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Local stability of a non-endoreversible Carnot refrigerator working at the maximum ecological function
Accepted Manuscript Local stability of a non-endoreversible Carnot refrigerator working at the maximum ecological function Wu Xiaohui, Chen Lingen, Ge Yanlin, Sun Fengrui PII: DOI: Reference:
S0307-904X(14)00466-1 http://dx.doi.org/10.1016/j.apm.2014.09.031 APM 10146
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Appl. Math. Modelling
Received Date: Revised Date: Accepted Date:
10 February 2011 20 December 2013 17 September 2014
Please cite this article as: W. Xiaohui, C. Lingen, G. Yanlin, S. Fengrui, Local stability of a non-endoreversible Carnot refrigerator working at the maximum ecological function, Appl. Math. Modelling (2014), doi: http:// dx.doi.org/10.1016/j.apm.2014.09.031
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Local stability of a non-endoreversible Carnot refrigerator working at the maximum ecological function
Wu Xiaohui1, 2, 3, Chen Lingen1, 2, 3, *, Ge Yanlin1, 2, 3, Sun Fengrui1, 2, 3 (1 Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033) (2 Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033) (3 College of Power Engineering, Naval University of Engineering, Wuhan 430033)
* To whom all correspondence should be addressed. E-mail address: [email protected] and [email protected] (Lingen Chen) Fax: 0086-27-83638709 Tel: 0086-27-83615046
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Abstract Local stability of a non-endoreversible Carnot refrigerator at the maximum ecological function is studied with Newton’s heat transfer law between working fluid and heat reservoirs. The steady state of the refrigerator working at the maximum ecological function is steady. It is derived that a general expression of relaxation time described as stability of the system refers to heat capacity C , total heat exchange area F , temperature ratio of heat reservoirs , the degree of internal irreversibility , heat transfer coefficients and . Distributing information of phase portraits of system is obtained. The results can provide some theoretical guidelines for the designs of practical refrigerator. Keywords: local stability, non-endoreversible Carnot refrigerator, temperature ratio of heat reservoirs, ecological optimization
1. Introduction Since Curzon and Ahlborn [1] proposed the efficiency at maximum power output, for an endoreversible Carnot heat engine with Newton’s heat transfer law, i.e. so-called CA efficiency in 1975, finite time thermodynamics (FTT) has seen tremendous progress by scientists and engineers [2-9]. Leff and Teeter [10] were the first to extend the Curzon–Ahlborn analysis method [1] to the analysis of refrigeration cycles. Rozonoer and Tsirlin [11] first derived the maximum coefficient of performance (COP) bound for fixed power input to a Newton’s law endoreversible Carnot refrigerator. Yan [12], Goth and Feidt [13] , Feidt [14], Philippi and Feidt [15], and Feidt [16] derived the optimal COP for the fixed cooling load, i.e. the fundamental optimal relation of Newton’s law endoreversible Carnot refrigerator. However, real refrigerators are usually devices
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with both internal and external irreversibilities. Chen et al. [17,18] established a generalized irreversible Carnot refrigerator model, which takes account for the effects of heat resistance, bypass heat leakage, and other internal irreversibility, and derived its fundamental optimal relation between cooling load and COP. Angulo-Brown [19] proposed an ecological criterion E P TL for finite-time Carnot heat engines, where TL is the temperature of cold heat reservoir, P is the power output and is the entropy generation rate. Yan [20] showed that it might be more reasonable to use E P T0 if the cold reservoir temperature TL is not equal to the environmental temperature T0 , the ecological function E represents a compromise between the power output P and
power loss T0 due to the entropy generation in the system. Furthermore, based on the viewpoint of exergy analysis, Chen et al. [21] provided a unified ecological optimization objective function for all of the thermodynamic cycles, that is E A / t T0 S / t A / t T0
(1)
where A is the exergy output of the cycle, S is the entropy generation of the cycle, and t is the cycle period. Equation (1) represents the best compromise between the exergy output rate and the exergy loss rate of the thermodynamic cycles. For heat engine cycles, the exergy output rate of the cycle is A / t P . For refrigeration cycles, the exergy output rate of the cycle is A / t QL (T0 / TL 1) QH (T0 / TH 1) , where QL is the rate of heat transfer supplied by the heat
source, QH is the rate of heat transfer released to the heat sink, and TH and TL are temperatures of the heat sink and heat source, respectively. Therefore, the ecological function of refrigeration cycles is E QL (T0 / TL 1) QH (T0 / TH 1) T0
2
(2)
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Chen et al. [22] , Zhu et al. [23] and Chen et al. [24] investigated the optimal ecological performance of generalized irreversible Carnot refrigerators considering the effects of heat resistance, heat leakage and internal irreversibility with Newton’s heat transfer law. [22] , the generalized radiative heat transfer law [23] and generalized convective heat transfer law [24]. Most of the previous works about the endoreversible and irreversible Carnot refrigerators have concentrated on the steady-state energetic properties of the systems. However, it is worthwhile to design the systems considering the local stability of the refrigerator’s steady state. Santillan et al. [25] firstly studied the local stability of a Curzon-Ahlborn-Novikov (CAN) engine working in a maximum- power-like regime considering the heat resistance and the equal highand-low-temperature heat transfer coefficients with Newton’s heat transfer law. Chimal-Eguia et al. [26] analyzed the local stability of an endoreversible heat engine working in a maximum-power-like regime with Stefan-Boltzmann heat transfer law. Huang et al. [27] analyzed the local stability of a non-endoreversible heat engine working between the maximum power output and the maximum efficiency with Newton’s heat transfer law. Huang and Sun [28, 29] studied the local stability of the endoreversible [28] and non-endoreversible [29] refrigerators operating at minimum input power for a given cooling load with Newton’s heat transfer law, and analyzed the effects of cooling load and heat transfer coefficients on the local stability of the system. He et al. [30] studied the local stability of the endoreversible Carnot refrigerators operating at maximum objective function of the product of the cooling load and COP with Newton’s heat transfer law. Chimal-Eguia et al. [31] studied the local stability of a non-endoreversible heat engine working at the maximum ecological function with Newton’s heat transfer law, and offered the phase portraits of the system. Nie et al. [32] studied the local stability
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of an irreversible heat engine with the losses of the heat leakage and heat resistance working at the maximum ecological function with Newton’s heat transfer law. Páez-Hernández et al. [33] analyzed the local stability of an endoreversible Curzon-Ahlborn-Novikov engine, using Van der Waals gas as a working substance and the corresponding efficiency for this engine working at temperatures within the maximum ecological regime. Arias-Hernandez et al. [34] studied the local stability of the refrigeration cycle working at the maximum ecological function with Newton’s heat transfer law, and concluded that temperature ratio of heat reservoirs represents a trade-off between stronger stability and better thermodynamic properties like power and efficiency for heat engine. On the bases of the Refs. [32, 50-51], this paper will analyze the local stability of a non-endoreversible refrigerator with Newton’s heat transfer law working at the maximum ecological function, discuss the effects of the degree of internal irreversibility ( ), heat reservoirs’ temperature ratio ( ), heat transfer coefficients ( and ) on the stability of the system, analyze the distribution information of phase portraits of the system.
2. Ecological performance of a non-endoreversible refrigerator Considering a model of a non-endoreversible Carnot refrigerator [22-24,34] as shown in Fig. 1, its working conditions are as follows: (1) The working fluid flows through the system in a time-invariant continuous way. The cycle consists of two isothermal and two adiabatic processes which are irreversible in general. (2) Because of the heat resistance, the working fluid’s temperatures ( x and y ) are different from the reservoirs’ temperatures ( TH and TL ) and the four temperatures are of the following decreasing order: x TH TL y . The heat transfer surface areas ( F1 and F2 ) of the high- and low-temperature side heat exchangers are finite. The overall heat transfer surface area ( F ) of the 4
two heat exchangers is assumed to be a constant: F F1 F2 . Assume that the heat transfer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
surface area ratio is f F1 / F2 , the working fluid’s temperature ratio is m x / y and the temperature ratio of heat reservoirs is TL / TH . Thus, 0 y x TL TH 1 . (3) There are irreversibilities in the system due to heat resistance and miscellaneous factors such as friction, turbulenceand non-equilibrium activities inside the refrigerator, etc.. Thus, when compared with an endoreversible Carnot refrigerator of the same cooling load, a larger power supply is needed. In other words, the rate ( QHC ) of heat rejected to the heat sink of the non-endoreversible Carnot refrigerator is higher than that the rate ( QHC ) of heat rejected to the heat sink of an endoreversible one. A constant coefficient is introduced in the following expression to characterize the additional internal miscellaneous irreversibility effect: ' QHC / QHC 1
(3)
When there is only the heat resistance loss, the second law of thermodynamics requires that ' QHC Q LC x y
(4)
The first law of thermodynamics gives the power input ( P ) of the cycle P QHC QLC
(5)
From Eqs. (3) - (5), the rate ( QLC ) of heat flow from the heat source to the working fluid and the rate ( QHC ) of heat flow from the working fluid to the heat sink can be expressed as QLC
y P x y
(6)
x P (7) x y Because heat transfer between working fluid and heat reservoirs obeys Newton’s heat transfer QHC
law, one has QHC F1 ( x TH )
(8)
QLC F2 (TL y )
(9)
where and are the overall heat transfer coefficients of the high- and low-temperature-side heat exchangers, respectively.
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When the heat transfer surface area ratio is f fopt fopt
(10)
the optimal ecological function ( Eopt ) at a certain temperature ratio ( m ) is [22-24] Eopt B(aL aH m)(TL mTH )
(11)
where B F [ (1 )2 ] , aL 2T0 TL 1 , and aH 2T0 TH 1 . Taking the derivative of Eopt with respective to m and setting it equal to zero ( dEopt / dm 0 ) yields that the optimal temperature ratio ( mE ) corresponding to the maximum ecological function and the maximum ecological function ( Emax ) are [22-24] m mE aH TL (aLTH )
(12)
Emax B( TL aL TH aH )2
(13)
3. The steady state of the refrigerator working at the maximum ecological function Assume that the working fluid’s temperatures of the steady state are x and y , respectively. (In this paper, the variables with over-bars denote the steady-state values and x y ). Considering the non-endoreversible refrigerator, the rates of heat flows can be given by J1 J2
x x y y
x y
P
(14)
P
(15)
where J1 and J 2 are rates of the steady-state heat flows from the refrigerator to hot working fluid and from cool working fluid to the refrigerator, respectively, P is steady-state power input. When the refrigerator operates in a steady state, it means that the rate ( QHC ) of heat flow from x to TH equals to J1 and the rate ( QLC ) of heat flow from TL to y equals to J 2 , i.e.
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J1 F1 x TH
(16)
J 2 F2 TL y
(17)
The steady-state COP of the non-endoreversibility refrigerator ( ) is
J2 y P x y
(18)
From Eqs. (10), (14)-(18), x and y can be expressed as 1 1 1 (1 )
x TH
(19)
1 1
1 (1 ) 1 1
y TL
(20)
From Eqs. (19) and (20), the temperatures of hot reservoir and cold reservoir ( TH and TL ) can be expressed as functions of x and y , respectively, i.e. TH x
1 1 1 1 (1 )
(21)
1
TL y
1
1 (1 ) 1
(22)
Because of the non-endoreversibility of the cycle, Eq. (18) becomes 1
1
1
x y y
x y
(23)
Thus, Eqs. (21) and (22) can be written as TH
TL
xy xy y x
xy xy y x
(24)
(25)
According to Eqs. (17), (18) and (25), one can obtain the steady-state power input P as a function of x and y P
F ( x y )( x y ) (1 )( x y )
7
(26)
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4. Local stability analysis of the non-endoreversible refrigerator 4.1 Linearization and stability analysis[34,35] Consider the dynamical system with respect to x and y
Let
dx dt f x, y
(27)
dy dt g x, y
(28)
x , y be the steady-state solutions of Eqs. (27) and (28) such that
f x , y 0 and
g x , y 0 . If x and y are close to their steady-state values but not too far away, one can
write x t x x t and y t y y t , where x and y are small perturbations. Substituting these into Eqs. (27) and (28), applying Taylor’s formula to f x, y and g x, y at steady state
x , y and neglecting the two and more degrees, one can obtain the following matrix
of linear differential equations d x t f dt x d y t g x dt
f y x t g y y t
(29)
where f x f x x , y , f y f y x , y , g x g x x , y and g y g y x , y . Assume that the matrix A is the first term of right of Eq. (29) A r r
(30)
where is the eigenvalue of the matrix A , and r ( x, y) . After solving the characteristic equation: det( A I ) 0
(31)
f x g y gx f y 0
(32)
i.e.
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one can get the eigenvalues 1 and 2 of the matrix A , and the corresponding eigenvectors u1 and u2 . And thus, the general form of the solution of Eq. (29) is
r c1e t u1 c2e t u2 1
2
(33)
In general, is a complex number. If the real part of is negative, the system operating at
x , y is steady and the perturbation decays exponentially. When the system is steady, the characteristic relaxation times can be defined as t1,2 1 1,2
(34)
The characteristic relaxation time reflects the speed at which the perturbation decays, so the smaller t1,2 are, the greater 1,2
are, that is to say, after a small perturbation the system returns
to the steady state is fast, and slow on contrary conditions. In a word, the smaller the relaxation time is, the better the stability of the system is.
4.2 Local stability analysis In order to analyze the local stability of a non-endoreversible refrigerator, assume that the temperatures corresponding to macroscopic objects with heat capacity C are x and y , respectively. The dynamical equations with respect to x and y are dx dt [ J1 F1 x TH ] C
(35)
dy dt [ F2 TL y J 2 ] C
(36)
respectively, where J1 and J 2 are rates of heat flows from the refrigerator to hot working fluid and from cool working fluid to the refrigerator, respectively. According to Eqs. (14) and (15), J1 and J 2 can be written as J1
x x y
9
P
(37)
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J2
y P x y
(38)
Substituting Eqs. (37) and (38) into Eqs. (35) and (36) yields dx dt [ xP x y F1 x TH ] C
(39)
dy dt [ F2 TL y yP x y ] C
(40)
4.3 Local stability analysis of the system working at maximum ecological function When the system works in the steady state of the maximum ecological function, the optimal temperature ratio of the working fluid with the case of T0 TL is given by Eq. (12) mE y x 2 2
(41)
The temperature ratio of heat reservoirs may be written as a function of x and y 1 4
1 y2 16 2 x 2
(42)
By using the Eqs. (19), (20), (23) and (41), the steady-state values x and y , as a function of TH , can be obtained x TH
y TH
2 2 2 2 (1 )
2 2 1
(43)
(44)
When the refrigerator works out of the steady state but not too far away, the power input of the refrigerator depends on x and y in the same way that it depends on x and y at the steady state, i.e. P x, y P x , y . Thus, the dynamic equations with respect to x and y can be written as
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f x, y
F x x y [ ( x TH )] C (1 ) x y
(45)
y x y F [TL y ] C (1 ) x y
(46)
g x, y
After solving the characteristic equation (32), the eigenvalues can be obtained
1
2
F 2C
F 2C
(a d (a d )2 4 b c )
(47)
(a d (a d )2 4 b c )
(48)
and the corresponding eigenvectors are d a (a d )2 4 b c u1 ,1 2c
(49)
d a (a d )2 4 b c u2 , 1 2c
(50)
where a , b , c and d are, respectively, a
2 1 (4 1) 2 2 1 2 1 (4 1)[ 2 1 ]2
b
c
d
(4 1)[ (2 1) ]2
2 (2 1) (4 1)[ (2 1) ]2
[ 2 (4 1) 2 ( 2 1) 2 1 ] (4 1)[ 2 1 ]2
(51)
(52)
(53)
(54)
From Eqs. (51)-(54) it is easy to prove that a , b , c and d are all positive and ad bc 0 for every values of 0.5 1 , 1 , 0 and 0 , and thus, both 1 and 2 are proved to be real numbers and negative ( 1 2 ). From Eq. (34), the characteristic relaxation times can be written as t1
2C
1
F a d (a d )2 4 b c
11
(55)
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t2
2C
1
F a d (a d )2 4 b c
(56)
Because both 1 and 2 are real numbers and negative, any perturbation decays exponentially to the steady state with time and the steady state of the refrigerator working at the maximum ecological function is steady. Eqs. (55) and (56) are general expressions of the characteristic relaxation times with respect to , and . It is shown that the characteristic relaxation times are proportional to heat capacity C , inversely-proportional to the overall heat transfer surface area F , relative to , and . Relaxation times of the system working at the maximum ecological function vs. heat reservoirs’ temperature ratio with different and / 1 are shown in Fig. 2. It can be seen that t1 t2 . t1 increases slowly as increases, but the change is small, the stability of the system is almost constant. t2 decreases as increases, the stability of the system improves. t1 is an increasing function of , t2 is a decreasing function of , but the changes are both small. It is noted that the internal irreversibility has little influence on the stability of the system. Relaxation times of the system working at the maximum ecological function vs. heat reservoirs’ temperature ratio with 1.0 , 1.1 , 1.2 and different / are shown in Figs. 3-5. It can be seen that t1 and t2 decrease as / increases, the stability of the system is improved. At a given value of / 1 , t1 increases slowly as increases, but the change is small, the stability of the system is almost constant. t2 is a decreasing function of , and thus, the stability of the system is improved as increases. If / 0 , t2 , the stability of the system is lost. When / 1 , t1 increases slowly as increases, but the change is small, the stability of the system is almost constant. t2 is a decreasing function of , t2 decreases slowly as increases, the stability of the system is improved.
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The three-dimension diagrams of relaxation times of the system working at the maximum ecological function vs. heat transfer coefficients and
with 0.88 and 1.1 are
shown in Fig. 6. It can be seen that both t1 and t2 decrease as and increase, and thus, the stability of the system is improved. It can be observed from Eqs. (55) and (56) that t1 t2 , i.e., t1 / t2 1 and the corresponding eigenvectors u1 and u2 can be described as fast eigendirection and slow eigendirection, respectively. According to numerical calculations by using the relaxation time ratio and corresponding eigenvectors, the phase portraits can be plotted and the distribution information of phase portraits of system (or the topological structure of the system) may be obtained. The phase portrait of x(t ) vs. y (t ) with / 1 , 0.88 and 1.1 is shown in Fig. 7. It is calculated that the relaxation time ratio is t1 / t2 0.60 and the eigendrections are u1 (1.23,1) and u2 (1.21,1) . There are two different linear trajectories named fast eigendirection and slow eigendirection, respectively. The phase portraits show that any perturbation on x and y values tend to approach the origin (steady-state point ( x , y ) ).
5. Conclusion The local stability of a non-endoreversible Carnot refrigerator with Newton’s heat transfer law working at the maximum ecological function is analyzed, the general expressions of the relaxation times with heat capacity C , overall heat transfer surface area F , heat reservoirs’ temperature ratio , the degree of internal irreversibility, heat transfer coefficients and are obtained. According to numerical calculations, the effects of and / on relaxation
times are discussed, and the phase portrait of the system is analyzed. It is shown that any
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perturbation decays exponentially with time to the steady state, and the steady state of the system working at the maximum ecological function is steady. It is noted that the internal irreversibility has little influence on the stability of the system, and the stability of the system is improved as
increases. The relaxation times t1 and t2 decrease as / (or and ) increase(s).
And thus, the local stability of the system is improved. There are two different linear trajectories named fast eigendirection and slow eigendirection, respectively. The phase portraits show that any perturbation on x and y values tend to approach the origin (steady-state point ( x , y ) ). The results can provide some theoretical guidelines for the designs of practical refrigerator.
Acknowledgements This paper is supported by the National Natural Science Foundation of P. R. China (Project No. 10905093). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.
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endoreversible heat engine with Stefan-Boltzmann heat transfer law working in maximumpower-like regime. Open Sys. Inform. Dynam., 2006, 13(1): 43-54 [27] Huang Y, Sun D, Kang Y. Local stability characteristics of a non-endoreversible heat engine
working in the optimum region. Appl. Therm. Engng., 2009, 29(2-3): 358-363 [28] Huang Y, Sun D. The effect of cooling load and thermal conductance on the local stability of
an endoreversible refrigerator. Int. J. Refri., 2008, 31(3): 483-489 [29] Huang Y, Sun D. Local stability analysis of a non-endoreversible refrigerator. Appl. Therm.
Engng., 2008, 28(11-12): 1443-1449 [30] He J, Miao G, Nie W. Local stability analysis of an endoreversible Carnot refrigerator.
Physica Scripta, 2010, 82(2): 025002. [31] Chimal-Eguia J C, Sanchez-Sala N. On the dynamic robustness of a nonendoreversible
Curzon-Ahlborn engine working in an ecological regime. Proceedings of ECOS 2009, 22nd International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, August 31 - September 3, 2009, Foz do Iguacu, Parana, Brazil, pp.107-116 [32] Nie W, He J, Zhou F. Local stability analysis for optimal performance of an irreversible
Carnot heat engine. J. Appl. Sci., 2007, 25(4): 418-423(in Chinese) [33] Páez-Hernández R, Portillo-Díaz P, Ladino-Luna D, et al. Local stability analysis of a
Curzon-Ahlborn engine considering the Van der Waals equation state in the maximum ecological regime. Proceedings of ECOS2012 - the 25th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Perugia, Italy, 26-29, june, 2012, 281.
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[34] Arias-Hernandez L A, Paez-Hernandez R T, Hilario-Ortiz O. Local stability of an irreversible
energy converter working in the maxima COP and ecological regimes. Proceedings of ECOS 2009, 22nd International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, August 31 - September 3, 2009, Foz do Iguacu, Parana, Brazil, pp.55-61 [35] Gao W. Stability of dynamical systems. Higher Education Press, Beijing, 1987.(in Chinese) [36] Strogatz, H.S. “Non Linear Dynamics and Chaos: With Applications to Physics, Chemistry
and Engineering”, Perseus, Cambridge MA (1994).
Figure Captions Figure 1 Schematic diagram of a non-endoreversible refrigerator Figure 2 Relaxation times vs. with different and / 2 Figure 3 Relaxation times vs. with different / and 1.0 Figure 4 Relaxation times vs. with different / and 1.1 Figure 5 Relaxation times vs. with different / and 1.2 Figure 6 Relaxation times vs. and with 0.88 and 1.1 Figure 7 Phase portrait of x(t ) vs. y (t ) with / 1 , 0.88 and 1.1
18
Figure 1 Schematic diagram of a non-endoreversible refrigerator
Figure 2 Relaxation times vs. with different and / 2
Figure 3 Relaxation times vs. with different / and 1.0
Figure 4 Relaxation times vs. with different / and 1.1
Figure 5 Relaxation times vs. with different / and 1.2
Figure 6 Relaxation times vs. with different /
Figure 7 Phase portrait of x(t ) vs. y (t ) with / 1 , 0.9 and 1.1
S0307-904X(14)00466-1 http://dx.doi.org/10.1016/j.apm.2014.09.031 APM 10146
To appear in:
Appl. Math. Modelling
Received Date: Revised Date: Accepted Date:
10 February 2011 20 December 2013 17 September 2014
Please cite this article as: W. Xiaohui, C. Lingen, G. Yanlin, S. Fengrui, Local stability of a non-endoreversible Carnot refrigerator working at the maximum ecological function, Appl. Math. Modelling (2014), doi: http:// dx.doi.org/10.1016/j.apm.2014.09.031
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Local stability of a non-endoreversible Carnot refrigerator working at the maximum ecological function
Wu Xiaohui1, 2, 3, Chen Lingen1, 2, 3, *, Ge Yanlin1, 2, 3, Sun Fengrui1, 2, 3 (1 Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033) (2 Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033) (3 College of Power Engineering, Naval University of Engineering, Wuhan 430033)
* To whom all correspondence should be addressed. E-mail address: [email protected] and [email protected] (Lingen Chen) Fax: 0086-27-83638709 Tel: 0086-27-83615046
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Abstract Local stability of a non-endoreversible Carnot refrigerator at the maximum ecological function is studied with Newton’s heat transfer law between working fluid and heat reservoirs. The steady state of the refrigerator working at the maximum ecological function is steady. It is derived that a general expression of relaxation time described as stability of the system refers to heat capacity C , total heat exchange area F , temperature ratio of heat reservoirs , the degree of internal irreversibility , heat transfer coefficients and . Distributing information of phase portraits of system is obtained. The results can provide some theoretical guidelines for the designs of practical refrigerator. Keywords: local stability, non-endoreversible Carnot refrigerator, temperature ratio of heat reservoirs, ecological optimization
1. Introduction Since Curzon and Ahlborn [1] proposed the efficiency at maximum power output, for an endoreversible Carnot heat engine with Newton’s heat transfer law, i.e. so-called CA efficiency in 1975, finite time thermodynamics (FTT) has seen tremendous progress by scientists and engineers [2-9]. Leff and Teeter [10] were the first to extend the Curzon–Ahlborn analysis method [1] to the analysis of refrigeration cycles. Rozonoer and Tsirlin [11] first derived the maximum coefficient of performance (COP) bound for fixed power input to a Newton’s law endoreversible Carnot refrigerator. Yan [12], Goth and Feidt [13] , Feidt [14], Philippi and Feidt [15], and Feidt [16] derived the optimal COP for the fixed cooling load, i.e. the fundamental optimal relation of Newton’s law endoreversible Carnot refrigerator. However, real refrigerators are usually devices
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with both internal and external irreversibilities. Chen et al. [17,18] established a generalized irreversible Carnot refrigerator model, which takes account for the effects of heat resistance, bypass heat leakage, and other internal irreversibility, and derived its fundamental optimal relation between cooling load and COP. Angulo-Brown [19] proposed an ecological criterion E P TL for finite-time Carnot heat engines, where TL is the temperature of cold heat reservoir, P is the power output and is the entropy generation rate. Yan [20] showed that it might be more reasonable to use E P T0 if the cold reservoir temperature TL is not equal to the environmental temperature T0 , the ecological function E represents a compromise between the power output P and
power loss T0 due to the entropy generation in the system. Furthermore, based on the viewpoint of exergy analysis, Chen et al. [21] provided a unified ecological optimization objective function for all of the thermodynamic cycles, that is E A / t T0 S / t A / t T0
(1)
where A is the exergy output of the cycle, S is the entropy generation of the cycle, and t is the cycle period. Equation (1) represents the best compromise between the exergy output rate and the exergy loss rate of the thermodynamic cycles. For heat engine cycles, the exergy output rate of the cycle is A / t P . For refrigeration cycles, the exergy output rate of the cycle is A / t QL (T0 / TL 1) QH (T0 / TH 1) , where QL is the rate of heat transfer supplied by the heat
source, QH is the rate of heat transfer released to the heat sink, and TH and TL are temperatures of the heat sink and heat source, respectively. Therefore, the ecological function of refrigeration cycles is E QL (T0 / TL 1) QH (T0 / TH 1) T0
2
(2)
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Chen et al. [22] , Zhu et al. [23] and Chen et al. [24] investigated the optimal ecological performance of generalized irreversible Carnot refrigerators considering the effects of heat resistance, heat leakage and internal irreversibility with Newton’s heat transfer law. [22] , the generalized radiative heat transfer law [23] and generalized convective heat transfer law [24]. Most of the previous works about the endoreversible and irreversible Carnot refrigerators have concentrated on the steady-state energetic properties of the systems. However, it is worthwhile to design the systems considering the local stability of the refrigerator’s steady state. Santillan et al. [25] firstly studied the local stability of a Curzon-Ahlborn-Novikov (CAN) engine working in a maximum- power-like regime considering the heat resistance and the equal highand-low-temperature heat transfer coefficients with Newton’s heat transfer law. Chimal-Eguia et al. [26] analyzed the local stability of an endoreversible heat engine working in a maximum-power-like regime with Stefan-Boltzmann heat transfer law. Huang et al. [27] analyzed the local stability of a non-endoreversible heat engine working between the maximum power output and the maximum efficiency with Newton’s heat transfer law. Huang and Sun [28, 29] studied the local stability of the endoreversible [28] and non-endoreversible [29] refrigerators operating at minimum input power for a given cooling load with Newton’s heat transfer law, and analyzed the effects of cooling load and heat transfer coefficients on the local stability of the system. He et al. [30] studied the local stability of the endoreversible Carnot refrigerators operating at maximum objective function of the product of the cooling load and COP with Newton’s heat transfer law. Chimal-Eguia et al. [31] studied the local stability of a non-endoreversible heat engine working at the maximum ecological function with Newton’s heat transfer law, and offered the phase portraits of the system. Nie et al. [32] studied the local stability
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of an irreversible heat engine with the losses of the heat leakage and heat resistance working at the maximum ecological function with Newton’s heat transfer law. Páez-Hernández et al. [33] analyzed the local stability of an endoreversible Curzon-Ahlborn-Novikov engine, using Van der Waals gas as a working substance and the corresponding efficiency for this engine working at temperatures within the maximum ecological regime. Arias-Hernandez et al. [34] studied the local stability of the refrigeration cycle working at the maximum ecological function with Newton’s heat transfer law, and concluded that temperature ratio of heat reservoirs represents a trade-off between stronger stability and better thermodynamic properties like power and efficiency for heat engine. On the bases of the Refs. [32, 50-51], this paper will analyze the local stability of a non-endoreversible refrigerator with Newton’s heat transfer law working at the maximum ecological function, discuss the effects of the degree of internal irreversibility ( ), heat reservoirs’ temperature ratio ( ), heat transfer coefficients ( and ) on the stability of the system, analyze the distribution information of phase portraits of the system.
2. Ecological performance of a non-endoreversible refrigerator Considering a model of a non-endoreversible Carnot refrigerator [22-24,34] as shown in Fig. 1, its working conditions are as follows: (1) The working fluid flows through the system in a time-invariant continuous way. The cycle consists of two isothermal and two adiabatic processes which are irreversible in general. (2) Because of the heat resistance, the working fluid’s temperatures ( x and y ) are different from the reservoirs’ temperatures ( TH and TL ) and the four temperatures are of the following decreasing order: x TH TL y . The heat transfer surface areas ( F1 and F2 ) of the high- and low-temperature side heat exchangers are finite. The overall heat transfer surface area ( F ) of the 4
two heat exchangers is assumed to be a constant: F F1 F2 . Assume that the heat transfer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
surface area ratio is f F1 / F2 , the working fluid’s temperature ratio is m x / y and the temperature ratio of heat reservoirs is TL / TH . Thus, 0 y x TL TH 1 . (3) There are irreversibilities in the system due to heat resistance and miscellaneous factors such as friction, turbulenceand non-equilibrium activities inside the refrigerator, etc.. Thus, when compared with an endoreversible Carnot refrigerator of the same cooling load, a larger power supply is needed. In other words, the rate ( QHC ) of heat rejected to the heat sink of the non-endoreversible Carnot refrigerator is higher than that the rate ( QHC ) of heat rejected to the heat sink of an endoreversible one. A constant coefficient is introduced in the following expression to characterize the additional internal miscellaneous irreversibility effect: ' QHC / QHC 1
(3)
When there is only the heat resistance loss, the second law of thermodynamics requires that ' QHC Q LC x y
(4)
The first law of thermodynamics gives the power input ( P ) of the cycle P QHC QLC
(5)
From Eqs. (3) - (5), the rate ( QLC ) of heat flow from the heat source to the working fluid and the rate ( QHC ) of heat flow from the working fluid to the heat sink can be expressed as QLC
y P x y
(6)
x P (7) x y Because heat transfer between working fluid and heat reservoirs obeys Newton’s heat transfer QHC
law, one has QHC F1 ( x TH )
(8)
QLC F2 (TL y )
(9)
where and are the overall heat transfer coefficients of the high- and low-temperature-side heat exchangers, respectively.
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When the heat transfer surface area ratio is f fopt fopt
(10)
the optimal ecological function ( Eopt ) at a certain temperature ratio ( m ) is [22-24] Eopt B(aL aH m)(TL mTH )
(11)
where B F [ (1 )2 ] , aL 2T0 TL 1 , and aH 2T0 TH 1 . Taking the derivative of Eopt with respective to m and setting it equal to zero ( dEopt / dm 0 ) yields that the optimal temperature ratio ( mE ) corresponding to the maximum ecological function and the maximum ecological function ( Emax ) are [22-24] m mE aH TL (aLTH )
(12)
Emax B( TL aL TH aH )2
(13)
3. The steady state of the refrigerator working at the maximum ecological function Assume that the working fluid’s temperatures of the steady state are x and y , respectively. (In this paper, the variables with over-bars denote the steady-state values and x y ). Considering the non-endoreversible refrigerator, the rates of heat flows can be given by J1 J2
x x y y
x y
P
(14)
P
(15)
where J1 and J 2 are rates of the steady-state heat flows from the refrigerator to hot working fluid and from cool working fluid to the refrigerator, respectively, P is steady-state power input. When the refrigerator operates in a steady state, it means that the rate ( QHC ) of heat flow from x to TH equals to J1 and the rate ( QLC ) of heat flow from TL to y equals to J 2 , i.e.
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J1 F1 x TH
(16)
J 2 F2 TL y
(17)
The steady-state COP of the non-endoreversibility refrigerator ( ) is
J2 y P x y
(18)
From Eqs. (10), (14)-(18), x and y can be expressed as 1 1 1 (1 )
x TH
(19)
1 1
1 (1 ) 1 1
y TL
(20)
From Eqs. (19) and (20), the temperatures of hot reservoir and cold reservoir ( TH and TL ) can be expressed as functions of x and y , respectively, i.e. TH x
1 1 1 1 (1 )
(21)
1
TL y
1
1 (1 ) 1
(22)
Because of the non-endoreversibility of the cycle, Eq. (18) becomes 1
1
1
x y y
x y
(23)
Thus, Eqs. (21) and (22) can be written as TH
TL
xy xy y x
xy xy y x
(24)
(25)
According to Eqs. (17), (18) and (25), one can obtain the steady-state power input P as a function of x and y P
F ( x y )( x y ) (1 )( x y )
7
(26)
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4. Local stability analysis of the non-endoreversible refrigerator 4.1 Linearization and stability analysis[34,35] Consider the dynamical system with respect to x and y
Let
dx dt f x, y
(27)
dy dt g x, y
(28)
x , y be the steady-state solutions of Eqs. (27) and (28) such that
f x , y 0 and
g x , y 0 . If x and y are close to their steady-state values but not too far away, one can
write x t x x t and y t y y t , where x and y are small perturbations. Substituting these into Eqs. (27) and (28), applying Taylor’s formula to f x, y and g x, y at steady state
x , y and neglecting the two and more degrees, one can obtain the following matrix
of linear differential equations d x t f dt x d y t g x dt
f y x t g y y t
(29)
where f x f x x , y , f y f y x , y , g x g x x , y and g y g y x , y . Assume that the matrix A is the first term of right of Eq. (29) A r r
(30)
where is the eigenvalue of the matrix A , and r ( x, y) . After solving the characteristic equation: det( A I ) 0
(31)
f x g y gx f y 0
(32)
i.e.
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one can get the eigenvalues 1 and 2 of the matrix A , and the corresponding eigenvectors u1 and u2 . And thus, the general form of the solution of Eq. (29) is
r c1e t u1 c2e t u2 1
2
(33)
In general, is a complex number. If the real part of is negative, the system operating at
x , y is steady and the perturbation decays exponentially. When the system is steady, the characteristic relaxation times can be defined as t1,2 1 1,2
(34)
The characteristic relaxation time reflects the speed at which the perturbation decays, so the smaller t1,2 are, the greater 1,2
are, that is to say, after a small perturbation the system returns
to the steady state is fast, and slow on contrary conditions. In a word, the smaller the relaxation time is, the better the stability of the system is.
4.2 Local stability analysis In order to analyze the local stability of a non-endoreversible refrigerator, assume that the temperatures corresponding to macroscopic objects with heat capacity C are x and y , respectively. The dynamical equations with respect to x and y are dx dt [ J1 F1 x TH ] C
(35)
dy dt [ F2 TL y J 2 ] C
(36)
respectively, where J1 and J 2 are rates of heat flows from the refrigerator to hot working fluid and from cool working fluid to the refrigerator, respectively. According to Eqs. (14) and (15), J1 and J 2 can be written as J1
x x y
9
P
(37)
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J2
y P x y
(38)
Substituting Eqs. (37) and (38) into Eqs. (35) and (36) yields dx dt [ xP x y F1 x TH ] C
(39)
dy dt [ F2 TL y yP x y ] C
(40)
4.3 Local stability analysis of the system working at maximum ecological function When the system works in the steady state of the maximum ecological function, the optimal temperature ratio of the working fluid with the case of T0 TL is given by Eq. (12) mE y x 2 2
(41)
The temperature ratio of heat reservoirs may be written as a function of x and y 1 4
1 y2 16 2 x 2
(42)
By using the Eqs. (19), (20), (23) and (41), the steady-state values x and y , as a function of TH , can be obtained x TH
y TH
2 2 2 2 (1 )
2 2 1
(43)
(44)
When the refrigerator works out of the steady state but not too far away, the power input of the refrigerator depends on x and y in the same way that it depends on x and y at the steady state, i.e. P x, y P x , y . Thus, the dynamic equations with respect to x and y can be written as
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f x, y
F x x y [ ( x TH )] C (1 ) x y
(45)
y x y F [TL y ] C (1 ) x y
(46)
g x, y
After solving the characteristic equation (32), the eigenvalues can be obtained
1
2
F 2C
F 2C
(a d (a d )2 4 b c )
(47)
(a d (a d )2 4 b c )
(48)
and the corresponding eigenvectors are d a (a d )2 4 b c u1 ,1 2c
(49)
d a (a d )2 4 b c u2 , 1 2c
(50)
where a , b , c and d are, respectively, a
2 1 (4 1) 2 2 1 2 1 (4 1)[ 2 1 ]2
b
c
d
(4 1)[ (2 1) ]2
2 (2 1) (4 1)[ (2 1) ]2
[ 2 (4 1) 2 ( 2 1) 2 1 ] (4 1)[ 2 1 ]2
(51)
(52)
(53)
(54)
From Eqs. (51)-(54) it is easy to prove that a , b , c and d are all positive and ad bc 0 for every values of 0.5 1 , 1 , 0 and 0 , and thus, both 1 and 2 are proved to be real numbers and negative ( 1 2 ). From Eq. (34), the characteristic relaxation times can be written as t1
2C
1
F a d (a d )2 4 b c
11
(55)
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t2
2C
1
F a d (a d )2 4 b c
(56)
Because both 1 and 2 are real numbers and negative, any perturbation decays exponentially to the steady state with time and the steady state of the refrigerator working at the maximum ecological function is steady. Eqs. (55) and (56) are general expressions of the characteristic relaxation times with respect to , and . It is shown that the characteristic relaxation times are proportional to heat capacity C , inversely-proportional to the overall heat transfer surface area F , relative to , and . Relaxation times of the system working at the maximum ecological function vs. heat reservoirs’ temperature ratio with different and / 1 are shown in Fig. 2. It can be seen that t1 t2 . t1 increases slowly as increases, but the change is small, the stability of the system is almost constant. t2 decreases as increases, the stability of the system improves. t1 is an increasing function of , t2 is a decreasing function of , but the changes are both small. It is noted that the internal irreversibility has little influence on the stability of the system. Relaxation times of the system working at the maximum ecological function vs. heat reservoirs’ temperature ratio with 1.0 , 1.1 , 1.2 and different / are shown in Figs. 3-5. It can be seen that t1 and t2 decrease as / increases, the stability of the system is improved. At a given value of / 1 , t1 increases slowly as increases, but the change is small, the stability of the system is almost constant. t2 is a decreasing function of , and thus, the stability of the system is improved as increases. If / 0 , t2 , the stability of the system is lost. When / 1 , t1 increases slowly as increases, but the change is small, the stability of the system is almost constant. t2 is a decreasing function of , t2 decreases slowly as increases, the stability of the system is improved.
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The three-dimension diagrams of relaxation times of the system working at the maximum ecological function vs. heat transfer coefficients and
with 0.88 and 1.1 are
shown in Fig. 6. It can be seen that both t1 and t2 decrease as and increase, and thus, the stability of the system is improved. It can be observed from Eqs. (55) and (56) that t1 t2 , i.e., t1 / t2 1 and the corresponding eigenvectors u1 and u2 can be described as fast eigendirection and slow eigendirection, respectively. According to numerical calculations by using the relaxation time ratio and corresponding eigenvectors, the phase portraits can be plotted and the distribution information of phase portraits of system (or the topological structure of the system) may be obtained. The phase portrait of x(t ) vs. y (t ) with / 1 , 0.88 and 1.1 is shown in Fig. 7. It is calculated that the relaxation time ratio is t1 / t2 0.60 and the eigendrections are u1 (1.23,1) and u2 (1.21,1) . There are two different linear trajectories named fast eigendirection and slow eigendirection, respectively. The phase portraits show that any perturbation on x and y values tend to approach the origin (steady-state point ( x , y ) ).
5. Conclusion The local stability of a non-endoreversible Carnot refrigerator with Newton’s heat transfer law working at the maximum ecological function is analyzed, the general expressions of the relaxation times with heat capacity C , overall heat transfer surface area F , heat reservoirs’ temperature ratio , the degree of internal irreversibility, heat transfer coefficients and are obtained. According to numerical calculations, the effects of and / on relaxation
times are discussed, and the phase portrait of the system is analyzed. It is shown that any
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perturbation decays exponentially with time to the steady state, and the steady state of the system working at the maximum ecological function is steady. It is noted that the internal irreversibility has little influence on the stability of the system, and the stability of the system is improved as
increases. The relaxation times t1 and t2 decrease as / (or and ) increase(s).
And thus, the local stability of the system is improved. There are two different linear trajectories named fast eigendirection and slow eigendirection, respectively. The phase portraits show that any perturbation on x and y values tend to approach the origin (steady-state point ( x , y ) ). The results can provide some theoretical guidelines for the designs of practical refrigerator.
Acknowledgements This paper is supported by the National Natural Science Foundation of P. R. China (Project No. 10905093). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.
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Figure Captions Figure 1 Schematic diagram of a non-endoreversible refrigerator Figure 2 Relaxation times vs. with different and / 2 Figure 3 Relaxation times vs. with different / and 1.0 Figure 4 Relaxation times vs. with different / and 1.1 Figure 5 Relaxation times vs. with different / and 1.2 Figure 6 Relaxation times vs. and with 0.88 and 1.1 Figure 7 Phase portrait of x(t ) vs. y (t ) with / 1 , 0.88 and 1.1
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Figure 1 Schematic diagram of a non-endoreversible refrigerator
Figure 2 Relaxation times vs. with different and / 2
Figure 3 Relaxation times vs. with different / and 1.0
Figure 4 Relaxation times vs. with different / and 1.1
Figure 5 Relaxation times vs. with different / and 1.2
Figure 6 Relaxation times vs. with different /
Figure 7 Phase portrait of x(t ) vs. y (t ) with / 1 , 0.9 and 1.1