Detailed analysis of an endoreversible fuel cell : Maximum power and optimal operating temperature determination

Producing useful electrical work in consuming chemical energy, the fuel cell have to reject heat to its surrounding. However, as it occurs for any other type of engine, this thermal energy cannot be exchanged in an isothermal way in finite time through finite areas. As it was already done for various types of systems, we study the fuel cell within the finite time thermodynamics framework and define an endoreversible fuel cell. Considering different types of heat transfer laws, we obtain an optimal value of the operating temperature, corresponding to a maximum produced power. This analysis is a first step of a thermodynamical approach of design of thermal management devices, taking into account performances of the whole system.


Introduction
The fuel cell (FC) is usually described as a system that directly converts into electricity the chemical energy provided by a process considered as a combustion [1]. Since it extracts a fraction of useful work from the energy provided by a fuel and a combustive, it could then be also viewed as a particular type of engine. Thereby, as any other engines and according to the second law of thermodynamics, all the provided chemical energy could not be converted into a useful form and a heat quantity is always rejected by the FC to its surrounding. The minimum thermal energy released, corresponding to the maximum produced work, is rejected by a reversible fuel cell that is in fact equivalent to a Carnot heat engine (CHE) [2,3]. * FEMTO-ST/ENISYS Institute, UMR CNRS 6174, University of Franche-Comte, Parc technologique, 2, avenue Jean Moulin, 90 000 Belfort, France. Phone : +33 (0)3 84 57 82 15 / Fax : +33 (0)3 84 57 00 32. email : philippe.baucour@univ-fcomte.fr However, as it was highlighted by Chambadal [4] and Novikov [5] and later by Curzon and Ahlborn [6], an reversible heat engine can only operate in exchanging heat as an infinitely slow process and therefore can not produce any practical useful power [7]. In fact, energy or mass transfers during finite durations or across finite areas are sources of entropy and leads to a decrease of whole system performances. Taking into account of these irreversibilities in the analysis of system performances is the scope of finite time thermodynamics [8,9]. A system producing entropy only because of irreversible exchange processes with its surrounding is said endoreversible [10]. The concept of endoreversibility have been successfully applied to a large scale of systems [8,11], including different types of engines [12,13], heat pumps [14], chemical systems [15,16], distillation devices [17] or pneumatic actuators [18].
As any other type of energy converter, the FC have to reject its own generated heat flux across finite areas or during finite times and can not produce any useful power if beeing entirely reversible. Taking into account the same limitations as for other systems, the FC could be considered as an endoreversible system, producing entropy only in rejecting heat flux in the ambiance [3].
In this article, we study and carry out the performances (produced work and energy efficiency) of an endoreversible fuel cell (EFC) considering different types of heat exchange laws (linear or not). We highlights the existence of an optimal operating temperature and discuss results dealing with different types of hydrogen fueled FC.
2 The endoreversible fuel cell

Fuel cell and heat engine
Let us consider an open and steady state system operating at a constant temperature T and a constant pressure p and housing the followed exothermic chemical reaction : with A i the chemical species and ν i their related stoichiometric coefficients. R and P are respectively the groups of reactives and products of reaction both considered as ideal gases. Besides molar quantities of reactants and products, it is assumed that the whole system exchanges work W and heat Q with its surrounding.
Considering a system noted 1 like e. g. a fuel cell, that directly converts into work the chemical energy provided by reaction (1), the energy and entropy balances are respectively expressed by : with ∆h and ∆s respectively the variations of enthalpy and entropy through reaction (1), δQ 1 the exchanged heat quantity, δ i S 1 the internal production of entropy, p the vector of partial pressures of both reactants and products and ξ the reaction progress coordinate defined by [19] : Combining (2) and (3), the variation of produced work can be expressed as : with ∆g < 0 the variation of the Gibbs energy g for chemical process (1). Logically, the maximum value of work δW 1 will be produced by a reversible system (δ i S 1 = 0) : Dividing this provided work by the reaction progress, we can express the molar reversible work w as : that corresponds to the quantity of useful energy provided by a reversible system with one mole of consumed chemical reactants. In the rest, the energy efficiency η will be defined as the fraction of useful energy w 1 on consumed energy, e. g. for the considered reversible system : Let us consider now an other system, noted 2, that only produces heat from chemical reaction (1), which is used as a hot source of a Carnot heat engine (CHE). First and second laws applied to this combustion system gives : with T * the temperature of the chemical process. Supposing it reversible (δ i S 2 = 0) and combining (9) and (10), we obtain the temperature T * , called sometimes entropic temperature [3,20] or equilibrium temperature [21] : This temperature corresponds to the one reached by a combustion process similar as (1) and both consuming and rejecting reactants and products at temperature T . Supplying a Carnot engine operating between temperatures T and T * with heat δQ 2 produced by previous ideal combustion system, we can produce a work quantity [2,3] : It can be seen that w 1,rev = w 2,rev and η 1,rev = η 2,rev and finally conclude on the thermodynamical equivalence of both reversible systems, i. e. of reversible fuel cell (RFC) and CHE [2,3]. Hence, a RFC could be viewed as a CHE operating with a heat source at temperature T * and a cold sink at is own temperature T (see figure 1 (a)).

Finite thermal conductance
As originally demonstrated by Chambadal and Novikov [4,5], an entirely reversible engine can exhange thermal energies only in a infinitely slowly process and finally can not produce any useful power. Practically, heat transfer can not occurs across finite exchange areas in an isothermal way and a finite difference of temperature is necessary to allow rejection of produced heat. In our model presented on figure 1 (b), the FC operates at temperature T in an ambiance at the cold temperature noted T c . Rejected molar heat quantity, noted q is now a function of temperatures T and T c . De Vos [13] showed that different heat transfer laws could be applied to this system. We will define the general heat transfer law as the form : where k is equivalent to a thermal conductance (usually defined as a product of heat transfer coefficient and heat transfer area), but divided by reaction progress as precedently did for molar work of relation (7). n is an integer representative of the type of heat transfer law [13]. Then, the heat exchange process will be called linear if n = 1 and nonlinear if n > 1. The equivalent heat engine of figure 1 (b) beeing internally reversible (i. e. endoreversible), its entropy balance can be written as : with q * (T ) = −∆h(T ) the heat delivered by an equivalent hot heat source (figure 1). Combining (12), (13) and (14), the molar work becomes : and related energy efficiency : Considering expression (15) of the molar work w, three cases have to be considered : 1. If T = T c , the efficiency η reaches a maximum equal to the Carnot's one of the whole system, in reason of the maximum value of difference temperature applied to the heat engine. Unfortunately, according to equation (13), no heat flux can be exchanged and then q = 0 lead up to w = 0.
2. If T = T * (p, T ) = T max , no difference temperature occurs in the equivalent heat engine of figure 1 (b) and according to equation (16), η = 0 brings to w = 0.
3. If T c < T < T max , the work w is a continuous and positive function of T which could be maximized.
Unlike other endoreversible engines [4][5][6][7][8], relationship (15) (i. e. the molar produced work) despicted a non linear link between entropic and operating temperatures, T * and T . Consequently, the maximum power correspondant temperature could not be expressed as the usual geometric mean of high and low temperatures √ T c · T max [7]. However, the optimal temperature T still corresponds to a maximum value of w : This relationship will be optimized by a Newton numerical algorithm using √ T c · T max as an initial point which turns out to be a good approximation (see figure 3).

Hydrogen fuel cell
As a practical example, let's considering the case of an endoreversible fuel cell (EFC) operating in consuming hydrogen as fuel. Chemical reaction (1) is : The surrounding temperature is fixed for example at T c = 298, 15 K. Thanks to experimental correlations published by the NIST [22], we have represented in figure 2 the evolution of the entropic temperature T * vs. reduced temperature : Two cases are studied : The first one uses pure oxygen (x O 2 = p O 2 /p • = 1) and the second one air as combustive (x O 2 = 0.21). The partial pressures of hydrogen and water are here always considered equal to unity. The maximum values of temperature are respectively T max (x O 2 = 1) ≃ 4 310 K and T max (x O 2 = 0, 21) ≃ 3 884 K. We can note a decrease of the entropic temperature with the operating one and with the molar fraction of oxygen. Using the relation (16), energy efficiencies for some pure oxygen and air as combustive are represented on figure 3. It can be observed a quasi linear decrease of η with the operating temperature T , leading to a zero value for T = T max . The following part of this study will only consider air as combustive agent.

Linear heat transfer law
Considering the molar work (15) produced with linear transfer law (n = 1), we can express it as : That corresponds to the work produced by a Carnot heat engine (CHE) operating between temperatures T c and T and consuming some molar heat q ′ = k 1 · (T * (p, T ) − T ). Numerical results are drawn of figure 4. We have presented in continuous line (curve 1) the evolution of reduced work w/ w vs. reduced operating temperature θ of relation (19). w is here the maximum value w regarding to temperature T . The second dashed line (curve 2), corresponds to the case where hot temperature is supposed constant and equal to the highest one T max and where expression of the molar work corresponds to the one of an usual endoreversible heat engine [4][5][6][7][8] : This engine is represented on figure 1 (b) by replacing variable high temperature T * by the constant one T max . In this particular case, it is easy to show that [4][5][6][7][8] : and the optimal operating temperature T also leads to a maximum work : and a related energy efficiency : The difference between curves 1 and 2 is a direct consequence of nonlinear link between the entropic temperature T * and the operating one T . Therefore, the value of the optimal temperature T corresponding to w = w is lightly modified and actually different from the one obtained by a geometric mean (22)   endoreversible system. The difference between curves due to nonlinearity of T * is here more significative, because of different values of maximum efficiencies. The previous curves are useful for making of the difference between Figure 5: Evolutions of the reduced molar work w/ w regarding to energy efficiency η for an endoversible FC (curve 1) and an endoreversible engine operating between T c and T max (curve 2).
low and high-temperatures FC. We have carried out on figure 6 the typical range of operating temperatures of protons exchange membrane fuel cells (PEMFC), i. e. 60 • C ≤ T ≤ 120 • C [1]. As previously presented on figure 3, a low-temperature fuel cell is characterized through high value of its energy efficiency. On a first hand, its low temperature difference with surrounding prevents to reject important heat quantity q, and according to the Carnot principle, to produce important work w like it is presented on figure 6. On a second hand, high-temperature fuel cells, such as solid oxyde fuel cells (SOFC), can easily evacuate generated thermal power, because of their high temperature differences with the ambiance, and are also able to produce high values of electrical power. On the right part of figure 6 is also drawn the typical operating temperature range of SOFC, i. e. 700 • C ≤ T ≤ 1 000 • C. However, this advantage is compensated by a lower energy efficiency, as shown on right part of figure 3. Table 2 presents the efficiency ranges and the proximity of operating temperatures with the optimal temperature for both type of FC. It can be observed that the maximum work temperature T is clearly within the operating temperature range of SOFC.
This first result is interesting but does not take into account the difference of heat transfer types between low and high temperature systems. Hence, high temperature heat transfer process are often driven by radiative effects.

Nonlinear heat transfer law
Let's consider now the case of a radiative heat transfer phenomenon (n = 4) between the FC and the surrounding, we can express the produced molar work as : As presented on figure 7 (full line), the nonlinear heat exchange law leads to a strong increase of the optimal temperature : T ≃ 2 910 K in the same chemical conditions as previously. This result could be explain by the fact that the same heat quantity needs a higher value of the temperature difference to be exchanged by a radiative way than thanks to convective phenomenon. The molar heat q corresponding to a maximum produced work could be released only beyond a temperature difference limit that is higher with some radiative exchanges than with only a convective phenomenon.  The efficiency-work curves of both linear and nonlinear heat transfers laws are presented on figure 8. Simultaneously to the increase of the maximum work operating temperature, we can note a strong decrease of related energy efficiency : η ≃ 25%. And yet, as presented on figure 3, an increase of the operating temperature corresponds to a decrease of its efficiency. Hence, the radiative heat transfer process appears to be unfavourable regarding to EFC performances because it moves away the optimal operating temperature from the one corresponding to convective heat exchange phenomena. Finally , a relevant thermal management system for high temperature FC will have to minimize radiative losses in order to decrease as much as possible the optimal temperature value T and to get the practical operating temperature T closed to it.

Conclusions
The formal equivalence between a reversible fuel cell (RFC) and a Carnot engine (CHE), both supplied by the same reversible combustion processes, allowed us to describe the first one with the finite time thermodynamics approach. The main result is the definition of an endoreversible fuel cell, operating in a reversible way but exchanging irreversibly heat with its surrounding, through finite thermal conductances. The optimization of the work produced regarding to the fuel cell operating temperature has led to an optimal EFC temperature, numerically calculated for a hydrogen-air reaction for standard conditions of pressure. The influence of heat transfer law type (linear or not) has been investigated for convective and radiative cases. The first one seems to be unfavourable on the fuel cell performances, because of the increase of the optimum temperature comparing to linear heat transfer case. At the same time, the efficiency of such system decreasing with temperature, the maximum produced work corresponding efficiency also has strongly decreased. We can conclude on the necessity for a relevant thermal management system to avoid radiative thermal effects and to favour the convective heat exchange phenomena. The present endoreversible fuel cell is based on an unique thermal finite conductance due to the heat flux exchange with surrounding. It would be Figure 7: Evolutions of reduced molar work w/ w vs. θ for n = 4 (continuous curve) and n = 1 (dashed curve).
significant to also consider a non reversible chemical reaction, using for example the results of chemical thermodynamics in finite time [8]. In the same way, different types of internal entropy production could be progressively taken into account. More relevant thermal management systems will have to carry out their's designs on the bases of the engines thermodynamics.
Moreover, design and optimization processes of fuel cell systems have also to take into account the fundamental Carnot principles. The rejected heat flux rejected of the system to the surrounding is a fundamental point and strongly influences the imaginable produced electrical power.