Optimization of the direct Carnot cycle

Applied Thermal Engineering 27 (2007) 829–839 www.elsevier.com/locate/apthermeng

Optimization of the direct Carnot cycle M. Feidt a, M. Costea b, C. Petre a

b,*

, S. Petrescu

c

L.E.M.T.A., U.R.M. C.N.R.S. 7563, University Henri Poincare´ of Nancy 1, Avenue de la Foreˆt de Haye B.P. 160, 54504 Vandoeuvre Cedex, France b Department of Applied Thermodynamics, Faculty of Mechanical Engineering, Polytechnic University of Bucharest, Splaiul Independentei, 313, Sector 6, 060042 Bucharest, Romania c Department of Mechanical Engineering, Bucknell University, Lewisburg, PA 17837, USA Received 12 October 2005; accepted 15 September 2006 Available online 16 November 2006

Abstract A model for the study and optimization of two heat reservoirs thermal machines is presented. The mathematical model basically consists of the First and Second Laws of Thermodynamics applied to the cycle and entire system, and the heat transfer equations at the source and sink. The internal and external irreversibilities of the cycle are considered by taking into account the entropy generation terms. Several constraints imposed to the system composed by the engine and the two heat reservoirs (namely, engine efficiency, or power output, or heat flux received by the engine, each of them together with imposed internal entropy generation and total number of heat transfer units of the machine heat exchangers) allow us to find the optimum operational conditions, as well as the limited variation ranges for the system parameters. Emphasis is put on coupling between various possible objective functions, namely thermal cost, useful effect, first law efficiency and whole system dissipation. It is for the first time to our knowledge when it has been proved that if one of the possible objective functions is fixed (as a parameter with imposed value), the optima of the other three always correspond to each other for the corresponding stationary state system, with a given optimum heat conductance allocation (one degree of freedom). Other interesting results are also reported in this paper. Some sensitivity studies were developed, too, with respect to various parameters of the model (engine performance, internal entropy generation, total number of heat transfer units). Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Optimization under constraints; Thermal engines; Non-linear heat transfer laws; Carnot direct cycle

1. Introduction Historically, Carnot [1] was the first who had developed a thermal model applied to an engine from an engineering point of view, being probably the father of all the forthcoming developments of the true Thermodynamics applied to machines, systems and processes, through his geometric and intuitive approach of what became the Second Law of Thermodynamics. Also, he probably was at the origin of the fundamental notion of efficiency that is strongly related to the Second Law, but remains a thermostatic limit.

*

Corresponding author. Tel.: +40 21 402 93 39. E-mail address: [email protected] (C. Petre).

1359-4311/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.09.020

Things remain unchanged until the paper of Curzon and Ahlborn [2] was published in 1975. The main point of their paper was that they had introduced what we call now, the time arrow, by considering finite durations of thermodynamic transformations inside the machine. Particularly, a finite power is derived for the corresponding machine. And the main result was that the efficiency associated to the optimum power is less than the one introduced by Carnot. In fact, this has already been proposed by Chambadal [3], and simultaneously by Novikov [4] in 1957. Nowadays, there is a come back to this global but renewed approach [5]. Since 1980s, approximately thirty books related to this subject have been published [17,27–43] and many papers have appeared in the literature. Comments on most of them could be found in two review papers [6,7,12,15].

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M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

Nomenclature A C_ cp K m_ NTU R S_ s_ T U Q_ q_ W_ w_

heat transfer area heat capacity rate, W K1 specific heat at constant pressure, J kg1 K1 heat transfer conductance, W K1 mass flow rate, kg s1 number of heat transfer units thermal resistance, W1 K entropy generation, W K1 non-dimensional entropy generation temperature, K overall heat transfer coefficient, W K1 m2 heat flux, W non-dimensional heat flux mechanical power, W non-dimensional mechanical power

In the same context Chen et al. [18] have presented a synthesis of several papers dealing with the influence of different irreversibility criteria on the Curzon–Ahlborn efficiency, emphasizing its link to real engine efficiency. Besides the efficiency, other optimization criteria have been studied, such as: maximum power density [19], thermoeconomic optimization [20] or ecological optimization [21]. In the design phase of the engine, the optimal distribution of heat transfer surfaces at maximum power output has been studied [13,44]. Also, other influences on the efficiency have been presented, such as engine speed [22] or particular forms of the heat transfer laws [23,24,26]. The final aim of these research studies is the control of operating engines [25]. The purpose of this paper is to present a historical introduction to this subject, by showing the growing interest for the proposed approach, to clarify the actual situation of the subject and to enlighten the progress direction of Engineering Thermodynamics, definitely escaping from Thermostatics to Irreversible Thermodynamics of machines and processes. To follow this direction some other branches of Physics are needed to complete the study (particularly heat and mass transfer, but not only). The present synthesis continues two previous ones [9,14] regarding the works performed on reversed cycle machines. Thus, we will focus here on Carnot thermal engine for which original results are presented. Also, they partially extend previous paper results [10,11]. The main originality of this paper with respect to the great majority of others, that have already been published, relies on considering the machine internal irreversibility instead of irreversibility ratio .The effect of the internal irreversibility on the machine performance clearly appears in our results, but using entropy analysis [17] and correlating all fundamental aspects one can get more complete and useful results from an engineering and practical point of view.

Greek symbols e heat exchanger effectiveness g engine efficiency h non-dimensional temperature Subscripts C cold gas H hot gas i internal or input L loss o output p at constant pressure SC related to the sink SH related to the source T total

2. Optimisation model of a Carnot engine 2.1. Engine description The schematic representation of a Carnot engine is given in Fig. 1 in a Temperature–Entropy diagram. The figure clearly shows that the two isothermal and the other two thermodynamic processes are irreversible for this engine. The irreversibility effect of the isothermal processes consists of reducing the thermal energy input from the source, while increasing the one released to the sink. The two heat transfers processes between the source and the sink clearly appear irreversible, too. From a global point of view, one can say that the engine has an internal irreversibility represented by Si, which is the sum of all internal irreversibilities of the engine all along the cycle. Generally, the heat reservoirs at source and sink are limited in size and consequently, limited in energy transfer. Accordingly, the heat exchanger finite size imposes a certain temperature variation during the heat transfer to and T

TSH, i TSH, o

TH

RL=

TC TSCo TSCi S Fig. 1. General representation of a Carnot engine.

1 KL

M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

from the working gas in the cycle. Thus, if a counter flow heat exchanger will model the heat transfer processes at the source, respectively the sink, then TSHi and TSHo will be the source temperatures, respectively TSCi and TSCo the sink ones. The limit case when the heat reservoirs are constant temperature ones will correspond to TSHi = TSHo = TSH at the source, and TSCi = = TSCo = TSC at the sink. The real operation of the engine located between the source and the sink also imposes a thermal short circuit between the hot and cold sides that is illustrated in Fig. 1 by the thermal resistance RL = 1/KL, where KL represents the loss conductance between the hottest temperature TSHi and the lowest temperature TSCo of the system. This thermal resistance represents the system thermal losses (equivalent resistance) globally and the corresponding thermal link between the three parts of the engine: source–cycle– sink.

To develop practical and simple algebra, we introduce here the assumption of a steady-state regime of the system (non-transient conditions, namely all properties are locally constant, but they can vary with the location in the system, as for example the temperatures), as this is the case of any device intended for continuous operation. Fluxes are associated to gradients of intensive physical properties. They are also constant in time. First Law of Thermodynamics applied to the source indicates the total heat flux delivered to the engine: Q_ SH ¼ Q_ H þ Q_ L

ð1Þ

Note that all heat fluxes from Eq. (1) are positive quantities. The corresponding balance equation at the sink yields: Q_ SC ¼ Q_ C  Q_ L

where Q_ SC and Q_ c are released by the system (see Fig. 2), so they are negative quantities. By using Eqs. (1) and (2), the energy balance equation for the engine becomes: W_ ¼ Q_ H þ Q_ C ¼ Q_ SH þ Q_ SC

ð2Þ

ð3Þ

According to the algebraic sign convention, W_ is considered a negative quantity. Second Law of Thermodynamics applied to the cycle gives: Q_ H Q_ C _ þ þ Si ¼ 0 TH TC

ð4Þ

The internal entropy generation term corresponds to a global approach that could be detailed in the future by using the entropy analysis [17]. Second Law of Thermodynamics applied to the system (source–cycle–sink) gives: T SHi T SCi Q_ L Q_ L C_ H ln þ C_ C ln þ  þ S_ T ¼ 0 T SHo T SCo T SHi T SCi

2.2. General equations and assumptions of the model

831

ð5Þ

where the external flow stream heat capacity rate at the hot side is given by: C_ H ¼ cp;H  m_ H , while the one at the cold side is: C_ c ¼ cp;c  m_ c . Adding the expression that defines the first law efficiency of a Carnot engine completes the preceding set of equations: g¼

_ W _QSH

ð6Þ

Note that the finite size of the source and sink adds to the model supplementary equations relating variables and parameters: Q_ H ¼ C_ H ðT SHi  T SHo Þ Q_ H ¼ eH C_ H ðT SHi  T H Þ

ð7Þ ð8Þ

where the heat exchanger effectiveness at the hot side of the engine is given by: eH ¼ 1  expðNTUH Þ

ð9Þ

where the number of heat transfer units of the hot side heat exchanger is given by NTUH ¼

K H U H AH ¼ : C_ H C_ H

The same reasoning can be applied to the cold side, namely: Q_ C ¼ C_ C ðT SCi  T SCo Þ Q_ C ¼ eC C_ C ðT SCi  T C Þ

ð10Þ ð11Þ

with the heat exchanger effectiveness at the cold side of the engine: eC ¼ 1  expðNTUC Þ

Fig. 2. Energy balance of the engine and system (algebraic convention).

ð12Þ

and NTUc ¼ KC_ c ¼ UC_c Ac . c c Note that the limiting flow stream (having C_ min ) in the heat exchangers at the hot side, respectively cold side, is the external one, due to the isothermal process performed

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M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

in the two heat exchangers by the internal fluid working in the engine. The model development in this paper will only consider constant temperature heat reservoirs, meaning that C_ H ; C_ c ! 1, so that TSHi = TSHo = TSH, TSCi = TSCo = TSC, Eqs. (7) and (10) disappear, and Eq. (5) becomes: Q_ H þ Q_ L Q_ C  Q_ L _ þ þ ST ¼ 0 T SH T SC

ð13Þ

3. Optimization of the engine characteristics subject to various constraints To render the model more general, a dimensionless form of the equation set is preferred, using different reference parameters (as described below). For the temperature, the reference parameter is set to be TSC, the sink temperature, as for an engine this is the temperature closest to the environment one. For the heat capacity rate or heat conductance, the _ p of the working reference parameter is chosen to be C_ ¼ mc fluid in the engine, in order to avoid cumbersome calculations. For the energy fluxes, the product C_  T SC is implicitly the reference as a result of the previous two choices. Then it results: W_ þ q_ H þ q_ C ¼ 0 q_ H q_ C þ þ s_ i ¼ 0 hH hC q_ H þ q_ L q_ C  q_ L þ s_ T ¼ 0 þ hSH 1 W_ ¼ g  q_ SH q_ SH ¼ q_ H þ q_ L

ð30 Þ ð40 Þ ð130 Þ ð60 Þ ð10 Þ

q_ H ð1  gÞ þ q_ C ¼ 0 q_ H q_ H ð1  gÞ  þ s_ i ¼ 0 hC hH

ð15Þ ð16Þ

These two constraints allow us determining the dimensionless temperatures hH and hC which are the model variables, when heat transfer laws at source and sink are defined, as well as the internal entropy generation. There are several objective functions (OF) that could be considered for the case where the engine efficiency is imposed, namely: OF 1 minimum thermal cost () q_ H to be minimized; OF 2 maximum useful effect () w_ ¼ g  q_ H to be maximized; OF 3 minimum dissipation for the whole system () to s_ T ¼ q_ H ðgc  gÞ to be minimized. Note that the three objective functions are actually proportional and the corresponding optima are interrelated and occur for the same optimum state parameters of the system if g is imposed: OPTq_ H

o

OPT w_

o

OPT s_ T

_ k c ¼ K C =C_ are If the design is imposed (k H ¼ K H =C; parameters), the state of the system is also imposed, namely the dimensionless temperatures hH and hC are fixed by the considered constraints. If the dimensionless heat conductances, kH and kC, are related by a total value (kH + kC = kT: allowed investment cost for the heat exchangers or maximum permitted geometry) to be distributed between the heat exchangers at the source and sink, there remains only one degree of freedom, and the optimization can be performed (see the following sections).

with 3.2. Model development when the engine dimensionless power output is imposed

q_ L ¼ k L ðhSH  1Þ k L ¼ K L =C_ hSH ¼ T SH =T SC : For the beginning we suppose hereafter that the engine has no heat losses, so that q_ L ¼ 0. 3.1. Model development when the engine efficiency is imposed When the efficiency of the engine is considered parameter (having as practical relevance the case when the engine is on continuous operation, so that an economical operation regime is sought), it is well known that its thermostatic upper boundary corresponds to the Carnot efficiency gC, given by: gC ¼ 1  1=hSH

ð14Þ

By combining Eqs. (6 0 ) and (3 0 ), respectively (6 0 ), (3 0 ), and (4 0 ), it yields the following constraints for the system:

Now, the engine dimensionless power output becomes a parameter. This is the practical case of a plane taking off, or a driven device requiring constant power. The two constraints considered in this case are: _ q_ C ¼ ðq_ H þ wÞ q_ H q_ H þ W_  þ s_ i ¼ 0 hC hH

ð17Þ ð18Þ

Similarly to the previous case, one can get the dimensionless temperatures hH and hC by combining Eqs. (17) and (18). The objective functions could be: OF 1 minimum thermal cost () q_ H to be minimized; OF 2 maximum efficiency () g ¼  q_w_H to be maximized; OF 3 minimum dissipation () s_ T ¼ gc  q_ H þ w_ to be minimized.

M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

The definitions of the preceding objective functions show that they are interrelated, namely: OPTq_ H

o

OPTg

o

OPT_sT

3.3. Model development when the dimensionless heat flux input is imposed When the dimensionless heat flux input becomes a parameter, it defines one of the two constraints of the mathematical model. Then the heat transfer law at the source allows us to determine the dimensionless temperature hH. Furthermore, one gets the dimensionless temperature hC from the second constraint given by Eq. (4 0 ). The three objective functions of the model are: OF 1 maximum dimensionless power output ()   w_ ¼ q_ H 1  hhHc þ hc  s_ i to be maximized; OF 2 maximum efficiency () g ¼ 1  hhHc  hq_cH_si ¼  q_w_H to be maximized.  _ _ OF 3 minimum dissipation ( ) s ¼ q gc  1 þ hhHc  T H  hc _si ¼ q_ H ðgc  gÞ to be minimized. q_ H Here again, due to the constraint, one gets: OPTw_

o

OPTg

o

OPT_sT

3.4. Conclusions of the paragraph For the three studied cases using different constraints, namely imposed efficiency g , or non-dimensional power output, w_ , or non-dimensional heat flux input, q_ H , together with some parameters, the non-dimensional temperatures hH and hC are determined at a given design of the engine. When we focus on the design or geometry optimization, the two dimensionless thermal conductances, kH and kC, become variables. However, they remain related by the total dimensionless thermal conductance, kH + kC = kT, so that there remains one degree of freedom for the optimization that yields optimal distribution of the heat transfer inventory between source and sink. Analytical or numerical calculations are needed to state the heat transfer laws at source and sink. The heat transfer, as well as mass transfer, is intimately related to thermodynamics of the engine. 4. Optimization with general heat transfer laws We suppose here that the heat transfer law is expressed as: q_ ¼ k  f ðhS ; hÞ

ð19Þ

so that q_ H ¼ k H  fH ðhSH ; hH Þ and q_ c ¼ k C  fC ðhSC ; hC Þ, with kH + kC = kT. The internal irreversibility, s_ i , of the engine is also considered. The assumption is that s_ i is a function of both dimensionless temperatures hH and hC:

833

s_ i ¼ fs ðhH ; hC Þ

ð20Þ

To establish the system of equations modeling this new approach, the variational calculus is used for the three cases previously considered (Section 3). 4.1. Model development when the engine efficiency is imposed The constraint (15) becomes: ð1  gÞk H  fH ðhH Þ þ ðk T  k H Þ  fc ðhc Þ ¼ 0

ðC1 Þ

The constraint (16) becomes: k H  fH ðhH Þ ðk T  k H Þ  fc ðhc Þ þ þ fs ðhH ; hc Þ ¼ 0 hH hc

ðC2 Þ

The optimum of the objective function corresponds to the optimum of q_ H ¼ k H  fH ðhH Þ, subjected to constraints C1 and C2. The corresponding Lagrange function is: Lg ðhH ; hC ; K H Þ ¼ k H fH ðhH Þ þ k  C 1 þ k0  C 2

ð21Þ

After performing some calculations, one gets the nonlinear system of equations consisting of three equations with three unknowns: 8 ð1  gÞk H  fH þ ðk T  k H Þ  fC ¼ 0 > > > > > < k H  fH þ ðk T  k H Þ  fC þ f ¼ 0 s hH hC ð22Þ > > 2 2 > f f f f f f H s;H C s;C > H C >  ¼  : fH;H  h2H k H fH;H fC;C  h2C ðk T  K H ÞfC;C Note that the notations in the above system of equations correspond to: ofH ; ohH ofs ¼ ohC

fH;H ¼ f s;C

f C;C ¼

ofC ; ohC

f s;H ¼

ofs ; ohH ð23Þ

For an endoreversible engine, the equality fs,C = fs,H = 0 restores results already reported in the literature [16]. 4.2. Model development when the engine dimensionless power output is imposed The two constraints given by Eqs. (17) and (18) become: k H fH þ ðk T  k H ÞfC þ w_ ¼ 0 k H  fH ðk T  k H Þ  fC þ þ fs ¼ 0 hH hC

ðC1 Þ ðC2 Þ

The optima of the objective functions correspond to the optimum of q_ H , subjected to the new constraints C1 and C2. The Lagrange function is: Lw_ ¼ k H fH þ k  C 1 þ k0  C 2

ð24Þ

After performing some calculations, one gets the three corresponding nonlinear equations that are to be solved:

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M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

8 k H  fH þ ðk T  k H Þ  fC þ w_ ¼ 0 > > > > > < k H  fH þ ðk T  k H Þ  fC þ f ¼ 0 s hH hC > > > fH2 fH fs;H fc2 fC fs;C > >  ¼  : 2 2 k f ðk  k H ÞfC;C fH;H  hH fC;C  hC H H;H T

ð25Þ

5.1. Particularization of the equation system

It appears that the only difference between the cases considered in Sections 4.1, respectively Section 4.2 is due to the expression of the constraint C1 that is changed. 4.3. Model development when the dimensionless heat flux input is imposed The parameter here is the dimensionless heat flux input, q_ H . Its imposed value will be denoted by q_ 0 and it represents the first constraint of the model: q_ 0 ¼ k H  fH

ðC1 Þ

The second constraint is given by: q_ 0 ðk T  k C Þ  fC þ þ fs ¼ 0 hH hC

ðC2 Þ

The optima of the objective functions correspond to the optimum of the dimensionless power output of the engine, subjected to the two constraints previously defined. The Lagrange function is: Lq_ ¼ k H fH þ ðk T  k H Þ  fC þ k  C 1 þ k0  C 2

ð26Þ

After performing some calculations, the nonlinear equation system becomes: 8 k H  fH  q_ 0 ¼ 0 > > > > k > < H  fH þ ðk T  k C Þ  fC þ f ¼ 0 s hH hC ð27Þ > 2 2 > > f f f f f f H s;H C s;C > H c >  ¼  : fH;H  h2H k H fH;H fC;C  h2C ðk T  k H ÞfC;C Once more, it appears that the only difference between the current case and those from Sections 4.1 and 4.2 is due to the constraint C1. 4.4. Conclusions of the paragraph Whatever the two constraints imposed to the Carnot engine are, one of the nonlinear equations of the model remains the same: fH2 fH fs;H fc2 fC fs;C  ¼  2 fH;H  hH k H fH;H fC;C  h2C ðk T  k H ÞfC;C

common case of heat transfer law form, namely the linear one. Furthermore, we will explore particularly the effect of the internal irreversibility of the engine through a sensitivity study.

ð28Þ

The other two equations are the constraints that were chosen. These three equations provide the state vector (hHopt, hCopt, kHopt, kCopt) corresponding to the desired optimum. 5. Illustration of the methodology The variety of cases to be treated is out of range for the present paper, so we restrict the illustration to the most

The linear heat transfer law has the general form: ð29Þ f ¼ ðhS  hÞ So, it comes that fH = hSH  hH; fC = 1  hC and consequently fH,H = fC,C = 1. The sensitivity analysis corresponding to the internal entropy generation was performed for three particular cases, the most commonly encountered in practice: (a) fs1 = si = constant, so that: fs1,H = fs1,C = 0. (b) fs2 ¼ s_ 0 ðhH  hC Þ linear variation with temperature, so that: fs2;H ¼ s_ 0 ; fs2;C ¼ _s0 . (c) fs3 = c Æ ln(hH/hC) logarithmic variation with temperature ratio, so that: fs3;H ¼ hcH ; fs2;C ¼  hcc . By using linear heat transfer laws, the third equation of the nonlinear system of equations given by (22) becomes:  2 hSH  hH hSH  hH  fs;H hH kH  2 hSC  hC hSC  hC ¼  fs;C ð30Þ hC kT  kH The particular form of the above equation for the three forms of the internal entropy generation will be given by: case ðaÞ hSH  hH hSC  hC ¼ hH hC

ð30aÞ

This equation is identical to the one obtained for the endoreversible engine [8] case ðbÞ  2 hSH  hH hSH  hH 0  Si hH kH  2 hSC  hC hSC  hC 0 ¼ þ S hC kT  kH i case ðcÞ  2 hSH  hH hSH  hH c  kH hH hH  2 hSC  hC hSC  hC c ¼ þ kT  kH hC hC

ð30bÞ

ð30cÞ

The corresponding forms of the constraints are given hereafter: Case where g is imposed: ð1  gÞk H ðhSH  hH Þ þ ðk T  k H Þð1  hC Þ ¼ 0

ð31Þ

k H ðhSH  hH Þ ðk T  k H Þð1  hC Þ þ þ fs ¼ 0 hH hC

ð32Þ

M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

835

Case where w_ is imposed: k H ðhSH  hH Þ þ ðk T  k H Þð1  hC Þ þ w_ ¼ 0

ð33Þ

k H ðhSH  hH Þ ðk T  k H Þð1  hC Þ þ þ fs ¼ 0 hH hC

ð34Þ

Case where q_ H is imposed ðq_ 0 Þ: k H ðhSH  hH Þ  q_ 0 ¼ 0

ð35Þ

k H ðhSH  hH Þ ðk T  k H Þð1  hC Þ þ þ fs ¼ 0 hH hC

ð36Þ

5.2. Results The results of computations based on this analysis are various and significant. They point out different operational regimes of the engine, namely maximum power output or minimum dissipation (entropy generation) dependently on the studied case. Also, they show limits of the variation range for the model variables, as well as reasons to choose the best solution between the physical ones. Illustration of some results is given in Figs. 3–16. 5.2.1. Case where g is imposed Fig. 3 illustrates the engine internal entropy generation versus the dimensionless temperature hH of the working fluid at the hot end for different values of the cycle efficiency. The curves clearly show a reverse proportionality between the internal entropy generation and the cycle efficiency. Then, for each imposed value of the efficiency, the dimensionless temperature hH has a specific variation range (imposed by the condition s_ i > 0) continuously decreasing as the efficiency increases. For example, at g = 0.2, hH can vary between 4.05 and 5, while at g = 0.7, its values vary from 4.58 to 5. Also, one can see that the efficiency is limited, its value of 0.8 being impossible because of the above mentioned condition regarding s_ i which is not accomplished.

0.12

η =0.2

NTUH=1.5

η =0.3

0.1

η =0.4

si [-]

0.08 .

η =0.5

0.06

η =0.6

0.04 0.02

η =0.7

0

η =0.8

4

4.2

4.4

θ H [-]

4.6

4.8

5

Fig. 3. Internal entropy generation versus hH for different values of the cycle efficiency; variation range limits for the physical values of hH (case with g imposed, hSH = 5, NTUT = 2).

Fig. 4. The two solutions of the dimensionless temperature hH as functions of NTUH (case with g and s_ i imposed as a constant, s_ i ¼ 0:005, hSH = 5, NTUT = 2).

Fig. 5. Engine performance for minimum dissipation regime corresponding to hH, solution 1 (case with g and imposed as a constant, s_ i ¼ 0:005, hSH = 5, NTUT = 2).

5.2.1.1. Case where g and s_ i are imposed. Physically, more interesting results are obtained when several parameters of the model are imposed. By solving the equation system corresponding to the case with cycle efficiency and internal entropy generation imposed, a second order equation in hH as a function of NTUHH yields, when s_ i ¼ constant. Its solutions are graphically represented in Fig. 4. One can see that solution 1 has high values, it is almost constant with NTUH, and very close to the source non-dimensional temperature, hSH = 5. This solution corresponds to minimum dissipation regime, due to the heat transfer across small temperature difference at heat reservoirs. The second one, with a higher variation with NTUH, corresponds to maximum power output of the engine, as shown in Fig. 6. The engine main performances for these two regimes are represented with respect to NTUH in Figs. 5 and 6. Obviously, there is a considerable difference between

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M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

Fig. 6. Engine performance for maximum power output regime corresponding to hH , solution 2 (case with g and s_ i imposed as a constant, s_ ¼ 0:005, hSH = 5, NTUT = 2).

Fig. 7. The dimensionless temperatures hH and hC corresponding to the optimum state of the minimum dissipation regime (_sT;min ), respectively, maximum power output regime (case with g and s_ i imposed as a constant, s_ i ¼ 0:005, hSH = 5, NTUT = 2).

the similar curves in the two figures corresponding to the operational regime: very reduced variation in Fig. 5, while all curves in Fig. 6 present maximum values. It results an optimum value for the variable NTUH, namely NTUH near of 1 for the chosen parameters of the model. By correlating the corresponding values of the two dimensionless temperatures, hH and hC, for the optimum state in each of the above mentioned regimes with imposed cycle efficiency, the four curves represented in Fig. 7 yield. In agreement with our expectations, the temperature difference between the hot and cold sides increases as the cycle efficiency increases. This remark is only valid for the maximum power output regime, w_ max . Once the temperature values of the optimum regimes are established, the performances of the engine could be illustrated as function of efficiency (Figs. 8 and 9). For solution 1, both dimensionless

Fig. 8. Engine performance for minimum dissipation regime as a function of efficiency (case with g and s_ i imposed as a constant, s_ i ¼ 0:005, hSH = 5, NTUT = 2).

Fig. 9. Engine performance for maximum power output regime as a function of efficiency (case with g and s_ i imposed as a constant, s_ i ¼ 0:005, hSH = 5, NTUT = 2).

_ sT;min Þ heat flux input, q_ H ð_sT;min Þ, and power output, wð_ increase with the efficiency, while the minimum total entropy production is almost constant. Also, note that all values are very low. For the maximum power output regime (Fig. 9), the already maximum power output of the engine represented versus the cycle efficiency shows a maximum value for g = 0.57. The results of a sensitivity study with respect to the internal entropy generation are illustrated in Fig. 10, for two values of the total number of heat transfer units corresponding to the heat exchangers of the engine. The curves help us to choose the engine operation regime according to internal entropy generation, efficiency and maximum power output/minimum dissipation. The upper curves show that maximum power output ðw_ max Þ increases with the efficiency, but decreases with s_ i .

M. Feidt et al. / Applied Thermal Engineering 27 (2007) 829–839

837

Fig. 12. Engine performance for maximum dissipation regime corresponding to hH , solution 2 (case with w_ and s_ i – ct imposed).

Fig. 10. Comparison of the non-dimensional power output of the engine for the two operational regimes when NTUT gets different values (case with g and s_ i imposed as a constant, hSH = 5, NTUT = 2 and 4).

Fig. 13. Engine performance for minimum dissipation regime corresponding to hH, solution 1 (case with w_ and s_ i – ct imposed).

Fig. 11. The two solutions of the dimensionless temperature hH as functions of NTUH (case with w_ and s_ i – ct imposed).

5.2.2. Case where w_ and s_ i are imposed Similar results are illustrated for the case when the dimensionless power output of the engine and the internal entropy generation are imposed. Now, the dimensionless temperature hH-solution 1 (Fig. 11) presents a more visible variation compared to the one from Fig. 4.

Fig. 14. Total entropy generation as function of NTUH, for different imposed values of the engine dimensionless power output (case with w_ and s_ i – linear imposed).

Both solutions for hH seem possible from the physical point of view. However, the engine performance variations

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when the imposed dimensionless power output decreases. Same behavior has the dimensionless heat flux input, represented in Fig. 15. Contrary, all cycle efficiency curves (Fig. 16) reach a maximum for values of NTUH that increase when the engine power output increases. Also, note that the maximum efficiency shown by the curves decreases when the engine power output increases. 6. Conclusions

Fig. 15. Dimensionless heat flux input as function of NTUH, for different imposed values of the engine dimensionless power output (case with w_ and s_ i – linear imposed).

Fig. 16. Cycle efficiency as function of NTUH for different imposed values of the engine dimensionless power output (case with w_ and s_ i – linear imposed).

illustrated for solution 2 in Fig. 12, namely low values of the efficiency and high values for the non-dimensional heat flux input and total entropy generation, make it an undesired choice. In the meantime, the engine performance for solution 1 (Fig. 13) looks interesting: high efficiency and low heat flux input, respectively total entropy generation, all of them being almost constant for NTUH = 0.5–1.5. When the internal entropy generation is imposed by a linear dependence with respect to the working fluid temperature difference ð_si ¼ s0 ðhH  hC ÞÞ, a third order equation yields when solving the system of equations corresponding to this case. From the three mathematical solutions, one is non-physical, and solution 3 provides similar behavior of the system as in the previous case (Fig. 12). Hence, the results are illustrated only for hH solution 2 in Figs. 14–16. The sensitivity study with respect to the dimensionless power output (Fig. 14) shows that a minimum value of the total entropy generation can be reached in each case _ The corresponding NTUH value decreases of imposed w.

The present paper proposes a model for a Carnot engine with heat leak, internal irreversibility and general heat transfer laws at source and sink of finite size. Emphasis is put on coupling between various possible objective functions (O.F.), namely thermal cost, useful effect, first law efficiency and whole system dissipation. For the first time to our knowledge it has been proved that if one of the possible O.F. is fixed (as a parameter with imposed value), the optima of the other three always correspond to each other for the corresponding steady-state operation of the system, with a given optimum heat conductance allocation (one degree of freedom), this result being sustained by analytical development and also by graphical results. The three cases (g imposed, w_ imposed, q_ H imposed) differ by the two constraints; only one general equation remains unchanged, whose general form is given in Section 4.4 whatever the heat transfer laws are; but heat transfer laws are intimately related to thermodynamics of the engine, as well as mass transfer. To fix an objective function is a current approach in practice. Often a certain quality of the engine is sought after, namely its efficiency. In this case the optimization is made with respect to the power output or heat input of the engine. If the power output of the engine is required, the optimization procedure looks for minimizing heat input (minimum fuel consumption) or maximizing the efficiency. For an engine whose heat input is imposed (e.g. a solar one), a third optimization case is chosen, namely maximizing the power output or the efficiency of the engine. The methodology illustration is reported for classical linear laws; a set of known results is restored, but new results are given, when s_ i is introduced as a constant or linearly dependent on the internal temperature difference of the working fluid (hH  hC), respectively Figs. 5,8 and 12. It clearly appears the limitation domain in the looped shape curve when g is a parameter (Figs. 3–10) and even more: only one physically acceptable configuration in a limited variation range if the power of the engine is imposed (Figs. 11–16). These results are new and our research group develops extensions of these in present. References [1] J.P. Maury, Carnot et la machine a` vapeur, Presses Universitaires de France, Paris, 1987.

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