Estimation of the critical temperature ratio
for thermoacoustic engines based on
adaptive control which maintains
closed-loop system at stability limit
Kazuaki Sakurai, Yasuhide Kobayashi and Noboru Yamada
Nagaoka University of Technology (Japan)
Background
Sound wave
Thermoacoustic engines
utilizing the thermoacoustic
phenomena increase the
efficiency on heat recovery
TH
TC
Resonator
Stack
(Regenerator)
It is important to know the critical temperature
ratio (CTR) in thermoacoustic systems
It is difficult to know it at the design stage when heat
exchangers and stacks have complex structures
Background
In order to estimate CTR, we have
proposed an experimental method
Wspk [W]
0.2
The tube pressure amplitude is
maintained to be a constant value
by an open-loop control
203 Pa
135 Pa
81 Pa
54 Pa
27 Pa
14 Pa
0 Pa
The open-loop control has a
manual tuning of driving signal
for loudspeaker
0.1
Power consumption of loudspeaker
becomes minimum at CTR
0
1
1.1
1.2
1.3
TH/TC
1.4
1.5
1.6
Background
In order to estimate CTR, we have
proposed an experimental method
To carry out the open-loop control, it is necessary
to previously measure the frequency responses
at various temperature ratios
The resonance frequency is calculated
in order to set the driving frequency
Labor saving of the measurement is desired
Objective
Propose an adaptive control system
Increases the tube pressure amplitude till a
reference value by destabilizing the closedloop system in order to hold the critical point
By using this method, there is no need to carry out the
frequency responses, since the resonance frequencies are
automatically determined by proposed control system
It will be shown that a relationship between the
adaptive gain and temperature ratio can be used an
alternative method to estimate the CTR
Experimental apparatus
Standing-wave thermoacoustic engine
A/D
p2
p1
Wspk
PA
loudspeaker
sensor2
565
sensor1
1312 mm
D/A
TC
164
・ The cold-side heat
exchanger’s temperature TC
is kept at room temperature
PC
u
502
・ The thermoacoustic core
is composed of a stack
and two heat exchangers
A/D
vs
vi
360
・ A loudspeaker and
two pressure sensors
are located
TH
core
Experimental apparatus
Standing-wave thermoacoustic engine
A/D
p2
p1
Wspk
PA
loudspeaker
sensor2
565
It has been found that the
CTR of the engine is
approximately 1.3
sensor1
1312 mm
D/A
TC
164
PC
u
502
A preliminary experiment
has been carried out to
obtain a rough estimation
of CTR
vs
vi
360
A/D
TH
core
Experiment method
The objective of the control system is to drive the loudspeaker
with as small amplitude as possible in driving signal so that
the amplitude of p2 matches to the reference value
・The reference value of p2 is set to 200 Pa.
・The heater temperature is adjusted so that TH / TC
varies from 1.00 to 1.35, 8 points with increment 0.05
The driving signal of the loudspeaker
u (t )  G (t ・) p 2(t τ)
(τ =11.5 ms )
This is known as a phase-delay controller which has been
conventionally used to suppress the thermoacoustic instabilities
Experiment method
Block diagram of the control system
・The absolute value signal of p2 goes through
to a low-pass filter and estimated pressure
amplitude P
＾2 is obtained
・The difference between ＾
P2 and the
reference value P2*are sent to the
PI controller
P2*
・The output of PI controller
- P
＾
2
itself is used as the
π/2
feedback gain G(t)
PI
LPF
G(t)
e-sτ
u
G(t)
|・|
At each setting of TH / TC, the steady-state
feedback gain G is measured
p2
Result
Time response
Pressure (Pa)
It is automatically adjusted
to achieve a constant
pressure amplitude at the
reference value
p
＾2
Pressure (Pa),Gain
As shown in blue line, the
absolute value of the gain
is increased so as to
oscillate the engine
ex. TH / TC= 1.0
p2
G
Time(s)
Result
Time response
ex. TH /TC= 1.0
It is observed that the
pressure amplitude
converges to 400 Pa
which implies the steadystate oscillation
Pressure (Pa)
Enlarged view of the time
response of p2 in steady state
400 Pa
Time(s)
Result
Feedback gain G
gain G
Linear relationship is observed
【TH / TC < 1.3】
0
・ Adjusted gain shows negative value
・ Loudspeaker acts to assist the
sound wave generation
-50
【TH / TC = 1.3】
・ Gain is almost zero
-100
【TH / TC = 1.35】
・ Positive gain is observed
・ Loudspeaker acts to suppress
1
1.1
1.2
1.3
the sound wave
T /T
If we assume the linear relationship, it is possible to
predict the CTR by performing measurement of gains
for several points in the region below the CTR
H
C
Result
Oscillation frequency
It is confirmed that the oscillation
frequency by adaptive control
has the same tendency with
frequency response experiment
adaptive control
frequency response
63
f (Hz)
The oscillation frequency has
been adjusted automatically
64
62
61
60
1
1.1
1.2
1.3
TH/TC
The frequency response experiment as in the conventional
method can be omitted with this proposed method
Conclusion
・We proposed an adaptive control system which maintains
the tube pressure amplitude to be a reference value.
The open-loop gain is automatically adjusted by destabilizing
the system around the critical point.
・The results show that the pressure amplitude is successfully
converged to the reference one.
This implies the closed-loop stability of the proposed
adaptive control system.
・The steady-state frequencies for various temperature ratios
are automatically obtained in similar values by manual control.
・A linear relation between the adaptive gain and temperature
ratio is also obtained, which provides an alternative method
to estimate the CTR.
Future work
・ Theoretical guarantee for the closed-loop
stability of the proposed adaptive
control system
・ Automatic adjustment of time delay
in controller