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Local stability analysis of an irreversible heat engine working in the maximum power output and the maximum efficiency
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Applied Thermal Engineering 28 (2008) 699–706 www.elsevier.com/locate/apthermeng
Local stability analysis of an irreversible heat engine working in the maximum power output and the maximum efficiency Wenjie Nie, Jizhou He *, Bei Yang, Xiaoxia Qian Department of Physics, Nanchang University, Nanchang 330047, PR China Received 1 February 2007; accepted 12 June 2007 Available online 23 June 2007
Abstract In this paper the local stability of an irreversible heat engine working in the maximum power output and the maximum efficiency is analyzed. At two optimal steady-states of the maximum power output and the maximum efficiency the expressions of the relaxation time of an irreversible heat engine are derived. It is found that the relaxation time is a function of the heat-transfer coefficient a and b, internal irreversibility factor /, heat leak rate q, heat capacity C, the total heat-transfer surface area F, and temperatures of the heat reservoirs TH and TL. The influence of heat resistance, internal irreversibility and heat leak rate on the relaxation time is discussed and some new results are gained. The local stability of the optimal steady-state of the maximum power output is better than that of the maximum efficiency. The results obtained here are more general and useful for an irreversible heat engine than an endoreversible heat engine. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Local stability analysis; Irreversible heat engine; Performance characteristics
1. Introduction Based on the finite-time thermodynamics (FTT) theory, many different endoreversible models of heat engines were established and more realistic limits for the performance of real engine models were obtained [1–10]. However, real heat engines are usually complex devices with both internal and external irreversibilities. Besides the irreversibility of finite-rate heat-transfer, there are also other sources of irreversibilities, such as heat leaks, dissipative processes inside the working fluid, and so on. In order to assess the effect of finite-rate heat-transfer, together with other major irreversibilities on the performance of heat engines, some new irreversible models of heat engines were established and the optimal performance characteristics were analyzed at the maximum power output, the maximum efficiency and the maximum ecological function [11–18].
*
Corresponding author. E-mail address: [email protected] (J. He).
1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.06.002
Most of the studies of heat engines have focused on their steady-state energetic properties. But the real heat engines are always in no-steady-state and there exists intrinsic cycle variability in the operation of the cycle, for example, incomplete combustion of fuel and other causes. Thus, it is necessary to analyze the effect of noisy perturbations on the stability of system’s steady-state. Recently, a sort of new design principles named as the local stability of energy-converting system have been put forward. Santillan et al studied the local stability of an endoreversible Curzon–Ahlborn–Novikov engine working in a maximum power output [19]. Guzman-vargas et al. analyzed the effect of heat-transfer laws and thermal conductance on the local stability of an endoreversible heat engine [20]. Pa´ezHerna´ndez et al. made the dynamic effects of the time delays and the effect of internal irreversibility on the local stability of non-endoreversible heat engine [21]. Following their works and considering the finite-rate heat-transfer, internal irreversibility and heat leak, we analyze the local stability of an irreversible heat engine working in the maximum power output and the maximum efficiency. Some general and useful results are obtained.
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W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
Nomenclature C F F1 F2 f J1 J2 J1 J2 m mP mg P P QH QL q TH TL t
heat capacity, W K1 total surface area of heat exchangers, m2 surface area of the hot-side heat exchanger, m2 surface area of the cold-side heat exchanger, m2 heat-transfer surface area ratio no-steady-state heat flows from warm working fluid to engine, W no-steady-state heat flows from engine to cold working fluid, W steady-state heat flows from warm working fluid to engine, W steady-state heat flows from engine to cold working fluid, W temperature ratio of the working fluid optimal temperature ratio of the working fluid at the maximum power output optimal temperature ratio of the working fluid at the maximum efficiency power output, W steady-state power output, W heat-transfer supplied by the hot reservoir, W heat-transfer released to the cold reservoir, W heat leak rate, W temperature of the hot reservoir, K temperature of the cold reservoir, K relaxation time, s
x y x y
no-steady-state temperature of the warm working fluid, K no-steady-state temperature of the cold working fluid, K steady-state temperature of the warm working fluid, K steady-state temperature of the cold working fluid, K
Greek letters a heat-transfer coefficient of the hot-side heat exchanger, W K1 b heat-transfer coefficient of the cold-side heat exchanger, W K1 / internal irreversibility factor g efficiency g0 efficiency of the irreversible engine without heat leak g0 steady-state efficiency of the irreversible engine without heat leak k complex number (the eigenvalue) s temperature ratio of heat reservoirs DS1 entropy difference in high-temperature isothermal process, J K1 DS2 entropy difference in low-temperature isothermal process, J K1
2. Performance characteristics of an irreversible heat engine
and heat leak rate (q) between the two reservoirs are assumed to be a constant
Consider a steady flow irreversible heat engine, including the irreversibilities of finite-rate heat-transfer between the working fluid and heat reservoirs, heat leak between the two reservoirs and internal irreversibility of the working fluid, shown in Fig. 1. The working fluid is alternately connected to a hot reservoir at constant temperature TH and to a cold reservoir at constant temperature TL, and its temperatures are x and y, respectively. Based on a linear phenomenological heat-transfer law of irreversible thermodynamics, the heat flow (QH) from the hot reservoir to the warm working fluid through the hot-side heat exchanger and the heat flow (QL) from the cold working fluid to the cold reservoir through the cold-side heat exchanger are
F1 þ F2 ¼ F
QH ¼ aF 1 ð1=x 1=T H Þ
ð1Þ
and QL ¼ bF 2 ð1=T L 1=yÞ;
and
q ¼ const:
ð3Þ
Besides the irreversibilities of the finite-rate heat-transfer and heat leak, there are other irreversibilities inside the heat engine, such as the friction, turbulence and non-equilibTH QH C, x
J1
P
q J2
ð2Þ
where a and b are the overall heat-transfer coefficients at the hot-and cold-side heat exchangers and they are all positive, F1 and F2 are the surface areas of the hot-side and cold-side heat exchangers, respectively. Both the total heat-transfer surface area (F) of the two heat exchangers
C, y
QL
TL Fig. 1. Schematic diagram of an irreversible Carnot heat engine.
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
rium activities. Thus, all the processes of the cycle are irreversible. The entropy of the working fluid in the two adiabatic processes increases. If the entropy differences of the working fluid in the two isothermal processes at temperatures x and y are, respectively, denoted by DS1 and DS2, then an internal irreversible parameter / is introduced to characterize the other irreversibilities [14–18], / ¼ DS 2 =DS 1 :
ð4Þ
According to the second law of thermodynamics and Eq. (4), for an irreversible heat engine, it requires QL =y /ðQH =xÞ ¼ 0:
ð5Þ
When there are no other irreversibilities except the irreversibility of the finite-rate heat-transfer, i.e. / = 1, the engine is an endoreversible one; When / > 1, the engine is a internal irreversible one. The parameter / depends on the property of the working fluid, friction, rate of the cycle operation. By the first law of thermodynamics, the power output of an irreversible heat engine is P ¼ QH QL :
ð6Þ
The efficiency is g ¼ ðQH QL Þ=ðQH þ qÞ:
ð7Þ
F 1 =F 2 ¼ m
P ¼ Bðm /Þð1=T L m=T H Þ
ðm /Þð1=T L m=T H Þ ; ð15Þ mð1=T L m=T H Þ þ q=B 2 pffiffiffiffiffiffiffiffiffiffiffi where B ¼ aF = m þ /a=b . Using Eqs. (14) and (15) and eliminating the temperature ratio of the working fluid, the optimal relation between the power output and efficiency may be obtained. Therefore, the power output versus efficiency is plotted for different values of the temperature ratio of heat reservoirs s = TL/TH at given TL = 300 K, / = 1.2, q = 150 W, b/a = 1 and aF = 4000 KW/ K, as shown in Fig. 2. The characteristic curve of P g is a loop shape passing the zero point and it is clearly shown that there are two optimal steady-state points of the engine. One is a maximum power output Pmax and the corresponding efficiency gP; the other is a maximum efficiency gmax and the corresponding power output Pg. Further, by using Eqs. (14) and (15) and the extremal conditions dP/dm = 0 and dg/dm = 0, the optimal temperature ratios of the working fluid at the maximum power output and maximum efficiency are obtained, respectively,
g¼
g0 ¼ ðQH QL Þ=QH ;
and
mg ¼
QH ¼
A þ 2/ þ As/ 1 þ /s þ 2As
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2/s 2a/ þ 2aA2 sÞ 4ðaA2 þ / þ 2aA/ þ a/A2 sÞð/s2 a 2aAs a/sÞ 2ða þ 2aAs þ a/s /s2 Þ
i.e. efficiency of the inner engine which is shown by dotted line in Fig. 1 is g0. According to Eqs. (4)–(6), the heat flows can be expressed as xP x /y
ð9Þ
ð10Þ
J1 ¼
Define the heat-transfer surface area ratio (f) and the temperature ratio of the working fluid (m)
and
f ¼ F 1 =F 2
J2 ¼
ð11Þ
ð16Þ
ð17Þ
pffiffiffiffiffiffiffiffiffiffiffi where A ¼ /a=b, a = qTL/aF and s = TL/TH. Consider the internal irreversible Carnot heat engine operates between temperatures x and y , where x and y represent the steady-state temperatures of working fluid. The irreversibility hypothesis means that the cycle is internally irreversible. Thus, the heat flows can be given by
and /yP QL ¼ : x /y
ð14Þ
and
mP ¼
2a/ 2aA2 s 2/s þ
ð13Þ
The power output and efficiency may be expressed, respectively,
The efficiency of the engine without heat leak may be expressed as ð8Þ
pffiffiffiffiffiffiffiffiffiffiffi b=/a:
701
x P x /y
ð18Þ
/y P; x /y
ð19Þ
and m ¼ x=y:
ð12Þ
By using Eqs. (1)–(7) and (11), (12) and the extremal conditions dP/df = 0 and dg/df = 0, the optimal surface area ratio is obtained [17,18]
where J 1 and J 2 are the steady-state heat flows from x to engine and from engine to y , respectively. P is the steadystate power output. The variables with overbars denote steady-state values. When the irreversible Carnot heat engine operates in a steady-state, this means that the heat
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W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
3. Local stability analysis of an irreversible heat engine
1.0
0.8
P/Pmax,φ=1,Ci=0
τ=0.4 τ=0.5 τ=0.6
1 2 3
When the heat engine works out of the steady-state, reservoirs x and y are not real heat reservoirs. It is assumed that the temperatures x and y correspond to macroscopic objects with finite heat capacity C. The differential equations for x and y are given by
Pmax 0.6
1
Pη
2
dx=dt ¼ ½aF 1 ð1=x 1=T H Þ J 1 =C
ð29Þ
0.4
and
3
dy=dt ¼ ½J 2 bF 2 ð1=T L 1=yÞ=C;
0.2
0.0 0.0
0.1
η
0.2
ηP
0.3
ηmax
0.4
Fig. 2. Plot of power output versus efficiency for different values of s = TL/TH at given TL = 300 K, b/a = 1, / = 1.2, q = 150 W and aF = 4000 KW/K.
where J1 and J2 are the heat flows from x to the engine and from the engine to y, respectively. By Eqs. (18) and (19) J1 and J2 are given by x J1 ¼ P ðx; yÞ ð31Þ x /y and J2 ¼
flow (QH) from TH to x equals to J 1 and the heat flow (QL) from y to TL equals to J 2 , i.e. J 1 ¼ aF 1 ð1=x 1=T H Þ
ð20Þ
and J 2 ¼ bF 2 ð1=T L 1=y Þ:
ð21Þ
According to Eqs. (6) and (9), the efficiency of an internal irreversible Carnot heat engine without heat leak is g0 ¼ P =J 1 ¼ 1 /y =x ¼ 1 /=m;
ð22Þ
By using Eqs. (13) and (18)–(22), it is possible to express the steady-state temperatures x and y as x ¼ T H
ðA þ mÞs 1 þ As
ð23Þ
A=m þ 1 : 1 þ As
ð24Þ
and y ¼ T L
ð30Þ
/y P ðx; yÞ: x /y
ð32Þ
It is worthy to note that the no-steady-state power output is always a function of temperatures of working fluid. Substitution of Eqs. (31) and (32) into Eqs. (29) and (30) leads to dx=dt ¼ ½aF 1 ð1=x 1=T H Þ xP ðx; yÞ=ðx /yÞ=C
ð33Þ
and dy=dt ¼ ½/yP ðx; yÞ=ðx /yÞ bF 2 ð1=T L 1=yÞ=C:
ð34Þ
Consider a Taylor expansion of no-steady-state power output P(x, y) about the point ðx; y Þ; P ðx; yÞ ¼ P ðx; y Þþ ðx xÞðoP =oxÞ þ ðy y ÞðoP =oyÞ þ Oððx xÞ2 ; ðy y Þ2 Þ þ . When the engine works out of the steady-state but not too far away, that is, the time derivate terms ðx xÞ and ðy y Þ are small enough to be neglected, one can assume that P ðx; yÞ P ðx; y Þ. It is useful to perform the local stability analysis [19–21]. The power output of an irreversible heat engine depends on x and y in the same way that it depends on x and y at the steady-state, i.e. x /y y sðx; yÞx : x þ Ay xy½1 þ Asðx; yÞ
When system works in the steady-state, from Eqs. (16) and (17), one may express s as function of x and y
P ðx; yÞ ¼ P ðx; y Þ ¼ aF
s ¼ sðx; y Þ:
Similarly, the temperature ratio s depends on x and y in the same way that it depends on x and y at the steady-state, i.e.
ð25Þ
Using Eqs. (13) and (22)–(24) one can obtain the temperatures of the hot and cold reservoirs and power output as a function of x and y 1
½A þ sðx; y Þ yx ; Ay þ x ½1 þ Asðx; y Þyx TL ¼ Ay þ x
TH ¼
ð26Þ ð27Þ
x /y y sðx; y Þx : x þ Ay xy ½1 þ Asðx; y Þ
ð28Þ
ð36Þ
The dynamic equations for x and y can be obtained by substituting Eqs. (13) and (35) into Eqs. (33) and (34), aF x 1 1 y xsðx; yÞ dx=dt ¼ ð37Þ C x þ Ay x T H xy½1 þ Asðx; yÞ and
and P ðx; y Þ ¼ aF
sðx; yÞ ¼ sðx; y Þ:
ð35Þ
aF dy=dt ¼ C
/x x þ Ay
½y xsðx; yÞ y 1 1 : x2 ½1 þ Asðx; yÞ Ax T L y ð38Þ
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
By solving Eqs. (16) and (17) the temperature ratio of heat reservoirs s at the maximum power output and maximum efficiency may be expressed as functions of x and y, respectively, sP ðx; yÞ ¼
Ay þ 2/y x /x þ 2Ax A/y
ð39Þ
and
sg ðx; yÞ ¼
703
where
2 mg þ A Asðmg þ AÞ ; 1 þ As mg ð1 þ AsÞ 2 mg þ A Asðmg þ AÞ b2 ¼ ; ðe2 mg þ s þ As2 Þ þ 1 þ As 1 þ As
2 / mg þ A /ðmg þ AÞ c2 ¼ ð1 þ As þ Ae2 mg Þ þ ; A mg ð1 þ AsÞ Amg ð1 þ AsÞ a 2 ¼ e2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi 1 2 2 2 2 2 ð2Ax /A y þ 2A xy þ /x Þa þ 2/xy a ðx þ AyÞ ð4A2 þ 4/A þ /2 Þax2 2ð2A /Þa/Axy þ a/2 A2 y 2 4/xð/y xÞ 2/x2
ð40Þ
Finally, based on linearization and stability analysis (a detailed content is given in Appendix A) [22], one can set aF x 1 1 y xsðx; yÞ f ðx; yÞ ¼ ð41Þ C x þ Ay x T H xy½1 þ Asðx; yÞ
2 mg þ A /ðmg þ AÞ ; d 2 ¼ /e2 1 þ As Að1 þ AsÞ 2
1 aA ð1 mg Þ A pffiffiffiffiffiffiffi e2 ¼ 3 þ ag2 mg mg / 2/ þ
and gðx; yÞ ¼
aF C
/x x þ Ay
½y xsðx; yÞ y 1 1 : x2 ½1 þ Asðx; yÞ Ax T L y ð42Þ
Using Eqs. (41) and (42) and solving the characteristic equation (A9), one can obtain that the characteristic eigenvalues at the maximum power output and maximum efficiency are, respectively, kP1;2 ¼
aF A þ A/s þ 2/ 2CT 2L 2ð/ þ AÞðmP þ AÞ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1 þ d 1 ða1 d 1 Þ2 þ 4b1 c1 ;
ð43Þ
g2 ¼ ½4aAð/ þ AÞ þ /ða/ þ 4Þm2g 2/mg ða/A þ 2aA2 þ 2/Þ þ a/2 A2 :
mP and mg in Eqs. (43) and (44) is given by Eqs. (16) and (17). The characteristic relaxation times can be obtained t1;2 ¼ 1=jk1;2 j;
tP1;2 ¼
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2CT 2L 2ð/ þ AÞðmP þ AÞ 2 a1 þ d 1 ða1 d 1 Þ þ 4b1 c1 ; A þ A/s þ 2/ aF
ð46Þ and tg1;2
" #2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2CT 2L ðmg þ AÞ2 2 ¼ a2 þ d 2 ða2 d 2 Þ þ 4b2 c2 : aF mg ð1 þ AsÞ
ð47Þ
and kg1;2
ð45Þ
i.e.
where 2 mP þ A 2Asð/ þ AÞ ; a1 ¼ e 1 1 þ As A þ A/s þ 2/ 2 mP þ A 2Asð/ þ AÞ b1 ¼ ; ðe1 mP þ s þ As2 Þ þ 1 þ As 1 þ /s þ 2As
2 / mP þ A c1 ¼ ð1 þ As þ Ae1 mP Þ A mP ð1 þ AsÞ 2/ð/ þ AÞ ; þ AðA þ 2/ þ A/sÞ 2 mP þ A 2/ð/ þ AÞ d 1 ¼ /e1 ; 1 þ As Að1 þ /s þ 2AsÞ 2 Aþ/ e1 ¼ 2 /mP þ 2AmP A/
rffiffiffiffiffi ðmg þ AÞ a ; ½a/A2 mg ða/A þ 2aA2 þ 2/Þ 3 2mg g2
" #2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aF mg ð1 þ AsÞ 2 ¼ a þ d ða2 d 2 Þ þ 4b2 c2 ; 2 2 2CT 2L ðmg þ AÞ2 ð44Þ
Eqs. (46) and (47) are the general expressions of the characteristic relaxation times as a function of s, /, q and b/a at the maximum power output and the maximum efficiency. It is easy to find that the characteristic relaxation time tP1;2 at the maximum power output is independent of the heat leak rate q. Relaxation times versus s are plotted for / > 1 and / = 1 at given TL = 300 K, b/a = 1 and aF = 4000 KW/K at the maximum power output, as shown in Fig. 3. It is found that both relaxation times decrease as s increases. So the stability of the system improves as the values of s increases. Otherwise, t2 increases more quickly than t1 as the value of / increases. Thus, the internal irreversibility made the stability of system decline. Relaxation times versus s are plotted for different b/a at given TL = 300 K, / = 1.2 and aF = 4000 KW/K at the maximum power output, as shown in Fig. 4. It is shown that the relaxation times
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W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706 0.6
0.2
0.15
φ=1.2 φ=1.0
t2 0.1
t1
t1 relaxation times (x1/2C)
relaxation times (x1/2C)
relaxation times (x1/2C)
φ=1.2 φ=1.0
0.4
t2
0.10
0.05
0.00 0.0
0.2
0.2
0.4
τ
0.6
0.
1.0
t1 0.0
0.0 0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
Fig. 3. Plot of relaxation times versus s for / > 1 and / = 1 at given TL = 300 K, b/a = 1 and aF = 4000 KW/K at the maximum power output.
0.6
1.0
0.15
β/α=1.0 β/α=0.5
q=150W q=200W
0.1
t1
0.4
t2 0.2
t1
relaxation times (x1/2C)
relaxation times (x1/2C)
relaxation times (x1/2C)
0.8
Fig. 5. Plot of relaxation times versus s for / > 1 and / = 1 at given TL = 300 K, b/a = 1, q = 150 W and aF = 4000 KW/K at the maximum efficiency.
0.2
t2
0.6
τ
τ
0.10
0.05
0.00 0.0
0.2
0.4
τ
0.6
0.8
1.0
t1 0.0
0.0 0.0
0.2
0.4
τ
0.6
0.8
1.0
0.0
0.2
0.4
τ
0.6
0.8
1.0
Fig. 4. Plot of relaxation times versus s for different b/a at given TL = 300 K, / = 1.2 and aF = 4000 KW/K at the maximum power output.
Fig. 6. Plot of relaxation times versus s for different heat leak at given TL = 300 K, b/a = 1, / = 1.2 and aF = 4000 KW/K at the maximum efficiency.
increases as b/a decreases at given s and the stability of system declines. Relaxation times versus s are plotted for different / and q at given TL = 300 K, b/a = 1 and aF = 4000 KW/K at the maximum efficiency, as shown in Figs. 5 and 6. It is found that both of the relaxation time t1 and t2 decrease as the possible values of s increase. The stability of system declines as the values of s approach zero and improves as s increases. The effect of internal irreversibility / and heat leak rate q on the relaxation times is very small. The relaxation times versus s are plotted for different b/a at given TL = 300 K, / = 1.2, q = 150 W and aF = 4000 KW/K, as shown in Fig. 7. The relaxation times decreases as b/a
increases. The effect of heat-transfer coefficient ratio b/a on the relaxation times is very large. Relaxation times versus s are plotted for the optimal steady-state of the maximum power output and the maximum efficiency at given TL = 300 K, b/a = 1, / = 1.2, q = 150 W and aF = 4000 KW/K, as shown in Fig. 8. It is shown that the relaxation time tP2 at optimal steady-state of the maximum power output is smaller than the relaxation time tg2 at optimal steady-state of the maximum efficiency, i.e. tg2 > tP2 . Thus, the stability of steady-state of the maximum power output is better than that of steadystate of the maximum efficiency.
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
Acknowledgements
0.2
relaxation times (x1/2C)
β/α=1.0 β/α=0.5
This work was supported by National Natural Science Foundation (No. 10465003) and Science and Technology Foundation of Jiangxi Education Bureau, People’s Republic of China.
t2
Appendix A
0.1
t1
0.0 0.0
0.2
0.4
τ
0.6
0.8
1.0
Fig. 7. Plot of relaxation times versus s for different b/a at given TL = 300 K, / = 1.2, q = 150 W and aF = 4000 KW/K at the maximum efficiency.
0.3 η
t2 > t2 relaxation times (x1/2C)
705
P
Pmax ηmax
Consider the dynamical system dx ¼ f ðx; yÞ ðA1Þ dt and dy ¼ gðx; yÞ: ðA2Þ dt Let ðx; y Þ be the steady-state solutions of Eqs. (A1) and (A2) such that f ðx; y Þ ¼ 0 and gðx; y Þ ¼ 0. If x and y are close to their steady-state values, we can write xðtÞ ¼ xþ dxðtÞ and yðtÞ ¼ y þ dyðtÞ, where dx(t) and dy(t) are small perturbations. By substituting this into Eqs. (A1) and (A2) and using the smallness of dx(t) and dy(t) to cut at first order in the Taylor series expansions, we obtain the following set of linear differential equations for perturbations dx(t) and dy(t): ddxðtÞ ¼ fx dxðtÞ þ fy dyðtÞ dt
0.2
t2
ðA3Þ
and ddyðtÞ ¼ gx dxðtÞ þ gy dyðtÞ; ðA4Þ dt og of og ; f ¼ ; g ¼ and g ¼ . where fx ¼ of y x y ox x;y oy ox x;y oy
0.1
t1
x;y
x;y
Assume that dx(t) and dy(t) are of the form dxðtÞ ¼ Z 1 ekt
0.0 0.0
0.2
0.4
0.6
0.8
1.0
τ Fig. 8. Plot of relaxation times versus s at of the maximum power output and the maximum efficiency at given TL = 300 K, b/a = 1, / = 1.2, q = 150 W and aF = 4000 KW/K.
4. Conclusions
and dyðtÞ ¼ Z 2 ekt ;
ðA6Þ
where k is a complex number to be determined. Substitution of Eqs. (A5) and (A6) into Eqs. (A3) and (A4) leads to the following set of homogeneous linear algebraic equations for Z1 and Z2 ðfx kÞZ 1 þ fy Z 2 ¼ 0
The local stability analysis of the irreversible heat engine working in a maximum power output and a maximum efficiency is presented in this paper. The general expressions of relaxation times for two optimal steady-states (the maximum power output or the maximum efficiency) are derived. The influence of heat resistance, internal irreversibility and heat leak rate on the relaxation times is discussed. Finally, we compare the stability of system at optimal steady-state of the maximum power output with the maximum efficiency. It is found that the former is better than the latter. The results obtained here are general and be useful for both determination of optimal operating conditions and design of real heat engines.
ðA5Þ
ðA7Þ
and gx Z 1 þ ðgy kÞZ 2 ¼ 0:
ðA8Þ
This set of equations has non-trivial solutions only if the determinant of the matrix of coefficients equals zero, i.e. ðfx kÞðgy kÞ gx fy ¼ 0;
ðA9Þ
which is called the characteristic equation and k is called the eigenvalue. The characteristic relaxation times can be defined as t¼
1 : jkj
ðA10Þ
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References [1] F.L. Curzon, B. Ahborn, Efficiency of a Carnot engine at maximum power output, Am. J. phys. 43 (1975) 22–24. [2] M.H. Rubin, B. Andresen, Optimal staging of endoreversible heat engines, J. Appl. Phys. 52 (1982) 1–7. [3] A. Bejan, Entropy generation minimization: the new thermodynamics of finite-size devices and finite-time processes, J. Appl. Phys. 79 (1996) 1191–1281. [4] A. De Vos, Efficiency of some heat engines at maximum power conditions, Am. J. Phys. 53 (1985) 570–573. [5] A. De Vos, Reflections on the power delivered by endoreversible engine, J. Phys. D: Appl. Phys. 20 (1987) 232–236. [6] L. Chen, Z. Yan, The effect of heat-transfer law on performance of two-heat-source endoreversible cycle, J. Chem. Phys. 90 (1989) 3740– 3743. [7] C. Wu, Power optimization of a finite time Carnot heat engine, Energy 13 (1988) 681–687. [8] L.B. Erbay, H. Yavuz, An analysis of an endoreversible heat engine with combined heat transfer, J. Phys. D: Appl. Phys. 30 (1997) 2841– 2847. [9] G. De Mey, A. De Vos, On the optimum efficiency of endoreversible thermodynamic processes, J. Phys. D: Appl. Phys. 27 (1994) 736–739. [10] J. Chen, C. Wu, Maximum specific power output of a two-stage endoreversible combined cycle, Energy 20 (1994) 305–309. [11] L. Chen, F. Sun, C. Wu, A generalized model of real heat engines and its performance, J. Inst. Energy 69 (1996) 214–222. [12] L. Chen, F. Sun, C. Wu, The influence of internal heat leak on the power versus efficiency characteristics of heat engines, Energy Convers. Manage. 38 (1997) 1501–1507.
[13] J.M. Gordon, M. Huleihil, General performance characteristics of real heal engines, J. Appl. Phys. 72 (1992) 829–837. [14] Z. Yan, L. Chen, Optimal performance of a generalized Carnot cycle for another linear heat transfer law, J. Chem. Phys. 9 (1990) 1994– 1998. [15] J. Chen, The maximum power output and maximum efficiency of an irreversible Carnot heat engine, J. Phys. D: Appl. Phys. 27 (1994) 1144–1149. [16] F. Moukalled, R.Y. Nuwayhid, N. Noueihed, The efficiency of endoreversible heat engines with heat leak, Int. J. Energy. Res. 19 (1995) 377–389. [17] L. Chen, F. Sun, C. Wu, Effect of a heat transfer law on the performance of a generalized irreversible Carnot-engine, J. Phys. D: Appl. Phys. 32 (1999) 99–105. [18] L. Chen, X. Zhu, F. Sun, C. Wu, Exergy-based ecological optimization of linear phenomenological heat-transfer law irreversible Carnot-engines, Appl. Energy 83 (2006) 573–582. [19] M. Santillan, G. Maya-Aranda, F. Angulo-Brown, Local stability analysis of an endoreversible Curzon–Ahborn–Nouikov engine working in a maximum-power-like regime, J. Phys. D: Appl. Phys. 34 (2001) 2068–2072. [20] L. Guzman-vargas, I. Reyes-Ramirez, N. Sanchez, The effect of heat transfer laws and thermal conductances on the local stability of an endoreversible heat engine, J. Phys. D: Appl. Phys. 38 (2005) 1282– 1291. [21] R. Pa´ez-Herna´ndez, F. Angulo-Brown, M. Santilla´n, Dynamic robustness and thermodynamic optimization in a non-Endoreversible Curzon–Ahlborn Engine, J. Non-Equilib. Thermodyn. 31 (2006) 173– 188. [22] S. Lang, Linear Algebra, Springer, New York, 1987.
Applied Thermal Engineering 28 (2008) 699–706 www.elsevier.com/locate/apthermeng
Local stability analysis of an irreversible heat engine working in the maximum power output and the maximum efficiency Wenjie Nie, Jizhou He *, Bei Yang, Xiaoxia Qian Department of Physics, Nanchang University, Nanchang 330047, PR China Received 1 February 2007; accepted 12 June 2007 Available online 23 June 2007
Abstract In this paper the local stability of an irreversible heat engine working in the maximum power output and the maximum efficiency is analyzed. At two optimal steady-states of the maximum power output and the maximum efficiency the expressions of the relaxation time of an irreversible heat engine are derived. It is found that the relaxation time is a function of the heat-transfer coefficient a and b, internal irreversibility factor /, heat leak rate q, heat capacity C, the total heat-transfer surface area F, and temperatures of the heat reservoirs TH and TL. The influence of heat resistance, internal irreversibility and heat leak rate on the relaxation time is discussed and some new results are gained. The local stability of the optimal steady-state of the maximum power output is better than that of the maximum efficiency. The results obtained here are more general and useful for an irreversible heat engine than an endoreversible heat engine. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Local stability analysis; Irreversible heat engine; Performance characteristics
1. Introduction Based on the finite-time thermodynamics (FTT) theory, many different endoreversible models of heat engines were established and more realistic limits for the performance of real engine models were obtained [1–10]. However, real heat engines are usually complex devices with both internal and external irreversibilities. Besides the irreversibility of finite-rate heat-transfer, there are also other sources of irreversibilities, such as heat leaks, dissipative processes inside the working fluid, and so on. In order to assess the effect of finite-rate heat-transfer, together with other major irreversibilities on the performance of heat engines, some new irreversible models of heat engines were established and the optimal performance characteristics were analyzed at the maximum power output, the maximum efficiency and the maximum ecological function [11–18].
*
Corresponding author. E-mail address: [email protected] (J. He).
1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.06.002
Most of the studies of heat engines have focused on their steady-state energetic properties. But the real heat engines are always in no-steady-state and there exists intrinsic cycle variability in the operation of the cycle, for example, incomplete combustion of fuel and other causes. Thus, it is necessary to analyze the effect of noisy perturbations on the stability of system’s steady-state. Recently, a sort of new design principles named as the local stability of energy-converting system have been put forward. Santillan et al studied the local stability of an endoreversible Curzon–Ahlborn–Novikov engine working in a maximum power output [19]. Guzman-vargas et al. analyzed the effect of heat-transfer laws and thermal conductance on the local stability of an endoreversible heat engine [20]. Pa´ezHerna´ndez et al. made the dynamic effects of the time delays and the effect of internal irreversibility on the local stability of non-endoreversible heat engine [21]. Following their works and considering the finite-rate heat-transfer, internal irreversibility and heat leak, we analyze the local stability of an irreversible heat engine working in the maximum power output and the maximum efficiency. Some general and useful results are obtained.
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W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
Nomenclature C F F1 F2 f J1 J2 J1 J2 m mP mg P P QH QL q TH TL t
heat capacity, W K1 total surface area of heat exchangers, m2 surface area of the hot-side heat exchanger, m2 surface area of the cold-side heat exchanger, m2 heat-transfer surface area ratio no-steady-state heat flows from warm working fluid to engine, W no-steady-state heat flows from engine to cold working fluid, W steady-state heat flows from warm working fluid to engine, W steady-state heat flows from engine to cold working fluid, W temperature ratio of the working fluid optimal temperature ratio of the working fluid at the maximum power output optimal temperature ratio of the working fluid at the maximum efficiency power output, W steady-state power output, W heat-transfer supplied by the hot reservoir, W heat-transfer released to the cold reservoir, W heat leak rate, W temperature of the hot reservoir, K temperature of the cold reservoir, K relaxation time, s
x y x y
no-steady-state temperature of the warm working fluid, K no-steady-state temperature of the cold working fluid, K steady-state temperature of the warm working fluid, K steady-state temperature of the cold working fluid, K
Greek letters a heat-transfer coefficient of the hot-side heat exchanger, W K1 b heat-transfer coefficient of the cold-side heat exchanger, W K1 / internal irreversibility factor g efficiency g0 efficiency of the irreversible engine without heat leak g0 steady-state efficiency of the irreversible engine without heat leak k complex number (the eigenvalue) s temperature ratio of heat reservoirs DS1 entropy difference in high-temperature isothermal process, J K1 DS2 entropy difference in low-temperature isothermal process, J K1
2. Performance characteristics of an irreversible heat engine
and heat leak rate (q) between the two reservoirs are assumed to be a constant
Consider a steady flow irreversible heat engine, including the irreversibilities of finite-rate heat-transfer between the working fluid and heat reservoirs, heat leak between the two reservoirs and internal irreversibility of the working fluid, shown in Fig. 1. The working fluid is alternately connected to a hot reservoir at constant temperature TH and to a cold reservoir at constant temperature TL, and its temperatures are x and y, respectively. Based on a linear phenomenological heat-transfer law of irreversible thermodynamics, the heat flow (QH) from the hot reservoir to the warm working fluid through the hot-side heat exchanger and the heat flow (QL) from the cold working fluid to the cold reservoir through the cold-side heat exchanger are
F1 þ F2 ¼ F
QH ¼ aF 1 ð1=x 1=T H Þ
ð1Þ
and QL ¼ bF 2 ð1=T L 1=yÞ;
and
q ¼ const:
ð3Þ
Besides the irreversibilities of the finite-rate heat-transfer and heat leak, there are other irreversibilities inside the heat engine, such as the friction, turbulence and non-equilibTH QH C, x
J1
P
q J2
ð2Þ
where a and b are the overall heat-transfer coefficients at the hot-and cold-side heat exchangers and they are all positive, F1 and F2 are the surface areas of the hot-side and cold-side heat exchangers, respectively. Both the total heat-transfer surface area (F) of the two heat exchangers
C, y
QL
TL Fig. 1. Schematic diagram of an irreversible Carnot heat engine.
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
rium activities. Thus, all the processes of the cycle are irreversible. The entropy of the working fluid in the two adiabatic processes increases. If the entropy differences of the working fluid in the two isothermal processes at temperatures x and y are, respectively, denoted by DS1 and DS2, then an internal irreversible parameter / is introduced to characterize the other irreversibilities [14–18], / ¼ DS 2 =DS 1 :
ð4Þ
According to the second law of thermodynamics and Eq. (4), for an irreversible heat engine, it requires QL =y /ðQH =xÞ ¼ 0:
ð5Þ
When there are no other irreversibilities except the irreversibility of the finite-rate heat-transfer, i.e. / = 1, the engine is an endoreversible one; When / > 1, the engine is a internal irreversible one. The parameter / depends on the property of the working fluid, friction, rate of the cycle operation. By the first law of thermodynamics, the power output of an irreversible heat engine is P ¼ QH QL :
ð6Þ
The efficiency is g ¼ ðQH QL Þ=ðQH þ qÞ:
ð7Þ
F 1 =F 2 ¼ m
P ¼ Bðm /Þð1=T L m=T H Þ
ðm /Þð1=T L m=T H Þ ; ð15Þ mð1=T L m=T H Þ þ q=B 2 pffiffiffiffiffiffiffiffiffiffiffi where B ¼ aF = m þ /a=b . Using Eqs. (14) and (15) and eliminating the temperature ratio of the working fluid, the optimal relation between the power output and efficiency may be obtained. Therefore, the power output versus efficiency is plotted for different values of the temperature ratio of heat reservoirs s = TL/TH at given TL = 300 K, / = 1.2, q = 150 W, b/a = 1 and aF = 4000 KW/ K, as shown in Fig. 2. The characteristic curve of P g is a loop shape passing the zero point and it is clearly shown that there are two optimal steady-state points of the engine. One is a maximum power output Pmax and the corresponding efficiency gP; the other is a maximum efficiency gmax and the corresponding power output Pg. Further, by using Eqs. (14) and (15) and the extremal conditions dP/dm = 0 and dg/dm = 0, the optimal temperature ratios of the working fluid at the maximum power output and maximum efficiency are obtained, respectively,
g¼
g0 ¼ ðQH QL Þ=QH ;
and
mg ¼
QH ¼
A þ 2/ þ As/ 1 þ /s þ 2As
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2/s 2a/ þ 2aA2 sÞ 4ðaA2 þ / þ 2aA/ þ a/A2 sÞð/s2 a 2aAs a/sÞ 2ða þ 2aAs þ a/s /s2 Þ
i.e. efficiency of the inner engine which is shown by dotted line in Fig. 1 is g0. According to Eqs. (4)–(6), the heat flows can be expressed as xP x /y
ð9Þ
ð10Þ
J1 ¼
Define the heat-transfer surface area ratio (f) and the temperature ratio of the working fluid (m)
and
f ¼ F 1 =F 2
J2 ¼
ð11Þ
ð16Þ
ð17Þ
pffiffiffiffiffiffiffiffiffiffiffi where A ¼ /a=b, a = qTL/aF and s = TL/TH. Consider the internal irreversible Carnot heat engine operates between temperatures x and y , where x and y represent the steady-state temperatures of working fluid. The irreversibility hypothesis means that the cycle is internally irreversible. Thus, the heat flows can be given by
and /yP QL ¼ : x /y
ð14Þ
and
mP ¼
2a/ 2aA2 s 2/s þ
ð13Þ
The power output and efficiency may be expressed, respectively,
The efficiency of the engine without heat leak may be expressed as ð8Þ
pffiffiffiffiffiffiffiffiffiffiffi b=/a:
701
x P x /y
ð18Þ
/y P; x /y
ð19Þ
and m ¼ x=y:
ð12Þ
By using Eqs. (1)–(7) and (11), (12) and the extremal conditions dP/df = 0 and dg/df = 0, the optimal surface area ratio is obtained [17,18]
where J 1 and J 2 are the steady-state heat flows from x to engine and from engine to y , respectively. P is the steadystate power output. The variables with overbars denote steady-state values. When the irreversible Carnot heat engine operates in a steady-state, this means that the heat
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W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
3. Local stability analysis of an irreversible heat engine
1.0
0.8
P/Pmax,φ=1,Ci=0
τ=0.4 τ=0.5 τ=0.6
1 2 3
When the heat engine works out of the steady-state, reservoirs x and y are not real heat reservoirs. It is assumed that the temperatures x and y correspond to macroscopic objects with finite heat capacity C. The differential equations for x and y are given by
Pmax 0.6
1
Pη
2
dx=dt ¼ ½aF 1 ð1=x 1=T H Þ J 1 =C
ð29Þ
0.4
and
3
dy=dt ¼ ½J 2 bF 2 ð1=T L 1=yÞ=C;
0.2
0.0 0.0
0.1
η
0.2
ηP
0.3
ηmax
0.4
Fig. 2. Plot of power output versus efficiency for different values of s = TL/TH at given TL = 300 K, b/a = 1, / = 1.2, q = 150 W and aF = 4000 KW/K.
where J1 and J2 are the heat flows from x to the engine and from the engine to y, respectively. By Eqs. (18) and (19) J1 and J2 are given by x J1 ¼ P ðx; yÞ ð31Þ x /y and J2 ¼
flow (QH) from TH to x equals to J 1 and the heat flow (QL) from y to TL equals to J 2 , i.e. J 1 ¼ aF 1 ð1=x 1=T H Þ
ð20Þ
and J 2 ¼ bF 2 ð1=T L 1=y Þ:
ð21Þ
According to Eqs. (6) and (9), the efficiency of an internal irreversible Carnot heat engine without heat leak is g0 ¼ P =J 1 ¼ 1 /y =x ¼ 1 /=m;
ð22Þ
By using Eqs. (13) and (18)–(22), it is possible to express the steady-state temperatures x and y as x ¼ T H
ðA þ mÞs 1 þ As
ð23Þ
A=m þ 1 : 1 þ As
ð24Þ
and y ¼ T L
ð30Þ
/y P ðx; yÞ: x /y
ð32Þ
It is worthy to note that the no-steady-state power output is always a function of temperatures of working fluid. Substitution of Eqs. (31) and (32) into Eqs. (29) and (30) leads to dx=dt ¼ ½aF 1 ð1=x 1=T H Þ xP ðx; yÞ=ðx /yÞ=C
ð33Þ
and dy=dt ¼ ½/yP ðx; yÞ=ðx /yÞ bF 2 ð1=T L 1=yÞ=C:
ð34Þ
Consider a Taylor expansion of no-steady-state power output P(x, y) about the point ðx; y Þ; P ðx; yÞ ¼ P ðx; y Þþ ðx xÞðoP =oxÞ þ ðy y ÞðoP =oyÞ þ Oððx xÞ2 ; ðy y Þ2 Þ þ . When the engine works out of the steady-state but not too far away, that is, the time derivate terms ðx xÞ and ðy y Þ are small enough to be neglected, one can assume that P ðx; yÞ P ðx; y Þ. It is useful to perform the local stability analysis [19–21]. The power output of an irreversible heat engine depends on x and y in the same way that it depends on x and y at the steady-state, i.e. x /y y sðx; yÞx : x þ Ay xy½1 þ Asðx; yÞ
When system works in the steady-state, from Eqs. (16) and (17), one may express s as function of x and y
P ðx; yÞ ¼ P ðx; y Þ ¼ aF
s ¼ sðx; y Þ:
Similarly, the temperature ratio s depends on x and y in the same way that it depends on x and y at the steady-state, i.e.
ð25Þ
Using Eqs. (13) and (22)–(24) one can obtain the temperatures of the hot and cold reservoirs and power output as a function of x and y 1
½A þ sðx; y Þ yx ; Ay þ x ½1 þ Asðx; y Þyx TL ¼ Ay þ x
TH ¼
ð26Þ ð27Þ
x /y y sðx; y Þx : x þ Ay xy ½1 þ Asðx; y Þ
ð28Þ
ð36Þ
The dynamic equations for x and y can be obtained by substituting Eqs. (13) and (35) into Eqs. (33) and (34), aF x 1 1 y xsðx; yÞ dx=dt ¼ ð37Þ C x þ Ay x T H xy½1 þ Asðx; yÞ and
and P ðx; y Þ ¼ aF
sðx; yÞ ¼ sðx; y Þ:
ð35Þ
aF dy=dt ¼ C
/x x þ Ay
½y xsðx; yÞ y 1 1 : x2 ½1 þ Asðx; yÞ Ax T L y ð38Þ
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
By solving Eqs. (16) and (17) the temperature ratio of heat reservoirs s at the maximum power output and maximum efficiency may be expressed as functions of x and y, respectively, sP ðx; yÞ ¼
Ay þ 2/y x /x þ 2Ax A/y
ð39Þ
and
sg ðx; yÞ ¼
703
where
2 mg þ A Asðmg þ AÞ ; 1 þ As mg ð1 þ AsÞ 2 mg þ A Asðmg þ AÞ b2 ¼ ; ðe2 mg þ s þ As2 Þ þ 1 þ As 1 þ As
2 / mg þ A /ðmg þ AÞ c2 ¼ ð1 þ As þ Ae2 mg Þ þ ; A mg ð1 þ AsÞ Amg ð1 þ AsÞ a 2 ¼ e2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi 1 2 2 2 2 2 ð2Ax /A y þ 2A xy þ /x Þa þ 2/xy a ðx þ AyÞ ð4A2 þ 4/A þ /2 Þax2 2ð2A /Þa/Axy þ a/2 A2 y 2 4/xð/y xÞ 2/x2
ð40Þ
Finally, based on linearization and stability analysis (a detailed content is given in Appendix A) [22], one can set aF x 1 1 y xsðx; yÞ f ðx; yÞ ¼ ð41Þ C x þ Ay x T H xy½1 þ Asðx; yÞ
2 mg þ A /ðmg þ AÞ ; d 2 ¼ /e2 1 þ As Að1 þ AsÞ 2
1 aA ð1 mg Þ A pffiffiffiffiffiffiffi e2 ¼ 3 þ ag2 mg mg / 2/ þ
and gðx; yÞ ¼
aF C
/x x þ Ay
½y xsðx; yÞ y 1 1 : x2 ½1 þ Asðx; yÞ Ax T L y ð42Þ
Using Eqs. (41) and (42) and solving the characteristic equation (A9), one can obtain that the characteristic eigenvalues at the maximum power output and maximum efficiency are, respectively, kP1;2 ¼
aF A þ A/s þ 2/ 2CT 2L 2ð/ þ AÞðmP þ AÞ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a1 þ d 1 ða1 d 1 Þ2 þ 4b1 c1 ;
ð43Þ
g2 ¼ ½4aAð/ þ AÞ þ /ða/ þ 4Þm2g 2/mg ða/A þ 2aA2 þ 2/Þ þ a/2 A2 :
mP and mg in Eqs. (43) and (44) is given by Eqs. (16) and (17). The characteristic relaxation times can be obtained t1;2 ¼ 1=jk1;2 j;
tP1;2 ¼
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2CT 2L 2ð/ þ AÞðmP þ AÞ 2 a1 þ d 1 ða1 d 1 Þ þ 4b1 c1 ; A þ A/s þ 2/ aF
ð46Þ and tg1;2
" #2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2CT 2L ðmg þ AÞ2 2 ¼ a2 þ d 2 ða2 d 2 Þ þ 4b2 c2 : aF mg ð1 þ AsÞ
ð47Þ
and kg1;2
ð45Þ
i.e.
where 2 mP þ A 2Asð/ þ AÞ ; a1 ¼ e 1 1 þ As A þ A/s þ 2/ 2 mP þ A 2Asð/ þ AÞ b1 ¼ ; ðe1 mP þ s þ As2 Þ þ 1 þ As 1 þ /s þ 2As
2 / mP þ A c1 ¼ ð1 þ As þ Ae1 mP Þ A mP ð1 þ AsÞ 2/ð/ þ AÞ ; þ AðA þ 2/ þ A/sÞ 2 mP þ A 2/ð/ þ AÞ d 1 ¼ /e1 ; 1 þ As Að1 þ /s þ 2AsÞ 2 Aþ/ e1 ¼ 2 /mP þ 2AmP A/
rffiffiffiffiffi ðmg þ AÞ a ; ½a/A2 mg ða/A þ 2aA2 þ 2/Þ 3 2mg g2
" #2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aF mg ð1 þ AsÞ 2 ¼ a þ d ða2 d 2 Þ þ 4b2 c2 ; 2 2 2CT 2L ðmg þ AÞ2 ð44Þ
Eqs. (46) and (47) are the general expressions of the characteristic relaxation times as a function of s, /, q and b/a at the maximum power output and the maximum efficiency. It is easy to find that the characteristic relaxation time tP1;2 at the maximum power output is independent of the heat leak rate q. Relaxation times versus s are plotted for / > 1 and / = 1 at given TL = 300 K, b/a = 1 and aF = 4000 KW/K at the maximum power output, as shown in Fig. 3. It is found that both relaxation times decrease as s increases. So the stability of the system improves as the values of s increases. Otherwise, t2 increases more quickly than t1 as the value of / increases. Thus, the internal irreversibility made the stability of system decline. Relaxation times versus s are plotted for different b/a at given TL = 300 K, / = 1.2 and aF = 4000 KW/K at the maximum power output, as shown in Fig. 4. It is shown that the relaxation times
704
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706 0.6
0.2
0.15
φ=1.2 φ=1.0
t2 0.1
t1
t1 relaxation times (x1/2C)
relaxation times (x1/2C)
relaxation times (x1/2C)
φ=1.2 φ=1.0
0.4
t2
0.10
0.05
0.00 0.0
0.2
0.2
0.4
τ
0.6
0.
1.0
t1 0.0
0.0 0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
Fig. 3. Plot of relaxation times versus s for / > 1 and / = 1 at given TL = 300 K, b/a = 1 and aF = 4000 KW/K at the maximum power output.
0.6
1.0
0.15
β/α=1.0 β/α=0.5
q=150W q=200W
0.1
t1
0.4
t2 0.2
t1
relaxation times (x1/2C)
relaxation times (x1/2C)
relaxation times (x1/2C)
0.8
Fig. 5. Plot of relaxation times versus s for / > 1 and / = 1 at given TL = 300 K, b/a = 1, q = 150 W and aF = 4000 KW/K at the maximum efficiency.
0.2
t2
0.6
τ
τ
0.10
0.05
0.00 0.0
0.2
0.4
τ
0.6
0.8
1.0
t1 0.0
0.0 0.0
0.2
0.4
τ
0.6
0.8
1.0
0.0
0.2
0.4
τ
0.6
0.8
1.0
Fig. 4. Plot of relaxation times versus s for different b/a at given TL = 300 K, / = 1.2 and aF = 4000 KW/K at the maximum power output.
Fig. 6. Plot of relaxation times versus s for different heat leak at given TL = 300 K, b/a = 1, / = 1.2 and aF = 4000 KW/K at the maximum efficiency.
increases as b/a decreases at given s and the stability of system declines. Relaxation times versus s are plotted for different / and q at given TL = 300 K, b/a = 1 and aF = 4000 KW/K at the maximum efficiency, as shown in Figs. 5 and 6. It is found that both of the relaxation time t1 and t2 decrease as the possible values of s increase. The stability of system declines as the values of s approach zero and improves as s increases. The effect of internal irreversibility / and heat leak rate q on the relaxation times is very small. The relaxation times versus s are plotted for different b/a at given TL = 300 K, / = 1.2, q = 150 W and aF = 4000 KW/K, as shown in Fig. 7. The relaxation times decreases as b/a
increases. The effect of heat-transfer coefficient ratio b/a on the relaxation times is very large. Relaxation times versus s are plotted for the optimal steady-state of the maximum power output and the maximum efficiency at given TL = 300 K, b/a = 1, / = 1.2, q = 150 W and aF = 4000 KW/K, as shown in Fig. 8. It is shown that the relaxation time tP2 at optimal steady-state of the maximum power output is smaller than the relaxation time tg2 at optimal steady-state of the maximum efficiency, i.e. tg2 > tP2 . Thus, the stability of steady-state of the maximum power output is better than that of steadystate of the maximum efficiency.
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
Acknowledgements
0.2
relaxation times (x1/2C)
β/α=1.0 β/α=0.5
This work was supported by National Natural Science Foundation (No. 10465003) and Science and Technology Foundation of Jiangxi Education Bureau, People’s Republic of China.
t2
Appendix A
0.1
t1
0.0 0.0
0.2
0.4
τ
0.6
0.8
1.0
Fig. 7. Plot of relaxation times versus s for different b/a at given TL = 300 K, / = 1.2, q = 150 W and aF = 4000 KW/K at the maximum efficiency.
0.3 η
t2 > t2 relaxation times (x1/2C)
705
P
Pmax ηmax
Consider the dynamical system dx ¼ f ðx; yÞ ðA1Þ dt and dy ¼ gðx; yÞ: ðA2Þ dt Let ðx; y Þ be the steady-state solutions of Eqs. (A1) and (A2) such that f ðx; y Þ ¼ 0 and gðx; y Þ ¼ 0. If x and y are close to their steady-state values, we can write xðtÞ ¼ xþ dxðtÞ and yðtÞ ¼ y þ dyðtÞ, where dx(t) and dy(t) are small perturbations. By substituting this into Eqs. (A1) and (A2) and using the smallness of dx(t) and dy(t) to cut at first order in the Taylor series expansions, we obtain the following set of linear differential equations for perturbations dx(t) and dy(t): ddxðtÞ ¼ fx dxðtÞ þ fy dyðtÞ dt
0.2
t2
ðA3Þ
and ddyðtÞ ¼ gx dxðtÞ þ gy dyðtÞ; ðA4Þ dt og of og ; f ¼ ; g ¼ and g ¼ . where fx ¼ of y x y ox x;y oy ox x;y oy
0.1
t1
x;y
x;y
Assume that dx(t) and dy(t) are of the form dxðtÞ ¼ Z 1 ekt
0.0 0.0
0.2
0.4
0.6
0.8
1.0
τ Fig. 8. Plot of relaxation times versus s at of the maximum power output and the maximum efficiency at given TL = 300 K, b/a = 1, / = 1.2, q = 150 W and aF = 4000 KW/K.
4. Conclusions
and dyðtÞ ¼ Z 2 ekt ;
ðA6Þ
where k is a complex number to be determined. Substitution of Eqs. (A5) and (A6) into Eqs. (A3) and (A4) leads to the following set of homogeneous linear algebraic equations for Z1 and Z2 ðfx kÞZ 1 þ fy Z 2 ¼ 0
The local stability analysis of the irreversible heat engine working in a maximum power output and a maximum efficiency is presented in this paper. The general expressions of relaxation times for two optimal steady-states (the maximum power output or the maximum efficiency) are derived. The influence of heat resistance, internal irreversibility and heat leak rate on the relaxation times is discussed. Finally, we compare the stability of system at optimal steady-state of the maximum power output with the maximum efficiency. It is found that the former is better than the latter. The results obtained here are general and be useful for both determination of optimal operating conditions and design of real heat engines.
ðA5Þ
ðA7Þ
and gx Z 1 þ ðgy kÞZ 2 ¼ 0:
ðA8Þ
This set of equations has non-trivial solutions only if the determinant of the matrix of coefficients equals zero, i.e. ðfx kÞðgy kÞ gx fy ¼ 0;
ðA9Þ
which is called the characteristic equation and k is called the eigenvalue. The characteristic relaxation times can be defined as t¼
1 : jkj
ðA10Þ
706
W. Nie et al. / Applied Thermal Engineering 28 (2008) 699–706
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