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CFD Analysis of a Supersonic Air Ejector. Part II:Relation between Global Operation and Local Flow

FeaturesAmel Hemidi, François Henry, Sébastien Leclaire, Jean-Marie Seynhaeve,

Yann Bartosiewicz

To cite this version:Amel Hemidi, François Henry, Sébastien Leclaire, Jean-Marie Seynhaeve, Yann Bartosiewicz.CFD Analysis of a Supersonic Air Ejector. Part II: Relation between Global Operationand Local Flow Features. Applied Thermal Engineering, Elsevier, 2009, 29 (14-15), pp.2990.�10.1016/j.applthermaleng.2009.03.019�. �hal-00589454�

Accepted Manuscript

CFD Analysis of a Supersonic Air Ejector. Part II: Relation between Global

Operation and Local Flow Features

Amel Hemidi, François Henry, Sébastien Leclaire, Jean-Marie Seynhaeve,

Yann Bartosiewicz

PII: S1359-4311(09)00096-9

DOI: 10.1016/j.applthermaleng.2009.03.019

Reference: ATE 2763

To appear in: Applied Thermal Engineering

Received Date: 6 August 2007

Revised Date: 16 March 2009

Accepted Date: 19 March 2009

Please cite this article as: A. Hemidi, F. Henry, S. Leclaire, J-M. Seynhaeve, Y. Bartosiewicz, CFD Analysis of a

Supersonic Air Ejector. Part II: Relation between Global Operation and Local Flow Features, Applied Thermal

Engineering (2009), doi: 10.1016/j.applthermaleng.2009.03.019

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

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ACCEPTED MANUSCRIPT

CFD Analysis of a Supersonic Air Ejector. Part

II: Relation between Global Operation and

Local Flow Features

Amel Hemidi, François Henry, Sébastien Leclaire,

Jean-Marie Seynhaeve and Yann Bartosiewicz ∗

Université catholique de Louvain UCL, Louvain School of Engineering EPL,

Mechanical Engineering Department, TERM Division, Place du Levant 2, B-1348,

Louvain-la-Neuve, Belgium. Tel: +32 10 47 22 06, Fax: +32 10 45 26 92

Abstract

This paper presents an original CFD analysis of the operation of a supersonic ejector.

This study is based on CFD and experimental results obtained in the first part

paper [1]. Results clearly demonstrates that a good predictions of the entrainment

rate, even over a wide range of operating conditions, do not necessarily mean a

good prediction of the local flow features. This issue is shown through the results

obtained for two turbulence models, and also raises the problem of their assessment.

In addition, an analysis based on the sonic line location in order to locate the choking

cross section is proposed in this paper. Based on this approach, this parameter

allowed to propose an explanation of such disparities between good predictions of

overall performances and significant discrepancies of local flow parameters.

Key words:

Supersonic Ejector, Air-Conditioning, CFD-Experiment Integration, Turbulence

models, Two-Phase compressible flows

Preprint submitted to Elsevier March 16, 2009

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1 Introduction

The problems of global warming, climate changes and ozone depletion, in con-

junction with the increasing use of air-conditioning systems in the southern

but also in the northern regions, stimulate new research in refrigeration sys-

tems. An interesting research path for moderate temperature refrigeration and

for air-conditioning is the use of vapor-jet refrigeration systems, because they

are simple, and they can be activated by low grade energy sources such as

wastes and thermal solar collectors (Fig. 1(a)). These systems rely mostly on

a simple mechanical device called an ejector. In the cycle represented figure

1(a), the ejector operates as a compressor in a classical vapor-compression cy-

cle. A typical ejector geometry is depicted figure 1(b). A primary flow at high

total pressure and temperature, coming from the boiler, is discharged into the

ejector through the primary nozzle. The resulting supersonic jet involves the

entrainment of a secondary stream, coming from the evaporator. Depending

on operating conditions, the secondary flow may reach also critical conditions

due to an area stretching between the primary jet core and the ejector walls.

The matching secondary mass flow rate is then maximum since sonic speed

is reached. In this case the total mass flow rate crossing the ejector is also

maximum because the primary flow is also choked (Fig. 4). Those operating

conditions are called "on-design" and rely on the occurrence of a critical cross

section for the secondary stream. This physical reasoning was the basis of the

∗ Corresponding author

Email address: yann.bartosiewicz@uclouvain.be (Yann Bartosiewicz).

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theory of Munday and Bagster [2]. If conditions are not favorable, the sec-

ondary stream may remain subsonic and its mass flow rate is then strongly

dependant on the back pressure (Fig. 4) imposed by the condenser conditions;

those conditions are called "off-design". A slight increase in this back pressure,

resulting from an increase of the ambient temperature, could involve a large

decrease of the secondary mass flow rate (Fig. 4) and thus significant decrease

of the cooling effect. Consequently, once the hot and cold sources temperatures

are known with the cooling capacity, the most important parameter is the en-

trainment rate ω of the ejector, which is defined as the ratio of the secondary

mass flow rate over the primary mass flow rate (Fig. 4).

Judging from recent studies [3–9], computational fluid dynamics becomes a

usual tool to investigate and predict ejectors global operation, understand the

complex local flow physics and its link with the overall performances, in order

to perform better design of this device or the cycle. However, in a previous

paper [1], authors pointed out some shortcomings usually done by using the

CFD approach. Particularly, the issue of turbulence modeling on the prediction

of the ejector operation was investigated. It was demonstrated that significant

discrepancies could appear between the classical k − ε and the k − ω − sst

models when the primary pressure is decreased. In addition, at high pressures,

when the agreement between both models was perfect in terms of entrainment

rate, it was raised that the local flow features inside the ejector could be

completely different, qualitatively and quantitatively. As the secondary mass

flow rate and then the entrainment rate depends primarily on its critical cross

section for given conditions [2], it is believed that the investigation of the

effective area location could help to understand this behavior. In their article

Sriveerakul et al. [8] claimed the difficulty to locate this section even with the

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CFD possibilities. In this work, a CFD analysis is performed to explain these

paradoxical results between both turbulence models investigated in [1] based

on the location of the critical coss-section.

2 CFD model, Validation

The choice of air as the working fluid is very useful because it avoids biased

results due to a two-phase aspect. In addition, future works will consist of flow

visualizations which often require an open access, which is easier with air. The

flow in the air-ejector is governed by the ideal gas compressible steady-state

axisymmetric form of the viscous fluid flow conservation equations. For vari-

able density flows, the Favre averaged Navier-Stokes (FANS) equations are

more suitable and will be used in this work. The total energy equation in-

cluding viscous dissipation is also included and coupled to the set with the

perfect gas law. The thermodynamics and transport properties for air are held

constant; their influence was not found to be significant during previous tests.

From the experimental point of view, the stand and the ejector have been

designed to be a "CFD-grade" experiment. This requires special care to have

a simple stand with a special focus on the geometrical and boundary condi-

tions in order to be able to match those applied in the CFD model. In this

regard, the ejector geometry has been built with a stagnation chamber at the

secondary inlet allowing to have stagnation conditions and an axisymmetric

flow at this inlet to match the CFD model conditions. Figure 2(a) represents

a 3D view of the experimental ejector geometry with a part of the suction

line, the stagnation chamber and the core of the ejector. Figure 2(b) depicts

the main dimension of the ejector where the geometry characteristics such as

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the distance between the primary nozzle and the inlet of the mixing chamber

is kept constant to ∆L = 3D∗, d∗ = 3.3mm, d = 4.5mm, D∗ = 7.6mm. The

total length of the ejector is 22.5mm. The axisymmetric computational do-

main (Fig. 3) includes the ejector from the secondary annular inlet where the

flow comes out of the stagnation chamber. In addition, inlet total pressure and

temperature are measured at a location where dynamics effects are negligible

juste before the ejector inlet in a larger cross section pipe. The secondary inlet

(suction) is taken from ambient conditions at atmospheric pressure. However

the pressure difference between the stagnation chamber and the suction cross

section is measured to take into account pressure losses along the suction line,

because a long pipe is required in order to have a fully developed flow to ac-

curately measure the secondary mass flow rate [1]. Indeed, mass flow rates

are measured through a diaphragm technique according ISO 5167 rules. At

the outlet, the pressure is measured beyond a diffuser to filter out dynamic

effects. The static pressure is then evaluated at the ejector outlet and set in

the CFD model. Few iterations are then required to match the same total

pressure obtained in experiment. This set of equations is solved by a control

volume approach in the CFD commercial package FLUENT 6.2. Although the

steady state is desired, the unsteady term is conserved since from a numerical

point of view, governing equations are solved with a time marching technique.

This allows to keep equations parabolic-hyperbolic for every Mach number.

The system is also time-preconditioned in order to overcome the problem of

numerical stiffness at low-mach numbers. The convection term is discretized

with a flux splitting method in order to capture shock accurately (second or-

der upwind), while the diffusive term uses a central difference discretization.

The coupled system is then solved by a block Gauss-Seidel method with an

algebraic multigrid acceleration algorithm. Details concerning the mesh con-

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vergence study or numerical accuracy can be found in the first part of the

current paper [1]. The chosen computational grid size was 25820 cells.

In the previous paper [1], a full validation procedure has been performed to

evaluate the widely used k − ε model and the k − ω − sst model which is

supposed to include more physics. Both models are expressed in their classical

version with published constants. The only correction is for the term taking

into account the compressibility effects due to vortex stretching phenomena.

The k− ε model uses the Sarkar formulation , which is applied over the whole

computational domain, while the k − ω − sst uses the Wilcox formulation

depending on the local flow conditions; both formulation and details can be

found in [3, 4]. This validation was done for the whole range of ejector oper-

ations (on-design, off-design) and for different driving pressures: P 0

1= 3bar,

P 0

1= 4bar, P 0

1= 5bar and P 0

1= 6bar. More details about the geometry,

boundary conditions and convergence can be found in [1]. Figure 5 depicts

the results for P 0

1= 4bar and P 0

1= 6bar. For the highest pressure, the agree-

ment between both models is almost perfect over the range of operation (Fig.

5(b)). On the contrary, for P 0

1= 4bar, the two models provide significant

differences in terms of entrainment rate for both the on-design and off-design

conditions (Fig. 5(a)). However, even though both models provide same re-

sults (Fig. 5(b)), the link with local flow dynamics inside the ejector is not

straightforward. Figure 6 illustrates the centerline Mach number for on-design

conditions in the case of P 0

1= 4bar and P 0

1= 6bar. From this plot, it is

clear for both primary pressures that the expansion-shock cells structures are

qualitatively and quantitatively different in the supersonic jet from the pri-

mary nozzle exit. The first cells are obviously more dissipated in the case of

the k − ε and the average Mach number is significantly larger in the mixing

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section for the k−ω − sst model. However, in terms of entrainment rate both

models provide exactly the same results along the whole range of operating

condition. This observation raises some doubts concerning the link between

the prediction of the local flow dynamics and the global ejector performance.

The next section is devoted to provide an explanation and attempts to find a

key parameter which could justify those similar results.

3 CFD analysis of the ejector operation

3.1 On-design operation: location of the sonic line as a key parameter

As noted in the previous section, even though both models predict the same

entrainment rate over the whole on-design conditions, local flow features such

as the Mach number may significantly differ. This clearly demonstrates that

global measurements are not enough to assess a given turbulence model. This

also contradicts a strong belief among the researchers linking a good predic-

tion of the local flow physics to a good prediction of the ejector overall per-

formances. This statement allows to understand why one-dimensional models,

providing a poor description of the local flow dynamics, may provide relatively

good results at on-design conditions. This reasoning means that despite differ-

ent local flow structures, both models provide a same key parameter involving

a same entrainment rate. From the theory on which Munday and Bagster [2]

relied, this key parameter could be the critical cross section and its location

where the ejector becomes choked. From the inviscid point of view, a duct is

said to be choked when the Mach number reaches the unity over a whole cross

section. At this point, the information cannot travel back through this section

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and the mass flow rate does not depend any more on the downstream pressure.

For a viscous flow, the phenomenon is the same but the Mach number cannot

reach the unity over the whole section due to the occurrence of a boundary

layer involved by the non-slip condition at walls. From this idea, it is believed

that knowing the sonic line location where M = 1 along the ejector, it could

help to find the location of this critical cross section and check whether or not

both turbulence models are in agreement for this parameter. Indeed, when

plotting the sonic line location along the ejector, this line should reach some

maxima, matching points where the distance between this line and a wall is

minimum. Therefore at those points, the supersonic part of the flow fills the

largest part of the given cross section.

Figure 7 illustrates the evolution of the Mach number field and the sonic line

location in the case P 0

1= 4bar for three decreasing back pressures (Fig. 7(a-

c)) around the critical point. The results are also depicted for both turbulence

models. For each figure, the upper-half Mach number field represents results

for the k−ω−sst model, while the results for the k− ε are shown in the lower

half Mach number field. The solid line overlapped in each field represents the

sonic line where M = 1. For each condition, the matching back pressure is

depicted by a single dot on the operation curve (lower left plots, Fig. 7(a-c))

and the sonic line location is plotted in more details along the mixing chamber

on the lower right plots, the upper wall being represented by a solid line.

The different Mach number fields clearly illustrate the different flow structures

provided by both models (Fig. 7(a-c)). The k−ω− sst model is obviously less

dissipative than k − ε: indeed more shock-cells are visible for the k − ω − sst

model and the gradients appear to be larger. In addition, for those conditions,

the first shock cell revealed a strong shock, the Mach number behind the shock

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being subsonic (sonic line) (Fig. 7(a-c)). For back pressures below Pb = 1.3bar,

both sonic lines are far from the wall as it is obtained for the k − ε model for

Pb = 1.3bar (Fig. 7(a)). This means that for the critical pressure, where the

on-design and off-design curves forms an angle, the maximum mass flow rate is

reached but the ejector does not work yet in choked conditions. This situation

is very clear for the k− ε model at Pb = 1.3bar and P 0

1= 4bar (Fig. 7(a)) and

to our knowledge this feature has never been noticed previously in literature.

However, for those same conditions, the ejector can be considered to be choked

for the k − ω − sst model: the sonic line is very close to the upper wall where

it reaches a maximum (Fig. 7(a)). This can be easily confirmed because when

the back pressure is further decreased (Fig. 7(b-c)), the sonic line location is

changing only downstream of this maximum for the k−ω−sst. Thus the axial

location of this maximum can be considered as the location of the critical cross

section. This also means that the secondary mass flow rate is built upstream

this section. If the back pressure is decreased to Pb = 1.25bar, the k−ε predicts

choking conditions but farther downstream than k−ω−sst, at the entrance of

the diffuser (Fig. 7(b)). In this case it is clear that the prediction of the critical

cross section location is very different according to the turbulence model.

However, it would be easy to think that a longer sub-critical area (obtained

for k−ε) could promote a higher entrainment rate. Indeed, as a result of energy

exchange between both streams, the total pressure of the secondary flow tends

to increase with length, which would induce a higher critical mass flow rate

for a given critical section. On the other hand, wall friction tend to dissipate

a part of the energy, levelling the total pressure increase and then the critical

mass flow rate. This two effect are in competition and further quantitative

analysis should be performed to clear out this issue providing entrainment rate

depends also on primary mass flow rate and on the effective critical section.

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However, this could explain why the k−ε model predicts an overall entrainment

rate lower than k − ω − sst. This critical section for the k − ε model is also

confirmed when the back pressure is decreased to Pb = 1.2bar (Fig. 7(c)):

the sonic is only changing dowstream its critical point previously noted figure

7(b). However, at this back pressure an interesting feature is observed for the

k − ω − sst model (Fig. 7(c)). For this pressure Pb = 1.2bar, the sonic line

location reaches new maximum roughly at the same location than the k − ε

cross section and farther downstream of the first observed from figure 7(a) for

the k − ω − sst model. It means that the predicted flow features two critical

sections from this back pressure. Consequently, the downstream information

(back pressure) cannot travel back any more upstream this new critical section

and it has been checked by decreasing again the outlet pressure. However, the

secondary mass flow rate was built up from the first critical section observed

figure 7(a). Multiple critical sections can be then observed in such ejectors

giving conclusions more difficult: to our knowledge, this point has never been

raised in literature.

Figure 8 illustrates the same choking process for P 0

1= 6bar. In this case, the

first shock is weaker than in the case P 0

1= 4bar, because the Mach number

behind the shock remain supersonic (no sonic line). At the critical point (Fig.

8(a)), the secondary maximum mass flow rate is reached, but the ejector is not

choked: the sonic lines are not very close to the wall, and the entrainment rate

is still sensitive to the back pressure. When the pressure is slightly decreased

to Pb = 1.65bar (Fig. 8(b)), the sonic lines location is significantly changed

to reach a maximum value closer to the wall. In addition, this maximum is

roughly located at the same axial position and its value is very similar for

both models. This means the critical section location predicted by the k − ε

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and k−ω− sst models is in good agreement. This is confirmed at Pb = 1.5bar

where the sonic line location upstream those maximum remains unchanged.

Beyond this point, other critical sections can be observed near the end of the

mixing section as it was the case for P 0

1= 4bar. Consequently, for P 0

1= 6bar,

both models predict a similar cross section location and also a similar effective

area value, providing the same entrainment rate even though the local flow

parameters are very different. This feature was not observed for P 0

1= 4bar

where the k − ε and k − ω − sst models predicted different results.

3.2 Off-design operation

The previous paper [1] showed that the different behavior according to the

turbulence model at off-design conditions is more sensitive to the back pres-

sure. Indeed at off-design conditions, the situation is more complex because

the secondary mass flow rate is not only connected to an effective cross sec-

tion, but depends also on the momentum transfers between both flows and

pressure losses along the whole ejector. In other words, the entrainment rate

would depend on the quality of the mixing, and this issue would deserve a

more detailed study. However, one of the possible reasons involving an overall

weaker entrainment rate for the k − ε is depicted figure 9. This figure repre-

sents the stream function for different back pressures at off-design operation

and for the case P 0

1= 6bar which gave the larger discrepancies between both

models [1]. As the back pressure is increased, a flow detachment occurs near

the wall of the secondary nozzle. When the back pressure is further increased,

this detachment is turned out into a large recirculation at Pb = 1.25bar. This

phenomenon involves high losses and reduces the effective area provided to

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the secondary flow. At Pb = 1.30bar, the recirculation takes the whole section

and the secondary mass flow rate falls to zero and even to negative values; in

this case the ejector does not operate correctly anymore. For the same condi-

tions, the k − ω − sst model does not predict any boundary layer separation

and recirculations, providing a much better entrainment rate than the k − ε

model. At this stage, there are two possible reasons for these recirculation

zones. On one hand, it is possible that the k − ε model predicts a weak mo-

mentum transfer to the secondary stream in the area of the secondary nozzle.

In this case, the secondary stream would not have a sufficient total pressure

to flow against the adverse pressure gradient imposed by the back pressure.

On the other hand, the wall treatment at walls which is different according

both models could involve such differences. Indeed a classical wall function

approach is used for the k − ε model while a one-dimensional low Reynolds

model smoothly matches the fully turbulent core flow. However a more de-

tailed study focussed on the off-design operation should be performed to solve

this issue. In addition, flow-field visualizations would be useful to identify such

structures.

4 Conclusion

This paper was mainly devoted to a numerical analysis of CFD-experiments

results presented in [1]. In this paper, two turbulence models were assessed in

terms of predictions of the operation curve of a supersonic ejector. For this

purpose, a new test rig was built with the focus of CFD-experiment integra-

tion. The required conditions for a good integration is a very good matching

between modeling and experiments in terms of geometry and boundary condi-

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tions. The obtained results were better than those which can be usually found

in literature and demonstrated the relative superiority of the k − ε model.

However, the question relative to the choice of the turbulence model is not yet

fully solved and understood. Indeed, it was demonstrated that the relative per-

formance of both tested models is strongly dependant of the primary pressure

and the operation mode (on-design, off-design). At high primary pressures,

the problem is even more complex since both models predict the same ejector

performance while providing very different local flow features. This indicates

that validations based on the entrainment rate is not sufficient for a correct

assessment. In addition, these results also raise the issue on the link between

a good prediction of the ejector operation (entrainment rate) and a good pre-

diction of the local flow physics. The former point is important in terms of

cycle operation, but the latter issue is very important for the understanding

and improvement of the ejector itself. In this paper, the analysis of the sonic

line location together with the Mach number field and operation curve was

proposed as a key parameter to understand the link between local flow features

and the entrainment rate. Furthermore, this parameter allowed to locate the

position of the critical section where the ejector becomes choked. This method

has never been proposed in previous CFD studies.

Acknowledgement

Author wish to acknowledge the General Directorate for Technology, Research

and Energy (D.G.T.R.E.) of the Ministry for Belgium’s Walloon Region to fi-

nancially support the PROFESSI project. Authors would also to thank all the

technical staff of the TERM division for its help in the set-up of the exper-

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imental stand and particularly François Vercheval for providing illustrations

for this paper.

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References

[1] A. Hemidi, F. Henry, S. Leclaire, J. M. Seynhaeve, Y. Bartosiewicz, Experimental

and numerical analysis of the operation curve of a supersonic ejector. single-phase

and two-phase operation, Submitted to Applied Thermal Engineering.

[2] J. T. Munday, D. F. Bagster, A new theory applied to steam jet refrigeration,

Ind. Eng. Chem. Process Res. Dev. 4 (1977) 442–449.

[3] Y. Bartosiewicz, Z. Aidoun, P. Desevaux, Y. Mercadier, Cfd-experiments

integration in the evaluation of six turbulence models for supersonic ejector

modeling, in: Integrating CFD and Experiments Conference, Glasgow,UK, 2003.

[4] Y. Bartosiewicz, Z. Aidoun, P. Desevaux, Y. Mercadier, Numerical and

experimental investigations on supersonic ejectors, International Journal of Heat

and Fluid Flow 26 (2005) 56–70.

[5] Y. Bartosiewicz, Z. Aidoun, Y. Mercadier, Numerical assessment of ejector

operation for refrigeration applications based on cfd, Applied Thermal

Engineering 26 (5-6) (2006) 604–612.

[6] E. Rusly, L. Aye, W. W. S. Charters, A. Ooi, Cfd analysis of ejector in a combined

ejector cooling system, International Journal of Refrigeration 28 (2005) 1092–

1101.

[7] T. Sriveerakul, S. Aphornratana, K. Chunnanond, Performance prediction of

steam ejector using computational fluid dynamics: Part 1. validation of the cfd

results, International Journal of Termal Sciences 46 (8) (2007) 812–822.

[8] T. Sriveerakul, S. Aphornratana, K. Chunnanond, Performance prediction of

steam ejector using computational fluid dynamics: Part 2. flow structure of a

steam ejector influenced by operating pressures and geometries, International

Journal of Termal Sciences 46 (8) (2007) 823–833.

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[9] K. Pianthong,

W. Seehanam, M. Behnia, T. Sriveerakul, S. Aphornratana, Investigation and

improvement of ejector refrigeration system using computational fluid dynamics

technique (2007).

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a)

b)

primary nozzle

primary flow Secondarynozzle

Mixing section b

Diffuser

2

1

Figure 1. A supersonic ejector in a solar air-conditioning cycle (a). A typical ejector

geometry (b)

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a)

b)

Figure 2. A 3D view of the ejector geometry with a secondary chamber to make the

flow axisymmetric (a). Main ejector dimensions (b)

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a)

b)

Figure 3. Computational mesh

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Figure 4. Ejector operation curve

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a)

1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

Pb/P

20

ω

P10 = 4 bar

Experimental datak−εk−ω−sst

b)

1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

Pb/P

20

ω

P10 = 6 bar

Experimental datak−εk−ω−sst

Figure 5. Comparison CFD-experiments for P01

= 4bar (a) and P01

= 6bar (b)

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a)0 20 40 60 80 100 120 140

0

0.5

1

1.5

2

2.5

x (mm)

Mac

h nu

mbe

r

P10= 4 bar − P

b/P

20 = 1.1 k−ε

k−ω−sst

b)0 20 40 60 80 100 120 140

0

0.5

1

1.5

2

2.5

x (mm)

Mac

h nu

mbe

r

P10= 6 bar − P

b/P

20 = 1.5 k−ε

k−ω−sst

Figure 6. Centerline Mach number at one on-design condition for P0

1= 4bar (a) and

P0

1= 6bar (b)

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a)

b)

c)

Figure 7. Mach number field, sonic line location in the mixing chamber for P01

= 4bar

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a)

b)

c)

Figure 8. Mach number field, sonic line location in the mixing chamber for P01

= 6bar

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k − ε Pb(bar) k − ω − SST

1.15

1.20

1.25

1.30

Figure 9. Stream functions inside the secondary nozzle (close to the primary nozzle

outlet) for P0

1= 6bar and P

0

2= 1bar

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Nomenclature

m [m.s−1] Mass flow rate

P [bar] Pressure

r [mm] Radial location

x [mm] Axial location from the throat of the primary nozzle

ω = m2

m1[−] Entrainment ratio or entrainment rate

Subscripts, superscripts:

0 Total, stagnation property

1 Primary flow inlet

2 Secondary flow inlet

b property at back (ejector outlet)

* Critical property

26