Mixing and Entrainment Characteristics in Circular Short Ejectors

Transcription

1 Mixing and Entrainment Characteristics in Circular Short Ejectors Ju Hyun Im 1 Agency for Defense Development, 4th R&D Institute Yuseong, P.O. Box 35, Daejeon , South Korea juhyunim@add.re.kr Seung Jin Song Department of Mechanical and Aerospace Engineering, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul , South Korea sjsong@snu.ac.kr Analytical and experimental investigations have been conducted to characterize the performance of short ejectors. In short ejectors, the core of primary (motive) flow still exists at the mixing duct exit, and nonuniform mixed flow is discharged from the mixing duct. Due to incomplete mixing, short ejector pumping performance is degraded and cannot be predicted by the existing long ejector models. The new analytical short ejector model presented in this paper is based on the control volume analysis and jet expansion model. The secondary (entrained) flow velocity and the corresponding shear layer (between the primary and the secondary flows) growth rate variations along the mixing duct are taken into account. In addition, measurements have been made in ejectors with length ratios (LRs) of two and three for an area ratio (AR) of 1.95; and a LR of two for an AR of Velocity profiles at the mixing duct inlet and exit, and static pressure distribution along the mixing duct have been measured with pitot probes and pressure taps. All of the tests have been carried out at a Reynolds number of 420,000. Comparison shows that the new ejector model can accurately predict flow characteristics and performance of short ejectors. For all of the test cases, the velocity profiles at the mixing duct inlet and exit are well predicted. Also, both predictions and measurements show pumping enhancement with increasing mixing duct length. The pumping enhancement is due to the increase in the static pressure difference between the mixing duct inlet and atmosphere as the mixing duct is lengthened. Furthermore, both measured and predicted static pressure distributions along the mixing duct show a kink. According to the analysis, the kink occurs when the outer shear layer reaches the mixing duct wall, and the secondary flow velocity decreases along the mixing duct upstream of the kink and increases downstream of the kink. Thus, the new ejector model can accurately predict not only the integral performance but also different mixing regimes in short ejectors. [DOI: / ] Introduction An ejector entrains external fluid, or the secondary flow, without moving parts as the motive flow, or the primary flow, is discharged into a mixing duct at high velocity. It is composed of a nozzle and a mixing duct (Fig. 1). Ejectors have several advantages, including its simplicity, reliability, and durability. Therefore, ejectors have been widely used in various applications. Ejectors are often used in air-conditioning and refrigeration systems to pump refrigerants [1,2]. Recently, ejectors have been also considered for fuel recirculation in fuel cell vehicles [3,4]. In these applications, ejectors have long mixing ducts whose length (L) is ordinarily longer than seven times the mixing duct diameter (D). Thus, they achieve complete mixing, or uniform exit flow. Ejectors have also been used in aviation to reduce noise and increase thrust efficiency of aircrafts [5] as well as to suppress helicopter infrared signals [6 8]. In these applications, ejectors have to be compact (short with a small cross-sectional area) to minimize weight and drag, and the mixing duct length is commonly much shorter than seven times the mixing duct diameter. For long ejectors, many experimental and analytical studies have been conducted. Experimentally, Keenan et al. [9,10] recommended a mixing duct length seven to eight times its diameter to achieve good pumping performance for ejectors with an AR of 256. Kastner and Spooner [11] also observed good pumping performance for mixing duct LRs of 7.5 for ejectors with additional long diffusers. To predict long ejector performance including pumping ( _m s = _m p ) and static pressure rise, several analytical models have been suggested. Keenan and Neumann [9] suggested a one-dimensional (1D) ejector model based on the conservation laws (mass, momentum, and energy) with assumptions of negligible friction on the mixing duct wall and adiabatic process in the ejector. Keenan et al. [10] extended Keenan and Neumann s model [9] to include isentropic diffusion effect and compared predictions with experimental results. Sokolov and Hershgal [12] suggested a design procedure for ejectors in refrigeration systems and introduced a real gas equation of state for refrigerants into Keenan s model. Eames and Aphornratana [2] added frictional loss terms to Keenan s model and predicted coefficient of performance in refrigeration cycles. Huang et al. [13] used an analytical model (similar to the 1D model of Eames) to predict ejector pumping performance. Huang s 1D analysis required empirical coefficients to match the performance of ejectors. Kim and Kwon [14] analyzed ejector performance to predict the pressure ratio 1 Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 18, 2014; final manuscript received December 13, 2014; published online February 2, Assoc. Editor: Kwang-Yong Kim. Fig. 1 Geometry of ejector Journal of Fluids Engineering Copyright VC 2015 by ASME MAY 2015, Vol. 137 /

2 between the primary flow and the secondary flow. The theoretical method used in the study was similar to Huang s 1D model [13]. Liao and Best [15] suggested a comprehensive 1D analytical model without the assumptions of constant-pressure and constantarea mixing which are used in previous long ejector models. Presz and Gritzer [16] analyzed a subsonic ejector operated in atmosphere, assuming complete mixing in the ejector. They need a control volume analysis (Fig. 2) to obtain a closed form equation to predict ejector pumping (Eq. (1)) " _m 2 # s T s A 2 s ffiffiffiffiffi p _m s T s þ1 þ 4 2 _m p T p A s _m p T p A s A p ¼ 0 (1) In Presz and Gritzer s model, pumping depends on the AR (A s /A p ) and temperature ratio (T s /T p ) but not on the mixing duct length. Thus, the ejector models stated above are valid only for long ejectors with complete mixing in the duct. Only a few studies have been conducted on ejectors with short mixing ducts. Quinn [17] measured pumping degradation in ejectors with decreasing mixing duct length, but he tested cases with an AR of 25.8 which were much larger than the values of two to four common in aerospace applications. Toulmay [18] measured pumping in ejectors with different mixing duct ARs for a fixed mixing duct LR of two and confirmed pumping performance deterioration. Analytical models to predict performance of ejectors with short mixing ducts have been suggested by Toulmay [18] and Millsaps and Markowicz [19]. Toulmay [18] and Millsaps and Markowicz [19] used the jet expansion model of Abramovich [20] and control volume analysis to predict flow nonuniformity at the mixing duct exit. In Abramovich s jet expansion model, the shear layer between the primary and the secondary flows grows linearly, and the growth rate depends on the velocity ratio of the primary and the secondary flows. However, flows in ejectors are different because the primary jet expansion is constrained by the mixing duct wall. Therefore, the secondary flow velocity and growth rate of the shear layer have to vary along the mixing duct to satisfy continuity. However, such effects were not considered by Toulmay and Millsaps and Markowicz. Thus, accurate analytical models for ejectors with short mixing ducts are still lacking, and there is a dearth of corresponding experimental data. Therefore, this paper presents a new analytical model for ejectors with short mixing ducts and presents corresponding measurements, including pumping and static pressure distribution. This paper focuses on ejectors with small ARs typically found in aircrafts and helicopters. The objectives of this research are: (1) to develop a new analytical model to predict short ejector performance and (2) to validate new the model s predictions with experiments. flow disappears, and the centerline velocity (velocity at the center of the mixed flow) decreases with mixing progress. In the short and medium ejectors, nonuniform mixed flow is discharged from the mixing duct exit due to incomplete mixing. Compared to the short and medium ejectors, the long ejectors achieve complete mixing and discharge uniform mixed flow from the mixing duct exit. In the short ejector model, the following assumptions are made. First, the flow is steady, incompressible, and 1D. The static pressure distribution at the mixing duct inlet plane is uniform. The secondary flow is entrained from, and the mixed flow is discharged to the atmosphere. Then, shear at the mixing duct wall is negligible. The input variables are as follows. Geometric variables include the primary flow area (A p ), the secondary flow area (A s ), and the mixing duct length (L). Flow variables include the primary flow velocity (u p ) and temperature (T p ). Atmospheric conditions (P atm, T atm ) are also needed in the model. The output variables include the radial locations of inner and outer shear layers (r 1, r 2 ), the secondary flow velocities at the mixing duct inlet and exit (u s,i, u s,e ), and the secondary flow temperature at the mixing duct exit (T s,e ). Subsequently, velocity and temperature profiles of the mixed flow at the mixing duct exit, pumping performance ( _m s / _m p ), and static pressure distribution along the mixing duct can be predicted by the short ejector model. To analyze short ejectors, control volume analysis and jet expansion theory are used. The control volume encloses the mixing duct inlet and exit and the mixing duct wall (Fig. 4), and analysis is based on the conservation laws. A modified form of the jet expansion model of Abramovich [20] is used to model mixing (shear layer width and radial location). The mixing duct constrains jet expansion; the secondary flow velocity varies with mixing progress; and the corresponding shear layer growth rate varies as well. The variation of the shear layer growth rate along the mixing duct is taken into account by using a discrete calculation method (Fig. 5). Short Ejector Model Description Based on the flow regimes of the mixed flow at the mixing duct exit, the ejectors are classified as short, medium, or long (Fig. 3). In the short ejectors, the core of the primary flow still exists at the mixing duct exit. In the medium ejectors, the core Fig. 2 Control volume applied to long ejector model Fig. 3 Classification of ejectors based on mixing duct length and flow regime: (a) short ejector, (b) medium ejector, and (c) long ejector / Vol. 137, MAY 2015 Transactions of the ASME

3 2 FðgÞ ¼ 1 g 3=2 (6) uðgþ ¼1 g 3=2 (7) where FðgÞ ¼ u p u u p u s (8) Fig. 4 Control volume of short ejector uðgþ ¼ T p T T p T s (9) In the short ejector analysis, a static pressure value at the mixing duct inlet (P s,0 ) is first assumed. Then, the secondary flow velocity at the mixing duct inlet u s,0 is determined from the assumed static pressure based on the Bernoulli equation. After setting of the secondary flow velocity at the mixing duct inlet, the analysis based on the modified jet expansion model begins with the velocity ratio at the mixing duct inlet u s,0 /u p. The modified jet expansion model is used to predict the shear layer locations (y 1, y 2 ) with conservation laws. The jet expansion in the mixing duct is calculated discretely along the mixing duct (Fig. 5). The shear layer thickness (b) and locations (y 1, y 2 )atnth location are calculated from the values at the previous (n 1th) location. m n 1 ¼ u s;n 1 u p (2) b n ¼ b n 1 þ c L 1 m n 1 n 1 þ m n 1 (3) y 1;n ¼ y 1;n 1 þ ðb n b n 1 Þ 0:416 þ 0:134m n 1 þ 0:021 b n b n 1 r 0 y 2;n ¼ y 2;n 1 þ ðb n b n 1 Þ þ 0:021 b n b n 1 r 0 1 þ 0:8m n 1 0:45m 2 n 1 (4) 0:584 þ 0:134m n 1 1 þ 0:8m n 1 0:45m 2 n 1 (5) In Eq. (3), c is an empirical constant and has a value of 0.27 in the short ejector model [20], and L=n is the element length, dx, in the discrete calculation. After obtaining the shear layer locations, y 1 and y 2, the secondary flow velocity and temperatures are obtained from conservation laws. At a given location, velocity and temperature in the primary flow core remain constant. The velocity and the temperature profiles in the shear region (between the inner and outer shear layers) are assumed to be similar [20,21]. The new ejector model uses the similarity functions for the velocity and the temperature profiles suggested by Abramovich [20]. They are expressed as g ¼ r 2 r r 2 r 1 (10) Then, these similarity equations are converted to where! r 2 r 3=2 2 ur ðþ¼u p u p u s 1 (11) r 2 r 1! r 2 r 3=2 Tr ðþ¼t p T p T s 1 (12) r 2 r 1 r 1 ¼ r 0 y 1 (13) r 2 ¼ r 0 y 2 (14) Therefore, when the primary flow velocity u p and temperature T p, and the similarity functions for velocity and temperature profiles in the shear region are given, the secondary flow velocity (u s ) and temperature (T s ) are obtained from the continuity (Eq. (15)) and energy conservation equation (Eq. (16)), respectively. Then, static pressure is calculated from the momentum conservation equation (Eq. (17)) _m p þ _m s ¼ _m core þ _m mix þ _m sec ¼ q p A core;n u p þ ð 2p ð r2;n _m p T t;p þ _m s T t;s;0 ¼ q p A core;n u p T t;p þ 0 qðþur r ðþrdrdh þ q s A s;n u s;n r 1;n ð 2p ð r2;n 0 (15) qðþur r ðþt t ðþrdrdh r r 1;n þ q s A s;n u s;n T t;s;n (16) q p A p u 2 p þ q sa s u 2 s;0 þ P s;0 A p þ A s ¼ qp A core;n u 2 p ð 2p ð r2;n þ qðþur r 2 rdrdh þ q s A s;n u 2 s;n þ P s;n A p þ A s r 1;n 0 (17) These calculation procedures (Eqs. (2) (17)) are repeated from the mixing duct inlet to the exit. Therefore, in the short ejector model, the flow characteristics inside as well as those at the mixing duct exit can be predicted. After completing the calculations, the predicted static pressure at the mixing duct exit is compared to the atmospheric pressure to verify the matching condition in the following equation: P s;e ¼ P atm (18) Fig. 5 Schematic of discrete calculation method for predicting shear layer in ejector If this pressure matching condition is not satisfied, the mixing duct inlet static pressure is re-assumed, and the analysis is repeated until the matching condition is satisfied. Once Eq. (18) is Journal of Fluids Engineering MAY 2015, Vol. 137 /

4 Fig. 6 Calculation process in short ejector model satisfied, the corresponding pumping, static pressure rise, velocity, and temperature profiles at the mixing duct exit can be obtained. Figure 6 shows the flow chart of the short ejector model, including the calculation procedure and the corresponding equation numbers. Seoul National University (SNU) Ejector Measurements New measurements have been made in short ejectors at SNU for comparison with the new model s predictions. Test Facility. The primary flow is supplied through a settling chamber by a 22 kw centrifugal blower. Flow speed is controlled by an inverter. Two mesh-screens and one honeycomb block are installed inside the settling chamber to reduce turbulence intensity of the flow. The freestream turbulence intensity in the primary flow is 0.45%. In this study, turbulence intensity has been measured using a hot-wire anemometry with a sampling rate of 10 khz. The ejector test section is composed of the primary flow nozzle and a mixing duct, both with circular cross sections. A schematic of the test section is shown in Fig. 7. During the tests, the Mach number and Reynolds number based on the nozzle diameter and the primary flow velocity have been set to 0.18 and , respectively. The test section and experimental equipment for ejector performance test are shown in Fig. 8. Fig. 7 Schematic of ejector test section and measurement locations / Vol. 137, MAY 2015 Transactions of the ASME

5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðP t P s Þ u ¼ q (19) Fig. 8 Test section and experimental equipment Here, density q is calculated from the atmospheric pressure and temperature. q ¼ P atm RT atm (20) Next, the corresponding mass flow rates are determined as _m p ¼ X qu p da p (21) _m s ¼ X qu s da s (22) Instrumentation. Pitot probes with a probe diameter of 2.4 mm have been traversed at the primary nozzle exit, secondary flow entrance, and mixing duct exit to obtain pressure and velocity distributions (Fig. 7). The measurement grids at the mixing duct inlet and exit are shown in Fig. 9. For the primary flow, velocity and pressure have been measured at 25 radial locations (2 mm resolution) for azimuthal locations of 0 deg, 90 deg, 180 deg, and 270 deg. Secondary flow has been measured at 16 and 23 radial points (2 mm resolution) at the same azimuthal locations for the ARs of 1.95 and 3.08, respectively. The exit flow has been measured at 29 and 34 radial points (3 mm resolution) at eight azimuthal locations for the ARs of 1.95 and 3.08, respectively. In addition, on the mixing duct walls, five to eight static pressure taps have been installed to measure axial pressure distributions in the mixing duct. In the experiments, pressure data have been acquired using a PSI 9116 Pneumatic Intelligent Pressure Scanner with a pressure range of 5.0 kpa and an accuracy of 60.15% full scale. Uncertainty of the pressure scanner has been separately checked in this study. Based on the measured 120 samples, the pressure data have 0.4% of uncertainty with a 95% confidence interval at gauge pressure of 1000 Pa [22]. The reference condition was selected based on the measured mass averaged total pressure at the mixing duct exit for an ejector with an AR of 1.95 and a LR of three. Pitot probes have been traversed in the measurement plane by a Velmex linear traverse system with a spatial resolution of mm. Atmospheric pressure and temperature have been measured using a barometer with a resolution of 100 Pa and an accuracy of 100 Pa, and a thermometer with a resolution of 0.1 C and an accuracy of 61 C. From the measurements, velocity profiles at the mixing duct inlet and exit, pumping, and static pressure distribution along the mixing duct are obtained [23]. Data Reduction and Uncertainty in Measurements. First, the velocities of the primary, secondary, and mixed flows are calculated from the pitot probe data. _m m ¼ X qu m da m (23) Then, ejector pumping is calculated as the mass flow rate ratio of the secondary flow to the primary flow [16]. Pumping ¼ _m s _m p (24) Static pressure (P s ) is converted into the pressure coefficient (C p )as C p ¼ P s P atm ð1=2þqu 2 p (25) where P s indicates the static pressure measured via static taps on the mixing duct. The uncertainties of velocity, ejector pumping, and static pressure coefficient are decided based on that of pressure data, and have values of 0.3%, 0.4%, and 0.8%, respectively, with a 95% confidence interval [22]. Test Matrix. For one primary flow nozzle, three mixing ducts of varying ARs and LRs have been tested. The primary flow nozzle has a diameter of 99 mm. The AR is defined as the ratio of the secondary flow area (A s ) to the primary flow area (A p ). The LR is defined as the mixing duct length (L) divided by its diameter (D). Ejectors with LRs of two and three for an AR of 1.95 and a LR of two for an AR of 3.08 have been tested. Results and Discussions Velocity Profiles. In Fig. 10, the predicted velocity profiles of the mixing duct inlet flows (primary and secondary flows) are compared to the measured profiles. The predicted velocity profiles of the primary and the secondary flows are uniform at the mixing duct inlet. Compared to the predicted values, the measured Fig. 9 Measurement grid (AR ): (a) mixing duct inlet and (b) mixing duct exit Journal of Fluids Engineering MAY 2015, Vol. 137 /

6 Fig. 10 Comparison of predicted and measured velocity profiles at the mixing duct inlet of SNU short ejectors at Re 5 420,000: (a)ar51.95, LR 5 2, (b) AR5 1.95, LR 5 3, and (c)ar53.08, LR 5 2 Fig. 11 Comparison of predicted and measured velocity profiles at the mixing duct exit of SNU short ejectors at Re 5 420,000: (a)ar5 1.95, LR 5 2, (b)ar5 1.95, LR 5 3, and (c) AR5 3.08, LR 5 2 velocity profiles of inlet flows are slightly nonuniform. The nonuniformity is due to: (1) the boundary layer near the rim of the primary nozzle and (2) the separation bubble at the mixing duct inlet for the secondary flow. 2 Due to the separation bubble, the secondary flow has its maximum at r/r ¼ 0.9. Despite small differences, the predicted profiles are well matched with the measured profiles. For the ejectors with an AR of 1.95, the increase in the secondary flow velocity is observed in both predicted and measured profiles (Figs. 10(a) and 10(b)) when the mixing duct LR increases from two to three. These increases in the secondary flow velocity explain the pumping enhancement with increasing mixing duct length. Figure 11 shows the comparison of the predicted and the measured velocity profiles of the mixed flows at the ejector exit. Near the center of the exit plane of the mixing duct (r/r ¼ 0), the core flow region is clearly visible in both the predicted and measured profiles. Also, for the ejectors with the AR of 1.95, the decreased radial extent of the core region is captured by the short ejector model (Figs. 11(a) and 11(b)) when the mixing duct LR increases from two to three. For all of the cases, the velocity profiles at the mixing duct exit are accurately predicted. 3 Figure 12 shows the mixing process between the primary and secondary flows. The predicted velocity profiles of the mixed flow at x/l ¼ 0, 0.33, 0.67, and 1 are plotted. As the mixing progresses, the radial extent of the core region is decreased and the shear region increased. The secondary flow velocity decreases until the shear layer reaches the mixing duct, and then increases. The 2 The separation bubble has been verified via flow-visualization. 3 The temperature profiles at the mixing duct exit have not been measured in this research because both primary and secondary flow temperatures were the same. However, the temperature scaling approximations can be found in Munk and Prim [23]. reason of the secondary flow velocity variation is discussed later in section Mixing Regimes in the Mixing Duct. Validation of Pumping Measurement and New Short Ejector Model Pumping. From velocity predictions, pumping can be evaluated and compared versus experimental data. First, however, the measured pumping from the current study is compared to the measurements of Toulmay [18] and McBean and Birk [24]. Toulmay measured pumping for ejectors with a fixed LR of two and three different ARs of 0.53, 1.47, and McBean and Birk [24] measured ejector pumping for a fixed AR of three and three different LRs of 2.04, 2.49, and For similar area and LRs, pumping from both experiments agree well. In addition, pumping predictions from the new short ejector model are also plotted in Fig. 13. Predictions and measurements from Toulmay, McBean, and Birk, and SNU all agree well. These accurate predictions prove the reliability of the new short ejector model. The measured and the predicted pumping performances are listed in Table 1. To compare the new short ejector model to Toulmay model, pumping performances have been predicted by new model for the test cases in Toulmay. Between the measurements and predictions listed in Table 1, the smaller discrepancies are found in the new model. Applying the effect of variations of the secondary flow velocity and growth rate of the shear layer to the analytical model enhances the accuracy of prediction. Figure 14 shows both the predicted and the measured pumping performance of SNU short ejectors plotted versus LR. In addition, the measured pumping performances in McBean and Birk [24] are plotted together in the figure. Even though the AR of ejectors shows small difference between SNU cases and McBean and / Vol. 137, MAY 2015 Transactions of the ASME

7 Fig. 12 Velocity profiles inside mixing duct at x/l 5 0, 0.33, 0.67, and 1 for SNU short ejectors at Re 5 420,000: (a) AR5 1.95, LR 5 2, (b)ar5 1.95, LR 5 3, and (c)ar5 3.08, LR 5 2 Fig. 13 Comparison of measured pumping from SNU, Toulmay [18], McBean and Birk [24], and predictions Birk, predictions and measurements from two studies agree well. The short ejector model can accurately (to within 5.4%) predict pumping performance of ejectors with LRs up to 3.1 for ejectors with an AR of 1.95; and with LRs up to 2.8 for ejectors with an AR of Beyond these LRs, the short ejector model is no longer applicable because the core flow disappears at the mixing duct exit. The values of the measured and predicted pumping are listed in Table 2. The short ejector model also captures the pumping enhancement with increasing mixing duct length. This pumping enhancement is consistent with the results of predicted secondary flow velocity profiles in Fig. 10. The mechanism of pumping enhancement is discussed next. Table 1 Comparison of the measured pumping performances of Toulmay [18] and McBean and Birk [24] and predicted by new short ejector model Geometry Prediction (discrepancy (%)) A s /A p L/D Measurement Toulmay model Short ejector model a a (þ2.20) b (þ1.00) a a ( 7.29) b ( 3.75) a a ( 3.20) b ( 2.21) c b (þ1.71) c b (þ2.10) c b (þ1.71) a Toulmay [18]. b Short ejector model. c McBean and Birk [24]. Fig. 14 Analytical and experimental pumping performance of SNU short ejectors at Re 5 420,000 Pumping Dependence on the Mixing Duct Length. In ejectors, the secondary flow is entrained by the static pressure difference at the mixing duct inlet. Then, the static pressure rises along the mixing duct as the mixing between the primary and secondary flows progresses and matches the atmospheric pressure at the mixing duct exit. As the mixing duct lengthens, more mixing, or higher static pressure rise, occurs in the mixing duct. However, the exit pressure remains atmospheric. Therefore, for longer ejectors, the static pressure at the mixing duct inlet has to decrease, thus entraining more secondary flow and enhancing ejector pumping (Fig. 15). To confirm pumping enhancement mechanism, the static pressure distribution has been predicted with the short ejector model and measured in SNU tests. Figure 16 shows the predicted and the measured static pressure distributions along the mixing duct of short ejectors with an AR of Near the mixing duct inlet (x/d ¼ 0), the differences between the predicted and the measured values arise mainly due to the presence of separation bubble at the Table 2 Measured and predicted pumping performance of SNU short ejectors A s /A p Geometry L/D Measurement (A) Prediction (B) Discrepancy (%) ((B A)/B) þ1.01 Journal of Fluids Engineering MAY 2015, Vol. 137 /

8 Fig. 15 Schematic of static pressure rise with mixing progress in the mixing duct inlet during experiment. This effect is not considered in the short ejector model. In such separation bubbles, a steep increase in static pressure has been also found by Cherry et al. [25] and Ruderich and Fernholz [26]. Except near the separation bubble region, the new ejector model predicts the static pressure distribution along the mixing duct well. For both predicted and measured static pressure distributions, the static pressure rises along the mixing duct and approaches the atmospheric pressure at the mixing duct exit. Also, the predicted static pressures at the mixing duct inlet decrease as the mixing duct is lengthened from two to three, validating the pumping enhancement mechanism suggested in Fig. 15. Mixing Regimes in the Mixing Duct. For the predicted static pressure distributions, kinks are observed in Fig. 16. The kink is much clearer at x/d ¼ 2.33 for LR ¼ 3. In the mixing duct, the static pressure rise pattern changes between upstream and downstream of the kink (Fig. 16). Upstream of the kink, the static pressure increases with ðd 2 P s =dx 2 Þ > 0 as the mixing progresses. Downstream of the kink, the static pressure increases with ðd 2 P s =dx 2 Þ < 0. According to the short ejector model, the kink occurs when the outer shear layer meets the mixing duct wall (Fig. 17). Upstream of the kink, the velocity profile variation with mixing is shown qualitatively in Fig. 18(a). The secondary flow velocity (u s in Fig. 17) has to decreases along the axial direction because the secondary flow is continuously entrained into the mixing region through the shear layer. The decrease of the secondary flow velocity is analytically verified by mass conservation constraint. Thus, the difference between the core flow velocity (which remains constant) and that of the secondary flow increases with mixing progress. Due to the increasing velocity difference, the static pressure rises at an increasing rate, or ðd 2 P s =dx 2 Þ > 0. Downstream of the kink, the schematic of velocity profile Fig. 16 Experimental and analytical static pressure distribution along the mixing ducts at Re 5 420,000 (AR ) Fig. 17 Schematic of static pressure rise patterns in ejector / Vol. 137, MAY 2015 Transactions of the ASME

9 Fig. 18 Regimes of secondary velocity variation relative to the kink: (a) upstream of kink and (b) downstream of kink variation with mixing progress is shown in Fig. 18(b). In this region, the secondary flow velocity (u s ) increases in the axial direction as mixing progresses via momentum transfer from the high velocity region. Consequently, the velocity difference of the core flow and the secondary flow decreases, and the static pressure rises at a decreasing rate, or ðd 2 P s =dx 2 Þ < 0. Measured static pressure distributions show a similar pattern (Fig. 16). Thus, the short ejector model can accurately predict the location where the outer shear layer reaches the mixing duct wall and also the different mixing regimes upstream and downstream of that location. In summary, the short ejector model can predict not only the velocity profiles at the mixing duct inlet and exit but also flow characteristics inside the mixing duct. Conclusions Short ejector has been defined and its performance has been investigated analytically and experimentally. From this study, a new analytical model for short ejectors has been developed based on the control volume analysis and jet expansion model. The development of new ejector model can be useful to design and analyze ejector devices with short mixing ducts such as the infrared suppressor attached to the helicopter and the exhaust nozzle of the aircraft. In the analytical model, the jet expansion model has been modified to consider the variation of the secondary flow velocity and the corresponding shear layer growth rate in the mixing duct. The short ejector model has been validated by comparison with experimental results. Short ejector model can accurately predict velocity profiles and ejector pumping. In short ejectors, pumping performance is enhanced with increasing mixing duct length because the static pressure difference between the mixing duct inlet and atmosphere increases, entraining more secondary flow. In addition, the kink in the static pressure distribution is captured by the short ejector model, and verified from the measured static pressure distribution. The kink occurs due to the attachment of the outer shear layer to the mixing duct wall. Upstream of the kink, the secondary flow velocity decreases with mixing; and downstream of the kink, the secondary flow velocity increases with mixing. The kink is a result showing the change of mixing regime. Acknowledgment The authors gratefully acknowledge financial support from the BK21 Program of Korean Government, Korea Aerospace Industries (KAI) (Project No ), and Institute for Advanced Machinery and Design (IAMD) of Seoul National University. Nomenclature A ¼ area AR ¼ area ratio (A s /A p ) b ¼ shear layer thickness C p ¼ static pressure coefficient P s P atm = ð1=2þqu 2 p d ¼ nozzle diameter D ¼ mixing duct diameter L ¼ mixing duct length LR ¼ mixing duct length ratio (L/D) m ¼ velocity ratio (u s /u p ) _m ¼ mass flow rate _m s = _m p ¼ pumping performance P s ¼ static pressure P t ¼ total pressure P atm ¼ atmospheric pressure r ¼ radial location from the center of mixing duct r 0 ¼ nozzle radius R ¼ Mixing duct radius Re ¼ Reynolds number based on the nozzle diameter and primary flow velocity (u p d=t) T ¼ temperature u ¼ velocity x ¼ distance from the mixing duct inlet y ¼ radial location of shear layer q ¼ density g ¼ ððr 2 rþ= ðr 2 r 1 ÞÞ Subscripts atm ¼ atmosphere core ¼ potential core region in mixing process e ¼ mixing duct exit i ¼ mixing duct inlet m ¼ mixed flow p ¼ primary flow s ¼ secondary flow sec ¼ secondary flow region in mixing process 1 ¼ inner shear layer 2 ¼ outer shear layer References [1] Elbel, S., and Hrnjak, P., 2008, Ejector Refrigeration: An Overview of Historical and Present Developments With Emphasis on Air Conditioning Applications, 12th International Refrigeration and Air Conditioning Conference at Purdue, Paper No Journal of Fluids Engineering MAY 2015, Vol. 137 /

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