Eleven International Conference on Thermal Engineering: Theory and Applications February 25-28, 2018, Doha, Qatar

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Local stability of a generalized irreversible heat engine with linear phenomenological heat transfer law working in an ecological regime

Lingen. Chen1, 2, 3, Xiaohui. Wu1, 2, 3, Xiaowei. Liu1, 2, 3

1 Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China; 2 Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China;

3 College of Power Engineering, Naval University of Engineering, Wuhan 430033, China. E-mail: lingenchen@hotmail.com.

Abstract Local stability analysis of a generalized irreversible Carnot heat engine working in an ecological regime with linear

phenomenological heat transfer law between working fluid and heat reservoir is performed. Effects of degree of internal irreversibility, heat leakage, temperatures of heat reservoirs and heat transfer coefficients on the local stability of the system are analyzed. The behavior of solutions of the system are presented qualitatively by sketching its phase portrait. Keywords: Finite time thermodynamics, Local stability, Irreversible Carnot heat engine, Ecological optimization.

1. Introduction The Most of the previous works of finite time

thermodynamics (FTT) [1,2] have concentrated on the steady-state energetic properties of the systems. Angulo-Brown [3] and Yan [4] proposed an ecological criterion for finite-time Carnot heat engines. Chen et al. [5, 6] established a generalized irreversible Carnot engine model [5] and investigated its optimal ecological performance with linear phenomenological heat transfer law [6]. Santillan et al. [7] studied the local stability of a Curzon-Ahlborn-Novikov engine working in a maximum-power-like regime with Newton’s heat transfer law. Then, Refs. [8-11] studied the dynamic robustness of endoreversible and non-endoreversible heat engines working in maximum-power-like regime and ecological regime with Newton’s heat transfer law. This paper will analyze the local stability of the generalized irreversible Carnot heat engine working at the maximum ecological point with linear phenomenological heat transfer law.

2. Ecological Performance of an Irreversible Heat Engine

Considering the model of a generalized irreversible Carnot heat engine [6] as shown in Fig. 1. There are finite rate heat transfer and a heat leakage which is a constant from the heat source to the heat sink. The rates of heat transfer from and to the high and low temperature side heat exchangers are HCQ and LCQ . The rate of heat transfer supplied by the heat source and the rate of heat transfer released to the heat sink are H HCQ Q q= + and

L LCQ Q q= + . A constant coefficient f is introduced to characterize the additional internal miscellaneous irreversibility: '/ 1LC LCQ Qf = ³ .

When there is only the heat resistance loss, one has / ( ) /LC HCQ y Q xf = . The power output of the cycle is

H L HC LCP Q Q Q Q= - = - . The rate of heat flow from the heat source to the working fluid and the rate of heat flow from the working fluid to the heat sink are

/ ( )HCQ Px x yf= - and / ( )LCQ P y x yf f= - . According to linear phenomenological heat transfer law, one has

1(1/ 1/ )HC HQ F T xa= - , 2(1/ 1/ )LC LQ F y Tb= - . When the heat transfer surface area ratio is

1 2/ / ( )f F F m b fa= = , the optimal ecological function at a working fluid’s temperature ratio (m ) is [7]

2 1 2 1( )( / 1/ ) ( )H LE B a m a m T T q a af= - - + - (1)

where 2/ ( / )B F ma fa b= + , 1 01 La T T= + ,

2 01 Ha T T= + , and 0T is the environmental temperature. The optimal temperature ratio and the maximum ecological function are [7]

1 2 1 1

1 2 2 1

( / / ) 2 // / 2 /H L L

H L H

a T a T A a Txmy a T a T a A T

f ff

+ += =

+ + (2)

22 1

max 2 12 1 1 1

( ) ( )4 ( )( )

H L

H L H L

F a T a TE q a aT T a A a T AT

a ff-

= - + -+ +

(3)

respectively, where 1A af b= .

3. Steady State of the Heat Engine Working in an Ecological Regime

Assume that the working fluid’s temperatures of the steady state are x and y . The rates of heat flows can

be given by 1 / ( )J Px x yf= - and 2 / ( )J P y x yf f= - ,

2

where 1J and 2J are rates of the steady-state heat flows from x to the engine and from the engine to y ,

respectively, P is steady-state power output. When the irreversible heat engine operates in a steady state, it means that the rate of heat flow from HT to x equals to

1J and the rate of heat flow from y to LT equals to 2J ,

( )1 1 1/ 1/HJ F T xa= - and ( )2 2 1/ 1/ LJ F y Tb= - . The steady-state efficiency of an irreversible heat engine without heat leakage is 0 1 1 1 /P J y x mh f f= = - = - . Then, the working fluid’s temperatures ( x and y ) can be expressed as 1 1( / )1 (1/ [ / ])Hx T A m A mt t+= + and

1 11 / (( ) / 1 )LT A my At+ += . The temperatures of reservoirs can be expressed as 1(1 ) / [ ( /H AT x x yt t+=

1)]A+ and 1 11 ) / (1( / )LT y y xA At+ += . 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 0 LT T= is

1

1

(2 1 ) 42 1 2 (1 )

Axmy A

ft t fft t t t

+ + += =

+ + + + (4)

Fig. 1. Schematic diagram of a generalized irreversible

heat engine

The working fluid’s temperatures are

11

1 1

(2 1 ) 4 )2 1 2 (1 )(

1HT Ax AA At ft t ft ft t t t

+ + +=

+ ++

+ ++ (5)

11

1 1

2 1 2 (1 )( ))(11 (2 1 ) 4

L Ay AA A

T ft t t tt ft t f

+ + ++

++

+ ++= (6)

The temperature ratio of heat reservoirs is

1 1 1

21 1 1

1 1

1

2 2 2

( 2 2 2 )8 ( 4 )

,4

( )

yA yA x x xA

x x xA yA yAxA x y yA

x yxA

f f

f ff

t

+ - - - +

+ + - -- - -

= (7)

Thus, one can obtain the steady-state power output

1 1

( )( )(

[ , ]( ),( )[ )1 , ]

F y x x y x yx yPxy x yA x y Aa t f

t- +

=-

+ + (8)

4. Local Stability Analysis of the Heat Engine in

an Ecological Regime Now, the reservoirs x and y are not real heat

reservoirs but macroscopic objects with a heat capacity C. Therefore, their temperature changes according to the

following equations [12]: ( )1[ 1/ 1/ Hdx dt F x Ta= -

1]J C- and ( )2 2[ 1/ 1/ ]Ldy dt J F T y Cb= - - .One

has 1 / ( )J Px x yf= - and 2 / ( )J P y x yf f= - . The power developed by the engine performing in a steady-state maximum ecological regime is given by Eq. (8). Only small deviations from the steady state are considered in a local stability analysis. Thus, it seems reasonable to assume that the power ( P ) related to the temperatures x and y in the same manner as the

power ( P ) depends on x and y in the steady-state maximum ecological regime [7]

1 1

[ , ]( ), ,( )

( )( ) ( )(1 ])[ ,

F y x x y x yP x y x yxy x yA y A

Px

a t ft

- + -+ +

== (9)

Therefore, one has

1 1

1 ( )[( 1) ]( ) (1 )H

F x y xC x yA T y A

dx dt a tt

- += - -

+ + (10)

1 1 1

1 1[ ( )](1 ) L

F x y yC x yA x A A

dy dty T

a f tt-

= - -+ +

(11)

where

1 1 1

21 1 1

11 1

[ 2 2 2

( 2 2 2 )] / (4 )

8 ( 4 )

yA yA x x xA

x x xA yA yAxA

xA x y yA

t f f

f ff

= + - - - +

+ + - -- - -

Based on linearization and stability analysis [15], let ( ),f x y and ( , )g x y be defined as

( )1 1

1 ( )[( 1) ]( (1 )

,) H

F x y xC x yA

f xT y A

y a tt

- += - -

+ + (12)

( )1 1 1

1 1[ ( )]( )

,1 L

F x y yC x yA x A A

g x yy T

a f tt-

= - -+ +

(13)

If x and y are close to their steady-state values

but not too far away, one has ( ) ( )x t x x td= + and

( ) ( )y t y y td= + , where xd and yd are small

perturbations. Applying Taylor’s formula to ( ),f x y and

( ),g x y at steady state ( ),x y , one can obtain the following matrix of linear differential equations

( )

( )( )( )

x y

x y

d x tf f x tdtg g y td y t

dt

dddd

æ öç ÷ æ öæ öç ÷ = ç ÷ç ÷ç ÷ è ø è øç ÷è ø

(14)

where ( ) ,x x yf f x= ¶ ¶ , ( ) ,y x yf f y= ¶ ¶ ,

( ) ,x x yg g x= ¶ ¶ and ( ) ,y x yg g y= ¶ ¶ .

The eigenvalues at maximum ecological function and the corresponding eigenvectors are

21 [ ( ) 4 ] / 2x y x y y xf g f g f gl = + - - + (15)

3

22 [ ( ) 4 ] / 2x y x y y xf g f g f gl = + + - + (16)

( )12[ ( ) 4 ] / ,12x y x y y x xf g f g f g gu - - - += (17)

( )22[ ( ) 4 ] / ,12x y x y y x xf g f g f g gu - + - += (18)

The eigenvalues are functions of C , F , a , b , t and LT . The calculations show that both eigenvalues are real and negative ( 1 2 0l l< < ). Therefore, any perturbation decays exponentially to the steady state with time and the steady state of the heat engine working at the maximum ecological function is steady, the characteristic relaxation times (which are defined as

1,2 1,21t l= ) can be written as

21 2 / [ ( ) 4 ]x y x y y xt f g f g f g= - + - - + (19)

22 2 / [ ( ) 4 ]x y x y y xt f g f g f g= - + + - + (20)

Relaxation times of the system working at the maximum ecological function vs. heat reservoirs’ temperature ratio t with / 1b a = , 300LT K= and different f are shown in Fig. 2. 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 may be obtained. The phase portrait of ( )x t vs. ( )y t with / 1b a = , 1.1f = , 0.5t = and

300LT K= is shown in Fig. 3. It is calculated that the relaxation time ratio is 1 2/ 0.51t t = and the eigendrections are 1 ( 0.59,1)u = - and 2 (6.25,1)u = .

Fig. 2. Relaxation times vs. t with different f

Fig. 3. Phase portrait of ( )x t vs. ( )y t with / 1b a = ,

1.1f = and 0.5t =

5. Conclusion After a small perturbation the system state

exponentially decays to steady state with either of two relaxation times. According to numerical calculations, both relaxation times 1t and 2t decrease as heat reservoirs’ temperature ratio t increases, and decrease as heat transfer coefficient ratio /b a increases, 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 steady-state point ( , )x y .

Acknowledgments This paper is supported by the Natural Science Fund

of China (Project No. 51576207).

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