Ind. Eng. Chem. Res. 1997, 36, 3937-3943
3937
Velocity and Temperature Fields of Dual Rectangular Jets Anthony S. Harpham and Robert L. Shambaugh* Department of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019
The velocity and temperature fields were measured below two parallel, rectangular air nozzles. The nozzles were closely spaced, and each nozzle had an l/w (length to width ratio) ranging from 55 to 170. The jets intersected at a 60° angle. Operating temperatures ranged from ambient to 321 °C. Measurements demonstrated that the temperature fields were analogous to the velocity fields. Hence, similar correlations were developed to describe either type of field. Introduction A recent paper by Harpham and Shambaugh (1996) describes velocity measurements below a pair of dual slot jets. This paper is a broad extension of this work to nonisothermal jets. Heated air jets are widely used in industry to transfer heat, mass, and momentum from the air to some other medium. Single-heated rectangular nozzles have been studied by Jenkins and Goldschmidt (1973), Sfier (1978), Sforza and Stasi (1979), and Marsters and Fotheringham (1980). The type of jet investigated by Harpham and Shambaugh (1996) is used in industry to produce melt blown fibers. For heated round nozzles, the goals of some recent work have included noise suppression for jet engines and mixing enhancement for reducing NOx in combustion chambers (Strykowski et al., 1993). For example, Lepicovsky and Brown (1989) used acoustic excitation of round jets to control mixing. Strykowski et al. (1993) placed an annular collar around a jet discharge. A suction was applied to the collar, and the resulting secondary counterflow increased the mixing of the jet. Lepicovsky (1992) did experimental studies on round nozzles for temperatures up to 900 K.
Figure 1. The experimental setup.
Experimental Equipment and Procedures The experimental equipment was similar to that used previously by Harpham and Shambaugh (1996). However, a 3 kW gas heater was added to the gas feed system; see Figure 1. Also, heating tapes and fiberglass insulation were added to maintain the temperature in the hot gas lines. Two 250 W cartridge heaters were used to control the die head temperature. The nozzle discharge temperature was controlled to within (1 °C during the experiments. The die had the configuration shown in Figures 2 and 3. The slot widths were adjustable; as a base condition, a slot width of b ) 0.65 mm was used. Each slot had a length of l ) 74.6 mm (2.94 in.). The h is the distance between the outer edges of the slots. Gas velocities were measured with a cylindrical impact tube and oil-filled manometer; see Harpham and Shambaugh (1996) for further details. Gas temperatures were measured with a fine iron-constantan thermocouple of the exposed-junction type. The junction diameter was 0.7 mm, and the junction end of the thermocouple was oriented parallel to the direction of gas flow. This arrangement minimized disturbance to the gas flow. The thermocouple was mounted on the same Velmex traverse unit that was used for impact * Author to whom correspondence should be addressed. S0888-5885(97)00145-0 CCC: $14.00
Figure 2. A side view of the air exit region of the die. A width setting of h ) 3.32 mm (the base setting) is shown.
tube positioning. This three-dimensional traverse unit permitted x, y, and z motions in 0.01 mm increments. Replicate thermocouple measurements had a standard deviation of about 0.8 °C (at z ) 1 mm) to 3.0 °C (for positions farther below the die). The air flow rate, at standard conditions of 21 °C and 1 atm of pressure, was maintained at either 1.67 × 10-3 or 3.33 × 10-3 m3/s (100 or 200 L/min). Gas temperatures of ambient (21 °C), 121, 221, and 321 °C were used. Equal slot widths of b ) 0.44, 0.65, and 1.35 mm were set for various parts of the experiments. The discharge gas velocityswhich depended on air flow, air temperature, and slot widthsranged from 16.6 to 70.0 m/s, and the Reynolds number ranged from 2170 to 7580. The coordinate system selected for the experiments is shown on Figure 3. The origin of the system is at the center of the face of the die. The y direction is parallel to the major axis of the nosepiece and slots, the x direction is transverse to the major axis of the © 1997 American Chemical Society
3938 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
Figure 3. A view of the face of the die. The z axis (not shown) is perpendicular to the plane of this drawing. Positive z values are located above the plane of the drawing.
Figure 5. Development of the temperature profile for positions near the die face. All values were measured at y ) 0.
Figure 4. Development of the velocity profile for positions near the die face. All values were measured at y ) 0.
Figure 6. Nondimensional velocity profiles at z ) 5 mm and for a range of values of vjo, θjo, and h.
nosepiece and slots, and the z direction is perpendicular to the plane of the drawing of Figure 3. The positive z axis is directed vertically downward (see Figure 1).
dividing by the jet half-width, which is the width at which the jet velocity drops to half of its maximum value. The velocity (ordinate) has been nondimensionalized by dividing by the maximum velocity at the z level (5 mm). The dotted line represents the empirical equation developed by Bradbury (1965) for rectangular jets, and the Bradbury equation is written on the figure. (A simpler equation can also be used to fit such data: see Figures 8 and 9.) Data are shown for excess temperatures of 0, 100, 200, and 300 °C. (Excess temperature is simply the difference between the actual temperature and the ambient temperature.) Also, three different slot widths (h values) are covered by the data, and initial velocities were also varied. In spite of temperature and slot width variations, all data essentially follow the empirical Bradbury equationsi.e., the profiles are well developed at the z ) 5 mm level. The data fit the Bradbury equation with an R2 of 0.994. Figure 7 shows the temperature data which corresponds to the velocity data on Figure 6. The x position (abscissa) has been nondimensionalized by dividing by the temperature half-width, which is the width at which the jet temperature falls to half of its maximum value. The temperature (ordinate) has been nondimensionalized by dividing by the maximum temperature at the z level (5 mm). The dotted line on the figure is the temperature analog of the Bradbury equation. Observe that all the data are nearly coincident. Also observe that the data diverge from the dotted line at positions far from the jet centerline. The data fit the Bradbury equation form with an R2 of 0.947.
Results and Discussion Harpham and Shambaugh (1996) showed that, for large l/w, a die of the type used in our experiments approximates an infinite-length die for positions away from the die ends. Hence, all measurements were run for positions in the bisecting plane (y ) 0; see Figure 3) of the die. Over 3000 individual velocity measurements and 7000 individual temperature measurements were made during this study. Profile Development. Figure 4 shows the development of the velocity profile for positions near the die face. At both 1.00 and 2.50 mm below the die, there are definitely two peakssan expected result because there are two slots. As z increases, the two peaks merge into one peak. Harpham and Shambaugh showed similar behavior for isothermal flow. Figure 5 shows the development of the temperature profile which corresponds to the velocity profile in Figure 4. The temperature profile exhibits a single, flattened hump at z ) 1.00 mm. Thus, the air in the core region is hot, though the air is not moving very rapidly (compare Figure 5 with Figure 4). As z increases, the temperature maximum decreases and the profile loses its blunt (flattened) look. Figure 6 shows the velocity profile at z ) 5 mm. The x position (abscissa) has been nondimensionalized by
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3939
Figure 7. Nondimensional temperature profiles corresponding to the velocity profiles of Figure 6.
Figure 9. Nondimensional temperature profiles at intermediate positions below the die.
Figure 8. Nondimensional velocity profiles at intermediate positions below the die.
Figure 10. Velocity profiles for positions far from the die. Data for three different die settings (h values) are given. The vjo values corresponding to air flow rates of 100 and 200 L/min.
Figure 8 shows the development of the velocity profile for positions from 5 to 60 mm below the die. Besides the Bradbury equation (the dotted line), a simpler exponential equation (the solid line) is included on the figure. This simpler Rajaratnam equation has been shown to fit data for slot jets (Rajaratnam, 1976), singlehole jets (Obot et al., 1984, 1986), annular jets (Uyttendaele and Shambaugh, 1989; Majumdar and Shambaugh, 1991), and rectangular arrays of jets (Mohammed and Shambaugh, 1993). The R2 values for the Rajaratnam and Bradbury equations are, respectively, 0.988 and 0.996. Figure 9 shows the temperature data which corresponds to the velocity data on Figure 8. The temperature analogs of the Bradbury and Rajaratnam equations are also given on the figure. Observe how, at low θ/θo, the data lie above the equation lines for low z values. By z ) 20 mm, however, the data have finally come down to the equation lines. This behavior is different than the situation with the velocity profiles on Figure 8. At low v/vo, the data reach the equation lines for z g 5 mm. The heating of ambient air near the die face could be the cause of the different behavior between the velocity and temperature profiles. The R2 values for the Rajaratnam and Bradbury equations are, respectively, 0.986 and 0.980. An experiment was run wherein the air flow was shut off, and the die was maintained at an excess temperature [θjo] of 100 °C. The thermocouple was placed at z ) 1, 3, and 5 mm below the die face, and excess temperatures of, respectively, 58, 33, and 17 °C were
measured. Free convection currents established by the heated die (and some minor radiation effects) undoubtedly caused the above ambient readings. In comparison, Figure 7 shows that, for large x/t1/2, the excess temperature is about 10 °C above ambient at z ) 5 mm. Thus, the air entrained by the jet (see Figure 17) keeps the thermocouple θjo at a significantly lower temperature than that caused by free convection (zero air flow rate). Furthermore, at positions where the air velocity is high (small x/t1/2), there is little perturbation of the θjo from the value predicted by the Bradbury equation (see Figure 7). Figure 10 shows the velocity profile at large distances from the die face. Data are given for ranges of excess temperatures, initial velocities, slot widths, and z positions. Under all conditions the fit of the data to either correlation is excellent: the data fit the Rajaratnam and Bradbury correlations with R2 values of 0.986 and 0.996, respectively. Hence, the more complex Bradbury equation is a slightly better fit. Figure 11 shows the temperature data which corresponds to the velocity data on Figure 10. As with the velocity data, the fit of all the data to either correlation is excellent: the data fit the Rajaratnam and Bradbury correlations with R2 values of 0.982 and 0.993, respectively. Also, as with the velocity profiles, the Bradbury equation provides a slightly better fit. Centerline Velocity and Temperature Decay. Harpham and Shambaugh (1996) used z/h as a dimensionless position in their isothermal plots of centerline
3940 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
Figure 11. Temperature profiles for positions far from the die.
Figure 12. Centerline velocity decay.
Figure 14. Centerline temperature decay.
velocity decay. For our nonisothermal work, the z position has been nondimensionalized by using the parameter
Z(h) ) [z/h][(F∞/Fo)1/2]
(1)
The use of the factor (F∞/Fo)1/2 was suggested by Majumdar and Shambaugh (1991) as a way to generalize the z position for nonisothermal conditions. Figure 12 shows the dimensionless centerline velocity for positions from z ) 5 to z ) 130 mm. The velocity decay for a two-dimensional jet can be described by the relation (Mohammed and Shambaugh, 1993; Schlichting, 1979)
vo ) c1vjoz-1/2
Figure 13. A replot of Figure 12, except that 2b is used as the characteristic length.
(2)
where c1 is an empirical constant. To account for temperature change, eq 2 was modified by replacing z with Z(h). The dotted line on Figure 12 is a leastsquares fit of the data to this modified eq 2; the data fit the dotted line with an R2 of 0.933. A power law expression (see the solid line) fits the data even better: the R2 is 0.977 for the fit of the data to this curve. The power law exponent is -0.624; this compares well with the exponent of -0.610 determined by Harpham and Shambaugh (1996) for isothermal flow. Figure 13 is a replot of the data of Figure 12 with 2b as the characteristic length used to nondimensionalize the abscissa values. Observe that the data have substantial spread when 2b changes. Hence, h is a much better correlating parameter. Majumdar and Shambaugh (1991) observed similar results when correlating
centerline velocity decay in annular jets: they found that the overall annulus diameter was a much better correlating parameter than the hydraulic diameter. Figure 14 is a plot of the centerline temperature decay as a function of Z(h). The dotted line is a fit to the data of a modified eq 2 (this modified equation is shown on Figure 14); the R2 of the fit is 0.955. The solid line on Figure 14 is a power law fit to the data; the R2 of this fit is 0.989. The power law exponent is -0.615, which is quite close to the exponent of -0.624 determined for the centerline velocity decay. A replot (not shown) of data of Figure 14 was done with 2b as the characteristic length. This replot looks almost identical to Figure 13: h is a much better correlating parameter than 2b. Spreading Characteristics. Figure 15 shows the increase in velocity half-width as a function of the distance below the die. Kotsovinos (1976) found that the spread of a rectangular jet can be described by a linear equation of the form
x1/2/h ) k1{(z/h) + k2}
(3)
In eq 3 the width w in the Kotsovinos equation has been replaced by h (as suggested by Harpham and Shambaugh [1996]). For the data of Figure 15, the leastsquares best fit to eq 3 gives k1 ) 0.119 and k2 ) 0.90 with an r2 of 0.991. For isothermal slot jets, Harpham and Shambaugh found similar values of k1 ) 0.118 and k2 ) 1.05. For nonisothermal annular jets, Majumdar and Shambaugh (1991) found k1 ) 0.112 and k2 ) 3.57 (with do, not h, as the characteristic length in eq 3). Majumdar and Shambaugh (1991) found that the temperature spread below an annular jet can be de-
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3941
Figure 15. The growth of the velocity half-width with increasing distance from the die.
Figure 17. The entrainment of air below the die.
Mohammed and Shambaugh (1993) give details of this integration procedure. The entrainment coefficient ψ is given by
ψ)
Figure 16. The growth of the temperature half-width with increasing distance from the die.
scribed by a linear equation of the form
t1/2/do ) m1{(z/do) + m2}
(4)
Majumdar and Shambaugh tested two annular dies and found m1 values of 0.109 and 0.113 and m2 values of 1.42 and 1.79. Figure 16 shows the increase in the temperature half-width as a function of the distance below the die. The dotted line on Figure 16 is a leastsquares best fit of eq 4 to the data (with the characteristic annular diameter do replaced by h). For this fit, m1 ) 0.198, m2 ) 0.97, and r2 ) 0.990. Thus, the rate of increase (slope) of temperature half-width is higher with slot jets than with annular jets. Because of the higher slope, the temperature halfwidth is always larger than the velocity half-width (compare Figures 15 and 16). This result corroborates what was previously stated by Schlichting (1979) and Reichardt (1942): for a two-dimensional jet the temperature distribution is broader than the velocity distribution. Mohammed and Shambaugh (1994) studied the flow field below a rectangular array of 165 air nozzles. For their array, m1 ) 0.247 at sufficient distances below the nozzle array. This compares well with the slope of 0.198 on Figure 16 since either a rectangular array of nozzles or a pair of slots approximate a single rectangular jet at sufficient distances below the die. Entrainment. At any position z below the die face, the mass flow rate M(z) can be calculated by integrating the velocity field over a plane defined by constant z.
M(z) - Mo Me ) Mo Mo
(5)
where Mo is the mass flow rate at the die discharge and Me is the mass flow rate of entrained air at some position z below the die. The Me is 0 at z ) 0. Figure 17 is a plot of ψ versus {z/h}{(F∞/F)1/2). Initial velocity vjo and initial temperature θjo have little effect on entrainment. However, the slot settings (h values) have a significant effect on the entrainment. For example, at {z/h}{(F∞/F)1/2} ) 16 the entrainment increases from about 1.5 to 4 as h is decreased from 4.73 to 2.91. Thus, narrower jetsswhich have higher vjo at the same total air flowrateshave higher entrainment. The entrainment levels shown in Figure 17 are somewhat less than the levels found for annular jets: at {z/ do}{(F∞/F)1/2} ) 16 Majumdar and Shambaugh (1991) determined that ψ ≈ 4. Heat Content of the Jets. At any axial position below the die, the heat content exiting from a segment of the die is given by
H(z) )
y)L/2 x)+∞ CpF(x,y,z)v(x,y,z)θ(x,y,z) dx dy ∫y)-L/2 ∫x)-∞
(6)
Equation 6 was evaluated numerically using the trapezoidal rule (see Obot et al., 1986; Majumdar and Shambaugh, 1991). A length L ) 74.6 mm (the die length) was selected. To permit integration over x, limits of x ) -x0.1 and x ) x0.1 were used. The heat content H(z) can be normalized by dividing through by Ho, the exit heat content of the jet. The Ho is defined as
Ho ) MoCpθjo
(7)
Figure 18 is a plot of H(z)/Ho versus dimensionless position below the die. From a value of 1 at the die face, the H(z)/Ho increases to values that are about 2.5-3.1. This increase is similar to the tripling of H(z)/Ho that was observed at large distances below a rectangular array of jets (see Mohammed and Shambaugh, 1994). Figure 18 shows that wider slots (larger h) result in smaller increases in H(z)/Ho. In order to incorporate the influence of localized heat and/or mass transfer characteristics, a fixed area was
3942 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
5. The h is the characteristic dimension for a dual slot die. 6. For distances extremely far from the die, the velocity and temperature fields should approach the fields of a single axisymmetric jet. However, even at the largest measured value of Z(h), both the velocity and temperature fields decay at a rate proportional to about [Z(h)]-0.5snot the [Z(h)]-1 dependence expected from an axisymmetric field. The motion of the ambient room air and the detection limit of the pitot tube and the thermocouple prevented us from taking significant measurements at larger Z(h). Acknowledgment
Figure 18. The variation of jet heat content at positions below the die.
We wish to thank the following organizations for their support: the National Science Foundation (Grant DDM9313694), the State of Oklahoma (project AR4-109/ 4844), the 3M Company, and Conoco/Du Pont. Nomenclature
Figure 19. Jet heat content for a limited region (-1.5 e x e +1.5 mm).
considered in the evaluation of the heat content of the jet. In eq 6, the lower and upper limits of integration were set at x ) -1.5 and x ) +1.5 mm, respectively. The results of these computations are shown in Figure 19. From a value of 1 at the die face, H/Ho rapidly increases to a value of about 1.4, and then H/Ho decreases as the distance from the die further increases. Similar calculations for a rectangular array of jets (Mohammed and Shambaugh, 1994) showed similar behavior, except that a maximum value of H/Ho ≈ 5 was achieved.
b ) slot width, mm c1 ) constant in eq 2, mm1/2 do ) outer diameter of annular orifice, mm h ) characteristic die dimension as defined on Figure 2, mm H ) heat content as defined in eq 6, W Ho ) heat content at die discharge, W k1 ) constant in eq 3 k2 ) constant in eq 3 l ) total length of a rectangular jet, mm L ) a length (section) of the die, mm m1 ) constant in eq 4 m2 ) constant in eq 4 M ) mass flow rate at a plane defined by constant z, kg/s Me ) entrained air mass flow rate as defined in eq 5, kg/s Mo ) discharge mass flow rate, kg/s t1/2 ) temperature half-width, °C v ) velocity, m/s vo ) maximum velocity of the jet at a specified z level, m/s vjo ) discharge velocity, m/s w ) width of a rectangular jet, mm x ) Cartesian coordinate defined on Figure 3, mm x1/2 ) jet velocity half-width, mm x0.1 ) position at which v ) 0.1vo, mm y ) Cartesian coordinate defined on Figure 3, mm z ) distance below the die, mm Z(2b) ) {z/2b}{(F∞/Fo)1/2} Z(h) ) {z/h}{(F∞/Fo)1/2} Greek Symbols
Conclusions and Recommendations 1. For nonisothermal flow from dual slot dies, the velocity field can be predicted from (a) the Bradbury equation, (b) the power law equation for centerline velocity decay (see Figure 12), and (c) the linear equation for velocity half-width (see Figure 15). 2. For nonisothermal flow from dual slot dies, the temperature field can be predicted from (a) the modified Bradbury equation, (b) the power law equation for centerline temperature decay (see Figure 14), and (c) the linear equation for temperature half-width (see Figure 16). 3. Because entrained air is undoubtedly warmed prior to entrainment, the temperature distribution develops more slowly than the velocity distribution. 4. Developed temperature profiles are broader than developed velocity profiles.
F∞ ) air density at ambient conditions, kg/m3 Fo ) air density along centerline downstream from nozzle, kg/m3 F ) mean value of air density at a z position, kg/m3 ψ ) entrainment coefficient θ ) excess temperature above ambient, °C θjo ) excess temperature at nozzle discharge, °C θo ) excess temperature along the centerline (the z axis), °C
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Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3943 Kotsovinos, N. E. A Note on the Spreading Rate and Virtual Origin of a Plane Turbulent Jet. J. Fluid Mech. 1976, 77 (2), 305311. Lepicovsky, J. An Experimental Investigation of Nozzle-Exit Boundary Layers of Highly Heated Free Jets. Trans. ASME: J. Turbomachinery 1992, 114 (4), 469-475. Lepicovsky, J.; Brown, W. H. Effects of Nozzle Exit Boundary Conditions on Excitability of Heated Free Jets. AIAA J. 1989, 27 (6), 712-718. Majumdar, B.; Shambaugh, R. L. Velocity and Temperature Fields of Annular Jets. Ind. Eng. Chem. Res. 1991, 30 (6), 1300-1306. Marsters, G. F.; Fotheringham, J. The Influence of Aspect Ratio on Incompressible, Turbulent Flows from Rectangular Slots. Aeronaut. Q. 1980, 37 (4), 285-305. Mohammed, A.; Shambaugh, R. L. Three-Dimensional Flow Field of a Rectangular Array of Practical Air Jets. Ind. Eng. Chem. Res. 1993, 32 (5), 976-980. Mohammed, A.; Shambaugh, R. L. Three-Dimensional Temperature Field of a Rectangular Array of Practical Air Jets. Ind. Eng. Chem. Res. 1994, 33 (3), 730-735. 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 SharpEdged Inlet Round Nozzles. Ind. Eng. Chem. Fundam. 1986, 25, 425.
Rajaratnam, N. Turbulent Jets; Elsevier: New York, NY, 1976. Reichardt, H. Gesetzmassigkeiten der freien Turbulenz. VDIForschungsh. 1942, 414, 1951. Schlichting, H. Boundary-Layer Theory, 7th ed.; McGraw-Hill: New York, NY, 1979. Sfier, A. Investigation of Three-Dimensional Turbulent Rectangular Jets. AIAA 11th Fluid and Plasma Dynamics Conference, Seattle, WA, July 10-12, 1978; paper 78-1185. Sforza, P. M.; Stasi, W. Heated Three-Dimensional Turbulent Jets. J. Heat Transfer 1979, 101 (5), 353-358. Strykowski, P. J.; Krothapalli, A.; Wishart, D. Enhancement of Mixing in High-Speed Heated Jets Using a Counterflowing Nozzle. AIAA J. 1993, 31 (11), 2033. Uyttendaele, M. A. J.; Shambaugh, R. L. The Flow Field of Annular Jets at Moderate Reynolds Numbers. Ind. Eng. Chem. Res. 1989, 28 (11), 1735.
Received for review February 10, 1997 Revised manuscript received April 28, 1997 Accepted May 1, 1997X IE970145N
X Abstract published in Advance ACS Abstracts, June 15, 1997.
