Open Sys. & Information Dyn. (2007) 14:411–424 DOI: 10.1007/s11080-007-9065-z

© Springer 2007

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime J. C. Chimal-Egu´ıa Centro de Investigaci´ on en Computaci´ on, CIC-IPN, Av. Juan de Dios B´ atiz s/n esq. Miguel Othon de Mendizabal, col. San Pedro Zacatenco, C. P. 07738, M´exico D. F.

I. Reyes-Ram´ırez and L. Guzm´ an-Vargas Unidad Profesional Interdisciplinaria en Ingenier´ıa y Tecnolog´ıas Avanzadas, Instituto Polit´ecnico Nacional, Av. IPN No. 2580, L. Ticom´ an, M´exico D.F. 07340, M´exico e-mail: [email protected]

(Received: January 15, 2007) Abstract. We present a local stability analysis of an endoreversible engine working in an ecological regime, for three common heat transfer laws. From our local stability analysis we conclude that the system is stable for every value of the heat conductivity g, the heat capacity C and the ratio of temperatures τ = T2 /T1 with T1 > T2 . After a small perturbation the system decays exponentially to the steady state determined by two different relaxation times. We observe that the stability of the system improves as τ increases whereas the steady-state energetic properties of the engine decline. Moreover, we compare the stability properties of the engine working in the ecological regime and under maximum power output. Finally, qualitative phase-space portraits for the evolution of the system are presented for representative cases.

1.

Introduction

In recent years, the finite-time operation of thermal engines has been characterized in the context of different optimization criteria. For instance, the maximization of power output (MP), the minimization of entropy generation (MEG), and the maximization of an ecological function (ME) [1 – 8]. The MP was used by Curzon and Ahlborn [1, 2] to describe the so-called Curzon-Alhborn engine (CA). The CA model was the starting point of a new branch called finite time thermodynamics (FTT). The CA engine model consists of an internal reversible cycle working between two internal temperatures and irreversible heat flows in the couplings of the working fluid with the hot and cold reservoirs. For this model, one finds that the efficiency that maximizes the power output is the well known Curzon-Alhborn
efficiency ηCA = 1 − T2 /T1 with T1 and T2 the temperatures of the hot and cold reservoirs, respectively. One of the goals of the CA engine model is to consider irreversible flows from the thermal sources to the working fluid for arriving to a more realistic efficiency. The ecological optimization criterion for the FTT-thermal cycles was proposed by Angulo-Brown [8]. This criterion considers the maximization of a function E, called the ecological function, which represents a compromise between high power

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J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas

output and low entropy production. The E function is given by E = P − T2 σ ,

(1)

where P is the power output of the cycle, σ the total entropy production per cycle and T2 is the temperature of the cold reservoir. One of the most important characteristics of a CA engine operating under maximum E conditions is that it produces around 80% of the maximum power and only 30% of the entropy produced in the maximum power regime [8, 9]. Another interesting property of the maximum-E regime is that the CA-engine’s efficiency in this regime, is given by ηE ≈ (ηC + ηCA )/2, where ηC is the Carnot efficiency [9]. The ME criterion has been successfully applied to describe energetic properties of both some realengine models and biological processes, such as biochemical reactions [12, 9]. In fact, between the efficiencies ηCA and ηC , one can find an infinity number of modes of performance for the CA-cycle. However, only some of them are interesting from a designing point of view. For example, in (1), the dissipative term T2 σ could be multiplied by an arbitrary coefficient α, but, such as it was shown in [9], the case α = 1 is which reaches the best trade off between high power output and low entropy production, at least for heat transfer laws concomitant with a parabolic behaviour of the function P = P (η) [9]. The function given by (1), was later generalized [10] for arbitrary heat transfer laws getting the form E = P − εT2 σ, where ε is a quantity depending of the particular heat transfer law used to model the heat exchange between thermal reservoirs and the working substance. Traditional FTT studies have focused on the steady-state energetic properties, which are important from the point of view of design. However, only few of these studies have dealt with the system’s dynamical properties. In this sense, if one needs to have a well designed system, it must has good energetic as well as dynamical properties (such as stable steady-states). For instance, in the operation of a real heat engine the variables involved in the description of the system frequently are influenced by external events or past states [11]. For this reason, it is important to evaluate the ability of the system to return to the steady-state regime in order to carry out its function well. In 2001, Santillan et al. [13] described an analysis on the stability of the CA engine performing in a maximum power regime. Later, Guzm´ an-Vargas et al. [14] generalized the Santillan’s ideas to different heat transfer laws but considering, as Santillan did, a maximum power-like regime. Following the ideas of these authors, we present a local stability analysis of a CA engine, but now considering an ecological optimization criterion for different heat transfer laws. Here, we address the question of how stable is the system when a particular regime of operation is considered and what are the advantages from the perspective of stability properties. The paper is organized as follows: In Sect. 2, we present the properties of the CA engine in steady-state conditions. In Sect. 3, the local stability analysis of a CA engine operating in an ecological regime for different heat transfer laws is presented. Finally in Sect. 4, we present some concluding remarks. 2.

Some Properties of the Steady State CA Engine

In this section we shall introduce some well-known results concerning the CA engine which are necessary for the rest of the paper. Consider an endoreversible

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime

413

T1 g

x

Q1 W Ca Q2 y g

T2 Fig. 1: Schematic representation of the CA engine.This engine consists of a Carnot engine, which in every cycle exchange heat in amounts Q1 and Q2 with the heat reservoirs T1 and T2 respectively (T1 > T2 ), by means of a certain kind of heat transfer law CA engine [1] pictured in Fig. 1. This engine works between two heat reservoirs with absolute temperatures T1 and T2 , (T1 > T2 ) respectively. We have two heat flows: from the hot reservoir (at temperature T1 ) to x, and from y to the cold reservoir ( at temperature T2 ), where x and y denote the temperatures of the working fluid along the isothermal branches of the internal reversible Carnot engine (see Fig. 1). We have used overbars to denote steady-state values of temperatures. The internal Carnot engine exchanges heat in amounts Q1 and Q2 with the heat reservoirs with temperatures x and y, respectively. The heat flows are governed by a certain kind of thermal transfer law through thermal resistances denoted by g in Fig. 1. The internal reversibility further implies  x  Q1 = W (2) x−y and

 Q2 =

y  W, x−y

(3)

where Q1 and Q2 are the steady-state heat flows from x to the engine and from the engine to y, respectively (see Fig. 1). The quantity W is the engine power output. Initially, we consider that the CA engine works in steady state conditions which means that the heat flow from the heat reservoir (at temperature T1 ) to x equals Q1 , and Q2 equals the heat flow from y to the second heat reservoir (at temperature T2 ), that is, Q1 = g(T1r − xr ) (4) and

Q2 = g(y r − T2r ) ,

(5)

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J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas

where the exponent r is a parameter related to different heat transfer laws and g is the thermal conductance at the high- and low-temperature branches of the cycle. For simplicity of the calculations, we have assumed that the heat exchanges take place in conductors with the same thermal conductance g. From (2)–(4) and the definition of efficiency, η =

W , Q1

(6)

we can express the temperature x as follows: xr =

T1r [(1 − η) + τ r ] , (1 − η) + (1 − η)r

(7)

with τ = T2 /T1 , and for y we obtain y = x(1 − η) .

(8)

After substituting (7) into (8) we get, yr =

T2r [(1 − η) τ −r + 1]

(9)

(1 − η)1−r + 1

and from (8) and (9) we have xr [(1 − η) + (1 − η)r ] (1 − η) + τ r

T1r = and T2r =

  yr (1 − η)1−r + 1 (1 − η) τ −r + 1

.

(10)

(11)

Using the fact that the cycle is internally reversible with efficiency given by (8), (10) and (11) can be written as T1r =

xr y + y r x y + xτ r

(12)

T2r =

xr y + yr x . yτ −r + x

(13)

and

From (4), (6) and (7), it is possible to express the power output as W = gT1r η



(1 − η)r − τ r  . (1 − η) + (1 − η)r

(14)

Finally, substituting (8) and (12) into (14) we obtain W =

g (x − y) (y r − xr τ r ) . y + xτ r

(15)

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime

3.

415

Numerical Stability Analysis of the CA Engine in an Ecological Regime for Different Heat Transfer Laws

Following Santillan et al. [13], we can derive a system of differential equations of the CA engine which provide information about the stability of the system. We assume that the working fluid is in thermal equilibrium with two intermediate heat reservoirs taken as two macroscopic objects with temperatures x and y, and both with the same heat capacity C (see Fig. 1). We now consider that these internal temperatures show a rate of change given by the following balance equations [13], 1 dx = [g (T1r − xr ) − Q1 ] dt C

(16)

and

1 dy (17) = [Q2 − g (y r − T2r )] , dt C where Q1 and Q2 are the heat flows from x to the working substance and from the reversible engine (Carnot’s engine) to y, respectively. Notice that our study is limited to a linear heat storage law given by Q = CdT /dt, with T the temperature of the intermediate reservoir. We can observe that in steady-state conditions (16) and (17) lead to (4) and (5). The endoreversible hypothesis further implies that heat flows are given by  x  W (18) Q1 = x−y 

and Q2 =

y  W. x−y

(19)

The substitution of (18) and (19) into (16) and (17) leads to

and

3.1.

 x   1 dx = g(T1r − xr ) + W dt C x−y

(20)

 dy 1  y  = W − g(y r − T2r ) . dt C x−y

(21)

Case r = 1

When r = 1, (4) and (5) become the Newton’s heat transfer law. For this case, the efficiency that maximizes the function E [15], is given by η = 1 − τ 3/4 .

(22)

The substitution of (22) and r = 1 into (14) results in W =

 gT1  1 − τ 1/4 − τ 3/4 + τ . 2

(23)

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J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas

The steady state values x and y, as functions of T1 and T2 , can be obtained after the substitution of (22) and r = 1, into (7) and (9), that is, x = and y=

T1 (1 + τ 1/4 ) 2

T1 τ (1 + τ −1/4 ) . 2

From (8) and (22), we obtain

(24)

(25)

 y 4/3

. (26) x Finally, the substitution of (26) into (23) gives the steady state power output in an ecological regime, g (x − y) (x1/3 − y 1/3 ) W = . (27) x1/3 + y 1/3 τ =

Using the assumption that, out of the steady state, the power of the CA engine depends on x and y in the same way as it depends on x and y in the steady state, it is possible to write the dynamic equations for x and y ((20) and (21) with r = 1) as [13] x1/3 − y 1/3  dx g (T1 − x) − x 1/3 (28) = dt C x + y 1/3 and

 dy g  x1/3 − y 1/3 y 1/3 − (y − T2 ) . = 1/3 dt C x +y

(29)

Next, we proceed following typical procedures for local stabilty analysis of a twodimensional nonlinear system which details are explained in references [16, 13, 14]. From (28) and (29), we arrive to an eigenvalue equation which gives the information about the stability properties of the steady state. After solving the corresponding eigenvalue equation, we find that both eigenvalues (λ1 and λ2 ) are function of g, C, and τ . The final expressions and the algebraic details are not shown because they are quite lengthy and can be easily reproduced with the help of a symbolic algebra package. Moreover, our calculations show that both eigenvalues are real and negative. Thus, the steady-state is a stable node because any perturbation would decay exponentially with time. The characteristic times (which are defined as t = 1/|λ|) as a function of τ are showed E and tM E are, the more quickly the in Fig. 2a. It is clear that, the smaller tM 1 2 E system returns to the steady state after a small perturbation. For the case tM 1 M E versus τ , we observe that as τ increases, t1 also increases until τ ≈ 0.2, above E this value it remains almost constant. For tM 2 , we observe that it is a decreasing function of τ , indicanting that the system’s stability improves as τ increases and approaches to one. For a comparison of the ME regime with the stability properties of a CA engine working under MP [13], we also plotted in Fig. 2a, the E is relaxation times obtained in the MP case [13, 14]. The relaxation time tM 1 M P M E M P greater than t1 while t2 is smaller than t2 . If we plot the ratio of relaxation

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime

1.75

Max. Power Max. Ecological

1.5 1.25 1 0.75 0.5 0.25

Relaxation times ratio t s/tf

Relaxation times (x g/C)

417

r=1

(a)

2

(b)

r=1

5

Max. Power Max. Ecological

4 3 2 1

0

0.2

0.4

t

0.6

0.8

1

Fig. 2: (a) Plot of relaxations times vs. τ for maximum-ecological and maximumpower regimes for r = 1. (b) Plot of relaxation times ratio ts /tf vs. τ . times defined as ts /tf vs. τ with ts = t2 and tf = t1 the characteristic times corresponding to the slow and fast eigendirection, respectively, we observe that the ratio in the ecological regime is smaller than the value under MP (see Fig. 2b). For a relaxation times ratio close to one, the approaches to the steady state are almost symmetrical with respect to the eigendirections. From these considerations, we remark that the system’s stability, in the ME regime, shows small differences with respect to the MP case. The characteristic time scales represent the relaxation times towards the steady state along each eigendirection. A comparison between both characteristic times E < tM E , that is, tM E corresponds to the fast eigendirection while shows that tM 1 2 1 M E t2 is related to the slow one. We calculate the corresponding eigenvector for each eigenvalue. With this information we are able to construct a qualitative phase portrait to illustrate the approach to the steady state after a small perturbation. In Fig. 3 a representative case is described for τ = 0.5, the system approachs the origin tangent to the slow eigendirection. 3.2.

Case r = −1

Now we consider the case when r = −1. Then, (4) and (5) represent the linear phenomenological heat transfer law of irreversible thermodynamics [9, 17]. Now, g

418

J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas

r=1 t=0.5

y on

cti

e dir

en

fast

low

eig

s

eigen

direc

tion

x

Fig. 3: Qualitative phase portrait of x(t) vs y(t) for r = 1 and τ = 12 . Both eigenvalues are negatives, that is, x(t) and y(t) decay exponentially to the origin (steady-state values x, y). In this case, a comparison between the relaxation times shows that t1 < t2 , that is, t1 corresponds to the fast eigendirection while t2 is related to the slow one. Thus, the system approaches the origin tangent to the slow eigendirection.

is negative and for this case the efficiency that maximizes the function E is given by [15] √ 3 + (2 − τ ) τ − 5 + 2τ + 6τ 2 + 2τ 3 + τ 4 . (30) η = 1 + 3τ The substitution of (30) and r = −1 into (14) produces gη  τ + η − 1  T1 τ (1 − η)2 + 1

W =

(31)

with η given by (30). The steady state values x and y, as functions of T1 and T2 respectively, in the ecological regime can be obtained by substituting (30) and r = −1, into (7) and (9). That is, τ T1  (1 − η)2 + 1  x = (32) 1 − η (1 − η) τ + 1 and y = T2

 (1 − η)2 + 1  (1 − η) τ + 1

.

(33)

From (8) and (30) we have 2

τe =

−3 − 2 xy + 3 xy 2 +



2

3

4

9 + 4 xy + 18 yx2 − 4 yx3 + 9 yx4 4 xy

.

(34)

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime

419

The substitution of r = −1 into (15) yields W = g

(x − y) (xτe − y) , xy (yτe + x)

(35)

where τe is given by (34). Using again the assumption that, out of the steady state, the power of the CA engine depends on x and y in the same way as it depends on x and y in the steady state, the dynamic equations for x and y are now of the form: dx g  1 1  1  xτe − y  = − (36) − dt C T1 x y yτe + x and

dy g  1  xτe − y   1 1  = − − . dt C x yτe + x y T2

(37)

For this case, after solving the corresponding eigenvalue equation, we also found that both eigenvalues (λ3 and λ4 ) are real and negative. As in the case r = 1, we see that any perturbation exponentially decays and thus the steady state is stable for every value of g, C, and τ . In Fig. 4a we plot the characteristic times as a function of τ . We observe that E shows a small decrement, that is, the relaxation time remains as τ increases tM 3 E almost constant for any value of τ in the interval 0 < τ < 1. For tM 4 , a decreasing function of τ is observed, indicating that the system’s stability improves when τ approaches to one. A comparison of the stability properties of ME and MP regimes E and tM E is also presented in Fig. 4a [14]. In this case, both relaxation times tM 3 4 are greater than the corresponding times in the MP regime. Moreover, we also plot the ratio rs /rf vs τ where ts = t4 and tf = t3 correspond to the slow and fast eigendirection, respectively (see Fig. 4b). Two regions are observed, above the value of τ ∗ ≈ 0.5, the (ts /tf )M E is smaller than (ts /tf )M P , indicating that the value of the times are closer each other than in the MP case. The opposite situation is observed below τ ∗ . In Fig. 5 we present a qualitative phase space of trajectories after a small perturbation about the steady state for τ = 0.5. As we can see, the trajectories are influenced by the value of the eigendirections. Close to the steady state, trajectories are tangent to the slow eigendirection which is almost parallel to the horizontal axis, whereas in backwards time the trajectories are parallel to the fast eigendirection, indicating that a small change in the temperature x corresponds to a large change in the temperature y. 3.3.

Case r = 4

Now we consider the case when r = 4. Then (4) and (5) are in the form of StefanBoltzman law. Using (14) we can write the Ecological function (1) as a function of the steady-state efficiency η,   (1 − η)4 − τ 4 4 E = gT1 η (38) − T2 σ . (1 − η) − (1 − η)4

J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas

Relaxation times ratio t s/tf

Relaxation times (x g/CT22)

420

(a)

12

r=-1

10

Max. Power Max. Ecological

8 6 4 2

(b)

8

r=-1 Max. Power Max. Ecological

6 4 2

0

0.2

0.4

t

0.6

0.8

1

Fig. 4: (a) Plot of relaxation times vs. τ for the CA engine operating under maximum-power and maximum-ecological conditions for r = −1. (b) Plot of the relaxation times ratio vs. τ . We observe that above τ ∗ = 0.5, the ME ratio is smaller than the MP ratio. For values below τ ∗ , the opposite situation is observed. where σ is the entropy production which is function of η and τ [9]. (38) represents a convex curve [9, 18] with a maximum value where dW /dη = 0, that is, for the η value satisfaying 2 (1 − η)5 [2 + (1 − η)3 ] + (1 − 4η) (1 − η)4 − 3τ (1 − η)4 −τ 5 [1 + 4 (1 − η)3 ] − τ 4 [1 + 2 (1 − η)3 (3η − 1)] = 0 .

(39)

We can observe that this eigth-degree equation has no√analytical solution. However, we can interpolate to a tenth-degree equation in τ given by √ η SB = 0.63430 − 0.07395 τ − 0.01998τ − 0.775541τ 2 + 0.06056τ 3 + 0.497525τ 4 − 0.251132τ 5 . (40) The steady state values x and y, as functions of T1 and T2 respectively, in the ecological regime can be obtained by substituting (40) and r = 4, into (7) and (9). That is, 

1/4 T14 (1 − η SB ) + τ 4 x= (41) (1 − η SB ) + (1 − η SB )4

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime

y

421

r=-1 t=0.5

slow eigendirection

x t fas tion rec

i end eig

Fig. 5: Qualitative phase portrait of x(t) vs y(t) for r = −1 and τ = 12 . Both eigenvalues are negatives, that is, x(t) and y(t) decay exponentially to the origin (steady-state values x, y). We can easily verify that 0 < t3 < t4 , that is, the corresponding eigenvectors can be described as the fast eigendirection and the slow eigendirection, respectively.

and

 y =



1/4 T14 (1 − η SB ) + τ 4

(1 − η SB ) + (1 − η SB )4

(1 − η SB ) .

(42)

From (8) and (40) we obtain
τSB = 0.9978+0.01682 η−1.45607η+1.2293η 2 −6.05376η 3 +13.697η 4 −12.3991η 5 , (43) where η = 1 − y/x. The substitution of τSB (x, y) into (15) with r = 4, produces  4 (x − y) y 4 − x4 τSB  W = g . (44) 4 +y xτSB As in the other cases, the dynamic equations for x and y are given by   4  4 y − x4 τSB g  4 dx 4 = T1 − x − x  4 dt C xτSB + y and

   4  4 y y 4 − x4 τSB dy g 4 + y − T2 . = −  4 dt C xτSB + y

(45)

(46)

We proceed as in the other cases and our calculations lead to two real and negative eigenvalues (λ5 and λ6 ). In Fig. 6a the behaviour of the characteristic

J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas 4

Relaxation times (x g/CT2 )

422

r=4

(a)

4

Max. Power Max. Ecological

3

2

1

Relaxation times ratio t s/tf

(b)

r=4

8

Max. Power Max. Ecological

6 4 2

0

0.2

0.4

t

0.6

0.8

1

Fig. 6: (a) Plot of the relaxation times versus τ for a CA engine operating under maximum-power and maximum-ecological regimes for r = 4. (b) Plot of the ratio ts /tf vs. τ . E times as a function of τ is presented. We observe that the relaxation time tM 5 is almost constant for every value of τ in the interval 0 ≤ τ ≤ 1, whereas the E decreases as τ increases, indicating that the system’s stability relaxation time tM 6 E diverges improves as τ approches one. In the opposite limit, when τ → 0, tM 6 and the stability is lost along one eigendirection. The comparison of the stability properties for the ME and MP regime are described in Figs. 6a and 6b. For this case, the relaxation times corresponding to the MP case are smaller than in the ME regime. It follows that, at least for this law, the system’s stability is better under MP than in ME conditions. In Fig. 7, the phase portrait for τ = 0.5 is described. In this case, eigendirections are almost parallel to the axis. The approximations to the steady state are tangent to the vertical axis and a small change in the temperature y corresponds to a large change in x, that is, the decay of x is almost instantaneous.

4.

Concluding Remarks

We performed a local stability analysis of a Curzon-Ahlborn engine working in an ecological regime under three different heat transfer laws. We found that the

Local Stability of an Endoreversible Heat Engine Working in an Ecological Regime

fast eigendirection

slow eigendirection

y

423

r=4 t=0.5

x

Fig. 7: Qualitative phase space portrait of x(t) vs. y(t) for r = 4 and τ = 1/2. After a small perturbation, the system approaches the origin tangent to the vertical axis while the decay of x is almost instantaneous.

system is locally stable in the three cases, that is, any perturbation would exponentially decay to the steady state. In particular, the characteristic times show dependency on τ . In the three cases (r = 1, −1, 4), one chacacteristic time remains almost constant for every value of τ whereas the second decreases as τ increases. This indicates that the stability of the system improves when τ is close to one. On the other hand, the power output and the efficiency depend on τ for the three cases under ME conditions, and both energetic quantities are decreasing functions of this parameter [9, 15, 18]. That is, the system’s stability moves in the opposite direction to that of the steady-state power output and efficiency as τ varies. This suggests a compromise between the energetic properties and the stability of the system driven by τ . We have compared the stability properties in the ME regime with those under MP conditions. Our results indicate that when one considers the case r = 1, the system’s stability in the ME regime shows small differences with respect to the MP case. In particular, both relaxation times in the ME regime are close each other indicating that the approximations to the steady state are more symmetrical. For the case r = −1, the system’s stability in the MP regime is better than under ME conditions. For r = 4, we observed that close to the value τ = 1, both regimes show similar stability properties but in the opposite limit (τ → 0), the stability in the ME case is lost. Finally, we remark that the stability properties of a CA engine working in a maximum ecological regime, except the case r = 4, are similar to those under maximum power. This is an interesting result because a wellknown phenomenon in control dynamics indicates that increased losses stabilize the dynamic systems [19]. As described at the beginning of this paper, a CA engine operating in an ecological regime produces around 80% of the maximum power and only 30% of the entropy produced in the maximum power regime [9]. The stability properties together with these energetic characteristics enhance the dynamic robustness of engines working under the so-called ecological regime.

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J. C. Chimal-Egu´ıa, I. Reyes-Ram´ırez and L. Guzm´ an-Vargas

Acknowledgments This work was supported in part by COFAA-IPN, EDI-IPN and CONACYT, M´exico. We thank F. Angulo-Brown, R. Hern´ andez-P´erez and L. Arias-Hern´andez for fruitful discussions and suggestions. Bibliography [1] F. L. Curzon and B. Alhborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43, 22 (1975). [2] M. Rubin, Optimal configuration of a class of irreversible heat engines, Phys. Rev. A. 53, 570 (1979). [3] A. De Vos, Endoreversible Thermodynamics of Solar Energy conversion, Oxford University Press, 1992; A. De Vos, Efficiency of some heat engines at maximum-power conditions, Am. J. Phys. 53, 570 (1985). [4] S. Sieniutycz and A. De Vos, eds., Thermodynamics of energy conversion and transport, Springer-Verlag, New York, 2000. [5] J. M. Gordon and M. Huleihil, On Optimizing maximun-power heat engines, J. Appl. Phys. 72, 829 (1992). [6] F. Angulo-Brown and R. Paez-Hern´ andez, Endoreversible thermal cycle with a nonlinear heat transfer law, J. Appl.Phys. 74, 2216 (1993). [7] B. Andresen, P. Salamon, and R. S. Berry, Thermodynamics in finite time, Physics Today, September 1984. [8] F. Angulo-Brown, An ecological optimization criterion for finite-time heat engines, J. Appl. Phys. 69, 7465 (1991). [9] L. A. Arias-Hern´ andez and F. Angulo Brown, A general property of endoreversible thermal engines, J. Appl. Phys. 81, 2973 (1997). [10] F. Angulo-Brown et al., Reply to “Comment on A general property of endoreversible thermal engines [J. Appl. Phys. 89, 1518 (2001)]”, J. Appl. Phys. 89, 1520 (2001). [11] Rocha-Mart´ınez et al., A simplified irreversible Otto engine model with fluctuations in the combustion heat, International Journal of Ambient Energy 27, 181 (2006). [12] F. Angulo-Brown, M. Santill´ an, and E. Calleja Quevedo, Thermodynamic optimality in some biochemical reactions, Il Nuovo Cimento 17D, 85 (1995). [13] M. Santill´ an, G. Maya, and F. Angulo-Brown, Local stability analysis of an endoreversible Curzon-Ahlborn-Novikov engine working in a maximum-power-like regime, J. Phys. D. 34, 2068 (2001). [14] L. Guzm´ an-Vargas, I. Reyes-Ram´ırez, and N. Sanchez, The effect of heat transfer laws and thermal conductances on the local stability of an endoreversible heat engine, J. Phys. D. 38, 1282 (2005). [15] L. A. Arias-Hern´ andez, MSc. Thesis, Instituto Polit´ecnico Nacional, M´exico, 1995. [16] S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering, Addison-Wesley Publishing Company, 1994. [17] L. Chen and Z. Yan, The effect of heat transfer law on the performance of a two-heat source endoreversible cycle, J. Chem. Phys. 90, 3740 (1989). [18] J. C. Chimal-Egu´ıa, M. A. Barranco, and F. Angulo-Brown, Stability analysis of an endorreversible heat engine with Stefan-Boltzmann heat transfer law working in maximum-power-like regime, Open Sys. Information Dyn. 13, 43 (2006); In this paper, the expression for the power output (W ) as function of η is correct but its corresponding plot is not. The correct plot is described in [9]. [19] Pa´ez-Hern´ andez, F. Angulo-Brown, and M. Santill´ an, J. Non-Equilib. Thermodyn. 31, 173 (2006).