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Thermoacoustic theory for cyclic flow regenerators. Part I: fundamentals
Thermoacoustic theory for cyclic flow regenerators. Part I: fundamentals J.H. Xiao Cryogenic Laboratory, Academia Sinica, PO Box 2711, Beijing 100080, Peoples' Republic of China
Received 19 February 1991; revised 19 February 1992 A thermoacoustic theory is presented for analysis of gas oscillations and time-averaged energy fluxes and energy transformations in cyclic flow regenerators. The fluctuations of pressure and velocity of the gaseous medium are transmitted along the longitudinal direction of cyclic flow regenerators in the form of acoustic wave motion. The working of cyclic flow regenerators relies on thermoacoustic effects, i.e., time-averaged energy effects caused by the thermal interaction of the oscillatory gaseous fluid medium and the porous solid-matrix medium. Regenerators used in regenerative heat engines are active components in which heat energy is transformed into acoustic energy to produce acoustic power (in prime movers) or acoustic energy is consumed and transformed into heat energy to pump heat (in refrigerators). The wave equations and energy-temperature equations to describe cyclic flow regenerators have been deduced, and a set of parameters to evaluate the regenerator performance and effectiveness is proposed. Keywords: regenerator; refrigerators; thermoacoustics Nomenclature Adiabatic sound velocity Cross-sectional area Heat transfer surface area per unit length AHT C Specific heat capacity COP Coefficient of performance E Second-order total energy flux Complex acoustic factor f g Real acoustic factor h Heat transfer coefficient Im Imaginary part of Imaginary unit J K Heat conductivity Pressure P a Second-order heat flux Re Real part of S Specific entropy of a gas t Time coordinate T Temperature U Volume velocity W Second-order work flux X Longitudinal coordinate Y Specific acoustic admittance Z Acoustic impedance a
r 5
A
Greek letters OL
,.¢
Flow viscous coefficient Thermal expansion coefficient Inertia additional mass coefficient Ratio of isobaric to isochoric specific heat
0011 - 2275/92/100895 - 07 © 1992 Butterworth - Heinemann
p
Ratio of actual to critical temperature gradient Penetration depth Efficiency Dynamic viscosity Density Power production or absorption rate Angular fluctuation frequency
Subscripts and superscripts b C C C cr f F H o q s s T w W x tz -
* '
Transverse Carnot's Flow capacitive Cold Critical Gas flow Flow inductive Hot Zero order Heat pump Solid matrix Entropy of gas Temperature Outer wall Work Thermal Viscous First-order fluctuation quantities Complex notation of fluctuation quantities Complex conjugate Per unit length
Ltd
Cryogenics 1992 Vol 32, No 10
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Thermoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao It is well known that the regenerators used in regenerative heat engines (refrigerators and prime movers) are of most importance to the performance and effectiveness of the engines 1.2. Conceptually, regenerators are very simple, being used for storing heat during one part of the cycle for re-use during another part. They usually comprise a finely divided metallic matrix (mostly wire-screen or spherical-bed matrices) which is subject to cyclic flows of working gas with a temperature gradient and near adiabatic outer walls. However, cyclic flow regenerator analysis is far from perfect, as the problem is considerably complicated. For instance, the flow and thermal processes in cyclic flow regenerators are mutually related, and an attempt to enhance the heat transfer performance would also cause a decrease in the flow performance. Another difficulty is that the flow process and thermal process are inhomogeneous (caused by the temperature gradient). All these factors make the problem hard to solve. Many theories have been proposed to analyse the thermal performance of cyclic flow regenerators 3-5. These theories usually make quasi-steady assumptions and do not consider the interaction between flow characteristics and thermal characteristics. They use steady flow heat transfer analysis methods to evaluate the heat transfer efficiency of cyclic flow regenerators, then by separately calculating various losses (pressure-drop loss, heat conduction loss, etc.), they determine the overall performance. It is known that these quasi-steady theories agree with experiments when the operation frequency is low, but the higher the frequency, the worse the theories perform. Experimental investigations have revealed that the flow process in cyclic flow regenerators has distinct wave features 6'7. The most significant evidence is: Pressure and velocity fluctuation are transmitted along the longitudinal direction of the regenerator in the form of wave propagation. There exists wave attenuation and the waves are influenced by the geometric arrangement, basic thermophysical parameters and the working conditions of the regenerators. 2 Resonance phenomena occur near a certain range of operating frequencies. At the range of resonant frequency, there is observed a noticeable rise in the amplitude of the oscillations compared with the amplitude at non-resonant frequencies. The wave motions in cyclic flow regenerators become important as the operating frequencies increase, and the well-established space averaged and quasi-steady state theories and experimental background are no longer valid. This calls for simultaneous consideration of flow dynamics, heat transfer and energy transformation. We have tried to deal with the problem by considering the mass, energy and momentum equations simultaneously and have proposed an analytical network model 8 to analyse the interrelated flow and thermal processes of cyclic flow regenerators based on linear wave theory and perturbation methods. We have developed a computational method and a simple personal computer program to calculate the flow and thermal characteristics of cyclic flow regenerators. Results
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Cryogenics 1992 Vol 32, No 10
show tha'~ this model reveals greater information about the working mechanism of cyclic flow regenerators and the proposed program is feasible for regenerator computation. However, the network model is far from perfect too. First, it is not a complete wave theory; we use mass flow rate rather than the velocity as a state variable to describe the motion of the gaseous fluid medium. Second, it does not consider the time averaged energy effects (for instance, work flux, heat flux etc.) and energy transformation in cyclic flow regenerators. It is well known that the oscillation of a compressible fluid is sound (acoustic oscillations). With this in mind, one can realize that regenerators rely on time-averaged energy effects caused by the thermal interaction of an oscillatory gaseous fluid medium and a solid medium, i.e., thermoacoustic effects. The most noticeable work on thermoacoustics in recent times has been carried out by Rott and Wheatley et al. Initially directed toward understanding Taconis oscillations, mainly in the 1970s, Rott and his coworkers established a sound theoretical foundation to describe how the acoustic energy and heat energy are mutually transformed in tubes where acoustic standing waves impose 9- ~. Based on Rott's theoretical work, in the 1980s, Wheatley and his co-workers developed the standing wave 'thermoacoustic engines', and extended Rott's theory to both thermoacoustic refrigerators and prime movers ~2-~4. This paper presents a thermoacoustic theory for regenerators with a porous solid matrix under cyclic flow working conditions. We will discuss wave propagation based on first-order (linear) wave theory and second-order time averaged energy effects (energy fluxes and energy transformations) in the regenerators. In Part II of the paper, we will present two case studies calculated by using the theoretical fundamentals described here.
Wave equations The basic assumptions for the theoretical analysis are: ! The problem is one-dimensional. 2 The solid materials (porous solid matrices and solid outer walls) are stationary and rigid. 3 The fluid medium is a simple compressible fluid, and mechanical energy and heat energy are the only two active work forms involved. 4 The regenerators are under periodically stationary working conditions, and the mean fluid velocity is zero.
5 The dimensions of the gaps in the porous matrix and the dimensions of porous solid matrix are smaller than the thermal penetration depth of the fluid 6,, 6, = (2Ko/poCpow) j/2, and the thermal penetration depth of the solid matrix material 6,, &s= ( 2 K J psoQo~o) ~/z, respectively, so we can use volume averaging to average the state variables to describe the flow and thermal characteristics of the regenerators. 6 The acoustic amplitudes are low enough to avoid turbulence, so that (po I O/Aflb,)/#o < 500 tS, b, = (2#0 ]19o60) I/2. 7 Let e = [O/Afl/27rao, which represents the ratio of particle displacement to the wavelength.
Therrnoacoustic theory for cyclic flow regenerators. Part I: J.H. Xiao Finally, it is assumed that e ,~ 1, so that we can assume complete linearization, with non-linear effects neglected. Using pressure, volume velocity, gas and solid temperature fluctuations as state variables, the basic equations for cyclic flow regenerators, under the above assumptions, can be written as t6
c]~ + P°a2°0(]_3/-- 1 hAHT(Ts- T) at
af
Ox
~o
Of+OdTo Ot
Ot
L/3o @ _
Ar dx
PoCpo Ot
C)7"s
hAHT
Ot
AsP~oC~o
(2)
0-/3, o(I
Po
Po Ox
Ot
Af
To3o
aO + A~ ap _
(I)
hA-,T (?s-- ¢3
(3)
0 dTo
( T - L)
(4)
jo~L-
f~u 1-@
jo,p
(5)
ft. (L - i3
To/3o 0oC~
+/3,) +
j¢opo (1 ZF-- ~
Otgo Ar
1 _ jwAf [ 1 + (3' - 1)fwT ] Zc poaZo (1 -
fwT
(10)
(b)fl H
j~oO(1 -- ~) + flH
Equations (9) are the longitudinal wave equations for cyclic flow regenerators, expressing the relationship between pressure and velocity fluctuations, from which acoustic attenuation and phase shift of pressure and velocity fluctuations can be predicted.
Afp o Cpo
Using complex notation for fluctuation quantities, Equations (3) and (4) can be rewritten as
jo~¢ 4 a~ dx
sional complex factor
= ~-
(?- L)
(6)
Thermoacoustic effects Thermoacoustic effects are time-averaged energy effects caused by interaction of the gas working medium and the solid working medium. The time-averaged energy effects concerned are: acoustic power (work flux and power production (absorption) rate), heat power (heat flux and heat absorption (production) rate), total energy flux etc. We simply write out equations for base waves (first harmonic). Higher order harmonics can be treated easily on the basis of the superposition principle.
Acoustic power The longitudinal acoustic work flux is
where • is the heat capacity ratio and flH is the characteristic heat exchange angular frequency ~b = AfpoCpo/(AfpoCpo +
AsP,oCso)
(7)
OH = hAHx/(AfPoCpo + AsP~oC~o)
(ll)
Wx = ½Re[Up*]
The transverse work flux is zero because the solid wall is rigid. The power production rate (when its value is negative, it represents the power absorption rate) per unit length along the direction of work flow is
Their solutions are joJcI,(1-~)+cI, fl. [To/3o
¢bfl.
?~ =j~(1
Z~-) +
[ To3o
fir,
LPoGo p
0
d 1 dTo E ~ v : ~ Wx---~ Re [Op*fw-r]flo -dx--
~]
1 ( 3 ' - l)Ar w l p l 2 g w K 1 Po 001012gwu(12) 2 poao 2 Af
o d o] jo~Ar dx
(8)
where _
Using complex notation for fluctuation quantities, Equations (1) and (2) with solution (8) can now be written as
gwK
dp
gwu -
Z~O
Ooao
(3" -- 1)Af60 Re
--- Re [JfwT ] (13)
Af
Poao
Re [ZF ] -
ot/zo
Poco
dx (9)
dU
1 /~ + fwT/3o dro 0
where ZF is the flow inductive impedance, Zc is the flow capacitive impedance and fWT is a non-dimen-
The first term in Equation (12) is caused by sound oscillations and temperature gradient; it can be positive or negative, depending upon the sound field and temperature distribution. The second and third terms are caused by finite heat capacity, finite heat transfer and flow resistance, respectively; they are always negative (absorbing acoustic power).
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Therrnoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao
Whether the acoustic power is produced or absorbed (the sign of ~;~v) depends upon the local temperature gradient (dTo/dx). There exists a power production critical temperature gradient (dTo/dx) w, such that: E~v = 0
(dTo~ w
(y
-~-]¢~ --
--
1 gwK + - 1)~o ( 3 ' - 1)
/3oao
I Y.
12gwo (14)
Re [ Y~WT]
We now can rewrite Equation (12) as 1 To/3oRe [ Op,fwr ]
1 dTo r w - 1 To dx Pw
(15)
where Pw = (dToldX)l(dToldX) w is the ratio of actual temperature gradient to the power production critical temperature gradient defined in Equation (14). Obviously, if the sign of
Ko
6~
151 ,gqx
(19)
where I O/AfoJI is the amplitude of the particle displacement in the cyclic flow regenerators. It is significant that in Equation (19) the thermal penetration depth of the gas appears in the denominator. Thus it is possible (and is usually the case) that the 'effective' coefficient of heat conductivity K~ can be orders of magnitude larger than the proper (molecular) conductivity Ko of the gas. The longitudinal conductive heat flux is: dTo
dTo
dTo
(20)
Q~ = - AfKo - - ~ - AsKso - - - A w l ( . o (ix dx
where the first and second terms are contributions due to gas and solid, respectively. Combining Equations (16) and (20), the total longitudinal heat flux can be written as
dTo To dx
I To(3oRe[Op,fwT]l
2
One can find
is positive (one can make this true by properly choosing the direction of the x-axis), for Pw > l, E~v > 0, acoustic power is produced; for Pw < 1, /~(v< 0, acoustic power is absorbed; and for Yw = l, I;(v = 0, work is neither produced nor absorbed.
Qx = - ~ To/3oRe[ OP*fqx ] - AfK~ dTo dx
(21)
where Ke is the total effective heat conductivity
Heat power
Heat flux is caused by the hydrodynamic transportation of entropy (carried by the oscillatory velocity) and heat conduction of gas and solid. The entropy transportation caused longitudinal heat flux is Q~ = 21 Re[poTog(J*] = 1 Re [poCpoI'O* - To/3o,O0*]
= _ _ 1 I 0 12 1 To/3oRe[~p,fq~ ] - 2 p o C p o - - g q x 2 ' Aro~
dTo dx (16)
where fqx =fw~
(17)
gqx = Re [jfwT ] The first term in Equation (16) is caused by sound oscillations; it can be positive or negative, depending upon the sound field. The second term is caused by irreversibility of heat exchange between the gas and solid media, and is proportional to the temperature gradient; its direction is always opposite to that of the temperature gradient, which means that it always flows from the hot end to the cold end. It is convenient to introduce an 'effective' coefficient of heat conduction by I 1012 K~ = ~ o,,Cpo A]w gqx
898
Cryogenics 1992 Vol 32, No 10
(18)
Ke = K~ + Ko + As/(so + A . Kwo Af Ar
(22)
The direction of longitudinal heat flux (the sign of Qx) is dependent upon local temperature gradient (dTo/dx) too. There exists a heat pump critical temperature gradient (dTo/dx)qr, such that Q~ = 0
¢'V ';'-'""1°12"°"°r--'',,,,1 ~-
cr =
~oao
Lr*
J
(23) Equation (21) can now be written as Q ~ = - ~ 1 To/3oRe[Op,fqx ]( 1 _ Fq)
(24)
where I'q = (dToldx)l(dToldx)q, is the ratio of actual temperature gradient to the heat pump critical temperature gradient defined in Equation (23). Obviously, for Pq > 1, Qx < 0, the direction of heat flux is opposite to that of the temperature gradient, i.e., heat flow from hot end to cold end; for Yq < 1, Qx > 0, the direction of heat flux is the same as that of the temperature gradient, i.e., heat flow from cold end to hot end, and there is a heat pumping effect; for rq ~---l, Qx = 0, the heat flux vanishes. The transverse heat flux is caused by heat exchange between the solid outer wall of the regenerator and its environment. It is reasonable to suppose that the transverse heat flux is proportional to the temperature
Thermoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao difference of the solid wall and its environment, so that the transverse heat flux per unit length reads
Q(, = hwU,~(To - T~)
(25)
where we choose the direction of heat flow from the regenerator wall to its environment as the positive direction. The acoustic energy absorbed by the working gas medium is transformed into heat energy, and vice versa, because no other energy form is active, and the total energy must be conserved. So the heat production rate (when its value is negative, it represents heat absorption rate) per unit length is E'q = - E"
(26)
Total energy The total energy flux of cyclic flow regenerators is caused by the hydrodynamic transportation of enthalpy (carried by the oscillatory velocity) and heat conduction of gas and solid. It is equal to the sum of acoustic energy and heat energy. The longitudinal total energy flux is 1
Ex = W~ + Q~ = ~ Re[Op*(1 - To/3ofqO] -A,Ke
dTo dx (27)
where the first term can be positive or negative, depending upon the sound field; the sign of the second term is always opposite to that of the temperature gradient. The transverse total energy flux is equal to the transverse heat flux. Since the total energy is conserved, it cannot be produced or absorbed. The increase of longitudinal total energy flux per unit length should be equal to the heat absorbed from the environment, so that dE~ _
.....
dx
Q~
(28)
Equations (27) and (28) express the relationship which the longitudinal total energy flux Ex and temperature distribution To should satisy. We can rewrite them as dT,, = t&Re[Op*(1 -To/3ofqx)] - Ex fix
dE~ dx
ArK e
- h,,, Uw(T~ - T,,)
their solutions, and other thermoacoustic relations, one can find out the performance and effectiveness of a cyclic flow regenerator. Using temperature gradient as the criterion, we find that the thermoacoustic effects in cyclic flow regenerators can be divided into the following three typical regions region I:
region II: region III:
\ - ~ / ] \ ~ }cr > 1 \--dx / l \ - ~ / " < 1 dT° q < cr
-<
\dx /c,
In region I (high temperature gradient ratio region), the actual temperature gradient is larger than the power production critical temperature gradient. The working media absorb heat energy (by decreasing the longitudinal heat flux) and transform it into acoustic power; part of this power is used to overcome the irreversible work dissipation, the remainder being used to amplify the longitudinal work flux. A net acoustic power is produced in regenerator; this is the case in a regenerative prime mover. In region II (low temperature gradient ratio region), the actual temperature gradient is smaller than the heat pump critical temperature gradient. The working media absorb some acoustic power (by decreasing the longitudinal work flux); part of this power is used to overcome the irreversible dissipation, the remainder being used to pump heat. A net heat flow from the cold end to the hot end is produced; this is the case in regenerative refrigerators (cryocoolers). In region III (middle temperature gradient ratio region), the actual temperature gradient is between the heat pump and power production critical temperature gradients. The working media transform acoustic power into heat, and heat flows from the hot end to the cold end. At the upper boundary, the actual temperature gradient is equal to the power production critical temperature gradient, the acoustic power produced is just used to overcome the irreversible work dissipation. At the lower boundary, the actual temperature gradient is equal to the heat pump critical temperature gradient. The heat flux produced by heat pumping effects is just used to overcome the heat flux from the hot end to the cold end caused by irreversible heat conduction. This case is not of use for regenerative heat engines.
(29)
We call Equations (29) the energy-temperature equations, from which the mean temperature and total energy flux fields can be predicted. The wave equations (9) and energy-temperature equations (29) form a close, complete thermoacoustic longitudinal equation group. Once we know the geometrical configurations, thermo-physical properties of the materials and longitudinal boundary conditions of a regenerator, the above equations can be solved. With
Regenerator performance and effectiveness From the point of view of energy transformation, regenerators in regenerative heat engines are active components in which heat energy is transformed into acoustic energy to produce acoustic power (in prime movers) or acoustic energy is transformed into heat energy to pump heat (in refrigerators). The following parameters are introduced to describe the performance and effectiveness of cyclic flow regenerators used in regenerative prime movers.
Cryogenics 1992 Vol 32, No 10
899
Thermoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao The acoustic power production rate r~,, which represents the acoustic power gain in a regenerator, is equal to the difference of output work flux and input work flux of the regenerator ~w = W ° U t - W~xn
(30)
2 The input heat flux Qx", which represents the heat energy input into the regenerator at the hot end. The regenerator transforms part of this heat flux into acoustic power. 3 The regenerator efficiency ~/, which is defined as the acoustic power gain in the regenerator divided by the heat energy input into the regenerator, i.e., the acoustic power production rate divided by the input heat flux at the hot end (31)
ri = r~./Q"x
4 The regenerator perfectness ~/p, which is defined as the regenerator efficiency divided by the Carnot's efficiency ~/c of a heat engine operating at the same temperature region
~lp
= r/ = [;, ~lc Q~x(T. - TD/T.
(32)
where the input heat multiplied by the Carnot's efficiency is equal to the maximum available work of the input heat, so the regenerator perfectness is equal to acoustic power gain divided by the maximum available work of the input heat flux. The following parameters are introduced to describe the performance and effectiveness of cyclic flow regenerators used in regenerative refrigerators. 1 The heat pumping rate Qx~, which represents the obtained heat flux into a regenerator at the cold end. 2 The acoustic power absorption rate r~,, which represents the acoustic energy consumed by the regenerator. The regenerator uses part of this acoustic energy to pump heat, and it is equal to the difference of input work flux and output work flux of the regenerator r., = W,x,- W~xu'
(33)
where the absorbed power multiplied by Carnot's coefficient of performance is equal to the maximum available heat pumping rate using the same amount of mechanical power, so the regenerator perfectness is equal to the obtained heat pumping rate divided by the maximum available heat pumping rate of the absorbed power. The above parameters for regenerator performance and effectiveness evaluation are very different from the 'traditional' ones. It is believed by the author that this set of parameters is more suitable for cyclic flow regenerators.
Conclusion A linear thermoacoustic theory has been developed and presented in this paper to analyse the gas oscillations and time averaged energy effects in cyclic flow regenerators. The working mechanism of cyclic flow regenerators relies on thermoacoustic effects, i.e., time averaged energy effects caused by the thermal interaction of the oscillatory gaseous fluid medium and the porous matrix solid medium. The fluctuations of pressure and velocity of the oscillatory gaseous medium are transmitted along the longitudinal direction of cyclic flow regenerators in the form of acoustic wave motion. There exist amplitude changes and phase shifts in the gas oscillations because the wave motion is influenced by the geometric arrangement, thermodynamic irreversibility, temperature gradient and operating conditions of regenerators. Regenerators in regenerative prime movers transform heat energy input at the hot end into acoustic energy to produce acoustic power, if the regenerator is working in the high temperature gradient ratio region, i.e., the actual temperature is larger than the power production critical temperature gradient, Regenerators in regenerative refrigerators consume acoustic energy and transform it into heat energy to pump heat from the cold end to the hot end, if the regenerator is working in low temperature gradient ratio region, i.e., the actual temperature gradient is smaller than the heat pump critical temperature gradient.
Acknowledgements
3 The regenerator coefficient of performance COP, which is defined as the obtained heat pumping effects divided by the acoustic power consumed by the regenerator, i.e., the heat pumping rate divided by the acoustic power absorption rate
This paper relies heavily on the author's PhD dissertation 16 under the supervision of Prof. C.S. Hong and Prof. F.Z. Guo.
(34)
1 Walker, G. Cryocoolers Plenum Press, New York, USA (1983) 2 Urieli, I. and Berchowitz, D.M. Stirling Cycle Engine Analysis Adam Hilger, Bristol, UK (1984) 3 Quale, E.B. and Smith, J.L. Jr. An approximate solution for the thermal performance of a Stirling engine regenerator Trans ASME A (1969) 91 109 4 Modest, M.F. and Tien, C.L. Thermal analysis of cyclic cryogenic regenerators lnt J Heat Mass Trans (1974) 17 37 5 Kohler, J.W.L., Stevens, P.F., de Jonge, A.K. and Beuzekom, D.C. Computation of regenerator used in regenerative refrigerators Cryogenics (1975) 15 521 6 Lee, S.Z. Experimental investigation of cyclic flow characteristics in wire-screen matrices Cryog Supercond (1984) 12 1 (in Chinese)
COP = Qc/F,,
The regenerator perfectness ~p, which is defined as the regenerator coefficient o f performance divided by the Carnot's coefficient of performance COPe of a refrigerator operating at the same temperature region r/p -
COP COPc
900
-
QC F,w T c / ( T . - Tc)
Cryogenics 1992 Vol 32, No 10
(35)
References
Therrnoacoustic theory for cyclic flow regenerators. Part I: J.H. Xiao 7 Guo, F.Z., Chou, Y.M., Lee, Z.S., Wang, Z.S. and Mao, W. Flow characteristics of cyclic flow regenerators Cryogenics (1987) 27 152 8 Xiao, J.H. and Guo, F.Z. Analytical network model on the flow and thermal characteristics of cyclic flow cryogenic regenerators Cryogenics (1988) 28 762 9 Rott, N. Damped and thermally driven acoustic oscillations in wide and narrow tubes ZAMP (1968) 20 230 10 Rott, N. Thermally driven acoustic oscillations, Part III: Second order heat flux ZAMP (1975) 26 43 11 Muller, U.A. and Rott, N. Thermally driven acoustic oscillations, Part VI: Excitation and power ZAMP (1983) 34 609
12 Wheatley, J.C., Holler, T., Swift, G.W. and Migliori, A. An intrinsically irreversible thermoacoustic heat engine J Acoust Soc Am (1983) 74(1) 153 13 Holler, T.J. Thermoacoustic refrigerator design and performance PhD dissertation Department of Physics, University of California at San Diego, USA (1986) 14 Swift, G.W. Thermoacoustic engines J Acoust Soc Am (1988) 84(4) 153 15 Merkli, P. and Thomann, H. Transition to turbulence in oscillating pipe flow J Fluid Mech (1975) 68 part 3 567 16 Xiao, J.H. Thermoacoustic effects and thermoacoustic theory for regenerative cryocoolers (heat engines) PhD dissertation Institute of Physics, Academia Sinica, China (1990) (in Chinese)
Cryogenics 1992 Vol 32, No 10
901
Received 19 February 1991; revised 19 February 1992 A thermoacoustic theory is presented for analysis of gas oscillations and time-averaged energy fluxes and energy transformations in cyclic flow regenerators. The fluctuations of pressure and velocity of the gaseous medium are transmitted along the longitudinal direction of cyclic flow regenerators in the form of acoustic wave motion. The working of cyclic flow regenerators relies on thermoacoustic effects, i.e., time-averaged energy effects caused by the thermal interaction of the oscillatory gaseous fluid medium and the porous solid-matrix medium. Regenerators used in regenerative heat engines are active components in which heat energy is transformed into acoustic energy to produce acoustic power (in prime movers) or acoustic energy is consumed and transformed into heat energy to pump heat (in refrigerators). The wave equations and energy-temperature equations to describe cyclic flow regenerators have been deduced, and a set of parameters to evaluate the regenerator performance and effectiveness is proposed. Keywords: regenerator; refrigerators; thermoacoustics Nomenclature Adiabatic sound velocity Cross-sectional area Heat transfer surface area per unit length AHT C Specific heat capacity COP Coefficient of performance E Second-order total energy flux Complex acoustic factor f g Real acoustic factor h Heat transfer coefficient Im Imaginary part of Imaginary unit J K Heat conductivity Pressure P a Second-order heat flux Re Real part of S Specific entropy of a gas t Time coordinate T Temperature U Volume velocity W Second-order work flux X Longitudinal coordinate Y Specific acoustic admittance Z Acoustic impedance a
r 5
A
Greek letters OL
,.¢
Flow viscous coefficient Thermal expansion coefficient Inertia additional mass coefficient Ratio of isobaric to isochoric specific heat
0011 - 2275/92/100895 - 07 © 1992 Butterworth - Heinemann
p
Ratio of actual to critical temperature gradient Penetration depth Efficiency Dynamic viscosity Density Power production or absorption rate Angular fluctuation frequency
Subscripts and superscripts b C C C cr f F H o q s s T w W x tz -
* '
Transverse Carnot's Flow capacitive Cold Critical Gas flow Flow inductive Hot Zero order Heat pump Solid matrix Entropy of gas Temperature Outer wall Work Thermal Viscous First-order fluctuation quantities Complex notation of fluctuation quantities Complex conjugate Per unit length
Ltd
Cryogenics 1992 Vol 32, No 10
895
Thermoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao It is well known that the regenerators used in regenerative heat engines (refrigerators and prime movers) are of most importance to the performance and effectiveness of the engines 1.2. Conceptually, regenerators are very simple, being used for storing heat during one part of the cycle for re-use during another part. They usually comprise a finely divided metallic matrix (mostly wire-screen or spherical-bed matrices) which is subject to cyclic flows of working gas with a temperature gradient and near adiabatic outer walls. However, cyclic flow regenerator analysis is far from perfect, as the problem is considerably complicated. For instance, the flow and thermal processes in cyclic flow regenerators are mutually related, and an attempt to enhance the heat transfer performance would also cause a decrease in the flow performance. Another difficulty is that the flow process and thermal process are inhomogeneous (caused by the temperature gradient). All these factors make the problem hard to solve. Many theories have been proposed to analyse the thermal performance of cyclic flow regenerators 3-5. These theories usually make quasi-steady assumptions and do not consider the interaction between flow characteristics and thermal characteristics. They use steady flow heat transfer analysis methods to evaluate the heat transfer efficiency of cyclic flow regenerators, then by separately calculating various losses (pressure-drop loss, heat conduction loss, etc.), they determine the overall performance. It is known that these quasi-steady theories agree with experiments when the operation frequency is low, but the higher the frequency, the worse the theories perform. Experimental investigations have revealed that the flow process in cyclic flow regenerators has distinct wave features 6'7. The most significant evidence is: Pressure and velocity fluctuation are transmitted along the longitudinal direction of the regenerator in the form of wave propagation. There exists wave attenuation and the waves are influenced by the geometric arrangement, basic thermophysical parameters and the working conditions of the regenerators. 2 Resonance phenomena occur near a certain range of operating frequencies. At the range of resonant frequency, there is observed a noticeable rise in the amplitude of the oscillations compared with the amplitude at non-resonant frequencies. The wave motions in cyclic flow regenerators become important as the operating frequencies increase, and the well-established space averaged and quasi-steady state theories and experimental background are no longer valid. This calls for simultaneous consideration of flow dynamics, heat transfer and energy transformation. We have tried to deal with the problem by considering the mass, energy and momentum equations simultaneously and have proposed an analytical network model 8 to analyse the interrelated flow and thermal processes of cyclic flow regenerators based on linear wave theory and perturbation methods. We have developed a computational method and a simple personal computer program to calculate the flow and thermal characteristics of cyclic flow regenerators. Results
896
Cryogenics 1992 Vol 32, No 10
show tha'~ this model reveals greater information about the working mechanism of cyclic flow regenerators and the proposed program is feasible for regenerator computation. However, the network model is far from perfect too. First, it is not a complete wave theory; we use mass flow rate rather than the velocity as a state variable to describe the motion of the gaseous fluid medium. Second, it does not consider the time averaged energy effects (for instance, work flux, heat flux etc.) and energy transformation in cyclic flow regenerators. It is well known that the oscillation of a compressible fluid is sound (acoustic oscillations). With this in mind, one can realize that regenerators rely on time-averaged energy effects caused by the thermal interaction of an oscillatory gaseous fluid medium and a solid medium, i.e., thermoacoustic effects. The most noticeable work on thermoacoustics in recent times has been carried out by Rott and Wheatley et al. Initially directed toward understanding Taconis oscillations, mainly in the 1970s, Rott and his coworkers established a sound theoretical foundation to describe how the acoustic energy and heat energy are mutually transformed in tubes where acoustic standing waves impose 9- ~. Based on Rott's theoretical work, in the 1980s, Wheatley and his co-workers developed the standing wave 'thermoacoustic engines', and extended Rott's theory to both thermoacoustic refrigerators and prime movers ~2-~4. This paper presents a thermoacoustic theory for regenerators with a porous solid matrix under cyclic flow working conditions. We will discuss wave propagation based on first-order (linear) wave theory and second-order time averaged energy effects (energy fluxes and energy transformations) in the regenerators. In Part II of the paper, we will present two case studies calculated by using the theoretical fundamentals described here.
Wave equations The basic assumptions for the theoretical analysis are: ! The problem is one-dimensional. 2 The solid materials (porous solid matrices and solid outer walls) are stationary and rigid. 3 The fluid medium is a simple compressible fluid, and mechanical energy and heat energy are the only two active work forms involved. 4 The regenerators are under periodically stationary working conditions, and the mean fluid velocity is zero.
5 The dimensions of the gaps in the porous matrix and the dimensions of porous solid matrix are smaller than the thermal penetration depth of the fluid 6,, 6, = (2Ko/poCpow) j/2, and the thermal penetration depth of the solid matrix material 6,, &s= ( 2 K J psoQo~o) ~/z, respectively, so we can use volume averaging to average the state variables to describe the flow and thermal characteristics of the regenerators. 6 The acoustic amplitudes are low enough to avoid turbulence, so that (po I O/Aflb,)/#o < 500 tS, b, = (2#0 ]19o60) I/2. 7 Let e = [O/Afl/27rao, which represents the ratio of particle displacement to the wavelength.
Therrnoacoustic theory for cyclic flow regenerators. Part I: J.H. Xiao Finally, it is assumed that e ,~ 1, so that we can assume complete linearization, with non-linear effects neglected. Using pressure, volume velocity, gas and solid temperature fluctuations as state variables, the basic equations for cyclic flow regenerators, under the above assumptions, can be written as t6
c]~ + P°a2°0(]_3/-- 1 hAHT(Ts- T) at
af
Ox
~o
Of+OdTo Ot
Ot
L/3o @ _
Ar dx
PoCpo Ot
C)7"s
hAHT
Ot
AsP~oC~o
(2)
0-/3, o(I
Po
Po Ox
Ot
Af
To3o
aO + A~ ap _
(I)
hA-,T (?s-- ¢3
(3)
0 dTo
( T - L)
(4)
jo~L-
f~u 1-@
jo,p
(5)
ft. (L - i3
To/3o 0oC~
+/3,) +
j¢opo (1 ZF-- ~
Otgo Ar
1 _ jwAf [ 1 + (3' - 1)fwT ] Zc poaZo (1 -
fwT
(10)
(b)fl H
j~oO(1 -- ~) + flH
Equations (9) are the longitudinal wave equations for cyclic flow regenerators, expressing the relationship between pressure and velocity fluctuations, from which acoustic attenuation and phase shift of pressure and velocity fluctuations can be predicted.
Afp o Cpo
Using complex notation for fluctuation quantities, Equations (3) and (4) can be rewritten as
jo~¢ 4 a~ dx
sional complex factor
= ~-
(?- L)
(6)
Thermoacoustic effects Thermoacoustic effects are time-averaged energy effects caused by interaction of the gas working medium and the solid working medium. The time-averaged energy effects concerned are: acoustic power (work flux and power production (absorption) rate), heat power (heat flux and heat absorption (production) rate), total energy flux etc. We simply write out equations for base waves (first harmonic). Higher order harmonics can be treated easily on the basis of the superposition principle.
Acoustic power The longitudinal acoustic work flux is
where • is the heat capacity ratio and flH is the characteristic heat exchange angular frequency ~b = AfpoCpo/(AfpoCpo +
AsP,oCso)
(7)
OH = hAHx/(AfPoCpo + AsP~oC~o)
(ll)
Wx = ½Re[Up*]
The transverse work flux is zero because the solid wall is rigid. The power production rate (when its value is negative, it represents the power absorption rate) per unit length along the direction of work flow is
Their solutions are joJcI,(1-~)+cI, fl. [To/3o
¢bfl.
?~ =j~(1
Z~-) +
[ To3o
fir,
LPoGo p
0
d 1 dTo E ~ v : ~ Wx---~ Re [Op*fw-r]flo -dx--
~]
1 ( 3 ' - l)Ar w l p l 2 g w K 1 Po 001012gwu(12) 2 poao 2 Af
o d o] jo~Ar dx
(8)
where _
Using complex notation for fluctuation quantities, Equations (1) and (2) with solution (8) can now be written as
gwK
dp
gwu -
Z~O
Ooao
(3" -- 1)Af60 Re
--- Re [JfwT ] (13)
Af
Poao
Re [ZF ] -
ot/zo
Poco
dx (9)
dU
1 /~ + fwT/3o dro 0
where ZF is the flow inductive impedance, Zc is the flow capacitive impedance and fWT is a non-dimen-
The first term in Equation (12) is caused by sound oscillations and temperature gradient; it can be positive or negative, depending upon the sound field and temperature distribution. The second and third terms are caused by finite heat capacity, finite heat transfer and flow resistance, respectively; they are always negative (absorbing acoustic power).
Cryogenics 1992 Vol 32, No 10
897
Therrnoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao
Whether the acoustic power is produced or absorbed (the sign of ~;~v) depends upon the local temperature gradient (dTo/dx). There exists a power production critical temperature gradient (dTo/dx) w, such that: E~v = 0
(dTo~ w
(y
-~-]¢~ --
--
1 gwK + - 1)~o ( 3 ' - 1)
/3oao
I Y.
12gwo (14)
Re [ Y~WT]
We now can rewrite Equation (12) as 1 To/3oRe [ Op,fwr ]
1 dTo r w - 1 To dx Pw
(15)
where Pw = (dToldX)l(dToldX) w is the ratio of actual temperature gradient to the power production critical temperature gradient defined in Equation (14). Obviously, if the sign of
Ko
6~
151 ,gqx
(19)
where I O/AfoJI is the amplitude of the particle displacement in the cyclic flow regenerators. It is significant that in Equation (19) the thermal penetration depth of the gas appears in the denominator. Thus it is possible (and is usually the case) that the 'effective' coefficient of heat conductivity K~ can be orders of magnitude larger than the proper (molecular) conductivity Ko of the gas. The longitudinal conductive heat flux is: dTo
dTo
dTo
(20)
Q~ = - AfKo - - ~ - AsKso - - - A w l ( . o (ix dx
where the first and second terms are contributions due to gas and solid, respectively. Combining Equations (16) and (20), the total longitudinal heat flux can be written as
dTo To dx
I To(3oRe[Op,fwT]l
2
One can find
is positive (one can make this true by properly choosing the direction of the x-axis), for Pw > l, E~v > 0, acoustic power is produced; for Pw < 1, /~(v< 0, acoustic power is absorbed; and for Yw = l, I;(v = 0, work is neither produced nor absorbed.
Qx = - ~ To/3oRe[ OP*fqx ] - AfK~ dTo dx
(21)
where Ke is the total effective heat conductivity
Heat power
Heat flux is caused by the hydrodynamic transportation of entropy (carried by the oscillatory velocity) and heat conduction of gas and solid. The entropy transportation caused longitudinal heat flux is Q~ = 21 Re[poTog(J*] = 1 Re [poCpoI'O* - To/3o,O0*]
= _ _ 1 I 0 12 1 To/3oRe[~p,fq~ ] - 2 p o C p o - - g q x 2 ' Aro~
dTo dx (16)
where fqx =fw~
(17)
gqx = Re [jfwT ] The first term in Equation (16) is caused by sound oscillations; it can be positive or negative, depending upon the sound field. The second term is caused by irreversibility of heat exchange between the gas and solid media, and is proportional to the temperature gradient; its direction is always opposite to that of the temperature gradient, which means that it always flows from the hot end to the cold end. It is convenient to introduce an 'effective' coefficient of heat conduction by I 1012 K~ = ~ o,,Cpo A]w gqx
898
Cryogenics 1992 Vol 32, No 10
(18)
Ke = K~ + Ko + As/(so + A . Kwo Af Ar
(22)
The direction of longitudinal heat flux (the sign of Qx) is dependent upon local temperature gradient (dTo/dx) too. There exists a heat pump critical temperature gradient (dTo/dx)qr, such that Q~ = 0
¢'V ';'-'""1°12"°"°r--'',,,,1 ~-
cr =
~oao
Lr*
J
(23) Equation (21) can now be written as Q ~ = - ~ 1 To/3oRe[Op,fqx ]( 1 _ Fq)
(24)
where I'q = (dToldx)l(dToldx)q, is the ratio of actual temperature gradient to the heat pump critical temperature gradient defined in Equation (23). Obviously, for Pq > 1, Qx < 0, the direction of heat flux is opposite to that of the temperature gradient, i.e., heat flow from hot end to cold end; for Yq < 1, Qx > 0, the direction of heat flux is the same as that of the temperature gradient, i.e., heat flow from cold end to hot end, and there is a heat pumping effect; for rq ~---l, Qx = 0, the heat flux vanishes. The transverse heat flux is caused by heat exchange between the solid outer wall of the regenerator and its environment. It is reasonable to suppose that the transverse heat flux is proportional to the temperature
Thermoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao difference of the solid wall and its environment, so that the transverse heat flux per unit length reads
Q(, = hwU,~(To - T~)
(25)
where we choose the direction of heat flow from the regenerator wall to its environment as the positive direction. The acoustic energy absorbed by the working gas medium is transformed into heat energy, and vice versa, because no other energy form is active, and the total energy must be conserved. So the heat production rate (when its value is negative, it represents heat absorption rate) per unit length is E'q = - E"
(26)
Total energy The total energy flux of cyclic flow regenerators is caused by the hydrodynamic transportation of enthalpy (carried by the oscillatory velocity) and heat conduction of gas and solid. It is equal to the sum of acoustic energy and heat energy. The longitudinal total energy flux is 1
Ex = W~ + Q~ = ~ Re[Op*(1 - To/3ofqO] -A,Ke
dTo dx (27)
where the first term can be positive or negative, depending upon the sound field; the sign of the second term is always opposite to that of the temperature gradient. The transverse total energy flux is equal to the transverse heat flux. Since the total energy is conserved, it cannot be produced or absorbed. The increase of longitudinal total energy flux per unit length should be equal to the heat absorbed from the environment, so that dE~ _
.....
dx
Q~
(28)
Equations (27) and (28) express the relationship which the longitudinal total energy flux Ex and temperature distribution To should satisy. We can rewrite them as dT,, = t&Re[Op*(1 -To/3ofqx)] - Ex fix
dE~ dx
ArK e
- h,,, Uw(T~ - T,,)
their solutions, and other thermoacoustic relations, one can find out the performance and effectiveness of a cyclic flow regenerator. Using temperature gradient as the criterion, we find that the thermoacoustic effects in cyclic flow regenerators can be divided into the following three typical regions region I:
region II: region III:
\ - ~ / ] \ ~ }cr > 1 \--dx / l \ - ~ / " < 1 dT° q < cr
-<
\dx /c,
In region I (high temperature gradient ratio region), the actual temperature gradient is larger than the power production critical temperature gradient. The working media absorb heat energy (by decreasing the longitudinal heat flux) and transform it into acoustic power; part of this power is used to overcome the irreversible work dissipation, the remainder being used to amplify the longitudinal work flux. A net acoustic power is produced in regenerator; this is the case in a regenerative prime mover. In region II (low temperature gradient ratio region), the actual temperature gradient is smaller than the heat pump critical temperature gradient. The working media absorb some acoustic power (by decreasing the longitudinal work flux); part of this power is used to overcome the irreversible dissipation, the remainder being used to pump heat. A net heat flow from the cold end to the hot end is produced; this is the case in regenerative refrigerators (cryocoolers). In region III (middle temperature gradient ratio region), the actual temperature gradient is between the heat pump and power production critical temperature gradients. The working media transform acoustic power into heat, and heat flows from the hot end to the cold end. At the upper boundary, the actual temperature gradient is equal to the power production critical temperature gradient, the acoustic power produced is just used to overcome the irreversible work dissipation. At the lower boundary, the actual temperature gradient is equal to the heat pump critical temperature gradient. The heat flux produced by heat pumping effects is just used to overcome the heat flux from the hot end to the cold end caused by irreversible heat conduction. This case is not of use for regenerative heat engines.
(29)
We call Equations (29) the energy-temperature equations, from which the mean temperature and total energy flux fields can be predicted. The wave equations (9) and energy-temperature equations (29) form a close, complete thermoacoustic longitudinal equation group. Once we know the geometrical configurations, thermo-physical properties of the materials and longitudinal boundary conditions of a regenerator, the above equations can be solved. With
Regenerator performance and effectiveness From the point of view of energy transformation, regenerators in regenerative heat engines are active components in which heat energy is transformed into acoustic energy to produce acoustic power (in prime movers) or acoustic energy is transformed into heat energy to pump heat (in refrigerators). The following parameters are introduced to describe the performance and effectiveness of cyclic flow regenerators used in regenerative prime movers.
Cryogenics 1992 Vol 32, No 10
899
Thermoacoustic theory for cyclic flow regenerators. Part h J.H. Xiao The acoustic power production rate r~,, which represents the acoustic power gain in a regenerator, is equal to the difference of output work flux and input work flux of the regenerator ~w = W ° U t - W~xn
(30)
2 The input heat flux Qx", which represents the heat energy input into the regenerator at the hot end. The regenerator transforms part of this heat flux into acoustic power. 3 The regenerator efficiency ~/, which is defined as the acoustic power gain in the regenerator divided by the heat energy input into the regenerator, i.e., the acoustic power production rate divided by the input heat flux at the hot end (31)
ri = r~./Q"x
4 The regenerator perfectness ~/p, which is defined as the regenerator efficiency divided by the Carnot's efficiency ~/c of a heat engine operating at the same temperature region
~lp
= r/ = [;, ~lc Q~x(T. - TD/T.
(32)
where the input heat multiplied by the Carnot's efficiency is equal to the maximum available work of the input heat, so the regenerator perfectness is equal to acoustic power gain divided by the maximum available work of the input heat flux. The following parameters are introduced to describe the performance and effectiveness of cyclic flow regenerators used in regenerative refrigerators. 1 The heat pumping rate Qx~, which represents the obtained heat flux into a regenerator at the cold end. 2 The acoustic power absorption rate r~,, which represents the acoustic energy consumed by the regenerator. The regenerator uses part of this acoustic energy to pump heat, and it is equal to the difference of input work flux and output work flux of the regenerator r., = W,x,- W~xu'
(33)
where the absorbed power multiplied by Carnot's coefficient of performance is equal to the maximum available heat pumping rate using the same amount of mechanical power, so the regenerator perfectness is equal to the obtained heat pumping rate divided by the maximum available heat pumping rate of the absorbed power. The above parameters for regenerator performance and effectiveness evaluation are very different from the 'traditional' ones. It is believed by the author that this set of parameters is more suitable for cyclic flow regenerators.
Conclusion A linear thermoacoustic theory has been developed and presented in this paper to analyse the gas oscillations and time averaged energy effects in cyclic flow regenerators. The working mechanism of cyclic flow regenerators relies on thermoacoustic effects, i.e., time averaged energy effects caused by the thermal interaction of the oscillatory gaseous fluid medium and the porous matrix solid medium. The fluctuations of pressure and velocity of the oscillatory gaseous medium are transmitted along the longitudinal direction of cyclic flow regenerators in the form of acoustic wave motion. There exist amplitude changes and phase shifts in the gas oscillations because the wave motion is influenced by the geometric arrangement, thermodynamic irreversibility, temperature gradient and operating conditions of regenerators. Regenerators in regenerative prime movers transform heat energy input at the hot end into acoustic energy to produce acoustic power, if the regenerator is working in the high temperature gradient ratio region, i.e., the actual temperature is larger than the power production critical temperature gradient, Regenerators in regenerative refrigerators consume acoustic energy and transform it into heat energy to pump heat from the cold end to the hot end, if the regenerator is working in low temperature gradient ratio region, i.e., the actual temperature gradient is smaller than the heat pump critical temperature gradient.
Acknowledgements
3 The regenerator coefficient of performance COP, which is defined as the obtained heat pumping effects divided by the acoustic power consumed by the regenerator, i.e., the heat pumping rate divided by the acoustic power absorption rate
This paper relies heavily on the author's PhD dissertation 16 under the supervision of Prof. C.S. Hong and Prof. F.Z. Guo.
(34)
1 Walker, G. Cryocoolers Plenum Press, New York, USA (1983) 2 Urieli, I. and Berchowitz, D.M. Stirling Cycle Engine Analysis Adam Hilger, Bristol, UK (1984) 3 Quale, E.B. and Smith, J.L. Jr. An approximate solution for the thermal performance of a Stirling engine regenerator Trans ASME A (1969) 91 109 4 Modest, M.F. and Tien, C.L. Thermal analysis of cyclic cryogenic regenerators lnt J Heat Mass Trans (1974) 17 37 5 Kohler, J.W.L., Stevens, P.F., de Jonge, A.K. and Beuzekom, D.C. Computation of regenerator used in regenerative refrigerators Cryogenics (1975) 15 521 6 Lee, S.Z. Experimental investigation of cyclic flow characteristics in wire-screen matrices Cryog Supercond (1984) 12 1 (in Chinese)
COP = Qc/F,,
The regenerator perfectness ~p, which is defined as the regenerator coefficient o f performance divided by the Carnot's coefficient of performance COPe of a refrigerator operating at the same temperature region r/p -
COP COPc
900
-
QC F,w T c / ( T . - Tc)
Cryogenics 1992 Vol 32, No 10
(35)
References
Therrnoacoustic theory for cyclic flow regenerators. Part I: J.H. Xiao 7 Guo, F.Z., Chou, Y.M., Lee, Z.S., Wang, Z.S. and Mao, W. Flow characteristics of cyclic flow regenerators Cryogenics (1987) 27 152 8 Xiao, J.H. and Guo, F.Z. Analytical network model on the flow and thermal characteristics of cyclic flow cryogenic regenerators Cryogenics (1988) 28 762 9 Rott, N. Damped and thermally driven acoustic oscillations in wide and narrow tubes ZAMP (1968) 20 230 10 Rott, N. Thermally driven acoustic oscillations, Part III: Second order heat flux ZAMP (1975) 26 43 11 Muller, U.A. and Rott, N. Thermally driven acoustic oscillations, Part VI: Excitation and power ZAMP (1983) 34 609
12 Wheatley, J.C., Holler, T., Swift, G.W. and Migliori, A. An intrinsically irreversible thermoacoustic heat engine J Acoust Soc Am (1983) 74(1) 153 13 Holler, T.J. Thermoacoustic refrigerator design and performance PhD dissertation Department of Physics, University of California at San Diego, USA (1986) 14 Swift, G.W. Thermoacoustic engines J Acoust Soc Am (1988) 84(4) 153 15 Merkli, P. and Thomann, H. Transition to turbulence in oscillating pipe flow J Fluid Mech (1975) 68 part 3 567 16 Xiao, J.H. Thermoacoustic effects and thermoacoustic theory for regenerative cryocoolers (heat engines) PhD dissertation Institute of Physics, Academia Sinica, China (1990) (in Chinese)
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