Method and apparatus for predicting current-voltage characteristics

Abstract

This invention provides a method for predicting current-voltage characteristics of a fuel cell equipped with a catalyst composed of any material, enabling easy and highly accurate prediction of its current-voltage characteristics. [Solution] The present invention provides a method for predicting current-voltage characteristics, comprising: a first step of performing RRDE measurement to obtain the measured value of the electrode current; a second step of solving equation (1) below, which relates to the concentration distribution of oxygen molecules and hydrogen peroxide molecules on the electrode surface of the fuel cell, to obtain the theoretical value of the electrode current; a third step of mathematically optimizing the theoretical value so that the difference between the measured value and the theoretical value is small; and a fourth step of substituting the mathematically optimized theoretical value into the Butler-Volmer equation to obtain the activation voltage under conditions where the electrode reaction is in equilibrium, and thereby determining the current-voltage characteristics. The theoretical value obtained in the second step is the sum of the electrode currents for each active site. TIFF2026075784000041.tif14170 (C is the concentration distribution, D is the diffusion coefficient, and K is the reaction rate constant.)

Description

[Technical Field]

[0001] This invention relates to a method and apparatus for predicting current-voltage characteristics. [Background technology]

[0002] Fuel cells are attracting attention as power generation devices that have a low environmental impact and high power generation efficiency. Fuel cells have the function of chemically reacting hydrogen and oxygen and directly converting them into electrical energy. Fuel cells are mainly composed of an electrolyte membrane, a catalyst layer, and a gas diffusion layer, and the performance of the catalyst contained in the catalyst layer in particular greatly affects the current-voltage characteristics of the fuel cell. Conventional fuel cells use platinum-supported catalysts (Pt / C) (Patent Document 1, etc.), but from the viewpoint of cost and resources, there is a demand for lower platinum content or even non-platinum content in the catalyst layer.

[0003] There are two main methods for evaluating the characteristics of fuel cells. One is the measurement using a membrane electrode assembly (MEA) and supplying hydrogen and oxygen (IV measurement). This measurement is similar to that of a real fuel cell, but the procedure is complicated and requires a large amount of sample. The other is the measurement using a rotating ring-disk electrode in an oxygen-saturated solution (RRDE measurement). The measurement procedure using a rotating ring-disk electrode is simple and can be performed with a small amount of sample, but it is difficult to perform with high accuracy. [Prior art documents] [Patent Documents]

[0004] [Patent Document 1] Japanese Patent Publication No. 2017-045549 [Overview of the project] [Problems that the invention aims to solve]

[0005] The present invention has been made in view of the above circumstances, and aims to provide a method and apparatus for predicting current-voltage characteristics that enables easy and highly accurate prediction of the current-voltage characteristics of a fuel cell equipped with a catalyst made of any material. [Means for solving the problem]

[0006] To solve the above problems, the present invention employs the following means.

[0007] [1] A method for predicting current-voltage characteristics according to one aspect of the present invention is a method for predicting current-voltage characteristics of a fuel cell, comprising: a first step of performing an RRDE measurement simulating the fuel cell and obtaining measured values ​​of the electrode current density of the disk electrode and the electrode current of the ring electrode generated by the electrode reaction; a second step of solving the following equation (1) relating to the concentration distribution of oxygen molecules and hydrogen peroxide molecules on the electrode surface of the RRDE measurement and obtaining theoretical values ​​of the electrode current density and electrode current from the obtained concentration distribution; a third step of mathematically optimizing the theoretical value so that the difference between the measured value and the theoretical value is less than or equal to a predetermined magnitude; and a fourth step of substituting the theoretical value after mathematical optimization into the Butler-Volmer equation to obtain the activation voltage under conditions in which the electrode reaction is in equilibrium and to obtain the current-voltage characteristics, wherein the theoretical value obtained in the second step is the sum of the electrode current density and electrode current, which correspond to the energy change for each active site.

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[0009] C is the concentration distribution of oxygen molecules and hydrogen peroxide molecules in the catalyst layer, D is the diffusion coefficient of oxygen molecules and hydrogen peroxide molecules, and K is the reaction rate constant of the catalyst layer formed on the electrode surface.

[0010] [2] In the method for predicting current-voltage characteristics described in [1] above, the concentration distribution C, the diffusion coefficient D, and the reaction rate constant K can each be expressed by the following equations (2) to (4).

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[0014] C O2 , C H2O2 These represent the concentration distributions of oxygen molecules and hydrogen peroxide molecules in the catalyst layer, respectively. O2 , D H2O2 K2, K3, and K4 represent the diffusion coefficients of oxygen molecules and hydrogen peroxide molecules, respectively. K2, K3, and K4 represent the reaction rate constants of the catalyst layer in the hydrogen peroxide molecule synthesis reaction, the water molecule synthesis reaction, and the hydrogen peroxide molecule decomposition reaction, respectively.

[0015] [3] In the method for predicting current-voltage characteristics described in either [1] or [2] above, it is preferable that the energy change for each active site is the energy change for two or more active sites with different activity levels in the catalyst layer.

[0016] [4] In the method for predicting current-voltage characteristics described in any one of [1] to [3] above, it is preferable to multiply the Butler-Volmer equation in the fourth step by the ratio of the amount of catalyst to the catalyst density.

[0017] [5] In the method for predicting current-voltage characteristics described in any one of [1] to [4] above, it is preferable that the mathematical optimization in the third step is performed such that the error index ΔS, expressed by the following equation (5), is -1.1 or less in the electrode current density of the disk electrode and within the range of -0.9 or less in the electrode current of the ring electrode.

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[0019] I M , I C These are the measured and theoretical values ​​of the electrode current density of the disk electrode or the electrode current of the ring electrode in the RRDE measurement, respectively. The sum is the potential condition E within the measured potential range. met To do so against.

[0020] [6] In the method for predicting current-voltage characteristics described in any one of [1] to [5] above, it is preferable that in the fourth step, the decrease due to concentration overvoltage and resistance overvoltage is subtracted from the activation voltage of the fuel cell.

[0021] [7] In the method for predicting current-voltage characteristics described in any one of [1] to [6] above, a first database may be created for the relationship between the amount of catalyst and the concentration overpotential, and the concentration overpotential may be predicted by referring to the first database.

[0022] [8] In the method for predicting current-voltage characteristics described in any one of [1] to [7] above, a second database may be created regarding the relationship between the ratio of the amount of ionomer to the amount of catalyst in the catalyst layer and the resistive overvoltage, and the resistive overvoltage may be predicted by referring to the second database.

[0023] [9] A current-voltage characteristic prediction device according to one aspect of the present invention is a current-voltage characteristic prediction device used in a current-voltage characteristic prediction method described in any one of [1] to [8] above, comprising: an input device that inputs measured values ​​of the electrode current density of the disk electrode and the electrode current of the ring electrode generated by the electrode reaction using the RRDE measurement; a first calculation device that solves equation (1) above and obtains theoretical values ​​of the electrode current density and electrode current from the obtained concentration distribution; a second calculation device that mathematically optimizes the theoretical values ​​so that the difference between the measured values ​​and the theoretical values ​​is less than or equal to a predetermined magnitude; and a third calculation device that substitutes the mathematically optimized theoretical values ​​into the Butler-Volmer equation and derives the relationship of current-voltage characteristics under conditions in which the electrode reaction is in equilibrium. [Effects of the Invention]

[0024] According to the present invention, it is possible to provide a method and apparatus for predicting current-voltage characteristics that enable easy and highly accurate prediction of the current-voltage characteristics of a fuel cell equipped with a catalyst composed of any material. [Brief explanation of the drawing]

[0025] [Figure 1] This is a process flow diagram of a method for predicting current-voltage characteristics according to one embodiment of the present invention. [Figure 2] (a) A perspective view of the rotating ring-disk electrode of the same embodiment. (b) A diagram illustrating measurement using the rotating ring-disk electrode of the same embodiment. [Figure 3] This figure illustrates the reaction-diffusion model used in the same embodiment. [Figure 4] (a) and (b) are graphs showing the current-potential curves (LSV) of the disk electrode and ring electrode obtained when the multi-active-point model is applied as Example 1. [Figure 5] (a) and (b) are graphs showing the LSV of the disk electrode and ring electrode obtained when the single-active-point model is applied as Comparative Example 1. [Figure 6](a) and (b) are graphs showing the LSV of the disk electrode and ring electrode obtained when applying the multi-active site model as Example 2. [Figure 7] (a) and (b) are graphs showing the LSV of the disk electrode and ring electrode obtained when the single-active-point model is applied, as Comparative Example 2. [Figure 8] (a) A graph showing the predicted and measured current-voltage characteristics of a fuel cell using a carbon alloy catalyst when Example 1 is applied. (b) A graph showing the predicted and measured current-voltage characteristics of a fuel cell using a platinum catalyst when Example 2 is applied. [Figure 9] (a) to (c) are graphs showing the current-voltage characteristics of fuel cells equipped with the catalyst layers of Examples 3 to 5, respectively. [Figure 10] (a) to (c) are graphs showing the current-voltage characteristics of fuel cells equipped with catalyst layers according to Examples 6, 3, and 7, respectively. [Modes for carrying out the invention]

[0026] The following describes in detail, with reference to the drawings, a method and apparatus for predicting current-voltage characteristics according to an embodiment to which the present invention is applied. Note that, for convenience, the drawings used in the following description may show enlarged versions of key features to make them easier to understand, and the dimensions, proportions, etc., of each component may not be the same as those in reality. Furthermore, the materials, dimensions, etc., exemplified in the following description are merely examples, and the present invention is not limited to them. It can be implemented with appropriate modifications without altering its essence.

[0027] <Method for predicting current-voltage characteristics> Figure 1 shows a flowchart of the steps involved in a method for predicting the current-voltage characteristics of a fuel cell according to one embodiment of the present invention. The method for predicting current-voltage characteristics mainly comprises a first, second, and third step of determining the reaction rate constant by mathematical optimization, and a fourth step of calculating the activation voltage using the determined reaction rate constant and determining the current-voltage characteristics.

[0028] (First step) Perform RRDE measurement simulating a fuel cell to obtain the measured values of the electrode currents (disk current I DM , ring current I RM ) generated by the electrode reaction. The measured values obtained by RRDE measurement can be measured, for example, at room temperature, etc., and may be measured at only one temperature, but in order to consider the influence of temperature, the experimental temperature may be changed and measured multiple times.

[0029] Fig. 2(a) is a perspective view of the rotating ring-disk electrode 100. The rotating ring-disk electrode 100 mainly includes a cylindrical disk electrode 101, a cylindrical ring electrode 102, and two insulating films 103, 104. The disk electrode 101 and the ring electrode 102 are arranged such that the center lines P are aligned and they are spaced apart from each other. The insulating film 103 is arranged so as to fill the space between the outer surface of the disk electrode 101 and the inner surface of the ring electrode 102. The insulating film 104 is arranged so as to cover the outer surface of the ring electrode 102.

[0030] Fig. 2(b) is a diagram for explaining the measurement using the rotating ring-disk electrode 100. The disk electrode 101 and the ring electrode 102 are brought into contact with the oxygen-saturated solution 105. A power source (not shown) is connected to each of the disk electrode 101 and the ring electrode 102. While rotating the rotating ring-disk electrode 100 about the center line P as an axis, the voltages (potentials) of the disk electrode 101 and the ring electrode 102 are controlled. The catalyst for the fuel cell to be evaluated can be attached to the surface of the disk electrode 101, etc.

[0031] The oxygen-saturated solution 105 contacting the surface of each electrode flows outward from the center of rotation due to the influence of the centrifugal force caused by rotation. Along with this, more oxygen-saturated solution 105 flows into the vicinity of the center of rotation. At the disk electrode 101, oxygen is reduced by the catalytic reaction to generate water and hydrogen peroxide, and at the same time, the disk current I DM flows. Also, at the ring electrode 102, a part of the hydrogen peroxide generated at the disk electrode 101 is oxidized, and at the same time, the ring current I RMA current flows. For each potential applied to the disk electrode 101 and the ring electrode 102, the disk current I that flows flows. DM and ring current I RM Measure and obtain the current-potential curve (LSV).

[0032] (Second process) Figure 3 is a diagram illustrating the reaction-diffusion model used in the analysis of the second step. The reaction-diffusion model is a model in which the catalyst layer has a finite thickness and introduces a diffusion process that occurs simultaneously with the reaction within the catalyst layer. 2,b , O 2,s , O 2,a These represent oxygen molecules located away from the catalyst layer (bulk), oxygen molecules on the surface of the catalyst layer, and oxygen molecules inside the catalyst layer, respectively. 2,b H2O 2,s H2O 2,a These represent hydrogen peroxide molecules located away from the catalyst layer (bulk), hydrogen peroxide molecules on the surface of the catalyst layer, and hydrogen peroxide molecules inside the catalyst layer, respectively. The reaction-diffusion equation reflecting this model is given by equation (1) below. In equation (1) below, O on the right-hand side represents a 2x1 zero matrix.

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[0034] In equation (1) above, C is the matrix of the concentration distribution of oxygen molecules and hydrogen peroxide molecules in the catalyst layer, and is expressed by equation (2) below. The component of C is C O2 , C H2O2 These represent the concentration distributions of oxygen molecules and hydrogen peroxide molecules in the catalyst layer, respectively.

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[0036] In equation (1) above, D is the matrix of diffusion coefficients for oxygen molecules and hydrogen peroxide molecules, and is expressed by equation (3) below. The components of D are DO2 , D H2O2 These represent the diffusion coefficients of oxygen molecules and hydrogen peroxide molecules, respectively.

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[0038] In equation (1) above, K is a matrix of reaction rate constants of the catalyst layer formed on the electrode surface, taking into account the sequential reaction via hydrogen peroxide, and is expressed by equation (4) below. The sequential reaction via hydrogen peroxide refers to a series of reactions in which hydrogen peroxide is produced by the reaction of hydrogen ions and oxygen, and water is produced by the reaction of this hydrogen peroxide with hydrogen ions. Each step of the sequential reaction via hydrogen peroxide is a two-electron reaction. The components K2, K3, and K4 of K represent the reaction rate constants of the catalyst layer in the reaction of hydrogen peroxide molecule synthesis, water synthesis, and hydrogen peroxide molecule decomposition, respectively.

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[0040] Solving the reaction-diffusion equation in equation (1) above for concentration yields a general solution relating to the concentration on the catalyst layer surface. Using this general solution, and setting a hypothetical reaction rate constant K, we can solve equation (5) or (9) below as boundary conditions, depending on the desired reactant concentration, to obtain the concentration C on the catalyst layer surface. i,O2,s , C i,H2O2,s This allows us to determine the following. As provisional values ​​for each reaction rate constant K2, K3, and K4, for example, we can use values ​​predicted in advance, or arbitrary values ​​generated randomly.

[0041] Here, i in equations (5) and (9) represents the i-th active site, assuming that there are m active sites in the catalyst layer exhibiting different activities. When m=1, all active sites in the catalyst layer exhibit the same activity, and there is only one type of reaction rate constant K represented by equation (4). When m=2 or greater, there are active sites in the catalyst layer exhibiting different activities, and there are m types of reaction rate constants K represented by equation (4). In this specification, the case where m=1 is called the single active site model, and the case where m=2 or greater is called the multi-active site model. When using the multi-active site model, the number of active sites m is not particularly limited as long as m=2 or greater; for example, m=2, or m=3 or greater. Equation (1) above is applied to each of the m active sites.

[0042] In actual electrodes, multiple active sites with different levels of activity exist due to differences in the microstructure of the catalyst layer. Therefore, using a multi-active-site model allows for the analysis of electrode reactions under conditions closer to those of the real environment. When using a multi-active-site model, a larger number of active sites allows for more detailed analysis and more accurate predictions, but it also increases the computational load. Therefore, the number of active sites can be set appropriately depending on the computer being used.

[0043] Equations (5) and (9) below are boundary conditions for determining the concentrations of oxygen and hydrogen peroxide on the catalyst surface, respectively. By applying them to the general solution of equation (1) above, the concentration C of each component on the catalyst surface can be determined. i,s , and a special solution to equation (1) above is obtained. By applying the obtained special solution to equation (1) above to the right-hand side of equations (13) and (14) below, the average concentration C of each component in the catalyst layer is obtained. i,a This can be obtained.

[0044] C in equations (5) and (9) below i,b , C i,s , C i These represent the concentration in the bulk, the concentration on the catalyst layer surface, and the concentration distribution within the catalyst layer at the active site i, respectively, and the concentration distribution within the catalyst layer C. i This is a function that has a distribution in the thickness direction of the catalyst layer. The concentration C in the bulk i,b L may be the saturation concentration in the solution used for RRDE measurement.cat This is the thickness of the catalyst layer in the RRDE measurement, K i Here, L is the reaction rate constant at the active site i. cat The amount of catalyst used is m cat ρ cat It can be found by dividing by . Z i,O2 , Z i,H2O2 These can be determined by equations (6) and (10) below. In equation (6) below, Z * O2 d c,O2 These are coefficients obtained by equations (7) and (8) below, respectively, and Z in equation (10) below * H2O2 d c,H2O2 These can be calculated using equations (11) and (12) below. d c d is the thickness of the boundary film, which depends on the rotation speed of the electrodes. See equations (6) and (10) below. i d is the distance between active sites that exhibit the same activity, and the distance d between active sites is the distance between active sites. i You can set any value as the initial value. D in equations (7) and (8) below O2 is the diffusion coefficient of oxygen, and D in equations (11) and (12) below. H2O2 is the diffusion coefficient of hydrogen peroxide. In equations (7) and (11) below, ν represents the bulk viscosity, and ω is the rotational speed of the electrode.

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[0055] Concentration C on the catalyst surface i,O2,s , C i,H2O2,s The average concentration C in the catalyst layer obtained by determining this is i,O2,a , C i,H2O2,a Substitute the reaction rate constants K2, K3, and K4 into equation (15) below to obtain the disk current density I DC We will determine the theoretical value of C. Also, the hydrogen peroxide concentration C on the catalyst layer surface. i,H2O2,s Substitute this into equation (16) below to obtain the ring current I RC We will find the theoretical value of . The sum of equations (15) and (16) below is performed for the number of active sites i=1 to m, respectively. That is, I DC , I RC These are the sum of the electrode current density and electrode current corresponding to the energy change at each active site, respectively. F is the Faraday constant, N is a constant determined by the electrode shape, A is the electrode area, and Z* H2O2 Z represents a constant determined by viscosity and diffusion coefficient. * H2O2 This can be determined from equation (11) above. K1 is the reaction rate constant corresponding to the four-electron reaction in which water is synthesized without the synthesis of hydrogen peroxide, but here it is set to 0.

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[0058] (Third step) In the first step, the measured values ​​of electrode current density and electrode current are compared with the theoretical values ​​of electrode current density and electrode current obtained in the second step. Mathematical optimization (fitting) of the theoretical values ​​is performed so that the difference between the measured and theoretical values ​​is less than or equal to a predetermined magnitude. Specifically, the absolute value of the difference between the measured and theoretical values ​​of disk current density |I DM -I DC | The absolute value of the difference between the measured and theoretical values ​​of the ring current | I RM -I RC | The reaction rate constant K and the distance d between active sites are such that each of them is less than or equal to a predetermined size. i Adjust the values ​​of the reaction rate constant K and the distance d between active sites. i Reset the settings and solve the reaction-diffusion equation (equation (1) above), and the disk current density I DC Ring current I RC The second step in calculating |I DM -I DC |, |I RM -I RC Repeat until the | symbol is less than or equal to a predetermined size.

[0059] The mathematical optimization of theoretical values ​​can be performed using evolutionary computation. This may involve performing a convergence check to determine if the evolutionary computation has approached a specific value, and then performing an error check to determine the difference between the theoretical value obtained by the evolutionary computation and the measured value.

[0060] When using evolutionary computation, convergence is determined to have been achieved when the coefficient of variation of the objective function r, expressed by equation (17) below, falls below a predetermined value. The coefficient of variation is obtained by dividing the standard deviation of the set of objective function values ​​at each search point in the evolutionary computation by the absolute value of the mean of the objective function values. For example, convergence can be determined when the coefficient of variation is 0.01 or less.

[0061] Evolutionary computation is an optimization method that mimics the evolutionary process of organisms in nature, based on Darwin's theory of evolution, and can utilize genetic algorithms. Specifically, it first creates a set of candidate search points randomly using methods such as the Latin rectangular grid, and calculates the objective function value. Next, evolutionary computation operations such as "variation and crossover" are performed to create a new set of candidate solutions based on the values ​​of the set of search points, and calculate the objective function value for this new set. Then, the corresponding search points and the new candidate solutions are compared one by one, and "selection" is performed so that the one with the smaller objective function value survives and the other is eliminated, determining the next set of search points. "Variation and crossover" and "selection" are repeated, and when the coefficient of variation of the objective function r for the set of search points falls within a predetermined range, it is determined that convergence has occurred and the calculation is terminated.

[0062] In equation (17) below, r is the objective function value, A is the electrode area, and c is the correction coefficient. Here, the correction coefficient c is the value obtained by dividing the ratio of the area under the line to the potential of the current-potential curves (LSV) of the disk electrode and the ring electrode by the capture rate, which represents the proportion of the solution that moves from the disk due to the centrifugal force of rotation and reaches the ring. The capture rate is a constant specific to the electrode, and a value obtained by calculation using the electrode shape factor or by actual measurement can be used as the capture rate.

[0063] E∈E in equation (17) below objis the set of potentials E used in the objective function obj (It means taking the sum over the potential E belonging to the potential condition E obj ) and can be appropriately set within the potential range measured in the first step. Specifically, calculation targets can be set at equal intervals for a part or all of the potential range measured in the first step, and E obj can be determined. Also, when plotting the measured values obtained by the measurement in the first step, in the potential range where the current value changes significantly, calculation targets should be set at finer intervals than in the range where the change is small, and E obj is preferably used. For example, when measuring from 0.02V to 1.2V in the first step, as E obj , a total of 20 points at intervals of 0.05V can be set in the range of 0.1V to 1.05V, or the range of 0.1V to 0.8V can be at intervals of 0.05V, the range of 0.8V to 0.9V can be at intervals of 0.01V, and the range of 0.9V to 1.05V can be a total of 28 points at intervals of 0.05V.

[0064] The error determination for judging the difference between the theoretical value obtained by evolutionary calculation and the measured value of RRDE measurement can be made by determining that the error index ΔS obtained by the following formula (18) is below a predetermined magnitude. For example, the error index ΔS D for the disk current density and the error index ΔS R for the ring current are such that ΔS D is -1.1 or less and ΔS R is - 0.90 or less. It is preferable that ΔS D is -1.2 or less and ΔS R is -1.0 or less. It is more preferable that ΔS D is -1.3 or less and ΔS R is -1.1 or less. It is even more preferable that both ΔS D and ΔS R are -1.3 or less. If within the above range, it can be judged that the theoretical value determined to converge by the following formula (17) has a small error from the measured value and is optimized, and the current-voltage characteristics of the fuel cell can be predicted accurately.

[0065] E ∈ E in the following formula (18) metThe set of potentials E used as error indicators met (Potential condition E met ) means taking the sum over the potential E belonging to it, and it can be appropriately set within the potential range measured in the first step. Specifically, the points obtained by equally dividing the range of 0.1 - 1.05V can be E met , and for example, a total of 20 points can be set at intervals of 0.05V in the range of 0.1V - 1.05V. Since it may not be possible to correctly evaluate the error if the number of potentials used in the calculation of the error indicator is small, it is preferable to divide the potential range measured in the first step into 20 or more points at equal intervals. The number of potentials used in the calculation of the error indicator is not particularly limited because even if it is large, it does not affect the evaluation of the error. The potential condition E of the objective function obj and the potential condition E of the error indicator met may be the same or different. [[ID=1�]]

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[0068] (Fourth step) The reaction rate constant K for each active site after mathematical optimization i,2 , K i,3 , K i,4 Substitute the theoretical values of into the Butler - Volmer equation respectively, and obtain the activation voltage V act under the condition that the electrode reaction reaches equilibrium. Since this method does not require information on the type of catalyst, it can be applied to any catalyst. By plotting the obtained activation voltage against the current, a theoretical IV curve showing the current - voltage characteristics can be obtained.

[0069] In this specification, in the Butler-Volmer equation described above, the relationship between the current density value obtained by each reaction and the reaction rate constant can be expressed as shown in equations (19) to (21) below. Furthermore, the relationship between the cell current density value under conditions where the electrode reaction is in equilibrium and the current density value obtained by each reaction can be described as shown in equation (24) below.

[0070] In equations (19) to (21) below, I i,2 , I i,3 , I i,4 is the exchange current density, and K i,2 , K i,3 , K i,4 C is the reaction rate constant after mathematical optimization. i,O2,fc , C i,H2O2,fc θ is the concentration of oxygen and hydrogen peroxide, respectively, n is the number of reaction electrons, α is the electrochemical mobility, R is the gas constant, T is the cell temperature, E 0 Here, C is the redox potential, and E is the electrode potential. Also, i is an integer from 1 to m representing the active site. i,O2,fc , C i,H2O2,fc This is calculated using equations (22) and (23) below. P in equations (22) and (23) below O2 This is the partial pressure of oxygen and can be calculated from the oxygen supply rate and the back pressure of the fuel cell.

[0071] In equations (19) to (21) below, m cat ρ represents the amount of catalyst in the electrode, cat Here, ρ represents the catalyst density. cat ρ is a value determined from the elemental ratio, pore volume, and porosity of the catalyst, and platinum-supported catalysts have a larger value compared to carbon alloy catalysts. cat m for cat The ratio is proportional to the thickness of the catalyst layer, and by multiplying this ratio by the reaction rate constant K, it becomes possible to calculate a rate constant that depends on the amount of catalyst.

[0072] In equation (24) below, I cell is the cell current density, and I i,2 , I i,3 , I i,4 This is the exchange current density at the active point i calculated by equations (19) to (21) above, and IxH2 is the hydrogen cross-leakage current density. The sum in equation (24) is performed for m different active sites. Here, the hydrogen cross-leakage current density can be a value obtained by methods such as calculating it from the hydrogen permeability of the electrolyte membrane or by measuring it using MEA.

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[0079] The cell voltage is calculated from the fuel cell's activation voltage, and the concentration overvoltage η is calculated from that. con It is preferable to subtract the reduction due to [the following factor]. Concentration overpotential is an energy loss caused by a deficiency of reactants in the electrode reaction, and it depends on the transport characteristics of the reactants and varies greatly depending on the electrode material and cell structure. Furthermore, the effect of concentration overpotential becomes larger in the high-current region. Concentration overpotential η con This can be calculated, for example, using equation (25) below. cell , I dl These represent the cell current density and the critical diffusion current density, respectively. αc This value is known as the transfer coefficient of cathode concentration overvoltage, and its range is 0 to 1. For specific values, see, for example, the reference (G. Inoue et al., Journal of Power Sources, 154 (2006), 18-34, DOI: 10.1016 / j.jpowsour.2005.03.216.).

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[0081] The cell voltage is calculated from the fuel cell's activation voltage, which is the resistive overvoltage η. ohm It is preferable to subtract the reduction due to the following. Resistive overpotential is the energy loss required for the movement of charge itself within the fuel cell, and its effect becomes larger in the intermediate range between the low current region and the high current region. Resistive overpotential η ohm This can be calculated, for example, using equations (26) to (29) below. Equation (29) is Bruggeman's equation.

[0082] In equation (26) below, t cat σ is the thickness of the catalyst layer of the MEA, and σ is the conductivity. In equation (26) below, f = e - This indicates that a catalyst is used as a filler, and f=H + This indicates the use of an ionomer as a filler. In equation (27) below, σ e- , σ e-,i , σ e-,s , T i T is the electron conductivity, a constant for the temperature-independent term related to electron conduction, a constant for the temperature-dependent term related to electron conduction, the reference temperature, and the cell temperature, respectively. In equation (28) below, σ H+ , σ H+,s , σ H+,iλ is the conductivity of protons, a constant for the humidity-dependent term related to proton conduction, a constant for the humidity-independent term related to proton conduction, and a humidity-dependent variable, water activity, respectively. In equation (29) below, V f , σ f These are the volume of the filler and the conductivity of the filler, respectively, and σ m This represents the conductivity of the matrix.

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[0087] Furthermore, regarding concentration overpotential, a first database may be created regarding the relationship between catalyst amount and concentration overpotential, and predictions may be made by referring to this first database. Also, regarding resistance overpotential, a second database may be created regarding the relationship between the ratio of ionomer amount to catalyst amount in the catalyst layer (I / C ratio) and resistance overpotential, and predictions may be made by referring to this second database.

[0088] <Current-voltage characteristic prediction device> The current-voltage characteristic prediction device used in the current-voltage characteristic prediction method described above mainly comprises an input device, a first calculation device, a second calculation device, and a third calculation device. The input device has the function of inputting measured values ​​of the electrode current density of the disk electrode and the electrode current of the ring electrode generated by the electrode reaction using RRDE measurement. The first calculation device has the function of solving equation (1) above and obtaining the theoretical values ​​of the electrode current density and electrode current from the obtained concentration distribution. The second calculation device has the function of mathematically optimizing the theoretical values ​​so that the difference between the theoretical values ​​and the measured values ​​is less than or equal to a predetermined magnitude. The third calculation device has the function of substituting the mathematically optimized theoretical values ​​into the Butler-Volmer equation and deriving the relationship of current-voltage characteristics under conditions in which the electrode reaction is in equilibrium.

[0089] As described above, the current-voltage characteristic prediction method of this embodiment determines the current-voltage characteristics of a fuel cell using a current calculated with a reaction rate constant that considers sequential reactions and a molecular concentration that considers three-dimensional diffusion. The calculation of the current does not require information about the catalyst material, nor does it require any special work or equipment. Therefore, the current-voltage characteristic prediction method of this embodiment can easily and accurately predict the current-voltage characteristics of a fuel cell equipped with a catalyst composed of any material.

[0090] Furthermore, in the current-voltage characteristic prediction method of this embodiment, a theoretical analysis formula exists, making it possible to easily and accurately detect the optimal conditions for maximizing the output of the fuel cell by adjusting various parameters. [Examples]

[0091] The effects of the present invention will be made clearer by the following examples. However, the present invention is not limited to the following examples and can be modified as appropriate without altering its essence.

[0092] The various measurement methods and fuel cell configuration used in this embodiment are shown.

[0093] [RRDE measurement] In the first step of the above embodiment, the measurement using a rotating ring-disk electrode (RRDE measurement) was evaluated using a rotating ring-disk electrode device (RRDE-3A rotating ring-disk electrode device ver.1.2, manufactured by BAS Corporation) and a dual electrochemical analyzer (CHI700C, manufactured by ALS Corporation). The working electrode of the electrode device is prepared by mixing fuel cell cathode catalyst and 5% Nafion® (Nafion perfluorinated ion exchange resin, 5% dispersion (product number: 510211), manufactured by Sigma-Aldrich) in slurry form, with a catalyst content of 0.1 mg / cm³ per unit area of ​​the electrode. 2 To achieve this, the catalyst was applied to a working electrode (RRDE-3A ring disk electrode, platinum ring-gold disk electrode, disk diameter 4 mm, manufactured by BAS Corporation) and dried to produce a working electrode on which the catalyst was supported. A platinum electrode (Pt counter electrode 23cm, manufactured by BAS Corporation) was used as the counter electrode, and a reversible hydrogen electrode (RHE) (accumulative reversible hydrogen electrode, manufactured by EC Frontier Corporation) was used as the reference electrode. A 0.1M perchloric acid aqueous solution was used as the electrolyte.

[0094] Linear sweep voltammetry was performed under nitrogen and oxygen atmospheres using a three-electrode rotating ring disk electrode apparatus having a working electrode equipped with the above catalyst, a platinum electrode as a counter electrode, and a reversible hydrogen electrode (RHE) as a reference electrode.

[0095] First, nitrogen bubbling was performed in the electrolyte for 10 minutes to remove oxygen from the electrolyte. Then, the electrode was rotated at a rotation speed of 1600 rpm, and the current density was recorded as a current-potential curve (LSV), which is a function of potential, when a potential sweep was performed at a sweep speed of 20 mV / sec.

[0096] Next, oxygen bubbling was performed for 10 minutes to fill the electrolyte with saturated oxygen. Then, the electrodes were rotated at a rotation speed of 1600 rpm, and the current density was recorded as a current-potential curve (LSV), which is a function of potential, when a potential sweep was performed at a sweep speed of 20 mV / sec.

[0097] The measured current-potential (LSV) curve of the oxygen reduction reaction was obtained by subtracting the LSV under a nitrogen atmosphere from the LSV under an oxygen atmosphere. The obtained LSV values ​​for the oxygen reduction reaction were assigned signs so that the reduction current was negative and the oxidation current was positive.

[0098] [IV Measurement (Actual Measurement Value)] IV measurements were performed to determine the actual current-voltage characteristics of the fuel cell. First, to prepare the battery electrodes, the catalyst for each electrode and 5% Nafion® (Nafion perfluorinated ion exchange resin, 5% dispersion (product number: 510211), manufactured by Sigma-Aldrich) were prepared into a slurry, and the area of ​​the gas diffusion layer ("29BC", manufactured by SGL Carbon) was 5 cm². 2 A catalyst layer was formed on the gas diffusion layer by applying it to the specified area and drying it. A platinum-supported catalyst (UNPC40-II, manufactured by Ishifuku Metal Industries Co., Ltd.) was used as the anode catalyst.

[0099] A polymer electrolyte membrane (MEA) was fabricated by placing a solid polymer electrolyte membrane (Dupont, "NAFION® 211") between the battery electrode containing the cathode catalyst and the battery electrode containing the anode catalyst, and then pressing them together. A pair of gaskets were attached to this MEA, and then sandwiched between a pair of separators to fabricate a fuel cell single cell.

[0100] As described above, power generation tests were conducted on the single fuel cell prepared using an automated fuel cell evaluation system (manufactured by Toyo Technica Co., Ltd.), and measured values ​​of the current-voltage characteristics were obtained. The power generation test involved heating a fuel cell single cell to 70°C, supplying 2.5 L / min of oxidizing gas (air) with 100% RH relative humidity to the positive electrode side at a back pressure of 70 kPa, and supplying 1.0 L / min of fuel gas (hydrogen) with 100% RH relative humidity to the negative electrode side. After measuring the open-circuit voltage for 5 minutes, the cell current density was measured at 4.0 A / cm². 2 From 0 A / cm 2 The cell voltage was measured after holding each current density for 3 minutes.

[0101] [Fuel cell configuration (predicted values)] The materials for each component of the fuel cell necessary for predicting the current-voltage characteristics were set as follows. Components not listed below were configured in the same way as those used for the measured values ​​described above. Electrolyte membrane: Solid polymer electrolyte membrane (Dupont, "NAFION® 211") Catalyst layer: Carbon (catalyst) + 5% Nafion® (ionomer) Gas diffusion layer: ("29BC", manufactured by SGL Carbon)

[0102] (Example 1) In a fuel cell using a carbon alloy catalyst, the current-voltage characteristics when applying a multi-active-site model were predicted as follows.

[0103] In the first step of the above embodiment, a carbon alloy catalyst was used as the catalyst for the working electrode, and the above RRDE measurement was performed to obtain the measured value of LSV. The carbon alloy catalyst was manufactured by the method described in International Publication No. 2017 / 209244, etc. The conditions were as follows: catalyst amount m cat : 0.1 mg / cm³ 2 , catalyst density ρ cat : 2.2g / cm 3 A rotating ring disk electrode with an I / C ratio of 0.4 and a capture rate of 0.424 was used.

[0104] In the second step of the above embodiment, the initial values ​​of the reaction rate constants and the distance between active sites in the respective catalyst layers for the synthesis reaction of hydrogen peroxide molecules, the synthesis reaction of water molecules, and the decomposition reaction of hydrogen peroxide molecules were set to arbitrary values, and the theoretical value of LSV was obtained by solving the reaction-diffusion equation represented by equation (1). Here, it was assumed that there were 2 active sites (in equations (15) and (16), m=2 was set and the sum was taken for i=1 to 2). Diffusion coefficient of oxygen D O2 to 2 × 10 -5 (cm 2 ( / sec), diffusion coefficient D of hydrogen peroxide H2O2 1.43 × 10 -5 (cm 2 The calculation was performed using the formula ( / sec).

[0105] In the third step of the above embodiment, using the measured values ​​obtained in the first step and the theoretical values ​​obtained in the second step, mathematical optimization using evolutionary computation was performed so that the coefficient of variation of the objective function r in equation (17) would be 0.01 or less, thereby obtaining the reaction rate constants in the catalyst layer for the hydrogen peroxide molecule synthesis reaction, the water molecule synthesis reaction, and the hydrogen peroxide molecule decomposition reaction. The value of the error index ΔS expressed in equation (18) was -1.80 for the disk electrode and -1.56 for the ring electrode. The range for summing equations (17) and (18) was set to 20 points divided into 0.05V intervals within the range of 0.1 to 1.05V. Figures 4(a) and (b) are graphs showing the LSV of the disk electrode and ring electrode after mathematical optimization, respectively.

[0106] In the fourth step of the above embodiment, the current-voltage characteristics were predicted by solving the Butler-Volmer equation in equilibrium using the theoretical value of the reaction rate constant obtained in the third step. Furthermore, based on the fuel cell configuration described above, concentration overpotential and resistance overpotential were calculated and subtracted to obtain more detailed current-voltage characteristics. Figure 8(a) is a graph showing the predicted and measured values ​​obtained when applying Example 1 to the current-voltage characteristics of a fuel cell using a carbon alloy catalyst.

[0107] (Comparative Example 1) Similar to Example 1, RRDE measurement was performed in the first step of the above embodiment. In the second step, a single-active-site model with one active site was used to determine the theoretical LSV from the reaction-diffusion equation represented by equation (1) above, and mathematical optimization was performed in the third step. The error index ΔS was -1.08 for the disk electrode and -0.90 for the ring electrode. Figures 5(a) and (b) are graphs showing the LSV of the disk electrode and ring electrode after mathematical optimization, respectively.

[0108] By comparing the graphs in Figures 4 and 5, it can be seen that when a carbon alloy catalyst is used as the working electrode, applying the multi-active-site model provides a higher accuracy fit for the LSV of the disk electrode and ring electrode compared to applying the single-active-site model. This is also evident from the value of the error index ΔS.

[0109] (Example 2) The procedure was carried out in the same manner as in Example 1, except that a commercially available platinum-supported catalyst for fuel cell cathodes (Pt / C) (platinum load: 50 wt%, support: carbon black) was used as the catalyst for the working electrode. The conditions were as follows: catalyst amount m cat : 0.1 mg / cm³ 2 , catalyst density ρ cat : 11.8g / cm³ 3 The I / C ratio was set to 0.4. In the third step of the above embodiment, the error index ΔS was -1.36 for the disk electrode and -1.39 for the ring electrode.

[0110] Figures 6(a) and 6(b) are graphs showing the LSV of the disk electrode and ring electrode after mathematical optimization, respectively. Figure 8(b) is a graph showing the predicted and measured values ​​obtained when applying Example 2 to the current-voltage characteristics of a fuel cell using a platinum-supported catalyst (Pt / C).

[0111] (Comparative Example 2) The procedure was carried out in the same manner as in Comparative Example 1, except that the catalyst used for the working electrode was the same commercially available platinum-supported catalyst (Pt / C) for fuel cell cathodes as in Example 2. The error index ΔS value in the third step of the above embodiment was -1.78 for the disk electrode and -0.03 for the ring electrode. Figures 7(a) and (b) are graphs showing the LSV of the disk electrode and ring electrode after mathematical optimization, respectively.

[0112] Comparing Figures 6 and 7, it can be seen that when using a platinum-supported catalyst (Pt / C), applying the multi-active-point model provides a more accurate fit to the LSV of the disk electrode and ring electrode compared to applying the single-active-point model. This is also evident from the value of the error index ΔS. Furthermore, when using the single-active-point model, the LSV of the ring electrode after mathematical optimization deviates significantly from the measured value, making it impossible to complete the third step and execute the fourth step. Therefore, it is impossible to predict the current-voltage characteristics of a fuel cell using a platinum-supported catalyst (Pt / C).

[0113] Figures 8(a) and (b) show that, in both cases—when a carbon alloy catalyst and when a platinum-supported catalyst (Pt / C) is used—the predicted curves overlap with the distribution of the plotted measured values, indicating that good predictions of the current-voltage characteristics can be made. These results demonstrate that the current-voltage characteristics prediction method using a multi-active-site model can predict the current-voltage characteristics of fuel cells equipped with catalysts composed of any material with high accuracy.

[0114] Furthermore, in the low-current region, the activation voltage obtained from the Butler-Volmer equation is found to be in good agreement. Therefore, it is possible to evaluate the performance of a cathode catalyst used in a fuel cell from its current-voltage characteristics in the low-current region. In the high-voltage region, the decrease due to resistive overpotential also becomes large, so by subtracting the resistive overpotential and concentration overpotential from the activation voltage, it is possible to evaluate the catalyst performance for all current ranges.

[0115] (Example 3) Catalyst amount m cat 1.5 mg / cm³ 2 The current-voltage characteristics of a fuel cell equipped with a catalyst layer having an I / C ratio of 0.7 were measured and predicted. The materials for each component of the fuel cell were set in the same way as in Example 1.

[0116] (Example 4) Catalyst amount m cat2.5 mg / cm³ 2 The current-voltage characteristics of a fuel cell equipped with a catalyst layer having an I / C ratio of 0.7 were measured and predicted. The materials for each component of the fuel cell were set in the same way as in Example 1.

[0117] (Example 5) Catalyst amount m cat 3.5 mg / cm³ 2 The current-voltage characteristics of a fuel cell equipped with a catalyst layer having an I / C ratio of 0.7 were measured and predicted. The materials for each component of the fuel cell were set in the same way as in Example 1.

[0118] (Example 6) Catalyst amount m cat 1.5 mg / cm³ 2 The current-voltage characteristics of a fuel cell equipped with a catalyst layer having an I / C ratio of 1.1 were measured and predicted. The materials for each component of the fuel cell were set in the same way as in Example 1.

[0119] (Example 7) Catalyst amount m cat 1.5 mg / cm³ 2 The current-voltage characteristics of a fuel cell equipped with a catalyst layer having an I / C ratio of 0.3 were measured and predicted. The materials for each component of the fuel cell were set in the same way as in Example 1.

[0120] Figures 9(a), (b), and (c) are graphs showing the measured and predicted current-voltage characteristics for Examples 3, 4, and 5. The horizontal axis of the graph represents current density (A / cm²). 2 The graph shows the current-voltage characteristics, with the vertical axis representing voltage (V). In all graphs, the predicted curve almost perfectly overlaps with the measured curve, indicating that the current-voltage characteristics prediction method can predict the measured current-voltage characteristics with high accuracy. Furthermore, from a comparison of the three graphs, it can be seen that the amount of catalyst m cat When the value is close to 1.5, the output is almost at its maximum, and the amount of catalyst m cat It can be seen that the greater the value of m exceeds 1.5, the lower the output will be from this maximum value. This result indicates that the amount of catalyst m catThis is thought to be due to the fact that increasing the value above 1.5 increases the diffusion resistance within the fuel cell.

[0121] Figures 10(a), (b), and (c) are graphs showing the measured and predicted current-voltage characteristics for Examples 6, 3, and 7, respectively. The horizontal and vertical axes of the graphs are the same as in Figures 9(a), (b), and (c). In all graphs, the predicted curves almost overlap with the measured curves, indicating that the current-voltage characteristic prediction method can predict the measured current-voltage characteristics with high accuracy. Furthermore, a comparison of the three graphs shows that when the I / C ratio is close to 0.7, the output is close to the maximum value, and as the I / C ratio moves further away from 0.7, the output becomes smaller than this maximum value. This result is thought to be due to the fact that when the I / C ratio is greater than 0.7, the electrical resistance in the fuel cell increases, and when the I / C ratio is less than 0.7, the ionic resistance in the fuel cell increases.

[0122] In both cases, using platinum as the catalyst and using a carbon alloy, the predicted curves overlap with the distribution of the plotted measured values, indicating that the current-voltage characteristics can be predicted well. These results show that the current-voltage characteristics prediction method can predict the current-voltage characteristics of fuel cells equipped with catalysts made of any material with high accuracy. [Explanation of Symbols]

[0123] 100... Rotating ring-disk electrode 101...Disk electrodes 102... Ring electrode 103, 104... insulating film 105...Oxygen-saturated solution P...center line I D ...Disk current I R ...ring current

Claims

1. A method for predicting the current-voltage characteristics of a fuel cell, The first step involves performing RRDE measurements simulating the aforementioned fuel cell to determine the measured values ​​of the electrode current density of the disk electrode and the electrode current of the ring electrode generated by the electrode reaction. The second step involves solving the following equation (1) relating to the concentration distribution of oxygen molecules and hydrogen peroxide molecules on the electrode surface of the RRDE measurement, and determining the theoretical values ​​of the electrode current density and the electrode current from the obtained concentration distribution. A third step involves mathematically optimizing the theoretical value so that the difference between the measured value and the theoretical value is less than or equal to a predetermined magnitude. The fourth step involves substituting the theoretical value after mathematical optimization into the Butler-Volmer equation to determine the activation voltage under the conditions in which the electrode reaction is in equilibrium, and then determining the current-voltage characteristics. A method for predicting current-voltage characteristics, characterized in that the theoretical value obtained in the second step is the sum of the electrode current density and the electrode current, which correspond to the energy change for each active site. [Math 1] (C is the concentration distribution of oxygen molecules and hydrogen peroxide molecules in the catalyst layer, D is the diffusion coefficient of oxygen molecules and hydrogen peroxide molecules, and K is the reaction rate constant of the catalyst layer formed on the electrode surface.)

2. The method for predicting current-voltage characteristics according to claim 1, characterized in that the concentration distribution C, the diffusion coefficient D, and the reaction rate constant K are each expressed by the following equations (2) to (4). [Math 2] [Math 3] [Math 4] (C Ｏ２ , C Ｈ２Ｏ２ These represent the concentration distributions of oxygen molecules and hydrogen peroxide molecules in the catalyst layer, respectively. Ｏ２ , D Ｈ２Ｏ２ These represent the diffusion coefficients of oxygen molecules and hydrogen peroxide molecules, respectively. ２ _K ３ _K ４ These represent the reaction rate constants of the catalyst layer in the synthesis reaction of hydrogen peroxide molecules, the synthesis reaction of water molecules, and the decomposition reaction of hydrogen peroxide molecules, respectively.

3. The method for predicting current-voltage characteristics according to claim 1 or 2, characterized in that the energy change for each active site is the energy change for two or more active sites with different activity levels within the catalyst layer.

4. A method for predicting current-voltage characteristics according to claim 1 or 2, characterized in that the Butler-Volmer equation in the fourth step is multiplied by the ratio of the amount of catalyst to the catalyst density.

5. The method for predicting current-voltage characteristics according to either claim 1 or 2, characterized in that the mathematical optimization in the third step is performed such that the error index ΔS, represented by the following equation (5), is -1.1 or less at the disk electrode and within the range of -0.9 or less at the ring electrode. [Math 5] (I Ｍ , I Ｃ are, respectively, the electrode current density of the disk electrode, the measured value, and the theoretical value of the electrode current of the ring electrode in the RRDE measurement. The sum is performed for the potential condition E ｍｅｔ within the measured potential.)

6. The method for predicting current-voltage characteristics according to either claim 1 or 2, characterized in that, in the fourth step, the reduction due to concentration overvoltage and resistance overvoltage is subtracted from the activation voltage of the fuel cell.

7. The method for predicting current-voltage characteristics according to claim 6, characterized in that a first database is created regarding the relationship between the amount of catalyst and the concentration overpotential, and the concentration overpotential is predicted by referring to the first database.

8. The method for predicting current-voltage characteristics according to claim 6, characterized in that a second database is created regarding the relationship between the ratio of the amount of ionomer to the amount of catalyst in the catalyst layer and the resistive overvoltage, and the resistive overvoltage is predicted by referring to the second database.

9. A current-voltage characteristic prediction device used in the current-voltage characteristic prediction method according to claim 1 or 2, An input device that inputs the measured values ​​of the electrode current density and electrode current of the disk electrode generated by the electrode reaction using the RRDE measurement described above, A first computing device that solves equation (1) above and determines the theoretical values ​​of the electrode current density and the electrode current from the obtained concentration distribution, A second computing device performs mathematical optimization of the theoretical value so that the difference between the measured value and the theoretical value is less than or equal to a predetermined magnitude. A current-voltage characteristic prediction device comprising: a third calculation device that substitutes the theoretical value after mathematical optimization into the Butler-Volmer equation to derive the relationship between current-voltage characteristics under conditions where the electrode reaction is in equilibrium; and

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