entropy Carnot-Like Heat Engines Versus Low-Dissipation Models Article

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1 entropy Article Carnot-Like Heat Engines Versus Low-Dissipation Models Julian Gonzalez-Ayala 1, *, José Miguel M. Roco 1,2, Alejandro Medina 1 and Antonio Calvo Hernández 1,2, * 1 Departamento de Física Aplicada, Universidad de Salamanca, Salamanca, Spain; roco@usal.es J.M.M.R.; amd385@usal.es A.M. 2 Instituto Universitario de Física Fundamental y Matemáticas IUFFyM, Universidad de Salamanca, Salamanca, Spain * Correspondence: jgonzalezayala@usal.es J.G.-A.; anca@usal.es A.C.H. Academic Editor: Micel Feidt Received: 20 Marc 2017; Accepted: 20 April 2017; Publised: 23 April 2017 Abstract: In tis paper, a comparison between two well-known finite time eat engine models is presented: te Carnot-like eat engine based on specific eat transfer laws between te cyclic system and te external eat bats and te Low-Dissipation model were irreversibilities are taken into account by explicit entropy generation laws. We analyze te matematical relation between te natural variables of bot models and from tis te resulting termodynamic implications. Among tem, particular empasis as been placed on te pysical consistency between te eat leak and time evolution on te one side, and between parabolic and loop-like beaviors of te parametric power-efficiency plots. A detailed analysis for different eat transfer laws in te Carnot-like model in terms of te maximum power efficiencies given by te Low-Dissipation model is also presented. Keywords: termodynamics; optimization; entropy analysis 1. Introduction A cornerstone in termodynamics is te analysis of te performance of eat devices. Since te Carnot s result about te maximum possible efficiency tat any eat converter operating between two eat reservoirs migt reac, te work in tis field is mainly focused on ow to fit real-life devices as close as possible to te main requirement beind te Carnot efficiency value, i.e., te existence of infinite-time, quasi-static processes. However, real-life devices work under finite-time and finite-size constraints, tus giving finite power output. Over te last several decades, one of te most popular models in te pysics literature to analyze finite-time and finite-size eat devices as been te so-called Carnot-like model. Inspired by te work reported by Curzon Alborn CA [1], tis model provides a first valuable approac to te beavior of real eat engines. In tis model, it is assumed an internally reversible Carnot cycle coupled irreversibly wit two external termal reservoirs endoreversible ypotesis troug some eat transfer laws and some penomenological conductances related wit te nature of te eat fluxes and te properties of te materials and devices involved in te transport penomena. Witout any doubt, te main result was te so-called CA-efficiency η 1 τ were τ T c /T is te ratio of te external cold and ot eat reservoirs. It accounts for te efficiency at maximum power MP conditions wen te eat transfer laws are considered linear wit te temperature difference between te external eat bats and te internal temperatures of te isotermal processes at wic te eat absorption and rejection occurs. Later extensions of tis model included te existence of a eat leak between te two external bats and te addition of irreversibilities inside of te internal cycle. Wit only tree main ingredients eat leak, external coupling, and internal irreversibilities te Carnot-like model Entropy 2017, 19, 182; doi: /e

2 Entropy 2017, 19, of 13 as been used as a paradigmatic model to confront many researc results coming from macroscopic, mesoscopic and microscopic fields [1 28]. Particularly relevant ave been tose results concerned wit te optimization not only of te power output but also of different termodynamic and/or termo-economic figures of merit and additionally te analysis on te universality of te efficiency at MP or on oter figures of merit as ecological type [29 34]. Complementary to te CA-model, te Low-Dissipation LD model, proposed by Esposito et al. in 2009 [35], consists of a Carnot engine wit small deviations from te reversible cycle troug dissipations located at te isotermal brances wic occur at finite-time. Te nature of te dissipations entropy generation are encompassed in some generic dissipative coefficients, so tat te optimization of power output or any oter figure of merit is made easily troug te contact times of te engine wit te ot and cold reservoirs [36 39]. In tis way, depending on te symmetry of te dissipative coefficients, it is possible to recover several results of te CA-model. In particular, te CA-efficiency is recovered in te LD-model under te assumption of symmetric dissipation. Recently, a description of te LD model in terms of caracteristic dimensionless variables was proposed in [40 42]. From tis treatment, it is possible to separate efficiency-power beaviors typical of CA-endoreversible engines as well as irreversible engines according to te imposed time constraints. If partial contact times are constrained, ten one obtains open parabolic endoreversible curves; oterwise, if total time is fixed, one obtains closed loop-like curves. Te objective of tis paper is to analyze in wic way te Carnot-like eat engines dependent on eat transfer laws and te LD models dependent on a specific entropy generation law are related and ow te variables of eac one are connected. Tis allows for an interpretation of te eat transfer laws, including te eat leak, in terms of te bounds for te efficiency at MP provided by te LD-model, wic, in turn, are dependent on te relative symmetries of te dissipations constants and te partial contact times. Te article is organized as follows: in Section 2, a correspondence among te variables of te two models for eat engines HE is made. In Section 3, we study te region of pysically acceptable values for te Carnot-like HE variables depending on te eat leak. In Section 4, te study of te MP regime is made, sowing tat a variety of results between bot descriptions can be recovered only in a certain range of eat transfer laws; in particular, we analyze te efficiency vs. power curves beaviors. Finally, some concluding remarks are presented in Section Correspondence between te HE s Variables of Bot Models A key point to establis te linkage between bot models is te entropy production. By equaling te entropy cange stemming from bot frameworks it is possible to give te relations among te variables tat describe eac model see Figure 1. In te LD case see Figure 1a, te base-line Carnot cycle works between te temperatures T c and T > T c, te entropy cange along te ot cold isotermal pat is S Q T S + Q T c and te times to complete eac isoterm processes are t and t c, respectively. Te adiabatic processes, as usual, are considered as instantaneous, toug te influence of finite adiabatic times as been reported in te LD-model in [43]. Te deviation from te reversible scenario in te LD approximation is considered by an additional contribution to te entropy cange at te ot and cold reservoirs given by S T S + Σ t, 1 S Tc S + Σ c t c, 2

3 Entropy 2017, 19, of 13 were Σ and Σ c are te so-called dissipative coefficients. Te signs + take into account te direction of te eat fluxes from toward te ot cold reservoir in suc a way tat Q c and Q are positive quantities. Ten, te total entropy generation is given by S tot Σ t + Σ c t c. 3 Figure 1. a Sketc of a low dissipation eat engine caracterized by entropy generation laws S T and S Tc ; b Sketc of an irreversible Carnot-like eat engine caracterized by generic eat transfers Q, Q c and Q L. At tis point, it is elpful to use te dimensionless variables defined in [40]: α t c /t, Σ c Σ c /Σ T and t t S/Σ T, were t t + t c and Σ T Σ + Σ c. In tis way, it is possible to define a caracteristic total entropy production per unit time for te LD-model as S tot S tot t S S tot t Σ T S 2 1 t [ 1 Σ c 1 α t + Σ ] c. 4 α t In te irreversible Carnot-like HE, te entropy generation of te internal reversible cycle is zero and te total entropy production is tat generated at te external eat reservoirs see Figure 1b. By considering te same sign convention as in te LD model Q T w S 0 and Q c T cw S 0, were S is te entropy cange in te ot isotermal branc of te reversible Carnot cycle, and a eat leak Q L 0 between te reservoirs T and T c, ten S T Q T Q L S + T S Tc Q c T c + Q L T c S + 1 a 1 a c 1 + τ Q L T c S Q L T c S S, 5 S, 6 were a T /T w 1 and a c T cw /T c 1. By introducing a caracteristic eat leak Q L Q L / T c S and a comparison wit Equations 1 and 2 gives te expressions associated wit te dissipations Σ t Σ c t c 1 a 1 τ Q L S, 7 a c 1 + Q L S. 8

4 Entropy 2017, 19, of 13 By assuming tat te ratio t c /t c + t is te same in bot descriptions, ten we introduce α 1/1 + t /t c into te Carnot-like model, and Σ c Σ c /Σ T and t t S/Σ T are Σ 1 c 1 + t α 1 α 1 a 1 τ Q L, 9 α a c 1 + Q L a c 1 + Q L [ α α 1 ], 10 1 a 1 τ Q L a c 1+ Q L wic are te relations between te caracteristic variables of te LD model and te variables of te Carnot-like HE. Tis is summarized in te following expression: 3. Pysical Space of te HE Variables Σ c α t a c 1 + Q L. 11 We stress tat all te above results between variables old for arbitrary eat transfer laws in te Carnot-like model. As a consequence, above equations provide te generic linkage between bot descriptions, and, from tem, useful termodynamic information can be extracted. In Figure 2a, te internal temperatures for te irreversible Carnot-like HE, contained in a and a c, are depicted wit fixed values τ 0.2, α 0.2 and Σ c 0.5. Notice tat, in order to obtain termal equilibrium between te auxiliary reservoirs and te external bats i.e., to acieve te reversible limit, it is necessary tat Q L 0. As soon as a eat leak appears, T w < T, meanwile T cw T c is always a possible configuration. As te eat leak increases in te HE, te internal temperatures get closer to eac oter until te limiting situation were T w T cw see contact edge in Figure 2a. Figure 2. a T w and T cw from Equation 9. Note ow, as te eat leak increases, te possible pysical combinations of T w and T cw become more limited; b t Q L, a c according to Equation 10. Te representative values α 5 1 τ, Σ c 2 1 ave been used, owever, te displayed beavior is similar for any oter combination of values. As a eat leak appears, te reversible limit t is no longer acievable. Tis is better reflected in Figure 2b, were we plot te total operation time t depending on a c and Q L see Equation 11. Only wen a c 1 and Q L 0 can large operation times be allowed. We can see in tis figure tat te existence of a eat leak imposes a maximum operational caracteristic time to te HE. Te total time is noticeably sorter as te eat leak increases, in agreement wit te fact tat, for t 1, te working regimes are dominated by dissipations. It could be said tat te eat leak beaves as a causality effect in te arrow of time of te eat engine.

5 Entropy 2017, 19, of 13 Notice tat, in Figure 2, tere is a region of proibited combinations of Q L and a c. Tis as to do wit te pysical reality of te engine negative power output and efficiency. In [41], te region of pysical interest in te LD model under maximum power conditions was analyzed. In te Carnot-like engine, some similar considerations can be addressed as follows: in a valid endoreversible HE, te internal temperatures may vary in te range a 1, τ 1 and a c 1, τ 1 a 1 in order to ave T c T cw T w T, a c τ 1 a 1 being te condition for T cw T w implying null work output and efficiency. From Equation 9, it is possible to obtain two conditions on Q L α, Σ c, a c, a, τ initially assumed to be 0 according to te values a 1 and a τ 1. For a 1, we obtain tat Q L a c 1 1 Σ c 1 Σ c Σ c + τ Σ c 1 α α 0, 12 wose only pysical solution is Q L 0. Ten, as long as tere is a eat leak in te device, te internal ot reservoir cannot reac equilibrium wit te external ot reservoir and te reversible configuration is not acievable. On te oter and as can be seen in Figure 2a, te largest possible eat leak i.e., te largest dissipation in te system as as an outcome tat T w T cw T c, tat is, a c 1 and a τ 1. In tat limit, Equations 9 and 11 give Q L,max 1 τ 1 α 1 Σ c Σ c + τ α 1 α α Σ c α t, 13 and, since in tis case all te input eat is dissipated to te cold external termal reservoir, te HE as a null power output. In Figure 3, we depict te range of possible values tat Q L can take from 0 up to Q L,max in terms of α and t. By means of Equation 13, it is establised a boundary condition for pysically acceptable values of te irreversible Carnot-like HE in terms of te LD variables, wic is t α 1 Σ c + Σ c τ 1 α. 14 α 1 α 1 τ Figure 3. Possible values of Q L as a function of te control parameters α and t. We used te values Σ c 0.8 and τ 0.2.

6 Entropy 2017, 19, of 13 Up to tis point, we ave proposed a generic correspondence between te variables of bot scemes: te LD treatment, based on a specific entropy generation law, and te irreversible Carnot-like engine based on eat transfer laws. In te following, we will furter analyze te connection given by Equation 11 wit te focus on different eat transfer laws and te maximum power efficiencies given by te low-dissipation model. 4. Maximum-Power Regime As is usual, te power output is given by P η Q t c + t. 15 In [41], it was sown tat, in te MP regime of an LD engine display, an open, parabolic beavior for te parametric P η curves wen α α P max is fixed and for Σ c [0, 1]. On te oter and, by fixing te value of t t P max, one obtains for te beavior of η vs. P loop-like curves see Figure 4 in [41]. In te irreversible Carnot-like framework, open η vs. P curves are caracteristic of endoreversible CA-type engines, and, wen a eat leak is introduced, one obtains loop-like curves. Te apparent connection between te beavior displayed by fixing t or α in te low dissipation context wit te presence of a eat leak, or te lack of it, is by no means obvious. A simple analysis of te MP regime in an irreversible Carnot-like engine in terms of te LD variables will sed some ligt on tis issue and will also provide us a better understanding of ow good te correspondence is between bot scemes Low Dissipation Heat Engine In terms of te caracteristic variables, te input and output eat are Q Q t Q c Q c t Q T c S Q c T c S giving a power output and efficiency as follows: P W t W T c S Σ T t S Σ T t S Σ T t S 1 1 Σ c 1 α t 1 Σ c α t [ 1 τ 1 1 τ 1 τ t, 16 1 t, 17 1 Σ c 1 α t Σ c α t ] 1 t, 18 η P W 1 τ 1 Σ c τ Σ c 1 α t α t η. 19 Q Q 1 1 Σ c 1 α t Te optimization of P t, α; Σ c, τ is accomplised troug te partial contact time α and te total time t, wose values are α P max Σ c, τ t P max Σ c, τ τ 1 Σ c, 20 τ Σ c 2 τ Σ c + 1 Σ c, 21

7 Entropy 2017, 19, of 13 wit an MP efficiency given by η P max Σ c, τ [ τ Σ 1 τ 1 + c [ ] 2 τ Σ 1 + c + τ 1 Σ c 1 Σ c ] 1 Σ c 1 Σ c. 22 One of te most relevant features of tis model is te capability of obtaining upper and lower bounds of te MP efficiencies witout any information regarding te eat fluxes nature. Tese limits are η P η C max 2 η P max τ, Σ c 2 η C η + P, 23 η C max corresponding to Σ c 1 and Σ c 0 for te lower and upper bounds, respectively. For te symmetric dissipation case, Σ c 1/2 Σ, te well known CA-efficiency η sym P max 1 τ η CA is recovered Carnot-Like Model witout Heat Leak Endoreversible Model Now, let us consider a family of eat transfer laws depending on te power of te temperature to model te eat fluxes Q and Q c see Figure 1b as follows: Q T k σ 1 a k t 0, 24 Q c Tc k σ c ac k 1 t c 0, 25 were k 0 is a real number, σ and σ c are te conductances in eac process and t and t c are te times at wic te isotermal processes are completed. Te adiabatic processes are considered as instantaneous, a common assumption in te two models. According to Equation 15, power output is a function depending on te variables a c, a and te ratio of contact times; k, τ and σ c are not optimization variables for tis model. Te endoreversible ypotesis S Tw S Tcw gives te following constriction upon te contact times ratio t c t σ c a c a τ 1 k 1 a k ac k, 26 1 were σ c σ /σ c. Since tere is no eat leak, te efficiency of te internal Carnot cycle is te same as te efficiency of te engine, ten a c a τ 1 η, and te dependence of a is substituted by η. Ten, in terms of α, Equation 26 is 1 ak cτ k α 1 α σ cτ k 1 η 1 η k ac k Te optimization of power output P a c, η; σ c, τ, T, k in tis case is acieved troug a c and η. Te maximum power is obtained by solving P a c η 0 for a c and P η 0 for η. From te first a c condition, we obtain P, wic is P η; σ c, τ, T, k σ T k η 1 η 1 η k τ k σc + 1 η k

8 Entropy 2017, 19, of 13 Tis function as a unique maximum corresponding to η Pmax, wic is te solution to te following equation [ ] [ σc 1 η τ k 1 η k 1 kη + 1 η k k η τ k 1 η k+1] 0, 29 and depends on te values σ c, τ and te exponent of te eat transfer law k as sowed in [9]. In Figure 4a, η Pmax is depicted for te limiting cases σ c {0, }. All of te possible values of η Pmax for different σ c s are located between tese two curves. It is well-known tat, for te Newtonian eat transfer law k 1, η Pmax η CA is independent of te σ c value. As te eat transfer law departs from te Newtonian case, te upper and lower bounds cover a wider range of efficiencies. Ten, te limits appearing in Equation 23 are fulfilled for a limited region of k values in te Carnot-like model. From Figure 4a, it is possible to see tat te results stemming from an endoreversible engine are accessible from an LD landmark only in te region k 1, 2.5] for oter values of k, tere are efficiencies outside te range given by Equation 23. By equaling tese efficiencies wit te LD one see Equation 22 and solving for Σ c, we obtain tose values tat reproduce te endoreversible efficiencies. Tis is depicted in Figure 4b. Notice also tat not all Σ c symmetries are allowed for every k 1, 2.5]. For example, wit a eat transfer law wit exponent k 1, all te possible values of te efficiency can be obtained if te parameter Σ c varies from 0 to 1, tat is, all symmetries are allowed. Meanwile, for k 1, only te symmetric case Σ c 1/2 is allowed, reproducing te CA efficiency. For k outside 1, 2.5], tere are efficiencies above and below te limits in Equation 23 wit no Σ c values tat migt reproduce tose efficiencies, tus limiting te eat transfer laws pysically consistent wit predictions of te LD model. Figure 4. a upper and lower bounds of te MP efficiency in terms of te exponent of te eat transfer law k of te Carnot-like eat engine; b te Σ c values tat reproduce te upper and lower bounds of te endoreversible engine. Inside te region were te LD model is able to reproduce te asymmetric limiting cases σ c {0, }, te correspondence between te two formalisms as not an exact fitting. In order to sow tis, we will address te symmetric dissipation case. As can be seen from Figure 4, in te endoreversible CA-type HE, for every k, tere is one σ c tat reproduces te CA efficiency. On te oter and, in te LD model, te symmetric dissipation is attaced to η CA. If we use te α and t values of MP of te LD model and calculate te values of a c and a associated wit tem instead of calculating tem according to Equations 28 and 29, we can see

9 Entropy 2017, 19, of 13 weter tey allow us to recover te correct value of σ c tat in te endoreversible model gives te CA efficiency or does not. Tat is, for Σ c 1/2, Equations 20 and 21 reduce to α sym P max τ 1 + τ, 30 t sym P max 1 + τ 1 τ. 31 From Equation 11, we obtain a c and wit te condition η η CA, wit T cw /T w a c a τ 1 η τ we calculate a, tus a sym c, P max 1 + τ 2 τ, 32 a sym, P max τ. 33 By using Equation 30, te ratio of contact times results in t c 1 α τ, and, by using te endoreversible ypotesis Equation 26, it is possible to obtain te value of σ c tat would produce te CA efficiency, being k k σ sym 1 + τ τ 2 2 k τ k c, P max 2 k 1 + τ, 34 k t α wic for k 1 gives σ sym τ and for k 1 gives σ sym 1/τ. Neverteless, c, P max c, P max by substituting Equation 34 into Equation 29, te MP efficiency is not exactly te CA one, as can be seen in Figure 5. Sowing tat te correspondence between bot models is a good approximation only in te range k [ 1, 1], and is exact only for k 1 and k 1. Anoter incompatibility of te two approaces comes up in te Newtonian eat excange k 1: meanwile, te Carnot-like sceme η CA is independent of any value of σ c, and, in terms of te LD model, η CA is strictly attaced to a symmetric dissipation Σ c 1/2 Σ. Ten, te only law tat as an exact correspondence for all values of Σ c and σ c is te law k 1. Figure 5. Maximum-power efficiency for te symmetric case Σ c 1/2, assuming te LD condition tat t c t τ and using te resulting σ c value tat fulfills te endoreversible ypotesis. Notice tat te matcing wit te CA efficiency is approximate for te interval k [ 1, 1] and is exact for k { 1, 1}, as can be seen in te zoom of tis region on te rigt side of te figure Carnot-Like Model wit Heat Leak Now, let us consider a eat leak of te same kind of te eat fluxes Q c and Q, tat is, Q L T k σ L 1 τ k t + t c 0, 35

10 Entropy 2017, 19, of 13 ten, te caracteristic eat leak is Q L Q L Tk σ L 1 τ k t + t c a c Q c Tc k σ c a k c 1 a c σ Lc 1 τ k t c α τ k ac k Te power output of te engine is te same tan in te endoreversible case; owever, a difference wit te previous subsection arises, and now te efficiency is given by te following expression: η Q + Q L Q c Q L Q + Q L 1 Q c + Q L QL Q c Q + Q L Q Q c + Q L Q c Q L Q + Q L, 37 were we ave used te fact tat T c S Q c to introduce te caracteristic eats in te last expression. From Equation 37, it can be derived tat, if te eat leak increases, te efficiency diminises. In Figure 6, we can observe ow te upper and lower bounds of te efficiency appearing in Figure 4 are affected by te introduction of a constant eat leak. Now, by using Equation 37 and te fact tat in te endoreversible case η 1 Q 1, we ave plotted te Q L values tat leads te efficiency η CA for k 1 of te endoreversible case down to te value η C /2 see point A in Figure 6, wic is Q L 1 τ ; and te value Q τ1+ τ L 1 τ 2τ tat lead to an efficiency η C/ 2 η C for k 1 of te endoreversible case to η C /2 see point B in Figure 6. Figure 6. Influence of te eat leak over te optimized efficiencies appearing in Figure 4. See te text for explanation. Notice in Figure 6 tat, for Q L 0.1, tere is a region around k 1 were η CA is outside te saded region of maximum power efficiencies. Ten, for tese values of k and Q L, te symmetric dissipation case, always attaced to te η CA efficiency, is out of reac. Additionally, tere is not a eat transfer law tat fulfills bot upper and lower bounds for maximum power efficiency given by te LD model as occurred for k 1 in te endoreversible case. Te additional degree of freedom caused by te appearance of te eat leak makes more complex te analysis of te validity of te correspondence between bot models, wic, in general, sould be andled numerically. 5. Conclusions We analyzed Carnot-like eat engines dependent on eat transfer laws and te LD models dependent on a specific entropy generation law and studied ow te variables of eac one are connected. We were able to provide an interpretation of te eat transfer laws, including te eat

11 Entropy 2017, 19, of 13 leak, in terms of te bounds for te efficiency at MP provided by te LD-model, wic, in turn, are dependent on te relative symmetries of te dissipations constants and te partial contact times. By comparing te entropy production of te low dissipation model and te Carnot-like model, we proposed a connection between te variables tat describe eac model. We sow tat, for an HE, te region of pysical interest is independent from te operation regime, being equivalent to tat for a maximum power LD-HE. Tat is, η t P, α, max Σ c, τ 0 defines te acceptable α values, and η t, α P, max Σ c, τ 0 tose of t gray saded areas in Figure 7a,b, respectively. Tese considerations on te LD model are recovered from pysical considerations on te Carnot-like HE. Tus, te difference in te performance of te HEs is not due to te pysical configurations of te system, but it comes from te approximations tat tese models rely on: one over te entropy and te oter over te eat fluxes. Figure 7. a Pysically well beaved region of te α Σ c variables. Te saded areas come from te LD model and te dased curves come from te Carnot-like model; b Te same for te t Σ c variables. Notice te agreement in bot models. In tese plots, we use τ 0.2. We sow tat te eat leak disappears wen fixing te partial contact time in te LD engine, leading to typical endoreversible open parabolic power vs. efficiency curves. However, te connection between te endoreversible case and te LD model is exact only for eat transfer laws wit exponents k 1 and k 1, and a good approximation in te region k 1, 1. On te oter and, te presence of a eat leak fixes te total operation time and te partial contact time is not constrained, tus, allowing te eat leak to act as an additional degree of freedom te same efficiency is acieved wit different combinations of partial contact time ratios and eat leaks. Te reversible limit is not accessible in tis case, a maximum operation time is establised, and wen te eat leak dissipation effects are important, te efficiency may be zero, wic is te origin of te loop beavior of te irreversible P vs. η curves. Te connection in te case wit eat leak is more complex and its validity depends on te value of Q L. Acknowledgments: Julian Gonzalez-Ayala acknowledges CONACYT-MÉXICO; José Miguel M. Roco, Alejandro Medina and Antonio Calvo-Hernández acknowledge te Ministerio de Economía y Competitividad MINECO of Spain for financial support under Grant ENE R. Autor Contributions: José Miguel M. Roco and Antonio Calvo Hernández conceived and made some simulations of te termodynamic models. Julian Gonzalez-Ayala and Alejandro Medina. complemented calculations, simulations and te elaboration of grapics. Julian Gonzalez-Ayala and Antonio Calvo Hernández wrote te paper. All autors ave read and approved te final manuscript. All autors ave read and approved te final manuscript. Conflicts of Interest: Te autors declare no conflict of interest.

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