Ecological Optimization of Thermoacoustic Engine with the
Characteristic Time
Jinhua Fei1,a, Feng Wu2,b, and Tuo Wang1,c
1
School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan, China
2
School of Science, Wuhan Institute of Technology, Wuhan, China
a
[email protected], [email protected], [email protected]
Keywords: Thermoacoustic engine; Characteristic time; Ecological function; Optimization
Abstract. This article used ecological function as an indicator to analyze the relationship between the
ecology target value and the characteristic time, obtained the characteristic time value in the
maximum ecology function value, made a numerical calculation between ecology target value and the
characteristic time. It shows that the ecological function value has a change process with the
characteristic time  k or  v that increases firstly and then decreases, which verifies the calculation
results. Simultaneously, it is analyzed the impact to ecology target value with the temperature gradient
and other factors. It could provide some help to future research.
Introduction
Thermoacoustic engine is a new type engine which is based on the thermoacoustic effect[1-5].
Compared with the conventional heat engines, it has some significant advantage, such as simple
structure, no moving parts, no pollution, high reliability, and so on. Therefore, it has become a
research focus in some fields like energy power and low-temperature engineering.
Recently, it has been a very active research in the field of finite time thermodynamics that
analyzing and optimizing the performance of the heat engine cycle with different target functions. The
ecological function is established based on the views of energy or exergy, it reflects the energy
consumption of the whole machine. Therefore, it can be a target to optimize the performance of heat
engine cycle. For the heat engine cycle, the ecological function is obtained as follow[6]:
E  P  Tc
(1)
Where, P is the power output, Tc is environmental temperature,  is the entropy generation rate.
In the thermoacoustic self-excited oscillation system, the characteristic time of the system 
reflects the energy exchange effect of the thermoacoustic engine. It determines the characteristic sizes
of the thermoacoustic stack or regenerator, such as the intrusion layer thickness and the resonance
tube length. Therefore, it is an important design parameter for thermoacoustic engine[7].
This article is based on Swift's thermoacoustic theory, and uses the ecological function as a target,
analyzes the relationship between the characteristic time of the thermoacoustic engine and ecology
function value, obtains the characteristic time when the target value is maximum, and makes
numerical calculation to verify the results, analyzes the influence to the target value of other factors
like the temperature gradient on target.
Characteristic Time of Thermoacoustic Engine
Thermal penetration depth and viscous penetration depth are defined as[2]
 k  2 K / 0 c p  2 / 
(2)
  2 / 0  2 / 
(3)
Where K is the thermal conductivity in the working fluid,  0 is the average density, c p is
constant pressure specific heat capacity,  is dynamic viscosity coefficient. Thermal diffusivity and
kinematic viscosity are defined as   K / 0c p and    / 0 .So there is   2 /  k 2 or   2v /  v 2 .
In thermoacoustic engine system, the contrast ratio, between the system’s eigenfrequency and the
horizontal entropy wave relaxation time, used to reflect the whole thermoacoustic engine system’s
energy exchange effect, is the characteristic time of thermoacoustic system (dimensionless). It is the
characteristic parameter which is determined the level of occurring the thermoacoustic effect and the
efficiency of energy transformation [1].
2
Thermal relaxation time and the viscosity relaxation time are defined as  k  r02 / 2 and   r0 / 2 .
Therefore, the characteristic times are
 k  r0 2 /  k 2
(4)
  r0 2 /  v 2
(5)
Namely,  k  r0 /  k ，   r0 / 
Heat Flux and Power Flux in the Standing Wave Sound Field
The thermoacoustic stack (regenerator) is the core component to achieve energy transformation in the
thermoacoustic system. The energy flow diagram of the thermoacoustic engine is shown in Fig.1 [8].
Fig.1 The energy flow of the thermoacoustic engine
Fig.2 The diagram of stack and flow
The length, width, and thickness of each plate of thermoacoustic stack respectively are L , L1 , 2L2 ,
and the working flow layer’s thickness is 2 y0 , it is shown in Fig.2.
The direction of vertical heat transfer is along the length direction of the stack (which is along the
direction of L ).In the thermoacoustic resonance tube, the acoustic wavelength is defined as  , and the
speed oscillation and the sound pressure oscillation of the standing wave are as follows:
U1s  U m sin(2 x /  )
（6
P1s  Pm cos(2 x /  )
（7
）
）
Consider the finite heat capacity of solid working medium, in a certain approximate conditions, the
system’s linearized control equation can be written as[2,9]:
 / t  0 (U1 / x)  0
（8
0T  s / t   U1  s / x    K   2T / y 2   K s   2Tc / y 2 
（9
）
）
0  U1 / t   P1 / x     2U1 / y 2 
（10
   0  0T   P1 / c 2
（11
）
）
Where U1 , P1 ,  , T and s are the speed oscillation, sound pressure oscillation, density oscillation,
temperature oscillation and entropy oscillation of micro-groups in working fluid.  ,  0 and c are the
specific heat capacity, isothermal expansion coefficient and adiabatic sound speed of working fluid.
 s , Cs and K s are the density, specific heat capacity and transverse thermal conductivity of solid
working medium.
With the above equations and considering the vertical heat leak between the hot side and cold side,
in the conditions of short-board similar, we take boundary layer approximation  k  y0 and a small
prandt1. Therefore, the heat flow and power flow can be obtained as follows[10]:
Q2 
L1r0T0  0 P1sU1s 1   
4  k 1   s  1  
0.5

 L1  y0 K x  L2 K sx 
dT0
dx
（12
）
 L1Lr0   1  P1s  1   
2
W2 
）
4  k 0c 2 1   s 

L1Lr00 U1s 
2
4 
（13
Where, Tcr  T0 0 P1s / ( 0C pU1s ) ，    dT0 / dx  / Tcr ，   C p / K ，  s  0c p K / (  s cs K s ) are
critical temperature gradient, specific temperature gradient, Prandt1 number and dimensionless
quantity. K x and K sx are respectively the vertical thermal conductivity of working fluid and solid
working medium.
Ecological Optimization of Characteristic Time
Heat engine efficiency, power, exergy loss rate, and ecological function can be respectively written as:
  W2 / QH  W2 / Q2
14）
（
P  W2
（15
）
Tc s  Tc (
QH QL
Q  W2 Q2
 )  Tc ( 2
 )
TH TL
TH
TL
（16
）
E  P  Tc s  W2  Tc (
T
Q2  W2 Q2
1 1
 )  Tc (  ) Q2  (1  c ) W2
TH
TL
TL TH
TH
（17
）
As P1s and U1s are approximately independent on acoustic circular frequency  , there is   1/  .
Define    , so ' is irrelevant with  . Therefore, Eqs.(14)~(17)can be rewritten as:
Q2  a1
1  
 a
2
 k
（18
）
P  W2  b1
  

 b2
 k
 v
（19
）

W2
b1 (  )  b2

Q2
  k
 v
a2  k  a1 (1   )
（20

）
E  [Tc (
T
T
1
1 a
  

1
1
 ) 1  (1  c )b1 ]
 b2 (1  c )
 Tc a2 (  )
TL TH 
TH
TH  v
TL TH
 k
（21
）
L L  (U s )2
L1r0T0 0 P1sU1s
L Lr (  1)( P1s ) 2
dT
, a2  L1 ( y0 K x  L2 K sx ) 0 , b1  1 0 2
, b2  1 0 1 .
dx
4
4(1   s )(1   )
4 0c (1   s )
When calculating in an engineering approximate situation, it can be taken T0  Tc  0.5L(dT0 dx) .
Appliances the extreme value conditions E ( k )  0 to Eq(21), and solves the equation and
discard the solution which does not conform the physical meaning, the result can be obtained as:
Where, a1 
 k 
m12  12m2 m3  m1
（22）
2m2
At this time, ecological target value E is maximum.
T b   b2
T
1
1
1
1
Where, m1  Tc (  )a1  (1  c )b1 , m2  (1  c ) 1
, m3  Tc (  )a1 k  .
TL TH
TH
TL TH
TH
k
Numerical Calculation
To illustrate the preceding analysis, a numerical example is provided. In the calculations, it is set as
follow:
Resonance tube inner diameter is 76mm , wall thickness is 3mm . Put the stainless steel
thermoacoustic stack in the tube, for the thermoacoustic stack, L  0.04m , 2L2  0.003m ,
2 y0  0.003m . Stainless steel’s parameters are  s  7.93 103 kg / m3 , cs  508J / (kg  K ) , and
Ks  16.2W / (m  K ) . Thermoacoustic device uses nitrogen as working fluid, the inflation pressure is
0.8MPa , the regenerator heat side and cold side’s temperatures are respectively taken as 450K and
350K . Taking the nitrogen’s physical parameters in the temperature 400K as follows:   1.4 ,
a  407.676m / s , 0  6.7235kg / m3 , K  32.51103W / (m  K ) , c p  1.0510kJ / (kg  K ) ,   21.67 106 Pa  s , K x  1.3
and K sx  1.3 . The environmental temperature is 300K ,define T0 0  1 , Pm  1104 Pa , take
r0  0.002m , take x and dT0 dx as: x   / 20 、 x   /10 、 x   / 8 ，and dT0 dx  1K / m ，
dT0 dx  2K / m ， dT0 dx  3K / m .
The relationships between the ecological function value and characteristic times are shown as
Fig.3 and Fig.4.
It can be seen from Fig.3 and Fig.4 that the ecological function value E has a change process with
the characteristic time  k or  v that increases firstly and then decreases. So there is a  k or  v to
make the ecological function maximum, just like shown in the figures. At this time, the characteristic
time is the optimal value with the control of ecological function. It is consistent with the previous
optimization results obtained by the calculation.
It can be seen from Fig.3 that the value of E is impacted with x. From Fig.4, it can been seen that,
with the same characteristic time, the greater the temperature gradient, the smaller the value of E. So
the temperature gradient should be considered to obtain best value of E.
Conclusions
Characteristic time is an important indicator to evaluate the thermoacoustic system performance. This
article uses ecological function as an indicator to analyze the relationship between the ecology target
value and the characteristic time, and obtains the characteristic time value in the maximum ecology
function value, then makes a numerical calculation between ecology target value and the
characteristic time, and learns that:(1) the ecological function value of E has a change process with
the characteristic time  k or  v that increases firstly and then decreases; (2) the value of E is
impacted with x, or decreases with the temperature gradient. It could be helpful for future research.
Acknowledgements
This paper is supported by the National Natural Science Fund, People’ Republic of China (Project
No. 51176143)
Fig.3.a Relationship between E and  k
Fig.3.b Relationship between E and  v
Fig.3 Relationship between value E and characteristic times（ dT0 dx  1K / m ）
Fig.4.a Relationship between E and  k
Fig.4.b Relationship between E and  v
Fig.4 Relationship between value E and characteristic times（ x   / 8 ）
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