Ind. Eng. Chem. Res. 2010, 49, 5877–5883
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Stagnation Point Offset of Two Opposed Jets Zhi-gang Sun, Wei-feng Li, and Hai-feng Liu* Key Laboratory of Coal Gasification of Ministry of Education, East China UniVersity of Science and Technology, Shanghai 200237, China
The axial velocity and the stagnation point offset of impinging streams from two opposed nozzles have been theoretically studied and experimentally measured by smoke wire visualization and hot-wire anemometry (HWA) techniques. The results show that, for small nozzle separation (L/D < 2), the equations based on Reynolds stress theory can be used to predict the stagnation point offset for two nozzles of the same size. For middle nozzle separation (2 e L/D e 12), the stagnation position is very sensitive to the exit velocity ratio and that it changes nonlinearly with increasing nozzle separation. For large nozzle separation (L/D > 12), the equations based on the free jet theory can be used to predict the stagnation point offset for two nozzles of the same and different sizes. The illustrated effects of the exit momentum ratio and the nozzle separation on the stagnation point offset are useful for defining the optimum conditions for many industrial applications of the opposed jets. 1. Introduction The impinging streams from opposed nozzles (opposed jets), responsible for rapid and effective mixing in a fluid reactor, have been the subject of many research projects in the past 20 years because of the increasing number of industrial applications such as coal gasification,1 mixing,2 absorption,3 liquid-liquid exaction,4 drying,5 and reaction injection molding (RIM)6 machines. Since the first comprehensive review of the research and development in opposed jets by Tamir,7,8 a wide range of nozzle separations have been employed in various applications; the nozzle separation (L) is often more than 20 times the nozzle diameter (D) in coal gasification,1 10.67D in RIM9 machines, 4.67D in confined impinging jets reactors (CIJRs),10 4-8D in SO2 absorption,3 and 1.5D in liquid-liquid extraction.4 An example for the need of unequal opposed jets is in mixing fluids at stoichiometric ratios, such as the polyol and isocyanate streams that are supplied at different flow rates in the polyurethane RIM. A fundamental study on the dynamic properties of such uneven flow is important for practical operations and applications; however, there are few to date. Champion and Libby11-13 in a study of two closely spaced opposed jets produced experimental results which verified the calculated mean velocity and the turbulence intensity based on the Reynolds stress theory. Ogawa et al.14,15 in studying turbulent opposed jets by HWA found that the impingement point of opposed jets was unstable; the stagnation point offset was predominantly affected by the exit momentum ratio with turbulence intensity a minor factor. Kind and Suthanthiran16 found that the position of the interaction zone of two opposed plane wall jets was also primarily dependent on the momentum ratio. Johnson17,18 reported an asymmetric flow field with the impingement point close to the low flow rate nozzle in the study of two laminar opposed jets in a confined cylindrical chamber at L ) 10.75D employing flow visualization, laser doppler anemometer (LDA) measurements, and three-dimensional numerical simulations. Li et al.19,20 studied the stagnation point offset of turbulent opposed jets with small and middle nozzle separations experimentally by HWA, the smoke-wire technique, and numerically by computational fluid dynamics (CFD) and * To whom correspondence should be addressed. Tel.: +86 21 64251418. Fax: +86 21 64251312. E-mail:
[email protected].
the effects of exit velocity ratio, nozzle separation, and the exit velocity profile on the stagnation point offset. Furthermore, the two nozzles may be unequal as a result of the low quality manufacturing process, blocked nozzle, and/or unequal abrasion in the service. There are yet few studies on the flow field of two opposed jets of unequal nozzles. Hosseinalipour and Mujumdar21,22 preformed numerical simulations of the flow, mixing and heat transport characteristics of the steady laminar two-dimensional confined opposed jets of the same exit velocity from equal and unequal nozzles; the streamline of the flow field was more bending when the nozzle width ratio was 1:2 and the stagnation point was much closer to the small nozzle. Devahastin and Mujumdar,23 in the numerical simulation of the two-dimensional laminar confined impinging streams of different geometric configurations, found that the mixing condition and the Reynolds numbers beyond which the flow becomes oscillatory and even random were affected by both the Reynolds numbers of the inlet jets and the system geometry. Wang et al.24 found that the mixing effectiveness of the laminar twodimensional opposed jets of two slot width ratios (2:1 and 3:1) decreased when the slot width radio increased. In the practice of the “opposed multi-burner coal-water slurry gasification process” of East China University of Science and Technology (Shanghai, China),1 the gasification rate and results are dependent on both the mass transport rate and the flow field, making it important to understand the fundamentals of the opposed jets. Motivated by those needs and to promote more beneficial applications, this research was conducted to study the axial velocity characteristics and such affecting factors as nozzle separation, the exit momentum ratio on the stagnation point offset of symmetric unconfined turbulent two opposed jets under equal or unequal inlet mass flow rates, and various nozzle separation conditions. The smoke-wire visualization and HWA techniques were experimentally employed to study the flow fields. 2. Experimental Setup 2.1. Two Opposed Jets. The opposed jets were produced by the apparatus shown in Figure 1. The air streams from the air cylinder were measured and then discharged from two axisymmetric opposed nozzles. Figure 2 depicts the cross section
10.1021/ie901056x 2010 American Chemical Society Published on Web 05/11/2010
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Figure 1. Schematic flowchart of the experimental system. 1, Roots fan; 2, air cylinder; 3, valve; 4, flow meter; 5, support; 6, nozzle; 7, halogen spotlight; 8, high-speed camera; 9, smoke-wire kit; 10, HWA kit.
a)
u1 u2
(4)
M1 Fu21D21 M) ) 2 2 M2 Fu2D2
Figure 2. Cross section of the nozzle.
(5)
where D1 and D2 are the diameters of nozzles I and II, respectively, and u1 and u2 are the bulk velocities at the exits of nozzles I and II, respectively. The exit velocity ratios ranged from 0.148 to 1.64, and the momentum ratios ranged from 0.160 to 6.25. The stagnation point offset is the distance from the stagnation point to the grid origin (∆x in Figure 3) and is defined as ∆x ) L/2 - x1 or ∆x ) x2 - L/2
Figure 3. Flow field of the two opposed jets.
of the nozzle; Figure 3 is a schematic presentation of the stagnation point offset of the two opposed jets system, which is the same as that of Li et al.19,20 The distance from the virtual origin of the free jet to the nozzle plane is labeled s0 in Figure 3. Three different sized (D ) 10, 20, and 30 mm) nozzles were employed in this study. The stagnation point is formed when two fluid elements of the same kinetic energy and opposite flow direction impinge. The jet Reynolds number and momentum at the nozzle exit are defined as DjujF µ
(1)
π 2 2 Fu D 4 j j
(2)
Rej )
Mj )
Where uj is the bulk velocity of the jet at the nozzle exit, and F and µ are, respectively, the density and dynamic viscosity of the air under normal conditions. Re1 and Re2 and M1 and M2 are the Reynolds numbers and momenta of the jet discharged from nozzles I and II, respectively. The Reynolds numbers of nozzle I ranged from 7131 to 68 773. j ), exit velocity ratio (a), and The average nozzle diameter (D momentum ratio (M) are defined: j ) (D1 + D2)/2 D
(3)
(6)
2.2. Hot-Wire Anemometer Measurement. The velocities of the flow field of the opposed jets were measured by HWA employing a DANTEC hot-wire anemometry system (DANTEC Dynamics A/S, Streamline 4) with a single wire probe (DANTEC, 55P11). The sampling frequency was set at 20 kHz, and the sampling duration was 10 s. The impingement stagnation point and the axial velocities were identified by the hot-wire probe, although it could not provide precise measurements of mean and RMS velocities in the vicinity of the stagnation plane.19,20,25 The probe was placed along the axis of the opposed jets. The stagnation point is experimentally defined as the position of the position of minimum mean velocity. The nozzle separation L ranged from 15 to 1500 mm, corresponding to 0.5 to 150 in normalized nozzle j ). Table 1 summarizes the experimental runs with separation (L/D the results of HWA measurements. The experiments were repeated twice at least, and the errors were less than 5%. 2.3. Smoke Wire Visualization Technique. The instantaneous flow field of two opposed jets was monitored by the smoke-wire visualization technique. A 0.1 mm diameter stainless steel wire was located approximately 1 mm downstream of the nozzle exit; the wire was coated with glycerin and connected to function as a resistor to an electric circuit. Then, the glycerin was heated to produce smoke by the sudden current when the circuit was energized. A high-speed camera (Fastcam of Photron Limited) was employed, along with a continuous 2000 W Table 1. Experimental Runs Employing HWA Technique D1 (mm)
D2 (mm)
j L/D
Re1
a
M
30 10 20 30 30
30 10 10 10 20
0.5-20 30-150 33.3, 66.7 33.3 33.3
23460 68773 7131-39747 16374-36843 12281-33322
0.700-1.00 0.700-1.00 0.242-1.25 0.148-0.833 0.267-1.64
0.490-1.00 0.490-1.00 0.234-6.25 0.198-6.25 0.160-6.08
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010 Table 2. Experimental Runs Employing the Smoke-Wire Visualization Technique D1 (mm)
D2 (mm)
Re1
a
M
j L/D
30
30
4962
0.6-1.0
0.36-1.0
0.5-8
halogen spot light, to record the fluid movement at 1000 frames per second with full resolution 1024 × 1024 pixels. The resolution of the pictures was 1024 × 1024. The high quality micro lens (Nikon, 200 mm f/4D ED-IF AF Micro-Nikkor) of the camera enabled clear full size image reproduction without extra photo processing. The camera was mounted on a tripod 1 m from the axis of the nozzle. The smoke wire visualization technique is applicable only to the cases of relatively low exit air velocity; otherwise the smoke is not dense enough to produce high quality photos. Table 2 presents the experimental runs with smoke wire visualization photos as the results for interpretation and discussion of the flow field. 3. Results and Discussion 3.1. Smoke Wire Visualization Photos. Figures 4 and 5 present the smoke wire visualization photos of the opposed jets from the two 30 mm nozzles at exit velocity ratios of a ) 1 and 0.97, respectively, for three separations of L/D ) 0.5, 1, and 2; Figure 6 presents the photos for the cases of L/D ) 2 at three exit velocity ratios of 0.9, 0.8, and 0.7. In all the visualization runs, u1 was kept constant at 2.36 m/s.
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Figures 4 and 5 show that, after the opposed jets impacted each other, the axial jets became radial jets, and the radial jets were perpendicular to the axis of the two nozzles at different nozzle separations when the exit velocities were nearly equal. Figures 5c and 6 show that, when the exit velocities differed slightly (a ) 0.97 and 0.9), the stagnation points moved away from the center of the two nozzles, the radial jets were still perpendicular to the axis of the two nozzles; the impingement region is shaped like a discus with a thick center and a thin edge. The impact plane became umbrella shaped with a greater curvature and moved closer to the nozzle of the weaker jet as the velocity difference increased (Figure 6). Additional photos like Figure 4 show that the resulting large vortices grew in size downstream, rotated, and moved away from the axis. At L/D ) 2, more effort was necessary to calibrate the flow rates to keep the stagnation point at the midpoint of the opposed nozzles. The less stable flow of opposed jets at L/D ) 2 is consistent with the findings of Li et al.20 Kostiuk and Libby26 also reported difficulties in balancing the flow rates to maintain the symmetrical flow field. At a ) 0.97 and a small nozzle separation L/D of 0.5 and 1, the stagnation point remained at the center of the axis; however, a larger separation had a greater effect on the position of the stagnation point, such as the 3% difference in exit velocities causing the stagnation point to deviate by 0.4D when L ) 2D (Figure 5c). Taking into account the different experimental
Figure 4. Photographs of opposed jets with various nozzle separation at a ) 1. (a) L ) 0.5D, (b) L ) 1D, (c) L ) 2D.
Figure 5. Photographs of the opposed jets at a ) 0.97. (a) L ) 0.5D, (b) L ) 1D, (c) L ) 2D.
Figure 6. Photographs of the opposed jets at three exit velocity ratios (L ) 2D).
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Figure 7. Distributions of normalized axial mean velocity and turbulence intensity of the free jet. (a) Axial mean velocity. (b) Turbulence intensity.
Figure 8. Axial mean velocity and turbulence intensity profiles of the free and opposed jets. (a) Axial mean velocity. (b) Turbulence intensity.
velocity, the observed results are consistent with the finding of Kostiuk and Libby26 of 0.075L offset at L ) 2D. 3.2. Axial Velocity Measurements of HWA. Figure 7 presents the HWA measured normalized axial mean velocity and turbulence intensity profiles at the nozzle exit. The results are consistent with those reported by Li et al.20 The top-hat shaped exit velocity profile shows the same mean velocity for about 80% of the nozzle diameter and that the thickness of the boundary layer was about 0.1D. The turbulence intensity was constant at about 5% except for the region of 0.1D from the nozzle wall. The exit air velocity was relatively uniform, and the axial velocities were almost symmetric around the stagnation point. Though the probe could not precisely identify the position of the stagnation point for all cases, the results were adequate considering the instability of the opposed jets. The normalized axial mean exit air velocity and turbulence intensity profiles of this study are similar to those reported by Cornaro et al.27 Li et al.20 also reported a similar distribution of normalized mean axial velocity for the same size nozzles (D1 ) D2 ) 30 mm) at four separations of L/D ) 1, 4, 12, and 20. Figure 8 presents the normalized mean axial velocity and j turbulence intensity of the free jet and the opposed jets at L/D ) 66.7 with unequal size nozzles and L/D ) 100 with equal size nozzles. The exit velocities were u0 ) 21.2 m/s for free jet and u1 ) 10.6 m/s and u2 ) 21.2 m/s for unequal nozzles and u1 ) u2 ) 21.2 m/s for equal nozzles, respectively. The Reynolds numbers and the momenta of the two-nozzle runs were equal. The axial velocity out of the impinging region of the opposed jets was the same as the free jet (Figure 8a); therefore, the opposed jets from the two nozzles entered the developed region in nearly same manner as the free jet before they impinged. The profiles were symmetric about the stagnation point in the region -0.3L to +0.3L. The stagnation point was the position with the smallest velocity. 3.3. The Stagnation Point Offset. Figure 9 represents normalized stagnation points offset at various exit velocity ratios and nozzle separations of the opposed jets of two equal nozzles (D ) 30 mm) for the HWA and the smoke wire runs. The results showing nearly the same stagnation point offset validated the use of these two experimental techniques. The stagnation point
Figure 9. Normalized stagnation point offsets at various exit velocity ratios and nozzle separations.
offsets in the range of 2D e L e 12D (especially at 4D e L e 8D) were larger than those of other nozzle separations at the same exit velocity ratio. The stagnation position was very sensitive to the exit velocity ratio in this range of nozzle separation. 3.3.1. Small Nozzle Separation (L < 2D). For the impinging streams from two opposed nozzles, as shown in Figure 3 using the coordinates of x and r for the velocity components ux and ur and assuming the velocity field is axisymmetric about x ) 0 and symmetric about the stagnation plane, the two jets from nozzles of small separations are on the potential core region, like the case of free jets, and the axial velocities may be assumed to be their respective exit velocities. Champion and Libby11-13 studied two closely spaced opposed jets of the same exit velocity and derived the axial velocity equation on the basis of the Reynolds stress theory. The mean momentum equation for turbulent flows involving high turbulent Reynolds numbers such that the molecular transport may be neglected is uk
∂ui ∂ 1 ∂P + uu ) ∂xk ∂xk i k F xi
The continuity equation is
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Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
∂ux 1∂ (ru ) + )0 r ∂r r ∂x
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(8)
For the unequal exit velocities of u1 and u2, the distances from stagnation point to nozzle I and nozzle II are x1 and x2, respectively. Using a similar scheme as Champion and Libby,11 the mean radial velocity in the proper form for the neighborhood of the axis is
{
ur ) (r/d)u1F'(ζ) x > 0 ur ) (r/d)u2F'(ζ) x < 0
(9)
The following result from substituting eq 9 into eq 8:
{
ux ) -2u1F(ζ) x > 0 ux ) -2u2F(ζ) x < 0
(10)
where F is a function form and ζ ≡ x/L. If we assume the mean pressure is of the form 1 P ) Ps - Fu20 P0(ξ) + (r/d)2 2
[
]
(11)
Substitute eqs 9-11 into eq 7, and the following are obtained for the boundary conditions: F(0) ) F'(x1 /L) ) F'(-x2 /L) ) 0 F(x1 /L) ) 1/2 F(-x2 /L) ) -1/2 giving the solutions for F(ζ):
{
( (
[ [
)] )]
L 1 L 2 ζ x1 2 x1 L 1 L 2 F(ζ) ) ζ + ζ x2 2 x2
F(ζ) ) ζ
ζ>0 ζ u1, x2 >x1, and that the axial velocities around the stagnation plane are symmetric, the absolute velocity at the point x ) -x1 is
[
( )
u ) -2u2F
() ]
-x1 -x1 L 1 L 2 x1 ) -2u2 ) u1 L L x2 2 x2 L
(13)
and x1 + x2 ) L
(14)
From eqs 13 and 14, the stagnation point offset at a < 1 is obtained: ∆x 1 1 ) L 2 2 - √1 - a
L/D < 2
(15)
At M < 1 and for the two same sized nozzles, it can be expressed as ∆x ) L
1 2 - √1 - √M
-
1 2
L/D < 2
(16)
That eq 16 can be used to estimate the stagnation point of the opposed jets from two same size nozzles of small separations, especially for L e 1D, is confirmed by the close agreements of the simulations with the experimental results, as
Figure 10. Normalized stagnation point offsets for opposed jets from nozzles of small separation.
illustrated in Figure 10 for the stagnation point offsets at various exit velocity ratios and L ) 0.5D and 1.0D. 3.3.2. Middle Nozzle Separation (2 e L/D e 12). The stagnation position is very sensitive to the exit velocity ratio at a middle nozzle separation (Figure 9); the sensitivity may be due to the changes in the mechanical energy and/or the static pressure of the two jets. At a middle nozzle separation, the jets from the two nozzles are in the potential core region, and the axial velocity is equal to the exit nozzle velocity u0, so the kinetic energy remains unchanged. The mean static pressure of the free jet decreases to negative values as the free jet develops.28 When the exit velocities from the two nozzles were slightly different, for example, at a ) u1/u2 ) 0.97 and Fu21/Fu22 ) 0.94, the stagnation point will move toward nozzle I. The decrease in static pressure from nozzle I is larger than that from nozzle II. At the stagnation plane, the mechanical energies of the fluid element from the two nozzles are equal: P1 + 1/2Fu21 ) P2 + 1/2Fu22 or P1 - P2 ) 0.03Fu22. The static pressure of the free jet decreases slowly with increases in the axial distance x/D;28 the stagnation point offset is very large, and the stagnation point will move closely to nozzle I, resulting in a higher P2 to balance the total pressure. At small nozzle separations, the impinging region is large relative to the separation, which results in a notable change in the static pressure along the axis upon a small shift in the stagnation point offset such that the exit velocity ratio does not strongly affect the offset. At large nozzle separations, the static pressure is close to the ambient pressure, and the change in the axial velocity which affects the kinetic head is much larger than that of the static pressure, so the stagnation point offset is also not sensitive to the exit velocity ratio. When the two nozzles are equal in size, the experimental stagnation point offset at 2D e L e 12D can be fit by the following equation: ∆x L ) (-0.00150 ln(a))0.207 L D
()
0.712
( DL ) )
(
1.91
exp -0.150
2 e L/D e 12 (17) Equation 17 can be expressed in terms of the momentum ratio as ∆x L ) (-0.00075 ln(M))0.207 L D
0.712
()
(
( DL ) )
exp -0.150
1.91
2 e L/D e 12 (18) According to eq 18, the normalized stagnation point offset will increase when the momentum ratio deviates from 1 and the relation of the stagnation point offset with the nozzle
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∆x/L )
(
)
s0 1 - √M 1 + 2 L 1 + √M
L/D > 12
(25)
Assuming t ) ln(M), with M ) e2t, where t is a variable, eq 25 becomes ∆x/L )
(
)
s0 1 - et 1 + 2 L 1 + et
(26)
When M is 0.2-5, t is -0.8 to +0.8, and using Taylor expansion with high t order terms neglected, eq 26 can be expressed as ∆x/L ≈ Figure 11. Correlation of the experimental results of ∆x/L at middle nozzle separations.
separations is nonlinear. Figure 11 presents the correlations of the experimental results of the two measuring techniques and the calculated ∆x/L for the nozzle separations; the high correlation coefficient (0.975) and small mean relative error (8.81%) suggest that those equations can indeed be used for estimating the offset. 3.3.3. Large Nozzle Separation (L/D > 12). With the distance from the virtual origin to the jet cross-section defined as x + s0 (Figure 3) and for nozzles with a constant s0 (Figure 2), the axial velocity at the developed region of a round free jet is29 umax ) const ×
1 × (x + s0)
MF
umax,1 ) const ×
1 × (x1 + s0)
M1 F
(20)
umax,2 ) const ×
1 × (x2 + s0)
M2 F
(21)
At a large nozzle separation, the static pressure is close to the ambient pressure, and the mechanical energy of the fluid element and the kinetic head is equal at umax,1 ) umax,2. So the stagnation point on the axis is the position at umax,1 ) umax,2. For two opposed nozzle jets, eqs 20 and 21 give x1 + s 0 ) √M x2 + s 0
(22)
The nozzle separation is L ) x 1 + x2
(23)
Combining eqs 22 and 23 produces x1 )
(L + s0)√M - s0 1 + √M
So the normalized stagnation point offset is
(24)
)
s0 1 ln(M) + × 2 L 4
L/D > 12
(27)
At large nozzle separations, eq 27 states that the stagnation point is in the center of the axis of the two nozzles (∆x/L ) 0) for two jets of equal momentum (M ) 1). Figure 12 shows the normalized stagnation point offsets of the opposed jets with two unequal nozzles at various exit j and 66.7D j . The stagnation momentum ratios for L ) 33.3D point moves to the nozzle with a smaller momentum continually as the exit nozzle momentum ratio increases for the two large nozzle separations; the stagnation point is in the center of the axis of the two nozzles at M ) 1. By fitting eqs 25 and 27 with the experimental results, s0 is estimated to be 104 mm. So the normalized stagnation point offset is ∆x/L )
(19)
For two opposed jets, the jets reach the developed region before the jets impinge when the nozzle separation is larger than 12D; the axial velocities from the two nozzles are
(
( 21 + 104L ) 11 +- √√MM
L/D > 12
(28)
104 1 1 + ln(M) L/D > 12 4 2 L
(29)
or ∆x/L ) -
(
)
Figure 12 presents the stagnation point offset values measured by HWA and those calculated using eq 29 for two unequal j ) 33.3-66.7 and opposed nozzles with opposed jets of L/D for two equal nozzles (D ) 10-30 mm) with opposed jets of L/D ) 16-150. The high correlation coefficient of 0.979 and the small relative error of about 8% have demonstrated that eq 28 or 29 can forecast the stagnation point offset precisely, for either two equal nozzles or unequal nozzles at large nozzle separations. 4. Conclusions The axial velocity and the stagnation point offset of impinging streams from two opposed nozzles have been theoretically studied and experimentally measured by smoke wire visualization and HWA techniques. The following conclusions are supported by the results: 1. The axial velocities of two opposed jets are symmetric around the stagnation point. 2. For small nozzle separation (L/D < 2), the equations based on the Reynolds stress theory can be used to predict the stagnation point offset for two nozzles of the same size. 3. For middle nozzle separation (2 e L/D e 12), the stagnation position is very sensitive to the exit velocity ratio, and it changes nonlinearly with the increasing nozzle separation.
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
Figure 12. Effects of momentum ratio M on the stagnation point offset.
4. For large nozzle separation (L/D > 12), the equations based on the free jet theory can be used to predict the stagnation point offset for two nozzles of the same and different sizes. 5. The illustrated effects of the exit momentum ratio and the nozzle separation on the stagnation point offset are useful for defining the optimum conditions for many industrial applications of the opposed jets. To achieve a full understanding of the opposed jets, additional studies are being conducted to determine the effects of turbulent intensity, the exit velocity profile and the exit Reynolds number on the stagnation point offset, and the instability of opposed jets at a middle nozzle separation. Acknowledgment This study was supported by grants from the China National Development Program of Key Fundamental Research (2010CB227004), China National Natural Science Foundation (50776033), Changjiang Scholar Program of Innovative Research Team in University (IRT0620), and New Century Excellent Talents in University Program of China Education Ministry (NCET-08-0775). Nomenclature a ) exit velocity ratio j ) nozzle diameter, mm D, D1, D2, D L ) nozzle separation, mm M1, M2 ) exit momentum, kg · m/s M ) exit momentum ratio Re ) exit Reynolds Number s0 ) the distance from origin to nozzle plane, mm u0, u1, u2 ) exit velocity, m/s x, x1, x2 ) the distance from stagnation point to nozzle plane, mm
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ReceiVed for reView July 1, 2009 ReVised manuscript receiVed April 12, 2010 Accepted April 23, 2010 IE901056X
