Optimization of a thermoacoustic engine with a complex heat transfer exponent. Entropy 2003

Abstract: Heat transfer between a thermoacoustic engine and its surrounding heat reservoirs can be out of phase with oscillating working gas temperature. The paper presents a generalized heat transfer model using a complex heat transfer exponent. Both the real part and the imaginary part of the heat transfer exponent change the power versus efficiency relationship quantitatively. When the real part of the heat transfer exponent is fixed, the power output P decreases and the efficiency increases along with increasing of the imaginary part. The Optimization zone on the performance of the thermoacoustic heat engine is obtained. The results obtained will be helpful for the further understanding and the selection of the optimal operating mode of the thermoacoustic heat engine. Keywords: Thermoacoustic engine, Complex heat transfer exponent, Optimization zone __________________________________________________________________________________ Introduction A thermoacoustic engine (prime mover and refrigerator) [1-4] is of the advantages of high reliability, low noise, simple construction, non-parts of motion, non-pollution, ability to self-start etc. It can utilize a wide variety of energy resources: solar, geothermal, industrial waste heat and marsh gas. It has very important significance for environmental protection and moderating the tense petroleum needs in the world. With this great potential, the thermoacoustic engine has captivated many engineers


Introduction
A thermoacoustic engine (prime mover and refrigerator) [1][2][3][4] is of the advantages of high reliability, low noise, simple construction, non-parts of motion, non-pollution, ability to self-start etc.It can utilize a wide variety of energy resources: solar, geothermal, industrial waste heat and marsh gas.It has very important significance for environmental protection and moderating the tense petroleum needs in the world.With this great potential, the thermoacoustic engine has captivated many engineers in the power and cryogenic engineering.
The power and efficiency be very important characteristics for energy conversion devices.Novikov [5], Chambadal [6], and Curzon and Ahlborn [7] extended the Carnot cycle by taking account of the irreversibility of finite-time heat transfer.Their work which is commonly referred to as finite-time thermodynamics (FTT) or entropy generation minimization, has been undertaken by many researchers.Some authors have assessed the effect of the heat transfer law on the performance of endoreversible [8,9] and irreversible [10,11]] heat engines and coolers.In these works, the heat transfer exponent is assumed to be real.
Modern heat transfer models account for many details of fluid motion, interaction between temperature field and acoustic field, and system geometry.For a thermoacoustec engine (or refrigerator), the interactions between the entropy wave and the oscillating flow produce a rich variety of thermoacoustic phenomena such as self-excited gas oscillation [12].A longitudinal pressure oscillating in the sound channel induces a temperature oscillating in time at angular frequency .In the circumstances the gas temperature is complex.The method of complex temperature often used in periodic conduction problems.It results in a heat transfer with complex exponent.ω A complex Nusselt number to heat transfer in a cylinder have been studied by Kornhauser [13] by using Newton's linear law of convection with complex temperature.In general, heat transfer is not necessarily linear.This paper will use a generalized heat transfer law , where n is complex, to find the power versus efficiency characteristics of the thermoacoustic heat engine.The effect of heat leakage, internal irreversibility is considered in this paper.

Thermoacoustic engine model
Essentially, a thermoacoustic engine consists of heat exchangers, thermoacoustic stack (or thermoacoustic regenerator) and working fluid gas.The energy flows in the apparatus is schematically illustrated in Fig. 1.Heat was supplied to the engine by hot heat exchanger, and waste heat was removed by cold heat exchanger, delivering acoustic power (W ) to the outside of system.The following assumptions are made for this model.(i)There exist external irreversibilities due to heat transfer in the hot and the cold heat exchangers between the engine and its surrounding heat reservoirs.Because of the heat transfer, the time average temperatures (T H0 and T L0 ) of the working fluid are different from the reservoir temperatures (T H and T L ).The second law of thermodynamics requires T H > T H0 > T L0 > T L .As a result of thermoacoustic oscillating, the temperatures (T Hc and T Lc ) of the working fluid can be expressed as complexes: (2) where T 1 and T 2 are the first-order acoustic quantities, 1 − = i , is the oscillating angular frequency.Here the reservoir temperatures (T ω H and T L ) are assumed as real constants, so they have no imaginary part.
(ii) Consider that the heat transfer between the engine and its surroundings follow a generalized law ( ), Then ) where is a complex heat transfer exponent, is the overall heat transfer coefficient and  3) and ( 4) can be rewritten as where and Q are the time average of and , respectively.
The heat-transfer surface areas (F 1 and F 2 ) of the hot and cold heat exchangers are finite.The total heat transfer surface area (F) of the two heat exchangers is assumed to be a constant 2 1 (iii)There exists a rate of heat leakage ( ) from the heat source to the heat sink [14].It is assumed as a constant [10].Thus where is rate of heat transfer supplied by the heat source and is rate of heat transfer released to the heat sink.
In addition to heat resistance between the working substance and the heat reservoirs, heat leakage between the heat reservoirs, there are internal irreversibilities in the system due to miscellaneous factors such as friction and non-equilibrium activities inside the engine.The power output produced by the irreversible thermoacoustic engine is less than that of the endoreversible thermoacoustic engine with the same heat input.In other words, the rate of heat flow ( Q ) from the cold working fluid to the heat sink for the irreversible thermoacoustic engine is larger than that for the endoreversible thermoacoustic engine ( ).The factor of internal irreversible degree to characterize the additional internal miscellaneous irreversibility effect is defined as follows

Optimal characteristics
For an endoreversible thermoacoustic engine, the second law of thermodynamics requires that Combining equations ( 10) and ( 11) The first law of thermodynamics gives that the power output of the engine is The efficiency of the engine is We define the heat transfer surface area ratio and working fluid temperature ratio as follows From equations ( 5)-( 9) and ( 12)-( 16), we obtain the complex power output ( ) and the complex efficiency ( P′ η′ ) of the engine The real part of the power output of the engine is { } Here and indicate the real part and imaginary part of complex number.The real part of the efficiency of the engine is Eqs. ( 19) and ( 20 both power output and efficiency P η approach optimal values { } Figure 4 gives the operation zones of the thermoacoustic heat engine for special case n 1 = 1 and n 2 =0.1.Both cases n 1 = -1 and n 1 = 4 are similar to case n 1 = 1 in the respect of the optimal zone.The curve OABO in Fig. 4 is the optimal region of the theoretical analysis given by equations ( 22) and ( 23).It is of practical signification to apply the theoretical results in this paper to the performance optimization and evaluation for the thermoacoustic heat engine.The performance as a whole for a practical thermoacoustic heat engine is the best when operation point lays on curve AB and is better when it lays in the zone and is worse when it lays in the zone in

Conclusion
For a thermoacoustic heat engine with temperature oscillating, an ordinary convective heat transfer model is incapable of predicting its thermal relaxation effect.The heat transfer exponent mast be complex number due to the thermal relaxation in heat transfer process.In summary, both the real part and the imaginary part of the heat transfer exponent change the power versus efficiency relationship quantitatively.The analysis includes the optimal performance characteristics and the optimal operation zone of the thermoacoustic heat engine with complex heat transfer exponent.The results are helpful for the selection of the optimal mode of operation of the thermoacoustic heat engine.
) indicate that both power output P and efficiency η of the engine are functions of the heat transfer surface area ratio (f) for given T H , T L , , , n α β 1 , n 2 , , and x.Taking the derivatives of and q & φ P η with respect to f and setting them equal zero (

Discussion 1 .in figure 3 .
The parameter equation defined by equations (22) and (23) gives the fundamental relation between the optimal power output and efficiency P η for given T H , T L , , , n α β 1 , n 2 , , , and F. It is the main result of this paper.q& φ Effect of Complex Heat Transfer Exponent From equations (22) and (23), the optimal power output versus efficiency P η characteristics are dependent on heat transfer exponent n for given .When the imaginary part n is fixed, The power output P decreases and the efficiency zone The P versus η characteristics of irreversible thermoacoustic heat engine is a loop-shaped curve, as shown by figure2and figure3.For all n 1 m

figure 4 .
Figure4gives the operation zones of the thermoacoustic heat engine for special case n 1 = 1 and n 2 =0.1.Both cases n 1 = -1 and n 1 = 4 are similar to case n 1 = 1 in the respect of the optimal zone.The curve OABO in Fig.4is the optimal region of the theoretical analysis given by equations (22) and (23).It is of practical signification to apply the theoretical results in this paper to the performance optimization and evaluation for the thermoacoustic heat engine.The performance as a whole for a practical thermoacoustic heat engine is the best when operation point lays on curve AB and is better when it lays in the zone and is worse when it lays in the zone in figure4.The zones , , are unstable regions because them is outside curve OABO and the zone don't exist because or max P P >