##### 1. Introduction

For formulating phenomena set within the cell, a group of sciences like electrochemical, physics, and mechanics intersect in a complex device called Fuel cell (FC). The practical contribution will be to lessen the cost of manufacturing and enhance the reliability and performance of the fuel cell by the proper realisation of these phenomena.

Empirical observation and modelling are the two most effective ways to realise these phenomena. In the case of the result’s accuracy, though the experimental technique provides more reliable products than modelling, the main problem is that it takes more effort and time along with its expensiveness. In addition, some pitfalls are also common here. For instance, despite its impact on the efficiency of the cell, a slight alteration in the current density is quite impossible to measure, which is stated in chapter 3 earlier. Moreover, based on the experimental results, a V-I curve can be drawn. Specifically, when FCs work in various operation conditions, sorting out the required parameters by which the curve shape will be determined is quite a challenging task. Conversely, the impact of the overall values of parameters can be replicated in the FCs modelling. It is helpful because, at a time, handy experiments are frequently confined to a few matters.

- Transfer of Photonic and Electronic charge
- Bunch transfer
- Cell reactions’ electrochemistry
- The stress of mechanics like internal compression pressure

There is a complication in the cell phenomena modelling – electricity and heat though it is a simple principle- oxygen and hydrogen react to forming water. Shortly, FC phenomena take place in:

- The structure level of material or the micrometre range
- The nanometre range which is modelled by density- molecular dynamics and functional theory
- The metre scale- the millimetre for charge transfer, heat, mass and flow dynamics through continuum mechanics.

The variation in time occurs due to the variation in the speed of the heat and masses transmission and the speed of the reactions of the electrochemical. Because of the complex nature of the FCs, it is not possible to deal with a single model of all the phenomena of the FC. It is true that for macroscopic level modelling, the quantum mechanics’ underlying physics can’t be applied in their accurate formulation. When assessing classical physics, an FC’s diversified phenomena are converted into a differential equation system that is nonlinear and complex. Furthermore, the cell’s performance is also affected by the cell’s thermal expansion, contact distribution of pressure and other mechanical stresses.

In order to account for existing changes drawn from cell and operating conditions’ changes along with considering the cell degradation, a time dimension should be included in an FC model. Even with the latest computers. It is impossible to solve this system in 3 dimensions, under handling diverse length scales and dynamic conditions ranging from macroscopic stack structure to catalyst particles of catalyst particles. Hence, some effects in certain primary conditions are focused on studying by an FC model. Unconventional instances are the cell’s electrochemistry modelling and analysing the mass transport with diversified field configurations’ flow. 0D or even 1D were the early models where the only parts of the FC were often studied (for instance, GDL and cathode electrode). Still, it has become quite typical of the 3D models with the development of computers and the entire cell geometry is possible to be modelled. [1]. For obtaining a solvable model, making simplifications and assumptions is still required. For instance, it is often necessary to exclude the anode mass transfer in the model due to performance limitations. Due to the absence of the theoretically derived values or experimental daily many parameter values are determined arbitrarily, which is another impediment. For example, in the electrochemical equations by which an FC model’s basis is formulated is the current density exchange, a vital parameter. Diversified forces like the temperature of the cell and electrodes’ amount of catalyst impact this parameter. Consequently, it is very critical to measure which is entirely dissimilar for every FC. The model’s outcome seems physically sensible and reasonable, and using the parameter fitting is often expected. Through selecting parameters, it is possible to conceal the modelling faults (further deep problems or easy numerical faults) for obtaining stunning results. Hence, whether the given predictions for the parameters will be feasible in reality or not can’t be guaranteed.

There are many problematic parameters like current density exchange, such as capillary pressure equations and evaporation and condensation rates in two-stage modelling. [2]. Different material properties that are not isotropic in reality, like contact resistances or GDL porosity among the cell’s several layers, are required to average when more concrete data is unavailable. In terms of the entire cell, these models’ results may be correct, but they can be grossly inaccurate in narrating the local phenomena.

##### 2. PEMFC Models’ Short Review

In this part, devolving the FC modelling from the analytic models of the first 0D to the 3D models, which are complex, will be reviewed in brief. The discussed studies of the model in this part are classified according to the modelling aim, dimension and method. Categorising the FC models is not an easy task, and models like spherical agglomeration modelling, which don’t match the categories, are also shown in this phase [3].

###### 2.1 Analytical models

The FC models use different computational tools like FEM (Finite Element Method). Equations that need to be resolved analytically are the basis of several models here. Generally, an FC model that is analytical is highly idealised and simplified. The model’s dimension is lessened to one or zero. The linear equations simplify the underlying physics, which couldn’t be resolved analytically otherwise. Especially at the larger current densities, the outcome isn’t entirely accurate. In order to perform small calculations on easy systems and gain approximate dependencies of current-voltage, it is possible to use the analytical models [3]. In ideal conditions, the cell operation’s basic view is given by analytical models.

###### 2.2 Semi-Empirical Modelling

It is often seen that people either can’t realise the FC phenomena’s physics correctly, or it becomes quite tough to incorporate it into the modelling for handy reasons. Instead of theoretically derived and more accurate ones, algebraic equations or differential obtained empirically may be resorted by the researchers for avoiding or solving these issues. Several FC models employ some empirical correlations. In single FC, one of the common modelling is semi-empirical modelling. In stack modelling, people frequently use semi-empirical models where too much computing capacity is required in the detailed models.

Due to the unavailability of better alternatives, it is often required to follow the semi-empirical relations whenever possible and try to avoid it. Firstly, to a specific parameter range, semi-empirical correlations can be used. The recognition failure of this may lead to erroneous conclusions. Secondly, the realisation of the physics underlying an FC may not be caused by applying the empirical correlations. We can be prevented from gathering knowledge regarding the new mechanisms which could be misunderstood for improving the performance of the FC by the empirical correlations resorting. In modelling materials and designs that are already underperforming, there are chances to be pretty accurate by the semi-empirical models. Still, it is not possible to get predictions regarding how to affect the cell operation by the new innovative design or new alternative materials.

###### 2.3 One-Dimensional Models

Through numerical algorithms and discretisation, the equations usually can be resolved. In case they aren’t linearised though, the models of 1D can be analytical in nature. This can make the results more acceptable compared to others. Nevertheless, in terms of voltage, current and other FC features, still, 1D models are confined to the entire correlations.

Beyond the 1D models, many local phenomena seem to have the identical feature of the cell in every direction, and then the variation between channels’ or ribs’ flow field plate can’t be taken into consideration. Part [3] presents the early models of FC, which typically were 1D with many assumptions which were simplified. Much focus is given on cathode flooding, water management and mass transport by them.

###### 2.4 Two-Dimensional Cell Models

The models of 2D are usually either channel cross-section models or channelling models. The information regarding how the flow field plates are conducted by the electrical current is given by the former one. On the other hand, the latter can be applied to study the reaction and reactant concentrations of the product as it consumes the reactants along the channel. All the 2D models share an infinite planar cell’s assumption. With the 2D models, it is not possible to study different 3D phenomena along with flow fields. Usually, GDLs and MEA consist of a 2D model which employs symmetry boundaries. For in3, it is required to exclude some cell components according to the study’s objective. The membrane and the anode side may be debarred to match to study the cathode mass transport. In order to form a *3D *model, if the computational power isn’t enough, then one of the valuable models can be the *2D* models. This case often happens when small details of geometry are included as a modelling domain, as the thin electrode layers. The model may be hard to solve in this type of case where the computational mesh or grid will be very well. Fresh information on the local fact may be found from the *2D *models. There are still widespread uses of the 2D models since either the *3D* models are impractical or unnecessary to study diverse phenomena of FC. In, e.g., [4, 5], some examples are presented. The transport phenomena in the current local distribution or GDLs and membrane are the main focusing areas of a conventional *2D* model.

###### 2.5 Three-Dimensional Cell Models

Though it still requires making more simplifications, it is possible to model the cell geometry less or more precisely where one of the most realistic models is the 3D models. On the other hand, more time and computing capacity will be taken to resolve these models. When assessing the phenomena that can’t be modelled adequately by the 2D like distribution and reactant flow, then the best result is found from the 3D models. For obtaining information on whether the cell’s existing distribution is uniform or the whole cell is distributed by the reactants, the best applicable model is the 3D model. The electrodes and the MEA are excluded from many models of 3D like the examples shown in [6, 7] since, in a 3D model, the computational grid’s size is greatly increased by these very slight layers. For taking the benefit of available symmetry and the repeating units, a serpentine channel and other parts of the flow field or the entire cell isn’t necessary to study. Different models often cover a small cell’s active area like, e.g., [7], where for analysing the impact of a diversified cross-sectional channel on the cell performance, it was required to model a 16 cm2 FC. There is a problem that even relatively small cells models, e.g., 49 cm2 in [8], are models of the large-scale models

For producing the actual amount of power, the much more extensive area that is active will be occupied by most real cells besides the tiniest applications. There lies a problem that the channel geometry that is suitable for the small cell may not be ideal for the larger cell, and the behaviour of the flow field isn’t scalable since when the size of the cell alters, the number of Reynolds doesn’t stay steady. However, for modelling cells that could be applied in stacks in place of these solely exercised in laboratories, it requires more effort to better the cells’ performance. The design of the flow field is a most vital one.

###### 2.6 Dynamic models

The time dimension included model is a dynamic model where inconsistency and time dependency of the cell operation is observed. Therefore, for predicting the responses at the system level, stack or cell to disturbance or change in operations, it is a valuable effort to use the dynamic models. It is a fruitful tool to assess the cell’s performance at the time of voltage cycling, shutdown and start-up [8].

The empirical correlations are the base of the dynamic models since more time and computing capacity is needed to resolve a model that is steady-state, for instance, through using an equal circuit which contains some electronic elements like capacitors and resistors connected in order that similar behaviour to an FC behaves in the circuit, the cell can be modelled [9]. By this, the models become simpler. Hence, as the focus is given on the system, sometimes it is not studying the cell itself [61]. Estimates regarding how rapidly the steady-state operation can be reached by the cell after operating conditions changes can be given by the dynamic models, which apply electrochemical and physical correlations [10, 11].

###### 2.7 Two-Phase Models

In cell operation and modelling, water management is PEMFCs’ one of the toughest challenges. For enhancing the membrane’s ionic conductivity, water is essential. That’s why, in the case of the performance of FC, water management is a vital task.

Multiplied model and multiphase mixture model are two methods by which the two-stage mass transport modelling is employed. The latter is considered a mixture that solves the equation [12, 13]. The solution of the mixture can be a source of necessary data for every stage. The multiplied model has an equation set in both phases, which are to be solved concurrently [14, 15]. Since it is pretty tough to obtain the convergence in the 1st one who needs a more competent solver and more computational capacity, more accurate predictions and results on the phenomena are given by it that is impossible to achieve following other methods. Solely with the flow of porous media, these two models are feasible. The water accumulation process should be assessed definitely in 3D with the entire flow field, not only with MEA and GDLs’ cross-section, which is a problem.

For assessing the FC’s accumulation and water transport, several experimental tactics like x-ray, EIS (electrochemical impedance spectroscopy) and neutron imaging technology, but in case of realising and improving the water management of PEMFC, it will be a useful method to use the two-stage models. Here for two-stage models, it is required to choose different parameters arbitrarily or obtain a result from the soil experiments [69].

Erroneous boundary conditions like the GDL and channel’s boundary with no liquid water [16, 17] are used by different two-stage models. Droplet channel formation has been shown in many experiments, which is inaccurate [18]. According to various studies, one possible solution to this problem can be the new boundary conditions, for instance, on the boundary; a basic model can be used for the droplet formation [19, 20].

Conversely, this boundary’s evaporation can be affected by the channels’ transport of liquid water, which is impossible to model accurately since the media don’t apply Darcy’s law in the FC’s two-stage models. In order to make the result trusted, many improvements are required in the two-stage modelling.

##### 3. Modelling in This Thesis

One of the sarcastic mission phenomena occurring in an FC is numerical modelling which isn’t an easy process. Non-linear equations systems’ coupling is involved in it. Another phenomenon that requires to be modelled is related to the phenomenon of acoustic emission. These include water content, heat transfer and species transport.

For resolving the non-linear equations of coupled transport through which the AE operation was described was used by a finite component based on COMSOL 4.4 (commercial Multiphysics numerical solver).

By applying the governing transport equation, it becomes easier to solve the species transportation problem. On the basis of the finite component into an algebraic equations system, it discredited the governing transport equations, and then an iterative algorithm solved it. Following the 2nd order Lagrangian approach, the computational domain of PEMFC is parted into a finite set of elements. These analyses use COMSOL electrochemical engineering module, and for resolving the electrochemical event, it uses the COMSOL electrochemical engineering’s physics application modes. Governing transport equations and other equations are followed. The respective dependant variables and governing transport equations are:

- To simulate the behaviour of the reactant fluid flow in porous media, Brinkman Equation (Darcy’s law) is coupled with the mode of incompressible Navier-Stokes;
- To simulate the behaviour of the fluid flow of Newtonian reactant in FC domain’s free media, application mode of incompressible Navier-Stokes;
- For charge balance in the domain of the fuel cell, application mode of conductive media DC;
- For transfer heat in the domain of the fuel cell by conduction and convection, application mode of general heat transfer;
- For products and reactants transport of species and the AE domain, a module of Maxwell-Stefan multi-element convection and diffusion.

It used UMFPACK (a direct linear solver) and a non-linear statistical solver, which is stationary since the problem was made non-linear by the charge conservation’s source terms. 1×10-4 was the set error criteria of the relative tolerance. The COMSOL parametric solver was applied for parametric assessment where with a particular analysis range, the variables of parametric were chosen. By following the COMSOL parametric solver by altering the cell potential, it obtained the curves of performance polarisation. For changing the values of the variables in a particular range, an iteration loop is used by the parametric solver.

The solution’s primary estimation is very highly sensitive to the nonlinear problem’s convergence behaviour. Consequently, to provide an excellent preliminary assessment of the reactant flow’s convective viscous properties and accelerate the non-linear problem’s numerical convergence, the Brinkman equation and Navier-Stokes equations are primarily resolved together. In order to evade from the convective terms’ probable numerical instability in gas flow distributor’s small channels, this solution tactic is widely followed since the flow behaviour of the numerical modelling in such type of tiny channels can drive to near the walls fluctuations which at the same time can drive to a simulation crash result and numerical instabilities. The artificial diffusion concept is followed here for minimising this type of small flow channels’ numerical instabilities. With a 0.25 tuning parameter, it uses streamline diffusion.

##### 4. Modelling Principles

One of the sarcastic mission phenomena occurring in an FC is numerical modelling which isn’t an easy process. Non-linear equations systems’ coupling is involved in it. Another phenomenon that requires to be modelled is related to the phenomenon of acoustic emission. These include water content, heat transfer and species transport.

For resolving the non-linear equations of coupled transport through which the AE operation was described was used by a finite component based on COMSOL 4.4 (commercial Multiphysics numerical solver).

By applying the governing transport equation, it becomes easier to solve the species transportation problem. On the basis of the finite component into an algebraic equations system, it discredited the governing transport equations, and then an iterative algorithm solved it. Following the 2nd order Lagrangian approach, the computational domain of PEMFC is parted into a finite set of elements. These analyses use COMSOL electrochemical engineering module, and for resolving the electrochemical event, it uses the COMSOL electrochemical engineering’s physics application modes. Governing transport equations and other equations are followed. The respective dependant variables and governing transport equations are:

- To simulate the behaviour of the reactant fluid flow in porous media, Brinkman Equation (Darcy’s law) is coupled with the mode of incompressible Navier-Stokes;
- To simulate the behaviour of the fluid flow of Newtonian reactant in FC domain’s free media, application mode of incompressible Navier-Stokes;
- For charge balance in the domain of the fuel cell, application mode of conductive media DC;
- For transfer heat in the domain of the fuel cell by conduction and convection, application mode of general heat transfer;
- For products and reactants transport of species and the AE domain, a module of Maxwell-Stefan multi-element convection and diffusion.

It used UMFPACK (a direct linear solver) and a non-linear statistical solver, which is stationary since the problem was made non-linear by the charge conservation’s source terms. 1×10-4 was the set error criteria of the relative tolerance. The COMSOL parametric solver was applied for parametric assessment where with a particular analysis range, the variables of parametric were chosen. By following the COMSOL parametric solver by altering the cell potential, it obtained the curves of performance polarisation. For changing the values of the variables in a particular range, an iteration loop is used by the parametric solver.

The solution’s primary estimation is very highly sensitive to the nonlinear problem’s convergence behaviour. Consequently, to provide an excellent preliminary assessment of the reactant flow’s convective viscous properties and accelerate the non-linear problem’s numerical convergence, the Brinkman equation and Navier-Stokes equations are primarily resolved together. In order to evade from the convective terms’ probable numerical instability in gas flow distributor’s small channels, this solution tactic is widely followed since the flow behaviour of the numerical modelling in such type of tiny channels can drive to near the walls fluctuations which at the same time can drive to a simulation crash result and numerical instabilities. The artificial diffusion concept is followed here for minimising this type of small flow channels’ numerical instabilities. With a 0.25 tuning parameter, it uses streamline diffusion.

Figure 3-3: COMSOL’s Solution Procedure

###### 4.1 Modelling Principles

In this part, a usual brief regarding the physics followed in the models here is given. Electrochemical reactions, charge, heat and mass transfer and finely AE consist in the modelled phenomena of FC.

With fitting boundary conditions and incomplete differential equations, these issues are modelled according to the model. The following section will display the partially coupled differential equations below.

**4.1.1 Assumptions**

The followings are the assumptions [125]:

- Within the channel, the condition of plug flow exists.
- The stable temperature includes the membrane, plates and electrodes, ranging from 20 oC to -80 oC.
- The behaviour of the gas mixture is quite ideal.
- Negligible heat transfer in the phase of gas by conduction of it
- In the structure of tiny droplets, it presents liquid water.
- It can neglect the electrode porous layer’s gas diffusion in the ‘ultrathin’ electrode layer.
- High conductivity is found in the current collectors; however, no voltage drop is found along the channel.
- The anode flow channel’s water activity initially determines the membrane’s water diffusion coefficient and electro-osmotic coefficient.

###### 4.2 Governing Equations

**4.2.1 Mass Balance**

The following equations show the y-direction’s average flux, which brings changes in every component’s moles number.

Equation (4.5)

The fuel cell’s local current density is I (x), and the Faraday constant is the F.

Along with the channel length, it changes the density of local currency as electrodes overvoltage and the membrane conductivity varies. The ratio of per proton flux to net water molecule is denoted by the parameter [126]. The following equation can be used to calculate it:

Equation (4.6)

The 1st term denotes the migration effect, and on the other hand, diffusion is represented by the 2nd one. The equation’s manipulation yields the expression:

Equation (4.7)

Assuming the variation in water concentration between anode and cathode can be pretty accurate by a linear difference of single-step.

Equation (4.8)

The thickness of the membrane is the time. Coefficient electro-osmotic (drag) is the parameter that is equivalent to the water molecules number that a photon carries. The membrane’s water content is the dependent area of this quantity, a function of water next to the membrane gas phase. At the current high densities and cathode, saturation, and anode, partial dehydration is likely to occur. Due to electro-osmosis’s higher rate of water transfer, this happens from the cathode to the anode compared to the water diffusion rate from anode to the cathode. This physically implies the lower anode side of the water content. That’s why, for calculating the coefficient of electro-osmotic crosswise the membrane, the anode side’s water activity can be used. It can be expressed the anode channel water activity’s function of electro-osmotic coefficient as follows:

Equation (4.9)

The computing of Eq. 4.8 can be done by the Dw parameter [126]. In the membrane, the water coefficient diffusion can be given by this quantity. As the coefficient electro-osmotic coefficient, the membrane’s water content is the dependent point of the water coefficient diffusion.

Equation (4.10)

As a task of the activity of respective water what is a water concentration in the electrodes is provided below:

Equation (4.11)

For the M;m; dry and k.m; dry subscript density can substitute either the cathode or anode, and respectively, an exchange of dry protons can have equivalent weight. The following are the cathode and anode’s activity of water.

Equation (4.14)

By substituting Eq. 4.15 in Eq. 4.1 – 4.3 can determine every reactant’s number of changes of moles where the consumed reactants are described in manners.

Equation (4.15)

The rates of condensation and evaporation [125] can determine every row in liquid water moles’ number of changes.

Equation (4.16)

Where respectively *d* and* h *are the height and width, water evaporation and condensation’s homogeneous constant rate is kc. The variation between water vapour pressure and partial pressure is proportional to the low channels’ liquid water. It means that if the vapour pressure is lower than the partial pressure of water vapour, it will condense the liquid water. In the same way, it will evaporate the liquid water if the vapour pressure is higher than the water vapour’s partial pressure if there liquid water is available. The nest equation will show the water vapour moles’ changes along with the ow channels:

Equation (4.17)

The flow channels’ water vapour amount has the following factors: (1) water is produced by the oxygen’s reaction with electron and proton at the cathode; (2) water vapour can be brought by protons which are migrated from anode to cathode; (3) due to difference of concentration, the anode in the membrane can use the produced water at the cathode; and (4) the vapour pressure and partial pressure’s variation is depending force of the water vapour condensation and liquid water evaporation. Water evaporation and condensation are the Eq. 4.17’s right side’s 1st term when water vapour’s net transport is the 2nd term across the membrane. The migrating protons carry water molecules and gradients or differences in pressure and concentration, whose net result is water transport.

**4.2.2 Balance of Energy **

It can be obtained the cathode and anode by the local temperature as follows:

Equation (4.18)

Here the overall coefficient of the heat transfer is U, where k represents either cathode or anode. Per unit length area of heat-transfer of the channel ow is defined by parameter an in equation 4.18 and by using the following way it can be computed [125]:

Equation (4.19)

Due to liquid water evaporation and water vapour condensation, the enthalpy change in the numerator is the first term of Eq. 4.18. In contrast, the second term denotes the transfer of heat between the fluid and the mass surface. Eq. 4.20 represents the latent heat as temperature’s function [125].

Equation (4.20)

**4.2.3 Electrochemistry Equations **

For covering potential, the variation between the losses and the cell voltage’s reversible thermodynamically can be expressed by the effective voltage of the cell through using the equation of Tafel and Nernst.

Equation (4.21)

The fuel cell’s open-circuit potential is Voc, where at one oxygen atmosphere’s current density exchange is I0 and the cathode stream’s partial oxygen pressure is PO2 (x). In Eq. 4.21, the potential over activation is the 2nd term, where the potential over ohmic is the 3rd term. To the current density, the voltage of the cell in Eq. 4.21 is proportional inversely. The membrane conductivity in Eq. 4.21 is parameter m which is the content of the water at the interface of the anode in the membrane.

Equation (4.22)

Table 4.1: Governing equations’ Summary for PEFC model

**4.3 Mass Transfer**

FC’s inert substances, reaction products and transport of reactants modelling are contained in the mass transfer modelling. Water is the reaction product of the cathode’s oxygen molecules and the anode’s hydrogen molecules, the reactants in a PEMFC. Nitrogen or an argon and nitrogen mixture is the inert part. Air has both of these gasses, and on-air the operation of FC is conducted in the cathode. Their participation in the reaction is not seen. Inside the cell, the mass transport is affected by them. In this study, as two-stage modelling, in vapour stage water is supposed due to the inaccuracies and complexity as the discussion of Section 3.1, which was away from this study’s scope. The models exclude the mass transport of anode, conversely, due to the augmented flooding risk and slower kinetics reaction. Usually, phenomena limiting performance on the cathode occur. In this paper, the key focusing area of the modelling of mass transport is on the distribution of the reactants on the electrode; on the contrary, the whole cell’s flow field studies an individual’s channels and ribs scale locally.

Through diffusion and convection, it transports the species on the cathode. Through the cell, a flow is created by the variation across the cell in an externally delivered pressure or free convection; the interpretation of density in the air caused by the differences of concentration and temperature produces the lift. The generated reaction products and consumed reactants on the electrode create the density gradient and the concentration gradients. Through the electrode, diffusion layer of porous gas and the channels, it passes the fluid flow. The water can be transported through the membrane, but it is resistant to gases.

With continuity equations and Navier–Stokes (4.2) and (4.1), it models the fluid flow in the channels. To model incompressible and laminar flow, there are several available standard equations. From turbulent flow to laminar, the change has no limit, but if Reynolds number remains under 2300, it considers the fluid flow as laminar. The FC environment is met with this condition where relatively slow flows. Darcy’s Law can be applied instead in the electrode and diffusion layer of porous gas.

The pressure is. The velocity vector is u, the porous medium’s permeability isk, the fluid’s average density is, every equation’s source term is S , and the fluid’s moderate viscosity is.

In a mass frame centre, the behaviour is modelled in equations (4.1 to 4.3). It is necessary to model the diffusion for considering the impact of concentration-driven flow. In an FC cathode, it is entirely incorrect to use the equation of simple diffusion like Fick’s Law; nevertheless, nitrogen, water and oxygen are the three components of the fluid that have diverse sources, diffusion constants and sinks. Therefore, it is essential to employ diffusion equations of Maxwell–Stefan or Maxwell–Stefan of multi-component (4.5) and (4.4).

Although for every component, it has to write the equation of Maxwell–Stefan, only two parts need to be solved since using these two components, the 3rd one can be calculated, and it also requires adding the mass fraction. Therefore, from oxygen and water, the nitrogen’s mass fraction can be derived.

Here, the i species’ mass fraction isi, the molar mass isMj, i species’ molar fraction isxj and the diffusion coefficient of Maxwell–Stefan isDij. For the molar fraction, which is quite the opposite to the fraction, it can write the equation of Maxwell–Stefan.

Equality is seen between the formulations of (Eq4.6) and (Eq4.4). This thesis has used both forms.

The cell reaction is the source for the terms Sin (4.6) and (4.1–4.4) equations, through which the water is produced and oxygen is consumed. Therefore, their existence is solely found in electrodes and elsewhere, they are zero. The source terms in continuity equations and Naiver–Stokes are the exceptions to this, and the flow field model only uses them in which the flow field is the sole component of the modelled cell, and as though in the channels, it occurs the reaction where it also models the reaction impacts.

Equation (4.5 – 4.11)

The FC’s active area isA, the number of electrons on the reaction isz, at the electrode, the reactant of current density isio, the Faraday number isF, the volume of the channel isVCh. In contrast, the water portion of the reaction product that leaves the FC is. Through diffusion from anode to cathode, it can drive a part of the water of the reaction product.

**4.4 Charges Transfer**

The movement of ions and electrons are composed in a PEMFC’s transfer of charge. Therefore, the potential of electrons and ions is required to be modelled separately.

Only electrodes and membranes hold the former, while GDLs and electrodes hold the latter. It can model the electronic or ionic charge transfer by:

Membrane humidity’s one of the functions is membrane , GDL connectivity. The electric potential is; the ionic potential is Si which is in the electrodes non-zero:

Per the equation (3.19) showed in chapter no 3, it can calculate the ic and ia current reaction densities at electrodes. Between the cell components, in reality, discontinuity of the variables is a charming aspect of FC’s transfer of charge and heat. The lifetime and performance of the cell can be influenced critically by the variation in the electric and thermal contact resistances.

**4.5 Transfer of Heat**

Compared to erstwhile FCs like SOFCs, in a lower temperature, PEMFCs run, but in terms of the cell’s performance, one of the vital factors is the heat transfer. As a consequence of current density and local variations of temperature, there occur hot spots. These don’t much influence the cell’s performance, but premature degradation of the membrane can be caused through the lifespan of the cell will be shortened. The straightforwardness of the heat transfer model is found. Conduction and convection are two common heat transfer mechanisms in the cell. Within a PEMFC, radiation’s heat transfer is insignificant since the relative lowness of the contact resistances and thermal bulk is observed. It models the transfer of heats applying the equation (4.12):

By ionic and electronic current by ohmic and reactions heating, heat is generated. In every region, the term ST is the thermal source:

Here, for catalyst and membrane, GDL’s ionic and electronic conductivity is, the electronic and ionic potential is ψe andi, the cathode and anode over-potentials are canda, the temperature isT, entropy’s change in the reaction is ΔS, cathode and anode’s reaction current densities are icand ia. asicin the equation is a negative one, and before the source terms, a negative sign is occupied by the source terms of the cathode side.

Both ions and electrons’ movements are composed in a PEMFC’s transfer of charge. It is pretty essential to model both the electronic and ionic potential individually.

Solely in the electrodes and membrane, here exists the former one wherein GDLs and electrodes, here exists the latter one.

The membrane humidity’s one of the crucial tasks is the membrane , GDL conductivity. In the electrodes, the non-zero terms are Se which represents the potential of electron and ionic potential is represented bySi:

In accordance with chapter 3’s equation (3.19), icand ia reactions of current densities are calculated in the electrodes. Charge and heat transfer in an FC is a fascinating aspect where diverse components of the cell are discontinuous in reality. This happens because of the resistances of the elements, for instance, between the electrodes and theGDL. Spatial variation is available in this resistance of the contact as variations of these components differ on the entire area under the channels and the ribs. The life span and the cell’s performance can be influenced dramatically by the variations between the electric and thermal resistances of the contact.