CN106848351A - The method that proton exchange film fuel battery performance forecast model is set up - Google Patents

Abstract

The invention discloses a kind of method that proton exchange film fuel battery performance forecast model is set up, the model of structure includes the quasi-three-dimensional model that one-dimensional model and 1+1+1 perpendicular to pole plate direction are tieed up, and the one-dimensional model for being constructed perpendicular to pole plate direction specifically includes four steps：Determine cell output voltage, determine ohmic loss, determine activation loss and water management；1+1+1 dimension quasi-three-dimensional models on the basis of the one-dimensional model of pole plate direction, increased along battery runner direction and the floor direction perpendicular to runner.Solve the mass-conservation equation of reactant and water, reactant concentration in average liquid water volume fraction and Catalytic Layer in each layer of battery for obtaining, thus ohmic loss and activation loss are obtained, the operating modes such as regulation current density, temperature, inlet relative humidity, can be predicted the output voltage of Proton Exchange Membrane Fuel Cells under different operating modes.The foundation of proton exchange film fuel battery performance forecast model, by effectively save exploitation funds and shortens the construction cycle.

Description

The method that proton exchange film fuel battery performance forecast model is set up

Technical field

The invention belongs to electrochemical fuel cell field, and in particular to a kind of proton exchange film fuel battery performance predicts mould The method that type is set up.

Background technology

Proton Exchange Membrane Fuel Cells (PEMFC) has the advantages that high power density, zero-emission, is considered as most being hopeful to answer For the power source of automobile power, the development of its technology is deep to be paid attention to by domestic and international researcher.To ensure that PEMFC can with stabilization The performance (the high-quality electron conductivity, the conveying of reactant and the discharge of product including exchange membrane etc.) leaned on, research PEMFC inside is each The water of layer, hydrogen and oxygen transmission process, so it is most important to providing good water management in battery, it is PEMFC performances Lifting provides guidance.

Fuel cell simulation modeling as fuel cell studies important means, not only to the suitable operating mode of battery and battery material Material carries out preliminary screening；Can also from battery the angle analysis battery performance difference of matter transportation and electrochemical reaction mechanism original Cause, for optimization battery design, lifts cell performance providing safeguard.

Anode of proton exchange membrane fuel cell supplies hydrogen, and negative electrode supplies oxygen, and reactant is highly concentrated in maintenance Catalytic Layer Degree, it is ensured that reacting gas efficient transportation to Catalytic Layer is to ensure that the key of battery performance.Except hydrogen and oxygen, appropriate water in battery Distribution is also particularly significant, and PEM should be made to have suitable humidity to ensure high-quality electron conductivity, also to avoid negative electrode The generation of water logging.Angle of the current Forecasting Methodology mostly from system control is predicted to the output voltage of fuel cell, suddenly Electric intracisternal material distribution and influence of the electrochemical reaction mechanism to battery performance are omited.Because this class model lacks matter transportation machine Reason, the operating condition parameter of many batteries and the material parameter for actually possessing cannot embody in a model, it is impossible to study operating mode Influence with cell design parameters to battery performance.

Set up accurate and effective and have the fuel cell mode of short execution cycle concurrently, by effectively save exploitation funds and shortening open The hair cycle.The Three-dimension Numerical Model of battery performance can be comprehensively studied, exists high to computing capability requirement, calculating cycle is long etc. asks Topic is, it is necessary to using the work station calculating of height calculating performance, only calculate the time that single operating mode is accomplished by one day, this is unfavorable for model In engineering application in practice, the battery design initial stage is required to mould of the fast prediction cell design parameters to its performance impact Type.According to electrochemical reaction mechanism, the analysis of battery inner transmission matter and water management method, can quick and precisely matter the invention provides one kind The model method of proton exchange film fuel cell performance.Both the one-dimensional analytic modell analytical model perpendicular to pole plate direction can have been set up, it can also be used to Set up 3 quasi- 3 D analysis models of one-dimensional superposition.Quasi-three-dimensional model is greatly shortened compared with Three-dimension Numerical Model calculating cycle, is widened Research contents is without increasing excessive amount of calculation.

The content of the invention

The purpose of the present invention is, according to electrochemical mechanism and battery inner transmission matter analysis theories, there is provided one kind is quick and precisely The method that prediction proton exchange film fuel battery performance model is set up, can be used to test various working and design parameter to battery The influence of performance.

Constructed model includes the quasi-three-dimensional model that one-dimensional model and 1+1+1 perpendicular to pole plate direction are tieed up.Wherein structure Build perpendicular to pole plate direction one-dimensional model its specifically include four steps：Determine cell output voltage, determine ohmic loss, really Determine activation loss and water management, be below explained.

(1) cell output voltage is determined

E_out=E_rev-η_ohm-η_act 1-1

Wherein E_outRepresent cell output voltage；E_revRepresent reversible voltage；η_ohmRepresent the ohmic loss of voltage；η_actRepresent The voltage loss caused by reactant concentration and water are lost is contained in the activation loss of voltage, ohmic loss and activation loss.

Reversible voltage is tried to achieve by Nernst equation：

Ohmic loss and activation loss two parts are only required to obtain, cell output voltage can be tried to achieve by 1-1.

(2) ohmic loss is determined

(2.1) ohmic loss includes the ohmic loss sum that pole plate, porous medium layer and PEM are caused, i.e.,：

Wherein η_{Ohm, P}、η_{Ohm, por}And η_{Ohm, m}Ohm that respectively pole plate, porous medium layer and PEM are caused is excessively electric Gesture；I is current density；Respectively each layer of runner pole plate and porous media transmits the surface resistance of electronics； The surface resistance of transmission proton respectively in Catalytic Layer and PEM.The solution formula of resistance：

Ω=L/ σ^eff 2-2

Wherein L is transmission range, also illustrates that thickness；σ^effIt is effective conductivity.

Next step needs to obtain electronics effective conductivity and Catalytic Layer and PEM endoplasm electron conductivity in each layer.

(2.2) electronics effective conductivity in porous medium layer

Variable virtual value is corrected frequently with Bruggemann in porous media, and correction factor uses 1.5：

For diffusion layer or microporous layers or Catalytic Layer：

In formulaRepresent the effective conductivity of electronics；σ_sIt is electronics intrinsic conductivity；ε is porosity.

(2.3) proton effective conductivity in PEM and Catalytic Layer

In formulaIt is proton effective conductivity in Catalytic Layer；X_mIt is Catalytic Layer Inner electrolysis matter Nafion volume fractions；σ_mFor The proton conductivity of PEM Nafion.

σ_mDepending on water content in Nafion：

Wherein λ is Nafion water content.

Wherein a is water activity,

For Catalytic Layer：

a_c1=RH+2s 2-7

Wherein RH is the relative humidity of gas in Catalytic Layer, and s is liquid water volume fraction in Catalytic Layer hole；

For PEM, water activity a_averIt is approximately equal to the average of water activity in anode catalyst layer and cathode catalysis layer Value：

(3) activation loss is determined

(3.1) analytic solutions of activation loss：

Wherein η_{Act, ano}, η_{Act, cat}Anode and activation of cathode overpotential are represented respectively；R is ideal gas constant；T is operating mode Temperature；α is electric charge transmission coefficient；N is the electron number of transmission in unit reaction；j_{0, ref}It is reference current density；Respectively It is oxygen concentration in density of hydrogen in anode catalyst layer under actual conditions and cathode catalysis layer；Respectively refer to Density of hydrogen and refer to oxygen concentration.

(3.2) gas concentration in Catalytic Layer：

Hydrogen, oxygen diffusion transport mode follow Fick's law in porous media structure in battery：

Anode and negative electrode are respectively respectively runner, diffusion layer, microporous layers, Catalytic Layer comprising four solution domains.

Anode catalyst layer density of hydrogen：

WhereinMicroporous layers, density of hydrogen at Catalytic Layer interface；It is Catalytic Layer, PEM friendship Interface density of hydrogen；It is hydrogen effective diffusion cofficient in anode catalyst layer, is corrected by BruggemannUnit is m²/s；δ_CLIt is Catalytic Layer thickness, unit is m.

Anode catalyst layer hydrogen mean concentration：

Cathode catalysis layer oxygen concentration：

WhereinMicroporous layers, oxygen concentration at Catalytic Layer interface；It is Catalytic Layer, PEM friendship Interface oxygen concentration；It is oxygen effective diffusion cofficient in cathode catalysis layer.

Cathode catalysis layer oxygen mean concentration：

Runner, diffusion layer, the reacting gas governing equation in microporous layers region can be similar to and list, then in conjunction with anode flow channel The boundary condition of oxygen concentration in density of hydrogen and cathode flow channels, can try to achieve Catalytic Layer reaction gases actual concentration.

(4) water management

Water transdermal delivery mode is pulled comprising electric osmose, three kinds of forms are spread in the diffusion of film state water and pressure difference.

Electric osmose pulls effect and shows as proton transdermal delivery, while a certain amount of water can be pulled from anode to negative electrode, electric osmose Pull coefficient n_dIt is the hydrone number with each proton by anode to negative electrode cross-film：

Film state water diffusion coefficient D_mComputational methods it is as follows：

For anode catalyst layer water conservation equation：

Wherein J_vapVapor transports flux；c_{Vap, MPL-CL}It is anode micro porous layer, Catalytic Layer interface water vapor concentration； c_{Vap, CL-PEM}It is Catalytic Layer, PEM interface water vapor concentration；It is vapor effective diffusivity in Catalytic Layer； ρ_dryIt is dry state film density；EW is the equivalent quality of PEM；λ_aclλ_cclRespectively anode and cathode catalysis layer mode water contains Amount；K_mIt is the permeability of film；Respectively anode and cathode catalysis layer liquid water pressure.

For cathode catalysis layer water conservation equation：

Wherein ρ_lIt is liquid water density；It is water molal weight；s_cclIt is cathode catalysis layer liquid water volume fraction；ε_ccl It is cathode catalysis layer porosity；K_{L, cl}It is the permeability of Catalytic Layer water；μ_lIt is the dynamic viscosity of water；Be cathode catalysis layer, The hydraulic pressure of PEM interface；It is cathode micro porous layer, the hydraulic pressure of Catalytic Layer interface；J_lFor aqueous water circulates Amount.

Diffusion layer, the water management equation in microporous layers region can be similar to be listed, by runner in hypothesis without aqueous water, with reference to sun Hydraulic pressure is equal to a boundary condition for atmospheric pressure at water vapor concentration and cathode flow channels and diffusion layer interface in the runner of pole, tries to achieve Each layer water vapor concentration of anode and each layer interface hydraulic pressure of negative electrode.

Capillary pressure p in porous media is drawn by Leverett equations_cWith the relation of liquid water volume fraction s：

P_c=P_g-P_l 4-7

WhereinSurface tension coefficient；θ is porous media contact angle, the hydraulic pressure P for trying to achieve_l, then obtain each portion in battery Divide liquid water volume fraction s.

Water distribution situation is brought into step (2), step (3) in the battery that step (4) is obtained, by aforementioned formula 2-1,3- 1 and 3-2 can obtain ohmic loss and activation loss, bring formula 1-1 into, and the battery predictive for finally trying to achieve the one-dimensional model is defeated Go out voltage.

1+1+1 quasi-three-dimensional models comprising x directions perpendicular to pole plate direction, y directions along runner direction, z directions perpendicular to stream Road and floor, three superpositions in direction.

The one-dimensional model of X-direction is that given electric current seeks voltage, after the voltage for trying to achieve the first Battery pack, is asked with this group of voltage Obtain the current density of the second Battery pack.

Quasi-three-dimensional model its specific steps method for building 1+1+1 dimensions includes：

(1) foundation of x directions vertical plate direction one-dimensional model is identical with 4 steps described in claim 1.

(2) y directions along runner direction one-dimensional model foundation, its specific method step includes：

Two Battery packs section is connected in parallel, output voltage is identical, and output current density is differed, battery is along runner direction point It is two parts.

Given first Battery pack section current density I_a, the first Battery pack can be tried to achieve using 4 steps of the claim 1 The output voltage E of section_{A, out}。

E_{B, out}=E_{A, out}5-1

Second Battery pack section current density I_b, tried to achieve by following steps：

η_{Act, cat}=E_rev-E_out-η_ohm-η_{Act, ano} 5-3

WhereinWhat is represented respectively is in the second Battery pack section anode catalyst layer in density of hydrogen and cathode catalysis layer Oxygen concentration.

Assuming that current density is I_assume,

Anode catalyst layer density of hydrogen：

Cathode catalysis layer oxygen concentration：

On boundary condition, the first Battery pack section anode export density of hydrogen is the second Battery pack section import density of hydrogen, the One Battery pack section cathode outlet oxygen concentration is the second Battery pack section import oxygen concentration.

WillBring 5-2 into and try to achieve η_{Act, ano}, by η_{Act, ano}Bring 5-3 into and try to achieve η_{Act, cat}, current density can be obtained by 5-4 I_solve。

WhenWhen, I_solveThe the second Battery pack section current density for as being solved.

(3) in the z-direction perpendicular to runner, the foundation of floor direction one-dimensional model, its specific method step includes：

Battery is divided into the 3rd Battery pack section below the first Battery pack section and floor below runner, first group in the z-direction Cell section and the 3rd Battery pack section are connected in parallel, and output voltage is identical.

The current density I of the first Battery pack section_a, its output voltage can be tried to achieve using 4 steps of the claim 1 E_{A, out},

E_{C, out}=E_{A, out} 6-1

The current density I of the 3rd Battery pack section_c, tried to achieve by following steps：

η_{Act, cat}=E_rev-E_out-η_ohm-η_{Act, ano} 6-3

WhereinWhat is represented respectively is in the 3rd Battery pack section anode catalyst layer in density of hydrogen and cathode catalysis layer Oxygen concentration.

Assuming that current density is I_assume,

3rd Battery pack section Catalytic Layer reaction gases include, by the section microporous layers diffusion of the 3rd Battery pack and the first Battery pack Section Catalytic Layer diffusion.

3rd Battery pack section anode catalyst layer density of hydrogen：

3rd Battery pack section cathode catalysis layer oxygen concentration：

WhereinThe 3rd Battery pack section microporous layers and catalysis are represented respectively The hydrogen and oxygen concentration of layer interface, Catalytic Layer and PEM interface；The 3rd Battery pack is represented respectively Hydrogen and oxygen mean concentration in section Catalytic Layer；Hydrogen and oxygen in the first Battery pack section Catalytic Layer are represented respectively Mean concentration.

First Battery pack section Catalytic Layer reaction gases concentration is tried to achieve by abovementioned steps (3.2), is known quantity.

WillBring 6-2 into and try to achieve η_{Act, ano}, by η_{Act, ano}Bring 6-3 into and try to achieve η_{Act, cat}, current density can be obtained by 6-4 I_solve。

WhenWhen, I_solveIt is the 3rd Battery pack section current density for being solved.

To try to achieve in each layer of battery reactant concentration in average liquid water volume fraction s and Catalytic Layer, according to proton exchange Membrane cell physical arrangement sets up computational fields, such as Fig. 1, using each layer interface as solution node.

The present invention is with hydraulic pressure p_lAs the solution parameter of aqueous water governing equation, hydraulic pressure p at the interface for first solving_l, then by Leverett equations try to achieve the liquid water volume fraction s of interface both sides.

Solve the mass-conservation equation of reactant and water, in each layer of battery for obtaining average liquid water volume fraction s with urge Change reactant concentration in layer, thus obtain ohmic loss and activation loss, and try to achieve the output voltage of prediction.Regulation electric current is close The operating modes such as degree, temperature, inlet relative humidity, can be predicted the output voltage of Proton Exchange Membrane Fuel Cells under different operating modes.Also may be used So that by regulating cell design parameter, such as the porosity of each layer porous media, hydrophobicity etc. studies battery structure and design parameter Influence to battery performance.

The features of the present invention and it is beneficial in that：

(1) Forecasting Methodology has high efficiency and accuracy, and Proton Exchange Membrane Fuel Cells is defeated under predictable different operating modes Go out voltage, can be used to probe into influence of the cell design parameters to performance.It is dense to real reaction gas under each operating mode by being then based on Degree and water management carry out voltage prediction, so its practical application is above Fuel Cell Control model.Due to being analytic modell analytical model Method, the calculating time can be greatlyd save than numerical model.

(2) method for providing can be used to develop accurate three of one-dimensional analytic modell analytical model and " 1+1+1 " dimension perpendicular to pole plate direction Dimension analytic modell analytical model.Quasi- 3 D analysis model can obtain the result of study of some threedimensional models again while computational efficiency is ensured.

(3) in water management solution, the governing equation of water is solution amount with hydraulic pressure, introduces " aqueous water rank at interface The concept of jump ", it is contemplated that because adjacent two layers are made up of Different structural parameters and hydrophobic porous media material, and cause phase The adjacent bed interface both sides discontinuous situation of liquid water volume fraction, can obtain by the different shadows to water management of layers of material parameter Ring.

Brief description of the drawings

Fig. 1 is perpendicular to the quasi-three-dimensional model physical arrangement schematic diagram of one-dimensional model and the 1+1+1 dimension in pole plate direction.

Fig. 2 is the governing equation iterative method block diagram of solution water in the present invention.

Fig. 3 is compareed using model prediction voltage of the present invention with experimental result.

Fig. 4 applications model of the present invention quantifies influence of the explanation battery operating temperature to battery performance.

Fig. 5 applications model of the present invention quantifies explanation Catalytic Layer and microporous layers hydrophobicity (contact angle) combination to battery performance Influence.

Specific embodiment

Design of the invention is further illustrated below in conjunction with accompanying drawing and by embodiment, it is necessary to explanation is embodiment It is the narrative explanation carried out for clear interpretation modeling procedure, protection scope of the present invention is not limited with this.

The method that proton exchange film fuel battery performance forecast model is set up, constructed model is included perpendicular to pole plate side To one-dimensional model and 1+1+1 tie up quasi-three-dimensional model.

Wherein being constructed perpendicular to the one-dimensional model in pole plate direction its specific steps method includes：

(1) cell output voltage E is determined_out

E_out=E_rev-η_ohm-η_act 1-1

Cell output voltage is equal to reversible voltage E_revCut the ohmic loss η of voltage_ohmWith the activation loss η of voltage_act, The voltage loss caused by reactant concentration and water are lost is contained in ohmic loss and activation loss.

E_revTried to achieve by Nernst equation：

In formula：Δ G- gibbs free energy changes；F- Faraday constants 96487C/mol；Δ S is Entropy Changes；R- perfect gases are normal Number 8.314J/mol K；T- working temperatures K；T_ref- reference temperature K；Respectively anode catalyst layer Hydrogen Vapor Pressure With cathode catalysis layer oxygen pressure.

Ohmic loss and activation loss two parts are tried to achieve, cell output voltage can be tried to achieve by 1-1.

(2) ohmic loss η is determined_ohm

(2.1) ohmic loss includes pole plate η_{Ohm, P}, porous medium layer η_{Ohm, por}The ohmic loss caused with PEM η_{Ohm, m}Sum, i.e.,：

η_ohmAlso referred to as ohm overpotential, I is current density A/m in formula²；Respectively runner pole plate and Each layer of porous media transmits the surface resistance of electronics；Transmission proton respectively in Catalytic Layer and PEM Surface resistance, the solution formula of resistance：

Ω=L/ σ^eff 2-2

Wherein L is transmission range, also illustrates that thickness；σ^effIt is effective conductivity.

Next step obtains electronics effective conductivity and Catalytic Layer and PEM endoplasm electron conductivity in each layer.

(2.2) electronics effective conductivity in porous medium layer

Porous medium layer includes diffusion layer, microporous layers and Catalytic Layer.In porous media variable virtual value frequently with Bruggemann is corrected, and correction factor uses 1.5：

For diffusion layer or microporous layers or Catalytic Layer：

In formulaRepresent the effective conductivity of electronics；σ_sIt is electronics intrinsic conductivity；ε is porosity.

(2.3) proton effective conductivity in PEM and Catalytic Layer

σ_mDepending on water content in Nafion：

Wherein λ is Nafion water content.

For Catalytic Layer, water activity is：a_c1=RH+2s, RH are the relative humidity of gas in Catalytic Layer, and s is Catalytic Layer hole Liquid water volume fraction in gap.For PEM, water activity is approximately equal to water in anode catalyst layer and cathode catalysis layer and lives The average value of degree：

Proton conductivity is related to battery inner conduit reason, tries to achieve water distribution situation in battery, can obtain proton conductivity.

(3) activation loss is determined

(3.1) analytic solutions of activation loss：

Due to gas and aqueous water are included in the space of porous media simultaneously, soWithRespectively anode catalyst layer Oxygen apparent concentration in interior hydrogen apparent concentration and cathode catalysis layer,

WhereinOxygen actual concentration in hydrogen actual concentration and cathode catalysis layer respectively in anode catalyst layer； ε_acl, ε_cclThe respectively porosity of anode catalyst layer and cathode catalysis layer；s_aclAnd s_cclRespectively anode catalyst layer and negative electrode is urged Change the liquid water volume fraction of layer.

Activation overpotential is tried to achieve, it is necessary to obtain water distribution in Catalytic Layer reaction gases actual concentration and battery.

(3.2) gas concentration in Catalytic Layer：

Analyzed by gas transport in battery, try to achieve under various operating modes density of hydrogen and cathode catalysis layer in anode catalyst layer Interior oxygen concentration.Hydrogen, oxygen follow Fick's law for diffusion transport mode in porous media structure in battery：

Anode catalyst layer density of hydrogen：

Anode catalyst layer hydrogen mean concentration：

Cathode catalysis layer oxygen concentration：

Cathode catalysis layer oxygen mean concentration：

Gas transport process is influenceed by water distribution in battery in battery, needs first to solve water distribution during solution, then bring into Solve gas transport equation.

(4) water management

As described in step (2) and step (3), in proton conductivity and porous medium layer gas transport with electric pool inner water Distribution is related, takes the gas-liquid water conservation equation solved with hydraulic pressure as variable, and then try to achieve water distribution in battery.

The existence form of electric pool inner water includes three kinds of vaporous water, aqueous water and film state water, is referred in aforementioned formula (1.3) Nafion water content is film state water.

Water transdermal delivery mode is pulled comprising electric osmose, three kinds of forms are spread in the diffusion of film state water and pressure difference.

Electric osmose pulls effect and shows as proton transdermal delivery, and electric osmose pulls coefficient n_dIt is by anode to the moon with each proton The hydrone number of pole cross-film：

According to the principle that balanced each other under stable state, aqueous water can be formed after vapor reaches saturation, by electrochemical reaction, negative electrode Reaction generates water, and electric osmose drag interaction takes water to negative electrode from anode in addition, and negative electrode vapor is constantly in saturated mode and exists Aqueous water, anode vapor is difficult to reach saturation, therefore anode without aqueous water.

Film state water diffusion coefficient D_mComputational methods it is as follows：

For anode catalyst layer water conservation equation：

For cathode catalysis layer water conservation equation：

Capillary pressure p in porous media is drawn by Leverett equations_cWith the relation of liquid water volume fraction s：

P_c=P_g-P_l4-7

Water distribution situation is brought into step (2), step (3) in the battery that step (4) is obtained, by aforementioned formula 2-1,3- 1 and 3-2 can obtain ohmic loss and activation loss, bring formula 1-1 into, and the battery predictive for finally trying to achieve the one-dimensional model is defeated Go out voltage.

1+1+1 quasi-three-dimensional models comprising x directions perpendicular to pole plate direction, y directions along runner direction, z directions perpendicular to stream Road and floor, three superpositions in direction, quasi-three-dimensional model its specific steps method for building 1+1+1 dimensions include：

(1) foundation of x directions vertical plate direction one-dimensional model is identical with foregoing 4 steps.

(2) y directions along runner direction one-dimensional model foundation, its specific method step includes：

Such as Fig. 1, two Battery packs section is connected in parallel, output voltage is identical, but output current density is differed, and battery is along stream Road direction is divided into two parts.

Given first Battery pack section a current densities I_a, using foregoing 4 steps, try to achieve the output electricity of the first Battery pack section Pressure E_{A, out}：

E_{B, out}=E_{A, out}5-1

Second Battery pack section b current densities I_b, tried to achieve by following steps：

η_{Act, cat}=E_rev-E_out-η_ohm-η_{Act, ano} 5-3

Solution amount is finally obtained by current density, and is solvedProcess needs to use current density, therefore first false If current density is I_assume,

Anode catalyst layer density of hydrogen：

Cathode catalysis layer oxygen concentration：

WillBring 5-2 into and try to achieve η_{Act, ano}, by η_{Act, ano}Bring 5-3 into and try to achieve η_{Act, cat}, current density can be obtained by 5-4 I_solve,

WhenWhen, I_solveThe the second Battery pack section current density for as being solved.

(3) in the z-direction perpendicular to runner, the foundation of floor direction one-dimensional model, its specific method step includes：

First Battery pack section a and the 3rd Battery pack section c are in parallel, and its output voltage E can be tried to achieve using foregoing 4 steps_{A, out}：

E_{C, out}=E_{A, out} 6-1

The current density I of the 3rd Battery pack section c_c, tried to achieve by following steps：

η_{Act, cat}=E_rev-E_out-η_ohm-η_{Act, ano} 6-3

3rd Battery pack section Catalytic Layer reaction gases include, by the section microporous layers diffusion of the 3rd Battery pack and the first Battery pack Section Catalytic Layer diffusion.

3rd Battery pack section anode catalyst layer density of hydrogen：

3rd Battery pack section cathode catalysis layer oxygen concentration：

WillBring 6-2 into and try to achieve η_{Act, ano}, by η_{Act, ano}Bring 6-3 into and try to achieve η_{Act, cat}, current density can be obtained by 6-4 I_solve,

WhenWhen, I_solveIt is the 3rd Battery pack section current density for being solved.

Specific implementation example

Physical model is set up according to Proton Exchange Membrane Fuel Cells physical arrangement, including it is assumed hereinafter that：

A gas flow is considered as one-dimensional stable laminar flow in () runner；

B aqueous water that () is set in runner as straight channel, i.e. runner quickly can be blown away by air inlet；

C each several part temperature and air pressure are accordingly to be regarded as identical in () battery, temperature is given operating temperature, and gas pressure is one Individual atmospheric pressure；

D (), due to relatively low operating temperature, the water of electrochemical reaction generation is aqueous water, film state water content in PEM It is balance state value；

E the material of each layer of () battery is considered as isotropic.

It is as follows that the present embodiment is related to major parameter：

Constant cell current work, temperature T=343.15K, current density I=10000A/m²Or 1A/cm², anode and cathode enters Mouth is 1atm, and anode hydrogen gas and cathode air supply ST=2 in given stoichiometric proportion mode, and anode and cathode air inlet is completely Humidification RH=100%, runner 0.1m long, polar plate area 2 × 10^-4m²。

Cell design parameters use Nafion212 including PEM, and thickness is 5 × 10^-5M, equivalent quality EW are 2.1kg/mol。

Diffusion layer, microporous layers, catalysis layer porosity are followed successively by 0.6,0.4,0.3, contact angle is the hydrophobic amount of reflection, 100 ° are followed successively by, 110 °, 100 °, anode and cathode catalysis layer electrolyte volume fraction are 0.2.

(1) cell output voltage

E_out=E_rev-η_ohm-η_act

(2) ohmic loss

Water content in battery is first tried to achieve, then tries to achieve proton conductivity, and then try to achieve ohmic loss.

(3) activation loss

Anode and cathode flow channels each lead into hydrogen and oxygen with certain humidity, water vapor concentration in air inlet

H in anode air inlet₂Concentration is

Cathode inlet is air, oxygen duty gas fractionIt is O in 0.21, therefore cathode inlet₂Concentration is

Fuel cell effective supply gives reactant stoichiometric proportion ST often, and anode and cathode induction air flow ratio can be tried to achieve accordingly V_in, exit gas concentration c_outWith average gas concentration c in runner_ch：

Anode hydrogen gas：

Cathode oxygen：

By Fick's lawCan obtainActivated electricity Gesture solution formula：

Wherein activation overpotential can be just tried to achieve comprising proton conductivity in Catalytic Layer, it is necessary to first try to achieve water content in battery.

(4) water management

Water conservation equation in anode catalyst layer：

Cathode catalysis layer water conservation equation：

The governing equation of solution by iterative method water is taken, iterative process such as Fig. 2, specific iterative process is as follows：

Iteration variable be in each layer of battery structure average liquid water volume fraction s, s value between 0-1, using 0.5 as respectively Layer in s initial value, the selection of initial value on final true value result without influence,

Wherein s_kTo assign the value of s after kth step iteration, and the initial value calculated as the step of kth+1 is substituted into, as old value； For the step of kth+1 calculates the value for solving s, s^k+1To assign the value of s after the step iteration of kth+1, as new value；Urf is relaxation factor, this Urf takes 0.1 in example；Equation the right Section 1The increment of as each iteration.

Often after step iteration, if new value is less than certain condition with old value difference, then it is assumed that equation is restrained, and the present embodiment is with residual errorUsed as convergence, now s values are the true value for meeting equation.Solve：

Water vapor concentration at each layer interface of anode：

Hydraulic pressure at each layer interface of negative electrode：

Air pressure is 101325Pa, capillary pressure at each layer interface of negative electrode everywhere in battery：

By

Try to achieve liquid water volume fraction at each layer interface of negative electrode：

{s_GDL-mpl, s_gdl-MPL, s_MPL-cl, s_mpl-CL, s_cl-pem}={ 0.299,0.145,0.213,0.413,0.416 }

Wherein s_GDL-mplIt is diffusion layer and microporous layers interface diffusion layer side liquid water volume fraction, s_gdl-MPLIt is diffusion layer With microporous layers interface microporous layers side liquid water volume fraction.

Average water vapor concentration or the average value that liquid water volume fraction is layer two ends are in each layer：

Average water vapor concentration in each layer of anode：

Relative humidity in each layer of anode：

Average liquid water volume fraction in each layer of negative electrode：

Water distribution in battery is substituted into (2), (3) obtain ohmic loss and activation loss respectively：

Anode catalyst layer water activity：a_ac1=RH_acl+2s_acl=0.7754

Cathode catalysis layer water activity：a_cc1=RH_ccl+2s_ccl=1.810

Water activity in PEM：

Film state water content：λ=14.0+1.4 (a-1)=14.41

Membrane conductivity

Ohmic loss：

Anode activation loses：

(5) cell output voltage is finally tried to achieve

Cell output voltage E_out=E_rev-η_ohm-η_act=0.429V

Fig. 3,4,5 are exactly the expected result of study drawn using this modeling method：

Fig. 3 is the comparing of a kind of battery operating parameter drag prediction and experiment value, model emulation result and experimental result Contrast has good uniformity.

Fig. 4 is influence of the battery operating temperature to battery performance.

Fig. 5 is the influence of Catalytic Layer and microporous layers hydrophobicity (contact angle) combination to battery performance

Claims (2)

1. the method that proton exchange film fuel battery performance forecast model is set up, it is characterized in that：Constructed model includes vertical In pole plate direction one-dimensional model and 1+1+1 tie up quasi-three-dimensional model, wherein be constructed perpendicular to pole plate direction one-dimensional model its Specific steps method includes：

(1) cell output voltage is determined

E_out=E_rev-η_ohm-η_act 1-1

Wherein E_outRepresent cell output voltage；E_revRepresent reversible voltage；η_ohmRepresent the ohmic loss of voltage；η_actRepresent voltage Activation loss, the voltage loss caused by reactant concentration and water are lost is contained in ohmic loss and activation loss,

Reversible voltage is tried to achieve by Nernst equation：

$E_{rev}=\frac{\mathrm{\Δ}G}{2F}+\frac{\mathrm{\Δ}S}{2F}(T-T_{ref})+\frac{RT}{2F}\[ln\left(P_{H_2,ano}\right)+\frac{1}{2}ln\left(P_{O_2,cat}\right)\]---1-2$

In formula：E_revIt is reversible voltage；Δ G is gibbs free energy change；F is Faraday constant；Δ S is Entropy Changes；R is preferable gas Body constant；T is working temperature；T_refIt is reference temperature；Respectively anode catalyst layer Hydrogen Vapor Pressure and negative electrode are urged Change layer oxygen pressure；

Ohmic loss and activation loss two parts are only required to obtain, cell output voltage can be tried to achieve by 1-1,

(2) ohmic loss is determined

(2.1) ohmic loss includes the ohmic loss sum that pole plate, porous medium layer and PEM are caused, i.e.,：

${\mathrm{\η}}_{ohm}={\mathrm{\η}}_{ohm,P}+{\mathrm{\η}}_{ohm,por}+{\mathrm{\η}}_{ohm,m}=({\mathrm{\Ω}}_{e^{-1},P}+{\mathrm{\Ω}}_{e^{-1},por}+{\mathrm{\Ω}}_{H^{+1},por}+{\mathrm{\Ω}}_{H^{+1},m})I---2-1$

Wherein η_ohm,P、η_ohm,porAnd η_ohm,mThe ohmic loss that respectively pole plate, porous medium layer and PEM are caused；I is Current density；Respectively each layer of runner pole plate and porous media transmits the surface resistance of electronics；The surface resistance of transmission proton, the solution formula of resistance respectively in Catalytic Layer and PEM：

Ω=L/ σ^eff 2-2

Wherein L is transmission range, also illustrates that thickness；σ^effIt is effective conductivity,

Next step needs to obtain electronics effective conductivity and Catalytic Layer and PEM endoplasm electron conductivity in each layer,

(2.2) electronics effective conductivity in porous medium layer

Variable virtual value is corrected frequently with Bruggemann in porous media, and correction factor uses 1.5：

For diffusion layer or microporous layers or Catalytic Layer：

In formulaRepresent the effective conductivity of electronics；σ_sIt is electronics intrinsic conductivity；ε is porosity,

(2.3) proton effective conductivity in PEM and Catalytic Layer

${\mathrm{\σ}}_{m,cl}^{eff}={\mathrm{\σ}}_m{X_m}^{1.5}---2-4$

In formulaIt is proton effective conductivity in Catalytic Layer；X_mIt is Catalytic Layer Inner electrolysis matter Nafion volume fractions；σ_mIt is proton The proton conductivity of exchange membrane Nafion,

σ_mDepending on water content in Nafion：

${\mathrm{\σ}}_m=(0.5139\mathrm{\λ}-0.326)\exp \[1268(\frac{1}{303.15}-\frac{1}{T})\]---2-5$

Wherein λ is Nafion water content.

$\mathrm{\&lambda;}=\left\{\begin{array}{cc}0.043+17.81a-39.85a^2+36.08a^3& 0\&le;a\&le;1\\ 14.0+1.4(a-1)& 1<a\&le;3\end{array}\right.---2-6$

Wherein a is water activity,

For Catalytic Layer：

a_cl=RH+2s 2-7

Wherein RH is the relative humidity of gas in Catalytic Layer, and s is liquid water volume fraction in Catalytic Layer hole；

For PEM, water activity a_averIt is approximately equal to the average value of water activity in anode catalyst layer and cathode catalysis layer：

$a_{aver}=\frac{a_{acl}+a_{ccl}}{2}---2-8$

(3) activation loss is determined

(3.1) analytic solutions of activation loss：

${\mathrm{\&eta;}}_{act,ano}=\frac{RT}{\mathrm{\&alpha;}nF}{\cosh }^{-1}\&lsqb;\frac{I^2}{4{\mathrm{\&sigma;}}_m^{eff2}\left(\frac{{\mathrm{\&sigma;}}_m^{eff}+{\mathrm{\&sigma;}}_s^{eff}}{{\mathrm{\&sigma;}}_m^{eff}\&CenterDot;{\mathrm{\&sigma;}}_s^{eff}}\right)\frac{RT}{\mathrm{\&alpha;}nF}{j}_{0,ref}^{ano}\left(\frac{c_{H_2}}{c_{H_2,ref}}\right)}+1\&rsqb;---3-1$

${\mathrm{\&eta;}}_{act,cat}=\frac{RT}{\mathrm{\&alpha;}nF}{\cosh }^{-1}\&lsqb;\frac{I^2}{4{\mathrm{\&sigma;}}_m^{eff2}\left(\frac{{\mathrm{\&sigma;}}_m^{eff}+{\mathrm{\&sigma;}}_s^{eff}}{{\mathrm{\&sigma;}}_m^{eff}\&CenterDot;{\mathrm{\&sigma;}}_s^{eff}}\right)\frac{RT}{\mathrm{\&alpha;}nF}{j}_{0,ref}^{cat}\left(\frac{c_{O_2}}{c_{O_2,ref}}\right)}+1\&rsqb;---3-2$

Wherein η_act,ano, η_act,catAnode and activation of cathode overpotential are represented respectively；α is electric charge transmission coefficient；N reacts for unit The electron number of middle transmission；j_0,refIt is reference current density；It is respectively dense with reference to density of hydrogen and with reference to oxygen Degree；Respectively with reference to density of hydrogen and oxygen concentration is referred to,

(3.2) gas concentration in Catalytic Layer：

Hydrogen, oxygen diffusion transport mode follow Fick's law in porous media structure in battery：

$J_i=-D_i^{eff}\&dtri;C_i---3-3$

Anode and negative electrode are respectively respectively runner, diffusion layer, microporous layers, Catalytic Layer, anode catalyst layer hydrogen comprising four solution domains Concentration：

$\frac{(c_{MPL-CL}^{H_{2}H}-c_{CL-PEM}^{H_2})D_{H_2,CL}^{eff}}{{\mathrm{\&delta;}}_{CL}}=\frac{I}{2F}---3-4$

WhereinMicroporous layers, density of hydrogen at Catalytic Layer interface；It is Catalytic Layer, PEM interface Place's density of hydrogen；It is hydrogen effective diffusion cofficient in anode catalyst layer, is corrected by Bruggemannδ_CLIt is Catalytic Layer thickness,

Anode catalyst layer hydrogen mean concentration：

$c_{CL}^{H_2}=\frac{c_{MPL-CL}^{H_2}+c_{CL-PEM}^{H_2}}{2}---3-5$

Cathode catalysis layer oxygen concentration：

$\frac{(c_{MPL-CL}^{O_2}-c_{CL-PEM}^{O_2})D_{O_2,CL}^{eff}}{{\mathrm{\&delta;}}_{CL}}=\frac{I}{4F}---3-6$

WhereinMicroporous layers, oxygen concentration at Catalytic Layer interface；It is Catalytic Layer, PEM interface Place's oxygen concentration；It is oxygen effective diffusion cofficient in cathode catalysis layer,

Cathode catalysis layer oxygen mean concentration：

$c_{CL}^{O_2}=\frac{c_{MPL-CL}^{O_2}+c_{CL-PEM}^{O_2}}{2}---3-7$

Runner, diffusion layer, the reacting gas governing equation in microporous layers region can be similar to and list, then in conjunction with hydrogen in anode flow channel The boundary condition of oxygen concentration in concentration and cathode flow channels, can try to achieve Catalytic Layer reaction gases actual concentration,

(4) water management

Water transdermal delivery mode is pulled comprising electric osmose, three kinds of forms are spread in the diffusion of film state water and pressure difference,

Electric osmose pulls effect and shows as proton transdermal delivery, while a certain amount of water can be pulled from anode to negative electrode, electric osmose is pulled Coefficient n_dIt is the hydrone number with each proton by anode to negative electrode cross-film：

$n_d=\frac{2.5\mathrm{\&lambda;}}{22}---4-1$

Film state water diffusion coefficient D_mComputational methods it is as follows：

$D_m=\left\{\begin{array}{cc}2.69266\&times;{10}^{-10}& \mathrm{\&lambda;}\&le;2\\ {10}^{-10}\exp \&lsqb;2416(\frac{1}{303}-\frac{1}{T})\&rsqb;\&lsqb;0.87(3-\mathrm{\&lambda;})+2.95(\mathrm{\&lambda;}-2)\&rsqb;& 2<\mathrm{\&lambda;}\&le;3\\ {10}^{-10}\exp \&lsqb;2416(\frac{1}{303}-\frac{1}{T})\&rsqb;\&lsqb;2.95(4-\mathrm{\&lambda;})+1.642(\mathrm{\&lambda;}-3)\&rsqb;& 3<\mathrm{\&lambda;}\&le;4\\ {10}^{-10}\exp \&lsqb;2416(\frac{1}{303}-\frac{1}{T})\&rsqb;(2.563-0.33\mathrm{\&lambda;}+0.0264{\mathrm{\&lambda;}}^2-0.00071{\mathrm{\&lambda;}}^3)& \mathrm{\&lambda;}>4\end{array}\right.---4-2$

For anode catalyst layer water conservation equation：

$\frac{D_{vap,CL}^{eff}(c_{vap,MPL-CL}-c_{vap,CL-PEM})}{{\mathrm{\&delta;}}_{CL}}=J_{vap}---4-3$

$\frac{{\mathrm{\&rho;}}_l}{M_{H_{2}O}}\frac{K_m}{{\mathrm{\&mu;}}_l}\frac{p_{CL}^{c,l}-p_{CL}^{a,l}}{{\mathrm{\&delta;}}_{CL}}+D_m\frac{{\mathrm{\&rho;}}_{dry}}{EW}\frac{({\mathrm{\&lambda;}}_{ccl}-{\mathrm{\&lambda;}}_{acl})}{{\mathrm{\&delta;}}_{PEM}}-\frac{n_{d}I}{F}=J_{vap}---4-4$

Wherein J_vapVapor transports flux；c_vap,MPL-CLIt is anode micro porous layer, Catalytic Layer interface water vapor concentration；c_vap,CL-PEM It is Catalytic Layer, PEM interface water vapor concentration；It is vapor effective diffusivity in Catalytic Layer；ρ_dryIt is dry state Film density；EW is the equivalent quality of PEM；λ_acl, λ_cclRespectively anode and cathode catalysis layer mode water content；K_mFor The permeability of film；Respectively anode and cathode catalysis layer liquid water pressure,

For cathode catalysis layer water conservation equation：

$\frac{{\mathrm{\&rho;}}_l}{M_{H_{2}O}}\frac{K_{l,cl}}{{\mathrm{\&mu;}}_l}\frac{p_{CL-PEM}^l-p_{MPL-CL}^l}{{\mathrm{\&delta;}}_{CL}}=J_l---4-5$

$\frac{n_{d}I}{F}+\frac{I}{2F}-\frac{{\mathrm{\&rho;}}_l}{M_{H_{2}O}}\frac{K_m}{{\mathrm{\&mu;}}_l}\frac{p_{CL}^{c,l}-p_{CL}^{a,l}}{{\mathrm{\&delta;}}_{CL}}-D_m\frac{{\mathrm{\&rho;}}_{dry}}{EW}\frac{({\mathrm{\&lambda;}}_{ccl}-{\mathrm{\&lambda;}}_{acl})}{{\mathrm{\&delta;}}_{PEM}}=J_l---4-6$

Wherein ρ_lIt is liquid water density；It is water molal weight；s_cclIt is cathode catalysis layer liquid water volume fraction；ε_cclIt is the moon Pole is catalyzed layer porosity；K_l,clIt is the permeability of Catalytic Layer water；μ_lIt is the dynamic viscosity of water；It is anode catalyst layer, matter The hydraulic pressure of proton exchange interface；It is cathode micro porous layer, the hydraulic pressure of Catalytic Layer interface；J_lIt is liquid water stream flux, Diffusion layer, the water management equation in microporous layers region can be similar to be listed, by runner in hypothesis without aqueous water, with reference in anode flow channel Hydraulic pressure is equal to a boundary condition for atmospheric pressure at water vapor concentration and cathode flow channels and diffusion layer interface, tries to achieve each layer of anode Water vapor concentration and each layer interface hydraulic pressure of negative electrode,

Capillary pressure p in porous media is drawn by Leverett equations_cWith the relation of liquid water volume fraction s：

P_c=P_g-P_l4-7

$P_c={\mathrm{\&sigma;}}_{lq}cos\mathrm{\&theta;}{\left(\frac{\mathrm{\&epsiv;}}{K}\right)}^{0.5}J\left(s\right)---4-8$

Wherein σ_lqSurface tension coefficient；θ is porous media contact angle, the hydraulic pressure P for trying to achieve_l, then obtain each several part liquid in battery State water volume fraction s,

Water distribution situation is brought into step (2), step (3) in the battery that step (4) is obtained, by aforementioned formula 2-1,3-1 and 3-2 can obtain ohmic loss and activation loss, bring formula 1-1 into, finally try to achieve the battery predictive output electricity of the one-dimensional model Pressure.

2. the method set up according to proton exchange film fuel battery performance forecast model described in claim 1, it is characterized in that：It is described 1+1+1 quasi-three-dimensional models comprising x directions perpendicular to pole plate direction, y directions along runner direction, z directions perpendicular to runner and floor, Three superpositions in direction, quasi-three-dimensional model its specific steps method for building 1+1+1 dimensions includes：

(1) foundation of x directions vertical plate direction one-dimensional model is identical with 4 steps described in claim 1,

(2) y directions along runner direction one-dimensional model foundation, its specific method step includes：

Two Battery packs section is connected in parallel, output voltage is identical, and output current density is differed, and battery is divided into two along runner direction Part,

Given first Battery pack section current density I_a, the first Battery pack section can be tried to achieve using 4 steps of the claim 1 Output voltage E_a,out,

E_b,out=E_a,out5-1

Second Battery pack section current density I_b, tried to achieve by following steps：

${\mathrm{\&eta;}}_{act,ano}=\frac{RT}{\mathrm{\&alpha;}nF}{\cosh }^{-1}\&lsqb;\frac{I^2}{4{\mathrm{\&sigma;}}_m^{eff2}\left(\frac{{\mathrm{\&sigma;}}_m^{eff}+{\mathrm{\&sigma;}}_s^{eff}}{{\mathrm{\&sigma;}}_m^{eff}\&CenterDot;{\mathrm{\&sigma;}}_s^{eff}}\right)\frac{RT}{\mathrm{\&alpha;}nF}{j}_{0,ref}^{ano}\left(\frac{c_{H_2}}{c_{H_2,ref}}\right)}+1\&rsqb;---5-2$

$\begin{array}{c}{\mathrm{\&eta;}}_{act,cat}=E_{rev}-E_{out}-{\mathrm{\&eta;}}_{ohm}-{\mathrm{\&eta;}}_{act,ano}---5-3\\ I=\sqrt{\left(\cosh \right({\mathrm{\&eta;}}_{act,cat}\frac{\mathrm{\&alpha;}nF}{RT})+1)4{\mathrm{\&sigma;}}_m^{eff2}\left(\frac{{\mathrm{\&sigma;}}_m^{eff}+{\mathrm{\&sigma;}}_s^{eff}}{{\mathrm{\&sigma;}}_m^{eff}\&CenterDot;{\mathrm{\&sigma;}}_s^{eff}}\right)\frac{RT}{\mathrm{\&alpha;}nF}{j}_{0,ref}^{cat}\left(\frac{c_{O_2}}{c_{O_2,ref}}\right)}---5-4\end{array}$

WhereinWhat is represented respectively is oxygen in density of hydrogen and cathode catalysis layer in the second Battery pack section anode catalyst layer Concentration, it is assumed that current density is I_assume,

Anode catalyst layer density of hydrogen：

$c_{CL}^{H_2}=\frac{c_{MPL-CL}^{H_2}+c_{CL-PEM}^{H_2}}{2}---5-6$

Cathode catalysis layer oxygen concentration：

$c_{CL}^{O_2}=\frac{c_{MPL-CL}^{O_2}+c_{CL-PEM}^{O_2}}{2}---5-8$

On boundary condition, the first Battery pack section anode export density of hydrogen is the second Battery pack section import density of hydrogen, first group Cell section cathode outlet oxygen concentration is the second Battery pack section import oxygen concentration, willBring 5-2 into and try to achieve η_act,ano, will η_act,anoBring 5-3 into and try to achieve η_act,cat, current density I can be obtained by 5-4_solve,

WhenWhen, I_solveThe the second Battery pack section current density for as being solved,

(3) in the z-direction perpendicular to runner, the foundation of floor direction one-dimensional model, its specific method step includes：

Battery be divided into the z-direction runner lower section the first Battery pack section and floor lower section the 3rd Battery pack section, the first Battery pack section and 3rd Battery pack section is connected in parallel, and output voltage is identical,

The current density I of the first Battery pack section_a, its output voltage E can be tried to achieve using 4 steps of the claim 1_a,out,

E_c,out=E_a,out 6-1

The current density I of the 3rd Battery pack section_c, tried to achieve by following steps：

${\mathrm{\&eta;}}_{act,ano}=\frac{RT}{\mathrm{\&alpha;}nF}{\cosh }^{-1}\&lsqb;\frac{I^2}{4{\mathrm{\&sigma;}}_m^{eff2}\left(\frac{{\mathrm{\&sigma;}}_m^{eff}+{\mathrm{\&sigma;}}_s^{eff}}{{\mathrm{\&sigma;}}_m^{eff}\&CenterDot;{\mathrm{\&sigma;}}_s^{eff}}\right)\frac{RT}{\mathrm{\&alpha;}nF}{j}_{0,ref}^{ano}\left(\frac{c_{H_2}}{c_{H_2,ref}}\right)}+1\&rsqb;---6-2$

η_act,cat=E_rev-E_out-η_ohm-η_act,ano 6-3

$I=\sqrt{\left(\cosh \right({\mathrm{\&eta;}}_{act,cat}\frac{\mathrm{\&alpha;}nF}{RT})+1)4{\mathrm{\&sigma;}}_m^{eff2}\left(\frac{{\mathrm{\&sigma;}}_m^{eff}+{\mathrm{\&sigma;}}_s^{eff}}{{\mathrm{\&sigma;}}_m^{eff}\&CenterDot;{\mathrm{\&sigma;}}_s^{eff}}\right)\frac{RT}{\mathrm{\&alpha;}nF}{j}_{0,ref}^{cat}\left(\frac{c_{O_2}}{c_{O_2,ref}}\right)}---6-4$

WhereinWhat is represented respectively is oxygen in density of hydrogen and cathode catalysis layer in the 3rd Battery pack section anode catalyst layer Concentration, the 3rd Battery pack section Catalytic Layer reaction gases include, by the section microporous layers diffusion of the 3rd Battery pack and the first Battery pack section Catalytic Layer spreads,

3rd Battery pack section anode catalyst layer density of hydrogen：

$\frac{(c_{HPL-CL,c}^{H_2}-c_{CL-PEM,c}^{H_2})D_{H_2,CL}^{eff}}{{\mathrm{\&delta;}}_{CL}}+\frac{(c_{CL,a}^{H_2}-c_{CL,c}^{H_2})D_{H_2,CL}^{eff}}{d}=\frac{I_{assume}}{2F}---6-5$

$c_{CL,c}^{H_2}=\frac{c_{MPL-CL,c}^{H_2}+c_{CL-PEM,c}^{H_2}}{2}---6-6$

3rd Battery pack section cathode catalysis layer oxygen concentration：

$\frac{(c_{MPL-CL,c}^{O_2}-c_{CL-PEM,c}^{O_2})D_{O_2,CL}^{eff}}{{\mathrm{\&delta;}}_{CL}}+\frac{(c_{CL,a}^{O_2}-c_{CL,c}^{O_2})D_{O_2,CL}^{eff}}{d}=\frac{I_{assume}}{4F}---6-7$

$c_{CL,c}^{O_2}=\frac{c_{MPL-CL,c}^{O_2}+c_{CL-PEM,c}^{O_2}}{2}---6-8$

WhereinThe 3rd Battery pack section microporous layers are represented respectively and Catalytic Layer is handed over The hydrogen and oxygen concentration of interface, Catalytic Layer and PEM interface；The 3rd Battery pack section is represented respectively to urge Change hydrogen and oxygen mean concentration in layer；It is average that hydrogen and oxygen in the first Battery pack section Catalytic Layer are represented respectively Concentration, the first Battery pack section Catalytic Layer reaction gases concentration is tried to achieve by (3.2) step in the claim 1,

WillBring 6-2 into and try to achieve η_act,ano, by η_act,anoBring 6-3 into and try to achieve η_act,cat, current density I can be obtained by 6-4_solve,

WhenWhen, I_solveIt is the 3rd Battery pack section current density for being solved.