Validation of cell voltage and water content in a PEM (polymer electrolyte membrane) fuel cell model using neutron imaging for different operating conditions

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es This is an Accepted Manuscript of an article published by Elsevier in Energy, Vol. 101, on April 2016, available at: https://doi.org/10.1016/j.energy.2016.02.006 Copyright 2016 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Validation of cell voltage and water content in a PEM fuel cell model using neutron imaging for different operating conditions Authors: J. Antonio Salvaa*, Alfredo Iranzoa, Felipe Rosab, Elvira Tapiaa a AICIA, School of Engineering,Seville, Spain. b Thermal Engineering Group, Energy Engineering Department, School of Engineering. University of Seville. Camino de los Descubrimientos s/n, 41092 Sevilla, Spain. *Corresponding author. Tel: +34 954482254; Fax: +34 954487247; E-mail address: [email protected] (J. A. Salva) Abstract This work presents a one dimensional analytical model developed for a 50 cm2 PEM fuel cell with five-channel serpentine flow field. The different coupled physical phenomena such as electrochemistry, mass transfer of hydrogen, oxygen and water (two phases) together with heat transfer have been solved simultaneously. The innovation of this work is that the model has been validated with two different variables simultaneously and quantitatively in order to ensure the accuracy of the results. The selected variables are the cell voltage and the water content within the membrane (MEA) and gas diffusion layers (GDL) experimentally measured by means of neutron radiography. The results show a good agreement for a comprehensive set of different operating conditions of cell temperature, pressure, reactants relative humidity and cathode stoichiometry. The analytical model has a relative error less than 3.5 % for the value of the cell voltage and the water content within the GDL+MEA for all experiments performed. This result presents a new standard of validation in the state of art of PEM fuel cell modeling where two variables are simultaneously and quantitatively validated with experimental results. The developed analytical model has been used in order to analyze the behavior of the PEM fuel cell under different values of relative humidity. Keywords: PEM fuel cell; model validation; water management; neutron imaging 1 Introduction Hydrogen is considered one of the future energy vectors that will have an important development in next years. Its applications as fuel present, compared with traditional fossil fuels, advantages from the point of view of the environment due to reduction of CO2, nitrogen oxides and sulfur emissions. Hydrogen can be used in chemical combustion processes and in internal combustion engines. However, the application that presents higher interest is the production of electrical energy in fuel cells (which are electrochemical devices able to convert chemical energy contained in a fuel into electrical energy with high performance as the conversion is not limited by Carnot´s cycle). Fuel cells have a theoretical performance around 80% and practical performance around 40-60 % [1], thus having a high potential of applications. PEM fuel cells are considered to play a fundamental role in the future energy sector. In this way, mathematical models and simulations are needed for the fuel cell design and optimization. In the literature exists many PEM fuel cell models [2-5]. From simplified models where only the cathode side is modeled [6] to complex models where a single cell is modeled completed [7, 8]. Two different techniques are normally used to modeling PEM fuel cells. The first one is using analytical models [9-11] while the second one is using Computational Fluid Dynamics techniques (CFD) [7, 8, 12-14]. CFD models always provide more information than analytical models. However, the computational time for CFD models are much higher than analytical models. To ensure the accuracy of the results, it is necessary to compare the results of the model with experimental results. In this way, the polarization curve usually is selected to compare experimental and model results. In the state of the art of PEM fuel cell modeling, there are several works where the polarization curve has been validated [6-14]. The polarization curve offers important information about the performance of the fuel cell. However, it is not enough to understand the real processes that occurs in it [2]. For this reason, it is advisable to develop properly validated models against more than one variable in order to analyzed and quantify what is happening inside the fuel cell to propose improvements and be able to increase the performance of the PEM fuel cell [2]. A relevant phenomenon to evaluate is the water management in PEM fuel cells [15-18]. To obtain an optimal PEM fuel cell performance it is necessary to achieve an adequate water balance to ensure that the membrane remains hydrated in order to reach a good electrical conductivity. Several authors have developed PEM fuel cell models where the water transport has been considered. However, there are only a few works where the water content has been validated qualitatively with experimental results [17-20]. For example, in the work developed by Carton et al [17] the water droplet accumulation in mini-channels was validated with experimental results, and in the work developed by Iranzo et al [18] the water distribution in the membrane was validated with experimental results for a comprehensive set of operating conditions. However, the cell voltage was not validated in this two works. Other possible interesting parameter to validate experimentally is the current density distribution in the membrane. Fink et al [21] validated qualitatively the current density distribution with experimental results obtaining the same behavior. In the case of the cell voltage, it was validated with good agreement but only in the ohmic losses region of the polarization curve. In conclusion, in the literature exists PEM fuel cell models where the polarization curve has been fully validated. In addition, there are some works where two variables have been validated simultaneously but just qualitatively and not quantitatively. In this work, a one dimensional analytical model has been developed and validated with experimental results with two variables simultaneously. The selected variables are the cell voltage and the water content within the GDL+MEA, measured with neutron radiographs technique. This technology allows to measure quantitatively the water content within the fuel cell and the results were used to validate the PEM fuel cell model providing more accuracy. The model has been validated for different operating conditions of temperature, pressure, relative humidity in anode and cathode, and cathode stoichiometry. This result presents a new standard of validation in the state of the art of PEM fuel cell modeling where two variables are simultaneously and quantitatively validated. 2 Modeling The PEM fuel cell has been modeled including electrochemistry and mass transport of water, oxygen and hydrogen within the cell. The modeling section has been divided into five different subsections. In the first subsection, all the hypothesis realized to constraint the model are described. After this, the electrochemical equations are shown in detail. In the third subsection, the transport equations of water, oxygen and hydrogen within the cell are presented. Then the heat transfer equations are shown, and finally the model parameters are listed. 2.1 Model assumptions The model presents the following assumptions: • The model has been developed in steady state • Gas properties of the mixture are modeled as ideal gas • Two phases of water has been modeled according to liquid-gas equilibrium • Hydrogen, oxygen and water vapor mass transfer within the GDL, MPL and CL have been modeled using the Fick diffusion equation • Liquid water transport has been modeled according to capillary pressure. • Water transport through the Nafion membrane is considered to be a combination of diffusion and electro-osmotic drag • Electrical and thermal conductivity of the BP, GDL, MPL and CL are considered to be constant with all the range of the used operating temperature (50-60ºC). • Radiation heat transfer is neglected. Only heat conduction and convection have been modeled. • Reactants and products temperature are assumed to be at the same temperature as the component they flow through (BP, GDL, MPL, CL, MEM) 2.2 Cell performance The measured of the fuel cell voltage (Vcell) can be expressed as Eq 1 [22, 23]. 𝑉𝑐𝑒𝑙𝑙=𝐸°(𝑇,𝑃)−𝜂𝑎,𝑎−|𝜂𝑎,𝑐|−𝜂𝑟−𝜂𝑚,𝑎−|𝜂𝑚,𝑐| (Eq 1) Where Eᵒ(T,P) is the theoretical equilibrium open-circuit potential of the cell, ηa,a and ηa,c are the activation overpotentials in anode and cathode, ηr is the ohmic overpotential and ηm,a and ηm,c are the concentration overpotentials in anode and cathode. The model equation of each term in equation 1 are described below. 2.2.1 Theoretical equilibrium open-circuit potential Theoretical equilibrium open-circuit potential can be calculated according to Nerst equation assuming the behavior of the ideal gases. The expression is shown in Eq 2[15, 16]. 𝐸°(𝑇,𝑃)=−∆𝐺°(𝑇) 𝑛𝐹 +𝑅𝑇 𝑛𝐹𝑙𝑛((𝑦𝐻2𝑃𝑎𝑛𝑜𝑑𝑒 𝑃°)(𝑦𝑂2𝑃𝑐𝑎𝑡ℎ𝑜𝑑𝑒 𝑃°)12 (𝑦𝐻2𝑂𝑃𝑐𝑎𝑡ℎ𝑜𝑑𝑒 𝑃𝑠𝑎𝑡(𝑇))) (Eq 2) Where ∆𝐺°(𝑇) is the Gibbs free energy (J), n is the number of electrons participating in a reaction (2 for anode side and 4 for cathode side), R is the constant of the ideal gases (J/mol.K), F is the Faraday constant (A.s/mol), Po is the reference pressure (Pa), Psat is the saturation pressure (Pa), Panode/cathode is the operating pressure in anode and cathode (Pa), yi is the mole fraction of hydrogen in anode stream and the mole fraction of water and oxygen in the cathode stream. All the parameters described in Eq 2 depend only of the operating conditions of the PEM fuel cell. 2.2.2 Activation overpotentials Activation overpotentials have been modeled according to Tafel simplification. Tafel approach is widely used in PEM fuel cell models. This approach allows to simplify the Butler-Volmer Model of Kinetics with little error [9]. The expressions are shown in Eq 3 and Eq 4 [22,23]. 𝜂𝑎,𝑎= R T 𝛼𝑎𝐹ln(𝑖 𝑖0,𝑎) (Eq 3) 𝜂𝑎,𝑐=RT 𝛼𝑐𝐹ln(𝑖 𝑖0,𝑐) (Eq 4) Where R is the constant of the ideal gases (J/mol.K), F is the Faraday constant (A.s/mol), T is the operating temperature (K), αa and αc are the charge transfer coefficient in anode and cathode, i is the current density (A/m2) and io,a and io,c are the exchange current density in anode and cathode. Exchange current density in anode and cathode can be calculated as show in Eq 5 [22]. 𝑖𝑜,𝑖=𝑖𝑟𝑒𝑓,𝑖(𝐶𝑖,𝐺𝐷𝐿−𝐶𝐿 𝐶𝑟𝑒𝑓 )𝛾𝑖𝑒𝑥𝑝 (−𝐸𝑎,𝑖 𝑅𝑇) (Eq 5) Where iref,I is the reference exchange current density with no concentration loss and at a reference temperature in anode or cathode (A/m2) ,Ci,GDL-CL is the hydrogen or oxygen concentration depending of anode or cathode side (mol/m3) , Cref is the reference concentration (mol/m3), Ea,i. are the activation energy in anode and cathode side (J/mol) and γi is the reaction order for the elementary charge transfer step. The reference exchange current density is a parameter that only depends on the particular electrode of the PEM fuel cell, and in this work it has been fixed to the value determined by Iranzo et al [7] for the same cell and MEA. 2.2.3 Ohmic overpotential Ohmic overpotential depends on the electrical conductivity and thickness of every material that form the fuel cell as the bipolar plates, GDLs, CLs, MPLs and the membrane. Furthermore, the contact resistance between the GDL and the bipolar plate has been included in the ohmic overpotential due to the impact that it has over the performance of the PEM fuel cell [24]. The contact resistance has been estimated according to a well defined technique [23]. The expression to calculate the ohmic overpotential is shown in Eq 6 [10]. 𝜂𝑟=𝑖(2𝑡𝐵𝑃 𝜎𝐵𝑃+2𝑡𝐺𝐷𝐿 𝜎𝐺𝐷𝐿+2𝑡𝑀𝑃𝐿 𝜎𝑀𝑃𝐿+2𝑡𝐶𝐿 𝜎𝐶𝐿+𝑡𝑀𝑒𝑚 𝜎𝑀𝑒𝑚+2𝑅𝑐𝑜𝑛𝑡𝑎𝑐𝑡) (Eq 6) Where ti and σi represent the thickness (m) and the electrical conductivity (1/Ωm) of the bipolar plate (BP), GDLs, MPLs, CLs and the MEA and Rcontact represents the contact resistance between the GDL and the bipolar plate (Ωm2). The electrical conductivity of the BP, GDLs, MPLs and CLs was obtained from the technical datasheet of each material [25-27]. However, the membrane protonic conductivity (Nafion 117) depends strongly on the water content inside the membrane. Therefore, depending on the cell operating conditions, the membrane protonic conductivity will be different. The expression to calculate the membrane protonic conductivity as a function of the water content is shown in Eq 7. The expression is widely used in PEM fuel cell models and it was developed by Springer [28]. 𝜎𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒=exp(1268(1 303−1𝑇))(0.005193𝜆−0.00326) (Eq 7) Where λ is the water content inside the membrane and T is the operating temperature (K). The calculation of λ is shown in the Section 2.3.2 Water transport. 2.2.4 Concentration overpotential Concentration overpotentials can be modeled according to Eq 8 and Eq 9 [22, 23]. 𝜂𝑚,𝑎=−RT n𝑎F∗ln(1− 𝑖 𝑖𝑙𝑖𝑚,𝑎) (Eq 8) 𝜂𝑚,𝑐=−RT n𝑐F∗ln(1− 𝑖 𝑖𝑙𝑖𝑚,𝑐) (Eq 9) Where R is the constant of the ideal gases (J/mol.K), T is the operating temperature(K), ni are the equivalent electrons per mole of reactant (2 for anode side and 4 for cathode side), i is the current density (A/m2) and ilim,I is the limiting current density in anode or cathode side (A/m2). The limiting current density can be in principle determined experimentally (the value of the current density where the voltage of the cell is 0 V). In this work, the lower voltage region has been modeled but not validated because no experiment was carried out in the low voltage region. In the model the limiting current density has been estimated as the value of current density when the reactant concentration at the catalyst layer is zero [22]. 2.3 Mass transport model Hydrogen, oxygen and water vapor mass transport within the GDL, MPL, CL and MEM have been modeled using the Fick diffusion model. For liquid water, capillary pressure force has been used. In the case of the membrane, a combined effect of diffusion and electro-osmotic drag has been considered. GDL, MPL and CL are considered porous medium and the Darcy´s Law has been used in order to considered its effects. The diffusion coefficient for each species has been correlated considering the effect of the liquid water accumulation inside the porous medium [29]. 2.3.1 Hydrogen and oxygen transport Hydrogen and oxygen are transported from the bipolar plate flow channels to the catalyst layer by diffusion mechanism. The amount of hydrogen and oxygen can be modeled according to the Faradays’s Law. The expressions are shown in Eq 10 and Eq 11 as a combination between the Fick and the Faradays’s Law [8]. 𝑁𝐻2=𝑖𝑐𝑒𝑙𝑙 2𝐹 =𝑄𝑎𝐵𝑃 𝐴𝑠(𝐶𝐻2 𝐵𝑃−𝐶𝐻2 𝐵𝑃−𝐺𝐷𝐿) (Eq 10) 𝑁𝑂2=𝑖𝑐𝑒𝑙𝑙 4𝐹 =𝑄𝑐𝐵𝑃 𝐴𝑠(𝐶𝑂2 𝐵𝑃−𝐶𝑂2 𝐵𝑃−𝐺𝐷𝐿) (Eq 11) Where NH2 and NH2O are the molar flow per area unit (mol/m2.s), icell is the current density (A/m2), F is the Faraday constant (A.s/mol), As is the contact area between the bipolar plate and the GDL (m2), Q is the volumetric flow introduced in the anode or cathode (m3/s), CH2 BP and CO2 BP is the molar concentration of hydrogen and oxygen in the bipolar plate channels and CH2 BP−GDL and CO2 BP−GDL are the molar concentration of hydrogen and oxygen in the interface between the bipolar plate channel and the GDL. The transport of hydrogen and oxygen through the GDL, MPL and CL are calculated directly by Fick´s Law as are shown in Eq 11 and Eq 12 [9]. 𝑁𝐻2=−𝐷𝑒𝑓𝑓,𝐻2 𝑖𝑡𝑖(∆𝐶𝐻2 𝑖) (Eq 11) 𝑁𝑂2=−𝐷𝑒𝑓𝑓,𝑂2 𝑖𝑡𝑖(∆𝐶𝑂2 𝑖) (Eq 12) Where ∆CH2 i and ∆CO2 i are the difference molar concentration of hydrogen or oxygen in the specific layer ``i´´ (GDL, MPL or CL) and ti is the thickness of the specific layer ``i´´. Finally, Deff,H2 i and Deff,O2 i are the effective diffusion coefficients in each layer of the PEM fuel cell. The expression formulated by Chan et al [29] has been used in this work. The expression is shown in Eq 13. 𝐷𝑒𝑓𝑓,𝑗 𝑖=[1−(2.76𝜀𝑖cosh(3𝜀𝑖−1.92)3(1−𝜀𝑖 3−𝜀𝑖))]𝐷𝑗(1−𝑠𝑖)2 (Eq 13) Where the index ``i´´ is referring to the different layers (GDL, MPL and CL) and the index ``j´´ is referring to the gas used in anode (air) or cathode (hydrogen). εi is the porosity of the layer and si is the liquid saturation inside the layer. 2.3.2 Water transport Water management plays a very important role in the performance of the PEM fuel cell. Water is necessary in order to hydrate the membrane increasing its electrical conductivity. However, an excess of water can flood the pores of the GDL and it makes difficult that the reactants can reach the active centers of the catalyst layer. In that way, an efficient water balance between generated water and removed water is necessary in order to obtain a good behavior in the PEM fuel cell. Liquid water transport can be calculated according to Darcy´s law as it is shown in equation 14 [19, 22]. 𝑢𝑙,𝑖=−𝑘𝑙,𝑖 𝜇𝑙∇𝑃𝑐,𝑖 (Eq 14) Where the index ``i´´ is referring to the different layers of the PEM fuel cell (GDL, CL and MPL). ul,i is the liquid water velocity inside the medium ``i´´ (m/s) that can be calculated according to continuity equation, kl is the liquid permeability of the porous media (m2), µ is the liquid water dynamic viscosity (kg/s.m) and Pc,I is the capillary pressure (Pa). Liquid permeability is related to the relative permeability and absolute permeability of the porous media according to the following expression (Eq 15). 𝑘𝑟,𝑖=𝑘𝑙,𝑖 𝑘𝑖 (Eq 15) Using the Wyllie model [19, 22] liquid water relative permeability can be formulated as a function of liquid saturation. The expression is shown in Eq 16. 𝑘𝑟,𝑖=𝑠𝑖3 (Eq 16) On the other hand, the capillary pressure can be calculated using the Leverett approach. The equations used in this model are shown in Eq 17 and Eq 18 [19, 22]. 𝑃𝑐=𝛾𝑐𝑜𝑠𝜃(𝜀𝑖 𝑘𝑖)0.5𝐽(𝑠) (Eq 17) 𝐽(𝑠)={1.417(1−𝑠𝑖)−2.12(1−𝑠𝑖)2+1.263(1−𝑠𝑖)3 𝑖𝑓 𝜃<90° (𝐻𝑦𝑑𝑟𝑜𝑝ℎ𝑖𝑙𝑖𝑐) 1.417𝑠𝑖−2.12𝑠𝑖2+1.263𝑠𝑖3 𝑖𝑓 𝜃>90° (𝐻𝑦𝑑𝑟𝑜𝑝ℎ𝑜𝑏𝑖𝑐) (Eq 18) Where θ is the contact angle (deg) and γ represents the surface tension (N/m) that it is only a function of the operating temperature (K) (Eq 19) [9]. 𝛾=−1.78∗10−4𝑇+0.1247 (Eq 19) The water transport through the membrane is assumed to be a combined effect of electroosmotic drag and diffusion. The combined effect is shown in Eq 21 [9, 15, 30]. 𝑁𝐻20 𝑀𝑒𝑚=2.5𝜆𝑖𝑐𝑒𝑙𝑙 22𝐹−𝜌𝑀𝑒𝑚 𝑀𝑊𝑀𝑒𝑚𝐷𝑒𝑓𝑓,𝐻2𝑂 𝑀𝑒𝑚 (𝜆)𝜕𝜆 𝜕𝑧 (Eq 21) Where 𝑁𝐻20 𝑀𝑒𝑚 is the water through the membrane (mol/m2.s), 𝜌𝑀𝑒𝑚is the dry density of the used membrane (kg/m3) and 𝑀𝑊𝑀𝑒𝑚 is the molecular weight of the membrane (mol/kg). λ is the water content in the membrane and can be calculated in function of the water activity ``a´´ according to the equation 22 and 23 [10, 22, 24, 31]. 𝜆=0.043+17.18𝑎−39.85𝑎2+36𝑎2 𝑖𝑓 𝑎<1 (Eq 22) 𝜆=14+14(𝑎−1) 𝑖𝑓 𝑎>1 (Eq 23) Where the water activity can be calculated according to the equation 24, 25 and 26 [4, 24, 31]. 𝑎𝑎=𝐶𝐻2𝑂,𝑎 𝑀𝐸𝑀𝑅𝑇 𝑃𝑠𝑎𝑡 (Eq 24) 𝑎𝑐=𝐶𝐻2𝑂,𝑐 𝑀𝐸𝑀𝑅𝑇 𝑃𝑠𝑎𝑡 (Eq 25) 𝑎=𝑎𝑎+𝑎𝑐 2 (Eq 26) Where 𝐶𝐻2𝑂,𝑎 𝑀𝐸𝑀 and 𝐶𝐻2𝑂,𝑎 𝑀𝐸𝑀 are the water concentration in the membrane in anode and cathode side (mol/m3), R is the constant of the ideal gases (J/mol.K),T is the operating temperature (K) and Psat is the water saturation pressure (Pa). Finally, the water diffusion coefficient (m2/s) inside the membrane can be calculated according to the Eq 27, that depends strongly on the value of water content in the membrane [30]. Equation 27 is widely used to calculate the water diffusion coefficient inside membranes made of Nafion [32]. 𝐷𝑒𝑓𝑓,𝐻2𝑂 𝑀𝑒𝑚 (𝜆) { 2.693𝑒−10 𝑖𝑓 𝜆≤2 1𝑒−10(exp(2436(1 303−1𝑇))(0.87(3−𝜆)+2.95(𝜆−2))) 𝑖𝑓 𝜆>2 𝑎𝑛𝑑 𝜆≤3 1𝑒−10(exp(2436(1 303−1𝑇))(2.95(4−𝜆)+1.645454(𝜆−3))) 𝑖𝑓 𝜆>3 𝑎𝑛𝑑 𝜆≤4 1𝑒−10(exp(2436(1 303−1𝑇))(2.563−0.33𝜆+0.0264𝜆2−0.000671𝜆3)) 𝑖𝑓 𝜆>4 (Eq 27) It is important to note the necessity to calculate the water transport in all the layers that form the PEM fuel cell in order to obtain the water concentration in the anode and cathode membrane sides. To calculate the water transport through the PEM fuel cell, it is important to take into account the amount of water generated in the cathode catalyst layer. The water transport from the bipolar plate to the GDL in anode and cathode side is shown in Eq 28 to Eq 33 where the term icell/2F represents the generated water in the cathode catalyst layer [9, 15]. 𝑁𝐻20,𝑎,𝑡𝑜𝑡𝑎𝑙=𝑁𝐻20,𝑎,𝑣𝑎𝑝𝑜𝑟+𝑁𝐻20,𝑎,𝑙𝑖𝑞𝑢𝑖𝑑=𝜌𝑀𝑒𝑚 𝑀𝑊𝑀𝑒𝑚𝐷𝑒𝑓𝑓,𝐻2𝑂 𝑀𝑒𝑚 (𝜆)𝜕𝜆 𝜕𝑧 (Eq 28) 𝑁𝐻20,𝑎,𝑣𝑎𝑝𝑜𝑟=𝑦𝑖,𝑎𝑁𝐻20,𝑎,𝑡𝑜𝑡𝑎𝑙=𝑄𝑎𝐵𝑃 𝐴𝑠(𝐶𝐻20,𝑎 𝐵𝑃 −𝐶𝐻20,𝑎 𝐵𝑃−𝐺𝐷𝐿) (Eq 29) 𝑁𝐻20,𝑎,𝑙𝑖𝑞𝑢𝑖𝑑=(1−𝑦𝑖,𝑎)𝑁𝐻20,𝑎,𝑡𝑜𝑡𝑎𝑙 (Eq 30) 𝑁𝐻20,𝑐,𝑡𝑜𝑡𝑎𝑙=𝑁𝐻20,𝑐,𝑣𝑎𝑝𝑜𝑟+𝑁𝐻20,𝑐,𝑙𝑖𝑞𝑢𝑖𝑑=(2.5𝜆𝑖𝑐𝑒𝑙𝑙 22𝐹+𝑖𝑐𝑒𝑙𝑙 2𝐹) (Eq 31) 𝑁𝐻20,𝑐,𝑣𝑎𝑝𝑜𝑟=𝑦𝑖,𝑐𝑁𝐻20,𝑐,𝑡𝑜𝑡𝑎𝑙=𝑄𝑐𝑎𝑡ℎ𝑜𝑑𝑒 𝐵𝑃𝐴𝑠(𝐶𝐻20,𝑐 𝐵𝑃 −𝐶𝐻20,𝑐 𝐵𝑃−𝐺𝐷𝐿) (Eq 32) 𝑁𝐻20,𝑐,𝑙𝑖𝑞𝑢𝑖𝑑=(1−𝑦𝑖,𝑐)𝑁𝐻20,𝑐,𝑡𝑜𝑡𝑎𝑙 (Eq 33) Where NH2O i,j is the water through the anode or cathode in liquid or vapor state (mol/m2.s), Qj is the volumetric flow introduced in anode or cathode side(m3/s), As is the contact area between the bipolar plate and the GDL (m2), 𝐶𝐻2𝑂,𝑗 𝐵𝑃 is the water molar concentration on the channels of the bipolar plate in anode and cathode side, 𝐶𝐻2𝑂 𝐵𝑃−𝐺𝐷𝐿 and 𝐶𝐻2𝑂 𝐵𝑃−𝐺𝐷𝐿 are the water molar concentrations in the interfaces between the channel of the bipolar plate and the GDL in anode and cathode side and yi is the water gas fraction calculated according to liquid-vapor equilibrium in each layer. The gas water transport trough the GDL, MPL and CL is calculated using the Fick´s Law (Eq 34) [9, 15]. 𝑁𝐻20,𝑔𝑎𝑠,𝑗=−𝐷𝑒𝑓𝑓,𝐻20,𝑗 𝑖𝑡𝑖(∆𝐶𝐻20,𝑗 𝑖) (Eq 34) Where the index ``i´´ is referred to the different layer (GDL, MPL and CL) and the index ``j´´ is referred to the anode or cathode side. ∆𝐶𝐻2𝑂,𝑗 𝑖 is the difference molar concentration of water (mol/m3), ti is the thickness of the specific layer (m) and 𝐷𝑒𝑓𝑓,𝐻2𝑂,𝑗 𝑖 is the effective diffusion coefficient in each layer of the PEM fuel cell. 2.4 Heat transfer model In order to obtain the temperature in each layer of the fuel cell, the energy balance must be solved completely. The developed heat transfer model solves both heat conduction (one dimensional in the through-plane direction) and heat convection on the outer walls of the cell. Radiation heat transfer has been neglected due to its minor contribution. The equations for each layer of the PEM fuel cell are shown in Eq 35 to Eq 44, where heat is generated in the cathode catalyst layer. Membrane heat balance: (𝑇𝑚𝑒𝑚−𝑇𝐶𝐿,𝑐) 𝑅𝑀𝑒𝑚−𝐶𝐿 +(𝑇𝑚𝑒𝑚−𝑇𝐶𝐿,𝑎) 𝑅𝑀𝑒𝑚−𝐶𝐿 +𝐴𝑙𝑎𝑡,𝑀𝑒𝑚 ℎ (𝑇𝑀𝑒𝑚−𝑇𝑒𝑛𝑣𝑖𝑟𝑜𝑚𝑒𝑛𝑡)=0 (Eq 35) Anode catalyst layer heat balance: (𝑇𝐶𝐿,𝑎−𝑇𝑀𝑒𝑚) 𝑅𝑀𝑒𝑚−𝐶𝐿 +(𝑇𝐶𝐿,𝑎−𝑇𝐺𝐷𝐿,𝑎𝑛) 𝑅𝐶𝐿−𝐺𝐷𝐿 +𝐴𝑙𝑎𝑡,𝐶𝐿 ℎ (𝑇𝐶𝐿,𝑎−𝑇𝑒𝑛𝑣𝑖𝑟𝑜𝑚𝑒𝑛𝑡)=0 (Eq 36) Figure 3. Validation results of the cell voltage and the water content within the MEA+GDL from the Test 1 to the Test 3 The experimental validation for the Test 4 is shown in Figure 4. It is observed that the analytical model predicts accurately both the cell voltage and the water content within the GDL+MEA. Figure 4. Validation results of the cell voltage and the water content within the MEA+GDL for the Test 4 In order to quantify the error committed by the analytical model versus experimental values, the relative error is defined in Equation 49. 𝐸𝑟𝑟𝑜𝑟 (%)=𝑋𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙−𝑋𝑚𝑜𝑑𝑒𝑙 𝑋𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑥100 (Eq 49) Where the variable X can be the cell voltage or the water content within the GDL+MEA. The maximum error is always less than 3.5 % from the Test 1 to the Test 4 for both the cell voltage and the water content within the GDL+MEA. The model validation against the Test 5 and the Test 6 are shown in Figure 4. The full polarization curve obtained with the analytical model accurately fits the experimental results, where the maximum error is less than 3% for both tests. Figure 5. Validation results of the polarization curve for the Test 5 and the Test 6 The presented results show that the one dimensional analytical model predicts with accuracy (relative error less than 3.5 %) both the cell voltage and the water content within the GDL+MEA. In addition, the model has been validated for different operating conditions of temperature, pressure, relative humidity and cathode stoichiometry. This result present a new standard of validation in the state of the art of PEM fuel cell modeling where two variables are simultaneously and quantitatively validated, providing a better confidence in the developed model. One additional important aspect is the computational time required for solving a specific operation point of the PEM fuel cell. The developed model requires less than 2 seconds to solve the full polarization curve (between 18 and 20 points). This presents an important advantage as it is possible to compare different operating conditions in a very short period of time in order to select the best operating conditions according to the application. On the other hand, the model can be used to understand the effects of the main operating conditions in the performance of the PEM fuel cell, not only in terms of polarization curve but also in terms of water management. 5 Applications As was discussed above, the developed analytical model can be used in order to analyze the behavior of the PEM fuel cell under different operating conditions. Furthermore, due to the fact that the analytical model has been validated simultaneously with both the cell voltage and the water content within the GDL+MEA, the model can provide information about performance and water management of the PEM fuel cell. In this section, a comprehensive analysis of the results from the Test 1 to the Test 3 is presented with the objective of understanding the effect of the relative humidity in the performance and in the water management of the PEM fuel cell. The analytical model allows to predict the different losses that exist in the PEM fuel cell such as activation, ohmic and concentration overpotentials. Thus, it is possible to evaluate the effect of the relative humidity in the performance of the PEM fuel cell not only in general terms but in terms of activation, ohmic and concentration overpotentials. This analysis is particularly important as it allows for the identification of weaknesses in the design of the PEM fuel cell. The Cell voltage and the activation, ohmic and concentration overpotentials obtained with the analytical model are shown in Figure 6. Figure 6. Cell voltage and activation, ohmic and concentration overpotentials obtained with the analytical model from the Test 1 to the Test 3 It is observed that the cell voltage increases when the relative humidity increases due to the ohmic overpotential decreases. In the cases of the activation and concentration overpotencials, it is not observed any important impact when the relative humidity increases. Focusing in the ohmic overpotential, the only variable that can be affected by the water management is the electrical conductivity of the membrane, according to Eq 7 where the electrical conductivity of the membrane depends on the water content [28]. The water content and the membrane electrical conductivity for different values of relative humidity are shown in Figure 7. Eliminado: Figure 6 Eliminado: Figure 7 Figure 7. Water content and electrical conductivity obtained with the analytical model from the Test 1 to the Test 3 It is observed that the water content increases when the relative humidity increases. Thus, the membrane electrical conductivity increases proportionally with the water content, so the ohmic overpotential decreases when the relative humidity increases. In the cases of activation and concentration overpotentials, these depend strongly on the oxygen concentration in the cathode catalyst layer. The oxygen concentration in the cathode catalyst layer and the liquid saturation and effective diffusion coefficient in GDL obtained with the analytical model are shown in Figure 8. Eliminado: Figure 8 Figure 8. Liquid saturation and effective diffusion coefficient in GDL and oxygen concentration in the GDL/MEA interface obtained with the analytical model from Test 1 to Test 3 It is observed that the liquid saturation increases when the relative humidity increases. This causes a decrease in the effective diffusion coefficient due to the water blocks the pores of the GLD. This combination causes that the oxygen concentration in the cathode catalyst layer decreases, thus causing an increase in the activation and concentration overpotentials. In this section, the developed analytical model has been used to analyze the effect of the relative humidity in the performance and in the water management of the PEM fuel cell. The analytical model has allowed to quantitatively determine the increases of the cell voltage when the relative humidity increases. The same exercise can be performed for the different operating variables such as temperature, pressure and the stoichiometry in cathode/anode. 6 Conclusions A one dimensional analytical model has been developed for a 50 cm2 single fuel cell with fivechannel serpentine flow field. The model has been validated for different experimental results modifying the main operating conditions such as temperature, pressure, relative humidity in anode and cathode, and cathode stoichiometry. The innovation of this work is the experimental validation of the cell voltage (usually the parameter to validate in state of the art PEM fuel cell models) and the water content within the GDL+MEA using the neutron imaging experiments. The validation presents a very good agreement between the experimental and the model results where the maximum relative error is less than 3.5 % for both the cell voltage and the water content within the GDL+MEA. This result presents a new standard of validation in the state of the art of PEM fuel cell modeling where two variable are simultaneously and quantitatively validated ensuring the quality of the model results. Finally, the developed analytical model has been used in order to analyze the behavior of the PEM fuel cell under different values of relative humidity. The model provides quantitative information about the different losses that exist in the PEM fuel cell such as activation, ohmic and concentration overpotentials, and about water management that allows to better assess and understand where the main losses are located. 7 Acknowledgements J. Antonio Salva gratefully acknowledges the support of INTA (National Institute for Aerospace Technology). This work was partly carried out with the support of the European Community. We appreciate the support of the European Research Infrastructure project “H2FC European Infrastructure” (funded under the FP7 specific programme Capacities, Grant agreement Number FP7-284522) and its partner Paul Scherrer Institut. Dr. Pierre Boillat and Johannes Biesdorf from Paul Scherrer Institut are gratefully acknowledged. 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