. The results of such computations for die head A are presented in Figure 22. As expected, there is a fall in the heat content of the jet with increasing axial distance due to the broadening of the temperature and velocity profiles of the jet. Consistent with our previous observations, no appreciable differences were observed between the two geometries considered. Conclusions From our experimental investigation of die head A and die head B, the following conclusions can be drawn: 1. The dimensionless centerline temperature and velocity profiles are independent of the exit temperature, Reynolds number, and length-to-diameter ratio of the orifice plate. In addition, the results with die head B show that annulus di has no effect. For all practical purposes, in the region of fully developed flow an annularjet can thus be treated as a jet emerging from a circular orifice plate. 2. For the region of fully developed flow, the dimensionless radial temperature and velocity profiles show no dependence on exit temperature, Reynolds number, and L / d o ratio. Furthermore, these profiles are also independent of annulus di. 3. Both the temperature and velocity half-widths show a linear increase with axial distance in the region of fully developed flow. These half-widths are also independent of the exit temperature, Reynolds number, L/doratio, and annulus di. The slopes of the correlating equations for the half-widths agree quite closely with those predicted by other investigators. The entrainment coefficient also agrees well with values found in the literature for similar nonisothermal jets. 4. Because of the wide diversity of geometries used in our studies, the correlations developed herein should have broad applicability. Acknowledgment We thank the following companies for their financial support: 3M, Dow Chemical U.S.A., Fina Oil and Chemical, Johnson & Johnson, Kuraray Limited, and Phillips Petroleum.
C, = specific heat, J kg-' K-' do = outer diameter of annular orifice, mm di = inner diameter of annular orifice, mm d h = hydraulic diameter of the ann& orifice = (do- di),mm H = heat content as defined in eq 11, W L = length of annular orifice, mm M = mass flow rate as defined in eq 8, kg/s Mo = discharge mass flow rate, kg/s Me = entrained air mass flow rate as defined in eq 9, kg/s r = radial distance from the jet axis, mm rlI2 = jet velocity half-width, mm r0,' = distance from jet axis to location where v = 0 . 1 ~ ~ Re = discharge Reynolds number, R e = P'vjodh/Pb t l j 2 = jet temperature half-width, mm u = velocity, m/s ujo = discharge velocity, m/s u, = maximum velocity of the jet at any specified axial pos-
ition, m/s axial position, mm a d o ) = z/d0(p,/po)'/2 Z(dh) = z/dh(pm/Po)'/2 z =
Greek Symbols 0 = excess air temperature above ambient, "C Ojo = excess temperature above ambient at nozzle exit, O C Bo = excess temperature above ambient along the centerline, "C pjo = air dynamic viscosity at discharge conditions, Pa s po = air density along the center line downstream from the
nozzle, kg/m3 air density at ambient conditions, kg/m3 p = mean value of air density at an axial position calculated by averaging the density values between r = 0 and r = ro.l, kg/m3 p' = air density at discharge conditions, kg/m3 = entrainment coefficient pm =
*
Literature Cited Abramovich, G. N. The Theory of Turbulent Jets; MIT Press: Cambridge, MA, 1963. Kayser, J. C.; Shambaugh, R. L. The Manufacture of Continuous Polymeric Filaments by the Melt Blowing Process. Polym. Eng. S C ~1990,30, . 1237-1251. 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, 581. Obot, N. T.; Graska, M. L.; Gandhi, F. Velocity and Temperature Fields in Turbulent Jets Issuing from Sharp-Edged Inlet Round Nozzles. Ind. Eng. Chem. Fuidam. 1986, -!E,4i5. Sforza, P. M.; Mons, R. F. Mass, Momentum, and Energy Transport in Turbulent Free Jets. Znt. J . Heat Mass Transfer 1978, 21, 371-384. Shambaugh, R. L. A MacroscopicView of the Melt-Blowing Process for Producing Microfibrs. Znd. Eng. Chem. Res. 1988,27,2363. Uyttendaele, M. A. J.; Shambaugh, R. L. The Flow Field of Annular Jets at Moderate Reynolds Numbers. Ind. Eng. Chem. Res. 1989, 28, 1735. Uyttendaele, M.A. J.; Shambaugh, R. L. Melt Blowing: General Equation Development and Experimental Verification. AZChE J . 1990, 36 (No. 2), 175.
Received for review June 21, 1990 Revised manuscript received January 9, 1991 Accepted January 21, 1991
