Method for calculating acoustic reflection and transmission coefficients of lithium-ion batteries

Abstract

The application discloses a method for calculating reflection coefficient and transmission coefficient of sound wave propagation in a lithium ion battery, step 1, the cell of the lithium ion battery stacked or wound is divided into multiple thin layers, and each thin layer is regarded as a uniform porous medium layer, and a generalized eigenvalue equation of sound wave propagation in a solid-liquid two-phase porous medium is derived according to a stress-strain expression; step 2, according to a Legendre series expansion method, a solution of displacement is expanded in the form of a Legendre series, and a Legendre series recursion form is combined; step 3, according to a boundary condition, a solving matrix of the reflection coefficient and the transmission coefficient of sound wave propagation in the lithium ion battery is obtained by means of an expression form of a global matrix. The application can calculate the reflection coefficient and the transmission coefficient of sound wave propagation in the lithium ion battery under different states of charge, and is used for studying the propagation characteristics of sound wave in the lithium ion battery.

Description

Technical Field

[0001] This invention belongs to the field of ultrasonic nondestructive testing technology, specifically relating to a method for calculating the acoustic reflection coefficient and transmission coefficient of lithium-ion batteries. Background Technology

[0002] As a representative of energy storage devices, lithium-ion batteries are widely used in electronic devices, new energy vehicles, aerospace, and other fields due to their advantages such as high operating voltage, high energy density, long cycle life, no memory effect, and no environmental pollution. However, during the cycling process of lithium-ion batteries, changes in the battery's state of charge (SOC) have the greatest impact on battery aging, and are therefore the main cause of lifespan limitations. To optimize battery performance and lifespan, accurate prediction of the battery's SOC is necessary to prevent overcharging, over-discharging, and other situations that may cause permanent damage to the battery's internal structure. Currently, commonly used methods for predicting battery SOC include: open-loop method, data-driven method, and ampere-hour integration method. Among them, the open-loop method uses the accumulation of current over a specific time to predict the battery's SOC. This method is relatively simple to operate, but requires the battery to be left undisturbed for a long time to achieve voltage stability. The data-driven method requires analyzing and training a large amount of battery SOC data to extract feature parameters for fitting and to achieve SOC prediction. However, the prediction results are highly dependent on the quantity and quality of the samples selected during training, making it difficult to guarantee the accuracy and robustness of this method. The measurement error of current in the ampere-hour integration method will lead to the deviation in the prediction of battery SOC. Over a long period of time, the deviation will become larger and larger.

[0003] Ultrasonic nondestructive testing (NDT) has attracted considerable attention from researchers due to its safety, efficiency, simplicity, accuracy, and real-time online detection capabilities, and is widely used in material performance testing. However, literature indicates a close relationship between changes in the state of charge (SOC) of lithium-ion batteries and variations in the parameters of the positive and negative electrode materials. Therefore, ultrasonic testing offers a novel approach for direct SOC detection, and several researchers have attempted it. However, these studies primarily utilize experimental measurements to obtain the relationship between ultrasonic signal characteristics and battery state parameters, lacking theoretical analysis of the ultrasonic reflection / transmission characteristics in the multilayer porous structure of lithium-ion batteries. Summary of the Invention

[0004] The purpose of this invention is to provide a method for calculating the acoustic reflection coefficient and transmission coefficient of lithium-ion batteries, which can provide a theoretical basis for the acoustic characterization of the state of charge of lithium-ion batteries.

[0005] The technical solution adopted in this invention is a method for calculating the acoustic reflection coefficient and transmission coefficient of a lithium-ion battery, which is implemented according to the following steps:

[0006] Step 1: Divide the stacked or wound lithium-ion battery cells into multiple thin layers, and regard each thin layer as a uniform porous medium layer. Based on the state vector matrix, write the expression for stress and strain, and derive the generalized eigenvalue equation for sound wave propagation in solid-liquid dual-phase porous medium.

[0007] Step 2: According to the Legendre series expansion method, the solution of displacement is expanded in the form of Legendre series, and the generalized eigenvalue equation is rewritten by combining the recursive form of Legendre series.

[0008] Step 3: Based on the boundary conditions, obtain the solution matrices for the acoustic reflection coefficient and transmission coefficient of the lithium-ion battery using the global matrix expression.

[0009] The invention is further characterized by:

[0010] The specific process of step 1 is as follows:

[0011] The lithium-ion battery cell, which is stacked or wound, is divided into multiple thin layers, and each layer is regarded as a uniform porous medium layer. Biot theory is introduced, and the wave control equation in the solid-liquid two-phase porous medium is given:

[0012] In Biot theory, the linear constitutive equations (stress-strain relationships) for the solid skeleton and the fluid are as follows:

[0013]

[0014] Where p is the number of layers. For the average solid force, s p e represents the average pore fluid pressure. p ε p The volumetric strains are those of the solid skeleton and the pore fluid, respectively. For the strain of the solid skeleton, u p U p δ represents the displacement of the solid framework and the pore fluid, respectively. ij Let Kroneckcr be the symbol, with subscripts i, j = 1, 2.

[0015] In addition, A p R p Q p N p , where is the Biot elastic coefficient, expressed as:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] Where, α p M p The Biot coefficient is used to characterize the compressibility of solid particles and pore fluids. The bulk modulus (Pa) of solid particles, solid framework, and pore fluid, respectively, λ p μ p Let ν be the Lamé coefficient. p Poisson's ratio. Variable modulus of solid particles. Modulus of solid skeleton The relationship between them is:

[0022]

[0023] Where, φ p =(1+ν p ) / [2(1-2ν p )).

[0024] Strain-displacement relationship under the assumption of small deformation:

[0025]

[0026] Assuming the pore fluid flow in the two-phase medium follows Darcy flow, the governing equations for the motion of the solid skeleton and pore fluid under no-body-force conditions are:

[0027]

[0028] and:

[0029]

[0030] in, These are the mass coupling coefficients for solids, fluids, and fluid-solids, respectively. These represent the solid framework and pore fluid density, respectively. b is the porosity. p The dissipation coefficient is related to porosity and Darcy's law, which is the permeability coefficient. and fluid viscosity η p related:

[0031]

[0032] Based on the state vector matrix, write the expressions for stress and displacement:

[0033]

[0034] Substituting Equation 7 into Equation 4 and neglecting the fluid-solid mass coupling coefficient, we can obtain the wave equation in state matrix form.

[0035]

[0036] In a two-phase medium, the wave equations for the solid and liquid phases are similar. Therefore, for ease of calculation, the stress and displacement in a two-phase medium are defined as follows:

[0037]

[0038] Substituting formula 9 into formulas 1 and 8, we get:

[0039]

[0040]

[0041] in, (i,j=1,2) represents the mechanical parameter matrix of the p-th layer, which can be expressed as:

[0042]

[0043]

[0044]

[0045]

[0046] coefficient matrix M p for:

[0047]

[0048] Substituting Equation 10 into Equation 11 and eliminating the stress term, we can obtain the generalized eigenvalue equation for acoustic waves in a two-phase porous medium:

[0049]

[0050] Step 2: According to the Legendre series expansion method, the solution of displacement is expanded in the form of Legendre series, and the generalized eigenvalue equation is rewritten by combining the recursive form of Legendre series.

[0051] The specific process of step 2 is as follows:

[0052] By choosing Legendre polynomials as orthogonal functions to expand the displacements in Equation 9, the displacement vector can be expressed as:

[0053]

[0054] Among them, P n(χ) is an n-order Legendre series, where N is the order of the Legendre polynomial cutoff term, provided that convergence is guaranteed. The p-th layer contains skeleton displacement u p With liquid displacement U p The amplitude can be expressed as:

[0055]

[0056] Considering that Legendre polynomials can only be expanded within the finite field [-1, 1], the thickness range of each thin layer in a lithium-ion battery is [h]. p-1 ,h p The normalization process is performed and transformed to the interval [-1, 1]. Let χ be the normalization variable, which can be expressed as:

[0057] χ=l p (x2-h p / 2),l p =2 / h p (1.15)

[0058] Formula 12 contains first-order and second-order partial differentials. Expanding using the recurrence relation of Legendre polynomials can eliminate the differential operations. Using the recurrence property of Legendre polynomials, we can obtain:

[0059]

[0060] in:

[0061]

[0062] Substituting equations 13 through 16 into equation 12, we obtain the generalized eigenvalue equations:

[0063]

[0064] in:

[0065]

[0066]

[0067] Step 3: Based on the boundary conditions, obtain the solution matrices for the acoustic reflection coefficient and transmission coefficient of the lithium-ion battery using the global matrix expression.

[0068] The specific process of step 3 is as follows:

[0069] The continuous boundary conditions for solid-liquid two-phase porous media are:

[0070] (1) The normal displacement of the solid skeleton is continuous and the tangential displacement is continuous:

[0071]

[0072] (2) Continuity of total normal and tangential stress:

[0073]

[0074] (3) The normal displacement of the pore fluid is continuous:

[0075]

[0076] (4) Continuous pore fluid pressure:

[0077]

[0078] In this context, the superscripts + and - represent the upper and lower interfaces of the thin layer, respectively.

[0079] Substituting Equations 10 and 13 into the above continuous boundary conditions, we get:

[0080]

[0081] in:

[0082]

[0083]

[0084] Combine Equations 17 and 22, and write the displacement term as a column vector. The generalized eigenvalue equation can be rewritten as:

[0085]

[0086] in, The coefficient matrix related to wavenumber and mechanical parameters can be represented as:

[0087]

[0088] In addition, for ease of representation, the definition is...

[0089] Boundary conditions at the interface between liquid load and solid-liquid two-phase porous media:

[0090] (1) The normal displacement of the pore fluid is continuous with the normal displacement of the liquid load:

[0091]

[0092] (2) The normal stress of the solid skeleton is continuous with the normal stress of the liquid load:

[0093]

[0094] (3) The tangential stress at the interface between the liquid load and the solid-liquid two-phase porous medium is zero:

[0095] τ 21 =0 (1.26)

[0096] (4) The tangential stress of the pore fluid is zero:

[0097] s 21 =0 (1.27)

[0098] (5) The normal stress of the pore fluid is continuous with the normal stress of the liquid load:

[0099]

[0100] The displacement and stress of the liquid load can be expressed as:

[0101] When x2 < 0, the displacement of the liquid load can be expressed as:

[0102]

[0103] When x2 > d, the displacement of the liquid load can be expressed as:

[0104]

[0105] Furthermore, considering the relationship between pressure and displacement in a liquid load:

[0106]

[0107] Therefore, when x2 < 0, the pressure of the liquid load can be expressed as:

[0108]

[0109] When x2 > d, the pressure of the liquid load can be expressed as:

[0110]

[0111] definition By applying the boundary conditions simultaneously and using the global matrix representation, the solution matrices for the acoustic reflection and transmission coefficients of a lithium-ion battery can be expressed as follows:

[0112]

[0113] in:

[0114]

[0115]

[0116]

[0117]

[0118]

[0119] The beneficial effects of this invention are:

[0120] This invention provides a method for calculating the acoustic reflection and transmission coefficients of lithium-ion batteries. By employing a state vector matrix and Legendre series expansion, and utilizing the global matrix expression, the acoustic reflection and transmission coefficients of lithium-ion batteries can be obtained. This method can calculate the propagation characteristics of sound waves under different states of charge of lithium-ion batteries. Furthermore, the results obtained by this method can provide useful references for experimental research and engineering applications of ultrasonic detection of the state of charge of lithium-ion batteries. In addition, by analyzing the reflection and transmission coefficients obtained from theoretical calculations, suitable acoustic excitation frequencies and incident angles can be selected for ultrasonic detection of the state of charge of lithium-ion batteries. Attached Figure Description

[0121] Figure 1 This is a schematic diagram of sound waves propagating in a lithium-ion battery.

[0122] Figure 2 shows the angular spectrum of the reflection coefficient and transmission coefficient.

[0123] Figure 3 shows the frequency spectrum of the reflection coefficient and the transmission coefficient. Detailed Implementation

[0124] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0125] This invention discloses a method for calculating the sound reflection and transmission coefficients of lithium-ion batteries, and describes the sound wave propagation process in lithium-ion batteries as follows: Figure 1 As shown, the solution process for the reflection coefficient and transmission coefficient is carried out according to the following steps:

[0126] First, input the parameters of each thin-layer dielectric material in the lithium-ion battery. The parameters for the negative electrode graphite are: density 2.2-2.485 g / cm³. 3 The Young's modulus is 29.94-89.16 GPa, the thickness is 77 μm, the Poisson's ratio is 0.32, and the porosity is 0.1-0.4; the parameters of the negative electrode current collector - copper: density is 8.0 g / cm³. 3 The Young's modulus is 100 GPa, the thickness is 8 μm, and the Poisson's ratio is 0.35; the parameters of the cathode lithium cobalt oxide are: density 4.99-5.047 g / cm³. 3The Young's modulus is 145.5-252.09 GPa, the thickness is 57 μm, the Poisson's ratio is 0.32, and the porosity is 0.3. The parameters of the positive electrode current collector-aluminum are: density 2.7 g / cm³. 3 The membrane has a Young's modulus of 70 GPa, a thickness of 12 μm, and a Poisson's ratio of 0.34; its density is 0.92 g / cm³. 3 The Young's modulus is 0.78 GPa, the thickness is 15 μm, the Poisson's ratio is 0.45, and the porosity is 0.3. The composition and parameters of the aluminum-plastic film are as follows: CPP parameters: density is 0.92 g / cm³. 3 The Young's modulus is 0.896 GPa, the thickness is 40 μm, and the Poisson's ratio is 0.41; the aluminum foil parameters are: density 2.7 g / cm³. 3 The Young's modulus is 70 GPa, the thickness is 40 μm, and the Poisson's ratio is 0.34; the nylon's parameters are: density 1.15 g / cm³. 3 The Young's modulus is 2 GPa, the thickness is 25 μm, and the Poisson's ratio is 0.4. Each layer is considered to be uniformly distributed. The structure of the lithium-ion battery cell is set as follows: aluminum-plastic film - separator - negative electrode graphite - copper - negative electrode graphite - separator - positive electrode lithium cobalt oxide - aluminum - positive electrode lithium cobalt oxide - separator - aluminum-plastic film. Substituting the above parameters into the solution matrix of reflection coefficient and transmission coefficient, the acoustic reflection coefficient and transmission coefficient of the lithium-ion battery can be obtained by solving. Figures 2(a) and 2(b) are the angular spectra of reflection coefficient and transmission coefficient, respectively, and Figures 3(a) and 3(b) are the frequency spectra of reflection coefficient and transmission coefficient, respectively.

[0127] As shown in the figure, the reflection and transmission coefficients of the sound waves exhibit corresponding regular changes with the change in the state of charge of the lithium-ion battery. The results obtained by this method can provide a useful reference for experimental research and engineering applications of ultrasonic detection of the state of charge of lithium-ion batteries.

[0128] Finally, it should be noted that the above embodiments are only used to illustrate the present invention and are not intended to limit the technical solutions described in the present invention. Therefore, although the present invention has been described in detail with reference to the above real-world examples, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the present invention. All technical solutions and improvements that do not depart from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A method for calculating the acoustic reflection coefficient and transmission coefficient of a lithium-ion battery, characterized in that: The specific steps are as follows: Step 1: Divide the stacked or wound lithium-ion battery cells into multiple thin layers, and regard each thin layer as a uniform porous medium layer. Based on the state vector matrix, write the expression for stress and strain, and derive the generalized eigenvalue equation for sound wave propagation in solid-liquid dual-phase porous medium. Step 2: According to the Legendre series expansion method, the solution of displacement is expanded in the form of Legendre series, and the generalized eigenvalue equation is rewritten by combining the recursive form of Legendre series. Step 3: Based on the boundary conditions, and using the global matrix representation, obtain the solution matrix for the reflection coefficient and transmission coefficient of sound waves propagating in the lithium-ion battery. The specific process of step 3 is as follows: The continuous boundary conditions for solid-liquid two-phase porous media are: (1) The normal displacement of the solid skeleton is continuous and the tangential displacement is continuous: (18) (2) Continuity of total normal and tangential stress: (19) (3) The normal displacement of the pore fluid is continuous: (20) (4) Continuous pore fluid pressure: (21) In this context, the superscripts + and - represent the upper and lower interfaces of the thin layer, respectively. Substituting Equations 10 and 13 into the above continuous boundary conditions, we get: (22) in: Combine Equations 17 and 22, and express the displacement term as a column vector E. 4N×1 =[ , ,… ] T The generalized eigenvalue equation can be rewritten as: (23) in, , , The coefficient matrix related to wavenumber and mechanical parameters is represented as follows: In addition, for ease of representation, the definition is... ; Boundary conditions at the interface between liquid load and solid-liquid two-phase porous media: (1) The normal displacement of the pore fluid is continuous with the normal displacement of the liquid load: (24) (2) The normal stress of the solid skeleton is continuous with the normal stress of the liquid load: (25) (3) The tangential stress at the interface between the liquid load and the solid-liquid two-phase porous medium is zero: (26) (4) The tangential stress of the pore fluid is zero: (27) (5) The normal stress of the pore fluid is continuous with the normal stress of the liquid load: (28) Among them, T w Let be the pressure of the liquid; furthermore, the displacement and stress of the liquid load are expressed as: When x2 < 0, the displacement of the liquid load is expressed as: (29) When x2 > d, where d is the total thickness of the multilayer porous medium, the displacement of the liquid load is expressed as: (30) Where A0 is the amplitude of the incident wave, A R Let A be the amplitude of the reflected wave. T Let α be the amplitude of the transmitted wave, and let α be the wavenumber component of the incident wave in the x2 direction, expressed as α = cotθ0, where θ0 is the incident angle. Furthermore, considering the relationship between pressure and displacement in a liquid load: (31) Among them, K w ρ is the compressibility coefficient of the liquid. w c is the density of the liquid. w Let u be the longitudinal wave velocity in water. w Displacement due to liquid load; Therefore, when x2 < 0, the pressure of the liquid load is expressed as: (32) When x2 > d, the pressure of the liquid load is expressed as: (33) Where c = c w / sinθ0; definition By combining the boundary conditions and using the global matrix representation, the solution matrices for the acoustic reflection coefficient and transmission coefficient of the lithium-ion battery are expressed as follows: (34) in: 。 2. The method for calculating the acoustic reflection coefficient and transmission coefficient of a lithium-ion battery according to claim 1, characterized in that, The specific process of step 1 is as follows: The lithium-ion battery cell, which is stacked or wound, is divided into multiple thin layers, and each layer is regarded as a uniform porous medium layer. Biot theory is introduced, and the wave control equation in the solid-liquid dual-phase porous medium is given: In Biot theory, the linear constitutive equations for the solid framework and the fluid are as follows: (1) Where p is the number of layers. For the average solid force, s p e represents the average pore fluid pressure. p ε p The volumetric strains are those of the solid skeleton and the pore fluid, respectively. For the strain of the solid skeleton, u p U p These represent the displacements of the solid framework and the pore fluid, respectively. The symbol is Kroneckcr, with subscripts i and g = 1, 2; In addition, A p R p Q p N p , where is the Biot elastic coefficient, expressed as: Where, β p M p The Biot coefficient is used to characterize the compressibility of solid particles and pore fluids. , , These represent the bulk modulus of solid particles, solid framework, and pore fluid, respectively. Let λ be the permeability coefficient of the solid framework. p μ p ν is the Lamé coefficient. p For Poisson's ratio, φ p Porosity; variable modulus of solid particles Modulus of solid skeleton The relationship between them is: (2) in, ; Strain-displacement relationship under the assumption of small deformation: (3) In a two-phase medium, the flow of the pore fluid follows Darcy flow; the governing equations for the motion of the solid skeleton and the pore fluid under no-body-force conditions are: (4) and: (5) in, , , These are the mass coupling coefficients for solids, fluids, and fluid-solids, respectively. , These represent the density of the solid framework and the density of the pore fluid, respectively, φ p b is the porosity. p The dissipation coefficient is related to porosity and Darcy's law, which is the permeability coefficient. and fluid viscosity η p related: (6) Based on the state vector matrix, write the expressions for stress and displacement: (7) Where k is the wave number, x1 represents the x1 direction of the coordinate axis, ω is the angular frequency, and t is time. Harmonic factors; Substituting Equation 7 into Equation 4, and neglecting the fluid-solid mass coupling coefficient, we obtain the wave equation in state matrix form. (8) Where x2 represents the x2 direction of the coordinate axis; In a two-phase medium, the wave equations for the solid and liquid phases are similar. The stress and displacement in a two-phase medium are defined as follows: (9) Substituting formula 9 into formulas 1 and 8, we get: (10) (11) in, The mechanical parameter matrix of the p-th layer, i, g = 1, 2, is represented as: coefficient matrix M p for: Substituting Equation 10 into Equation 11 and eliminating the stress term, we obtain the generalized eigenvalue equation for acoustic waves in a two-phase porous medium: (12)。 3. The method for calculating the acoustic reflection coefficient and transmission coefficient of a lithium-ion battery according to claim 1, characterized in that, The specific process of step 2 is as follows: Choosing Legendre polynomials as orthogonal functions to expand the displacements in Equation 9, the displacement vector is expressed as: (13) Among them, P n (χ) is an n-order Legendre series, where N is the order of the Legendre polynomial cutoff term under the premise of ensuring convergence; The p-th layer contains skeleton displacement u p With liquid displacement U p The amplitude is expressed as: (14) Considering that Legendre polynomials can only be expanded within the finite field [-1, 1], the thickness range of each thin layer in a lithium-ion battery is [h]. p-1 ,h p ], normalize and transform to the interval [-1, 1]; define χ as the normalized variable, expressed as: (15) Among them, h p This represents the thickness of the p-th layer in a multilayer porous medium. Formula 12 contains first-order and second-order partial differentials, which are eliminated by expansion using the recurrence relation of Legendre polynomials; the recurrence relation of Legendre polynomials is expressed as: (16) in: Substituting equations 13 through 16 into equation 12, we obtain the generalized eigenvalue equations: (17) in: 。

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