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Numerical study for the influences of primary steam nozzle distance and mixing chamber throat diameter on steam ejector performance
International Journal of Thermal Sciences 132 (2018) 509–516
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Numerical study for the influences of primary steam nozzle distance and mixing chamber throat diameter on steam ejector performance
T
Weina Fu, Zhongliang Liu∗, Yanxia Li, Hongqiang Wu, Yongzhi Tang College of Environmental and Energy Engineering, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing, 100124, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Steam ejector CFD Primary steam nozzle distance Mixing chamber throat diameter Entrainment ratio
The geometry and operation parameters have important influences on the performance of steam ejectors. An axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector has been established in this paper to investigate the flow characteristics inside steam ejectors aiming at optimizing primary steam nozzle exit section distance and mixing chamber throat diameter simultaneously. The results show that there are an optimum value for the primary steam nozzle distance ratio and an optimum diameter ratio of mixing chamber throat to primary nozzle throat at which the steam ejector acquires its best entrainment performance under the given design conditions. With the optimized primary steam nozzle distance, the optimization of the diameter ratio of the mixing chamber throat to primary nozzle throat is significant. In our case, the improvement of the entrainment ratio is as large as 25%. Deviation from its optimized value may result in a serious degeneration in the ejector performance. Therefore, to ensure the steam ejector performance, the primary steam nozzle distance should be designed at its optimum value and the diameter ratio of the mixing chamber throat to primary nozzle throat be within a narrow vicinity of its optimum value.
1. Introduction Energy shortage and environmental pollution have become the bottleneck for sustainable economy and society development. Energy saving and emission reduction have thus received increasing attention from all over the world. Steam ejector is a kind of fluid machinery that uses high-pressure steam to pump low-pressure steam [1]. It has been widely used in thermal evaporation system [2,3], vacuum [4,5] and refrigeration [6,7], due to its convenient operation, simple structure and obvious energy saving effect. With the rapid development of CFD software, many scholars [8–11] have studied the ejector numerically. Adiabatic acceleration flow and expansion of the primary steam inside Laval nozzle (the primary steam nozzle) leads to the formation of supersonic and low-pressure region at the nozzle outlet section, and a pressure difference between this low-pressure region and the entrained steam is thus established and pushes the low-pressure entrained steam into the ejector. As one could well understand, the geometry of the primary steam nozzle has great influences on flow pattern and thus the ejectors' performance. Installing swirl vanes at the nozzle outlet section could improve the ejector entrainment performance and reduce the frictional loss [12]. Compared with circular, elliptical, rectangular nozzle ejector, the performance of cross nozzle ejector was the best [13]. Fu et al. [14] studied the influences of primary steam nozzle
∗
Corresponding author. E-mail address: [email protected] (Z. Liu).
https://doi.org/10.1016/j.ijthermalsci.2018.06.033 Received 6 June 2017; Received in revised form 25 June 2018; Accepted 26 June 2018
1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
outlet diameter on the flow characteristics and entrainment ratio of a steam ejector. Their results showed that the supersonic jet in the overexpanded wave state flowed more smoothly after leaving the nozzle, and therefore, the energy loss was relatively small and the ejector achieved a good performance. The primary steam nozzle position is also an important structural parameter for steam ejectors. The critical entrainment ratio of an ejector at different nozzle positions were experimentally studied by Chen et al. [15]. Their results showed that the entrainment ratio increased with the nozzle outlet position at first and then remained unchanged within the range covered, which meant that there was an optimum nozzle-exit position (NXP) at which the entrainment ratio would acquire its maximum value. It was found that the entrainment ratio was fully decided by the so-called double choking phenomenon that appeared in the converging section, i.e., only when both the primary and the entrained steam were at their critical state (maximum mass flowrate state) simultaneously, could the ejector obtain its largest entrainment ratio. It should be pointed out however, their experimental result was somewhat based on the assumption that the entrained flow velocity in the converging cone and the mixing flow velocity in the constant area section were both smaller than the sound velocity. Sag and Ersoy [16] studied the influence of the primary steam nozzle on the performance of a steam ejector used for refrigeration and they found experimentally that the ejector system that used the
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conditions. The effects of mixing chamber length and convergence angle on the performance of steam ejectors were investigated by Wu et al. [27], the shock wave and vortex inside the steam ejector under different conditions were discussed. Yang et al. [28] studied the influences of the nozzle structure on fluid streamline distribution and vortex structure inside the mixing chamber. Although many efforts have been made so far concerning the structure optimization of steam ejectors, however, little systematic research work has been reported in the literature for optimizing the primary steam nozzle position and the mixing chamber throat diameter simultaneously. As we have discussed earlier, these two structural parameters have direct and very strong influences on the ejector performance and may affect each other's optimum value, i.e. the change in one parameter may result in significant change in the optimum value of another. In this paper, an axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector has been established to explore and analyze the influences of primary steam nozzle distance and mixing chamber throat diameter on the ejector performance.
optimum primary steam nozzle throat diameter exhibited a higher COP than the classic system. Sun [17] established a thermodynamic model for ejector refrigeration systems and they found that using variable geometry (including NXP) ejector would enhance the performance of ejector refrigeration system. Dong et al. [18] set up a prototype steamejector refrigeration system and the effects of nozzle exit position (NXP) and the diameter of the constant area section on the working performance of steam ejector were investigated. Chunnanond and Aphornratana [19] constructed an experimental steam ejector refrigerator and the experiments were carried out to examine the influences of the operating conditions and the geometry including NXP on the system performance. They found that using smaller primary nozzle or retracing the nozzle out of mixing chamber could both increase the COP and the cooling capacity of the system. Riffat and Omer [20] carried out an experimental and numerical study of an ejector refrigeration system using methanol as the working fluid. 4 different NXPs were chosen to investigate the effect of the relative position of the primary nozzle exit within the mixing chamber on the performance of the ejector and the numerical results were used for optimizing ejector geometry. Their results showed that positioning the nozzle exit at least 0.21 length of the mixing chamber throat's diameter upstream of the entrance of the mixing chamber gave better performance than pushing it into the mixing chamber. Varga et al. [21] numerically studied the influences of area ratio between the nozzle and constant area section, nozzle exit position (NXP) and constant area section length on the ejector performance. Their results indicated that for ejectors obtaining optimum performance, all these three geometrical parameters should be optimized. Zhu et al. [22] optimized numerically of primary nozzle exit position and converging angle of mixing section. They found that these two geometry parameters both had very strong influence on ejector performance. The optimum NXP was obtained and was found to be 1.7–3.4 length of the mixing section throat diameter upstream of the start of the mixing section. One may understand that changing the primary steam nozzle structure and its relative position will change the vortex structure formed in the downstream of the primary steam nozzle as it has been reported in literature, therefore, there should be an optimum structure and position to ensure the ejector to obtain its best performance. Most researches used ideal gas model and the assumption of constant physical property. Wet steam models were also established by some researchers to study the flow inside ejectors. Ding et al. [23] studied nonequilibrium condensation process of water vapor in moist air expanding through a sonic nozzle by numerical simulation. They found that the flow rate of the sonic nozzle was affected by both homogeneous and heterogeneous nucleation. Abadi et al. [24] studied the unsteady supersonic flow of wet steam through a high-pressure thermo-compressor (steam ejector) with consideration of non-equilibrium homogeneous condensation numerically. Two-fluid multiphase flow formulations were used and their results were validated against the industrial data, which proved the effectiveness of the model for modeling steam condensing flows within high-pressure steam ejectors. However, due to the obvious difficulties and the less reliability of the physical, mathematical and numerical model in taking condensation process into consideration, most of the numerical optimization work reported in the literature concerning steam ejectors simply used ideal gas model. The performance of a steam ejector depends largely on the interactions between the high-pressure and the low-pressure steam. Therefore, during this process, very complex physical phenomena may occur, including transonic flow, shockwaves, mixing, phase change (condensation and evaporation), boundary separation and so on. The exploration of the mixing process inside an air steam ejector was carried out by Bouhanguel et al. [25] experimentally. The flow pattern, shock wave position and turbulence structure were observed visually and the analyses of the observed phenomena were presented. Ariafar et al. [26] studied the fluid mixing and the pressure driving effects on the entrainment ratio of steam ejectors under various operation
2. Physical and mathematical model The mathematical descriptions for the flow onside the steam ejector including the governing equations and boundary conditions used were exactly same as that given by Fu et al. [14]. The detailed CFD model, the validation of the grid independence, the physical and mathematical and the CFD model were all discussed in full details in Ref. [14] and are simply omitted here for conciseness. A schematic view of a typical supersonic ejector is shown in Fig. 1. Based on the assumption of one-dimensional steady state flow and constant-pressure mixing, the entrainment ratio and the main structural dimensions were calculated under the operation conditions (the primary steam pressure pp = 600 kPa, the entrained steam pressure ps = 15 kPa, and the mixed steam pressure pc = 40 kPa, which are something for the so-called low-temperature MEE desalination systems) and listed in Table 1. The main designed geometrical parameters of the ejector are listed in Table 2. Comparing with the supersonic jet of the nozzle exit, the initial velocity at the nozzle inlet may well be neglected. The lateral inlet of the entrained steam is simplified as a uniformly-distributed axial circular-inlet and thus the axisymmetric two-dimensional transonic compressible flow model could be used, as shown in Fig. 2. According to the shape of flow channel, the map-style block partition mesh is adopted to divide the flow domain structure of the ejector into 8 quadrilateral subdomains that are same as that in Ref. [14]. Second-order up-winding formulation is used to discretize convection terms and second-order central differencing for diffusion terms. To further ensure that the mathematical and numerical model can produce a reliable prediction of the performance of ejector, just like that in Ref. [14], a simulation was carried out to investigate the influence of mixing steam pressure pc on the performance of the steam ejector. Fig. 3 presents the simulated entrainment ratio ω as a function of the outlet mixing steam pressure pc under the given operation conditions (pp = 600 kPa, ps = 15 kPa). As one can see from this figure, the
Fig. 1. Schematic diagram of steam ejector. 510
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ejectors and has been proved by experiments. Therefore, this result proves that the mathematical and numerical model used here predicts the ejector performance correctly [1,29].
Table 1 Design operation parameters. Parameter primary steam pressure primary steam temperature entrained steam pressure entrained steam temperature mixed steam pressure mixed steam temperature
Symbol
Value
Unit
pp Tp ps Ts pc Tc
600 432.15 15 327.15 40 349.15
kPa K kPa K kPa K
3. Results and discussion 3.1. Optimization of primary steam nozzle distance The flow in primary steam nozzle is transonic, i.e., the flow in the convergence section is subsonic, in the throat section sonic and supersonic in the divergence section and viscous effect could not be neglected. Therefore, the performance of the primary steam nozzle has a direct influence on the flow characteristics and the entrainment performance and thus on the structure optimization of steam ejectors. The primary steam nozzle location is determined by the so-called primary steam nozzle distance lc that is defined as the distance between the outlet section of primary steam nozzle and the inlet section of cylindrical mixing chamber. To investigate the influence of the primary steam nozzle distance on ejector entrainment performance, the primary distance ratio γlc is defined,
Table 2 Geometrical parameters of the steam ejector. Parameter
Symbol
Value
Unit
The Dimensionlessa
Nozzle throat Diameter Nozzle entrance diameter Nozzle exit diameter Nozzle throat length Nozzle convergent section length Nozzle divergent section length Nozzle distance Mixing chamber entrance diameter Mixing chamber throat diameter Mixing chamber length Mixing chamber throat length Diffuser exit diameter Diffuser length
dp* dp dp1 s2 s1 s3 lc d2
3.70 12.00 10.00 5.00 10.00 25.00 62.90 23.20
mm mm mm mm mm mm mm mm
1.00 3.24 2.70 1.53 3.06 7.65 17.00 6.27
d3 l1 l2 dc l3
15.80 62.90 48.00 28.40 102.00
mm mm mm mm mm
4.27 19.24 14.68 7.68 27.57
a
γlc =
lc dp ∗
(9)
where dp* is the nozzle throat diameter. The entrainment ratio ω is calculated for various γlc for the given value of the nozzle throat diameter under the operation condition of the primary steam pressure pp = 600 kPa, the entrained steam pressure ps = 15 kPa, and the mixed steam pressure pc = 40 kPa. The designed outlet section of the primary steam nozzle is coincident with the conical inlet section of the mixing chamber, and the ratio of the primary steam nozzle distance to the primary steam nozzle throat diameter γlc is 17. Therefore, if γlc < 17, then the primary steam nozzle is actually placed inside the conical mixing section of the mixing chamber; and if γlc > 17 then it is in the upstream of the conical mixing section of the mixing chamber. Fig. 4 presents the entrainment ratio as a function of the primary steam nozzle distance ratio γlc for 3 primary steam nozzle exit diameter ratios (γd = 2.0, 2.5, and 3.0; where γd is defined as the ratio of the primary steam nozzle outlet diameter to the primary steam nozzle throat diameter). One can see from this figure, the entrainment ratio variation with the primary steam nozzle distance ratio for 3 primary steam nozzle outlet diameter ratios follows a similar pattern, i.e., the
Normalized by the nozzle throat diameter.
Fig. 2. Axisymmetric simplified model of steam ejector.
Fig. 3. Entrainment ratio ω as a function of outlet mixing steam pressure pc.
simulation predicts a critical pressure pcr for the outlet mixing steam pressure in the steam ejector. For the cases that the pc is less than pcr, entrainment ratio ω remains unchanged and the steam ejector is in a stable working state (double choking state). However, once pc exceeds pcr, the entrainment ratio ω decreases sharply as pc increases, and if pc is large enough, reverse flow may appear. The phenomenon of the double choking state is a direct result of thermodynamic theory for steam
Fig. 4. Variation of entrainment ratio as a function of primary steam nozzle distance ratio of different nozzle exit diameter ratios. 511
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entrainment ratio ω increases first and then decreases sharply with γlc. At the primary steam nozzle distance of 19.0 the steam ejector obtains the maximum entrainment ratio and the best entrainment performance. This indicates that there exists an optimum distance from the primary steam nozzle exit section to the cylindrical mixing chamber inlet section. The entrainment ratio ω increases with γlc if γlc is smaller than its optimum value, although this increase is not significant when γlc is close to its optimum value. Take the case γd = 3.0 whose variation with γlc is the most significant among the three as an example. The calculated ω is 0.3191 at γlc = 7.0, and as γlc increases to 15.0, ω is 0.4643 which means a 45.5% increase over that at γlc = 7.0; As γlc is further increased to 19 which is the optimum value, ω is 0.4805 which means a 3.5% increase only compared with that at γlc = 15.0. However, once γlc exceeds its optimum value, the ejector performance deteriorates sharply (say, for example, in the case of γd = 3.0, the calculated ω is 0.3867 at γlc = 21.0 and 0.2155 at γlc = 23.0, which means a reduction of 19.5% and 55.2%, respectively, from its maximum value of 0.4805 at γlc = 19.0). This tells us that ensuring γlc is not bigger than its optimum value is crucial for maintaining the ejector at good working state. Fig. 4 discloses that the nozzle outlet diameter ratio γd also has an influence on both the optimum primary steam nozzle distance ratio γlc and the maximum entrainment ratio. As γd increases the maximum entrainment ratio corresponding to the optimum primary steam nozzle distance ratio also increases, although only slightly. However, at the same time, the increase in γd results in the decrease in the optimum primary steam nozzle distance ratio range. This means that the influence of the primary steam nozzle distance ratio γlc on the entrainment ratio ω is becoming stronger for large γd. The reason is that the primary steam expands greatly in the jet core due to the increased primary steam nozzle outlet diameter ratio (i.e., the increased cross-sectional area of the primary steam nozzle outlet). The effective flow area for the entrained steam which is the annular region between the primary stream jet core and the wall is decreased, and thus the entrainment ratio becomes smaller. To understand the variation of the entrainment ratio with the primary steam nozzle distance ratio further, the velocity contours of internal flow field are shown in Fig. 5 at the diameter ratio 2.5 for various γlc. It can be seen from these contours that the supersonic primary steam continues to expand and accelerate after passing through the primary steam nozzle outlet section. The sudden increase of the primary steam nozzle outlet channel leads to large disturbances and generates oblique shock waves at the outlet edge of the primary steam nozzle. The oblique shock waves pass through the fluid, reach the mixed boundary layer and become expansion wave after being reflected by the mixed boundary layer. At the mixed boundary layer, the expansion wave is then converted to compression wave. During the reflection process, the shock wave energy dissipates gradually until it disappears as the reflections increase. A shock wave chain is formed after multiple reflections inside the ejector. When the primary steam nozzle distance ratio is small, the primary steam from the primary steam nozzle even could crash into the mixing chamber directly, which avoids the collision between the primary steam and the convergence wall. However, the shock wave chain is formed due to the alternate appearance of complex expansion and compression wave at the primary steam nozzle outlet section, and the shock wave chain even extends to the inlet section of the diffuser, then the second oblique shock wave and first shock wave fused. This phenomenon may lead to the formation of local reverse flow of the mixed steam and this local reverse flow may in turn influence the upstream flow. If this effect is strong enough, then it may even cause the local backpressure of the primary and entrained streams exceeds their critical value and the primary and entrained streams, most likely the entrained steam, could no longer maintain their corresponding choking state. As we know, double-chocking state is important for the ejector maintaining the designed or expected performance (the entrainment ratio acquires its largest value under the given operation conditions). The
Fig. 5. Velocity contours inside the ejector of various primary steam nozzle distance ratios (γd = 2.5).
second shock wave causes the flow fluctuation. In other words, the pressure and the velocity fluctuate strongly, and thus the flow frictional resistance increases and the mechanical energy is reduced. The fluctuation in flow parameters are significant. Take γlc = 7.0, γd = 2.5 as an example. As the Mach number at the nozzle axis increases to its largest value (3.44) at x = 66.56 mm (x/dp* = 23.94) from the entrance, the Mach number starts to fluctuate: at x = 93.74 mm (x/dp* = 25.34), it acquires its first minimum value (2.83), then increases to 3.45 at x = 119.30 mm (x/dp* = 32.24). After that, the fluctuation continues: at x = 131.90 mm (x/dp* = 35.65), the Mach number is 3.11, then it increases to 3.32 at x = 148.10 mm (x/dp* = 40.03). The fluctuations almost extend over the whole mixing chamber and ejector throat section. The strong fluctuations will certainly lead to strong irreversibility and therefore energy dissipation is inevitable. The energy dissipation causes poor steam ejector performance when the primary steam nozzle distance ratio is small. Choking of the entrained steam appears in the convergence section of the mixing chamber and of the mixed steam in the throat section [1]. As the primary steam nozzle distance ratio increases, i.e. the primary steam nozzle is moved upstream, the jet core circular area of the primary steam increases at the nozzle outlet section. Then the primary steam will get fully expanded and form a low enough pressure at the nozzle outlet section. Thus, the flow rate of the entrained steam increases due to this increased pressure difference (the entrained steam pressure is kept unchanged). The choking position of the entrained steam moves to the upstream of the mixing chamber convergence 512
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section. Meanwhile, the effective cross-sectional area for the entrained steam also becomes bigger, which allows more entrained steam be sucked into the mixing chamber. And this explains at least in part why the entrainment ratio increases with γlc. Once γlc approaches to its optimum value, the shock wave at the diffuser inlet section tends to be relatively steady. This means that the mixing process in the mixing chamber of the primary and the entrained steam is more complete and the velocity difference is relatively small. It could be seen from the contours in Fig. 5 that the number of shock waves of a shock chain is also reduced, so the entrainment ratio of the steam ejector is increased. Once γlc acquires its optimum value, the ejector can operate at a higher outlet pressure and thus a larger compression ratio, and at the same time maintain the entrainment ratio at its maximum value. It can be seen from Fig. 5 that as γlc exceeds its optimum value (=19.0), the primary steam nozzle is actually located far away from the mixing chamber inlet section, i.e. it is moved to the upstream. The position of second shock wave moves to the upstream of the diffuser, the critical back pressure (outlet pressure) value of the steam ejector is decreased if you compare Fig. 5d) with Fig. 5c). It should be pointed out, as the primary steam nozzle is shifted upstream gradually, the cross-sectional area for the primary steam that flows out of the primary steam nozzle increases. Thus, the interaction of the primary steam with the wall of the mixing chamber convergence section becomes stronger and even a direct impingement to the wall may be the case and thus results in reflection. A portion of the reflected fluid may possibly flow out from the suction chamber. Fig. 6 shows the wall pressure distribution at various primary steam nozzle distance ratios. Once γlc exceeds its optimum value, the pressure near the mixing chamber wall is becoming higher. This larger pressure enhances the interactions between the steam and the mixing chamber wall and thus results in greater kinetic energy loss. The flow of the primary steam will there interfere severely with the entrained steam and directly affects the entrainment flow. In addition, the Mach number (velocity) in the mixing chamber is reduced and the pressure is raised, therefore the pressure difference, the driving force for pumping the entrained steam into the ejector is decreased. This will certainly cause the entrainment ratio decreases dramatically. Say, for the caseγd = 2.5, ω is reduced from its peak value of 0.48 to 0.32 as γlc increases from its optimum value of 19.0 to 23.0, which means a 33.33% reduction in the entrainment ratio. If the distance continues increases, as the outlet section of primary steam nozzle is too far away from the inlet section of the
Fig. 7. Mach number along the nozzle axis under various primary steam nozzle distance ratios (γd = 2.5).
cylindrical mixing chamber, i.e. the distance between the nozzle outlet and the inlet of the mixing chamber is too big to maintain the entrainment function of the steam ejector and the ejector simply does not work at all. Fig. 7 presents variation of Mach number along the nozzle axis under various primary steam nozzle distance ratios. It can be seen from this figure, that the primary steam of a low initial speed experiences an adiabatic expansion in the Laval nozzle, the velocity along the axial direction increases sharply, the flow changes from subsonic to supersonic and the pressure energy of the fluid is converted into kinetic energy. Meanwhile, the shock wave chain at the downstream of the primary steam nozzle outlet section moves downstream further as γlc decreases, and even spreads to the diffuser. There is also an increase for the core shock wave. With the alternation of expansion wave and compression wave, the viscous effects of the primary steam and entrained steam in the shear mixing process lead to the so-called diamond wave weakening gradually. In other words, the pressure decreases and the velocity increases when the steam is in the expansion wave state, and vice versa for the compression wave. It therefore could be concluded from the above analysis that the axial Mach number of the mixed steam presents an alternating wave-like pattern, that is, it increases first and then decreases and repeats this pattern later on. The mixed steam is discharged from the steam ejector after it is compressed and decelerated in the diffuser. It should be stressed, the Mach number in the mixing chamber decreases obviously as γlc increases. After that, once it enters the diffuser, the fluctuations disappear and the flow becomes ‘stable’ and into the subsonic, the pressure is recovered to its outlet pressure which is greater than that of the entrained steam but smaller than that of the primary steam. As one can understand, the divergence section length of the primary steam nozzle with the given throat and the outlet cross-sectional area will also influence the ejector performance. To include this parameter, the ratio γs of the divergence section length to the throat diameter of the primary steam nozzle is defined. Numerical simulations are carried out to find out the γs influences on the optimum nozzle distance ratio and the main results are summarized in Fig. 8. As one can see from this figure, the entrainment ratio ω increases slowly first and then follows by a sharp decrease with γlc. And for 3 different divergence section length to the throat diameter ratios, the optimum γlc is the same and equals to 19.0. It should also be stressed that the larger the primary steam nozzle divergence length to the throat diameter ratio, the smaller
Fig. 6. Wall pressure along nozzle axis under various primary steam nozzle distance ratios. 513
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Fig. 8. Variation of entrainment ratio with primary steam nozzle distance of 3 typical divergent section length-to- throat diameter ratios γs (γd = 2.5).
Fig. 9. Variation of entrainment ratio with mixing chamber throat diameter ratio.
the entrainment ratio, i.e. the increase in the primary steam nozzle divergence length will result in the deterioration in the ejector performance. And the largest difference among the entrainment ratios for three γs is that γlc is in the range of 15.0 and 21.0. However, even this largest difference still could be simply neglected, since it is actually very small: Taking γlc 19.0 as example, the entrainment ratio for γs = 10.0 and γs = 13.0 is only 2% and 3.7% smaller, respectively, compared with that of γs = 7.0 that is the largest.
γl2 = 11. Firstly, one can see from Fig. 10 that for small γd3, the length of the oblique shock wave at the primary steam nozzle outlet section does not extend to the mixing chamber throat, and there are little influences of the second throat on the shock wave structure. If the throat section area, i.e. γd3 is too small, say for example in the case of γd3 = 3.8, the flow of the mixed steam is somehow restricted due to the small cross-sectional area, this will certainly increase the flow resistance and cause severe energy loss. As an extremity, if the throat section area is further reduced to its limit value of 0.0, then the flow inside the steam ejector is nothing but the impingement of the enclosed space. In this case, the outlet of the ejector is actually closed, the fluid can only return and flow out from the entrained steam inlet, and the entrainment ratio will be negative. Of course, once this mode of operation occurs, then actually, the steam ejector plays no function at all. As γd3 increases, the shock wave chain moves gradually to the mixing chamber throat, say for example that of γd3 = 4.4 and 4.8. When γd3 increases to about 4.8, the shock wave chain generated by the primary steam at the primary steam nozzle outlet section just passes through the throat, and the entrainment ratio acquires its maximum value as shown in Fig. 10 for γd3 = 4.8. If γd3 is too large, i.e. the throat section area is too large (see Fig. 10 for γd3 = 5.0), then the high-speed jet flow from the primary steam nozzle is something like an impinging flow into an open space. The velocity of the primary steam reduces quickly in the wall region and could not produce the low-pressure needed for forming the choking flow of the entrained steam. Therefore, in this case, the ejector could not maintain the double-choking mode which is essential for its good performance and this explains why the entrainment ratio reduces dramatically from its peak value of 0.60 to 0.32 in the case of γl2 = 11. It can be seen from Fig. 11 that the wall pressure decreases with the increase of γd3 if γd3 is smaller than its optimum value. This explains why the entrainment ratio increases with γd3 for small γd3. On the other hand, once γd3 exceeds its optimum value, then the wall pressure of mixing chamber increases with the increase of γd3, which means that the kinetic energy has been transformed into the pressure energy. The higher wall pressure means the smaller driving force for the entrainment of the entrained steam. As a result, the entrainment ratio is decreased sharply with the increase of γd3 for large γd3. It can also be seen from the wall pressure distribution that the wall pressure increases almost monotonously along axial direction once the diameter ratio of throat section exceeds its the optimum value (see for the cases γd3 = 4.9 and 5.0 in Fig. 11). This also explains the dramatic decrease of the
3.2. Optimization of the diameter ratio of mixing chamber throat to primary nozzle throat The mixing chamber throat diameter, as one expects, also plays an important role in determining the ejector performance and should be optimized. To be specific, it is the throat diameter ratio γd3 of the mixing chamber throat section diameter d3 to the primary steam nozzle throat section diameter dp* matters once all the other structural parameters of the ejector are fixed. Of course, with a given diameter, the length of the mixing chamber throat section l2 also has an important effect. To include this parameter, the mixing chamber throat section length ratio γl2 is defined as l2/dp*. A series of numerical computations for various γd3 and γl2 were carried out to optimize the throat diameter ratio of the mixing chamber throat to the primary nozzle throat under the condition that the primary steam nozzle throat section diameter dp* remains at its design value. In completing these computations, the primary steam nozzle distance ratio is set at its optimal value (γlc = 19.0) and the primary steam, the entrained steam and the mixed steam pressure are set at 600 kPa, 15 kPa and 40 kPa, respectively. Fig. 9 depicts the entrainment ratio as a function of γd3 and γl2. One can see clearly from this figure that there exists an optimum throat diameter ratio γd3 at which the entrainment ratio acquires its maximum value. However, it should be noted, the influence of γl2 is important only if γd3 is relatively smaller or significantly larger than its optimum value. What is more impressive, the optimum value of γd3 is same (=4.85) and is regardless of γl2 in the range covered in this simulation. The optimization is significant: at the design mixing chamber throat section diameter ratio (4.27, see Table 2) the entrainment ratio is 0.52, while the optimized value is 0.60 at γd3 = 4.85, which means an increase of 15.4%. To understand the above results better, Figs. 10 and 11 display the velocity contour and wall pressure distribution of different diameter ratios for the case of the mixing chamber throat section length ratio 514
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Fig. 11. Wall pressure distribution under various mixing chamber throat diameter ratio.
in the relative distance between the outlet section of the primary steam nozzle and the inlet section of the mixing chamber leads to the location change of shock wave. This change in shock wave may either enhance or deteriorate the entrainment performance of the steam ejector. Our results showed that there is an optimum value for the primary steam nozzle distance (the primary steam nozzle distance ratio) at which the steam ejector acquires its best entrainment performance under the given design conditions. The entrainment ratio increases slowly with the primary steam nozzle distance when the primary steam nozzle distance is less than its optimum value, and decreases sharply once the primary steam nozzle distance exceeds this optimum value. The mixing chamber throat section plays a very important role on boosting pressure. It directly affects the distribution of internal flow field and shock wave structure. For given operation parameters and other structural parameters, there is an optimum throat diameter ratio corresponding to the biggest entrainment ratio. Actually, with the optimized primary steam nozzle distance, the optimization result of mixing chamber throat diameter is significant. In our case, the improvement of the entrainment ratio is as large as 25%. Deviation from the optimized value of the diameter ratio γd3 may result in a serious degeneration of the ejector performance. In extreme situations, the wrong mixing chamber throat diameter ratio may cause the ejector simply does not work. Therefore, to ensure the good performance of steam ejectors, the diameter ratio γd3 of the mixing chamber throat section must be designed within a narrow vicinity of its optimum value. In our case, it should be ranged from 4.8 to 4.85.
Fig. 10. Velocity contours inside ejector of various mixing chamber throat diameter ratios (γl2 = 11).
entrainment ratio when γd3 exceeds its optimum value (=4.85). If one compares the present optimized entrainment ratio which is 0.60 and the one we obtained in Section 3.1 that is 0.48, then it can be shown that the effect of the optimization of mixing chamber throat diameter is significant: it leads to a 25.0% improvement over the result of the optimization of primary steam nozzle distance.
Conflicts of interest The author declares no conflict of interest.
4. Concluding remarks Acknowledgements
An axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector is established and a series of numerical simulations have been carried out to study the influences of the primary steam nozzle distance and mixing chamber throat diameter on the steam ejector performance. The effect of shock wave and choking on the internal flow characteristics of steam ejector is also discussed in detail. It is found that the primary steam nozzle distance has a significant influence on the entrainment performance of steam ejector. The change
This work is supported by the Beijing Municipal Science & Technology Plan Project (No. Z111100058911006).
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. doi.org/10.1016/j.ijthermalsci.2018.06.033. 515
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References [17]
[1] G. Sokoloff, Ejector. Huang Q Y, Trans, Science Press, Beijing, 1977 280. [2] B. Han, Z. Liu, H. Wu, et al., Experimental study on a new method for improving the performance of thermal vapor compressors for multi-effect distillation desalination systems, Desalination 344 (2014) 391–395. [3] O. Samaké, N. Galanis, M. Sorin, Thermodynamic study of multi-effect thermal vapor-compression desalination systems, Energy 72 (2014) 69–79. [4] N.H. Aly, A. Karameldin, M.M. Shamloul, Modelling and simulation of steam jet ejectors, Desalination 123 (1999) 1–8. [5] S.K. Gupta, R.P. Singh, R.S. Dixit, A comparative parametric study of two theoretical models of a single-stage, single-fluid, steam jet ejector, Chem. Eng. J. 18 (1979) 81–85. [6] G. Chen, R. Zhang, D. Zhu, et al., Experimental study on two-stage ejector refrigeration system driven by two heat sources, Int. J. Refrig. 74 (2017) 293–301. [7] J. Yan, G. Chen, C. Liu, et al., Experimental investigations on a R134a ejector applied in a refrigeration system, Appl. Therm. Eng. 110 (2017) 1061–1065. [8] F. Giacomelli, G. Biferi, F. Mazzelli, et al., CFD modeling of the supersonic condensation inside a steam ejector, Energy Procedia 101 (2016) 1224–1231. [9] S. Croquer, S. Poncet, N. Galanis, Comparison of ejector predicted performance by thermodynamic and CFD models, Int. J. Refrig. 68 (2016) 28–36. [10] Z. Ding, L. Wang, H. Zhao, et al., Numerical study and design of a two-stage ejector for subzero refrigeration, Appl. Therm. Eng. 108 (2016) 436–448. [11] V.V. Chandra, M.R. Ahmed, Experimental and computational studies on a steam jet refrigeration system with constant area and variable area ejectors, Energy Convers. Manag. 79 (2014) 377–386. [12] J.J. Yan, D.T. Chong, X.Z. Wu, Effect of swirling vanes on performance of steam–water jet injector, Appl. Therm. Eng. 30 (2010) 623–630. [13] X. Yang, X. Long, X. Yao, Numerical investigation on the mixing process in a steam ejector with different nozzle structures, Int. J. Therm. Sci. 56 (2012) 95–106. [14] W. Fu, Y. Li, Z. Liu, et al., Numerical study for the influences of primary steam nozzle on steam ejector performance, Appl. Therm. Eng. 106 (2016) 1148–1156. [15] S. Chen, G. Chen, L. Fang, An experimental study and 1-D analysis of an ejector with a movable primary steam nozzle that operates with R236fa, Int. J. Refrig. 60 (2015) 19–25. [16] N.B. Sag, H.K. Ersoy, Experimental investigation on motive nozzle throat diameter
[18]
[19]
[20]
[21]
[22]
[23] [24]
[25] [26]
[27] [28] [29]
516
for an ejector expansion refrigeration system, Energy Convers. Manag. 124 (2016) 1–12. D.W. Sun, Variable geometry ejectors and their applications in ejector refrigeration systems, Energy 21 (1996) 919–929. J. Dong, X. Chen, W. Wang, C.L. Kang, H.B. Ma, An experimental investigation of steam ejector refrigeration system powered by extra low temperature heat source, Int. Commun. Heat Mass Tran. 81 (2017) 250–256. K. Chunnanond, S. Aphornratana, An experimental investigation of a steam ejector refrigerator: the analysis of the pressure profile along the ejector, Appl. Therm. Eng. 24 (2004) 311–322. S.B. Riffat, S.A. Omer, CFD modelling and experimental investigation of an ejector refrigeration system using methanol as the working fluid, Int. J. Energy Res. 25 (2001) 115–128. S. Varga, A.C. Oliveira, B. Diaconu, Influence of geometrical factors on steam ejector performance – a numerical assessment, Int. J. Refrig. 32 (7) (2009) 1694–1701. Y.H. Zhu, W.J. Cai, C.Y. Wen, Y.Z. Li, Numerical investigation of geometry parameters for design of high performance ejectors, Appl. Therm. Eng. 29 (2009) 898–905. H. Ding, C. Wang, C. Chen, Non-equilibrium condensation of water vapor in sonic nozzle, Appl. Therm. Eng. 71 (2014) 324–334. S.M.A.N.R. Abadi, R. Kouhikamali, K. Atashkari, Non-equilibrium condensation of wet steam flow within high-pressure thermo-compressor, Appl. Therm. Eng. 81 (2015) 74–82. A. Bouhanguel, P. Desevaux, E. Gavignet, Flow visualization in supersonic ejectors using laser tomography techniques, Int. J. Refrig. 34 (7) (2011) 1633–1640. K. Ariafar, D. Buttsworth, G. Al-Doori, et al., Mixing layer effects on the entrainment ratio in steam ejectors through ideal gas computational simulations, Energy 95 (2016) 380–392. H.Q. Wu, Z.L. Liu, B. Han, et al., Numerical investigation of the influences of mixing chamber geometries on steam ejector performance, Desalination 353 (2014) 15–20. X. Yang, X. Long, X. Yao, Numerical investigation on the mixing process in a steam ejector with different nozzle structures, Int. J. Therm. Sci. 56 (2012) 95–106. J.M. Cardemil, S. Colle, A general model for evaluation of vapor ejectors performance for application in refrigeration, Energy Convers. Manag. 64 (2012) 79–86.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Numerical study for the influences of primary steam nozzle distance and mixing chamber throat diameter on steam ejector performance
T
Weina Fu, Zhongliang Liu∗, Yanxia Li, Hongqiang Wu, Yongzhi Tang College of Environmental and Energy Engineering, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing, 100124, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Steam ejector CFD Primary steam nozzle distance Mixing chamber throat diameter Entrainment ratio
The geometry and operation parameters have important influences on the performance of steam ejectors. An axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector has been established in this paper to investigate the flow characteristics inside steam ejectors aiming at optimizing primary steam nozzle exit section distance and mixing chamber throat diameter simultaneously. The results show that there are an optimum value for the primary steam nozzle distance ratio and an optimum diameter ratio of mixing chamber throat to primary nozzle throat at which the steam ejector acquires its best entrainment performance under the given design conditions. With the optimized primary steam nozzle distance, the optimization of the diameter ratio of the mixing chamber throat to primary nozzle throat is significant. In our case, the improvement of the entrainment ratio is as large as 25%. Deviation from its optimized value may result in a serious degeneration in the ejector performance. Therefore, to ensure the steam ejector performance, the primary steam nozzle distance should be designed at its optimum value and the diameter ratio of the mixing chamber throat to primary nozzle throat be within a narrow vicinity of its optimum value.
1. Introduction Energy shortage and environmental pollution have become the bottleneck for sustainable economy and society development. Energy saving and emission reduction have thus received increasing attention from all over the world. Steam ejector is a kind of fluid machinery that uses high-pressure steam to pump low-pressure steam [1]. It has been widely used in thermal evaporation system [2,3], vacuum [4,5] and refrigeration [6,7], due to its convenient operation, simple structure and obvious energy saving effect. With the rapid development of CFD software, many scholars [8–11] have studied the ejector numerically. Adiabatic acceleration flow and expansion of the primary steam inside Laval nozzle (the primary steam nozzle) leads to the formation of supersonic and low-pressure region at the nozzle outlet section, and a pressure difference between this low-pressure region and the entrained steam is thus established and pushes the low-pressure entrained steam into the ejector. As one could well understand, the geometry of the primary steam nozzle has great influences on flow pattern and thus the ejectors' performance. Installing swirl vanes at the nozzle outlet section could improve the ejector entrainment performance and reduce the frictional loss [12]. Compared with circular, elliptical, rectangular nozzle ejector, the performance of cross nozzle ejector was the best [13]. Fu et al. [14] studied the influences of primary steam nozzle
∗
Corresponding author. E-mail address: [email protected] (Z. Liu).
https://doi.org/10.1016/j.ijthermalsci.2018.06.033 Received 6 June 2017; Received in revised form 25 June 2018; Accepted 26 June 2018
1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.
outlet diameter on the flow characteristics and entrainment ratio of a steam ejector. Their results showed that the supersonic jet in the overexpanded wave state flowed more smoothly after leaving the nozzle, and therefore, the energy loss was relatively small and the ejector achieved a good performance. The primary steam nozzle position is also an important structural parameter for steam ejectors. The critical entrainment ratio of an ejector at different nozzle positions were experimentally studied by Chen et al. [15]. Their results showed that the entrainment ratio increased with the nozzle outlet position at first and then remained unchanged within the range covered, which meant that there was an optimum nozzle-exit position (NXP) at which the entrainment ratio would acquire its maximum value. It was found that the entrainment ratio was fully decided by the so-called double choking phenomenon that appeared in the converging section, i.e., only when both the primary and the entrained steam were at their critical state (maximum mass flowrate state) simultaneously, could the ejector obtain its largest entrainment ratio. It should be pointed out however, their experimental result was somewhat based on the assumption that the entrained flow velocity in the converging cone and the mixing flow velocity in the constant area section were both smaller than the sound velocity. Sag and Ersoy [16] studied the influence of the primary steam nozzle on the performance of a steam ejector used for refrigeration and they found experimentally that the ejector system that used the
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conditions. The effects of mixing chamber length and convergence angle on the performance of steam ejectors were investigated by Wu et al. [27], the shock wave and vortex inside the steam ejector under different conditions were discussed. Yang et al. [28] studied the influences of the nozzle structure on fluid streamline distribution and vortex structure inside the mixing chamber. Although many efforts have been made so far concerning the structure optimization of steam ejectors, however, little systematic research work has been reported in the literature for optimizing the primary steam nozzle position and the mixing chamber throat diameter simultaneously. As we have discussed earlier, these two structural parameters have direct and very strong influences on the ejector performance and may affect each other's optimum value, i.e. the change in one parameter may result in significant change in the optimum value of another. In this paper, an axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector has been established to explore and analyze the influences of primary steam nozzle distance and mixing chamber throat diameter on the ejector performance.
optimum primary steam nozzle throat diameter exhibited a higher COP than the classic system. Sun [17] established a thermodynamic model for ejector refrigeration systems and they found that using variable geometry (including NXP) ejector would enhance the performance of ejector refrigeration system. Dong et al. [18] set up a prototype steamejector refrigeration system and the effects of nozzle exit position (NXP) and the diameter of the constant area section on the working performance of steam ejector were investigated. Chunnanond and Aphornratana [19] constructed an experimental steam ejector refrigerator and the experiments were carried out to examine the influences of the operating conditions and the geometry including NXP on the system performance. They found that using smaller primary nozzle or retracing the nozzle out of mixing chamber could both increase the COP and the cooling capacity of the system. Riffat and Omer [20] carried out an experimental and numerical study of an ejector refrigeration system using methanol as the working fluid. 4 different NXPs were chosen to investigate the effect of the relative position of the primary nozzle exit within the mixing chamber on the performance of the ejector and the numerical results were used for optimizing ejector geometry. Their results showed that positioning the nozzle exit at least 0.21 length of the mixing chamber throat's diameter upstream of the entrance of the mixing chamber gave better performance than pushing it into the mixing chamber. Varga et al. [21] numerically studied the influences of area ratio between the nozzle and constant area section, nozzle exit position (NXP) and constant area section length on the ejector performance. Their results indicated that for ejectors obtaining optimum performance, all these three geometrical parameters should be optimized. Zhu et al. [22] optimized numerically of primary nozzle exit position and converging angle of mixing section. They found that these two geometry parameters both had very strong influence on ejector performance. The optimum NXP was obtained and was found to be 1.7–3.4 length of the mixing section throat diameter upstream of the start of the mixing section. One may understand that changing the primary steam nozzle structure and its relative position will change the vortex structure formed in the downstream of the primary steam nozzle as it has been reported in literature, therefore, there should be an optimum structure and position to ensure the ejector to obtain its best performance. Most researches used ideal gas model and the assumption of constant physical property. Wet steam models were also established by some researchers to study the flow inside ejectors. Ding et al. [23] studied nonequilibrium condensation process of water vapor in moist air expanding through a sonic nozzle by numerical simulation. They found that the flow rate of the sonic nozzle was affected by both homogeneous and heterogeneous nucleation. Abadi et al. [24] studied the unsteady supersonic flow of wet steam through a high-pressure thermo-compressor (steam ejector) with consideration of non-equilibrium homogeneous condensation numerically. Two-fluid multiphase flow formulations were used and their results were validated against the industrial data, which proved the effectiveness of the model for modeling steam condensing flows within high-pressure steam ejectors. However, due to the obvious difficulties and the less reliability of the physical, mathematical and numerical model in taking condensation process into consideration, most of the numerical optimization work reported in the literature concerning steam ejectors simply used ideal gas model. The performance of a steam ejector depends largely on the interactions between the high-pressure and the low-pressure steam. Therefore, during this process, very complex physical phenomena may occur, including transonic flow, shockwaves, mixing, phase change (condensation and evaporation), boundary separation and so on. The exploration of the mixing process inside an air steam ejector was carried out by Bouhanguel et al. [25] experimentally. The flow pattern, shock wave position and turbulence structure were observed visually and the analyses of the observed phenomena were presented. Ariafar et al. [26] studied the fluid mixing and the pressure driving effects on the entrainment ratio of steam ejectors under various operation
2. Physical and mathematical model The mathematical descriptions for the flow onside the steam ejector including the governing equations and boundary conditions used were exactly same as that given by Fu et al. [14]. The detailed CFD model, the validation of the grid independence, the physical and mathematical and the CFD model were all discussed in full details in Ref. [14] and are simply omitted here for conciseness. A schematic view of a typical supersonic ejector is shown in Fig. 1. Based on the assumption of one-dimensional steady state flow and constant-pressure mixing, the entrainment ratio and the main structural dimensions were calculated under the operation conditions (the primary steam pressure pp = 600 kPa, the entrained steam pressure ps = 15 kPa, and the mixed steam pressure pc = 40 kPa, which are something for the so-called low-temperature MEE desalination systems) and listed in Table 1. The main designed geometrical parameters of the ejector are listed in Table 2. Comparing with the supersonic jet of the nozzle exit, the initial velocity at the nozzle inlet may well be neglected. The lateral inlet of the entrained steam is simplified as a uniformly-distributed axial circular-inlet and thus the axisymmetric two-dimensional transonic compressible flow model could be used, as shown in Fig. 2. According to the shape of flow channel, the map-style block partition mesh is adopted to divide the flow domain structure of the ejector into 8 quadrilateral subdomains that are same as that in Ref. [14]. Second-order up-winding formulation is used to discretize convection terms and second-order central differencing for diffusion terms. To further ensure that the mathematical and numerical model can produce a reliable prediction of the performance of ejector, just like that in Ref. [14], a simulation was carried out to investigate the influence of mixing steam pressure pc on the performance of the steam ejector. Fig. 3 presents the simulated entrainment ratio ω as a function of the outlet mixing steam pressure pc under the given operation conditions (pp = 600 kPa, ps = 15 kPa). As one can see from this figure, the
Fig. 1. Schematic diagram of steam ejector. 510
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ejectors and has been proved by experiments. Therefore, this result proves that the mathematical and numerical model used here predicts the ejector performance correctly [1,29].
Table 1 Design operation parameters. Parameter primary steam pressure primary steam temperature entrained steam pressure entrained steam temperature mixed steam pressure mixed steam temperature
Symbol
Value
Unit
pp Tp ps Ts pc Tc
600 432.15 15 327.15 40 349.15
kPa K kPa K kPa K
3. Results and discussion 3.1. Optimization of primary steam nozzle distance The flow in primary steam nozzle is transonic, i.e., the flow in the convergence section is subsonic, in the throat section sonic and supersonic in the divergence section and viscous effect could not be neglected. Therefore, the performance of the primary steam nozzle has a direct influence on the flow characteristics and the entrainment performance and thus on the structure optimization of steam ejectors. The primary steam nozzle location is determined by the so-called primary steam nozzle distance lc that is defined as the distance between the outlet section of primary steam nozzle and the inlet section of cylindrical mixing chamber. To investigate the influence of the primary steam nozzle distance on ejector entrainment performance, the primary distance ratio γlc is defined,
Table 2 Geometrical parameters of the steam ejector. Parameter
Symbol
Value
Unit
The Dimensionlessa
Nozzle throat Diameter Nozzle entrance diameter Nozzle exit diameter Nozzle throat length Nozzle convergent section length Nozzle divergent section length Nozzle distance Mixing chamber entrance diameter Mixing chamber throat diameter Mixing chamber length Mixing chamber throat length Diffuser exit diameter Diffuser length
dp* dp dp1 s2 s1 s3 lc d2
3.70 12.00 10.00 5.00 10.00 25.00 62.90 23.20
mm mm mm mm mm mm mm mm
1.00 3.24 2.70 1.53 3.06 7.65 17.00 6.27
d3 l1 l2 dc l3
15.80 62.90 48.00 28.40 102.00
mm mm mm mm mm
4.27 19.24 14.68 7.68 27.57
a
γlc =
lc dp ∗
(9)
where dp* is the nozzle throat diameter. The entrainment ratio ω is calculated for various γlc for the given value of the nozzle throat diameter under the operation condition of the primary steam pressure pp = 600 kPa, the entrained steam pressure ps = 15 kPa, and the mixed steam pressure pc = 40 kPa. The designed outlet section of the primary steam nozzle is coincident with the conical inlet section of the mixing chamber, and the ratio of the primary steam nozzle distance to the primary steam nozzle throat diameter γlc is 17. Therefore, if γlc < 17, then the primary steam nozzle is actually placed inside the conical mixing section of the mixing chamber; and if γlc > 17 then it is in the upstream of the conical mixing section of the mixing chamber. Fig. 4 presents the entrainment ratio as a function of the primary steam nozzle distance ratio γlc for 3 primary steam nozzle exit diameter ratios (γd = 2.0, 2.5, and 3.0; where γd is defined as the ratio of the primary steam nozzle outlet diameter to the primary steam nozzle throat diameter). One can see from this figure, the entrainment ratio variation with the primary steam nozzle distance ratio for 3 primary steam nozzle outlet diameter ratios follows a similar pattern, i.e., the
Normalized by the nozzle throat diameter.
Fig. 2. Axisymmetric simplified model of steam ejector.
Fig. 3. Entrainment ratio ω as a function of outlet mixing steam pressure pc.
simulation predicts a critical pressure pcr for the outlet mixing steam pressure in the steam ejector. For the cases that the pc is less than pcr, entrainment ratio ω remains unchanged and the steam ejector is in a stable working state (double choking state). However, once pc exceeds pcr, the entrainment ratio ω decreases sharply as pc increases, and if pc is large enough, reverse flow may appear. The phenomenon of the double choking state is a direct result of thermodynamic theory for steam
Fig. 4. Variation of entrainment ratio as a function of primary steam nozzle distance ratio of different nozzle exit diameter ratios. 511
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entrainment ratio ω increases first and then decreases sharply with γlc. At the primary steam nozzle distance of 19.0 the steam ejector obtains the maximum entrainment ratio and the best entrainment performance. This indicates that there exists an optimum distance from the primary steam nozzle exit section to the cylindrical mixing chamber inlet section. The entrainment ratio ω increases with γlc if γlc is smaller than its optimum value, although this increase is not significant when γlc is close to its optimum value. Take the case γd = 3.0 whose variation with γlc is the most significant among the three as an example. The calculated ω is 0.3191 at γlc = 7.0, and as γlc increases to 15.0, ω is 0.4643 which means a 45.5% increase over that at γlc = 7.0; As γlc is further increased to 19 which is the optimum value, ω is 0.4805 which means a 3.5% increase only compared with that at γlc = 15.0. However, once γlc exceeds its optimum value, the ejector performance deteriorates sharply (say, for example, in the case of γd = 3.0, the calculated ω is 0.3867 at γlc = 21.0 and 0.2155 at γlc = 23.0, which means a reduction of 19.5% and 55.2%, respectively, from its maximum value of 0.4805 at γlc = 19.0). This tells us that ensuring γlc is not bigger than its optimum value is crucial for maintaining the ejector at good working state. Fig. 4 discloses that the nozzle outlet diameter ratio γd also has an influence on both the optimum primary steam nozzle distance ratio γlc and the maximum entrainment ratio. As γd increases the maximum entrainment ratio corresponding to the optimum primary steam nozzle distance ratio also increases, although only slightly. However, at the same time, the increase in γd results in the decrease in the optimum primary steam nozzle distance ratio range. This means that the influence of the primary steam nozzle distance ratio γlc on the entrainment ratio ω is becoming stronger for large γd. The reason is that the primary steam expands greatly in the jet core due to the increased primary steam nozzle outlet diameter ratio (i.e., the increased cross-sectional area of the primary steam nozzle outlet). The effective flow area for the entrained steam which is the annular region between the primary stream jet core and the wall is decreased, and thus the entrainment ratio becomes smaller. To understand the variation of the entrainment ratio with the primary steam nozzle distance ratio further, the velocity contours of internal flow field are shown in Fig. 5 at the diameter ratio 2.5 for various γlc. It can be seen from these contours that the supersonic primary steam continues to expand and accelerate after passing through the primary steam nozzle outlet section. The sudden increase of the primary steam nozzle outlet channel leads to large disturbances and generates oblique shock waves at the outlet edge of the primary steam nozzle. The oblique shock waves pass through the fluid, reach the mixed boundary layer and become expansion wave after being reflected by the mixed boundary layer. At the mixed boundary layer, the expansion wave is then converted to compression wave. During the reflection process, the shock wave energy dissipates gradually until it disappears as the reflections increase. A shock wave chain is formed after multiple reflections inside the ejector. When the primary steam nozzle distance ratio is small, the primary steam from the primary steam nozzle even could crash into the mixing chamber directly, which avoids the collision between the primary steam and the convergence wall. However, the shock wave chain is formed due to the alternate appearance of complex expansion and compression wave at the primary steam nozzle outlet section, and the shock wave chain even extends to the inlet section of the diffuser, then the second oblique shock wave and first shock wave fused. This phenomenon may lead to the formation of local reverse flow of the mixed steam and this local reverse flow may in turn influence the upstream flow. If this effect is strong enough, then it may even cause the local backpressure of the primary and entrained streams exceeds their critical value and the primary and entrained streams, most likely the entrained steam, could no longer maintain their corresponding choking state. As we know, double-chocking state is important for the ejector maintaining the designed or expected performance (the entrainment ratio acquires its largest value under the given operation conditions). The
Fig. 5. Velocity contours inside the ejector of various primary steam nozzle distance ratios (γd = 2.5).
second shock wave causes the flow fluctuation. In other words, the pressure and the velocity fluctuate strongly, and thus the flow frictional resistance increases and the mechanical energy is reduced. The fluctuation in flow parameters are significant. Take γlc = 7.0, γd = 2.5 as an example. As the Mach number at the nozzle axis increases to its largest value (3.44) at x = 66.56 mm (x/dp* = 23.94) from the entrance, the Mach number starts to fluctuate: at x = 93.74 mm (x/dp* = 25.34), it acquires its first minimum value (2.83), then increases to 3.45 at x = 119.30 mm (x/dp* = 32.24). After that, the fluctuation continues: at x = 131.90 mm (x/dp* = 35.65), the Mach number is 3.11, then it increases to 3.32 at x = 148.10 mm (x/dp* = 40.03). The fluctuations almost extend over the whole mixing chamber and ejector throat section. The strong fluctuations will certainly lead to strong irreversibility and therefore energy dissipation is inevitable. The energy dissipation causes poor steam ejector performance when the primary steam nozzle distance ratio is small. Choking of the entrained steam appears in the convergence section of the mixing chamber and of the mixed steam in the throat section [1]. As the primary steam nozzle distance ratio increases, i.e. the primary steam nozzle is moved upstream, the jet core circular area of the primary steam increases at the nozzle outlet section. Then the primary steam will get fully expanded and form a low enough pressure at the nozzle outlet section. Thus, the flow rate of the entrained steam increases due to this increased pressure difference (the entrained steam pressure is kept unchanged). The choking position of the entrained steam moves to the upstream of the mixing chamber convergence 512
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section. Meanwhile, the effective cross-sectional area for the entrained steam also becomes bigger, which allows more entrained steam be sucked into the mixing chamber. And this explains at least in part why the entrainment ratio increases with γlc. Once γlc approaches to its optimum value, the shock wave at the diffuser inlet section tends to be relatively steady. This means that the mixing process in the mixing chamber of the primary and the entrained steam is more complete and the velocity difference is relatively small. It could be seen from the contours in Fig. 5 that the number of shock waves of a shock chain is also reduced, so the entrainment ratio of the steam ejector is increased. Once γlc acquires its optimum value, the ejector can operate at a higher outlet pressure and thus a larger compression ratio, and at the same time maintain the entrainment ratio at its maximum value. It can be seen from Fig. 5 that as γlc exceeds its optimum value (=19.0), the primary steam nozzle is actually located far away from the mixing chamber inlet section, i.e. it is moved to the upstream. The position of second shock wave moves to the upstream of the diffuser, the critical back pressure (outlet pressure) value of the steam ejector is decreased if you compare Fig. 5d) with Fig. 5c). It should be pointed out, as the primary steam nozzle is shifted upstream gradually, the cross-sectional area for the primary steam that flows out of the primary steam nozzle increases. Thus, the interaction of the primary steam with the wall of the mixing chamber convergence section becomes stronger and even a direct impingement to the wall may be the case and thus results in reflection. A portion of the reflected fluid may possibly flow out from the suction chamber. Fig. 6 shows the wall pressure distribution at various primary steam nozzle distance ratios. Once γlc exceeds its optimum value, the pressure near the mixing chamber wall is becoming higher. This larger pressure enhances the interactions between the steam and the mixing chamber wall and thus results in greater kinetic energy loss. The flow of the primary steam will there interfere severely with the entrained steam and directly affects the entrainment flow. In addition, the Mach number (velocity) in the mixing chamber is reduced and the pressure is raised, therefore the pressure difference, the driving force for pumping the entrained steam into the ejector is decreased. This will certainly cause the entrainment ratio decreases dramatically. Say, for the caseγd = 2.5, ω is reduced from its peak value of 0.48 to 0.32 as γlc increases from its optimum value of 19.0 to 23.0, which means a 33.33% reduction in the entrainment ratio. If the distance continues increases, as the outlet section of primary steam nozzle is too far away from the inlet section of the
Fig. 7. Mach number along the nozzle axis under various primary steam nozzle distance ratios (γd = 2.5).
cylindrical mixing chamber, i.e. the distance between the nozzle outlet and the inlet of the mixing chamber is too big to maintain the entrainment function of the steam ejector and the ejector simply does not work at all. Fig. 7 presents variation of Mach number along the nozzle axis under various primary steam nozzle distance ratios. It can be seen from this figure, that the primary steam of a low initial speed experiences an adiabatic expansion in the Laval nozzle, the velocity along the axial direction increases sharply, the flow changes from subsonic to supersonic and the pressure energy of the fluid is converted into kinetic energy. Meanwhile, the shock wave chain at the downstream of the primary steam nozzle outlet section moves downstream further as γlc decreases, and even spreads to the diffuser. There is also an increase for the core shock wave. With the alternation of expansion wave and compression wave, the viscous effects of the primary steam and entrained steam in the shear mixing process lead to the so-called diamond wave weakening gradually. In other words, the pressure decreases and the velocity increases when the steam is in the expansion wave state, and vice versa for the compression wave. It therefore could be concluded from the above analysis that the axial Mach number of the mixed steam presents an alternating wave-like pattern, that is, it increases first and then decreases and repeats this pattern later on. The mixed steam is discharged from the steam ejector after it is compressed and decelerated in the diffuser. It should be stressed, the Mach number in the mixing chamber decreases obviously as γlc increases. After that, once it enters the diffuser, the fluctuations disappear and the flow becomes ‘stable’ and into the subsonic, the pressure is recovered to its outlet pressure which is greater than that of the entrained steam but smaller than that of the primary steam. As one can understand, the divergence section length of the primary steam nozzle with the given throat and the outlet cross-sectional area will also influence the ejector performance. To include this parameter, the ratio γs of the divergence section length to the throat diameter of the primary steam nozzle is defined. Numerical simulations are carried out to find out the γs influences on the optimum nozzle distance ratio and the main results are summarized in Fig. 8. As one can see from this figure, the entrainment ratio ω increases slowly first and then follows by a sharp decrease with γlc. And for 3 different divergence section length to the throat diameter ratios, the optimum γlc is the same and equals to 19.0. It should also be stressed that the larger the primary steam nozzle divergence length to the throat diameter ratio, the smaller
Fig. 6. Wall pressure along nozzle axis under various primary steam nozzle distance ratios. 513
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Fig. 8. Variation of entrainment ratio with primary steam nozzle distance of 3 typical divergent section length-to- throat diameter ratios γs (γd = 2.5).
Fig. 9. Variation of entrainment ratio with mixing chamber throat diameter ratio.
the entrainment ratio, i.e. the increase in the primary steam nozzle divergence length will result in the deterioration in the ejector performance. And the largest difference among the entrainment ratios for three γs is that γlc is in the range of 15.0 and 21.0. However, even this largest difference still could be simply neglected, since it is actually very small: Taking γlc 19.0 as example, the entrainment ratio for γs = 10.0 and γs = 13.0 is only 2% and 3.7% smaller, respectively, compared with that of γs = 7.0 that is the largest.
γl2 = 11. Firstly, one can see from Fig. 10 that for small γd3, the length of the oblique shock wave at the primary steam nozzle outlet section does not extend to the mixing chamber throat, and there are little influences of the second throat on the shock wave structure. If the throat section area, i.e. γd3 is too small, say for example in the case of γd3 = 3.8, the flow of the mixed steam is somehow restricted due to the small cross-sectional area, this will certainly increase the flow resistance and cause severe energy loss. As an extremity, if the throat section area is further reduced to its limit value of 0.0, then the flow inside the steam ejector is nothing but the impingement of the enclosed space. In this case, the outlet of the ejector is actually closed, the fluid can only return and flow out from the entrained steam inlet, and the entrainment ratio will be negative. Of course, once this mode of operation occurs, then actually, the steam ejector plays no function at all. As γd3 increases, the shock wave chain moves gradually to the mixing chamber throat, say for example that of γd3 = 4.4 and 4.8. When γd3 increases to about 4.8, the shock wave chain generated by the primary steam at the primary steam nozzle outlet section just passes through the throat, and the entrainment ratio acquires its maximum value as shown in Fig. 10 for γd3 = 4.8. If γd3 is too large, i.e. the throat section area is too large (see Fig. 10 for γd3 = 5.0), then the high-speed jet flow from the primary steam nozzle is something like an impinging flow into an open space. The velocity of the primary steam reduces quickly in the wall region and could not produce the low-pressure needed for forming the choking flow of the entrained steam. Therefore, in this case, the ejector could not maintain the double-choking mode which is essential for its good performance and this explains why the entrainment ratio reduces dramatically from its peak value of 0.60 to 0.32 in the case of γl2 = 11. It can be seen from Fig. 11 that the wall pressure decreases with the increase of γd3 if γd3 is smaller than its optimum value. This explains why the entrainment ratio increases with γd3 for small γd3. On the other hand, once γd3 exceeds its optimum value, then the wall pressure of mixing chamber increases with the increase of γd3, which means that the kinetic energy has been transformed into the pressure energy. The higher wall pressure means the smaller driving force for the entrainment of the entrained steam. As a result, the entrainment ratio is decreased sharply with the increase of γd3 for large γd3. It can also be seen from the wall pressure distribution that the wall pressure increases almost monotonously along axial direction once the diameter ratio of throat section exceeds its the optimum value (see for the cases γd3 = 4.9 and 5.0 in Fig. 11). This also explains the dramatic decrease of the
3.2. Optimization of the diameter ratio of mixing chamber throat to primary nozzle throat The mixing chamber throat diameter, as one expects, also plays an important role in determining the ejector performance and should be optimized. To be specific, it is the throat diameter ratio γd3 of the mixing chamber throat section diameter d3 to the primary steam nozzle throat section diameter dp* matters once all the other structural parameters of the ejector are fixed. Of course, with a given diameter, the length of the mixing chamber throat section l2 also has an important effect. To include this parameter, the mixing chamber throat section length ratio γl2 is defined as l2/dp*. A series of numerical computations for various γd3 and γl2 were carried out to optimize the throat diameter ratio of the mixing chamber throat to the primary nozzle throat under the condition that the primary steam nozzle throat section diameter dp* remains at its design value. In completing these computations, the primary steam nozzle distance ratio is set at its optimal value (γlc = 19.0) and the primary steam, the entrained steam and the mixed steam pressure are set at 600 kPa, 15 kPa and 40 kPa, respectively. Fig. 9 depicts the entrainment ratio as a function of γd3 and γl2. One can see clearly from this figure that there exists an optimum throat diameter ratio γd3 at which the entrainment ratio acquires its maximum value. However, it should be noted, the influence of γl2 is important only if γd3 is relatively smaller or significantly larger than its optimum value. What is more impressive, the optimum value of γd3 is same (=4.85) and is regardless of γl2 in the range covered in this simulation. The optimization is significant: at the design mixing chamber throat section diameter ratio (4.27, see Table 2) the entrainment ratio is 0.52, while the optimized value is 0.60 at γd3 = 4.85, which means an increase of 15.4%. To understand the above results better, Figs. 10 and 11 display the velocity contour and wall pressure distribution of different diameter ratios for the case of the mixing chamber throat section length ratio 514
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Fig. 11. Wall pressure distribution under various mixing chamber throat diameter ratio.
in the relative distance between the outlet section of the primary steam nozzle and the inlet section of the mixing chamber leads to the location change of shock wave. This change in shock wave may either enhance or deteriorate the entrainment performance of the steam ejector. Our results showed that there is an optimum value for the primary steam nozzle distance (the primary steam nozzle distance ratio) at which the steam ejector acquires its best entrainment performance under the given design conditions. The entrainment ratio increases slowly with the primary steam nozzle distance when the primary steam nozzle distance is less than its optimum value, and decreases sharply once the primary steam nozzle distance exceeds this optimum value. The mixing chamber throat section plays a very important role on boosting pressure. It directly affects the distribution of internal flow field and shock wave structure. For given operation parameters and other structural parameters, there is an optimum throat diameter ratio corresponding to the biggest entrainment ratio. Actually, with the optimized primary steam nozzle distance, the optimization result of mixing chamber throat diameter is significant. In our case, the improvement of the entrainment ratio is as large as 25%. Deviation from the optimized value of the diameter ratio γd3 may result in a serious degeneration of the ejector performance. In extreme situations, the wrong mixing chamber throat diameter ratio may cause the ejector simply does not work. Therefore, to ensure the good performance of steam ejectors, the diameter ratio γd3 of the mixing chamber throat section must be designed within a narrow vicinity of its optimum value. In our case, it should be ranged from 4.8 to 4.85.
Fig. 10. Velocity contours inside ejector of various mixing chamber throat diameter ratios (γl2 = 11).
entrainment ratio when γd3 exceeds its optimum value (=4.85). If one compares the present optimized entrainment ratio which is 0.60 and the one we obtained in Section 3.1 that is 0.48, then it can be shown that the effect of the optimization of mixing chamber throat diameter is significant: it leads to a 25.0% improvement over the result of the optimization of primary steam nozzle distance.
Conflicts of interest The author declares no conflict of interest.
4. Concluding remarks Acknowledgements
An axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector is established and a series of numerical simulations have been carried out to study the influences of the primary steam nozzle distance and mixing chamber throat diameter on the steam ejector performance. The effect of shock wave and choking on the internal flow characteristics of steam ejector is also discussed in detail. It is found that the primary steam nozzle distance has a significant influence on the entrainment performance of steam ejector. The change
This work is supported by the Beijing Municipal Science & Technology Plan Project (No. Z111100058911006).
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx. doi.org/10.1016/j.ijthermalsci.2018.06.033. 515
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References [17]
[1] G. Sokoloff, Ejector. Huang Q Y, Trans, Science Press, Beijing, 1977 280. [2] B. Han, Z. Liu, H. Wu, et al., Experimental study on a new method for improving the performance of thermal vapor compressors for multi-effect distillation desalination systems, Desalination 344 (2014) 391–395. [3] O. Samaké, N. Galanis, M. Sorin, Thermodynamic study of multi-effect thermal vapor-compression desalination systems, Energy 72 (2014) 69–79. [4] N.H. Aly, A. Karameldin, M.M. Shamloul, Modelling and simulation of steam jet ejectors, Desalination 123 (1999) 1–8. [5] S.K. Gupta, R.P. Singh, R.S. Dixit, A comparative parametric study of two theoretical models of a single-stage, single-fluid, steam jet ejector, Chem. Eng. J. 18 (1979) 81–85. [6] G. Chen, R. Zhang, D. Zhu, et al., Experimental study on two-stage ejector refrigeration system driven by two heat sources, Int. J. Refrig. 74 (2017) 293–301. [7] J. Yan, G. Chen, C. Liu, et al., Experimental investigations on a R134a ejector applied in a refrigeration system, Appl. Therm. Eng. 110 (2017) 1061–1065. [8] F. Giacomelli, G. Biferi, F. Mazzelli, et al., CFD modeling of the supersonic condensation inside a steam ejector, Energy Procedia 101 (2016) 1224–1231. [9] S. Croquer, S. Poncet, N. Galanis, Comparison of ejector predicted performance by thermodynamic and CFD models, Int. J. Refrig. 68 (2016) 28–36. [10] Z. Ding, L. Wang, H. Zhao, et al., Numerical study and design of a two-stage ejector for subzero refrigeration, Appl. Therm. Eng. 108 (2016) 436–448. [11] V.V. Chandra, M.R. Ahmed, Experimental and computational studies on a steam jet refrigeration system with constant area and variable area ejectors, Energy Convers. Manag. 79 (2014) 377–386. [12] J.J. Yan, D.T. Chong, X.Z. Wu, Effect of swirling vanes on performance of steam–water jet injector, Appl. Therm. Eng. 30 (2010) 623–630. [13] X. Yang, X. Long, X. Yao, Numerical investigation on the mixing process in a steam ejector with different nozzle structures, Int. J. Therm. Sci. 56 (2012) 95–106. [14] W. Fu, Y. Li, Z. Liu, et al., Numerical study for the influences of primary steam nozzle on steam ejector performance, Appl. Therm. Eng. 106 (2016) 1148–1156. [15] S. Chen, G. Chen, L. Fang, An experimental study and 1-D analysis of an ejector with a movable primary steam nozzle that operates with R236fa, Int. J. Refrig. 60 (2015) 19–25. [16] N.B. Sag, H.K. Ersoy, Experimental investigation on motive nozzle throat diameter
[18]
[19]
[20]
[21]
[22]
[23] [24]
[25] [26]
[27] [28] [29]
516
for an ejector expansion refrigeration system, Energy Convers. Manag. 124 (2016) 1–12. D.W. Sun, Variable geometry ejectors and their applications in ejector refrigeration systems, Energy 21 (1996) 919–929. J. Dong, X. Chen, W. Wang, C.L. Kang, H.B. Ma, An experimental investigation of steam ejector refrigeration system powered by extra low temperature heat source, Int. Commun. Heat Mass Tran. 81 (2017) 250–256. K. Chunnanond, S. Aphornratana, An experimental investigation of a steam ejector refrigerator: the analysis of the pressure profile along the ejector, Appl. Therm. Eng. 24 (2004) 311–322. S.B. Riffat, S.A. Omer, CFD modelling and experimental investigation of an ejector refrigeration system using methanol as the working fluid, Int. J. Energy Res. 25 (2001) 115–128. S. Varga, A.C. Oliveira, B. Diaconu, Influence of geometrical factors on steam ejector performance – a numerical assessment, Int. J. Refrig. 32 (7) (2009) 1694–1701. Y.H. Zhu, W.J. Cai, C.Y. Wen, Y.Z. Li, Numerical investigation of geometry parameters for design of high performance ejectors, Appl. Therm. Eng. 29 (2009) 898–905. H. Ding, C. Wang, C. Chen, Non-equilibrium condensation of water vapor in sonic nozzle, Appl. Therm. Eng. 71 (2014) 324–334. S.M.A.N.R. Abadi, R. Kouhikamali, K. Atashkari, Non-equilibrium condensation of wet steam flow within high-pressure thermo-compressor, Appl. Therm. Eng. 81 (2015) 74–82. A. Bouhanguel, P. Desevaux, E. Gavignet, Flow visualization in supersonic ejectors using laser tomography techniques, Int. J. Refrig. 34 (7) (2011) 1633–1640. K. Ariafar, D. Buttsworth, G. Al-Doori, et al., Mixing layer effects on the entrainment ratio in steam ejectors through ideal gas computational simulations, Energy 95 (2016) 380–392. H.Q. Wu, Z.L. Liu, B. Han, et al., Numerical investigation of the influences of mixing chamber geometries on steam ejector performance, Desalination 353 (2014) 15–20. X. Yang, X. Long, X. Yao, Numerical investigation on the mixing process in a steam ejector with different nozzle structures, Int. J. Therm. Sci. 56 (2012) 95–106. J.M. Cardemil, S. Colle, A general model for evaluation of vapor ejectors performance for application in refrigeration, Energy Convers. Manag. 64 (2012) 79–86.