- 1 - Optimum Positioning of Ports in the Limaçon Gas Expanders 1 Ibrahim A. Sultan Carl G. Schaller School of Science and Engineering The University of Ballarat PO Box 663 Ballarat 3353 VICTORIA, Australia Email: [email protected]ABSTRACT Positive displacement expanders are quickly gaining popularity in the fields of micro power generation and refrigeration engineering. Unlike turbo-machines, expanders can handle two-phase flow applications at low speed and flow rate levels. This paper is concerned with a simple-design positive displacement expander based on the limaçon of Pascal. The paper offers an insight into the thermodynamic workings of the limaçon gas expander, and presents a mathematical model to describe the manner in which the port locations affect the expander performance. A stochastic optimisation technique is adopted to find the locations, for the expander ports, which produce best expander performance for given chamber dimensions. The operating speed and other parameters will be held constant during the optimisation procedure. A case study is offered in the paper to prove the validity of the presented approach; and comments are given on how various operating parameters affect system performance in the limaçon design. 1 Sultan, I. A. and Schaller, C. G., “Optimum Positioning of Ports in the Limaçon Gas Expanders,” ASME Journal of Engineering for Gas Turbines and Power: vol 133(10), pp. 103002-1 - 103002-11, 2011.
Transcript
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Optimum Positioning of Ports in the Limaçon Gas Expanders 1
Positive displacement expanders are quickly gaining popularity in the fields of micro
power generation and refrigeration engineering. Unlike turbo-machines, expanders
can handle two-phase flow applications at low speed and flow rate levels. This paper
is concerned with a simple-design positive displacement expander based on the
limaçon of Pascal. The paper offers an insight into the thermodynamic workings of
the limaçon gas expander, and presents a mathematical model to describe the manner
in which the port locations affect the expander performance. A stochastic
optimisation technique is adopted to find the locations, for the expander ports, which
produce best expander performance for given chamber dimensions. The operating
speed and other parameters will be held constant during the optimisation procedure.
A case study is offered in the paper to prove the validity of the presented approach;
and comments are given on how various operating parameters affect system
performance in the limaçon design.
1 Sultan, I. A. and Schaller, C. G., “Optimum Positioning of Ports in the Limaçon Gas Expanders,” ASME Journal of Engineering for Gas Turbines and Power: vol 133(10), pp. 103002-1 - 103002-11, 2011.
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INTRODUCTION
Positive displacement expanders acquired popularity as prime movers for micro-
power generation systems which are usually based on the Organic Rankine Cycle
(ORC) as pointed out by Mathias et at [1]. These plants employ an organic fluid as a
work medium and their boilers are intended to capture the low-grade heat obtainable
from such sources as solar collectors or exhaust gases. Lemort et al [2] point out that
ORC power systems often feature high pressure differentials and low flow rates
occurring under two-phase conditions. As such, positive displacement expanders are
more suitable for these applications than turbo-machines. Saitoh et al [3]
demonstrated an interesting solar-fuelled micro-ORC prototype which employs a
scroll positive displacement expander and produces both mechanical power and heat
(i.e. cogeneration).
Expanders are also gaining popularity in the field of cooling and air conditioning as a
result of the move to replace the Ozone-depleting gases, traditionally used in
refrigeration units, by CO2. Unfortunately, refrigeration units based on CO2 are
thermally inefficient unless the energy lost to the throttling process is captured and
fed back into the system. This motivated Smith and Stosic [4] to investigate the
concept of replacing the throttle valve by a gas expander which was used to drive a
small auxiliary compressor. The expander-compressor unit, often referred to as the
expressor, has been based on the screw design. In fact, other geometric designs are
also available for positive displacement expanders, and many of these designs have
been studied by researchers interested in the domain of energy production. For
example, Mathias et at [1] use two types of gas expanders, gerotor and scroll,
arranged in series in an ORC as they present an interesting discussion on the ideal
isentropic energy extraction of the working medium. Without the need for a
dedicated inlet valve, the gerotor and scroll expanders are both capable of entrapping
an amount of fluid in an expanding chamber. This may explain the interest which has
been given, particularly, to the scroll expander by a number of researchers such as
Xiaojun et al [5] who, mathematically, discuss the inherent leakage problem these
expanders suffer. This problem was experimentally highlighted by Lemort et al [6] as
they make reference to the sensitivity of the expander performance to shaft speed and
pressure ratio. It is interesting that even though Wang et al [7] opt for using a special
scroll expander design, which is meant to reduce leakage, they still had to maintain
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the shaft speed above 2500 rpm to reduce the system losses. This may be attributed to
leakage effects which are more prominent at low speeds.
Nagata et al [8] describe a compressor-expander unit constructed, of a dual-sided
scroll, for use in CO2-based refrigeration units. They report an estimated increase in
the system's coefficient of performance (COP) by 30%. On the other hand, Yang et al
[9], who use a rotary vane-type expander, measure a COP increase of 23%. They also
highlight leakage and friction losses which the vane expander exhibits. Similar to the
scroll expander, the vane-type expander offers isentropic expansion without requiring
the use of an inlet valve. Another vane-type design which may be able to offer more
efficient alternative has been detailed by Ertesvag [10], but no experimental
investigation has been reported on this design.
Li et al [11] present the result of experimental work undertaken to investigate the
performance of an expander based on the concept of rolling piston. They suggest a
10% improvement in the COP of the refrigeration unit used for testing as they offer a
brief discussion on the inlet valve employed to entrap the working fluid in the
expanding chamber. The effects of inlet valve control parameters (e.g. time of
response) on the performance of a reciprocating piston expander have been
insightfully discussed by Baek et al [12]. Concerned by the drawbacks of the
mechatronic valve designs, Zhang et al [13] opt for using an inbuilt mechanical valve,
directly attached to the reciprocating piston, to control the flow inside the work
chamber. Vanyashov and Kovalenko [14] discuss the effects of a spring-operated
valve on the performance of the reciprocating expander. In fact inlet valves are
necessary for some expander designs if isentropic expansion is desired. The dynamic
characteristics of these valves may be found in work published by authors interested
in the field of automatic control, e.g. Wang et al [15] and Wu et al [16]. However,
these authors often discuss pneumatic systems which mainly use air as the working
medium, e.g. Yang et al [17]. Besides their suitability for control applications,
Samuel and Miska [18] suggest that air-operated positive displacement expanders
may conveniently be employed to drive power tools. In their paper, Samuel and
Miska also express concerns in regards to the reliability and efficiency of currently
available positive displacement machines and call for further research and
development effort in this area.
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The work presented in this paper, features an insight into the performance of the
limaçon-based expander. Since highly reliable and sufficiently fast control valves are
not yet available for these machines, only the ported expanders will be considered in
the paper. Early work on the geometric aspects of limaçon machines and their
mechanical designs has been published by Sultan [19]. These machines have been
patented since as far back as the later years of the 19th century; however, problems
with rotor-housing interference inhibited further investment into their development.
Sultan [20] presented solutions for this problem, and as such industry interest in these
machines is expected to develop. In the current paper, a theoretical simulation is
presented to estimate the angular locations of the input and discharge ports which will
optimise the isentropic and volumetric performance indices. Adoption of air as the
work medium will result in a considerably simplified thermodynamic model for the
simulation. This is meant to eliminate detraction from the main focus of the paper
which is the optimisation problem. A loss function designed to maximise the
performance indices will be employed in a procedure based on the approach of
Simultaneous Perturbation Stochastic Approximation (SPSA). This approach is
efficient and suited for such an intricate optimisation application as abundantly
explained in literature, e.g. the excellent paper by Spall [21] and the interesting
application tackled by Kothandaraman and Rotea [22]. In the work presented here,
the stochastic optimisation algorithm employs the particulars of ports as the design
parameters whilst the machine dimensions and operating conditions are kept constant
during the optimisation procedure
For the benefit of the reader, a brief background on the limaçon fluid processing
technology will be presented in the next section.
BACKGROUND ON THE LIMAÇON TECHNOLOGY
Sultan [19] presented a discussion on the limaçon positive displacement machines and
reflected on their suitability for fluid processing. The main geometric features of
these machines are detailed in Figure 1. As shown in the figure, the rotor (of length
2L ) is made to rotate and slide about the limaçon pole (point o ), which is the origin
of a stationary XY-frame. In so doing, the centre of the rotor, point m , stays
kinematically attached to a circle of radius r . This circle is referred to as the base
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circle of the limaçon. The limaçon is in fact the curve traced by the apices, pl and
pt , on the rotor centreline, which is also known as the limaçon chord. The radial
distance, hR , measured along the chord from the centre o to any point on the housing
is given by;
2 sin 1hR L b (1)
where 0,2 is the angle rotated by the chord as measured in a right-hand sense
from the positive X-axis, and /b r L is referred to as the limaçon aspect ratio. For
the limaçon to be free of looping and dimples, b has to be bounded by a maximum
value of 0.25.
By virtue of the geometric properties of the limaçon curve, the two apices are always
touching the housing wall, and slide on it in the directions defined by the tangents at
the points of contact. This ensures good sealing action and smooth contact
conditions. To prevent the rotor-housing interference, Sultan [20] proposes applying
a clearance, rC , to the rotor flank; and/or employing a slightly larger rotor base circle
radius, rr , than the one used for the housing base circle. As shown in Figure 1, the
XrYr-frame is rigidly attached to the rotor and moves with it. The radial distance, rR ,
which defines the profile of the rotor lower lobe, is given in this frame as follows;
2 sin 1r rr
r CR L
L L
(2)
where the angle ,2 is measured in a right-hand sense from the positive Xr-
direction. The rotor lower lobe is simply mirror-imaged about the Xr-axis to produce
the upper lobe of the rotor profile.
The cross-sectional area, bA , of the space available for fluid processing below the
rotor can now be calculated by the following integral;
2 22 21
2b h rA R d R d
(3)
Therefore, the control volume (CV), bV , available for fluid processing below the rotor
can be calculated as follows;
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b b rV A H (4)
where rH is the axial depth of the rotor measured perpendicular to the page.
Based on the above geometric model, Sultan [23] presents the following expression
for the size of control volume at any rotor angle ;
22 2
2
11 4 1 4 cos
2r r r r r
b rr C C r C
V H L b bL L L LL
(5)
The derivative, bdV d , influences the thermodynamic performance of the limaçon
gas expander. This derivative can be expressed as follows;
24 sinb rdV d bL H (6)
A thermodynamic model will be presented for this control volume where the
instantaneous values obtained for the crank torque and the chamber pressure are
expected to be repeated, at a phase-shift of 180 , on the other side of the rotor.
The next section introduces the model presented in this paper to calculate the
instantaneous port area as seen from the control volume being studied.
THE ROTOR-PORT INTERACTION
The limaçon machine has two ports; one for fluid admission and one for discharge.
The concepts presented in this section will apply to both ports without discrimination;
and as such, a specific reference will not be made to either port in the section.
Figure 2 depicts a port on a limaçon machine with two position vectors, lR and tR ,
used to define the radial positions of the port's leading and trailing edges respectively.
The leading edge is the one which the rotor's leading apex, lp , meets first during one
cycle of operation. The port edge vectors can be expressed as follows;
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ˆ(2 sin 1) and
ˆ(2 sin 1)
l l l
t t t
R L b R
R L b R
(7)
where the angular locations of the leading and trailing edges are, respectively, defined
by the angles, l and t . Usually, l is assigned first and then a port angular width,
p , is determined. The angular position of the trailing edge is then calculated from
t l p . If t is found be greater than 2 , then 2 has to subtracted from the
calculated value of t .
The unit vectors pointed out in equation 7, ˆlR and ˆ
tR , are given as follows;
T
T
ˆ cos sin 0 and
ˆ cos sin 0
l l l
t t t
R
R
(8)
The length of the port, pL , is normally assigned within the constraints imposed by the
rotor depth. The width, W , is calculated from the two edge vectors as follows;
t lW R R (9)
Taking the port ends to be semicircular, the full area, fA , of the port may be
expressed as follows;
2 14f pA L W W
(10)
The rotor-port interaction is studied here in relation to the location of the leading apex
of the rotor with respect to the port edges. For this purpose we define the position of
the rotor's leading apex using the vector lP as follows;
ˆ(2 sin 1)l rP L b X (11)
where the unit vector ˆrX is given as follows;
Tˆ cos sin 0rX (12)
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The relative locations of the rotor leading apex with respect to the port edges can now
be defined using the signs of two scalar quantities, ls and ts , which are given as
follows;
ˆ ˆ ˆ ˆ( ) ( ) and
ˆ ˆ ˆ ˆ( ) ( )
l l r r r
t t r r r
s R X X Y
s R X X Y
(13)
where the unit vector r̂Y is given as follow;
Tˆ sin cos 0rY (14)
The port area is now calculated in relation to the following four cases;
Case I, 0ls and 0ts , implies that the port is fully open to the control volume. In
this case, the port area, PA , is simply set equal to fA .
Case II, 0ls and 0ts , implies that the port is progressively being opened to the
control volume. In this case, the port area may be approximated by the following
simple form;
lp f
WA A
W (15)
where lW may be calculated from the following equation;
l l lW P R (16)
Case III, 0ls and 0ts , which suggests that the port is progressively being shut off
the control volume. In this case, the port area may be approximated as follows;
tp f
WA A
W (17)
where the width, tW , can be calculated from the following equation;
t t tW P R (18)
and tP is given as follows;
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ˆ(2 sin 1)t rP L b X (19)
Case IV, 0ls and 0ts , implies that the port is totally shut off the control and the
port area is set equal to 0 (i.e. 0pA ).
The next section offers an insight into the thermodynamic model used for the limaçon
air expander.
THERMODYNAMIC MODEL
The rate, cdm dt , at which the mass inside the control volume varies in relation to
density and available volume, bV , can be calculated from the following equation,
24 sinc b c c rdm dt V d d bL H (20)
where cm and c are the mass and density of fluid in the control volume respectively
at any crank angle . In the model presented here, d dt signifies derivative with
respect to time and is a constant rotor velocity given in / secrad . Realising that
the change occurring to the control volume mass is manifested in the inlet and outlet
flows, the continuity equation can be written in the following form;
21/ 4 sinc b c d i pi i r d o po o od d V C A U b L H C A U (21)
where leakage past a usually well-sealed rotor is negligible and the subscripts i and
o refer to the inlet and outlet respectively. In the above equation, d iC and doC
denote constant discharge coefficients; and inlet and outlet port areas, piA and poA ,
respectively are calculated in accordance with the cases given in the section above.
Moreover, signifies fluid density and U represents the velocities at which the fluid
enters or leaves the control volume. The values of iU and oU are calculated by
approximating the inlet and outlet ports as adiabatic nozzles. As such, iU is given as
follows;
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2 i i c c i c ri
c i c r
P P if P PU
RT if P P
(22)
where cP and iP are the chamber and inlet pressures respectively; and cT is the
chamber temperature. In the equation, r is the critical pressure ratio for the working
fluid, is its adiabatic exponent and R is its universal gas constant. In a like
manner, oU can be calculated as follows;
0
2 c c o o c o r
o c o r
P P if P PU
RT if P P
(23)
where oP and oT are pressure and temperature, respectively, on the downstream side
of the discharge port.
Taking the energy transfer to and from the control volume as an adiabatic process will
result in the following equation;
i o c c c bdH dt dH dt d m e dt P dV dt (24)
where iH and oH are the enthalpies moving in and out of the control volume,
respectively, and ce is the specific internal energy available in the control volume.
Assuming that the rates at which the temperatures vary are small compared to the
rates at which other variables change, it will be possible to manipulate (24) into the
following form;
24 sinc b d i c pi i i d o o po o c c rdP d R V C A U T C A U T P bL H R
(25)
Now the differential equations in (21) and (25) can be solved numerically to find
instantaneous values for cP and c at corresponding values for the crank angle, .
At every iteration, the corresponding control volume temperature can be obtained
using the equation of state (i.e. c c cT P R ). Also, the temperature, oT , on the
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downstream side of the discharge port can be obtained from the adiabatic nozzle
process. In such a case, o can be obtained from the equation of state with oP and oT
both being known. The instantaneous value of shaft torque, c , which results from
the fluid pressure has been given by Sultan [19] as follows;
24 sinc c rbP L H (26)
The iterative approach proceeds by calculating the thermodynamic model at small
intervals in the range 2li li , where the li defines the angular position of
the inlet port leading edge. The values, ( )c liP and ( )c li , assumed for the chamber
pressure and density, respectively, at the start of the cycle are compared to the
corresponding values, ( 2 )c liP and ( 2 )c li , calculated at the end of the cycle
using the following dimensionless error expression;
2 2( ) ( 2 ) ( ) ( 2 )c li c li c li c li
c c
P P
P
(27)
where cP and c are, respectively, pressure and density values used for error
calculations. These values as obtained follows;
( ) ( 2 )and
2( ) ( 2 )
2
c li c lic
c li c lic
P PP
(28)
If the outcome of equation (27) is larger than a small predefined value, ( )c liP and
( )c li are set, respectively, equal to ( 2 )c liP and ( 2 )c li to repeat the
procedure again over the 2 range of . This is iterated until the calculated error,
, falls within an acceptable range to reflect the cyclical nature of the
thermodynamic process described as has been suggested by Peng et al [24]. Once
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convergence has been achieved, the total energy, expE , obtained from the expander in
one cycle is calculated using numerical integration as follows;
0
exp0
22
Nc cN
c nn
E
(29)
where is the size of the angular interval, n is a counter for successive points on
the curve being integrated and N is the total number of intervals on the curve. In a
like manner, the total mass flow through the expander in one cycle, expM , is
calculated from the following expression;
0
exp0
22
N c pi i c pi iNd i c pi i n
n
A U A UM C A U
(30)
It is worthy of noting here that the expressions in (29) and (30) feature a
multiplication by 2 in order to account for the two sides of the rotor.
OPTIMISATION PROCEDURE
Generally, the expander performance depends on many design and operating
variables, such as the chamber and rotor dimensions, the values set for the inlet and
outlet pressures, the rotational speed of the rotor and the port locations and sizes.
During the optimisation procedure, however, all these variables will be maintained
unchanged except for the port-related aspects which will be used as the model design
parameters. For the inlet port, these parameters are li and pi , which are the
angular position of the leading edge and the port angular width respectively. The
corresponding design parameters for the outlet port are lo and po . This definition
results in a four-element design vector, , given as follows;
T
li pi lo po (31)
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The performance aspects which are required to be optimised in this paper are the
isentropic efficiency, i , and the filling factor, v . In the context of the simulation
presented in this paper, i is calculated as follows;
1
exp exp100 1i p i o iE M C T P P
(32)
where pC is the specific heat at constant pressure and iT is the inlet temperature.
On the other hand, the filling factor is defined as the actual volume of gas (measured
at the inlet conditions) induced into the chamber in one cycle divided by the swept
volume. This factor is given by the following expression;
exp
2 ( ) (0)vi b b
M
V V
(33)
Based on the two performance indices described above, the following loss function,
pf , is proposed for minimisation;
2
21 1
100i
p v v if w w
(34)
where vw and iw are positive weighting values assigned subjectively to reflect the
importance of each performance criterion for a particular design.
As revealed in the introduction, the approach of SPSA is adopted here for the
optimisation procedure. In accordance with this approach, the updated values of the
design parameter, 1kj , which occupies the position number j in the design vector is
calculated at the end of iteration step number k as follows;
1 2k k k k k k kj j k p k p k k ja f C f C C (35)
where is a four-element vector whose entries are randomly assigned the values of
either 1 or 1 as generated, at every iteration, by a binary Bernoulli distribution.
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The parameters ka and kC in equation (35) are the sequence gains which are
calculated at iteration number k as follows;
0.602
0.101
k
k
a a B k
C c k
(36)
Spall [21] points out the guidelines which should be followed to select numerical
values for the constants, a and c , in equation (36). For the work presented here,
which features a low-noise application, c has been set equal to 0.001 and a is set
equal to 0.5. The value of B is calculated as 10K , where K is the maximum
allowable number of iterations set at the start of the procedure. The limits imposed on
the values of the design parameter j (i.e. maxj and
minj ) are incorporated in
the procedure, as suggested by Kothandaraman and Rotea [22], as follows;
1
max max1
1
min min
kj j jk
j kj j j
if
if
(37)
A flowchart of the computational procedure presented in this paper is depicted in
Figure 3, and a case study is given in the next section to demonstrate the
implementation of this procedure.
CASE STUDY
For this case study, it is required to optimise the performance of a given expander
running at 2000 rpm by carefully locating its outlet and input ports. The speed value
in the case study is randomly set for demonstration purposes only and does not
suggest any speed constraints imposed on the limaçon design. In a real life
application, the speed can take any value dictated by the mechanical aspects of the
drive, the fluid properties and the power system requirements. The dimensions of the
expander are given as follows;
7 mmrr r , 63.17 mmL , 0.3mmrC and 80mmrH .
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The constant lengths, piL and poL , of inlet and discharge ports are given respectively
as 45mm and 55mm . During iterations, the inlet temperature has been kept constant
at o80 C ; and the inlet and discharge pressures were set equal to 5bar and 1bar ,
respectively. The discharge coefficients, d iC and doC , depend on geometric and
thermodynamic aspects which vary during operation. To maintain simplicity as a
theme for the simulated model whilst observing a conservative perspective, these two
coefficients have been assigned here as 0.35 and 0.5 respectively. The values set for
the ports angular locations at the start of the procedure were as follows;
o8li , o6pi , o170lo and o10po
At the end of iterations the ports angular locations were found be;
o15.7li , o11.9pi , o160lo and o13.35po
These new port locations increased the isentropic efficiency from 47% to 53.2%; and
improved the filling factor from 0.88 to 1.02. The numerical values obtained for these
two performance figures during the course of iterations are shown in Figures 4 and 5
respectively.
For the optimised expander, the mass flow into the control volume is graphed against
the crank angle in Figure 6 and compared to the mass flow of the expander which had
been set before the optimisation procedure has been implemented. It is obvious from
the figure that the mass flow inside the control volume is more effective for the
optimised expander. Faster admission of fluid has been allowed into the control
volume during the intake stroke and faster discharge has been achieved during the
scavenging stroke. This does reflect favourably on the pressure profile as shown in
Figure 7. The higher pressure level reached in the chamber during the intake stroke
results in higher energy values obtained per cycle as suggested by the PV-diagram
graphed in Figure 8. This improved energy level is also evident by Figure 9 which
demonstrates the torque-angle profile during one cycle of the expander operation.
The next section is intended to provide an insight into how operating conditions affect
the expander performance.
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SENSITIVITY TO OPERATING PARAMETERS
As pointed out above, during the optimisation procedure, all operating parameters are
kept constant except for the port locations and widths. However, in real life,
variations are expected to occur and the optimised expander is not expected to operate
at these exact values set for the constant parameters. In this section, the sensitivity of
the expander performance to variations occurring to the rotor rotational speed, the
inlet pressure and the pressure ratio will be studied. For this purpose, a number of
expanders have been optimised at different operating parameters as given in Table 1.
All optimised expanders are assumed to run at a constant inlet temperature of o80 C ;
and they all have the same dimensions given in the case study demonstrated above.
Each optimised expander was run at a series of operating conditions and performance
criteria were calculated and plotted to provide an insight into the workings of these
expanders.
Table 2 is intended to study the effects produced by the pressure ratio variations on
the expander filling factor. The four expanders given in the table were initially run at
an inlet pressure of 8 bars and a speed of 2000 rpm. It is obvious from the table that
the pressure ratio has a very little impact on the filling factor of the expander.
However, the expanders that had been optimised at a speed of 500 rpm obviously
exhibit reduced filling factor values than those which had been optimised at 2000
rpm. This suggests that the speed at which the optimisation process has been
performed does have a considerable impact on the expander performance. This same
numerical experiment was repeated at an inlet pressure of 18 bar, and the results are
also given in Table 2. Comparing the results for the two values set for inlet pressure,
it is safe to suggest that the operating inlet pressure doesn't reflect on the filling factor
of the expander. However, the results in Table 2 also suggest that the pressure ratio
employed in the optimisation procedure should be set equal to the minimum value
expected in real life. This is concluded by noting the values obtained for the filling
factor of the expander A 2000.
The manner in which the operating speed affects the filling factor is demonstrated in
Figure 10. The figure features two expanders; 500 B which was optimised at a speed
of 500 rpm and 2000 B which was optimised at 2000 rpm as given in Table 1. It is
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clear from the figure that each expander exhibits a filling factor close to unity in the
vicinity of the speeds which were used in their respective optimisation procedures.
Moreover, the performance curves demonstrate a level of flatness near these speeds
which suggests a possible utilisation range.
The effects of changes, occurring to pressure ratio on the expander isentropic
efficiency, are shown in Figure 11. All expanders featured in the figure have been
operated at a speed of 2000 rpm and an inlet pressure of 8 bar. It is clear from the
figure that those expanders which had been optimised at a speed of 2000 rpm are not
as severely affected by the applied pressure ratio as those which had been optimised at
500 rpm. This again highlights the importance of observing the speed used in the
optimisation procedure in relation to the expected operating speed of the expander.
Figure 12 shows the results of the same experiment except the inlet pressure was set
equal to 18 bar instead of 8 bar. Comparing the plots in Figures 11 and 12, it is
possible to suggest that the applied inlet pressure does not reflect on the isentropic
efficiency of the expander. However, the two figures also suggest that high values
used for inlet pressure during the optimisation procedure are likely to produce an
expander which is somewhat less sensitive to pressure ratio and speed variations.
Figure 13 suggests that the rotor speed has the most profound effect on the expander
isentropic efficiency. The figure features two expanders; 500 B which was optimised
at a speed of 500 rpm and 2000 B which was optimised at 2000 rpm. Like Figure 10,
this figure suggests that the optimised expander could be operated in the vicinity of
the speed used for optimisation.
The above discussion reveals that the speed used for the optimisation procedure
should be selected in relation to the speed range expected to be applied in real life.
The obvious influence of speed may be attributed to the fact that the control volume is
momentarily exposed to both the inlet and discharge ports simultaneously. This
overlap results in a blow-by flow which depends mainly on the time available for the
inlet-outlet exposure. The effect of the inlet pressure on this flow is constrained by
the thermodynamic properties of the ports which behave like convergent nozzles. The
optimisation procedure endeavours to minimise the blow-by flow for a given speed by
- 18 -
proper port allocation. In so doing, the procedure produces an efficient expander
which is also immune to variation in pressure ratio. This highlights the importance of
undertaking an optimisation procedure before a limaçon expander is designed for a
specific application. The optimisation procedure should be conducted at the operating
speed expected for the application (as per Table 2 and Figures 11, 12 and 13),
minimum expected pressure ratio (as per Table 2), and maximum expected inlet
pressure (as per Figures 11 and 12).
CONCLUSIONS
Positive displacement expanders are gaining popularity in the areas of micro power
generation and refrigeration. Limaçon expanders offer a simplified positive
displacement design with geometry that lends itself easily to mathematical modelling
and optimisation procedure. This paper presented a thermodynamic model for these
expanders combined with a stochastic optimisation technique implemented to place
the expanders ports at the locations which produce the best performance at given
operating conditions. The results of a case study prove the validity of the proposed
approach for the intended application where considerable improvements have been
obtained for the selected expanded performance indices. A discussion was presented
on the effects of operating conditions variations on the performance of optimised
expanders. It was found that the speed has the most prominent effect on this
performance and recommendations have been made to ensure that the expander
performance will remain within an acceptable range during operation.
Nomenclature
L half the limaçon chord length
o the pole of the housing limaçon
m midpoint on the limaçon chord
r radius of the housing base circle
XY frame fixed to the housing
pl point on the housing limaçon
- 19 -
pt point on the housing (opposite pl )
hR radial distance of the housing
angle rotated by the chord
b limaçon aspect ratio (r/L)
rC radial clearance of rotor profile
rr radius of the rotor base circle
XrYr frame attached to the rotor
rR radial distance on the rotor
angle used to draw the rotor
bV Control Volume (CV)
bA Cross-sectional area of CV
rH the rotor depth
lR radial distance to the port leading edge
tR radial distance to the port trailing edge
ˆlR unit vector in the direction of lR
ˆtR unit vector in the direction of tR
l position of port leading edge
t position of port trailing edge
p the port angular width
pL the port length
W total width of port
fA area of a fully opened port
lP position vector of pl
pL the port length
lW width of the opening port
pA area of the port
ls scalar used to calculate pA
ts scalar used to calculate pA
- 20 -
tP position vector of pt
tW width of the shutting port
cm mass inside CV
c density inside CV
t time
angular velocity of rotor
d iC discharge coefficient (inlet port)
doC discharge coefficient (outlet port)
piA instantaneous area (inlet port)
poA instantaneous area (outlet port)
iU velocity of fluid entering CV
oU velocity of fluid leaving CV
cP pressure of fluid in CV
cT temperature of fluid in CV
R universal gas constant
r critical pressure ratio
adiabatic gas constant
iP pressure in the inlet manifold
oP pressure in the outlet manifold
oT temperature in the outlet manifold
o density in the outlet manifold
iH enthalpy of fluid entering CV
oH enthalpy of fluid leaving CV
ce specific internal energy in CV
c shaft torque
li position of leading edge (inlet port)
error used for solving (21) and (25)
cP pressure used for solving (21) and (25)
- 21 -
c density used for solving (21) and (25)
expE total energy per cycle
n counter for solving (21) and (25)
N number of intervals
expM mass flow per cycle
angular size of intervals
pi angular width (inlet port)
po angular width ( outlet port)
lo position of leading edge (outlet port)
design vector used for optimisation
i isentropic efficiency
v filling factor
pC specific heat at constant pressure
iT temperature in the inlet manifold
Θpf loss function
vw and iw weighting factors
k iteration number: optimisation procedure
Bernoulli distribution
K allowable number of iterations
a , B and c constants used for optimisation
ka and kC updated values of a and c
- 22 -
REFERENCES
[1] Mathias, J.A., Johnson Jr., J.R., Cao, J., Priedeman, D.K. and Christensen,
R.N., 2009, "Experimental Testing of Gerotor and Scroll Expanders Used in,
and Energetic and Exergetic Modelling of, an Organic Rankine Cycle,"
ASME J. Energy Resources Tech. Vol. 131(3), pp. 012201-1-9
[2] Lemort, V., Quoilin, S., Cuevas, C. and Lebrun, J., 2009, "Testing and
Modelling A Scroll Expander Integrated Into An Organic Rankine Cycle,"
Applied Thermal Engineering, vol. 29, pp. 3094-3102.
[3] Saitoh, T., Yamada, N. and Wakashima, S.-i., 2007, "Solar Rankine Cycle
System Using Scroll Expander," J. Environment and Engineering, vol 2(4),
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[4] Smith, I.K. and Stosic, N.R., 1995, “The Expressor: An Efficiency Boost to
Vapour Compression Systems by Power Recovery from the Throttling
Process,” ASME Heat Pump and Ref. Sys. Des. Vol. 34, pp. 173-181.
