Electrothermal analysis method and system for contact of thermoelectric coating with any performance gradient and electrode

Abstract

The invention discloses an electrothermal analysis method and system for contact between a thermoelectric coating with any performance gradient and an electrode, and the method comprises the steps: dispersing a functional gradient thermoelectric coating into a plurality of uniform hot electron layers in the thickness direction, and constructing a multilayer thermoelectric material model; establishing a thermoelectric coupling control equation in each uniform thermoelectron layer in the multi-layer thermoelectric material model, and setting boundary and interface conditions; solving a thermoelectric coupling control equation by utilizing Fourier transformation, and deducing to obtain a Cauchy singular integral equation satisfied by normal current and normal energy flow in combination with an interlayer continuity condition; and dispersing and solving the Cauchy singular integral equation by using a collocation method to obtain a coating performance index. According to the method, the problem of thermoelectric field coupling when the multi-layer and functional gradient thermoelectric coating is in contact with the electrode can be accurately solved, and the influence rules of the conductivity gradient index, the thermal conductivity gradient index change, the layer number, the coating thickness and other parameters on the current density distribution, the energy flow distribution and the intensity factor can be quantitatively analyzed.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of thermoelectric material and electrode contact electrothermal analysis, and particularly relates to a method and system for electrothermal analysis of thermoelectric coating with arbitrary performance gradient and electrode contact. BACKGROUND

[0002] Thermoelectric devices can directly utilize electrical energy to achieve refrigeration, or conversely, directly convert thermal energy into electrical energy. These devices play a key role in home heating and refrigeration systems, energy production, and environmental energy resource management. The performance of thermoelectric devices is not only affected by the inherent properties of thermoelectric materials, but also by metal interconnects (including electrodes). Diffusion of metals at the contact interface can change over time and degrade the performance of thermoelectric materials, thereby reducing the efficiency of the device. In order to improve the output power of thermoelectric devices and expand their operating temperature range, a strategy of varying parameters has been adopted. Functionally graded thermoelectric (FGTE) materials are designed to gradually change (increase, decrease, or non-monotonic change) their composition or microstructure according to pre-set standards. Therefore, their thermal and electrical properties also change continuously, meeting specific non-uniform performance requirements.

[0003] In the study of thermoelectric coating and electrode contact, when multilayer thermoelectric and gradient thermoelectric coatings are in contact with electrodes (rigid flat press heads), it is necessary to accurately analyze their electrical and thermal properties. The existing technology lacks adaptability to arbitrary performance changes when dealing with multilayer thermoelectric coatings and functionally graded thermoelectric materials, and lacks effective semi-analytical solution and numerical calculation methods. For example, in the analysis of thermoelectric coating contact problems, it is difficult to reasonably transform the contact problem into a singular integral equation and accurately solve it. For multilayer periodic thermoelectric coatings and performance-graded thermoelectric coatings, there is a lack of accurate analysis of the variation of key parameters such as current density and energy flux at the contact edge, which cannot efficiently verify the accuracy of the method and determine the influence of parameters, hindering the accurate design and performance optimization of thermoelectric coatings in practical applications. Therefore, a method for electrothermal analysis of thermoelectric coating with arbitrary performance gradient and electrode contact is needed to solve the above technical problems. SUMMARY

[0004] The present application aims to solve the deficiencies of the prior art and provides the following solutions:

[0005] A method for electrothermal analysis of thermoelectric coating with arbitrary performance gradient and electrode contact, comprising the following steps:

[0006] Discretizing the functionally graded thermoelectric coating into multiple uniform thermoelectric layers along the thickness direction to construct a multilayer thermoelectric material model;

[0007] Establishing thermoelectric coupling control equations in each of the uniform thermoelectric layers in the multilayer thermoelectric material model and setting boundary and interface conditions;

[0008] The Fourier transform is used to solve the thermoelectric coupling control equation, and the Cauchy singular integral equation satisfied by the normal current and the normal energy flow is derived by combining the interlayer continuity condition.

[0009] The Cauchy singular integral equation is discretized and solved by using the collocation method, and the coating performance index is obtained.

[0010] Preferably, the constructed multi-layer thermoelectric material model comprises:

[0011] The electrode with a width of 2a is in contact with the multi-layer thermoelectric coating with a thickness of h, the multi-layer thermoelectric coating is composed of N thermoelectric elastic sub-layers, and is bonded to the thermoelectric half-plane substrate.

[0012] Preferably, the method for establishing the thermoelectric coupling control equation comprises:

[0013] The constitutive relation of the thermionic layer and the thermoelectric substrate is constructed:

[0014]

[0015] wherein, represents the current density vector in the i-th layer, x represents the horizontal direction coordinate, and y represents the vertical direction coordinate, represents the gradient operator, γ i represents the conductivity coefficient, F i represents the electrochemical potential function, h represents the thickness of the thermoelectric coating, h i represents the thickness of the i-th thermionic layer, represents the energy flow vector in the i-th layer, λ i represents the thermal conductivity coefficient, T i represents the temperature;

[0016] Based on the constitutive relation, the thermoelectric coupling control equation is constructed:

[0017]

[0018] wherein, represents the Laplace operator.

[0019] Preferably, the boundary and interface conditions comprise:

[0020] For the electric field, the mixed boundary condition of the surface of the multi-layer thermoelectric coating is:

[0021]

[0022] F i (x, h i ) = F i+1 (x, h i, i = 1,..., N, -∞ < x < ∞

[0023] j iey (x, h i ) = j (i+1)ey (x, h i ), i = 1,..., N, -∞ < x < ∞

[0024]

[0025] where j (N+1)ey (x, 0) denotes the normal current density at the contact surface, denotes the unknown surface normal current density in the contact region, a denotes half of the contact region width, F N+1 (x, 0) denotes the electrochemical potential function at the contact surface, F i (x, i ) denotes the value of the electrochemical potential function of the i-th layer at y = h i (x, h iey ) denotes the value of the normal current density of the i-th layer at y = h i (x, h i ) denotes the value of the normal current density of the i-th layer at y = h e0 denotes the total current load applied;

[0026] For the temperature field, the mixed boundary conditions on the surface of the multilayer thermoelectric coating are:

[0027]

[0028] T i (x, h i ) = T i+1 (x, h i ), i = 1,..., N, -∞ < x < ∞

[0029] j iuy (x, h i ) = j (i+1)uy (x, h i ), i = 1,..., N, -∞ < x < ∞

[0030]

[0031] where j (N+1)uy (x, 0) denotes the normal energy flux at the contact surface, denotes the unknown surface normal energy flux in the contact region, T N+1 (x, 0) denotes the temperature at the contact surface, T i (x, h i ) denotes the value of the temperature of the i-th layer at y = h i (x, h iuy(x, hi) represents the value of the normal energy flux of the i-th layer at y = h i u0 J represents the total energy flux applied.

[0032] Preferably, the method for obtaining the Cauchy singular integral equation comprises:

[0033] Applying Fourier transform to the thermoelectric coupling control equation with respect to x, the general solution of the electric potential function and temperature in each thermoelectric layer is obtained:

[0034]

[0035] wherein F i (x, y) represents the general solution of the electric potential function in the i-th thermoelectric layer, A i1 (s) and A i2 (s) represent coefficients to be solved according to the boundary conditions, s represents the Fourier transform variable, T i (x, y) represents the general solution of the temperature, T Hi (x, y) represents the homogeneous solution of the temperature in the i-th thermoelectric layer;

[0036] Based on the general solution, the singular integral equation is obtained:

[0037]

[0038] wherein k linf and k 2inf represent limit values, k 11 (r, x) and k 22 (r, x) represent kernel functions, and r represents the integral variable.

[0039] Preferably, the method for solving the Cauchy singular integral equation comprises:

[0040] Introducing normalized variables and Converting the interval (-a, a) to (-1, 1), and rewriting the singular integral equation as:

[0041]

[0042] wherein and represent normalized variables belonging to (-1, 1), represent unknown normal current density, represent unknown normal energy flux;

[0043] The unknown normal current density and unknown normal energy flux in the equation are approximated by the following series:

[0044] ​

[0045] wherein, represents a function to be solved related to the normal current density by series expansion, represents a function to be solved related to the normal current density by series expansion, represents the first kind of Chebyshev polynomial, d j and c j represents a to-be-determined coefficient, j=0,1,...,M-1;

[0046] The series is substituted into the rewritten singular integral equation, and a collocation method is used to obtain a solution result:

[0047]

[0048]

[0049] wherein, and represent collocation points.

[0050] The application further provides an electrothermal analysis system for a thermal electrocoating layer with an electrode contact with an arbitrary performance gradient, and the system applies the method and comprises a model construction module, a first equation construction module, a second equation construction module and a solution analysis module.

[0051] The model construction module is used for discretizing the functionally gradient thermal electrocoating layer into multiple layers of uniform thermal electron layers along the thickness direction, and constructing a multilayer thermal electro material model.

[0052] The first equation construction module is used for establishing a thermal electro coupling control equation in each of the uniform thermal electron layers in the multilayer thermal electro material model, and setting boundary and interface conditions.

[0053] The second equation construction module is used for solving the thermal electro coupling control equation by using Fourier transformation, combining an interlayer continuous condition, and deducing a Cauchy singular integral equation satisfied by the normal current and the normal energy flow.

[0054] The solution analysis module is used for discretizing and solving the Cauchy singular integral equation by using a collocation method, and obtaining a coating performance index.

[0055] Compared with the prior art, the application has the beneficial effects that:

[0056] The technical scheme of the application provides an electrothermal analysis technology of any performance gradient thermoelectric coating in contact with an electrode, by means of discretization modeling, control equation solving, singular integral equation derivation and numerical analysis and other technical means. On the one hand, the thermal-electric field coupling problem of the multi-layer and function gradient thermoelectric coating in contact with the electrode can be accurately solved, the Cauchy singular integral equation of the normal current and energy flow is derived, the numerical solution is solved by combining the collocation method, the current density intensity factor and the energy flow intensity factor at the electrode end are accurately obtained, and the complex contact electrothermal characteristic analysis problem is solved. On the other hand, the influence law of the gradient index of the electrical conductivity, the gradient index of the thermal conductivity, the number of layers, the coating thickness and other parameters on the current density distribution, the energy flow distribution and the intensity factor can be quantitatively analyzed, reliable data support is provided for the optimization design of the thermoelectric coating, and the target of optimizing the electrothermal performance of the coating by parameter control is realized. BRIEF DESCRIPTION OF DRAWINGS

[0057] In order to more clearly illustrate the technical scheme of the application, the drawings needed in the embodiments are briefly introduced as follows. Obviously, the drawings in the following description are only some embodiments of the application, and other drawings can be obtained by those skilled in the art without creative labor.

[0058] Figure 1 The method flowchart of the embodiment of the application is shown in the figure.

[0059] Figure 2 The multi-layer thermoelectric material model structure of the embodiment of the application is shown in the figure.

[0060] Figure 3 The dimensionless normal current density j ey (x, 0) / j ey0 in the multi-layer thermoelectric coating of the embodiment of the application is shown in the figure. ey0 = J e0 / 2a, h / a = 1, γ0 = 1 x 10 5 S / m);

[0061] Figure 4 The periodic thermoelectric coating and electrode contact problem of the embodiment of the application is shown in the figure.

[0062] Figure 5 The dimensionless normal current density j ey (x, 0) / j ey0 in the periodic multi-layer thermoelectric coating model of the embodiment of the application is shown in the figure, where (a) is the ratio of different coating thicknesses and contact widths, (b) is the three-point x / a = 0.0314, x / a = 0.5621, x / a = 0.9409 in the contact area.

[0063] Figure 6Figure 1 is a schematic diagram of the dimensionless normal current density j in a periodic multilayer thermoelectric coating model of embodiments of the present application uy (x, 0) / j uy0 where (a) is the ratio of different coating thicknesses to contact width, and (b) is the dimensionless normal current density at three points in the contact region, x / a = 0.0314, x / a = 0.5621, and x / a = 0.9409.

[0064] Figure 7 Figure 2 is a schematic diagram of the conductivity gradient along the coating thickness in embodiments of the present application.

[0065] Figure 8 Figure 3 is a schematic diagram of the dimensionless normal current density j in an arbitrary property gradient thermoelectric coating model of embodiments of the present application ey (x, 0) / h ey0 where (a) is the ratio of different coating thicknesses to contact width, and (b) is the dimensionless normal current density at three points in the contact region, x / a = 0.0314, x / a = 0.5621, and x / a = 0.9409.

[0066] Figure 9 Figure 4 is a schematic diagram of the dimensionless normal energy flux j in an arbitrary property gradient thermoelectric coating model of embodiments of the present application uy (x, 0) / j uy0 where (a) is the ratio of different coating thicknesses to contact width, and (b) is the dimensionless normal current density at three points in the contact region, x / a = 0.0314, x / a = 0.5621, and x / a = 0.9409. DETAILED DESCRIPTION

[0067] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of protection of the present application.

[0068] In order to make the above objectives, characteristics and advantages of the present application more apparent, the present application will be further described in detail below with reference to the drawings and specific embodiments.

[0069] Embodiment One

[0070] In this embodiment, as shown in Figure 1, a method for analyzing the electro-thermal properties of an arbitrary property gradient thermoelectric coating in contact with an electrode includes the following steps: Figure 1 S1. Discretize the functionally graded thermoelectric coating into multiple uniform thermoelectric layers along the thickness direction, and construct a multilayer thermoelectric material model.

[0071]

[0072] ​A functionally graded thermoelectric (FGTE) coating is discretized into multiple uniform thermoelectric (TE) sub-layers along the thickness direction to construct a multi-layer thermoelectric material model. Each sub-layer can be independently set with material properties (e.g., electrical conductivity, thermal conductivity, etc.) to adapt to complex cases with different thermal conductivity gradient exponents and electrical conductivity gradient exponents. Specifically, the constructed multi-layer thermoelectric material model includes: an electrode with a width of 2a in contact with a multi-layer thermoelectric coating with a thickness of h, the multi-layer thermoelectric coating is composed of N thermoelectric elastic sub-layers, and is bonded to a thermoelectric half-plane substrate. The coordinate system x-y is placed at the center of the electrode. A total current load J e0 and a total energy flow load J u0 .

[0073] S2. Establish thermoelectric coupling control equations in each uniform thermoelectric sub-layer in the multi-layer thermoelectric material model, and set boundary and interface conditions.

[0074] In each uniform thermoelectric sub-layer, thermoelectric coupling control equations are established to describe the heat flow and current transmission rules (covering physical processes such as heat conduction and electro-thermal conversion). The method for establishing the thermoelectric coupling control equations includes:

[0075] Construct the constitutive relationship of the thermoelectric sub-layers and the thermoelectric substrate:

[0076]

[0077] where, represents the current density vector in the i-th layer, x represents the horizontal direction coordinate, and y represents the vertical direction coordinate, represents the gradient operator, γ i represents the electrical conductivity coefficient, F i represents the electrical potential function, h represents the thickness of the thermoelectric coating, h i represents the thickness of the i-th thermoelectric sub-layer, represents the energy flow vector in the i-th layer, λ i represents the thermal conductivity coefficient, T i represents the temperature; for the thermoelectric substrate, the subscript i is equal to 1; for the thermoelectric coating sub-layers, i is equal to 2 to N+1.

[0078] Based on the constitutive relationship, the thermoelectric coupling control equations are constructed:

[0079]

[0080] where, represents the Laplace operator.

[0081] The boundary and interface conditions include:

[0082] For the electric field, the mixed boundary condition on the surface of the multi-layer thermoelectric coating is:

[0083]

[0084]

[0085] where j (N+1)ey (x, 0) represents the normal current density at the contact surface, i.e., the value of the normal current density of the N+1th layer at y = 0, represents the unknown surface normal current density in the contact region, a represents half of the width of the contact region, F N+1 (x, 0) represents the electrochemical potential function at the contact surface, i.e., the value of the electrochemical potential of the N+1th layer at y = 0; it is assumed that the electrochemical potential function and the normal current density remain continuous at the interface of each layer, i.e.,

[0086] F i (x, h i ) = F i+1 (x, h i ), i = 1,..., N, -∞ < x < ∞

[0087] j iey (x, h i ) = j (i+1)ey (x, h i ), i = 1,..., N, -∞ < x < ∞

[0088] where F i (x, h i ) represents the value of the electrochemical potential function of the ith layer at y = h i , and j ieY (x, h i ) represents the value of the normal current density of the ith layer at y = h i .

[0089] The equilibrium equation is:

[0090]

[0091] where J e0 represents the total current load applied.

[0092] For the temperature field, the mixed boundary conditions, the continuity conditions and the equilibrium equation of the surface of the multilayer thermoelectric coating are:

[0093]

[0094] T i (x, h i ) = T i+1 (x, h i ), i = 1,..., N, -∞ < x < ∞

[0095] jiuy (x, h i ) = j (i+)uy (x, h i ), i = 1,..., N, -∞ < x < ∞

[0096]

[0097] where j (N+1)uy (x, 0) represents the normal energy flux at the contact surface, i.e., the value of the normal energy flux of the N+1 layer at y = 0, represents the unknown surface normal energy flux in the contact region, T N+1 (x, 0) represents the temperature at the contact surface, i.e., the value of the temperature of the N+1 layer at y = 0, T i (x, h i ) represents the value of the temperature of the i-th layer at y = h i (x, h iuY ) represents the value of the normal energy flux of the i-th layer at y = h i (x, h i ), J u0 represents the total energy flux applied.

[0098] S3. Using Fourier transform to solve the thermoelectric coupling control equation, combined with the interlayer continuity condition, the Cauchy singular integral equation satisfied by the normal current and the normal energy flux is derived.

[0099] The method for obtaining the Cauchy singular integral equation comprises:

[0100] Applying Fourier transform to the thermoelectric coupling control equation with respect to x, the general solution of the electrochemical potential function and the temperature in each thermoelectric layer is obtained, specifically:

[0101] First, analyze the electric field, apply Fourier transform to the thermoelectric coupling control equation with respect to x, and obtain the general solution of the electrochemical potential function in each thermoelectric layer:

[0102]

[0103] And the electrochemical potential function in the thermoelectric base is:

[0104]

[0105] where F i (x, y) represents the general solution of the electrochemical potential function in the i-th thermoelectric layer, A i1 (s), A i2 (s) and A 11(s) represents the coefficient to be solved according to the boundary condition, and s represents the Fourier transform variable. Considering the constitutive relation of the thermionic layer and the thermoelectric base, for the thermionic layer, the normal current density can be written as:

[0106]

[0107] where j iey (x, y) represents the rewritten normal current density of the thermionic layer; for the thermoelectric base, the normal current density can be written as:

[0108]

[0109] where j 1ey (x, y) represents the rewritten normal current density of the thermoelectric base. Through the conversion matrix method, the relationship between the surface unknowns and the coefficients can be established. The equation in the Fourier transform domain is written in matrix form, and the following is obtained:

[0110] f i (s, y) = W i (s, y)a i (s)

[0111]

[0112] where f i (s, y) represents the column vector composed of the potential function and the normal current density in the transform domain as above, W i (s, y) represents the coefficient matrix, a i (s) represents the coefficient matrix to be solved, represents the potential function in the Fourier transform domain in the i-th layer, represents the normal current density in the Fourier transform domain in the i-th layer; when i = 1:

[0113]

[0114] a i (s) = A 11 (s)

[0115] At this time, W i (s, y) is a 2 × 1 vector, and a i (s) is a 1 × 1 vector; when i = 2,..., N + 1:

[0116]

[0117] At this time, W i (s, y) is a 2 × 2 coefficient matrix, and a i(s) is a 2x1 vector. At the surface of the multilayer thermoelectric coating, the equation can be rewritten as:

[0118] f N+1 (s,0) = W N+1 (s,0)a N+1 (s)

[0119] According to the continuity condition, it can be expressed in matrix form:

[0120] f i (s,h i ) = f i+1 (s,h i ), i = 1,..., N

[0121] After some algebraic operations and simplification, from the equation, we can get:

[0122]

[0123] where a N+1 (s) represents the coefficient vector in the N+1th thermionic layer, and represents the continuous multiplication symbol. Substitute the obtained a N+1 (s) into f N+1 (s,0), and considering the mixed boundary conditions at the surface of the multilayer thermoelectric coating, we can obtain the relationship between the transformed domain surface electric potential function and the unknown normal current density in the transformed domain as:

[0124]

[0125] where, represents the transformed domain surface electric potential function, represents the unknown normal current density in the transformed domain, w1(s) and w2(s) represent the coefficients, and F N+1 (x,0) represents the surface electric potential function; this reflects the influence of the thickness and electrical conductivity of the thermionic layer and the thermoelectric substrate. Applying the inverse Fourier transform to the relationship equation between the transformed domain surface electric potential function and the unknown normal current density in the transformed domain, we get:

[0126]

[0127] Taking the derivative with respect to the horizontal coordinate x, performing asymptotic analysis, and considering the equation, we can obtain the singular integral equation for the normal current density as:

[0128]

[0129] where k linf represents the limit value, and k 11(r, x) represents the kernel function, r represents the integral variable; after solving the singular integral equation by using the collocation method, the unknown function A can be determined completely i1 (s), A i2 (s) and A 11 (s). In the derivation of the relationship equation between the transformed domain surface electrochemical potential function and the unknown normal current density in the transformed domain, the unknown function A 11 (s) is:

[0130]

[0131] The unknown function A i1 (s) and A i2 (s) is:

[0132]

[0133] Based on the above equation, the physical quantity in the electric field can be completely determined from the general solution of the electrochemical potential function.

[0134] Secondly, the temperature field is analyzed, and the solution process of the temperature field is similar to that of the electric field. Fourier transform is performed on the thermoelectric coupling control equation with respect to x to obtain the general solution in each thermoelectron layer:

[0135]

[0136] Wherein, T i (x, y) represents the general solution of the temperature, T Hi (x, y) represents the homogeneous solution of the temperature in the i-th thermoelectron layer; and the temperature in the thermoelectric base is:

[0137]

[0138] Wherein, T H1 (x, y) represents the homogeneous solution of the temperature field in the first thermoelectric layer, i.e. the thermoelectric half-plane, B i1 (s), B i2 (s) and B 11 (s) represents the unknown coefficient. Considering the constitutive relation, for the normal energy flow of the thermoelectron layer, it can be written as:

[0139]

[0140] Wherein, j iuy (x, y) represents the rewritten normal energy flow of the thermoelectron layer; for the thermoelectric base, the normal energy flow can be written as:

[0141]

[0142] Wherein, j 1uv(x, y) represents the rewritten normal current density of the thermoelectric substrate. Position function B il (s), B i2 (s) and B 11 The determination process of (s) and the unknown function A i1 (s), A i2 (s) and A 11 The process of determining (s) is similar.

[0143] Based on the general solution and boundary conditions, the singular integral equation satisfied by the normal energy flow is obtained:

[0144]

[0145] Among them, k 2inf represents the limit value, k 22 (r, x) represents the kernel function. The coefficients g1(s) and g2(s) are calculated as:

[0146]

[0147] When i=1:

[0148]

[0149] At this time, G1(s, y) is a 2×1 vector; when i=2,...,N+1:

[0150]

[0151] At this time, G i (s,y) is a 2×2 coefficient matrix. In addition, the unknown normal energy flow in the singular integral equation that satisfies the normal energy flow is The equilibrium equations need to be satisfied.

[0152] S4. Use the collocation method to discretize and solve the Cauchy singular integral equation to obtain the coating performance indicators.

[0153] Methods for solving Cauchy's singular integral equations include:

[0154] Introducing normalized variables and Convert the interval (-a,a) to (-1,1) and rewrite the singular integral equation as:

[0155]

[0156] in, and represents a normalized variable belonging to (-1,1), represents the unknown normal current density, unknown normal energy flux; unknown normal current density and unknown normal energy flux in the equation are approximated by the following series:

[0157]

[0158] where, represents the unknown function related to the normal current density expanded by the series, represents the unknown function related to the normal current density expanded by the series, represents the first kind of Chebyshev polynomials, d j and c j represent the undetermined coefficients, j = 0, 1,..., M - 1; the series is substituted into the rewritten singular integral equation, and the collocation method is used to obtain the solution:

[0159]

[0160] where, and represent the collocation points.

[0161] Result analysis: To verify the developed method, this example first provides a comparison of the normal current density with that obtained in literature 1 (Tian XJ, Zhou YT, Zhang CZ. On the imperfect interface of a functionally graded thermoelectric layered structure. Composite Structures 2023; 322: 117394). Literature 1 studies a functionally graded thermoelectric coating with exponentially varying electrical and thermal conductivities, with the property variation expressed as:

[0162] γ(y) = γ1e δy , 0 ≤ y < h

[0163] Since the position and direction of the coordinate system in literature 1 are different from the current proposal, γ1represents the electrical conductivity of the thermoelectric substrate in Figure 2 . The property at the interface y = h is continuous. Here, the functionally graded thermoelectric coating is composed of N thermoelectric layers. The material properties of each thermoelectric layer are represented by the properties at the surface y coordinate of the specific layer.

[0164] Figure 3 The dimensionless normal current density j ey (x, 0) / j ey0 ( j ey0 = J e0Comparison of the results of the present model and the model of / 2a). The conductivity gradient parameter δh is chosen to be -0.4, -0.2, 0, 0.2, 0.4. The thickness of the thermoelectric coating is fixed to h = 0.1 mm, the contact width is a = 0.1 mm, and the conductivity of the thermoelectric substrate is γ1= 1 x 10 5 S / m. 15 thermal electron layers are chosen in the modeling of the functionally graded thermoelectric coating. From the figures, one can see a good agreement. Table 1 gives the convergence analysis of the number of thermal electron layers. Table 1 shows the dimensionless normal current density at the three contact points, and the relative error in percentage for different number of thermal electron layers. The relative error is defined as follows:

[0165]

[0166] The number of collocation points M is chosen to be 50. When the number of thermal electron layers N is 15, the relative error of the dimensionless normal current density is less than 0.03%. Therefore, choosing the number of collocation points to be 50 and the number of thermal electron layers to be 15 can ensure the convergence and accuracy of the results.

[0167] Table 1

[0168]

[0169] The first issue of interest in this example is the periodic thermoelectric multilayer contact problem, the geometry of which is shown in Figure 4 A thermoelectric device with a multilayer structure composed of layers with different metal contents has a high thermoelectric figure of merit. Antimony telluride (Sb2Te3) and bismuth telluride (Bi2Te3) are two types of traditional thermoelectric materials that have low thermal conductivity and high electrical conductivity. The periodic thermoelectric coating is composed of alternating layers of Sb2Te3 and Bi2Te3 stacked on a Bi2Te3 thermoelectric substrate. There are 8 layers of Sb2Te3 and 7 layers of Bi2Te3. The thickness of each thermal electron layer is equal and depends on the thickness of the thermoelectric coating. The values of the electrical conductivity and the thermal conductivity are shown in Table 2.

[0170] Table 2

[0171]

[0172] Figure 5 The effect of different coating thickness to contact width ratios h / a on the dimensionless normal current density j ey (x, 0) / j ey0 (x, 0) is shown. The contact width is a = 0.1 mm. From Figure 5 (a) and Figure 5 (b), one can see that at the center of the contact area (x / a = 0.0314, x / a = 0.5621), the dimensionless normal current density j ey (x, 0) / j ey0increases significantly, and then fluctuates around a stable value. However, at the contact boundary (x / a = 0.9409), j ey (x, 0) / j ey0 decreases first and then stabilizes. Figure 6 (a) and Figure 6 (a) and uy (x, 0) / j uy0 ( j uy0 = J u0 / 2a) are given. The variation trend is similar to that of the dimensionless normal current density in Figure 5 (a) and Figure 5 (b), but its variation and fluctuation are smaller.

[0173] The second issue of interest in this example is the contact of a performance- arbitrarily varying graded thermoelectric coating with an electrode. The thickness of the thermoelectric coating is h. Unlike in Literature 1, Literature 2 (Tian XJ, Zhou YT, Wang LH, Ding SH. Surface contact behavior of functionally graded thermoelectric materials indented by a conducting punch. Applied Mathematics and Mechanics (English Edition) 2021; 42:649-64.), the current multilayer thermoelectric model does not require the thermal conductivity gradient parameter and the electrical conductivity gradient parameter to be equal when solving the temperature field. As shown in Figure 2 , the electrical conductivity and thermal conductivity can be expressed as following power-law functions:

[0174]

[0175] where the exponents η1and η2are called power-law exponents. The material performance of the functionally graded thermoelectric coating exhibits non-proportional variation. The coating is considered to consist of Sb2Te3at the surface y = 0 and Bi2Te3at y = h. The graded thermoelectric coating is bonded to a uniform Bi2Te3substrate, thus ensuring the continuity of the performance at the interface. The performance parameters of Sb2Te3and Bi2Te3are shown in Table 2.

[0176] Figure 7 The distribution of the electrical conductivity in γ(y) at different exponents η1is given. It can be seen that η1= 1 describes a linear gradient of electrical conductivity, η1→ ∞ corresponds to the electrical conductivity of the material being γ0, and η1→ 0 corresponds to the electrical conductivity of the material being equal to γ hThe four types of power law exponents used in numerical analysis are shown in Table 3.

[0177] Table 3

[0178]

[0179] Figure 8 The distribution of the non-dimensional normal current density j ey (x, 0) / j ey0 is shown for three different functional gradient thermoelectric coatings and a uniform Sb2Te3coating. From Figure 8 (a) in FIG. 6, as the conductivity power law exponent η1 increases, the j ey (x, 0) / j ey0 slightly monotonically decreases in the middle of the contact region, which is consistent with the trend observed in Figure 8 (b) in FIG. 6 (x / a = 0.0314, x / a = 0.5621). As shown in Figure 8 (b) in FIG. 6, at the contact region boundary (x / a = 0.9409), the influence of the thermoelectric coating performance (characterized by a smaller η1) on the normal current density j ey (x, 0) / j ey0 is less than that of a uniform thermoelectric coating (characterized by a relatively large η1value, such as 10 or 15). Figure 9 (a) in FIG. 6 and Figure 9 (b) in FIG. 6 show the influence of the thermal conductivity power law exponent on the non-dimensional normal energy flux, and the influence trends are consistent with Figure 8 (a) in FIG. 6 and Figure 8 (b) in FIG. 6.

[0180] Example Two

[0181] In this example, an electro-thermal analysis system for an arbitrary performance gradient thermoelectric coating in contact with an electrode includes a model construction module, a first equation construction module, a second equation construction module, and a solution analysis module.

[0182] The model construction module is configured to discretize the functional gradient thermoelectric coating along the thickness direction into multiple layers of uniform thermoelectronic layers, and construct a multi-layer thermoelectric material model. The first equation construction module is configured to establish a thermoelectric coupling control equation in each uniform thermoelectronic layer in the multi-layer thermoelectric material model, and set boundary and interface conditions. The second equation construction module is configured to solve the thermoelectric coupling control equation using Fourier transform, and derive Cauchy singular integral equations satisfied by the normal current and the normal energy flux in combination with the interlayer continuity conditions. The solution analysis module is configured to discretize and solve the Cauchy singular integral equations using a collocation method, and obtain coating performance indicators.

[0183] The above described embodiments are only to illustrate the preferred modes of the present application, and are not intended to limit the scope of the present application. Any modification and improvement made by those skilled in the art to the technical solutions of the present application without departing from the design spirit of the present application shall fall within the protection scope of the present application as defined by the claims.

Claims

1. An electrothermal analysis method for contact between an arbitrary performance gradient thermoelectric coating and an electrode, characterized in that: The following steps are involved: The functionally graded thermoelectric coating is discretized into multiple uniform thermoelectron layers along the thickness direction to construct a multilayer thermoelectric material model. Establishing thermoelectric coupling control equations in each of the uniform hot electron layers in the multilayer thermoelectric material model, and setting boundary and interface conditions; The thermoelectric coupling control equation is solved by Fourier transform, and the Cauchy singular integral equation satisfied by the normal current and normal energy flow is derived in combination with the interlayer continuity condition; The Cauchy singular integral equation is discretized and solved using the collocation method to obtain coating performance indicators.

2. The electrothermal analysis method of the contact between an arbitrary performance gradient thermoelectric coating and an electrode according to claim 1, characterized in that: The multilayer thermoelectric material model constructed includes: An electrode with a width of 2a is in contact with a multilayer thermoelectric coating with a thickness of h, which is composed of N thermoelectric elastic sublayers and is bonded to a thermoelectric semi-planar substrate.

3. The electrothermal analysis method of the contact between an arbitrary performance gradient thermoelectric coating and an electrode according to claim 1, characterized in that: The method for establishing the thermoelectric coupling control equation includes: Construct the constitutive relations of the hot electron layer and the thermoelectric substrate: in, represents the current density vector in the i-th layer, x represents the horizontal coordinate, y represents the vertical coordinate, represents the gradient operator, γ i Indicates the conductivity coefficient, F i represents the electrochemical potential function, h represents the thickness of the thermoelectric coating, h i represents the thickness of the i-th hot electron layer, represents the energy flow vector in the i-th layer, λ i represents the thermal conductivity coefficient, T i Indicates temperature; Based on the constitutive relationship, the thermoelectric coupling control equation is constructed: in, Represents the Laplace operator.

4. The electrothermal analysis method of the contact between an arbitrary performance gradient thermoelectric coating and an electrode according to claim 3, characterized in that: The boundary and interface conditions include: For the electric field, the mixed boundary condition on the surface of the multilayer thermoelectric coating is: F i (x,h i )=F i+1 (x,h i ),i=1,...,N,-∞<x<∞ j iey (x,h i )=j (i+1)ey (x,h i ),i=1,...,N,-∞<x<∞ Among them, j (N+1)ey (x, 0) represents the normal current density at the contact surface, represents the unknown surface normal current density in the contact area, a represents half the width of the contact area, and F N+1 (x, 0) represents the electrochemical potential function at the contact surface, F i (x,h i ) represents the electrochemical potential function of the i-th layer at y = h i The value at j iey (x,h i ) represents the normal current density of layer i at y = h i The value of J e0 represents the total current load applied; For the temperature field, the mixed boundary conditions on the surface of the multilayer thermoelectric coating are: T i (x,h i )=T i+1 (x,h i ),i=1,...,N,-∞<x<∞ j iuy (x,h i )=j (i+1)uy (x,h i ),i=1,...,N,-∞<x<∞ Among them, j (N+1)uy (x, 0) represents the normal energy flow at the contact surface, represents the unknown surface normal energy flow in the contact area, T N+1 (x, 0) represents the temperature at the contact surface, T i (x,h i ) indicates that the temperature of layer i is at y = h i The value at j iuy (x,h i ) indicates that the normal energy flow of layer i is at y = h i The value of J u0 represents the total energy flow applied.

5. The electrothermal analysis method of the contact between an arbitrary performance gradient thermoelectric coating and an electrode according to claim 4, characterized in that: The method for obtaining the Cauchy singular integral equation includes: Applying Fourier transform to the governing equation of thermoelectric coupling with respect to x, we obtain the general solution of the electrochemical potential function and temperature in each thermoelectric layer: Among them, F i (x, y) represents the general solution of the electrochemical potential function in the i-th hot electron layer, A i 1(s) and A i2 (s) represents the coefficient to be determined based on the boundary conditions, s represents the Fourier transform variable, T i (x, y) represents the general solution of temperature, T Hi (x,y) represents the homogeneous solution of the temperature in the i-th hot electron layer; Based on the general solution, the singular integral equation is obtained: Among them, k 1inf and k 2inf represents the limit value, k 11 (r, x) and k 22 (r, x) represents the kernel function, and r represents the integration variable.

6. The electrothermal analysis method of the contact between an arbitrary performance gradient thermoelectric coating and an electrode according to claim 5, characterized in that: The method for solving the Cauchy singular integral equation includes: Introducing normalized variables and Convert the interval (-a,a) to (-1,1) and rewrite the singular integral equation as follows: in, and represents a normalized variable belonging to (-1,1), represents the unknown normal current density, represents the unknown normal energy flow; The unknown normal current density and unknown normal energy flux in the equation are approximated using the following series: in, represents the function to be solved related to the normal current density expanded by series, represents the function to be solved related to the normal current density expanded by series, represents the first kind of Chebyshev polynomials, d j and c j represents the undetermined coefficient, j=0,1,...,M-1; Substitute the series into the rewritten singular integral equation and use the collocation method to obtain the solution: in, and Indicates a configuration point.

7. An electrothermal analysis system in which an arbitrary performance gradient thermoelectric coating contacts an electrode, the system applying the method according to any one of claims 1 to 6, characterized in that: include: A model building module, a first equation building module, a second equation building module and a solution analysis module; The model building module is used to discretize the functionally gradient thermoelectric coating into multiple layers of uniform thermoelectric layers along the thickness direction to build a multi-layer thermoelectric material model; The first equation building module is used to establish a thermoelectric coupling control equation in each uniform hot electron layer in the multilayer thermoelectric material model and set boundary and interface conditions; The second equation building module uses Fourier transform to solve the thermoelectric coupling control equation, and combines the interlayer continuity condition to derive the Cauchy singular integral equation satisfied by the normal current and normal energy flow; The solution analysis module discretizes and solves the Cauchy singular integral equation using a collocation method to obtain coating performance indicators.

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