Proceedings of the ASME 2009 Dynamic Systems and Control Conference
DSCC2009
October 12-14, 2009, Hollywood, California, USA
DSCC2009-2749
OPTIMAL EFFICIENCY-POWER RELATIONSHIP FOR AN AIR
MOTOR-COMPRESSOR IN AN ENERGY STORAGE AND REGENERATION SYSTEM
Caleb J. Sancken and Perry Y. Li
Center for Compact and Efficient Fluid Power
Department of Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota 55455
Email: [email protected], [email protected]
ABSTRACT
Compressing air from atmospheric pressure into high pressure storage and expanding the compressed air in reverse is
a means of energy storage and regeneration for fluid power
systems that can potentially improve energy density by an order of magnitude over existing accumulators. This approach,
known as the “open accumulator” energy storage concept, as
well as other applications such as compressed air powered cars,
rely on the availability of efficient and power-dense air motor/compressors. Increasing power is typically accompanied by
reducing efficiency with the trade-off being determined by the
heat transfer capability. In this paper, the authors present the
Pareto optimal trade-off between the efficiency and power for a
given heat transfer capability and ambient temperature in an air
motor/compressor to achieve a given pressure ratio. It is shown
that the optimal frontier is generated by an air motor/compressor
that compresses and expands the air via a sequence of adiabatic,
isothermal, and adiabatic processes. For the same efficiency of
80%, such an optimal volume trajectory achieves 3-5 times increased power over ad-hoc volume trajectories. It is also shown
that approximating the infinitely fast adiabatic portions by finite
time processes do not significantly reduce the effectiveness of the
optimal operating strategy.
Figure 1.
A DIAGRAM OF A SIMPLE ENERGY STORAGE AND RE-
GENERATION SYSTEM USING COMPRESSED AIR.
pressure. A key element of the concept is that air is compressed
and expanded between the atmosphere and high pressure storage.
The increased energy density is achieved by the increased pressure ratio (350 versus 2 or 3 at 35 MPa) and by needing to only
contain the compressed air rather than the expanded air volume
and the displaced oil. Because of the increased energy density,
compressed air is also being considered as a clean power source
for vehicles [2, 3] as well as for energy storage for large power
plant during off-peak periods ( [4]).
A simplified system that stores energy using compressed air
consisting of a single-stage air motor/compressor coupled to a
reservoir, is shown in Fig.(1). The open accumulator system
differs from Fig. (1) in that it has some augmentations to enable constant pressure operation and to accommodate hydraulic
loads, loads that may be transiently high. For these applications,
1 INTRODUCTION
In an attempt to significantly increase the energy storage
density for fluid power, a novel “open accumulator” concept was
recently proposed [1] which has an order of magnitude increase
in fluid power energy storage density over existing closed accumulators (which contain a fixed amount of gas) at the same
1
Copyright © 2009 by ASME
the power capability and efficiency of the air motor/compressor
used for energy extraction or addition are critical. Power can
be increased by increasing the operating frequency, but such as
change is generally accompanied by a decrease in efficiency.
This trade-off is a function of the heat transfer capability within
the motor/compressor. Thus, improving heat transfer is key to increasing power capability while maintaining efficiency. Yet, how
heat transfer exactly affects this trade-off is not exactly known.
In this paper, the effect of heat transfer on air motor/compressor
power and efficiency is determined. Specifically, for a given heat
transfer capability and environmental temperature, the optimal
compression and expansion volume trajectories–such as to optimize efficiency for a given power, or to optimize the power for a
given efficiency–are determined.
Some authors do not consider heat transfer as important at
relatively small compression ratios (like ∼ 4.5 in [5]) and [6].
For high pressure ratios, such as for high density energy storage, heat transfer becomes important ( > 14 in [7]; see also simple analysis in [5]). There has been research into modeling heat
transfer in air compressors ( [5] contains a good summary), but
the formulations are not simple to use.
Work in [8] focuses on minimizing work in a multi-stage
compressor with “polytropic” stages by varying the stage compression ratios. The authors of [9, 10] seek to maximize efficiency or work (negative work defined as into the compressor)
by changing air motor or compressor geometry, assuming adiabatic operating conditions. Similar to the present work, [11, 12]
consider maximizing average output power (or minimize input
power) by changing the chamber volume trajectory. However,
[11] does not take heat transfer into account but focuses on valve
operation. The author in [12] does include heat transfer as well
as losses in the modeling of an Otto cycle, but the complexity
of the model requires that it be solved numerically rather than
analytically. In this paper, a lumped ideal gas model is assumed
and heat transfer is modeled by a heat transfer coefficient. This
allows the optimal control problem to be solved analytically and
the trade-off between efficiency and power to be easily evaluated.
The rest of the paper is organized as follows. System model
and key definitions are presented in section 2. The optimal compression and expansion volume trajectories results and implications are presented in Section 3. Section 4 applies the results
for an energy storage and regeneration system design example.
Modifications of the optimal compression and expansion volume
trajectories are examined in section 5. Section 6 contains concluding remarks.
Figure 2.
DIAGRAM OF COMPRESSION AND EXPANSION PRO-
FILES ON THE P-V PLANE.
ter isobaric cooling to T0 in the air storage reservoir, energy is
extracted by expanding the air back to P0 and a temperature of
T0 − ∆T . A sample diagram of these processes is in Fig. (2).
The assumption that the air completely cools before the energy
is extracted is conservative; however, it allows the expansion process to be independent of the compression process, and it at least
partially makes up for non-conservative assumptions elsewhere
in the model. The heat transfer coefficient h in expansion and
compression is assumed to be constant, as in [13] (in [5]). Total heat transfer during the expansion and compression processes
depends on how the air is expanded or compressed.
There are multiple of ways of compressing and expanding
air from one state to another. Each is determined by a path on
the P-V diagram and the heat transfer characteristic. The actual
time trajectory of the pressure and volume, ζ(t) = (P(t),V (t)),
can be discovered from
nCv Ṫ = Q(T, T0 ) − Pu,
V̇ = u,
and
P=
nRT
(1)
V
where Cv is the constant volume heat capacity of air, Q(T, T0 )
is the heat transfer (rate) given mean air temperature T and a
constant wall temperature of T0 , n is the number of moles of air,
P is pressure, V is volume, R is the ideal gas constant, n is moles
of gas, and u is the “control variable.” For simplicity, a linear
heat transfer model is used:
Q(T, T0 ) ≡ hA(T0 − T ).
2 SYSTEM DESCRIPTION AND MODEL
In the subsequent analysis, the motor/compressor is assumed to consist of one stage with a constant area A over which
heat transfer occurs. To store energy, air, which is assumed to
be an ideal gas, is compressed from an ambient pressure of P0
and an ambient temperature of T0 to a pressure of rP0 and a temperature of T0 + ∆T , where ∆T is a change in temperature. Af-
(2)
For compression, it is desired that
1. the work input for one mole of air,
Win (ζ) = −
2
Z
tf1
0
(P(t) − P0)V̇ (t)dt + nR∆T
(3)
Copyright © 2009 by ASME
is minimized. In (3), t f 1 is the compression time such that
P(t f 1 ) = rP0 . The last term in (3) is the isobaric flow work
associated with cooling the air back to ambient which occurs
in the gas storage reservoir.
2. the compression time (time it takes to compress to rP0 ), t f 1 ,
is minimized. Notice that the time it takes for the isobaric
compression is not included, since this takes place outside
of the compression chamber.
The motoring power is defined as
Powe (ζe ) ≡
1. the work output for one mole of compressed air given by
Wout (ζe ) =
0
(P(t) − P0)V̇ (t)dt
3 OPTIMAL COMPRESSION OR EXPANSION TRAJECTORIES
Different compression/expansion trajectories lead to different works and cycle times. There is a trade-off between W in and
t f 1 and between Wout and t f 2 . The ratio between Wout and Win
is the regenerative efficiency; whereas t f 1 and t f 2 are inversely
related to power. Fixing efficiency fixes work done per cycle;
then, minimizing the compression or expansion times t f 1 and t f 2
maximizes power. Thus, there is a trade-off between efficiency
and power density. The key to the model will be to determine the
state trajectories that yield the Pareto optimal trade-off between
power and efficiency.
(4)
is maximized. Here the initial pressure P(t = 0) = rP0 and
t f 2 is the expansion time such that P(t f ) = P0 .
2. the expansion time (time it takes to expand to P0 ), t f 2 , is
minimized.
A unit of compressed air in the open accumulator at pressure
rP0 and T0 has the same internal energy as the same amount of air
at P0 and T0 . Thus, strictly speaking, energy has not been stored.
However, the compressed air has the potential to do work. We
define the amount of energy stored as the maximum work available by expanding the compressed air to P0 . With an environment
temperature of T0 , it can be shown that the maximum work per
mole of compressed gas at (rP0 , T0 ) is obtained via isothermal
expansion and is given by
1
E/n ≡ RT0 ln(r) − 1 +
r
Theorem 1. (Compression) Let P0 be initial pressure, T0 be
the initial and environment temperature, and rP0 , with r > 1,
be the desired final pressure. The compression trajectory
ζ∗c (∆T ≥ 0) consisting of an instantaneous adiabatic compression from
(P0 , T0 ), followed by an isothermal compression at
T1 = T0 (T0 + ∆T ), in turn followed by adiabatic compression
to the desired pressure (rP0 , T0 + ∆T ) is a Pareto optimal
trajectory with respect to input work and compression time; i.e.
there does not exist a compression trajectory ζ c with the same
initial state and final pressure such that
(5)
The compressor efficiency for a compression trajectory
ζc (·), the motoring efficiency for an expansion trajectory ζ e (·),
and the overall regenerative efficiency are, respectively,
E
ηc (ζc ) ≡
Win (ζc )
Wout (ζe )
ηe (ζe ) ≡
E
Wout (ζe )
ηoverall (ζc , ζe ) ≡
= ηc ηe
Win (ζc )
Win (ζc ) < Win (ζ∗c ) and t f 1 (ζc ) < t f 1 (ζ∗c ).
(Expansion) Let rP0 , with r > 1, be the initial pressure, T0 be the
initial and ambient temperature, and P0 be the environment and
final pressure. The expansion trajectory ζ ∗e (∆T ≥ 0) consisting
of an instantaneous adiabatic expansion from (P,
T ) = (rP0 , T0 ),
followed by an isothermal expansion at T1 = T0 (T0 − ∆T ),
in turn followed by adiabatic compression to pressure P0 is a
Pareto optimal expansion trajectory; i.e. there does not exist a
compression trajectory ζe with the same initial state and final
pressure such that
(6)
(7)
(8)
Thus, efficiencies will be improved by minimizing W in and
maximizing Wout . For compression, since storing E amount of
energy requires a time of t f 1 , the energy storage power is given
by
Powc (ζc ) ≡
E
.
tf1
(10)
Strictly speaking, some time is needed for the filling phase as
well. However, since filling time is not limited by heat transfer,
the topic of investigation right now, it can, theoretically, be made
as quick as possible if the orifice through which the gas flows is
made sufficiently large.
Similarly, for motoring, it is desired that
Z tf2
Wout (ζe )
.
tf2
Wout (ζe ) < Wout (ζ∗e ) and t f 2 (ζe ) < t f 2 (ζ∗e ).
Thus, the frontiers for the optimal trade-off between compression/expansion efficiencies and energy storage/motoring power
is generated by ζ∗c (∆T ) or ζ∗e (∆T ) with different ∆T ≥ 0.
(9)
Sketch of the proof:
3
Copyright © 2009 by ASME
Figure 3.
3. Successively
replacing
each
isothermal-adiabaticisothermal step with an adiabatic-isothermal-adiabatic
step results in a trajectory with the same end points and
work done as the initial, arbitrary trajectory, but successively
less time is needed for the adiabatic-isothermal-adiabatic
curves. Hence any trajectory can be completed with the
same work but less time using a trajectory consisting
of adiabatic-isothermal-adiabatic steps. Thus, we need
only consider adiabatic-isothermal-adiabatic trajectories as
candidates for Pareto optimal trajectories.
4. Index the ζ ∗ curve so that 0-1 is first adiabatic curve; 1-2, the
isothermal curve; and 2-3, the adiabatic curve. To find the
optimal ζ∗ between an initial state and final pressure, minimize work input such that the work and heat transfer done
during the isothermal curve 1-2 balance. For compression,
work in Eq. (3) may be written as
DIAGRAM OF ISOTHERMAL-ADIABATIC-ISOTHERMAL (ζ)
AND ADIABATIC-ISOTHERMAL-ADIABATIC (ζ∗ ) PROFILES.
γ 1 T0 1−γ
γnR
(T3 − T0 ) + nRT1 ln
+
Win = −
1−γ
r T3
1
,
nRT0 1 −
r
1. Any arbitrary curve can be uniformly approximated by a series of isothermal-adiabatic-isothermal steps.
2. A cartoon of an adiabatic-isothermal-adiabatic trajectory–
that is, a trajectory made up of three pieces: an adiabatic
section followed by an isothermal section followed by an
adiabatic section–is shown in Fig. (3) as A-E-F-D (ζ ∗ ); a
cartoon of an isothermal-adiabatic-isothermal trajectory is
shown in Fig. (3) as A-B-C-D (ζ). For a prescribed work
level, it can be shown that there is an adiabatic-isothermaladiabatic trajectory that takes less time than an isothermaladiabatic-isothermal trajectory with the same endpoints. To
show this, let
rI ≡
and cycle time may be written as
γ nRT1
1 T0 1−γ
ln
.
tf1 =
hA(T0 − T1 )
r T3
PF
PB PD
·
=
PA PC
PE
Thus, for compression, the optimization problem may be
written as
be the total isothermal compression ratio, and let q be such
that
PB
q
= rI
PA
and
min Win
T1 ,T3
PD
(1−q)
= rI
.
PC
with
t f 1 − t f = 0,
where t f is a constant. Please see the appendix for expansion work and time. Using the standard Lagrange multiplier method, the works in Eq. (3) and (4) can be optimized to find that each adiabatic-isothermal-adiabatic trajectory can be fully defined by the ambient temperature, T 0 ,
and the temperature at the end of the last adiabatic step,
T0 ± ∆T (i.e., T3 ). The optimal isothermal temperatures are
T0 (T0 ± ∆T )–where the “plus” corresponds to compression and “minus,” expansion.
Setting W (ζ∗ ) = W (ζ) results in
t f (ζ) − t f (ζ∗ )
t f (ζ∗ )
1
TA
1
−
= qT0 1 −
≥ 0.
TE
TA − T0 TD − T0
s(ζ, ζ∗ ) =
Thus, an adiabatic-isothermal-curve can be shown to be optimal.
Q.E.D.
The Pareto optimal Win , Wout , t f 1 , and t f 2 as functions of ∆T
are given in the proof above and in the Appendix and are plotted in Figs. (4-5) for the case of r = 350 and T0 =300 K. Notice
that both optimal power and optimal efficiency decrease as functions of final temperature deviation from the environment temperature. The Pareto optimum relationship between power and
For the case of rI > 1, TE > T0 . If, further, 0 ≤ q ≤ 1, the
terms inside (·) and [·] are of the same sign so that s(ζ, ζ ∗ ) ≥
0. If p < 0 or p > 1, then TA − T0 < 0 or TD − T0 < 0 respectively, the quantity inside (·) > 0, and the [·] takes the sign
of p. Hence s(ζ, ζ∗ ) > 0 when rI > 1. A similar result holds
for rI < 1.
4
Copyright © 2009 by ASME
1
4
Optimum
Sinusoid
Linear
0.9
0.8
3
10
Efficiency, ηc
Scaled Power, Pow/(hA) [K]
10
2
10
0.7
0.6
0.5
1
10
Compressing
Motoring
0
100
200
300
Temperature Difference, ∆T [K]
0.4
400
0
10
1
10
2
3
10
10
Scaled Power, Powc/(hA) [K]
4
10
5
10
Figure 4. POWER INCREASES WHEN THE DIFFERENCE BETWEEN
THE INITIAL AND FINAL TEMPERATURE INCREASES (FINAL PRES-
Figure 6.
SURE FOR FIGURE IS 35 MPa).
OPTIMAL CURVE RATHER THAN AN ARBITRARY CURVE SUCH AS
A COMPRESSOR CAN ABSORB MORE POWER USING AN
SINUSOIDAL OR LINEAR.
1
Compressing
Motoring
0.9
0.8
0.8
Efficiency, ηe
Efficiency, η
Optimum
Sinusoid
Linear
0.9
0.7
0.6
0.7
0.6
0.5
0.5
0.4
0.4
0
100
200
300
Temperature Difference, ∆T [K]
400
0
10
Figure 5. EFFICIENCY DECREASES AS THE DIFFERENCE BETWEEN THE INITIAL AND FINAL TEMPERATURE INCREASES (FINAL
Figure 7.
PRESSURE FOR FIGURE IS 35 MPa).
1
10
2
3
10
10
Scaled Power, Powe/(hA) [K]
4
10
5
10
AN AIR MOTOR CAN PRODUCE MORE POWER USING AN
OPTIMAL CURVE RATHER THAN AN ARBITRARY CURVE SUCH AS
SINUSOIDAL OR LINEAR.
efficiency for compression and expansion for r=350 and T 0 =300
K are shown in Figs. (6) and (7). Comparisons with the tradeoffs using sinusoidal and linear profiles are also shown. Power
is scaled by heat transfer parameters so the plot applies for any
geometry and heat transfer coefficient.
Efficiency decreases with power which is consistent with the
simulations in [1] in which a decrease in efficiency is observed
when the heat transfer coefficient h is decreased, but, in this paper, efficiency decreases as the time allowed for the compression
or expansion decreases. In either case, heat transfer is less at
higher powers and lower efficiencies. For compression and at an
efficiency of 80%, the average power absorbed during compression of a gas using an optimal volume trajectory is 2 times the
average power absorbed during compression of a gas using a sinusoidal volume trajectory, and using the optimal volume trajectory can increase power absorption 5 times compared to a linear
volume trajectory.
For different powers or efficiencies the shape of the optimal volume trajectory changes, as shown on the V-t plane in Fig.
(8). Common non-optimal trajectories, such as linear and sinusoidal profiles, do scale as power or efficiency change. A gas
5
Copyright © 2009 by ASME
1
400
Opt. (ηc=80%)
300
0.7
0.9
Compressor Volume (L)
350
0.8
Sinusoid
Linear
Opt. (ηc=99%)
Opt. (ηc=50%)
V/Vmax
0.6
0.5
0.4
0.3
V vs. h
Volume for h=100 W/(m2K)
Target Compressor Volume
250
200
150
100
0.2
50
0.1
0
0
0
0.2
0.4
0.6
Fraction of a Cycle
0.8
0
1
500
1000
1500
2000
Heat Transfer Coefficient [W/(m2K)]
Figure 8. OPTIMAL CURVES IN THE VOLUME-TIME DOMAIN,
Figure 9.
SHOWN WITH SINUSOIDAL AND LINEAR CURVES FOR COMPARI-
TRANSFER.
COMPRESSOR VOLUME DECREASES WITH HEAT
SON. TIME IN THE PLOT IS SCALED BY THE TOTAL TIME FOR COMPRESSION. FOR DIFFERENT POWERS AND EFFICIENCIES, THE
energy storage and regeneration system is 260. L, which may be
too big to be practical, depending on the application.
To reduce system volume, the heat transfer coefficient or
geometry could be modified, for instance, by inducing a higher
level of turbulence in the compressor or by changing the expansion/compression chamber aspect ratio, respectively. Say that
the approach of increasing the heat transfer coefficient is taken
and that the compressor volume should be less than 12 L; a heat
transfer coefficient of 809 W/(m 2 K) is required, as shown in Fig.
(9). Since such a high heat transfer coefficient is unusual for
air, trying to increase the heat transfer coefficient alone would
probably not be enough to decrease volume; geometry changes
would also need to be considered. For example, the surface area
increase associated with the use of multiple small liquid pistons,
as described in [15], could sufficiently decrease compressor volume.
It should be noted that coming up with a heat transfer
coefficient to use in the model is generally not trivial. However,
a heat transfer coefficient may be chosen from a range of typical
heat transfer coefficients for convection in air. Doing calculations with a low and a high value will bound design parameters.
SHAPE OF THE OPTIMAL CURVE CHANGES; I.E., THE CURVES DO
NOT SCALE WITH TIME.
motor/compressor should, then, have a way to change compression and expansion trajectories if it is to absorb or regenerate
power at a variety of levels optimally–something not possible
with typical “crank-slider” type designs. A “liquid piston” type
design–described in [1], [14], and [15]–is one way to accomplish
the trajectory changes.
4 APPLICATION
For a design example, imagine using compressed air energy storage system where compression ratio, energy storage efficiency, power, energy storage capacity, and compression chamber geometry are specified, and the design objective is to find the
volume of the system for particular heat transfer coefficient h.
If an 80% efficiency is desired, then an optimum trajectory
corresponding to ∆T =137 K should be used, as can be seen from
Fig. (5). If the compressor should have a storage power of 15
kW, dQ/dT = hA must be 274 W/K. For a heat transfer coefficient h of 100 W/(m 2 K), the area over which heat transfer occures A should be 2.74 m 2 . Assume that the geometry of the
compression chamber is hemispherical because it is constructed
using a diaphragm, that heat transfer occurs through an area
equal to the minimum diaphragm area, and that the chamber’s
aspect ratio (radius to height) is unity so that the relationship between the chamber’s maximum volume and heat transfer area is
V = 0.0563A 3/2. The volume of the compressor is then 255 L.
If the desired energy storage capacity is 800 kJ, then it can be
determined from Eq. (5) and the ideal gas law that the required
(ideal) energy storage volume is 4.64 L. The total volume of the
5 DISCUSSION
In the previous work, it is assumed that the adiabatic portion of the compression/expansion process can take place in zero
time. This is true in the sense that this portion is not heat transfer
limited. In real situations, there are physical limitations to the
compression and expansion rate. This situation is now considered.
To find the non-zero time associated with each adiabatic section, assume that the gas is expanded or compressed at a constant
6
Copyright © 2009 by ASME
1
1
0.9
0.8
Efficiency, ηc
Efficiency, ηc
0.95
0.9
0.85
p=2
p=4
p=8
p=16
p=32
p=∞
0.8
0.75
0.7
0
10
Figure 10.
increasing p
0.7
0.6
0.5
Optimum (tadi ≠ 0)
0.4
Adi.−Iso.
Sinusoid
Linear
0
10
1
10
Scaled Power, Powc/(hA) [K]
2
1
2
10
10
Scaled Power, Powc/(hA) [K]
3
10
10
Figure 11.
EFFICIENCY-POWER RELATIONSHIP FOR THE NON-
ZERO-TIME ADIABATIC TRAJECTORY AND THE ADIABATICISOTHERMAL TRAJECTORY PLOTTED ALONG WITH THE
THE OPTIMAL EFFICIENCY-POWER CURVE FOR A VARI-
ETY OF p. THE LARGER p IS, THE LARGER THE MAXIMUM TIME
RATE CHANGE OF VOLUME IS. LARGER p ALLOW FOR LARGER
EFFICIENCY-POWER RELATIONSHIP FOR SINUSOID AND LINEAR CURVES ( p = 4 AND r
POWERS FOR ANY EFFICIENCY. FOR THE PLOT, r = 350. THE
CURVES QUICKLY CONVERGE, MAKING IT DIFFICULT TO DISTIN-
= 350).
GUISH THE CURVES FOR LARGE p.
total time (tadi + tiso ) decreases and power capability increases
for any particular efficiency, with the largest difference being for
high powers.
rate, so that, for compression
tadi =
V0 − V1 V2 − V3
+
.
V̇max
V̇max
(11)
With non-zero time for the adiabatic sections and restrictions
on the maximum time rate of change of volume, the relationship
between power and efficiency is shown in Fig. (11) with r = 350,
T0 = 300 K, and p = 4 along with results for sinusoid and linear
simulations. As expected, the power with a finite adiabatic portion is lower than the unlimited case. However, the difference
is small for high efficiency operation. The curve is conservative because efficiencies would actually be higher, a phenomenon
caused by the fact that heat transfer would actually occur during
the parts of the trajectory that should be adiabatic.
(Recall that indices 0 to 1 and 2 to 3 correspond to the adiabatic sections of the adiabatic-isothermal-adiabatic optimal trajectory.) Define p as the ratio of the rate of change of volume
during the adiabatic sections to the rate of change of volume during the isothermal sections:
V̇max = p max max V̇iso
T1
t
p≥1
(12)
At 80% efficiency, the scaled power for the case with restricted time rate change of volume is 12.9% less than the scaled
power for the unrestricted (ideal) case. The product hA would
need to increase 14.8% to meet the requirements for the energy
storage application described in the previous section.
where V̇iso is the time rate change of volume during the isothermal section. In parallel to Eq. (9), power for a case where adiabatic sections take nonzero time can be written as
E
Pow =
,
tadi + tiso
A fast change in the volume trajectory, even if not instantaneous, at high pressure may not be possible. If the
high-pressure adiabatic phase is not possible, then it may be
skipped without completely losing the power advantage over
using sinusoid or linear volume trajectories, as can be seen in
Fig. (11); in compression, using an adiabatic-isothermal trajectory allows absorption of 33.5% less power than using the
adiabatic-isothermal-adiabatic trajectory at 80% efficiency, but
the adiabatic-isothermal trajectory has a 50% and 230% power
advantage over using sinusoid and linear trajectories, respectively
(13)
where tiso = t f 1 for compression. Eq. (13) implies an assumption
that it is possible to have an adiabatic process in compression or
expansion that takes finite time.
The effect of the p factor on the efficiency-power relationship is shown in Fig. (10). The change in power for the zero-time
adiabatic case versus the non-zero-time adiabatic case is due to
the extra time needed for the non-zero-time case. As p increases,
7
Copyright © 2009 by ASME
6 CONCLUSIONS
A trajectory that compresses or expands an ideal gas in the
sequence adiabatic-isothermal-adiabatic yields maximum work
(least work in, most work out) for a particular time (or minimum time for a particular work level). As a result, the adiabaticisothermal-adiabatic trajectory yields the highest efficiency for a
particular power level. The optimum volume trajectory found in
this paper match well with [12]. In [12], the volume trajectory for
an internal combustion chamber is numerically optimized, and
the authors get a “fast-slow-fast” type of profile. The optimum
volume trajectory results characterize the efficiency and power
trade-off of an air motor/compressor with a given heat transfer
capability. Compared to ad-hoc trajectories, a power increase
up to 5 times is possible. The characterization can also be used
to determine the required heat transfer to achieve a desired efficiency and power density. Knowledge of the optimum trajectory
for expansion and compression may be helpful for engineers developing compact, high-efficiency motor/compressors.
In the future, the results will be compared to experiments
and/or detailed simulations (e.g., computational fluid mechanic
models). Extension of the results to include losses, variable heat
transfer, and multiple stages are also being conducted.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
ACKNOWLEDGMENT
This work is performed within the ERC for Compact and
Efficient Fluid Power, supported by the National Science Foundation under Grant No. EEC-0540834.
[15]
REFERENCES
[1] Li, P., Van de Ven, J., and Sancken, C., 2007. “Open accumulator concept for compact fluid power energy storage”.
In Proceedings of the 2007 ASME-IMECE, ASME. Paper
number IMECE2007-42580.
[2] MDI Enterprises,
2009.
Air compressed
cars.
On the WWW, March.
URL:
http://www.theAirCar.com/.
[3] Liu, H., Tao, G., and Chen, Y., 2008. “Study on the twostage expansion air-powered engine”. In Proceedings of
the 7th JFPS International Symposium on Fluid Power,
TOYAMA 2008, pp. 803–808. Paper number P2-39.
[4] Schainker, R. B., Mehta, B., and Pollak, R., 1993.
“Overview of CAES technology”. In Proceedings of the
American Power Conference, Vol. 55, Illnois Institute of
Technology, pp. 992–997.
[5] Brok, S., Touber, S., and van der Meer, J., 1980. “Modelling of cylinder heat transfer - large effort, little effect?”.
In Proceedings of teh 1980 Purdue Compressor Technology
Conference, Purdue Research Foundation, pp. 43–50.
[6] Brown, R. N., 1997. Compressors: Selection and Sizing.
Gulf Publishing Co., Houston, TX.
[7] Recktenwald, G. W., Ramsey, J. W., and Patankar, S. V.,
1986. “Predictions of heat transfer in compressor cylin-
ders”. In Proceedings of the 1986 International Compressor
Engineering Conferenence – at Purdue, J. F. Hamilton and
R. Cohen, eds., Vol. 1, Purdue University, pp. 159–174.
Lewins, J. D., 2003. “Optimising an intercooled compressor for an ideal gas model”. International Journal of Mechanical Engineering Education, 31(3), July, pp. 189–200.
Kerr, S., Hoare, R., and MacLaren, J., 1980. “Optimum design of reciprocating compressors to meet thermodynamic
criteria”. In Proceedings of the 1980 Purdue Compressor
Technology Conference, W. Soedel, ed., Purdue Research
Foundation, pp. 8–14.
Liu, L., and Yu, X.-L., 2006. “Optimal design of ideal cycle
in air powered engine”. Journal of Zhejiang University,
40(10), Oct, pp. 1815–1818.
Liu, L., and Yu, X.-L., 2006. “Optimal piston trajectory
design of air powered engine”. Journal of Zhejiang University, 40(12), Dec, pp. 2107–2111.
Mozurkewich, M., and Berry, R. S., 1982. “Optimal paths
for thermodynamic systems: The ideal Otto cycle”. Journal
of Applied Physics, 53(1), Jan, pp. 34–42.
Kollmann, K., 1931. “Der waermeuebergang im luftkompressor”. Forschungsheft(348).
Lemofouet, S., and Rufer, R., 2005. “Hybrid energy storage
systems based on compressed air and supercapacitors with
maximum efficiency point tracking”. In Proceedings of the
IEEE 2005 European Conference on Power Electronics and
Applications, IEEE, p. 1665393.
Van de Ven, J., and Li, P. Y., 2009. “Liquid piston gas compression”. Applied Energy, 86(10), Oct, pp. 2183–2191.
A Appendix A: Formulae for optimal work and time
for expansion
Work for expansion:
Wout
γ T 1−γ
nR
(T3 − T0 ) + nRT1 ln r 0
+
=
1−γ
T3
T
nR 0 − T3 ,
r
(14)
with
T1 =
Time for expansion:
T0 T3 .
γ nRT1
T0 1−γ
ln r
.
tf2 =
hA(T0 − T1 )
T3
(15)
(16)
Substituting the above relations and similar relations for compression into Eq. (9) and Eq. (10), one can see that power is independent
of the initial quantity gas (i.e., n or V0 ) and that it is proportional to
the product of the heat transfer coefficient and the area over which heat
transfer occurs.
8
Copyright © 2009 by ASME