Ind. Eng. Chem. Res. 1994,33, 730-735
730
Three-Dimensional Temperature 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 temperature field was measured below a representative section of a rectangular array of 165 air nozzles. The array was arranged in a rectangular pattern with three side-by-side columns. On the basis of energy conservation considerations, a single equation was developed that fits the data and can be used to predict the temperature a t any position below the nozzle array. At large distances from the array, the temperature field approximates the field of a two-dimensional jet. An important use of this field study is in the design and improvement of devices used for producing melt blown fibers. Introduction
Y
A recent paper by Mohammed and Shambaugh (1993) describes velocity field measurements on a rectangular array of nozzles. Their array consisted of three columns of nozzles. Each column contained 55 nozzles; thus, there were 165 total nozzles. Each nozzle had a sharp-edged inlet, and each nozzle discharged air through an area that approximated an annulus. On the basis of the theoretical equation originally developed by Baron and Alexander (1951), Mohammed and Shambaugh fit 4600 velocity measurements with the following equation:
See the Nomenclature section for definitions of the symbols in this equation. The u2 in eq 1 is defined as u2 = r2
+ (ns)' + (mh')2- 2[(ns)' +
-1
(mh')2]1/2 ( r ) cos[y - tan- 1 mh' ns
0"
4
Otl'
f0'
9
It2,O'
$-I'
0""
6"t
0 1 1
ON
(2)
The variable h' is h' = -m,z
+h
(3)
See Figure 1for the coordinate system used in developing the above equations. The three empirical constants in these equations are C,, A,, and m,. Mohammed and Shambaugh determined the following best-fit values for these constants: C, = 0.2733, A , = 0.2019 mm2, and m, = 0.2538. For large distances from the nozzle array, Mohammed and Shambaugh determined that the velocity field approached the expected profiles from a twodimensional jet. They suggested equations that should be used in place of eq 1when the distance from the array is large. Refer to Mohammed and Shambaugh (1993)for a survey of the literature on velocity fields of single- and multiplehole jets. Studies on temperature fields are less numerous, particularly for the case of multiple jets. Corrsin (1944) investigated the temperature field downstream from a twodimensional grid formed by parallel rods. Makarov and Khudenko (1965)studied an array of five plane turbulent jets where the flows in two of the five jets were heated. Simonich (1986) studied the temperature field below a
* Author to whom correspondence should be addressed.
Figure 1. Cylindrical-coordinatesystem and hole-numbering system used in developing the model equations for the Schwarz die; s = 2.15 mm and h = 3.25 mm. This figure is reproduced from Mohammed and Shambaugh (1993).
pair of round holes. No work has been reported on the temperature fields of rectangular arrays of jets. Investigation of these fields was the goal of the research described in this paper. Furthermore, this paper concerns itself with the study of practical nozzles with sharp-edged inlets. Examination of single practical nozzles has been done by Obot et al. (1984, 1986), Uyttendaele and Shambaugh (19891,and Majumdar and Shambaugh (1991). Experimental Equipment and Procedures The experimental apparatus (see Figure 2) was a modification of the equipment used by Mohammed and Shambaugh (1993). A 4-kW gas heater was added downstream from the rotameter. The Schwarz die was kept
0 S S ~ - ~ ~ ~ 5 / 9 4 / 2 6 3 3 - 0 7 3 0 $ 0 4 . 5 00 1 01994 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 731
HETERING VALVE
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JET AMY
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Figure 4. Hole geometry and spacing in the Schwarz die. This figure is reproduced from Mohammed and Shambaugh (1993). REWE REWTU? Figure 2. The experimental apparatus.
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Temperatures below the die were measured with a fine iron-constantan thermocouple of the exposed junction type. Details of this thermocouple are given in Majumdar and Shambaugh (1991). To minimize flow disturbance, the axis of the thermocouple was always oriented parallel to the z axis (see Figure 2). Similar to the procedure followed by Mohammed and Shambaugh (1993), measurements were taken below a center section of the die (specifically, below the nine holes in rows 26-28). There were an average of 425 temperature measurements taken at z = 1.27,2.54,5.08, and 7.62 mm (1700 total measurements). No air temperature measurements were made at z = 0 because of interference from the metal die. For z > 7.62 mm, relatively few temperature 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 2.01 X m3/s (at standard conditions of 21 "C and 1-atm pressure). The average gas discharge temperature was maintained at 176 "C, and the average gas discharge velocity was 100 m/s. No polymer was fed to the die during the runs.
Results and Discussion
Y
52 53
a
I
X Figure 3. View of the face of the Schwarz die. The die contains 165 holes in three columns. This figure is reproduced from Mohammed and Shambaugh (1993).
heated with four 400-W cartridge heaters. In addition, the gas line between the gas heater and the nozzles was heated with three 104-W heating tapes. Temperatures were controlled to within f l O C of the set point during the experiments. The Schwarz die (jetarray) is a 165-holedie suitable for small-scale commercial production of melt blown fibers. Figures 3 and 4 illustrate the hole arrangement in the die. Further details on this die are given in Mohammed and Shambaugh (1993).
Experimental Results for Positions Near the Die. Figures 5-8 show the progressive development of the temperature profiles below the die. The temperature used in these figures is the excess temperature, 8,the difference between the measured temperature and the ambient temperature. Though the presence of the nine separate jets is quite apparent at z = 1.27 mm (Figure 5), apparent merging of the individual jets has already begun. This contrasts with the situation with the velocity field: Mohammed and Shambaugh (1993) observed no velocity field merging at this level. However, part of this apparent merging of the jets is undoubtedly due to heating caused by the nearness of the thermocouple to the hot face of the die. Evidence of this is the fact that, when a general equation was fit to the data, the data at the z = 1.27 mm level was poorly fit by the general equation (see the next section). Figures 6,7, and 8 show the progressive merging of the jets for z = 2.54 mm, z = 5.08 mm, and z = 7.62 mm. By z = 7.62 mm, there is little evidence of the individual jets. A similar intensity of merging occurs at a somewhat larger z value (at about z = 12.7 mm) for the case of a velocity field (see Mohammed and Shambaugh, 1993). Model Equation for the Temperature Field. Equation l is based on the additivity of momentum fluxes from
732 Ind. Eng. Chem.
Res., Vol. 33, No. 3, 1994
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.
Figure 5. Experimental temperature profile at z = 1.27 mm; Sp = 155 O C , uj. = 100 mls, and I?, = 149.8 O C (these values for 0, up, and 8. are averages of the data from the nine holes investigated).
Figure 7. Experimental temperature profile at I = 5.08 mm; 0, = 155 “C. up = 100 mls, and 0. = 120.3 O C .
Figure 6. Experimental temperature profile at z = 2.54 mm; Sp = 155 O C , up = 100 mls, and 8. = 142.8 O C .
Figure 8. Experimental temperature profile at z = 1.62 mm; 0p = 155 “C,up = 100 mls, and I?. = 114.6 OC.
point sources. Similarly, heat fluxes from point sources should also be additive. As suggested by Baron and Alexander (1951), and using the format developed by Mohammed and Shamhaugh (1993) for a 165-hole die, the temperature distribution below our 165-holedie should be predictable with an equation of the form
Mohammed and Shambaugh (1993): C, = 0.2733, A, = 0.2019 mm2, and m. = 0.2538. The least-squares best-fit values for the two constants, Ce and As, were determined tobe 0.3409 and 0.7617 mm2,respectively. Thecoefficient of determination, R2, for the fit of the predicted temperature to the experimental data is 0.8459. However, the model equation can be shown to fit the data even better if it is recognized that this Rzmeasures both the fit of the equation to the data and the hole-to-hole variances in the die. The region over which the temperature data were collected (the region below the nine holes) has 12-fold symmetry; refer to Figure 4. For example, one of these 12 zones is bounded by the following: (a)a ray which extends in the +x direction from the center nozzle (row 27, center column), (b) a line which extends in the +y direction from thecenter nozzle toapoint “P-halfway betweenthecenter nozzle and the next nozzle (row 26, center column), and (c) a ray extending in the +x direction from point ’P”. The data from the 12 symmetric zones were averaged to produce temperature data where the hole-bhole variations were minimized. For example, Figures 9 and 10show the averaged temperature data for the z = 2.54 mm and
where a2 is the same as defined in eq 2. The jet shift constant, m., was assumed to be equal to 0.2538, the value determined from momentumcalculations (see Mohammed and Shamhaugh, 1993). Hence, the only unknown parameters in eq 4 were ce,the thermal spreading coefficient, and Ae, the thermal discharge area. The temperature data taken at z = 2.54 mm, z = 5.08 mm, and z = 7.62 mm were fit with eq 4. The data at z = 1.27 mm were not used because, as described above, these data were affected by the heated die face. The velocity values needed in the fitting procedure were calculated with eq 1 with the constants determined by
Ind. Eng. Chem. Res.,Vol. 33, No.3,1994 733
Figure 9. Average measured temperature profile at 2 = 2.54 mm; E, = 155 'C, vp = 100 mls, and 8, = 142.8 'C.
Figurell. Calculatedtemperatureprofileati=2.54mm:Ei.- 155 "C.u, = 100 mls, and E, = 120.2 'C.
A
Figure 10. Average measured temperature profile at z = 5.08 mm: 8, = 155 OC, up = 100 mls. and 8. = 120.3 'C.
z = 5.08 mm levels (Figures 6 and 7 show the unaveraged data at these same levels). In comparison, Figures 11and 12 show the calculated temperature profiles a t the same levels. This calculation involves the model equation (eq 4) with the following best-fit values for the parameters: C, = 0.3346 and A, = 0.6990 mm2. Though the simulated profiles retain evidence of the individual nozzles longer (i.e., for larger z values) than the averaged profiles, the R2 for the fit of the model to the averaged data is 0.8683, which is quite good. This value for R2is a more legitimate estimate of the fit of the model to the data. TemperatureFieldsat DistancesFarther from the Die. Forz 2 7.62 mm,the temperature profileslookmuch like the expected profiles from a two-dimensional jet; see Figure 13, which shows the experimental temperature profiles for z = 7.62 m m to z = 127.0 mm. Observe the suhstantialflatteningandbroadeningatlargez. InFigure 14thecenterlinetemperaturedecayisplottedasafunction of z1l2. The centerline temperature decay for a twodimensional jet is similar to that for velocity (Sfeir, 1976) and can be described by the relation
Figure 12. Calculated temperature profile at z = 5.08mm; 8, = 155 "C,u, = 100 m/s, and E. = 104.6 OC.
(5) where dl and do are empirical constants. On the basis of thecenterlinetemperature measurementaforz 2 25.4mm, the hest-fitvaluesfordlanddoare 1.789mm1~2and0.1571, respectively. The dashed line on Figure 14 is a plot of eq 5 with these values for the constants. Alsoshown on Figure 14 is the centerline temperature predidion based on eq 4. Equation 4 should be used for small z (where there in fact are multiple peaks), while eq 5 is suitable for large z. For intermediate z (for approximately 5.08 m m < z < 25.4 mm), a weighted average of the estimated 0, values from both equations gives an adequate estimate of the true e.. Figure 15 shows the growth of the temperature halfwidth as a function of z. The solid line on the figure is a plot of the half-width predicted from eq 4. For z 2 5.08 mm, eq 4 predicts values of tlp that are too high. The dashed line is a least-squares best fit of the data for z 2 25.4 mm to the equation
734 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
I
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105 0.8
90
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0
75
.0.6
v
a 60 0.4
45
0.2 30 UNO
.
.
(mml Figure 13. Experimental temperature profiise for z 2 7.62 mm. x
t
o
bosed on eq 4
W l / 2
Figure 16. Temperature profiles at various z levels. &
Experimental temperature profile profile from eq 4 Temperature profile from eq 5
o
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-Temperature
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- emitting heat content of gas from nine holes I
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Figure 14. Decay of centerline temperature: comparison of experiment with predictions from eq 4 and eq 5. o
/ *-*.*-70
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30
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50
60
70
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(mm) Figure 15. Growth of jet temperature half-width comparison of experiment with predictions from eq 4 and eq 6. t1/2
= d#
30
40
50
60
,
,
70
(mm> Figure 17. Variation of jet heat content aa a function of axial (2) position.
Heat Content of the Jet. At any axial position below the die, the heat content per length of the die is given by
Experimental temperature spread Temperature rpread from eq 4 Temperature rpread from eq 6
.--
20
(6)
The best fit value for d2 is 0.2466, and the r2 for the fit of the data is 0.9933. Figure 16 shows the dimensionless velocity profile plotted as a function of dimensionless width. All the profiles fit the exponential relation shown on the figure; this exponential relation has also been used to fit temperature data for single-hole jets (Obot et al., 1986) and annular jets (Majumdar and Shambaugh, 1991). Also shown on the figure is the prediction of eq 4; this eq 4 prediction is indistinguishable froni the prediction of the exponential equation.
As suggested by Obot et al. (1986) and Majumdar and Shambaugh (19911, eq 7 was evaluated numerically by use of the trapezoidal rule. The length L used was 3s = 6.45 mm. Instead of x = --m and x = +-m, the boundary planes x = - x o . ~ and x = x0.1 were used. As shown in Figure 17, H ( z ) / L is independent of downstream location for z > 5.08 mm. This behavior is consistent with the criteria for the constancy of heat flow at consecutive cross sections (Abramovich, 1963). Because of heating of gas (the gas that will be entrained in the jet) near the die face, H ( z ) / L increases in the range 0 I z I 5.08 mm. The value of H ( z ) / Lat z = 0 was determined from the heat content of the discharge gas. In order to incorporate the influence of localized heatand/or mass-transfercharacteristics,the heat content was also evaluated over a fixed area. Specifically, the lower and upper limits of integration on x in eq 7 were changed tox =-1.2mmandx = +le2mm,respectively. Theresults of these computations are shown in Figure 18. As expected, there is a fall in the heat content with increasing z due to the broadening of the temperature and velocity profiles of the jets.
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 735
A
7
- emitting heat content of gas from three holes
0
\ J 20
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0
lot
d"
O
10
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z (mm>
Figure 18. Jet heat content for alimited region (-1.2mm d x d +1.2 mm) as a function of axial position.
Conclusions and Recommendations 1. Equations 4-6 and Figure 16 can be used to predict the temperature 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 4-6 can be used to correlate the experimental data. Relatively few temperature measurements need to be made in order to determine the appropriate constants ( A Band CB). 3. By extrapolation, these equations can also be used to predict the temperature field of an array of other than three columns. 4. Besides predicting behavior near the center of the array, eq 4 can predict the temperature fields at the ends of the array.
Acknowledgment We thank the following companies for their financial support: 3M, Dow Chemical U.S.A., Fina Oil and Chemical, Kuraray Limited, and Phillips Petroleum. The Schwarz melt blowing die was donated by Biax-Fiberfilm Corporation, Neenah, WI.
Nomenclature a = coordinate parameter defined by eq 2, mm A, = momentum discharge area of a single orifice, mm2 A8 = thermal discharge area of a single orifice, mm2 C, = spreading coefficient for momentum transfer C, = specific heat, J kg-1 K-'
C8 = spreading coefficient for heat transfer do = constant in eq 5 d l = constant in eq 5, mm1/2 dz = constant in eq 6 h = distance between adjacent holes in the x direction, mm h' = modified distance between adjacent holes in the x
direction, mm
H = heat content as defined in eq 7, W L = a length of the die in the y direction, mm m = column position in jet array m, = slope of jet shift M, = discharge flow rate from entire die, kg/s n = row position in jet array r = cylindrical coordinate, mm s = distance between adjacent holes in the y direction, mm t l p = jet temperature half-width, mm v = velocity, m/s vio = discharge velocity, m/s u, = maximum velocity of the jet at any specifiedaxial position, m/s x = Cartesian coordinate, mm X O J = distance from jet axis to location where v = 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 pj, = air density at discharge conditions, kg/m3
0 = excess temperature above ambient, O C Oj, = excess temperature above ambient at nozzle exit, "C 0, = maximum excess temperature above ambient at any
specified axial position, "C
Literature Cited Abramovich, G. N. The Theory of Turbulent Jets; MIT Press: Cambridge, MA, 1963. Baron, T.; Alexander, L. G. Momentum, Mass, and Heat Transfer in Free Jets. Chem. Eng. Prog. 1951, 47 (4), 181. Corrsin, S.Investigation of the Behavior of Parallel Two-Dimensional Air Jets. NACA W-90, November 1944. Majumdar, B.; Shambaugh, R. L. Velocity and Temperature Fields of Annular Jets. Znd. Eng. Chem. Res. 1991,29, 1300. Makarov, I. S.;Khudenko, B. G. A System of Plane Turbulent Jets. J. Eng. Phys. 1965,9 (2), 125. Mohammed, A.; Shambaugh, R. L. Three-Dimensional Flow Field of a Rectangular Array of Practical Air Jets. Ind. Eng. Chem.Res. 1993, 32 (5), 976. 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. Znd. Eng. Chem. Fundam. 1986, 25, 425. Sfeir, A. A. The Velocity and Temperature Fields of Rectangular Jets. Znt. J . Heat Mass Transfer 1976,19, 1289. Simonich, J. C. Isolated and Interacting Round Parallel Heated Jets. AZAA Paper 86-0281, January 1986. Uyttendaele, M. A. J.; Shambaugh, R. L. The Flow Field of Annular Jets at Moderate Reynolds Numbers. Znd. Eng. Chem. Res. 1989, 28, 1735.
Received for review August 9, 1993 Revised manuscript received November 18, 1993 Accepted December 7, 1993' Abstract published in Advance ACS Abstracts, February 1, 1994.
