9th Annual International Energy Conversion Engineering Conference AIAA 2011-5804

31 July - 03 August 2011, San Diego, California

An Ejector Transient Performance Model for Application in

a Pulse Refrigeration System

S. du Clou1 and M.J. Brooks2

University of KwaZulu-Natal, Durban, 4000, South Africa

W.E. Lear3 and S.A. Sherif4

University of Florida, Gainesville, FL 32611-6300

and

E.E. Khalil5
Cairo University, Cairo, Egypt

This paper addresses the design and performance of an ejector model under two-phase,
transient flow conditions, as encountered in a recently proposed, pump-free, pulse
refrigeration system, or PRS. In the PRS, a cooling effect is derived by supplying the
primary inlet of an ejector with unsteady pressure pulses from a bank of constant volume
boilers. The interaction of the resulting transient flow with the ejector is predicted using an
analytical model implemented in MATLAB and incorporating real vapor data sub-routines.
The software consists of two parts, a design code and a performance analysis code. The
design code permits optimization of the ejector geometry for a given set of steady flow
conditions, whereas the performance code permits investigation of the given geometry with
unsteady, two phase flow. A quasi-steady approach is followed to map the performance of
the device through the transient flow regime. A comparison is made to experimental results
in the literature and a revised ejector test facility is proposed for experimental comparison.

Nomenclature
A = area m2
a = sonic velocity m/s
CD = converging-diverging nozzle
d = diameter m
h = enthalpy J/kg
ID = inner diameter
k = specific heat ratio, Cp/Cv
L3 = length of constant area mixing chamber m
M = Mach number
= mass flow rate g/s
P = pressure bar
PRS = pulse refrigeration system
Pr = pressure ratio, Pe/P0
s = entropy J/kg.K

1
Graduate Student, School of Mechanical Engineering, UKZN, Durban, South Africa. Non-member of AIAA.
2
Senior Lecturer, School of Mechanical Engineering, UKZN, Durban, South Africa. Member of AIAA
3
Associate Professor, Mechanical and Aerospace Engineering, University of Florida, PO Box 116300, Gainesville,
FL 32611-6300. Associate Fellow of AIAA.
4
Professor, Mechanical and Aerospace Engineering, University of Florida, PO Box 116300, Gainesville, FL 32611-
6300. Associate Fellow of AIAA.
5
Professor, Mechanical Engineering, Cairo University, Cairo, Egypt. AIAA Deputy Director (International).
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Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
V = flow velocity m/s
x = quality
Greek letters
ρ = density kg/m3
ω = entrainment ration,
ψ = ejector compression ratio, Pc/Ps0
Subscripts
e = converging diverging nozzle exit property
is = isentropic
mix = property after mix
p = primary stream property
s = secondary stream property
t = converging-diverging nozzle throat property
x = conditions before shock
y = conditions after shock
0 = total stagnation property
1 = property downstream of CD nozzle
Superscripts
* = critical point

I. Introduction

E JECTORS are frequently used in the design of recirculating fluid systems to reduce power consumption and
exploit renewable energy sources. A variety of ejector-based refrigeration and cooling systems have been
proposed or developed. However, most either rely on a pump to circulate the fluid through the system, or only
operate in the presence of gravity. Huang et al. 1, 2 described cooling systems that utilize an ejector as the mixing and
compression device. The benefits include reduced mass and simplified design. Such systems can also be powered by
solar thermal energy, with the only significant pitfall being a relatively low coefficient of performance. A solar
integrated thermal management and power (SITMAP) cycle was investigated by Nord et al. 3 that utilized an ejector
in the cooling loop. The ejectors of these designs rely on steady flow operation and provide the compression in the
refrigeration part of the cycle.
A Pulsed Refrigeration System (PRS) that operates independently of gravity, and which is therefore suitable for
both terrestrial and space systems, was recently proposed by Brooks et al. 4. In the thermally pumped refrigeration
loop, variable flow is generated by a bank of constant volume boilers, such as those in the pulse thermal loop
proposed by Weislogel et al.5. Previous work by Brooks et al.4 and du Clou et al.6 aimed to incorporate an ejector
into the PRS and suggested that an ejector design based purely on geometric scaling ratios from the literature is
inadequate. This is due to the unsteady operation of the device which is fed by a depressurizing boiler and which
blows down through the ejector to a condenser. The complexity introduced by transient two phase flow resulted in
the test ejector operating in off-design mode, where it failed to provide meaningful entrainment.
Although much research has been carried out analytically 3, 7-12, computationally13 and experimentally14, 15 on
ejectors, there is still not a well defined method to design such a device for operation under transient conditions,
especially where two-phase flow is involved. Sun10 investigated the effects of various operating conditions on the
geometry of an ejector in a refrigeration system. The author concluded that the geometry of the ejector should
ideally be variable in order to provide constant cooling capacity at various generator, evaporator and condenser
operating conditions.
This paper describes a design approach for a PRS-based, transient flow ejector experiencing two-phase
conditions. The approach utilizes two ejector models: a design model is first used to calculate the initial ejector
geometry and a performance model is then used to investigate the effects of operating the proposed geometry under
transient conditions.

II. Ejector Design

The ejector is a mechanically simple mixing device which provides compression to a secondary stream and
which may be analyzed using compressible flow theory. The ejector depicted in Fig. 1 is comprised of four sections:
the converging-diverging (CD) nozzle at the primary inlet, the suction chamber housing the secondary inlet, the
constant-area mixing chamber and the recovery diffuser. Ideally, the primary flow expands isentropically and
accelerates through the CD nozzle to reach supersonic velocity (process 0-e). At the outlet of the nozzle a low
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pressure region forms in the suction chamber that entrains a secondary flow (process e-1) that is at some
intermediate pressure. The secondary flow undergoes Fabri chocking, as described by Munday et al.16, due to the
hypothetical converging duct formed between the primary flow stream and the ejector wall. The primary and
secondary flows mix during the constant pressure process (process 1-mix). A shock wave forms in the constant area
chamber if the ejector is operating under choked conditions (critical operation). The resulting stream regains
pressure in the diffuser (process 2-C).
An ejector is defined using geometric ratios, pressure ratios and the entrainment ratio. The entrainment ratio (ω)
is the ratio of the entrained secondary flow rate to the primary flow rate. The ejector compression ratio (ψ) is the
ratio of the compressed downstream pressure to the entrained secondary pressure and is inversely proportional to the
entrainment ratio.
In a PRS system with a large enough feed boiler, the flow through the ejector would be considered steady. The
PRS has a limited volume boiler and condenser, however, and the ejector is therefore subject to a transient, blow-
down effect characterized by an initially strong but decaying pressure pulse. The driving pressure decreases while
the back pressure in the condenser increases, which introduces unsteady flow and limits entrainment at the
secondary inlet.
Suction chamber Constant area chamber, Lm Diffuser

Secondary
flow

Nozzle
 s0
s1 (Hypothetical throat)
Primary flow 
 p0  e  1  mix  2  c
Pp0, Tp0, hp0, t
to condenser
ρp0,

Figure 1. Ejector design schematic

III. Nozzle Theory

The primary nozzle within an ejector converts enthalpy into kinetic energy. Figure 2 depicts the operating modes
of a converging-diverging (CD) nozzle. In conventional nozzle theory the pressure ratio (Pr) is adjusted by reducing
the back pressure at the nozzle exit to induce flow. For application in a PRS, the nozzle Pr is affected by changes in
both the upstream and the back pressure, the former more so than the latter. The ejector will initially have a very low
Pr and it will gradually increase to unity, where the back pressure equals the inlet pressure.
The third critical isentropic curve in Fig. 2 resembles the design condition. The flow is choked at the throat, is
supersonic for the entire diverging section and expands perfectly to meet the back pressure at the nozzle exit, Pe. For
a Pr less than the third critical point (Pr < 3rd critical) the flow is under-expanded. A slight increase in the Pr from
this point (3rd < Pr < 2nd critical) results in the flow being over-expanded with a complex series of supersonic wave
motions, or non isentropic oblique shocks, outside the nozzle. The oblique shocks are weaker than normal shocks
and the angle of the shock adjusts to produce the required pressure rise. If the oblique shock is weak the resulting
flow will still be supersonic. Stronger oblique shocks will result in subsonic flow. For a CD nozzle operating at its
second critical point, a normal shock is located at the exit plane resulting in a pressure increase that is precisely
required to meet the higher back pressure. For an operating Pr higher than the second critical but less than the first
critical (1st < Pr < 2nd critical) a normal shock locates itself inside the diverging section such that the pressure change
before the shock, across the shock and downstream of the shock will result in the exit pressure being equal to the
back pressure. The first critical point represents flow that is choked at the throat with a Mach number of one and is
subsonic for both the converging and diverging sections. Any Pr above the first critical point (Pr > 1 st critical) will
result in subsonic flow throughout the CD nozzle. In a choked nozzle any disturbance downstream cannot be
transmitted upstream because the fluid velocity is greater than or equal to the velocity of the pressure wave
propagation. For application in a PRS-based ejector, the flow in the nozzle should be perfectly expanded to
achieve best performance.

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 P0 Pe

Venturi
1.0
1st critical
Pressure ratio, P/P0

Normal shocks

2nd critical (exit shock)

Over-expansion (oblique shocks)

0 3rd critical
Entrance Throat Exit Under-expansion
Nozzle position
Figure 2. Operating modes of a converging-diverging nozzle

IV. Analytical Model

Two software tools have been developed to investigate ejector transient flow; a design model and a performance
model. These are coded in MATLAB with the refrigerants’ thermodynamic properties referenced using NIST
RefProp (Ver. 7.0) subroutines. The local geometric characteristics and fluid properties are iteratively calculated at
various locations along the ejector axis using compressible flow theory with the conservation of mass, energy, and
momentum described by Eqs. (1), (2), and (3). The sonic velocity for a real gas is calculated using Eq. (4) and the
Mach number is found using Eq. (5). The area and diameter can then be calculated using Eqs. (6) and (7).

(continuity) (1)

(energy) (2)

(momentum) (3)

(4)

(5)

(6)

(7)

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The following assumptions are made for both models:
1. For steady flow, the fluid properties are constant across the cross-section at any given x-coordinate.
2. Due to the isentropic curve crossing the saturated vapor line into the two phase region of the pressure-
enthalpy diagram, certain fluids (like R134a) condense when they expand. The resulting mixture quality is
high, x > 0.95, which gives a low liquid volume faction. Under these conditions it is reasonable to assume
that the two phase mixture is homogeneous with no slip between phases.
3. To account for non ideal losses due to friction and mixing, isentropic efficiency coefficients are included in
the code. When set to 1, adiabatic flow is assumed everywhere except across shocks where there is an
entropy rise.
4. The thickness of the shock is negligible.
5. Kinetic energy at the primary and secondary reservoir is negligible.
6. The mixing of the primary and secondary streams is assumed to occur at constant pressure after the
secondary stream has expanded to the hypothetical throat.
7. Flow separation from the boundary is ignored, assuming small angles and polished surfaces.
8. The ejector walls are adiabatic.

A. Design Model
The design model generates the optimum geometry of INPUTS
an ejector given the required entrainment ratio and user- 1) Fluid
defined steady operating conditions (inputs) at the primary
2) Pp0, Pr, xp0, , ηis, Ps0, xs0, Pc, xc, ω
inlet, secondary inlet and the outlet. The nozzle is set to
expand perfectly with the pressure ratio, Pr. 3) Inlet and outlet tube diameters
The code is implemented using the logic diagram in
Fig. 3. Routine application of the governing equations CD THROAT
solves the flow at each point along the ejector axis. The Reduce Pt until Mt=1 At
non-isentropic normal shock wave is modeled by
iteratively increasing the pressure, and finding the local CD EXIT
fluid properties using conservation of mass and Ae
Expand to Pr
momentum for the control volume surrounding the shock,
until the calculated density after the shock is equal to or
greater than the reference density. SECONDARY FLOW
As1
Expand until Ms1=1
B. Performance Model
The performance model solves for the operating MIXING Am
parameters of a given ejector that is fed by a Pe=P1=Pm
depressurizing boiler initially containing a fixed mass. The
unsteady flow is assumed to be quasi-steady with the
SHOCK
instantaneous flow properties being a function of time. At
Increase Py until
each incremental time step, a fixed mass of refrigerant
ρcalc > ρreference
leaves the boiler resulting in a decrease in feed pressure,
temperature and density. Although the boiler block in a
PRS would operate at a constant temperature, the rapid EJECTOR EXIT
blow-down reduces the pressure of the refrigerant inside Pc
Increase Pc until Ac
the vessel faster than thermal conduction and convection
from the block to the refrigerant can take place. Therefore
the boundary for the control volume of the fluid in the WRITE TO FILE
boiler is considered to be adiabatic, and it depressurizes
Figure 3. Design model logic diagram for
isentropically.
solution of optimal ejector geometry under
The temporal operating modes of the CD nozzle in the
steady conditions
ejector are solved to examine the dependence of normal
and oblique shock waves on the transient flow. Ideally a variable geometry nozzle is required for maximum
performance, for unsteady flow. Adequate entrainment could be achieved for a narrow range of operating conditions
for a fixed geometry nozzle. For these reasons the performance code is focused on solving the flow for a CD nozzle.
Once the flow through the nozzle is fully characterized, the rest of the ejector can be modeled. Oblique shocks
cannot be accurately simulated in this one dimensional approach. The code is implemented using the logic diagram
in Fig. 4.
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 INPUTS
1) Fluid
2) Pp0, xp0, (m0 & V0 for constant volume boiler), Pp1, ηis
3) dt, de, dm, and the inlet and outlet tube diameters
4) Time step for quasi-steady analysis

INITIALISE
New boiler conditions for reduced mass (isentropic blow-down)
3rd critical isentropic calculated for given CD nozzle

SUPERSONIC SUBSONIC
Pr < 1st critical Pr > 1st critical

CD THROAT C-D NOZZLE THROAT

Reduce Pt until Mt=1 Reduce Pt for Mt < 1

CD EXIT C-D NOZZLE EXIT

Reduce Pe to Ae and compare to P1 Set Pe=P1

Is Pe < P1? No
CD EXIT
Yes Free expansion to P1

OVER-EXPANDED 1st <Pr < 3rd critical

2nd < Pr < 3rd critical

BEFORE SHOCK
Guess Px C-D NOZZLE EXIT (e)
 Reduce Pe isentropically until Ae
AFTER SHOCK reached (3rd crit)
Increase Py until ρcalc > ρreference
AFTER OBLIQUE SHOCK (1)
 Reduce to P1 isentropically
1st < Pr < 2nd critical Is Py > P1? assuming Ae=A1
No Yes

CD EXIT
Pe=P1 WRITE TO
Yes FILE
Is Ae_calc > Ae
No
No
TERMINATE LOOP
Is Pp0 < P1 ?

Figure 4. Performance model logic diagram for the analysis of unsteady flow through an ejector
converging-diverging nozzle
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 V. Model Validation

A. Design Model
The design model is validated by comparing results with experimental data and the theoretical design model of
Huang et al.7. Although the models are similar, the fluid properties here are solved using the governing equations
rather than ideal-gas isentropic relations. Using refrigerant R141b, different operating pressures and the entrainment
ratio as inputs, the area ratio (Am/At) of the designed ejector is determined. Table 1 compares the theoretical results
obtained in the current model with the computed results of Huang et al.7. Table 2 compares the theoretical values
obtained in the current model with the experimental results of Huang et al.7. Three generator pressures; 400 kPa, 465
kPa, and 537 kPa (saturated vapor) are investigated. For a given generator pressure, three entrainment ratios are
modeled resulting in the corresponding optimal area ratio. For the cases considered the evaporator (secondary inlet)
pressure was kept constant at 40 kPa saturated vapor. The error is calculated using Eq. (8) to compare the current
model with the work of Huang et al.7. The model under predicts the required area ratio for the theoretical and
experimental comparisons with a mean error of 8.7% and 6% respectively.

(8)

Figure 5 graphs the area ratio results of the new model (solid lines) and the model of Huang et al.7 (dashed lines)
against the experimental area ratios of Huang et al.7. For each primary pressure three entrainment ratios result in
three specific area ratios. The trends are similar with small systematic error in the current model. This may be due to
the empirical loss coefficient that is not included in the model. The area ratio is confirmed to be proportional to the
entrainment ratio.

Table 1. Design model vs. Huang et al.7 model results

Input Pressure Theoretical Huang et al.7 Design model Error, %
(kPa) entrainment ratio, ω Theoretical area ratio, Am/At Theoretical area ratio, Am/At
604 0.46 10.87 10.24 -5.8
0.3 8.57 7.86 -8.3
0.19 7.05 6.32 -10.4
538 0.42 9.28 8.61 -7.2
0.24 7.03 6.3 -10.3
0.22 6.74 6 -11.1
465 0.52 9.34 8.72 -6.6
0.37 7.68 7.02 -8.6
0.29 6.79 6.1 -10.2
Average Error. 8.7%

Table 2. Design model vs. Huang et al.7 experimental results

Input Pressure Experimental Huang et al.7 Design model Error, %
(kPa) entrainment ratio, ω Experimental area ratio, Am/At Theoretical area ratio, Am/At
604 0.44 10.64 9.88 -7.2
0.28 8.28 7.61 -8.1
0.2 6.77 6.5 -4
538 0.44 9.41 8.96 -4.8
0.3 7.73 7.14 -7.7
0.22 6.44 6.11 -5.1
465 0.54 9.41 8.91 -5.3
0.39 7.73 7.22 -6.6
0.29 6.44 6.1 -5.3
Average Error. 6%

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 Huang experimental ejector area 11
10.5 Fluid = R141b
10 PS0 = 40 kPa
604 kPa
= 100 g/s
9.5 Huang
538 kPa
ratio, A3/At

9 model
8.5 465 kPa
8 604 kPa
7.5 Current
538 kpa
7 model
465 kPa
6.5
6
6 7 8 9 10 11
Theoretical ejector area ratio, Am/At

Figure 5. Area ratio for the design model is compared to the experimental results of Huang et al. 7 (solid
lines), and as a comparison, the referenced authors’ theoretical results are plotted against their experimental
results (dashed lines). 1 bar = 100 kPa

B. Performance Model
The performance model is validated using an ideal gas, Nitrogen, and comparing the computed results with the
ideal-gas isentropic relations at each transient time step. The isentropic Mach relation Eqs. (9-13) and the non-
isentropic shock relations Eqs. (14-17) are computed for the comparison. Figures 6, 7, and 8 compare the real to the
ideal results for the case where a 100 cm3 boiler containing 8 g of pressurized Nitrogen is expanded through a CD
nozzle having a 0.8 mm throat and 1.6 mm exit diameter. The back pressure for the CD nozzle is set at 40 kPa to
simulate the required ejector suction pressure. The output pressures, Mach number and the mass flow rate converge
to similar results. The isentropic free expansion wave for under-expanded flow and oblique shocks for over-
expanded flow downstream of the CD nozzle cannot be modeled accurately using one-dimensional theory.

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

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 1800
Fluid = Nitrogen Boiler Pressure
1600
Mass = 8 g Ptt
P
1400
Volume = 100 cm3 Pt_is
P t,is
1200
Pressure, kPa

P1 = 40 kPa Pee
P
1000 dt = 0.8 mm Px
Px
800 de = 1.6 mm Px_is
P x,is
600 Pyy
P
400 Py_is
P y,is
200
0
0 0.5 1 1.5 2 2.5
Time, s
Figure 6. Static pressure trace during transient blow-down. Real solution, solid line, compared with the ideal
solution of Eqs. (9) and (15), dotted line.

3
Mthroat
M t
2.5 Mx
M x

2 My
M y
Mach

1.5 My_is
M y,is

Mee
M
1
M1
M 1
0.5
0
0 0.5 1 1.5 2 2.5
Time, s
Figure 7. Mach number plot during transient blow-down. Real solution, solid line, compared with the ideal
solution of Eq. (14), dotted line.

4
Mass flow rate, g/s

m_dot_p
3 m_dot_p_is
2

0
0 0.5 1 1.5 2 2.5
Time, s
Figure 8. Mass flow rate reduced as boiler empties during blow-down. Real solution, solid line, compared
with the ideal solution of Eq. (13), dotted line.

VI. Results

A. Design Model
Before the model is implemented, the user must consider the working fluid and the application or system for
which the ejector is designed. In this study, the PRS is driven by pulse thermal loop (PTL) boilers which requires the
ejector to operate with a varied primary pressure ranging from 8 bar to 20 bar6. The vapor quality and operating flow
rates must also be anticipated, refer to the logic diagram in Fig. 3 for all inputs.
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 Tables 3 and 4 give an example set of results for an R134a ejector operating with an 18 bar primary and 4 bar
secondary saturated vapor inlet pressure, primary mass flow rate of 5 g/s, an entrainment ratio of 0.2, and a CD
nozzle Pr of 0.1. The inputs to the model are shaded in grey. The tube diameters upstream and downstream of the
ejector are constrained by the system in which it will operate. Here the primary inlet and outlet diameters are 1/4”
(4.5 mm ID) with a 1/8” (1.8 mm ID) secondary inlet. The output to the model is the ejector geometry and the fluid
properties at various locations along the ejector axis. One input variable (Pp0, Ps0, Pc, x0, xs, or ) can be varied
while keeping the others constant to obtain design graphs as shown in Figs. 9 and 10. Here the secondary inlet is
maintained at 4 bar whilst the primary pressure is adjusted from 18 bar down to 10 bar to investigate the effect on
the ejector geometry.
Figure 9 shows that a larger primary pressure requires a smaller geometry ejector to enable the working fluid to
expand fully, to the given Pr. Referring to the 18 bar results in Fig. 10 (red trace), the cross section diameter at
different axial locations along the ejector is plotted against the static pressure at that location. The working fluid
initially expands from 4.5 mm to the 0.92 mm throat to reach sonic velocity and the flow is choked. The flow
continues to expand to the CD nozzle outlet with a diameter of 1.51 mm. The secondary flow is entrained by the low
pressure region and is choked by the hypothetical throat that is formed between the core flow and the ejector wall.
The primary and secondary flow then mix at constant pressure and the diameter of the flow increases to 1.76 mm. A
shock wave in the constant area section (shown by vertical lines in the plot) raises the pressure of the working fluid
and prevents the downstream condenser from communicating with the secondary inlet to the ejector. The diameter
increases to the 4.5 mm diffuser outlet, resulting in further pressure recovery. The 16 bar results can be interpolated
between the 18 bar and 14 bar contours. Figure 11 illustrates the static pressure and the Mach contour along the
ejector axis. The normal shock is clearly evident by the immediate increase in static pressure with a decrease in
Mach number. The straight line plots in Figs. 10 and 11 are due to the limited sample points solved by the model;
realistically the pressure and Mach number plots follow a curve.

Table 3. Sample results for 1800 kPa primary inlet at 5 g/s, shaded regions denotes user specified inputs to
the model. 1 bar = 100 kPa
P T h s ρ x Mach A d
POSITION (kPa) (K) (J/kg) (J/kg.K) (kg/m3) (g/s) (mm2) (mm)
Inlet (p0) 1800 336 427356.5 1700.7 94.7 1 5 0 15.9 4.5
Throat (t) 1076.4 315.3 417326.5 1700.7 54.2 0.98 5 1 0.67 0.92
Exit (e) 180 260.4 382007.8 1700.7 9.5 0.96 5 2.1 1.8 1.51
Mix (m) 180 260.4 388198.3 1724.5 9.2 0.99 6 1.9 2.43 1.76
Before Diffuser (2) 635.94 305.8 420802.8 1747.4 29.3 1 6 0.6 2.43 1.76
Exit (c) 752.953 312 424472.9 1747.4 34.6 1 6 0.1 15.9 4.5
Secondary Inlet (s0) 400 282.1 403719.4 1722.6 19.5 1 1 0 3.14 2
Secondary exit (s1) 215.6 265 391258.2 1722.6 10.9 0.99 1 1 0.63 0.9

Table 4. Key design features for 1800 kPa example

EJECTOR DESIGN RESULTS (1800 kPa) R134a
Nozzle area ratio, Ae/At 2.69
Nozzle diameter ratio, de/dt 1.64
Nozzle pressure ratio, Pe/Pp0 0.1
Ejector area ratio, Am/At 3.64
Ejector diameter ratio, dm/dt 1.91
Ejector compression ratio, ψ 1.88
Ejector entrainment ratio, ω 0.2
Isentropic efficiency of nozzle 0.95
Isentropic efficiency of secondary inlet 0.85
Isentropic efficiency of diffuser 0.95

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 5
4.5 i Pr = 0.1 vi 18 bar
4 Ps0 = 4 bar 14 bar
= 5 g/s
Diameter, mm

3.5
ω = 0.2 10 bar
3
2.5 iv v
iii
2 i CD inlet
1.5 ii ii CD throat
iii CD exit
1
iv,v constant area
0.5 vi diffuser exit
0
0 1 2 3 4 5
Ejector axis (not to scale)

Figure 9. Ejector geometry designs for different primary inlet pressures

2000
1800 Pr = 0.1 18 bar
Ps0 = 4 bar 14 bar
1600
= 5 g/s
Pressure, kPa

1400 10 bar
ω = 0.2
1200
1000 i throat position
800 ii constant pressure
600 iv mixing
400 i iii shock wave
200 ii location
0 iv diffuser pressure
iii
0 1 2 3 4 5
Diameter, mm
Figure 10. Static pressure plot at different cross section diameter locations along the ejector in response to
primary inlet pressure for a constant flow rate.

i ii iii v vi
2000 2.5 18 bar
1800 Pr = 0.1 14 bar
Ps0 = 4 bar 10 bar
1600 2
= 5 g/s
Mach Number

1400 Mach
Pressure, kPa

ω = 0.2 Mach
1200 1.5
1000 Mach
800 1 i CD inlet
600 ii CD throat
400 0.5 iii CD exit
200 iv constant area
0 0 v normal shock
iv vi diffuser exit
0 1 2 3 4
Ejector axis (not to scale)

Figure 11. Static pressure and Mach number contours at different ejector axis locations for given steady
operating pressures.
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B. Performance Model
Since the under-expanded free expansion wave and the over-expanded shock train cannot be captured using one-
dimensional theory, only the CD nozzle at the inlet of the ejector is modeled for the transient regime. Investigating
the performance of a nozzle fed by a decreasing pressure pulse, similar to the pulses in the PRS will result in an
improved understanding of the flow dynamics during transient operation and will help lead to an improved ejector
design. For an ejector to provide entrainment, the CD nozzle must fully expand and avoid oblique and normal
shocks that produce entropy.
The results presented here are based on the geometry from the 16 bar design results, interpolated from Fig. 9,
giving a 1 mm throat and 1.6 mm exit diameter for the CD nozzle. Simulating an 18 bar pulse with 8 grams of
R134a from a 100 cm3 boiler, the flow is expected to initiate below the third critical point (under-expanded). This
will ensure that the design condition is met at some time during the blow-down where the flow is perfectly expanded
to the third critical Pr. To ensure that the flow is supersonic for a greater portion of the blow-down, a smaller exit
diameter of 1.4 mm is modeled.
Figure 12 graphs the operating modes during the transient blow-down; note the similarity to Fig. 2. The quasi-
steady approach tracks the shock location as it moves up the nozzle towards the throat. The cross section diameter at
the shock can be read off of the x-axis. At 2.8 seconds the normal shock is located at the 1.04 mm cross section. The
oblique shocks plotted, brown and yellow trace, are not an accurate representation of the flow because of one-
dimensional flow assumptions.

1
0.9 1st critical
Pressure ratio P/P0

0.8 3rd critical

0.7 Mass = 8 g
Volume = 100 cm3 Below 3rd critical 0s
0.6
Pp0 = 18 bar 1s
0.5 2nd to 3rd critical
0.4 P1 = 2 bar 2.15 s
0.3 dt = 1 mm 2nd critical 2.2 s
0.2 de = 1.4 mm
∆t = 0.05 s 1st to 2nd critical 2.4 s
0.1 Shock regression
0 2.8 s
Inlet
0.4 0.6 0.8 1 1.2 1.4 1.6
Throat Exit
Cross section diameter for transient shock location, mm

Figure 12. Ejector operating modes and shock location in response to an increasing pressure ratio during
transient blow-down operation
The Mach number at the throat, before the shock, after the shock, at the exit, and downstream of the CD nozzle
is plotted in Fig. 13. The flow downstream of the CD nozzle (M1) remains supersonic for 1.3 seconds (45 % of the
blow-down) which will result in secondary entrainment. This occurs when the pressure ratio is low (0.1 < Pr < 0.2)
as the flow is able to expand completely. Progressively stronger oblique shocks occur from 0.45 seconds to 2.15
seconds, reducing the downstream Mach number, followed by a normal shock wave at the exit of the CD nozzle at
2.2 seconds. As the normal shock regresses up the nozzle, the Mach number before the shock (Mx) reduces. The
shocks are progressively weaker resulting in increased Mach numbers after the shock. Although the Mach number
after subsequent shocks increases (My), the diverging diffuser reduces the kinetic energy thus increasing the exit
temperature and pressure. The resulting Mach number downstream of the CD nozzle (M1) decreases during the
transient in response to the emptying boiler, and the Pr increases to unity.
The flow rate decreases as the boiler empties, shown in Figure 14. The time scale for the transient blow-down is
directly proportional to the choked mass flow rate at the throat. A smaller throat (or a larger diffuser exit) diameter
will result in a longer transient but a reduced period of supersonic flow and an increased period of shocks. A larger
throat (or smaller diffuser exit) diameter would reduce the period of internal shocks but the required Pr may not be
achieved. Entrainment can be maximized by investigating different geometry ejectors for a set of operating
conditions.

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American Institute of Aeronautics and Astronautics
 Under-expanded Weaker Stronger Normal
flow oblique shocks oblique shocks shocks Venturi
2.5
Mthroat
M t
2
Mx
M x

1.5 My
M
Mach

1 Me
M e

M1
M 1
0.5

0
0 0.5 1 1.5 2 2.5
Time, s
Figure 13. Mach number at different locations along the nozzle as a function of time during blow-down

6
Mass flow rate, g/s

5 m_dot_p
4
3
2
1
0
0 0.5 1 1.5 2 2.5
Time, s
Figure 14. Mass flow rate during transient blow-down

VII. Proposed ejector test facility

A revised ejector test facility is proposed to investigate the transient blow-down operation of an ejector. Results can
then be experimentally validated for a set of test ejectors. The schematic in Figure 15 highlights the key components
which include temperature and pressure sensors, sample cylinders and the test ejector.

Evaporator

Pressure
transducer
Ejector

Thermocouple Boiler Solenoid Quick connect Condenser Charge port

valve

Figure 15. Revised ejector test facility

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American Institute of Aeronautics and Astronautics
 VIII. Conclusions
The design and performance of an ejector for use under transient flow conditions is detailed using a two part,
analytical, two-phase model. The design model accurately describes the geometry required for steady flow
conditions. Different operating parameters were imposed to achieve specific ejector geometries and the results were
comparable with the literature. A selected design was then tested using the performance code to investigate transient
off-design conditions.
The performance model was not extended to include the mixing of the transient primary and secondary stream
because the free expansion wave and the oblique shock wave could not be modeled accurately using one-
dimensional theory. The CD nozzle code was validated using an ideal gas and then tested with R134a. The
performance results provided insight into the operation of an ejector during transient blow-down. It is evident that
only a portion of the blow down would result in entrainment during a narrow range of pressure ratios. For an
increasing pressure ratio, the flow rapidly becomes sub sonic and results in zero entrainment. Designed correctly, an
ejector that operates in a transient system can provide entrainment for a finite period.

IX. Acknowledgements
Sven du Clou and Michael Brooks appreciate the support of Prof. Lance Roberts, Prof. Jeff Bindon (UKZN) and
the Centre for Renewable and Sustainable Energy Studies (CSRES) at the University of Stellenbosch. This work is
supported by Eskom TESP and the National Research Foundation of South Africa.

X. References
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11
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