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nano heat engine
Physics Letters A 372 (2008) 1168–1173 www.elsevier.com/locate/pla
Performance analysis of a thermosize micro/nano heat engine Wenjie Nie, Jizhou He ∗ Department of Physics, Nanchang University, Nanchang 330047, People’s Republic of China Received 24 May 2007; received in revised form 13 September 2007; accepted 14 September 2007 Available online 26 September 2007 Communicated by C.R. Doering
Abstract In a recent paper [A. Sisman, I. Muller, Phys. Lett. A 320 (2004) 360] the thermodynamic properties of ideal gases confined in a narrow box were examined theoretically. The so-called “thermosize effects” similar to thermoelectric effects, such as Seebeck-like thermosize effect, Peltier-like thermosize effect and Thomson-like thermosize effect, were analyzed. Like the thermoelectric generator, based on the thermosize effects we have established a model of micro/nano scaled ideal gas heat engine cycle which includes two isothermal and two isobaric processes. The expressions of power output and efficiency of this cycle in the two cases of reversible and irreversible heat exchange are derived and the optimal performance characteristics of the heat engine is discussed by some numerical example. The results obtained here will provide theoretical guidance for the design of micro/nano scaled device. © 2007 Elsevier B.V. All rights reserved. PACS: 05.70.-a; 05.90.+m; 05.70.Ln Keywords: Thermosize effects; Thermoelectric effects; Thermodynamic cycle; Performance characteristics
1. Introduction In micro/nano mechanical systems, the mean thermal wave length of gas atoms may be comparable with the size of the system, and the Casimir-like size effect becomes important one. In recent years, the boundary effect on the ideal classical gases confined in a rectangular box or spherical and cylindrical geometries is studied theoretically in [1,2]. The dependence of the boundary effect on ideal quantum gases is analyzed in [3–5]. In general, the expressions of the thermodynamic quantities of the ideal gases in micro/nano mechanical systems are the appropriate conventional terms plus correction terms. These global thermodynamic quantities are non-additive and depend strong on the size of the box or containers. When atoms pass from one box to another one of different size through a permeable wall, the size difference creates “thermosize effects” similar to thermoelectric effects [1]. With the rapid development of the nanotechnology it makes us possibly to design the micro/nano scaled devices [6–8]. Therefore, it is necessary to study further the thermosize effects (Seebeck-like thermosize effect, Peltier-like thermosize effect and Thomson-like thermosize effect) and design a micro/nano scaled ideal gas device based on these effects. In this Letter, similar to the thermoelectric generator [9–12], we establish the thermodynamic cycle model of a thermosize micro/nano heat engine in which is composed of two isothermal and two isobaric processes. The power output and efficiency of the heat engine are calculated in the two cases of reversible and irreversible heat exchange. Furthermore, the influence of Fourier’s heat flow on the performance of the cycle is analyzed. Some performance characteristic curves are plotted by numerical example and the results obtained here will provide theoretical guidance for the design of micro/nano scaled device.
* Corresponding author.
E-mail address: [email protected] (J. He). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.09.041
W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
Fig. 1. Schematic diagram of a thermosize micro/nano heat engine.
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Fig. 2. The T –p diagram of a thermosize micro/nano heat engine.
2. A micro/nano scaled heat engine model Consider a rectangular box with the micro/nano dimensions of Lx , Ly and Lz . The box is divided into two different scale portions by an adiabatic wall, respectively, called narrow (Lnx ) and wide (Lw x ). The walls located in the bottom and top of the boxes are permeable for the atoms of the gas and the width of permeable walls is δ. The ideal gases are filled in this box. In particular, the permeable wall is a wall, which contains many small holes without disturbing the stationary de Broglie wave patterns in each box. Schematic diagram of a micro/nano scaled ideal gas heat engine is shown in Figs. 1–2, which is composed of the two isothermal and two isobaric processes. This cycle is a microscopic analog of the Ericsson heat engine cycle. TH and TL are the temperatures of the hot reservoir and the cold reservoir, respectively. T1 and T2 are the temperatures of the gas fluid in two isothermal processes, respectively. And there is a relation, TH > T1 > T2 > TL . In the two isobaric processes, a regenerator is often used to improve the performance of the heat engine cycle. According to the Seebeck-like thermosize effect in micro/nano mechanical system [1], when the upper permeable wall is close but the lower permeable wall is open, the boxes can exchange atoms through the lower permeable wall. Then the top of the narrow and wide boxes have different chemical potential. Because of the different chemical potentials, there is a “potential for a gas fluid” across the wall and a gas fluid would occur. This chemical potential difference is called “thermosize potential” and the expression is given by T1 Vth-size = T2
kb √ π
1 1 − w n Lx Lx
Lc (T ) dT , Lc (T2 ) − 2
(1)
√ where Lc (T ) = h/ 8mkb T is one half of the most probable de Broglie wave length of the particles at temperature T , h is Planck’s constant, m is the atomic mass and kb is Boltzmann’s constant. Furthermore, when the upper and the lower permeable wall are open, a gas fluid would occur continuously. The mechanical system may be regarded as a micro/nano scaled ideal gas heat engine based on the thermosize effects. It is the analogues of thermoelectric generator based on thermoelectric effect [9–12]. Now assume that a steady gas fluid N˙ passes through the narrow and wide rectangular boxes. When the gas fluid passes through the upper permeable wall from state a to state b and maintains the constant temperature T1 , the gas flow will absorb heat quantity from the hot reservoir, kb Lc (T1 ) 1 1 ˙ ˙ − w , Q1P = NT1 √ Lnx Lx 2 π
(2)
which is called as Peltier-like thermosize effect. Similarly, when the gas fluid passes through the lower permeable wall from state c to state d and maintains the constant temperature T2 , the gas fluid will reject heat quantity to the cold reservoir, kb Lc (T2 ) 1 1 ˙ ˙ Q2P = NT2 − w . √ Lnx Lx 2 π
(3)
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W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
When the gas fluid passes along the wide box from state b to state c and maintains the constant pressure Pw , the gas fluid will reject heat quantity to the regenerator, Q˙ T w = N˙
T2 T1
kb Lc (T ) 1 1 1 5 + + kb + dT , √ 2 Lw Ly Lz 4 π x
(4)
which is called Thomson-like thermosize effects. Similarly, when the gas fluid passes along the narrow box from state d to state a and maintains the constant pressure Pn , the gas fluid will absorb heat quantity from the regenerator, Q˙ T n = N˙
T2 T1
kb Lc (T ) 1 1 1 5 + + kb + dT . √ 2 Lnx Ly Lz 4 π
(5)
˙ T w and Q ˙ T n are positive. Except Peltier-like heat flow Q ˙ P and Thomson-like heat flow Q ˙ T in ˙ 1P , Q˙ 2P , Q All heat quantities Q the cycle, there is additional Fourier’s heat flow between the upper and lower of the box, and it is proportional to the temperature gradient Q˙ F = κ(T1 − T2 ),
(6)
where κ is the thermal conductance of the ideal gas fluid. 3. The power output and the efficiency 3.1. The case of reversible heat exchange When the heat exchanges between the gas fluid and the two heat reservoirs are reversible, this means T1 = TH and T2 = TL . Using Eqs. (2)–(6), one can obtain the power output, 1 1 ˙ ˙ ˙ ˙ ˙ − w ( TH − TL ), P = Q1P − Q2P + QT n − QT w = 2C0 N (7) n Lx Lx kb h is a constant and is only dependent of the property of ideal gas fluid. From Eqs. (4) and (5), one can find that where C0 = 4√2πmk b the net amount of heat exchange between the gas fluid and the regenerator during the two isobaric processes is determined by
˙Tn − Q ˙ T w. Q˙ T = Q
(8)
˙ T w flowing into the regenerator in one regenerative (isobaric) process is smaller than that Due to Q˙ T > 0, the amount of heat Q ˙ of QT n flowing from the regenerator in the other regenerative (isobaric) process. The inadequate heat in the regenerator per cycle must be compensated from the hot reservoir in a timely manner. This will result in the increase of the amount of heat absorbed from the hot reservoir. If not, the temperature of the regenerator would be changed such that the regenerator would not operate normally. Thus, the net amounts of the heat absorbed from the hot reservoir and rejected to the cold reservoir are obtained, respectively, 1 1 ˙ ˙ ˙ ˙ ˙ − w (2 TH − TL ) + κ(TH − TL ), QH = Q1P + QF + QT = C0 N (9) n Lx Lx and ˙ 2P + Q ˙ F = C0 N˙ Q˙ L = Q
1 1 TL + κ(TH − TL ). − w Lnx Lx
(10)
Define a length ratio (λ) λ=
Lnx , Lw x
(11)
where 0 < λ < 1. The dimension Lx of the rectangular box is assumed to be a constant, Lnx + Lw x = Lx = const.
(12)
Thus, the power output, the amounts of the heat absorbed from the hot reservoir and rejected to the cold reservoir can be rewritten as P = 2Z( TH − TL ), (13)
W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
and
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˙ H = Z(2 TH − TL ) + κ(TH − TL ), Q
(14)
˙ L = Z TL + κ(TH − TL ), Q
(15)
˙ where Z = CL0xN λ is dependent of the gas fluid N˙ and the length ratio λ for given ideal gas fluid. It is an important parameter of a thermosize micro/nano heat engine cycle and its unit is W/K 1/2 . Using Eqs. (13) and (14), one can obtain the efficiency 1−λ2
η=
1−τ P √ √ = < ηc , √ √ ˙ 1 QH 1 + 2 τ (1 − τ ) + κ 2ZTL (1 − τ )( 1+√τ τ )
(16)
where τ = TL /TH is the temperature ratio of the hot and cold reservoirs, ηc = 1 − τ is the efficiency of Carnot heat engine. It is easily found from Eqs. (13) and (16) that the power output and efficiency are all the monotonically increasing function of Z. When Fourier’s heat flow can be neglected comparing with the Peltier-like and the Thomson-like heat flows, i.e. κ = 0, the efficiency is simplified as √ 1− τ . η0 = (17) √ 1 − τ /2 3.2. The case of finite-rate heat exchange When the heat exchanges between the gas fluid and the two heat reservoirs are irreversible and governed by Newtonian linear ˙ H ) from the hot reservoir to the gas fluid and the heat flow (Q˙ L ) from the gas fluid to the cold heat-transfer law, the heat flow (Q reservoir should be ˙ H = α(TH − T1 ) = Z(2 T1 − T2 ) + κ(T1 − T2 ), Q (18) and
˙ L = β(T2 − TL ) = Z T2 + κ(T1 − T2 ), Q
(19)
respectively, where α is the thermal conductance between the hot reservoir and the warm gas fluid, β is the thermal conductance between the cold reservoir and the cold gas fluid. The power output and the efficiency are ˙ L = 2Z( T1 − T2 ), ˙H −Q P =Q (20) and η=
P = Q˙ H
1+
1 2
T2 T1 (1 −
1 − T2 /T1 T1 T2
)+
√ κ T1 T2 T2 (1 − )(1 + 2Z T1 T1
,
(21)
)
respectively. By the numerical computation method, T1 and T2 can be solved from Eqs. (18) and (19). Substituting T1 and T2 into Eqs. (20) and (21), one can obtain the expressions of the power output and the efficiency as a function of α, β, TH , TL , κ and Z. When the Fourier’s heat flow can be neglected, i.e. κ = 0, Eqs. (18) and (19) can be simplified as
2
Z Z Z Z TL Z T1 = + TL + + + − , (22) α α 2β 2β τ α and T2 =
Z 2β
2 + TL +
Z . 2β
(23)
Substituting Eqs. (22) and (23) into Eqs. (20) and (21), one can obtain the detail expressions of the power output and efficiency,
2 2
Z Z Z 2β + α Z Z TL Z P = 2Z (24) + TL + + TL − + + − , α α 2β 2β τ 2β α 2β and η=1− 2 Zα ( Zα +
Z 2β
(Z/2β)2 + TL + Z/2β , TL Z 2 Z 2 Z 4β+α + ( 2β ) + TL ) + τ − ( 2β ) + TL − α ( 2β )
(25)
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W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
√ respectively. Define the dimensionless heat conductivity K (K = κ TL /Z), for given TL = 290 K, α = β = 1 × 10−12 W/K and τ = 0.9, the characteristic curves between the dimensionless power output and efficiency and the reversion of the dimensionless heat conductivity 1/K at different thermal conductance κ > 0 can be plotted by numerical computation method, as shown in Figs. 3–4, where (1/K)P and (1/K)η are the corresponding values of 1/K at the maximum power output and the maximum efficiency respectively. It is clearly seen from Figs. 3–4 that the maximum power output Pmax , the corresponding (1/K)P , the maximum efficiency ηmax and the corresponding (1/K)η increase as κ decreases. Thus, in order to obtain the power output and efficiency as large as possible, one must decrease thermal conductance κ. When κ = 0, by Eqs. (24) and (25) and the numerical calculation we can attain that the power output is not the monotonically increasing function of Z and exists maximum values (see Fig. 5), but the efficiency is the monotonically decreasing function of Z. The characteristic curves of the dimensionless power output versus efficiency are plotted for κ > 0 and κ = 0 at given TL = 290 K, τ = 0.9, α = β = 1 × 10−12 W/K, as shown in Fig. 5. When the Fourier’s heat flow is not neglected, i.e. κ > 0, the characteristic curve of the dimensionless power output versus thermal efficiency is a loop line passing the zero point. On such a curve, there are two maximum points. One is the maximum power output (Pmax ) and the corresponding efficiency (ηP ); the other is the maximum efficiency (ηmax ) and the corresponding power output (Pη ). But when the Fourier’s heat flow can be neglected, i.e. κ = 0, the characteristic curve of the dimensionless power output versus efficiency is not a loop line. There exists the maximum
Fig. 3. The dimensionless power output P /Pmax versus 1/K curves for different thermal conductance at given TL = 290 K, α = β = 1 × 10−12 W/K and τ = 0.9.
Fig. 4. The thermal efficiency η versus 1/K curves for different thermal conductance at given TL = 290 K, α = β = 1 × 10−12 W/K and τ = 0.9.
Fig. 5. The dimensionless power output P /Pmax versus thermal efficiency η curves for different thermal conductance at given TL = 290 K, τ = 0.9, α = β = 1 × 10−12 W/K.
W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
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Table 1 Optimal values of main performance parameters for different thermal conductance κ (×10−14 ) (W/K)
Pmax (×10−14 ) (W)
ηP
(1/K)P
ηmax
Pη (×10−14 ) (W)
(1/K)η
(1/K)max
0.00 3.00 5.00 8.00
41.87 39.03 38.11 34.53
0.051 0.046 0.041 0.038
– 0.893 0.538 0.323
0.098 0.062 0.054 0.046
0.00 23.84 27.92 28.21
– 0.327 0.252 0.179
– 1.785 1.089 0.649
power output (Pmax ) and corresponding efficiency (ηP ). These performance characteristic curves in a thermosize micro/nano heat engine are similar to those in thermoelectric generator [10]. Optimal values of main performance parameters for different thermal conductance are listed in Table 1. For the cases of κ > 0, it is found that the maximum power output and the corresponding (1/K)P , and the maximum efficiency and the corresponding (1/K)η increase as κ decreases. Moreover, the maximum value (1/K)max of the dimensionless parameter 1/K also increases as κ decreases. Especially, the maximum value (1/K)max for both the power output and the efficiency is same. 4. Conclusions In the present Letter, a thermosize micro/nano heat engine model is established based on the thermosize effects. It is analyzed that the thermosize micro/nano heat engine composed of two isothermal processes and two isobaric processes can operate normally and the heat engine can product the power output. Moreover, the expressions of the power output and efficiency were derived in two cases of reversible and irreversible heat exchange. Some optimal performance characteristic curves are plotted and the optimal values of main performance parameters are obtained by the numerical calculation. The results obtained in here are useful and provide the basis for designing a micro/nano scaled heat exchange device. Acknowledgements This work was supported by National Natural Science Foundation (Nos. 10465003, 10765004) and Science and Technology Foundation of Jiangxi Education Bureau, People’s Republic of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Sisman, I. Muller, Phys. Lett. A 320 (2004) 360. A. Sisman, J. Phys. A: Math. Gen. 37 (2004) 11353. H. Pang, W.S. Dai, M. Xie, J. Phys. A: Math. Gen. 39 (2006) 2563. R.M. Ziff, G.E. Uhlenbeck, M. Kac, Phys. Rep. C 32 (1977) 169. W.S. Dai, M. Xie, Phys. Lett. A 311 (2003) 340. M. Reith, Nano-Engineering in Science and Technology, World Scientific, New Jersey, 2003. G. Timp, Nanotechnology, Springer, New York, 1999. J.M. Saleh, Fluid Flow Handbook, McGraw–Hill, New York, 2002. J. Markku, R. Lampinen, J. Appl. Phys. 69 (1991) 4318. J.M. Gordon, Am. J. Phys. 59 (1991) 6. J. Chen, Z. Yan, J. Appl. Phys. 79 (1996) 11. D.C. Agrawal, V.J. Menon, J. Phys. D: Appl. Phys. 30 (1997) 357.
Performance analysis of a thermosize micro/nano heat engine Wenjie Nie, Jizhou He ∗ Department of Physics, Nanchang University, Nanchang 330047, People’s Republic of China Received 24 May 2007; received in revised form 13 September 2007; accepted 14 September 2007 Available online 26 September 2007 Communicated by C.R. Doering
Abstract In a recent paper [A. Sisman, I. Muller, Phys. Lett. A 320 (2004) 360] the thermodynamic properties of ideal gases confined in a narrow box were examined theoretically. The so-called “thermosize effects” similar to thermoelectric effects, such as Seebeck-like thermosize effect, Peltier-like thermosize effect and Thomson-like thermosize effect, were analyzed. Like the thermoelectric generator, based on the thermosize effects we have established a model of micro/nano scaled ideal gas heat engine cycle which includes two isothermal and two isobaric processes. The expressions of power output and efficiency of this cycle in the two cases of reversible and irreversible heat exchange are derived and the optimal performance characteristics of the heat engine is discussed by some numerical example. The results obtained here will provide theoretical guidance for the design of micro/nano scaled device. © 2007 Elsevier B.V. All rights reserved. PACS: 05.70.-a; 05.90.+m; 05.70.Ln Keywords: Thermosize effects; Thermoelectric effects; Thermodynamic cycle; Performance characteristics
1. Introduction In micro/nano mechanical systems, the mean thermal wave length of gas atoms may be comparable with the size of the system, and the Casimir-like size effect becomes important one. In recent years, the boundary effect on the ideal classical gases confined in a rectangular box or spherical and cylindrical geometries is studied theoretically in [1,2]. The dependence of the boundary effect on ideal quantum gases is analyzed in [3–5]. In general, the expressions of the thermodynamic quantities of the ideal gases in micro/nano mechanical systems are the appropriate conventional terms plus correction terms. These global thermodynamic quantities are non-additive and depend strong on the size of the box or containers. When atoms pass from one box to another one of different size through a permeable wall, the size difference creates “thermosize effects” similar to thermoelectric effects [1]. With the rapid development of the nanotechnology it makes us possibly to design the micro/nano scaled devices [6–8]. Therefore, it is necessary to study further the thermosize effects (Seebeck-like thermosize effect, Peltier-like thermosize effect and Thomson-like thermosize effect) and design a micro/nano scaled ideal gas device based on these effects. In this Letter, similar to the thermoelectric generator [9–12], we establish the thermodynamic cycle model of a thermosize micro/nano heat engine in which is composed of two isothermal and two isobaric processes. The power output and efficiency of the heat engine are calculated in the two cases of reversible and irreversible heat exchange. Furthermore, the influence of Fourier’s heat flow on the performance of the cycle is analyzed. Some performance characteristic curves are plotted by numerical example and the results obtained here will provide theoretical guidance for the design of micro/nano scaled device.
* Corresponding author.
E-mail address: [email protected] (J. He). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.09.041
W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
Fig. 1. Schematic diagram of a thermosize micro/nano heat engine.
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Fig. 2. The T –p diagram of a thermosize micro/nano heat engine.
2. A micro/nano scaled heat engine model Consider a rectangular box with the micro/nano dimensions of Lx , Ly and Lz . The box is divided into two different scale portions by an adiabatic wall, respectively, called narrow (Lnx ) and wide (Lw x ). The walls located in the bottom and top of the boxes are permeable for the atoms of the gas and the width of permeable walls is δ. The ideal gases are filled in this box. In particular, the permeable wall is a wall, which contains many small holes without disturbing the stationary de Broglie wave patterns in each box. Schematic diagram of a micro/nano scaled ideal gas heat engine is shown in Figs. 1–2, which is composed of the two isothermal and two isobaric processes. This cycle is a microscopic analog of the Ericsson heat engine cycle. TH and TL are the temperatures of the hot reservoir and the cold reservoir, respectively. T1 and T2 are the temperatures of the gas fluid in two isothermal processes, respectively. And there is a relation, TH > T1 > T2 > TL . In the two isobaric processes, a regenerator is often used to improve the performance of the heat engine cycle. According to the Seebeck-like thermosize effect in micro/nano mechanical system [1], when the upper permeable wall is close but the lower permeable wall is open, the boxes can exchange atoms through the lower permeable wall. Then the top of the narrow and wide boxes have different chemical potential. Because of the different chemical potentials, there is a “potential for a gas fluid” across the wall and a gas fluid would occur. This chemical potential difference is called “thermosize potential” and the expression is given by T1 Vth-size = T2
kb √ π
1 1 − w n Lx Lx
Lc (T ) dT , Lc (T2 ) − 2
(1)
√ where Lc (T ) = h/ 8mkb T is one half of the most probable de Broglie wave length of the particles at temperature T , h is Planck’s constant, m is the atomic mass and kb is Boltzmann’s constant. Furthermore, when the upper and the lower permeable wall are open, a gas fluid would occur continuously. The mechanical system may be regarded as a micro/nano scaled ideal gas heat engine based on the thermosize effects. It is the analogues of thermoelectric generator based on thermoelectric effect [9–12]. Now assume that a steady gas fluid N˙ passes through the narrow and wide rectangular boxes. When the gas fluid passes through the upper permeable wall from state a to state b and maintains the constant temperature T1 , the gas flow will absorb heat quantity from the hot reservoir, kb Lc (T1 ) 1 1 ˙ ˙ − w , Q1P = NT1 √ Lnx Lx 2 π
(2)
which is called as Peltier-like thermosize effect. Similarly, when the gas fluid passes through the lower permeable wall from state c to state d and maintains the constant temperature T2 , the gas fluid will reject heat quantity to the cold reservoir, kb Lc (T2 ) 1 1 ˙ ˙ Q2P = NT2 − w . √ Lnx Lx 2 π
(3)
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W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
When the gas fluid passes along the wide box from state b to state c and maintains the constant pressure Pw , the gas fluid will reject heat quantity to the regenerator, Q˙ T w = N˙
T2 T1
kb Lc (T ) 1 1 1 5 + + kb + dT , √ 2 Lw Ly Lz 4 π x
(4)
which is called Thomson-like thermosize effects. Similarly, when the gas fluid passes along the narrow box from state d to state a and maintains the constant pressure Pn , the gas fluid will absorb heat quantity from the regenerator, Q˙ T n = N˙
T2 T1
kb Lc (T ) 1 1 1 5 + + kb + dT . √ 2 Lnx Ly Lz 4 π
(5)
˙ T w and Q ˙ T n are positive. Except Peltier-like heat flow Q ˙ P and Thomson-like heat flow Q ˙ T in ˙ 1P , Q˙ 2P , Q All heat quantities Q the cycle, there is additional Fourier’s heat flow between the upper and lower of the box, and it is proportional to the temperature gradient Q˙ F = κ(T1 − T2 ),
(6)
where κ is the thermal conductance of the ideal gas fluid. 3. The power output and the efficiency 3.1. The case of reversible heat exchange When the heat exchanges between the gas fluid and the two heat reservoirs are reversible, this means T1 = TH and T2 = TL . Using Eqs. (2)–(6), one can obtain the power output, 1 1 ˙ ˙ ˙ ˙ ˙ − w ( TH − TL ), P = Q1P − Q2P + QT n − QT w = 2C0 N (7) n Lx Lx kb h is a constant and is only dependent of the property of ideal gas fluid. From Eqs. (4) and (5), one can find that where C0 = 4√2πmk b the net amount of heat exchange between the gas fluid and the regenerator during the two isobaric processes is determined by
˙Tn − Q ˙ T w. Q˙ T = Q
(8)
˙ T w flowing into the regenerator in one regenerative (isobaric) process is smaller than that Due to Q˙ T > 0, the amount of heat Q ˙ of QT n flowing from the regenerator in the other regenerative (isobaric) process. The inadequate heat in the regenerator per cycle must be compensated from the hot reservoir in a timely manner. This will result in the increase of the amount of heat absorbed from the hot reservoir. If not, the temperature of the regenerator would be changed such that the regenerator would not operate normally. Thus, the net amounts of the heat absorbed from the hot reservoir and rejected to the cold reservoir are obtained, respectively, 1 1 ˙ ˙ ˙ ˙ ˙ − w (2 TH − TL ) + κ(TH − TL ), QH = Q1P + QF + QT = C0 N (9) n Lx Lx and ˙ 2P + Q ˙ F = C0 N˙ Q˙ L = Q
1 1 TL + κ(TH − TL ). − w Lnx Lx
(10)
Define a length ratio (λ) λ=
Lnx , Lw x
(11)
where 0 < λ < 1. The dimension Lx of the rectangular box is assumed to be a constant, Lnx + Lw x = Lx = const.
(12)
Thus, the power output, the amounts of the heat absorbed from the hot reservoir and rejected to the cold reservoir can be rewritten as P = 2Z( TH − TL ), (13)
W. Nie, J. He / Physics Letters A 372 (2008) 1168–1173
and
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˙ H = Z(2 TH − TL ) + κ(TH − TL ), Q
(14)
˙ L = Z TL + κ(TH − TL ), Q
(15)
˙ where Z = CL0xN λ is dependent of the gas fluid N˙ and the length ratio λ for given ideal gas fluid. It is an important parameter of a thermosize micro/nano heat engine cycle and its unit is W/K 1/2 . Using Eqs. (13) and (14), one can obtain the efficiency 1−λ2
η=
1−τ P √ √ = < ηc , √ √ ˙ 1 QH 1 + 2 τ (1 − τ ) + κ 2ZTL (1 − τ )( 1+√τ τ )
(16)
where τ = TL /TH is the temperature ratio of the hot and cold reservoirs, ηc = 1 − τ is the efficiency of Carnot heat engine. It is easily found from Eqs. (13) and (16) that the power output and efficiency are all the monotonically increasing function of Z. When Fourier’s heat flow can be neglected comparing with the Peltier-like and the Thomson-like heat flows, i.e. κ = 0, the efficiency is simplified as √ 1− τ . η0 = (17) √ 1 − τ /2 3.2. The case of finite-rate heat exchange When the heat exchanges between the gas fluid and the two heat reservoirs are irreversible and governed by Newtonian linear ˙ H ) from the hot reservoir to the gas fluid and the heat flow (Q˙ L ) from the gas fluid to the cold heat-transfer law, the heat flow (Q reservoir should be ˙ H = α(TH − T1 ) = Z(2 T1 − T2 ) + κ(T1 − T2 ), Q (18) and
˙ L = β(T2 − TL ) = Z T2 + κ(T1 − T2 ), Q
(19)
respectively, where α is the thermal conductance between the hot reservoir and the warm gas fluid, β is the thermal conductance between the cold reservoir and the cold gas fluid. The power output and the efficiency are ˙ L = 2Z( T1 − T2 ), ˙H −Q P =Q (20) and η=
P = Q˙ H
1+
1 2
T2 T1 (1 −
1 − T2 /T1 T1 T2
)+
√ κ T1 T2 T2 (1 − )(1 + 2Z T1 T1
,
(21)
)
respectively. By the numerical computation method, T1 and T2 can be solved from Eqs. (18) and (19). Substituting T1 and T2 into Eqs. (20) and (21), one can obtain the expressions of the power output and the efficiency as a function of α, β, TH , TL , κ and Z. When the Fourier’s heat flow can be neglected, i.e. κ = 0, Eqs. (18) and (19) can be simplified as
2
Z Z Z Z TL Z T1 = + TL + + + − , (22) α α 2β 2β τ α and T2 =
Z 2β
2 + TL +
Z . 2β
(23)
Substituting Eqs. (22) and (23) into Eqs. (20) and (21), one can obtain the detail expressions of the power output and efficiency,
2 2
Z Z Z 2β + α Z Z TL Z P = 2Z (24) + TL + + TL − + + − , α α 2β 2β τ 2β α 2β and η=1− 2 Zα ( Zα +
Z 2β
(Z/2β)2 + TL + Z/2β , TL Z 2 Z 2 Z 4β+α + ( 2β ) + TL ) + τ − ( 2β ) + TL − α ( 2β )
(25)
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√ respectively. Define the dimensionless heat conductivity K (K = κ TL /Z), for given TL = 290 K, α = β = 1 × 10−12 W/K and τ = 0.9, the characteristic curves between the dimensionless power output and efficiency and the reversion of the dimensionless heat conductivity 1/K at different thermal conductance κ > 0 can be plotted by numerical computation method, as shown in Figs. 3–4, where (1/K)P and (1/K)η are the corresponding values of 1/K at the maximum power output and the maximum efficiency respectively. It is clearly seen from Figs. 3–4 that the maximum power output Pmax , the corresponding (1/K)P , the maximum efficiency ηmax and the corresponding (1/K)η increase as κ decreases. Thus, in order to obtain the power output and efficiency as large as possible, one must decrease thermal conductance κ. When κ = 0, by Eqs. (24) and (25) and the numerical calculation we can attain that the power output is not the monotonically increasing function of Z and exists maximum values (see Fig. 5), but the efficiency is the monotonically decreasing function of Z. The characteristic curves of the dimensionless power output versus efficiency are plotted for κ > 0 and κ = 0 at given TL = 290 K, τ = 0.9, α = β = 1 × 10−12 W/K, as shown in Fig. 5. When the Fourier’s heat flow is not neglected, i.e. κ > 0, the characteristic curve of the dimensionless power output versus thermal efficiency is a loop line passing the zero point. On such a curve, there are two maximum points. One is the maximum power output (Pmax ) and the corresponding efficiency (ηP ); the other is the maximum efficiency (ηmax ) and the corresponding power output (Pη ). But when the Fourier’s heat flow can be neglected, i.e. κ = 0, the characteristic curve of the dimensionless power output versus efficiency is not a loop line. There exists the maximum
Fig. 3. The dimensionless power output P /Pmax versus 1/K curves for different thermal conductance at given TL = 290 K, α = β = 1 × 10−12 W/K and τ = 0.9.
Fig. 4. The thermal efficiency η versus 1/K curves for different thermal conductance at given TL = 290 K, α = β = 1 × 10−12 W/K and τ = 0.9.
Fig. 5. The dimensionless power output P /Pmax versus thermal efficiency η curves for different thermal conductance at given TL = 290 K, τ = 0.9, α = β = 1 × 10−12 W/K.
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Table 1 Optimal values of main performance parameters for different thermal conductance κ (×10−14 ) (W/K)
Pmax (×10−14 ) (W)
ηP
(1/K)P
ηmax
Pη (×10−14 ) (W)
(1/K)η
(1/K)max
0.00 3.00 5.00 8.00
41.87 39.03 38.11 34.53
0.051 0.046 0.041 0.038
– 0.893 0.538 0.323
0.098 0.062 0.054 0.046
0.00 23.84 27.92 28.21
– 0.327 0.252 0.179
– 1.785 1.089 0.649
power output (Pmax ) and corresponding efficiency (ηP ). These performance characteristic curves in a thermosize micro/nano heat engine are similar to those in thermoelectric generator [10]. Optimal values of main performance parameters for different thermal conductance are listed in Table 1. For the cases of κ > 0, it is found that the maximum power output and the corresponding (1/K)P , and the maximum efficiency and the corresponding (1/K)η increase as κ decreases. Moreover, the maximum value (1/K)max of the dimensionless parameter 1/K also increases as κ decreases. Especially, the maximum value (1/K)max for both the power output and the efficiency is same. 4. Conclusions In the present Letter, a thermosize micro/nano heat engine model is established based on the thermosize effects. It is analyzed that the thermosize micro/nano heat engine composed of two isothermal processes and two isobaric processes can operate normally and the heat engine can product the power output. Moreover, the expressions of the power output and efficiency were derived in two cases of reversible and irreversible heat exchange. Some optimal performance characteristic curves are plotted and the optimal values of main performance parameters are obtained by the numerical calculation. The results obtained in here are useful and provide the basis for designing a micro/nano scaled heat exchange device. Acknowledgements This work was supported by National Natural Science Foundation (Nos. 10465003, 10765004) and Science and Technology Foundation of Jiangxi Education Bureau, People’s Republic of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Sisman, I. Muller, Phys. Lett. A 320 (2004) 360. A. Sisman, J. Phys. A: Math. Gen. 37 (2004) 11353. H. Pang, W.S. Dai, M. Xie, J. Phys. A: Math. Gen. 39 (2006) 2563. R.M. Ziff, G.E. Uhlenbeck, M. Kac, Phys. Rep. C 32 (1977) 169. W.S. Dai, M. Xie, Phys. Lett. A 311 (2003) 340. M. Reith, Nano-Engineering in Science and Technology, World Scientific, New Jersey, 2003. G. Timp, Nanotechnology, Springer, New York, 1999. J.M. Saleh, Fluid Flow Handbook, McGraw–Hill, New York, 2002. J. Markku, R. Lampinen, J. Appl. Phys. 69 (1991) 4318. J.M. Gordon, Am. J. Phys. 59 (1991) 6. J. Chen, Z. Yan, J. Appl. Phys. 79 (1996) 11. D.C. Agrawal, V.J. Menon, J. Phys. D: Appl. Phys. 30 (1997) 357.