Ind. Eng. Chem. Res. 1992,31, 37+389 with an average concentration of 40% SO2and a methane concentration of 2% to 18%. The initial reaction rates will be less than those shown in Figure 9,where the highest SO2 level was 30%. A few kinetic tests need to be performed with 40-45% SO2 concentrations.
Summary Kinetic tests in a thermal balance reactor showed that the reaction of CuS04/A1203with CH, is strongly inhibited by the product gases, particularly SO2. The regeneration reaction is adversely affected by CuO in the spent solids, since this converts to CuS03 before being reduced to Cu. The reactions of CuS04 and CuS03 with CHI do not go to completion when SO2 is present, but there are not enough data to clearly define the equilibrium limits. More laboratory tests are needed to develop kinetic correlations that can be used to help interpret pilot-plant data and determine the best conditions for sorbent regeneration. Acknowledgment We gratefully acknowledge Robert Navadauskas and Charles Brodd of the Pittsburgh Energy Technology Center for their efforts in performing the microbalance tests. Registry No. CaO, 1317-380;SOz,7446-09-5;NO,,11104-93-1; NHB,7664-41-7;Alto3,1344-281; CuS04,7758-98-7;CHI, 7482-8.
379
Literature Cited Fogler, H. Scott. Elements of Chemical Reaction Engineering; Prentice-Hak Englewood Cliffs, NJ, 1986; p 245. Harriott, P. Kinetic Tests of CuSO, Regeneration. Memo to C. J. Drummond; Pittsburgh Energy Technology Center, Pittsburgh, PA, Sept 30,1988. McCrea, D. H.; Forney, A. J.; Myers, J. G. Recovery of Sulfur from Flue Gases Using a Copper Oxide Absorbent. JAPCA 1970,20, 819-824. McCrea, D. H.; Myers, J. G., Forney, A. J. Evaluation of Solid Absorbents for Sulfur Oxides Removal from Stack Gases. Proceedings Second International Clean Air Congress; Academic: New York, 1971;p 922. Plantz, A. R.; Drummond, C. J.; Hedges, S. W.; Gromicko, F. N. Performance of the Fluidized-Bed Copper Oxide Process in an Integrated Test Facility. Presented at the 79th Annual Meeting APCA, Minneapolis, MN; June 1986. Williamson, R. R.; Movici, J. A.; Lacrosse, T. L. “Sorbent Life-Cycle Testing Fluidized-Bed Copper Oxide Process”; Project Report for the Department of Energy; UOP Des Plaines, IL, 1988. Yeh, J. T.; Demski, R. J.; Strakey, J. P.; Joubert, J. I. Combined S02/N0, Removal from Flue Gas. Enuiron. Prog. 1985, 4, 223-228. Yeh, J. T.; Strakey, J. P.; Joubert, J. I. SO2Absorption and Regeneration Kinetics Employing Supported Copper Oxide. Unpublished paper; Pittsburgh Energy Technology Center, Pittsburgh, PA, 1987. Received for review December 18, 1990 Revised manuscript received July 22, 1991 Accepted August 19,1991
Characterization of the Melt Blowing Process with Laser Doppler Velocimetry Tien T. Wu and Robert L. Shambaugh* Department of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
Laser Doppler velocimetry (LDV) was used to study the process of melt blowing. For an array of positions below the melt blowing die, LDV measurements were made on the fiber velocities in three-space. This information quantifies the existence of an expanding “cone” of fibers below the melt blowing die. Through the use of a correlation function developed in this work, one can determine the actual rate of fibers passing through a specified area below the melt blowing die. An LDV system can be calibrated to measure the mass flux of fibers a t any point in space. This mass flux information is particularly useful for rapid, on-line prediction of the basis weight of a melt blown web of fibers.
Introduction Melt Blowing. In the melt blowing process, a fine polymeric stream is extruded into a high-velocity gas stream. The force of the gas rapidly attenuates the polymer into filaments of very small diameters; see Figure 1 for a diagram of the process. Melt blown fibers typically range from 30 pm in diameter down to filaments as fine as 0.1 pm in diameter. This extreme fineness gives melt blown fibers advantages in uses such as insulation, absorbent material, and filters. An overview of the melt blowing process is given in the article by Shambaugh (1988). The performance characteristics of various melt blowing geometries are given by Kayser and Shambaugh (1990). Uyttendaele and Shambaugh (1989)and Majumdar and Shambaugh (1991)examined the gas velocity and temperature fields in melt blowing, while Majumdar and Shambaugh (1990)measured the drag coefficient between
* Author to whom correspondence should be addressed.
the gas and the filament. Uyttendaele and Shambaugh (1990)developed a mathematical model for melt blowing. Laser Measurements in Fiber Science. Since the construction of the fmt laser in 1960 (Hieftje et al., 1981), the use of lasers in science, industry, and commerce has risen exponentially. The use of lasers in the fiber industry is no exception. For example, a single laser beam, when shined upon a fiber, will produce a forward scatter pattern that is dependent on the fiber diameter. Fouda et al. (1988)showed that, as a method of off-line fiber size measurement, use of this pattern is more accurate and much faster than obtaining diameters by optical microscopy or scanning electton microscopy. This forward scatter technique can also be used to measure the diameter of moving filaments. Thus, for example, the procedure can be used to monitor the diameter of a fiber during conventional melt spinning (Hamza et al., 1980). In addition, the backscattered radiation from a moving (or stationary) filament can be used to measure refractive index (Presby, 1974;Hamza et al., 1980;Wilkes, 1982).
0888-5885/92/2631-0379$03.00/00 1992 American Chemical Society
380 Ind. Eng. Chem. Res., Vol. 31, No. 1,1992
to the signals from a sphere [see Farmer (1972)l. However, for a cylinder with random orientation in space, prediction of the expected signals is mathematically intractable.
POLYYER
” \
’ i \ - / ‘ I ’ /‘ \ /‘ ZONE B
ZoNEC
.....................
7 Collection Screen
Figure 1. Melt blowing process. Zones of fiber motion are shown below the melt blowing die.
Besides the analysis of single filaments, single-beam lasers can also be used to examine defects in yarns and fabrics. The randomness/ texture of the surfaces causes the formation of a diffraction (speckle) pattern from which information on defects can be determined (Toba, 1980). Another laser configuration that has found use in fiber science is laser Doppler velocimetry, or LDV. LDV is commonly used to measure fluid velocities by examining the Doppler shift in the laser light scattered by small particles moving with the fluid. In the usual dual-beam LDV system, two laser beams are focused so as to intersect and form a small measuring volume, or mv. Particles which pass through this mv produce Doppler bursts of scattered light. An LDV system is noninvasive and can measure velocities in the range of 10 pm/s to 1 km/s. General information on LDV can be found in Adrian (1983) and Durst et al. (1981). In the textile area, LDV has been used for the measurement of the velocity of textile surfaces (Ringens et al., 1988) and the measurement of fiber velocities in a fiber conduit during open-end friction spinning (Brockmanns et al., 1987; Bauer et al., 1989). In the common application of LDV to a single-phase system (e.g., air or water), the fluid must often be artificially seeded with fine particles (e.g., oil droplets or a fine powder) in order for the laser light to be scattered and measured. The seed concentration is low and the seed sizes are small: 0.5-5-pm diameten are typical. A more complex problem is the application of LDV to two-phase flows such as air bubbles in water or large liquid drops in air (Durst and Zar6, 1975). In these two-phase flows, the size and concentrationof the dispersed phase are much greater than the seed concentration in a single-phasesystem (of course, a seeded system is actually a two-phase system). Surfaces can be 80 large, in fact, that the mv intemction of the laser beams cannot illuminate the entire surface at one time. For the particular case of spherical particles of one phase dispersed in a second phase, exact mathematical treatments are possible for relating the Doppler signals to the particle sizes (Farmer, 1972,1974; Durst and Eliason, 1975; Wigley, 1978; Durst and Ruck, 1987). For cylinders, the expected signals can be much more complex. If the cylinder moves with its major axis perpendicular to the plane of the crossing beams,the expected signals are very similar
The Zone Concept We define three zones of fiber motion at the exit of a melt blowing die; see Figure 1. Zone A is located adjacent to the die exit. In zone A, the fiber(s) is (are) predominately oriented in the z direction and fiber motion is also predominately in the z direction. In zone C the fibers are nearly randomly oriented in space and fiber motion ccurs in all directions. However, there is a net velocity in the +u, direction, since fiber continues to build on the collection screen. Also, since the fiber cone continues to expand, there is a net velocity in the +u, direction. Zone B is a transition zone between zone A and zone C. Some arbitrary, quantitative measure of randomness of fiber orientation can be used to define the boundary between zone A and zone B and the boundary between zone B and zone C. Melt blowing zones should not be confusedwith the melt blowing regions defined by Shambaugh (1988). In region I, the mass loading (kg of air/kg of polymer) is low and the fibers are continuous. As mass loading is further increased, region 11is entered, fibers are discontinuous, and undesirable polymer lumps, or ‘shots”, are present. Finally, as mass loading is increased further, region I11 is entered. In region 111,large and undesirable shots are not present, and the fibers are very fine. For a given set of operating parameters, a melt blowing system will be in only one region of operation. However, the three zones of melt blowing exist for each region of operation. Of course, the spatial location of the zones is a function of the region of operation. For example, the transition from zone A to zone B might occur much closer to the die for operation in region I11 versus operation in region I. Experimental Equipment and Details Melt Blowing Equipment and Polymers. The melt blowing equipment used in our experiments was the same as that described in previous work [see Kayser and Shambaugh (199011. The polymer capillary used in our work had an inside diameter of 0.533 mm, an outside diameter of 0.826 mm, and a length of 15.9 mm. The orifice plate had a straight bore (s-type),a diameter of 1.656 mm, and a length-to-diameter ratio of 1.92. E c e p t where noted, the polymer used in our studies was 54 MFR (melt flow rate) Fina Dypro polypropylene with an M, of 40 700 and an M, of 157OOO. Small amounts of Dow Aspun polyethylene and Kodak Kodapak 9899U polyester were also used. Laser Doppler Velocimeter. For our experiments we used a one-dimensional, frequency shift, fiber optic LDV system; see Figure 2. The bulk of this system was supplied by TSI Incorporated. The laser was a 15-mW He-Ne laser built by Spectra Physics. A Bragg cell provided frequency shifting for measuring flow reversals. The measuring volume (mv) was produced by a backscatter probe that was at the end of a 10-m-long fiber optic cable. This probe and cable system allowed us to keep the LDV equipment remote from the hazards of the melt blowing equipment. The small, lightweight probe was 14 mm in diameter, 100 mm long, and had a working distance of 60 mm. A TSI data analysis system and FIND software package were used to acquire, analyze, and display data on a 80286-based personal computer. A Tektronis oscilloscope (Model 2445B) was used to observe the Doppler bursts.
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 381
>
e l:
'
+
9 . 5 3 lllD 6.35
""4
F i b e r tapod t o edge of wheel
'(
Down Mlxu
Figure 2. LDV equipment. F i h r croming
/M.a.uring
Dual Laser ackscatter
f m t a t i l g Aluminum Calibration Wheel
Figure 4. Cross-sectional view of calibration wheel; a single fiber is taped to the rim of the wheel.
/
F i b e r Cable with Transmitting and Receiving F i b e r s
Figure 3. Calibration wheel. The speed range of the wheel is 30-2000 rpm; the wheel diameter d = 171.5 mm.
The laser probe was mounted on a Velmex 3-D traverse system which was secured on top of a steel support stand. The Velmex unit permitted x , y, and z motions in 0.001-in. increments. The signal processor was of the counter type (TSI Model 199OC-1).This processor type was selected because the concentration of fibers is a low seeding condition. Also, this counter type gives more details on the flow than any other type of signal processor. The resolution of the 199OC-1counter is 1 ns. Details on LDV operating conditions are given by Adrian et al. (1988). Calibration Wheel. Because of the constraints on fiber motion in zone A, one would expect the LDV signals produced for zone A measurements to be fairly easy to interpret. Such is not the case for zone C. As discussed in the Introduction, the interpretation of the signals produced by a randomly oriented cylinder is very difficult. Furthermore, the fibers are of variable diameters. Because of this complexity, an empirical approach to the problem was selected: correlations were developed to predict the signal based on the fiber orientation and diameter. (Measurements of zone B can be considered a combination of signals of the zone C type and the zone A type.) The correlations were developed with a device we call a calibration wheel; see Figure 3. The calibration wheel consisted of an aluminum cylinder rotating with a known rim velocity. A dc motor with solid-state control allowed the wheel speed to be varied continuously from 30 to 2000 rpm; this translated to a surface speed of 0.27-18.0m/s. A digital strobe light was used to measure rotational speed. For each set of correlation experiments, a single fiber was taped on the rim of the wheel; see Figure 4. The LDV backscatter probe was situated such that the fiber passed through the laser mv as the wheel rotated. The five major experimental variables in this process are as follows:
a=
0'
Calibration Wheel
a=
45'
Calibration Wheel
Figure 5. Calibration wheel rim as seen by the backscatter probe when B = 90°.
Figure 6. Fiber coordinate system.
1. fiber angle (a):the angle that the major axis of the fiber makes with a line that is both on the surface of the rim and parallel to the rotational axis of the calibration wheel. See Figures 5 and 6. 2. probe angle (B): the angle made between the major axis of the probe and the fiber axis. See Figure 6. 3. fiber-mu crossover (y?: the location within the mv where the fiber intersects the mv. See Figure 7.
382 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
A
0
.
A
W
2.4 mm
=
]
photogmphicolly PP fiber velocitiea by LDV PP/PET fiber velocitier by LOV
.
W
A
PP fiber wlocltiea determined
.
7-
2a
+Y’
I
0.00
I
I
I
I
,
,
, I I I I I I I I
0.d5
I I I I I I I I I
0.io
I
I
I
I
1
I
I
0.45
I
I
,
I
0.20
I
I
I
I
I
0.25
Axial Distance z (m)
1 I
I
dct
0.14 mm = 2b Figure 7. Measuring volume (mv) of the fiber optics probe. Laser wavelength = 0.6328 pm; fringe spacing = 5.36 pm; number of fringes = 30; working distance = 60 mm.
4. motion of fiber rehtiue t o the fringes: the angle that the fiber motion vector makes with respect to the optical fringes of the mv. 5. fiber diameter ( 6 ) . The above five variables are the “natural” variables for our particular experimental apparatus. In the Development of Burst Algorithm section below the first two of theee experimental variables will be shown to be equivalent to the orientation of a fiber described in traditional spherical coordinates. For the calibration wheel experiments the laser probe was mounted on the Velmex unit mentioned previously. The probe angle B (see Figure 6) was varied by moving the entire probe support stand and using a protractor to measure the probe angle to within *lo. Note that = 90° was where the probe had a “head-on” view of the rim of the calibration wheel (see Figures 5 and 6). After a change was made in the probe angle, the Velmex unit was used to make the fine adjustments required to bring the measuring volume into the desired position relative to the fiber. This position could be varied from the point where the fiber just touched either end of the measuring volume to a position where the fiber went through the center of the measuring volume. A diagram of the measuring volume is given in Figure 7.
LDV Compared with High-speed Photography Uyttendaele and Shambaugh (1990) used high-speed photography to take on-line measurements of fiber diameter during melt blowing. This photographic method only works well with the relatively slow fiber velocities in region I melt blowing. Because of the speed of the fibers in regions I1 and 111, fiber photographs tend to be blurred. Furthermore, the rapid sideways motion of fibers in regions I1 and I11 makes it difficult to keep the fibers within the field of the lens. Figure 8 compares the fiber velocities measured by LDV with fiber velocities determined photographically. Beaides comparison data for pure polypropylene (PP), LDV data for a polypropylene/polyester (PP/PET) blend are given.
Figure 8. Comparison of fiber velocities determined by LDV &th fiber velocities determined by high-speed photography. The region I conditions for the polypropylene were gas rate = 4.855 g/min, uqo = 90 m/s, polymer rate = 0.436 g/min, T p= 400 O C , and T,= 395 OC. The conditions for the PP/PET blend were the same as the conditions for the polypropylene. The polypropylene was FINA 54 MFR PP, the polyester was Kodak KODAPAK 9899U, and the blend was 5% by weight PET.
Table I. LDV Equipment Settings” comparison of calibration LDV with wheel expta high-speed and on-line Photo@ zone expta low-limit filter, MHz 0.03 0.3 high-limit filter, MHz 1.0-5.0 5-10 counter mode continuous TBC frequency shift, MHz 0 2-5 number of samples 256 up to 1024 number of cycles 8 8 comparison, % 1 1 0.5-0.8 0.8-1.2 gain Refer to Adrian et al. (1988) for details of LDV operation.
The same photographic technique used by Uyttendaele and Shambaugh (1990) was used to measure fiber diameters. Then, by continuity considerations, the fiber velocity was caldted. For the LDV technique the velocities were measured with the LDV operating conditions listed in Table I. Figure 8 shows that, up to about 5 cm, the photographic and LDV technique give nearly identical results (only the PP was used for comparison tests). However, beyond 5 cm, the photographic technique predicta a constant fiber velocity, while the LDV technique shows a decreasing fiber velocity. The photographic technique is undoubtedly the technique in error for z > 5 cm. The photographic technique indirectly calculates the velocity with the assumption of continuity (v = 4Qp/?rS2). Thus, once the fiber stops attenuating (6 = constant), then the fiber velocity is calculated to be constant. However, at z > 5 cm the fiber begins to whip about and is no longer primarily oriented in the z direction (i.e., the fiber enters zones B and C). Hence, the net velocity in the z direction is less than that calculated by a simple form of the continuity relation. Figure 9 is similar to Figure 8, except that Figure 9 is for polyethylene (PE) melt blowing. Also shown on Figure 9 is the predicted centerline air velocity for the melt blowing system. Observe that the air velocities and the (correct) LDV fiber velocities reach the same values for z > 6 cm. This seems reasonable: fine filamenta should reach the air velocities at large distances from the die. LDV data rate is a measure of the frequency of valid laser measurements. Figure 10 gives the data rates cor-
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 383 1 2 .
'
O a PE fiber velocitier determined photographically A PE fiber velocities by LDV
E
.
i 1.0
- Predicted Air Velocity
?50\
Q)
(ro,l
v
0
m m
a Calibration Wheel at 500 RPU A Calibration Wheel at 900 RPM
-
--
Probe An le 90' Fiber Angle = -45' 0.
. Frinqe Angle .
E0.6
I A
A
0
B
I
I,
I
0
o'o-!.O
-d.8
4 . 6
-d.4
-d.2
fiber-mv Figure 9. Fiber velocity comparison for polyethylene melt blowing. = 62 m/s, The region I conditions were gas rate = 3.884 g/min, polymer rate = 0.484 g/min, Tp = 305 "C, and Tg= 305 O C . The polymer was DOW ASPUN 6814 polyethylene. Also shown is the gas velocity predicted by the correlation of Majumdar and Shambaugh (1991).
180
16-
'i
m
0
0
PP doto rote PP/PET doto rote PE doto rote
14-
v
120, .cI
0 10-
E
Figure 10. LDV data rate corresponding to the velocities in Figures 8 and 9.
responding to the LDV velocity measurements in Figures 8 and 9. Besides pure PP and PE, velocity and data rate information are given (on Figures 8 and 10) for a PP/PET blend. Surface imperfections, solid particles or bubbles within the polymer stream, or interfaces within the polymer stream can all cause a Doppler signal to be created when these objects pass through the LDV measuring volume. As Figure 10 shows, different polymer systems have different data rates. Additives in the PE may be the cause of the PE data rate being higher than the PP data rate. The very high PP/PET data rate is probably due to the interface between the phases. Data rate determines how quickly a laser measurement can be made. For example, with a typical sample number setting of 256 (see Table I), it took less than 20 s to make a velocity determination for the highest data rate on Figure 10 (15 s-l for the PP/PET system). However, several minutes were required to collect the required data for some of the points on Figure 10. (Note: for Figures 8-10 all LDV settings were kept constant except for the gain setting. The gain sometimes had to be reduced to prevent overload a t high data rates.)
Use of the Calibration Wheel On the basis of some preliminary experiments with the melt blowing machine and with the calibration wheel, a
0.b
0:2
0.;
O.b
0.b
110
crossover (mm)
Figure 11. Effect of calibration wheel speed and mv position on data/passage. A single 28-pm nylon filament was attached to the wheel.
set of standard operating conditions were established for LDV measurements on fibers of arbitrary orientation (i.e,, measurements in zone C). Naturally, it was desirable to have the same settings for both calibration wheel tests and actual melt blowing tests (so the results would be comparable). The final settings selected are given in Table I. Only one fiber was taped to the calibration wheel rim for each experiment. Ideally, each revolution of the calibration wheel should give the same LDV burst or group of bursts, regardless of the calibration wheel speed. This is indeed the case, as Figure 11 illustrates. The ordinate value, data/passage, is the number of LDV bursts counted by the LDV system for each passage of the fiber through the mv (Le., for each revolution of the calibration wheel). The following five types of fiber were used in our calibration wheel experiments: 76-pm Du Pont industrial nylon, 28-pm Du Pont industrial nylon, 17-pm Dacron polyester, 10-pm polypropylene, and 7-pm polypropylene. The 10- and 7-pm fibers were produced by drawing 20-pm, region I melt blown fibers made from 54 MFR Fina Dypro polypropylene. Though all fibers were translucent, their opacities and refractive indices were different. These differences were not considered in the correlation of the LDV signal data. Thus, the five variables of interest are those stated in the experimental section: fiber angle, probe angle, fiber-mv crossover, fringe angle, and fiber diameter. In our experiments the probe angles /3 were set at 90°, 75O, 60°, 45O, 30°,and 20°. The fiber angles a! on the calibration wheel were set at Oo, 15O, 30°, 45O, 60°, and 75O. We varied fiber-mv crossover at selected fiber angles (a = Oo, 45O, and 75O). The angle between fiber motion vector and the fringea was tested at Oo, 4 5 O , and 60°, and the fiber diameter ranged from 7 to 76 pm. Over a thousand separate combinations of conditions were tested experimentally. With data/passage as the ordinate value, typical results showing the effects of fiber angle, probe angle, fiber-mv crossover, and fiber diameter are given in Figures 11-14. The angle between the fiber motion vector and fringes was found to not affect the data/passage. To describe mathematically the effects of the four relevant variables, the function h* was defined as follows: data/passage = h*(a!,B,y',6)
(1)
After a number of attempts, the following empirical function was selected to approximate h*: h* (a!,&y',6) = A*B* CD
(2)
384 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 4.0
where
A* = al exp(a2a)/(a3 + a) B* = B b l exp(b,8)
C = 1/[c,
-predicted doto/ oaroga rneaaured datoppoaaogr
3.5
o
3.0 e,
+ czyf2 exp(c3y?]
0 0 2.5
d -0.01 7 rnrn 6 =90‘
v)
D = (6/27.6)d1 The constants (al, a2, a3, bl, bz, cl, c2, c3, and dl) in the above equation were determined by minimizing the following quantity: C [h*(a,B,Y’,6)-
In Ez.0 \
y’=O rnrn
50 1.5 -0 1 .o
all data points
0.5
bursts per passage for a given point], (3) This minimization was carried out using the Gauss-Newton numerical scheme (SASInstitute, 1984). The best fit values for the empirical constants were found to be a1 = 0.192, a2 = 1.719, a3 = 0.286, bl = 1.818, bz = 4.0900, CX = 1.385, c2 = 1.O00, c3 = 1.46, and dl = 0.254. The solid lines on Figures 12-14 are predicted data/passage based on the above-described correlation. The coefficient of determination, P, for the fit of all the data to the correlation is 0.69. With consideration of the many variables and conditions involved in the calibration wheel experiments, this r2 is adequate. Use of the h* function is a valuable tool for the semiquantitative interpretation of the electronic signals produced during the u8e of LDV in a melt blowing system (see the next section).
Development of Burst Algorithm Our motivation for developing h* was to allow the measurement of fiber concentrations in zone C (and zone B). In zone C the fibers are randomly oriented. Since the h* function gives the expected signal for a known fiber orientation, a probability analysis is needed to convert h* to a form applicable to zone C. Figure 15 has a spherical coordinate system overlaid on a Cartesian coordinate system. Consider a fiber that passea through the origin of the coordinate system and is randomly oriented in space. The probability that the fiber is located in a small differential “patch” on the surface of the sphere is
The h* function discussed in the last section can be converted to the spherical coordinate system of Figure 15 by observing that /3 = ~ / -20 and a = 9 - 7r/2. We will call this converted function h. Thus h(@,O,y’,6)= h*(a,By’,6) (5) The converted correlation for h is h(@,Q’,6)= ABCD (6) where A = al exp[a2(9 - ?r/2)]/(a3 + @ - 7/21
B = ( ~ / 2- e)bl exp[bz(*/2 - e)] The C and D parameters and all empirical constants are the same as previously defined for h*. Because the fibers in zone C usually have a distribution of sizes, let us define a probability density j ( 6 ) such that (7) The function j ( 6 ) can be determined from methods such as (a) off-line measurements on the melt blown mat or (b) predictions of the computer model of Uyttendaele and Shambaugh (1990).
I
0.0
10
20
30
40
50
60
70
80
fiber angle (degrees)
1.6 0) 1.4 m 0 VI 1.2 :1,0 \ 0 0.8 0 mO.6 0.4 1.8
I
prcdiclad doto/ osrogr a rnmosured datopposaogm d-0.010 rnrn a 160. y’-0 rnrn
v)
U
0.2
-
2.5 -
-pndicted doto/ oaaoge a maoaured dato&oaaoge
a -75’ y’=0.610 rnm
0.5
-
’ 0
We can now combine h, j , and our probability considerations. The average LDV signal expected in zone C of a melt blowing process is bursts/passage = { = j(6) dB d 9 dy ’ d6 (8)
The term dy‘/L has been added to the integrand to account for fibers which pass through the LDV measuring volume at locations other than the center. Also, the limits of integration for the fiber sizes have been reduced to the range [p - 2.5a, p + 2.501. As long as (p - 2 . 5 ~ )> 0, this
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 385 DIE C X
+ve
+
z Figure 15. Coordinate system for examining the random orientation of fibers in space. Table 11. Expected Bursts/Parrage Calculated from Eq 8 fiber size distribution ( = bursts/passage 0.2784 p = 5 pm, monotonic 0.2771 p = 5pm, u = 1pm p = 5 pm, u = 2 pm 0.2716 0.3963 p = 15 pm, monotonic 0.3777 p = 15pm, u = 1pm p = 15 pm, u = 5 p m 0.3628 0.5355 p = 40 pm, monotonic 0.4722 p = 40pm, u = 1 pm 0.4692 p = 40pm, u = 5 p m 0.4687 p = 40pm, u = 10pm 0.4 126O p = 24.3 pm, u = 6.5 pm 0.4411a p = 31.4 pm, u = 7.5 pm
-
dr
Figure 16. Cylindrical and Cartesian coordinate systems for examining fibers below melt blowing nozzle.
.-
Z
E
W
3
6.0
1
8 inchea overage value
OTheae fiber size distributions were produced with the operating conditions listed on Figure 17. The fiber sizes were measured with optical microscopy.
range avoids the problem of negative fiber diameter. Since, with the assumption of a normal distribution, 99% of all fiber diameters are contained in this range, little accuracy is lost. With a known-or assumed-fiber size distribution, we can use eq 8 to calculate the expected number of LDV bursta/passage. Table I1 lists the results of a number of these calculations. Observe that smaller diameter fibers give less bursts/passage. Also, a monotonic ('spike") distribution gives a somewhat higher bursts/passage than does a broader distribution. The last two lines of the table correspond to fiber size distributions produced by melt ~ and 73 m/s (see blowing polypropylene with u ~=, 118 Figure 17 for the other operating conditions).
LDV Measurements in Zones B and C With a singlehole, annular melt blowing die, the fibers appear to move in roughly a 'cone" of space with the apex of the cone near the die capillary and the base of the cone at the collection screen. Cylindrical symmetry would be expected in the fiber cone. Figure 16 shows an appropriate cylindrical coordinate system for our single-hole melt b l o w , thiscoordinate system is overlaid on the Cartesian coordinate system appropriate to our LDV traverse system. With the same LDV settings used in the calibration wheel experiments (see Table I), an extensive series of melt blowing experiments were run wherein LDV was used to measure fiber velocity and fiber concentration in zones B and C. Figure 17 shows typical results of our LDV measurements of u,. Because of the cylindrical symmetry of the melt blowing nozzle, we expected that u, should be a function of r and not B. To test this hypothesis, u, values were taken for different r values and for B = Oo, 90°, NOo, and 270°,or, in terms of the Cartesian system, traverse
1
Figure 17. Typical LDV measurements of u,. The measurements are compared at four symmetrically equivalent positions. The standard deviation of the data points taken at u%o= 118 m/s ranged from 0.890 to 1.131 m/s. For uaP = 73 m/s, the standard deviation ranged from 0.704 to 1.017 m/s. The operating conditions for the 73 m/a case were gas rate = 3.923 g/min, polymer rate = 0.436 g/min, Tp = 395 O C , and Tg= 400 OC. For the 118 m/s case, the conditione were gas rate = 6.273 g/min, polymer rate = 0.436 g/min, Tp = 395 OC, and Tg= 400 "C.
in the + x , +y, -x, and -y directions, respectively. As Figure 17 shows, there was little apparent dependence on 8. For the 118 m/s curve, a Student t test was done on the points of widest spread at r = 0.4 in. (i.e., the +y direction and +x direction points). Even for these points, the u, values are so close that the t statistic shows insignificance even at a 20% probability level. The lines on Figure 17 (- and - -1 are simply lines drawn between the averages of the u, values at each radial position. Figure 18 shows u, values for two different gas velocities and four different z values (planes) below the die. The symboled points are averages of the velocities taken at multiple B positions. For both the 118 and 73 m/s gas velocities, the trends are the same: fiber velocities near the die are large and have a "peaked" distribution; fiber velocities far from the die are low and the distribution is flat. Because of the previously described presence of a fiber cone, measurements of fiber velocity are not given for large r and small z. Simply put, there were not any fibers present in these positions. Quantatively, this lack of data shows up as negligible data rate on the LDV equipment. Figure 19 shows typical values of data rate for measurements of u,. As can be observed, data rate (i.e., the
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386 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
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