976
I n d . Eng. Chem. Res. 1993,32,976-980
Three-Dimensional Flow Field of a Rectangular Array of Practical Air Jets Abdeally Mohammed and Robert L. Shambaugh' Department of Chemical Engineering and Materials Science, The University of Oklahoma, Norman, Oklahoma 73019
The velocity field was measured below a rectangular array of 165air nozzles. The array was arranged in a rectangular pattern with three side-by-side columns. All nozzles were identical. Each nozzle had a sharp-edged inlet, and each nozzle discharged air through an area that approximated an annulus. Based on momentum conservation considerations, a single equation was developed that fits the data well and can be used t o predict the velocity at any position below the nozzle array. At large distances from the array, the velocity field can be approximated as the field of a two-dimensional jet.
Introduction A number of recent investigators have examined the flow fields of practical nozzles used in industry. Obot et al. (1984, 1986) examined the velocity and temperature distributions in the fields produced by air exiting from straight, sharp-edgedentrance orificeplates. Uyttendaele and Shambaugh (1989) and Majumdar and Shambaugh (1991)examined the velocityand temperature fields below sharp-edged annular nozzles. These past investigators considered only single nozzles. This present paper concerns the velocity field below an array of practical nozzles. A common industrial use of such an array is the production of fine polymeric fibers by the process known as melt blowing (Shambaugh, 1988). Some past work has been done on air fields produced by a linear array of rectangular nozzles. Corrsin (1944) investigated the velocity field downstream from a twodimensional grid formed by parallel rods. Laurence and Benninghoff (1957) examined the noise-suppression characteristics of a row of rectangular nozzles. Miller and Comings (1960) examined, both theoretically and experimentally, the convergence of unventilated dualjets. Later work by Tanaka (1970,1974) also concerned unventilated dual jets. Marsters (1979) examined the flow fields of linear arrays of ventilated rectangular jets (both threeand five-nozzle arrays were tested). Krothapalli et al. (1980) studied a row of five ventilated rectangular jets; thrust augmentation was the goal of this research. The same apparatus was used for further studies by McLachlan and Krothapalli (1984). Ventilated and unventilated dual jets were recently studied by both Lund et al. (1986) and Lin and Shieu (1991). Ng and Grandmaison (1989)studied a row of three parallel, rectangular jets where the center jet had about twice the mass throughput of either outer jet. The flow field below a row of round holes was examined by Knystautas (1964) and Pani and Dash (1983). (Pani and Dash also examined the flow field below a row of rectangular nozzles.) Simonich (1986) studied the flow field below a pair of round holes; noise reduction was the goal of this work. Ragunathan and Reid (1981) examined the velocity profile, thrust, and noise reduction character of flow fields produced by one, five, seven, and nine round holes placed in a circular pattern.
* Author to whom all correspondence should be addressed. 0888-5885/9312632-0976$04.O0/0
No work has been reported on rectangular arrays (rows and columns) of jets. Also, no work has been reported on multiple jets produced with annular or near-annular nozzles.
Experimental Equipment and Procedures The experimental setup is shown schematicallyin Figure 1. Air at 480 kPa (70 psig) was fed through a pressure regulator and rotameter. Gasvelocitiesbelow the die were measured with a cylindrical impact (pitot) tube. Details of this pitot tube assembly are given by Uyttendaele and Shambaugh (1989). The impact tube was positioned with a Velmex 3-D traverse system. The Velmex unit permitted x , y, and z motions in 0.01-mm increments. The Schwarz die (jet array) is an actual commercial die manufactured by Biax-Fiberfilm Corporation (Neenah, WI). This die is suitable for small-scale production of melt-blown fibers. The die has the hole arrangement shown in Figures 2 and 3. From each of the 165 nozzles, air exits through a zone that is approximately annular. The inside of each zone is bounded by a0.635-mm (outside diameter) metal capillary; the outside of each zone is bounded by the edges of a stack of thin metal plates. The configuration of the plates is shown in Table I. To reduce the amount of data collectionand to minimize die end effects, measurements were primarily taken below a center section of the die. Specifically, measurements were concentrated in the zone below the nine holes in rows 26, 27, and 28 (see Figure 2). The origin of the coordinate system was placed at the center hole in row 27. The x and y directions were as shown in Figure 2; the z direction was positive downward from the die face (see Figure 1). For spatial positions near the die face (0 Iz I12.7 mm), velocities were measured over a fine grid at selected z positions. An average of 762 measurements were made at z = 0.0, 1.27, 2.54, 5.08, 7.62, and 12.7 mm (4574 total measurements). For z > 12.7 mm, fewer velocity measurements needed to be made because (a) the changes in measurements were much more gradual and (b) the effects of the individual orifices were no longer apparent. For all runs the air flow rate to the 165-hole die was maintained at 8.27 X m3/s (at standard conditions of 21 "C and 1-atm pressure). This converts to an air loading of 3.62 (g/min)/hole, which is in the range of typical commercial melt-blowing operations. Under these loading conditions, the average maximum discharge velocity (at z = 0) of the nine holes was uj, = 190.8 m/s. On the basis of the hydraulic diameter of the annular holes (at layer 5; 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 977 ,-3.?jrn~
Figure
larmar m m l 1. The experimental apparatus.
41 nn
IeAVrnlWG ulll
-
IIM
Figure 3. Hole geometry and spacing in the Schwarz die.
Table 1. Arrangement of Metal Plates at t h e Fa- of the Sehwarz Die. layer 1 2
3 4 5
layer thickness mm (in.)
hole type
0.254(0.010) 0.254 (0.010) 0.254 (0.010) 0.381 (0.015) 0.381 (0.015)
triangular triangular triangular round round
hole diameter or length of side of triangle mm (in.) 1.100 (0.043) 1.100 (0.043) 1.100 (0.043) 1.400 (0.055) 1.016 (0.040)
a The major axes of the capillaries are perpendicular to the face of the stack of metal plates. The 0.635-mm (outside diameter) by 0.330-mm (inside diameter) cnpillaries are recessed 0.254 mm (0.010 in.) below the surface of layer 5. Layer 5 is on the outside (discharge) face of die.
Figure 2. View of the face of the Schwarz die. The die contains 165 holes in three columns.
see Table I), the discharge Reynolds number was 4,800. For all runs, the air temperature was maintained a t ambient (21 "C). Results and Discussion Experimental Results for Positions Near the Die. At z = 0, 1.27, 2.54, 5.08, 7.62, and 12.7 mm, velocity measurements were made below the die as a function of
x and y (for z > 12.7 mm, measurementa were only made as a function of x). Figures 4 and 5 are examples of these
measurements; these figures show the progressive development of the velocity profiles below the die. Figure 4, the profile at z = 1.27 mm, shows that there is very little jet interaction at this level. Observe that not all holes are equal in terms of the velocity profile. Small irregularities
Figure 4. Experimental velocity profile a t z = 1.21 mm; uj. = 190.8 mls and uo = 132.7 mis (these values for up and uo are averages of the data from the nine holes investigated).
in the machining and assembly of the die cause these holeto-hole differences. Figure 5 shows that, at z = 7.62 mm, substantial merging of the jets has occurred. The profile in Figure 5 looks much like the profile from a two-dimensional jet (more about this will be discussed below). Model Equation for the Velocity Field. To best interpret the data of the type shown in Figures 4 and 5, it is desirable to fit the data to an equation or equations. Furthermore, an equation or equations would allow rapid predictions of die behavior both within and beyond the range of our experiments. Preferably, this equation or equations should have some theoretical basis. On the basis of previous work by
978 Ind. Eng. Chem. Res., Vol. 32,No.5,1993
Y
4
Figure 5. Experimental velocity profile at z = 7.62 mm; uj, = 190.8 mi8 and vu = 47.2 mia.
Reichardt (1941),BaronandAlexander (1951)theoretically examined the fluxes of momentum, heat, and mass in free jets. Since the applicable differential equations are linear in the various fluxes, Baron and Alexander suggested that the mathematical solutions for the distributions from a number of point sources are superimposable. Later, Knystautas (1964) used this summation technique to model the discharge from a series of holes in a line. For a single axisymmetric jet, Reichardt (1941)developed the following solution:
where u and p are, respectively, the axial velocity and density at a position below the orifice, uj. and p j , are, respectively, the velocity and density at the discharge of the orifice, r and z are cylindrical coordinates, A, is the cross-sectional area of the orifice, and C, is the spreading coefficient for momentum transfer. Figure 6 shows the Schwarz die face with a cylindricalcoordinate system overlaid on the Cartesian system. If each hole is considered independently, then the velocity distribution due to each hole can be described by eq 1 with a small modification: r is replaced by a, where a is the distance between the hole of interest and the coordinate position of interest. For the center hole, a is simply r. In general, a is defined by the relation a2 = r2
+ (ns)' + (mh)'-
~((RP)'
+
(mh)2)1'2rcos[ y - tan-'
$1
Figure6. Cylindrical-mordinatq&.em and hole-numberingsystem used in developing the model equations for the Schwarz die; s = 2.15 mm and h = 3.25 mm.
All velocity measurementstaken helow the Schwarz die with 1.27 mm 5 z 5 12.7 mm (3952measurements) were fit to eq 3. The best values for the two constants, C, and A,, were determined to he 0.3383 and 0.2781 mm2, respectively, by a least-squares best fit technique. The coefficient of determination, R2, for the fit of the data to the equation is 0.5546. It was suspected that this poor fit was because the model equation inadequately predicted the pulling in of the maximum velocities of the outer columns of holes toward the center column. By examining contour plots of our velocity distributions, it was determined that the maximums of the outer holes appeared to linearly shift toward the center row of holes with complete overlap occurring at about z = 12.9 mm. (In his examination of the interference of two slot jets, Tanaka (1970)used segments of arcs to define the movement of the jet maximums toward each other.) To account for the shifting of the maximums, a new hole-to-hole spacing h' was defined as
+
(2)
where s and h are hole-to-hole spacings and n and mare, respectively, the row and column positions of the hole. Superimposing solutions of the form of eq 1 for the entire 165 holes in the Schwarz die gives (see Baron and Alexander, 1951)
In his modeling of a series of holes in a line, Knystautas (1964)developed a one-dimensional form of the above equation.
h' = -m,z h (4) for m q 5 h. For m q > h, h' = 0. With h' used instead of h, then eq 3 had three parameters to fit: C, A,, and ma. The best-fit values of the three parameters to the data are C, = 0.2707,A, = 0.2277 mm2, and m. = 0.2545. The R2for this fit is 0.7588. This is a betterR2than the 0.5546 value calculated for the two-parameter model. However, an even higher R2 is preferable. In fact, the threeparameter model can be shown to fitthe data even better if one recognizes that the R2 measures both the fit of the model to the data and the hole-to-hole variances in the die. The three-dimensional velocity data collected for the Schwarz die has 12-fold symmetry; see Figure 3. The data
Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993 979 I
(mm) v. ( m / 4 o 12.7 41.3 0 19.05 39.3 A 25.4 34.7 o 38.1 26.9 + 50.8 25.6 x 63.5 22.9
1 .o
0.8
-12
I
I
-4
4
I
12
I 20
x (mm)
I
0.4
0.8
1.2
1.6
I
2.0
X/X1/2
Figure 7. Experimental velocity profiles for L 1 12.7 mm.
Figure 8. Axial velocity profiles: comparison of experiment with eq 3 and with the exponential decay of a two-dimensional jet.
from these 12 zones were averaged to produce velocity data where the hole-to-hole variations were minimized. The three-parameter model (eq 3) was fit to these averaged data; the best-fit values for the parameters are C, = 0.2733, A, = 0.2019 mm2, and m, = 0.2538. Quantitatively, the R2for the fit of the model to the averaged data is 0.8656. This R2is a more legitimate estimate of the fit of the model. Velocity Fields at Distances Further from the Die. For z 1 12.7 mm, the velocity profiles look much like the expected profiles from a two-dimensional jet. Figure 7 shows the experimental velocity profiles for z = 12.7 mm to z = 127.0 mm. The velocity profiles show substantial flattening and broadening at large z. As described by Schlichting (1979), the decay of centerline velocity of a two-dimensional jet can be described by the relation
u, =
-3lI2 (I 2 Ka2
112
(5)
where K is the kinematic momentum and Q is an empirical constant. Since K = Auj, at the jet discharge, then where c1 is an empirical constant. On the basis of the centerline velocity measurements for z > 50 mm, the bestfit value for c1 is 0.9929 mm1I2. Equation 3 should be used for small z-where, in fact, there are usually multiple peaks rather than a single peak (actually, a ridge)-while eq 6 is suitable for large z. For intermediate z (for approximately 15 mm < z < 50 mm) a weighted average of the estimated uovalues from both equations gives an adequate estimate of the true u,. For z > 12.7 mm, eq 3 does a poor job of predicting the jet half-width x1/2. For this region, the data is well-fit by the simple equation = CZZ (7) where c2 is a constant; this linear equation fits the data quite well for all z 1 12.7 mm (i.e., the linear assumption is good even for intermediate z ) . The best-fit value for c2 is 0.1412, and the r2 for the fit of the data is 0.9369. (As described by Schlichting (1979),x 1 p is linearly related to z in a two-dimensional jet.) Figure 8 shows the dimensionless velocityprofile plotted as a function of dimensionless width. In this format, the profiles are essentially identical for z > 12.7 mm. The exponential equation shown on the figure (the solid line) has been shown to fit data for slot jets (Rajaratnam, 19761, single-holejets (Obot et al., 1984,19861, and annular jets (Uyttendaele and Shambaugh, 1989; Majumdar and Shambaugh, 1991). Also shown on the figure is the XljZ
Figure 9. Entrainment coefficient as a function of axial position.
prediction of eq 3; this eq 3 prediction is indistinguishable from the prediction of the exponential equation. Entrainment. At any axial location z below the die, the mass flow rate per length L of the die is given by the following integral:
The mass flow of entrained air is given by
M M(z) M, --e @ )--_L L L
(9)
Similar to the procedure used by previous researchers (Obot et al., 1986;Uyttendaele and Shambaugh, 19891,eq 9 was evaluated numerically by use of the trapezoidal rule. The length L used was 35 (6.45 mm). Instead of x = --m and z = +m, the boundary planes x = -zo.1 and x = x0.1 were used. These boundary planes define where the velocity equals 1/10 of the centerline ( x = 0) velocity. Similar to the work of previous researchers (Obot et al., 1986; Uyttendaele and Shambaugh, 1989), and entrainment coefficient \E is defined as
Figure 9 is a plot of \E versus z for our experimental data. If 2h = 6.5 mm is considered to be the characteristic dimension of the jet array, then Q reaches a value of about 4 at z/2h = 10. Axisymmetric annular jets (Majumdar and Shambaugh, 1991)have similar levels of entrainment.
980 Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993
Conclusions and Recommendations 1. Equations 3, 6, and 7 and Figure 8 can be used to predict the velocity field of an actual three-column array of practical jets (i-e.,the field produced by the Schwarz die). 2. For any three-column array of jets (an array other than the Schwarz die used in our experiments), equations of the form of eqs 3,6, and 7 can be used to correlate the experimental data. Relatively few velocity measurements need to be made in order to determine the appropriate constants (Ao,m,,and Cm). 3. By extrapolation, the form of these equations can also be used to predict and correlate the velocity field of an array of other than three columns. Acknowledgment We thank the following companies for their financial support: 3M, Dow Chemical U.S.A.,Conoco/DuPont, Fina Oil and Chemical, Kuraray Limited, and Phillips Petroleum. The Schwarz melt blowing die was donated by BiaxFiberfilm Corporation, Neenah, WI. Nomenclature a = coordinate parameter defined by eq 2, mm A, = discharge area of a single orifice, mm2 c1 = constant in eq 6,mm1/2 c2 = constant in eq 7 C, = spreading coefficient for momentum transfer h = distance between adjacent holes in the x direction, mm h’ = modified distance between adjacent holes in x direction, mm K = kinematic momentum, m4/s2 L = a length of the die in the y direction, mm m = column position in jet array (see Figure 6) m , = slope of jet shift M = mass flow rate as defined in eq 8, kgls Me = entrained air mass flow rate as defined in eq 9,kg/s Mo = discharge mass flow rate, kg/s n = row position in jet array (see Figure 6) r = cylindrical coordinate, mm s = distance between adjacent holes in y direction, mm u = velocity, mls ujo = discharge velocity, mls u, = maximum velocity of the jet at any specified axial position, rnls x = Cartesian coordinate, mm x l i 2 = jet velocity half-width, mm ~ 0 =.distance ~ from jet axis to location where u = O.lu,, mm y = Cartesian coordinate, mm z = axial coordinate, mm Greek Symbols y = cylindrical coordinate, rad p
= air density at any location, kg/m3
air density at discharge conditions, kg/m3 empirical constant in eq 5 i& = entrainment coefficient
pjo =
u=
Literature Cited Baron, T.; Alexander, L. G. Momentum, Mass, and Heat Transfer in Free Jets. Chem. Eng. Prog. 1961,47(4),181. Corrsin, S.Investigation of the Behavior of Parallel Two-Dimensional November 1944. Air Jets. NACA W-90, Knystautas, R. The Turbulent Jet from a Series of Holes in Line. Aeronaut. Q. 1964,15,1. Krothapalli, A.; Baganoff, D.; Karamcheti, K. Development and Structure of a Rectangular Jet in a Multiple Jet Configuration. AIAA J . 1980,18 (8),945. Laurence, J. C.; Benninghoff, J. M. Turbulence Measurements in Multiple Interfering Air Jets. NACA T N 4029,December 1957. Lin, Y. F.; Sheu, M. J. Interaction of Parallel Turbulent Plane Jets. AIAA J . 1991,29 (9),1372. Lund, T. S.; Tavella, D. A.; Roberts, L. Analysis of Interacting Dual Lifting Ejector Systems. AIAA Paper 86-0478,January 1986. Majumdar, B.; Shambaugh, R. L. Velocity and Temperature Fields of Annular Jets. Ind. Eng. Chem. Res. 1991,29,1300. Marsters, G. F. Measurements in the Flow Field of a Linear Array of Rectangular Nozzles. AIAA Paper 79-0350,January 1979. McLachlan, B. G.; Krothapalli, A. Effect of Mach Number on the Development of a Subsonic Multiple Jet. AZAA Paper 84-1656, June 1984. Miller, D. R.; Comings,E. W. Force-Momentum Fields in a Dual-Jet Flow. J . Fluid Mech. 1960,7, 237. Ng, S.; Grandmaison, E. W. Mixing Indices in CoflowingPlane Jets: High Momentum Conditions. Can. J . Chem. Eng. 1989,67,898. 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. Fundarn. 1986,25,425. Pani, B.; Dash, R. Three-Dimensional Single and Multiple Free Jets. J . Hydraul. Eng. 1983,109,254. Raghunathan, S.;Reid, I. M. A Study of Multiple Jets. AIAA J . 1981,19 (l),124. Rajaratnam, N. Turbulent Jets; Elsevier, New York, NY, 1976. Reichardt, H. h r eine neue Theorie der freien Turbulenz. 2.Angew. Math. Mech. 1941,21,257. (Translation: On a New Theory of Turbulence. R. Aeronaut. SOC. J . 1943,47,167.) Schlichting, H. Boundary-Layer Theory,7th ed.; McGraw-Hill: New York, NY, 1979. Shambaugh, R. L. A Macroscopic View of the Melt-Blowing Process for Producing Microfibers. Ind. Eng. Chem. Res. 1988,27,2363. Simonich, J. C. Isolated and Interacting Round Parallel Heated Jets. AIAA Paper 86-0281,January 1986. Tanaka, E. The Interference of Two-Dimensional Parallel Jets (1st Report). Bull. JSME 1970,13,272. Tanaka, E. The Interference of Two-DimensionalParallel Jets (2nd Report.) Bull. JSME 1974,17,920. Uyttendaele, M. A. J.; Shambaugh, R. L. The Flow Field of Annular Jets at Moderate Reynolds Numbers. Ind. Eng. Chem. Res. 1989, 28, 1735.
Received for review August 26, 1992 Revised manuscript received February 18,1993 Accepted March 1, 1993
