Download Optimal Control Experimentation of Compression

Proceedings of the ASME 2013 Heat Transfer Summer Conference
HT2013
July 14-19, 2013, Minneapolis, MN, USA
HT2013-17613
OPTIMAL CONTROL EXPERIMENTATION OF COMPRESSION TRAJECTORIES FOR
A LIQUID PISTON AIR COMPRESSOR
Farzad A. Shirazi, Mohsen Saadat, Bo Yan, Perry Y. Li, and Terry W. Simon
Department of Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota 55455
Email: [email protected]
ABSTRACT
corresponding to each compression time. The resulted efficiency
versus power density of optimal profiles is then compared with
ad-hoc constant flow rate profiles showing up to %4 higher efficiency in a same power density or %30 higher power density in
a same efficiency in the experiment.
Air compressor is the critical part of a Compressed Air Energy Storage (CAES) system. Efficient and fast compression of
air from ambient to a pressure ratio of 200-300 is a challenging
problem due to the trade-off between efficiency and power density. Compression efficiency is mainly affected by the amount
of heat transfer between the air and its surrounding during the
compression. One way to increase heat transfer is to implement
an optimal compression trajectory, i.e., a unique trajectory maximizing the compression efficiency for a given compression time
and compression ratio. The main part of the heat transfer model
is the convective heat transfer coefficient (h) which in general
is a function of local air velocity, air density and air temperature. Depending on the model used for heat transfer, different
optimal compression profiles can be achieved. Hence, a good
understanding of real heat transfer between air and its surrounding wall inside the compression chamber is essential in order to
calculate the correct optimal profile. A numerical optimization
approach has been proposed in previous works to calculate the
optimal compression profile for a general heat transfer model.
While the results show a good improvement both in the lumped
air model as well as Fluent CFD analysis, they have never been
experimentally proved. In this work, we have implemented these
optimal compression profiles in an experimental setup that contains a compression chamber with a liquid piston driven by a water pump through a flow control valve. The optimal trajectories
are found and experimented for different compression times. The
actual value of heat transfer coefficient is unknown in the experiment. Therefore, an iterative procedure is employed to obtain h
1
Introduction
Gas compression and expansion has many applications in
pneumatic and hydraulic systems, including in the Compressed
Air Energy Storage (CAES) system for offshore wind turbine
that has recently been proposed in [1, 2]. In the proposed CAES
system, high pressure (∼20-30MPa) compressed air is stored in a
dual chamber storage vessel with both liquid and compressed air.
Since the air compressor/expander is responsible for the majority
of the storage energy conversion, it is critical that it is efficient
and sufficiently powerful. This is challenging because compressing/expanding air 200-300 times heats/cools the air greatly, resulting in poor efficiency, unless the process is sufficiently slow
which reduces power [3, 4]. There is therefore a trade-off between efficiency and power.
Most attempts to improve the efficiency or power of the air
compressor/expander aim at improving the heat transfer between
the air and its environment. One approach is to use multi-stage
processes with inter-cooling [5]. Efficiency increases as the number of stages increase. To improve the efficiency of the compressor/expander with few stages, it is necessary to enhance the heat
transfer during the compression/expansion process. A liquid piston compression/expansion chamber with porous material inserts
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Copyright © 2013 by ASME
has been studied in [3]. The porous material greatly increases the
heat transfer area and the liquid piston prevents air leakage [6].
Numerical simulation studies of fluid flow and enhanced heat
transfer in round tubes filled with rolled copper mesh are studied
in [7]. Application of porous inserts for improving heat transfer
during air compression has also been investigated [8].
In addition, the compression/expansion trajectory can be
optimized and controlled to increase the efficiency for a given
power or to increase power for a given efficiency. For high compression/expansions ratios 200:1-350:1, such Pareto optimal trajectories have been shown, theoretically, to increase the power
of the compressor/expander by 200-500% at the same efficiency,
over ad-hoc trajectories such as linear and sinusoidal trajectories [3, 9, 10]. In [3, 9], the optimal trajectories were derived
analytically based on simple heat transfer models and by considering thermodynamic losses alone. For example in [3], the
product of the heat transfer coefficient and heat transfer surface
area hA is assumed to be constant; and in [9], hA is allowed to
vary with air volume to take into account the decrease in surface area as the porous material is submerged in the liquid piston. In both cases, the optimal trajectories consist of fast adiabatic portions at the beginning and at the end. For the constant hA case [3], the middle portion is isothermal resulting in
an Adiabatic-Isothermal-Adiabatic (AIA) trajectory, whereas for
the volume dependent hA(V ) case [9], the temperature
difference
√
from the ambient is inversely proportional to hA(V ) leading to
an Adiabatic-Pseudo-Isothermal-Adiabatic (APIA) trajectory.
Optimal trajectories have also been numerically obtained
for cases with varying heat transfer coefficient, and considering the effect of liquid friction and flow rate constraints of the
system [10]. Application of such optimal trajectories has been
verified for a liquid piston compressor/expander using CFD and
the results have been compared with AIA and APIA trajectories.
While the theoretical improvement in power/efficiency with
the optimal trajectories for air compression over conventional
linear and sinusoidal trajectories has been validated analytically
and numerically, this paper provides the experimental validation
of the usefulness of such optimal trajectories. One issue with
implementing the optimal trajectories experimentally is the uncertainty of the heat transfer model. The transient heat transfer
coefficient depends in reality on many factors. Even if a constant heat transfer coefficient h is assumed, its value is often difficult to access a-priori. If the estimate of h is inaccurate, there
are two consequences: 1) the pressure-volume or temperaturevolume trajectories of the air being compressed, which determine
the thermodynamic performance cannot be tracked properly; 2)
the “optimal” trajectory itself cannot be defined correctly.
To overcome these issues, a temperature-volume trajectory
tracking controller that adaptively estimates the unknown heat
transfer coefficient h was developed and presented in our previous paper [11]. This controller is used to track an estimated optimal temperature-volume profile. The achieved thermodynamic
profile is then used to compute an average heat transfer coefficient. This value is then used to compute the next estimated
optimal temperature-volume profile. With this adaptive-iterative
optimal control strategy, the compression efficiency successively
improves for a given power-density.
The rest of the paper is organized as follows. Experimental setup capable of 10:1 compression ratio is discussed in the
next section. A theoretical background is introduced in Section 3
on optimal trajectories used in experiments. Section 4 discusses
the experimental tracking results. The iterative procedure to obtain the optimal compression trajectories is discussed in Section
5. The experimental results of constant flow rate and optimal
trajectories are presented and discussed in Section 6. Section 7
concludes the paper.
2
Experimental Setup
In order to investigate the performance of the optimal trajectories for air compression an experimental setup was built
at power fluid control lab at University of Minnesota shown in
Figure 1. A water pump circulates water inside the circuit and
provides the required flow rate to compress air inside the compressor chamber. Two pressure transducers are mounted in the
setup to measure the upstream water pressure and downstream
air pressure. An Omega FTB-1412 turbine flow-meter and an Endress+Hauser Promass 80 Coriolis flow-meter are used to measure the water volume and flow rate. In one hand, the Coriolis
meter measurement has a time lag of ∆t = 71 ms and on the other
hand, the turbine meter is not accurate at small flow rates. Therefore, we combine the volume information of two meters to obtain
a more accurate measurement of the water volume flowing inside
the chamber and consequently the volume of the air under compression. The water and air volume at each instance of time are
determined as follows.
Vw (t) = VCor (t − ∆t) +VT M (t) −VT M (t − ∆t)
(1)
V (t) = V0 −Vw (t)
where V0 is the initial volume of the air. We add the turbine
meter volume difference between time t and t − ∆t to the volume measured from Coriolis meter to compensate for the existing time-lag. A relief valve is mounted in the circuit to keep
the pressure of the circuit below 160 psi. Filters are also included to remove unwanted particles from flowing water. A 16bit PCI-DAS1602/16 multifunction analog and digital I/O board
is used to acquire the system data. This board is linked with
MATLAB/Simulink via xPC Target to send and receive the control and measured signals. The sampling frequency is chosen to
be 4000 Hz. The main reason for the high sampling frequency is
to correctly capture the output signal of Coriolis and turbine meters. A Burkert 6223 Servo-assisted proportional control valve
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compression time and air volume at which the desired pressure
ratio is obtained. While the viscous friction energy loss is relatively large for small diameter tubes and large flow rates, such a
loss can be neglected due to the large diameter of the compression tube used in this experiment (d = 5.04cm). In Eq. (3), term
V̇(t) is the control input while T (air temperature) is a dynamic
state for which, the dynamic equation comes from the first law
of Thermodynamics as
Ṫ(t) =
FIGURE 1.
T(t)
h(t) A(t)
(TW − T(t) ) + (1 − γ )
V̇
mCv
V(t) (t)
(4)
EXPERIMENTAL SETUP
where Cv and γ are the heat capacity of air at constant volume
and the heat capacity ratio of air, respectively. h is the heat transfer coefficient between air and solid boundary which, in general, varies with time during compression based on air properties (h(t) = h(T(t) ,V(t) , V̇(t) )). Tw is the wall temperature which is
assumed to be constant (at ambient temperature) during the compression process (due to large heat capacity of the solid boundary
compared to air). Finally, A(t) is the active heat transfer area between the air and cylinder wall, tube’s cap and liquid piston.
is employed to manipulate the flow rate inside the chamber. The
temperature of the air inside the compressor is obtained from the
ideal gas law using the pressure and volume measurements as
follows.
T=
PV
T0
P0V0
(2)
where P and V are the measured air pressure and volume, and
P0 and T0 are initial pressure and temperature of the air, respectively. In each experiment, the nut on the cap of the compressor
is opened to make the initial air pressure equal to the ambient.
Water can be added to the column to set the initial air volume at
the desired value. The maximum available flow-rate in the setup
is 302cc/s. It is noted that we only use the data up to the pressure
ratio of 10.
A(t) =
4V(t) π d 2
+
d
2
(5)
While the air is initially at T0 , V0 and P0 (at t = 0), it is desired to
compress it to a final compression ratio of r in a final compression time of tc . Therefore, there exists a final manifold for the
related optimal control problem that relates the final volume and
temperature as follows.
3
Theoretical Background on Optimal Trajectories
In this section, a brief background on optimal compression
trajectories is presented. The interested reader is referred to [10]
for a detailed mathematical modeling. Calculating the optimal
compression profile can be addressed as a functional optimization problem (or optimal control problem) for which the cost
function is the input work defined as follows.
∫ tc
J=
|
0
(
(rP0 − P0 )V f
|
{z
}
(6)
In summary, finding optimal compression profile which results
the maximum compression efficiency is equivalent to minimizing the cost function defined by Eq. (3), subject to dynamic constraint of Eq. (4) and final manifold of Eq. (6). Additional inequality constraints due to hardware limitations can also be imposed. For example, here we have considered the liquid pump
limitation so that the liquid piston flow rate is limited.
)
∫ tc
mRT(t)
− P0 V̇(t) dt +
Γ(V(t) ,V̇(t) )V̇(t) dt
V(t)
0
|
{z
}
{z
}
Compression Work
+
T(tc )
T0
−r = 0
V(tc )
V0
Friction Work
(3)
|V̇(t) | 6 V̇max
Isobaric Cooling Work
where m, T and V are the mass, temperature and volume of air
under compression, R is the specific gas constant of air, Γ is the
viscous friction power (between liquid piston and its surrounding solid wall), r is the final pressure ratio, and tc and V f are the
(7)
The continuous optimal control problem is then parameterized
as a finite dimensional problem and solved numerically by standard algorithms for constrained parameter optimization. Once
the control input is defined over the time range, air temperature
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100
200
150
100
50
0
450
90
350
300
0
0.5
1
1.5
Optimal Profile
Constant Flow
Adiabatic
95
400
Efficiency %
250
Temperature (K)
Flow Rate (cc/s)
300
2
50
100
Time (s)
150
200
250
300
Volume (cc)
85
5
x 10
80
Temperature (K)
Pressure (psi)
10
8
6
4
2
450
75
400
350
70
1
2
10
10
3
0
Power Density (kW/m )
300
50
100
150
200
250
300
Volume (cc)
0
0.5
1
1.5
2
Time (s)
FIGURE 3. COMPRESSION EFFICIENCY VS. POWER DENSITY
FOR r = 10 AND h = 10 mW
2 .K
FIGURE 2. SAMPLE OPTIMAL COMPRESSION PROFILE
(FLOW RATE-TIME, TEMPERATURE-VOLUME, PRESSUREVOLUME AND TEMPERATURE-TIME) FOR r = 10, tc = 2s AND
h = 16 mW
2 .K
460
440
420
tc=3s
Temperature (K)
can be found by solving Eq. (4). Figure 2 illustrates sample flow
rate, temperature-volume and pressure-volume optimal trajectories obtained from the optimization problem for r = 10, tc = 2.5s,
h = 10 mW
2 .K , V0 = 300cc and V̇max = 280cc/s. The efficiency
of trajectories is obtained from dividing the isothermal process
work over the input compression work for the same pressure ratio of r as follows.
400
tc=5s
tc=7s
380
360
340
320
300
Wiso = P0V0 ln(r)
(8)
Win = −
(9)
η=
∫ Vf
V0
Wiso
Win
(P − P0 )dV + (rP0 − P0 )V f
100
150
200
250
300
350
400
Air Volume (cc)
450
500
550
600
FIGURE 4. EXPERIMENTAL T-V TRACKING CURVES OF OPTIMAL TRAJECTORIES FOR COMPRESSION TIMES OF 3s, 5s
AND 7s
(10)
In calculation of both compression works the isobaric ejection
work is also included. The improvement of compression efficiency for a fixed compression power (or vice versa) resulted by
optimal compression trajectories is shown in Figure 3 compared
to constant flow rate profiles. As a rule of thumb, this improvement is more noticeable at lower power densities and higher pressure ratios. As mentioned earlier, the heat transfer coefficient h
is a complex function of air properties, flow regime and the compression chamber geometry. As the result, it is usually difficult
to have an accurate model of h during the compression process.
In this work, we use an iterative procedure to estimate a constant
average value for h to design the optimal compression profiles.
4
Optimal Trajectory Tracking Results
Figure 4 illustrates the tracking of the temperature-volume
optimal trajectory for compression times of 3s, 5s and 7s. The
reader is referred to [11] for more details on design and implementation of the controller employed to track the optimal trajectories. The experimental temperature profiles for compression
times of 1.5s and 2s are also shown for optimal and constant
flow-rate trajectories in Figure 5. It can be seen that constant
flow-rate trajectories result in higher temperatures at the second
half of the process that makes an important effect on decreasing
the compression efficiency compared to optimal trajectories.
4
Copyright © 2013 by ASME
12
500
Optimal for tc=1.5s
11.5
Constant flow−rate for tc=1.5s
450
11
Optimal for tc=2s
Constant flow−rate for tc=2s
hi (W/m2.K)
10.5
T(K)
400
hguess
10
hactual
9.5
9
350
8.5
8
300
1
2
3
4
Iteration number
250
0
0.5
1
t(s)
1.5
FIGURE 6. CONVERGENCE OF GUESS AND ACTUAL VALUES
OF hi FOR tc = 1.5s AND V0 = 300cc AFTER 4 ITERATIONS
2
FIGURE 5. EXPERIMENTAL TEMPERATURE PROFILES OF
OPTIMAL AND CONSTANT FLOW-RATE COMPRESSION TRAJECTORIES FOR COMPRESSION TIMES OF 1.5S AND 2S
90
80
2
h =12 W/m K
1
70
2
h2=14 W/m K
5
Iterative Calculation of Optimal Trajectories
The optimal trajectories are designed for constant values of
h. Here, we consider an iterative algorithm that takes the advantage of both experimental data and theory to obtain optimal
compression trajectories for the experimental setup. In this algorithm, for a given compression time tc and a maximum flow rate
V̇max , an initial value of heat transfer coefficient h0 is assumed
and the optimal compression trajectory, e.i., temperature versus
volume Ti (V ) is determined using the optimization code. This
trajectory is used in the experiment for tracking. The experimental data is acquired and analyzed to obtain the corresponding
efficiency ηi , compression time tci and an average value of actual
hi . This calculated hi is again inserted in the optimization code to
obtain the next optimal trajectory. This iterative procedure is carried on until the efficiency is not increased anymore and a final
value of hN is obtained, where N is number of iterations. Figure 6
shows a sample convergence curve of h for tc = 1.5s. Following
the procedure explained above, an initial guess of h0 = 8 mW2 K converges to h4 = 12 mW2 K after 4 iterations that results in the same experimental compression time. Figure 7 shows the h profiles versus volume for optimal trajectories experimented in successive
iterations of the iterative procedure for tc = 3s and V0 = 608cc.
It is observed from this figure that the constant h assumption is
valid for a major part of the compression process. The main difference appears at the end of the compression where h increases
sharply resulting in lower final temperatures in the actual experiment (Figure 4) than the designed optimal trajectories. The final
results of the iterative procedure are listed in Table 1 for different compression times. The efficiencies are obtained for optimal
h3=12.3 W/m2K
60
2
h (W/m2K)
h =13 W/m K
4
50
40
30
20
10
0
100
150
200
250
300
350
Air Volume(cc)
400
450
500
FIGURE 7. COMPARISON OF HEAT TRANSFER COEFFICIENT
PROFILES DURING COMPRESSION FOR SUCCESSIVE ITERATIONS OF AVERAGE h value FOR tc = 3s AND V0 = 608cc
trajectories corresponding to the converged h values.
6
Experimental Results
As explained before, the trade-off between efficiency and
power density of air compression plays an important role in
CAES systems. In order to obtain a benchmark for powerefficiency characteristics of our experimental setup we first conducted a series of experiments with different constant flow-rate
linear volume trajectories. Table 2 summarizes the compression time and efficiency values for the tested constant flow rates.
Figure 8 shows the comparison of pressure-volume curves for
5
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SUMMARY OF RESULTS FOR OPTIMAL TRAJECTO-
PV curves
150
tc (s)
V0 (cc)
h( mW2 K )
η (%)
10
608
16
91.56
7
608
15.75
88.43
5
608
15
85.70
3
608
13
80.94
3
300
18
82.90
2
300
16
79.57
1.5
300
12
77.28
140
Pressure (psi)
TABLE 1.
RIES
120
Optimal for tc=3s
100
Isothermal
Adiabatic
Constant flow rate for tc=3s
80
60
40
20
TABLE 2. COMPRESSION TIME AND EFFICIENCY VALUES
FOR THE TESTED CONSTANT FLOW RATES
tc (s) V0 (cc) V̇ (cc/s) η (%)
715
65.0
86.86
6.01
715
106.0
84.24
4.43
715
144.1
81.99
3.54
715
179.8
80.14
3.03
715
216.2
79.63
2.62
715
250.0
77.99
2.36
715
280.7
77.79
2.19
715
301.9
77.18
1.21
300
201.9
76.20
0
50
100
150
Volume (cc)
200
250
300
FIGURE 8. COMPARISON OF PRESSURE-VOLUME CURVES
FOR OPTIMAL, ISO-THERMAL, ADIABATIC AND CONSTANT
FLOW-RATE TRAJECTORIES FOR tc = 3s AND V0 = 300cc
Adiabatic
tc=3s
140
tc=5s
tc=7s
120
Isothermal
100
Pressure (psi)
9.80
0
optimal, iso-thermal, adiabatic and constant flow-rate trajectories for a compression ratio of 10, tc = 3s and V0 = 300cc. The
isothermal curve corresponds to an infinitely slow, 100% efficient process while the lowest efficiency of 70.4% is obtained
from the adiabatic infinitely fast compression. It is observed
that the optimal trajectory P-V curve is found to be shifted towards the isothermal curve compared to the constant flow-rate
process that results in up to 4% higher efficiency for a same
power density. Figure 9 also illustrates the P-V curves for 3s,
5s and 7s optimal trajectories and their comparison with isothermal and adiabatic compressions. A summary of efficiency versus power density of optimal and constant flow rate trajectories
is shown in Figure 10. It is observed that for a same value of
power density the efficiency of optimal trajectories can be higher
up to 4%. This difference is more significant at higher efficiencies that is consistent with the simulation results discussed in
Section 3. On the other hand, for a same efficiency the power
density can be increased up to 30% for a pressure ratio of 10.
The actual temperature-volume tracks the desired T-V optimal
trajectory closely. Therefore, the efficiency of each process is
80
60
40
20
100
200
300
400
Air Volume (cc)
500
600
FIGURE 9. EXPERIMENTAL P-V CURVES OF OPTIMAL TRAJECTORIES FOR COMPRESSION TIMES OF 3s, 5s and 7s
obtained to be approximately the same as its theoretical value as
shown in Figure 11. The uncertainty in volume measurement can
affect the efficiency calculation. The percentage of error in volume measurement is about ±1% based on the specification of the
flow-meters used in the experiment. Numerical sensitivity analysis shows ±1% error in efficiency due to ∓1% error in volume
measurement throughout the experiment.
6
Copyright © 2013 by ASME
perimental trajectory tracking results and efficiency-power density calculations were presented. It was shown that for a same
power density and a relatively low pressure ratio of 10 the optimal trajectories can improve the efficiency up to 4% compared
to linear constant flow rate trajectories.
92
Optimal Trajectory
Linear trajectory
90
Efficiency (%)
88
86
84
REFERENCES
[1] P. Y. Li, E. Loth, T. W. Simon, J. D. Van de Ven and
S. E. Crane, “Compressed Air Energy Storage for Offshore
Wind Turbines,” in Proc. International Fluid Power Exhibition (IFPE), Las Vegas, USA, 2011.
[2] M. Saadat and P. Y. Li, “Modeling and Control of a Novel
Compressed Air Energy Storage System for Offshore Wind
Turbine,” in Proc. American Control Conference, pp. 30323037, Montreal, Canada, 2012.
[3] C. J. Sancken and P. Y. Li,“Optimal Efficiency-Power Relationship for an Air Motor/Compressor in an Energy Storage
and Regeneration System,” in Proc. ASME Dynamic Systems
and Control Conference, pp. 1315-1322, Hollywood, USA,
2009.
[4] M. Nakhamkin, E. Swensen, R. Schainker and R. Pollak,
“Compressed Air Energy Storage: Survey of Advanced CAES
Development,” in Proc. ASME International Power Generation Conference, pp. 1-8, San Diego, USA, 1991.
[5] J. D. Lewins, “Optimising and intercooled compressor for an
ideal gas model,” Int. J. Mech. Engr. Educ., vol. 31, Issue 3,
pp. 189-200, 2003.
[6] J. D. Van de Ven and P. Y. Li, “Liquid Piston Gas Compression,” Applied Energy, vol. 86, Issue 10, pp. 2183-2191, 2009.
[7] M. Sozen and T. M. Kuzay, “Enhanced Heat Transfer in
Round Tubes with Porous Inserts,” Int. J. Heat and Fluid
Flow, vol. 17, Issue 2, pp. 124-129, 1996.
[8] A. Rice, Heat Transfer Enhancement in a Cylindrical Compression Chamber by Way of Porous Inserts and the Optimization of Compression and Expansion Trajectories for Varying
Heat Transfer Capabilities, Master thesis, University of Minnesota, 2011.
[9] A. T. Rice and P. Y. Li, “Optimal Efficiency-Power Tradeoff for an Air Motor/Compressor with Volume Varying Heat
Transfer Capability,” in Proc. ASME Dynamic Systems and
Control Conference, Arlington, USA, 2011.
[10] M. Saadat, P. Y. Li and T. W. Simon, “Optimal Trajectories
for a Liquid Piston Compressor/Expander in a Compressed
Air Energy Storage System with Consideration of Heat Transfer and Friction,” in Proc. American Control Conference, pp.
1800-1805, Montreal, Canada, 2012.
[11] F. A. Shirazi, M. Saadat, B. Yan, P. Y. Li and T. W. Simon,“Iterative Optimal Control of a Near Isothermal Liquid
Piston Air Compressor in a Compressed Air Energy Storage
System,” to appear in American Control Conference, Washington, DC, 2013.
82
80
78
76
0
20
40
60
80
100
120
140
160
180
Power Density (kW/m3)
FIGURE 10. COMPARISON OF EFFICIENCY VS POWER DENSITY CURVES OF OPTIMAL AND CONSTANT FLOW RATE
COMPRESSION TRAJECTORIES FOR r = 10
92
Optimal Trajectory Experiment
Optimal Trajectory Theory
90
Efficiency (%)
88
86
84
82
80
78
76
20
40
60
80
100
120
140
160
Power Density (kW/m3)
FIGURE 11. COMPARISON OF THEORETICAL AND EXPERIMENTAL EFFICIENCY VS POWER DENSITY CURVES OF OPTIMAL TRAJECTORIES FOR r = 10
7
Conclusions
This paper presented the experimental investigation of optimal trajectories for air compression. The experimental setup
and procedure were discussed in detail. An adaptive nonlinear
controller designed in [11] is employed to command the desired
flow-rate to the compressor chamber and track the temperaturevolume optimal trajectories. Since the actual value of h is not
known, an iterative procedure was introduced to obtain the optimal trajectory using the experimental data. For a given compression time and compression ratio, the final optimal trajectory was
determined based on the converged average value of h. The ex-
7
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