Article Reconsideration of Criteria and Modeling in Order to Optimize the Efficiency of Irreversible Thermomechanical Heat Engines

The purpose of this work is to precise and complete one recently proposed in the literature and relative to a general criterion to maximize the first law efficiency of irreversible heat engines. It is shown that the previous proposal seems to be a particular case. A new proposal has been developed for a Carnot irreversible thermomechanical heat engine at steady state associated to two infinite heat reservoirs (hot source, and cold sink): this constitutes the studied system. The presence of heat leak is accounted for, with the most simple form, as is done generally in the literature. Irreversibility is modeled through , created internal entropy rate in the converter (engine), and , total created entropy rate in the system. Heat transfer laws are represented as general functions of temperatures. These concepts are particularized to the most common heat transfer law (linear one). Consequences of the proposal are examined; some new analytical results are proposed for efficiencies.


Introduction
Carnot is without doubt the precursor of the development in the field of Equilibrium Thermodynamics applied to machines, systems and processes; he introduced the concept of cycle, mainly the "Carnot cycle". Finally he can be credited with being the originator of the notion of efficiency, an important concept in today's world [1].

OPEN ACCESS
Finite Time Thermodynamics (F.T.T.) is associated with the work of Curzon and Ahlborn [2]; in that work, the authors observed that the efficiency at MAX(-W) becomes less than the Carnot limit. But this was first pointed out by Chambadal [3] and by Novikov [4] in 1957: the Chambadal-Novikov-Curzon-Ahlborn efficiency differs from that given by Carnot. These works have been continuously completed since; see for example [5][6][7][8][9].
It is well know that the first law efficiency of the Carnot engine in the equilibrium thermodynamics limit is: The CNCA efficiency delivered by an endoreversible Carnot engine in contact with two infinite heat reservoirs at T SH (hot source) and T SC (cold sink) is: Numerous works are concerned with heat leak model [12][13][14], internal irreversibility model [15][16][17], and irreversible model with heat resistance, heat leakage and internal irreversibility with different heat transfer laws [18][19][20][21].
Models have been developed to account for the loop shaped form [see Figure 1(b)]: it includes heat losses, or irreversibility trough an entropy ratio [22], or more recently through a created entropy rate . s [23]. This second method is preferred because connected to the entropy analysis [24]. A I MAX η , maximum efficiency according to first law appears, that is smaller than C η , and occurs at point C, thus reducing the high efficiency zone of the engine. In a recent paper [25] Aragon-Gonzales et al. propose a general criterion to maximize efficiencies of several heat engines (Brayton engine; Carnot engine). This criterion seems to be a particular one that cannot be applied as a general criterion to all kind of heat engines. We intend to demonstrate this fact, in the present paper, by two ways: first, we apply the proposal of the authors, to some cases of irreversible Carnot cycle, as done in Section 4 of the referenced paper [25], and we will observe that, it does not fit the hypothesis.
Secondly, we develop always for the irreversible Carnot cycle, a general model, under the same main conditions as used in Aragon-Gonzales et al. paper. We deduce the new general method relative to first law efficiency of a Carnot engine optimization.
We apply this last method to some particular and realistic models corresponding to what is done in the literature [26]. This provides some new analytical results concerning first law efficiency upper bound that will be commented: the optimum efficiency differs from the one relative to minimum of total created entropy rate.

General Model of the CARNOT Irreversible Engine
The Carnot irreversible engine could be defined as a thermomechanical engine (converter) in contact with two infinite heat reservoirs (hot and cold thermostats); the engine and the two thermostats constitute the studied system. The authors of Reference [25] consider a finite-time thermodynamics approach, that could be discussed; we however prefer to use steady state modeling, but also with irreversibilities and heat losses. The scheme of the system is proposed in Figure 2. It can be seen that the thermal loss used by the authors of Reference [25] corresponds to general actual modelling: a thermal short circuit, between the hot source at temperature T SH , and the cold sink at temperature T SC , generally the ambient temperature. The corresponding heat flux is: where: K SL , generalized heat loss conductance, f SL (T SH , T SC ), function characterizing the heat transfer.
It is to be noted that, in the material of the engine too, it must exist conductive heat loss, between the hot part and the cold part: But, we can suppose that these heat losses correspond to internal dissipation in the engine converter, and participate of the internal entropy flux of the engines i s .
[see Relation (11) hereafter]. According to this assumption, we could summarize saying that the heat losses in the system are assumed to occur between the maximum temperature T SH , and the minimum temperature T SC of the system, through an equivalent heat transfer conductance K SL [see Relation (3)].
Using thermodynamical convention (see Figure 2), it comes for the used heat rate at the source SH q , and rejected heat rate at the sink: In fact, these two definitions are completed by the two heat transfer laws between source and hot side of the engine at temperature T H , and between cold side of the engine at T C and the sink [26]: Applying first law of thermodynamics to the system it becomes: Applying second law of thermodynamics to the system it becomes: where T s . represents the total entropy rate of the system due to all irreversibilities. It differs from the one use by Aragon-Gonzales et al. [25], that corresponds to the entropy balance applied to the converter as: (11) Further , the authors use, as is traditional in the literature, an irreversibility ratio I (parameter); the interest of using, the entropy rate i s has been exposed in recent paper [10], and we move to this entropy flux method preferably; for generality, we choose to express T s as a function of temperatures as a function of (T H , T C ): The justification of the form of T s . , is easily obtained, using (10,11,12,13). Effectively it appears that:

Maximum Efficiency Criterion
We focus here on the system first law efficiency IS η : It is easy to eliminate . w − through (9): In the same way, it is possible to obtain the converter (or engine) first law efficiency; it corresponds to: Using the entropy balance of the engine, it is easy to show that IE η could also be expressed as: correspondingly, IS η could be expressed as: The two last relations suggest that the first law efficiency of the system is bounded by the classical Carnot efficiency associated to equilibrium thermodynamics, whereas the converter efficiency is bounded by the endoreversible efficiency, whatever the link with the source and sink. In case of a perfect (reversible) link, we recover the Carnot first law efficiency. Generally the published paper consider more the engine aspect, than the system one: we focus here on the system one.
Relation (18) indicates that the system efficiency depends on two functions T s . and SH q .
. So, for a designed system K H , K C , T SH , T SC are parameters, and T H , T C (or X H = T SH -T H , X C = T SC -T C ) natural (generic) variables. But the two variables are connected through the entropy balance (10). There is only one degree of freedom as supposed in the paper of [25].

Developing the Criterion with the Degree of Freedom x
By derivation of (18), it comes: x q

Developing the Criterion with Variationnal Calculus
The lagragian of the system L (T H , T C ) is: By derivation, we obtain the equations system to solve with respect to (T H , T C , λ ), and eliminating λ , the two equations in T H , T C , given hereafter: (20) with the notation The expression of the optimum system efficiency, is calculated using T H *, T C *, solutions of the given system in:  [25] Are the conditions 0 Choosing T H as independent variable it comes:

Example 2: Generalized Convective Laws
In this case we move from variable T i , to X i according to: and the entropy constraint: With X H independent variables, it comes: The optimum of IS η is associated to a given value x* of the degree of freedom according to: We renew that the system entropy balance suppose that: The second derivative is expressed as: If the bracket is negative, the optimum of efficiency coincides with a maximum, and the condition is: This general condition differs essentially from the one proposed in [25]. We remark too, that in the linear case generally studied in papers, the last condition proposed is not strictly satisfied. There is no optimum, and the corresponding first law efficiency of the system is:  (21) It results that, if the system is a reversible one  T  T  f  T  T   T  T  K  T  T   T  T  K  s  s  ,   2  2 0 . .
Consequently   Using Equation (21), it appears that second law of thermodynamics implies that f ST must be a decreasing function of variable T H , and an increasing function of T C ; these conditions are not consistent with the common (linear, phenomenological) laws [see Relation (23)]: these laws are only compatible for the engine (converter) according respectively to: linear law: ( )