Effective performance prediction method for finite-size thermoelectric plate containing any plurality of round inclusions

Abstract

The invention discloses an effective performance prediction method for a finite-size thermoelectric plate containing any multiple round inclusions, and the method comprises the steps: 1) building series expressions of an electric field complex potential function and a thermal field complex potential function in the inclusions and a matrix material according to the shape of the plate and the size and position of the inclusions; 2) establishing an interface inner boundary equation and a plate edge outer boundary equation which need to be satisfied by the electric field complex potential function; 3) establishing a coefficient matrix according to an electric field boundary condition, and solving to obtain an electric field complex potential function in a matrix and inclusions; 4) establishing a temperature field interface continuous condition and a plate edge outer boundary condition, and constructing a satisfied thermal field complex potential function in inclusions and matrixes of the inner boundary equation set and the outer boundary equation set; 5, the obtained electric field complex potential function and the obtained thermal field complex potential function are used for thermoelectric semiconductor coupling temperature field calculation and effective thermoelectric performance evaluation.The method is directly used for thermoelectric composite material energy conversion efficiency evaluation and prediction, and research, development and application of efficient thermoelectric devices are promoted.

Description

Technical Field

[0001] The invention relates to a method for predicting equivalent parameters of composite materials, in particular to a method for predicting effective thermoelectric performance of a thermoelectric plate of finite size containing any number of circular inclusions. Background Art

[0002] Thermoelectric materials are a new type of functional material that can directly realize the mutual conversion of electrical energy and thermal energy on a solid-state basis. Thermoelectric devices made of them have no moving parts, noise and pollutant emissions, which makes them very attractive in the field of renewable energy, such as waste heat recovery, solar power generation, carbon emission reduction and semiconductor refrigeration. High-performance thermoelectric materials require materials to have a high Seebeck coefficient, high electrical conductivity and low thermal conductivity. However, these seemingly simple requirements are difficult to meet at the same time. For example, metals have high electrical conductivity, but at the same time they are also good conductors of heat, making the thermoelectric effect of metal materials extremely weak. The mutually coupled electric-thermal transport processes within the material make it very challenging to improve the conversion efficiency of a single, homogeneous thermoelectric material. Therefore, in the process of developing high-performance thermoelectric materials, a lot of work is done to form composite materials by introducing inclusions / fibers into single-phase thermoelectric materials, thereby improving the overall energy conversion efficiency of the composite system.

[0003] On the other hand, thermoelectric devices are inevitably subjected to large temperature differences, large currents, and non-uniform heat flow during service. If the physical properties of the introduced reinforcement phases such as inclusions or fibers are quite different from those of the matrix material, the thermoelectric composite materials in service will inevitably experience severe thermal current accumulation and thermal stress concentration at the inclusion-matrix interface. In fact, most thermoelectric materials are brittle semiconductors with poor mechanical properties. Excessive heat accumulation and thermal stress concentration may cause local melting and cracking of the material, which in turn causes premature failure of the entire thermoelectric device. Accurately solving the electric field and temperature field in thermoelectric composite materials containing inclusions under the combined action of strong current and large temperature difference is crucial for the performance optimization and reliability analysis of thermoelectric composite materials. However, under the conditions of strong current and large temperature difference, the internal electrical transport and heat conduction processes of thermoelectric materials have strong nonlinear coupling effects, which leads to great mathematical difficulties in predicting the effective performance of thermoelectric materials containing inclusions. The only theoretical research work is also limited to the simple case of a single inclusion, that is, it is assumed that the size of the inclusion is much smaller than the matrix material, and the matrix material can be regarded as infinite in mathematics. Although this simplified model is well applicable to thermoelectric composite materials with a small inclusion volume fraction, when the inclusion volume fraction increases, the mutual influence between inclusions cannot be ignored. Continuing to use this simplified model cannot accurately evaluate the effective performance of the thermoelectric composite system. Summary of the invention

[0004] The purpose of the present invention is to overcome the defects of the prior art and provide a method for predicting the effective performance of a finite-size thermoelectric plate containing any number of circular inclusions, which is used to analyze the temperature distribution in a finite-size thermoelectric plate containing any number of circular inclusions at any position and to evaluate the effective thermoelectric performance. It is directly used for the effective performance evaluation of thermoelectric composite materials and the optimization design of the microstructure of thermoelectric composite materials, and has great value in promoting the research and development and application of high-efficiency and high-reliability thermoelectric composite materials.

[0005] The object of the present invention is achieved by: a method for predicting the effective performance of a thermoelectric plate of finite size containing any number of circular inclusions, comprising the following steps:

[0006] 1) According to the plate shape and inclusion size and position, establish the series expressions of the electric field complex potential function and thermal field complex potential function in the inclusion and matrix material;

[0007] 2) According to the electric field continuity conditions on the interface between inclusions and matrix and the electric charge on the outer edge of the thermoelectric plate, the precise electric field inner boundary conditions of the inclusion-matrix interface and the electric field outer boundary conditions of the thermoelectric plate outer edge are established;

[0008] 3) According to the boundary conditions, a coefficient matrix is ​​established, and the Lagrange multiplier method is used to construct a linear equation system that is exactly satisfied by the inner boundary and approximately satisfied by the outer boundary. The electric field complex potential function h0(z) in the matrix and the electric field complex potential function h0(z) in the inclusion are obtained by solving the equation. j Semi-analytical solution of (z);

[0009] 4) According to the temperature field interface continuity conditions and boundary conditions, establish the inclusion thermal field potential function g j (z) and the matrix thermal field potential function g0(z) satisfy the inner boundary equation group and the outer boundary equation group; the Lagrange multiplier method is used to obtain the least square solution of the temperature field that exactly satisfies the inner boundary and approximately satisfies the outer boundary;

[0010] 5) The obtained electric field complex potential function and thermal field complex potential function are used to represent the physical quantities inside the material and to evaluate the energy conversion efficiency of thermoelectric semiconductors.

[0011] As a further limitation of the present invention, the step 1) specifically comprises:

[0012] According to the properties of the analytical function, the area S occupied by the jth inclusion j The electric field complex potential function and thermal field complex potential function inside the inclusion have the following series form:

[0013]

[0014]

[0015] Where N is the number of inclusions, A j,0 and B j,0,j=1,2,…N are 2N real constants, which are related to the equilibrium potential and uniform temperature of each inclusion; A j,k and B j,k ,j=1,2,…,N;k=1,2,…+∞is the unknown complex constant to be determined by the boundary conditions; z 0j is the coordinate of the center of the jth circle inclusion, R j is the radius of the jth circular inclusion; the area S0 occupied by the matrix is ​​the area inside the area surrounded by the outer boundary L0 and the circular inclusion S j The electric field complex potential function and thermal field complex potential function in the matrix material have the following series form:

[0016]

[0017]

[0018] Among them, A 0,0 and B 0,0 are two real constants, which are related to the equilibrium potential and uniform temperature on the substrate; A 0,k , B 0,k , A j,-k and B j,-k are unknown complex coefficients; constants z0 and R0 are two constants related to the finite matrix size.

[0019] As a further limitation of the present invention, the step 2) specifically comprises:

[0020] Since the inclusion and the matrix are perfectly connected, the matrix material domain S0 and the inclusion domain S j Interface L j The thermoelectric potential, net current, temperature, and net energy flow on both sides are continuous, so the inner boundary conditions are:

[0021]

[0022]

[0023] Where U (0) (z) is the thermoelectric potential on the substrate side, U (j) (z) is the thermoelectric potential on the inclusion side; is the net current on the substrate side, is the net current on the mixed side; T (0) (z) is the temperature on the substrate side, T (j) (z) is the temperature on the inclusion side; is the net energy flow on the substrate side, is the net energy flow on the inclusion side; the superscripts (0) and (j) indicate that the physical quantity is on the matrix side and the inclusion side, respectively;

[0024] The outer boundary conditions are established. On the outer edge of the matrix material, the electric potential and temperature of the upper and lower boundaries are given constants, so the upper and lower boundary conditions are:

[0025] U (0) (z) = φ 01 +ε0T 01 ,T (0) (z) = T 01 ,z∈CD

[0026] U (0) (z) = φ 02 +ε0T 02 ,T (0) (z) = T 01 ,z∈AB

[0027] Where ε0 is the Seebeck coefficient of the matrix material, φ 01 and T 01 are the potential and temperature of the lower boundary, respectively, 02 and T 02 are the electric potential and temperature of the upper boundary respectively; CD and AB are the upper and lower outer boundaries of the finite plate;

[0028] The left and right boundaries of the matrix material are in contact with the air and are considered to be electrically and thermally insulated; the normal current component and normal heat flow component on the left and right boundaries of the matrix material are zero, that is, the net current density and net energy flow density are both zero, so the left and right boundary conditions are:

[0029]

[0030] Among them, DA and BC are the left and right outer boundaries of the finite plate.

[0031] As a further limitation of the present invention, the step 3) specifically comprises:

[0032] According to the electric field boundary conditions in the boundary conditions, solve the electric field potential function in the matrix and the inclusion. The electric field complex potential function h0(z) in the matrix and the electric field complex potential function h0(z) in the inclusion are solved. j (z), j = 1, 2, ..., N; On the inner boundary, the thermoelectric potential continuity and current conservation conditions on both sides of the matrix and inclusion interface require:

[0033]

[0034] Among them, the inner boundary L j Take office a little satisfaction zz 0j =R j e iθ =R j σ, where σ = e iθ Denotes any point on the unit circle that satisfies

[0035] Substituting the truncated form of the electric field complex potential function in the matrix and the inclusion with a finite number of terms into the inner boundary conditions of the electric field, we obtain:

[0036]

[0037] where M is the number of series terms intercepted. Using the mathematical software Maple, the constant terms on both sides of the equation are separately taken out, and the coefficients of σ 0 are made equal to obtain a system of N real coefficient equations:

[0038]

[0039] When j = 1 to N, the coefficients of the same power terms of σ k , k = ±1, ±2,... ±M on both sides of the equation are made equal, and a complex coefficient system of equations with 2MN in number about the complex coefficients A 0,k , A j,-k , A j,k j = 1, 2,... N; k = 1, 2,... M is obtained; separating the real and imaginary parts of the complex coefficients, a system of 4MN real coefficient linear equations is obtained; the total of N + 4MN real coefficient linear equations is called the inner boundary equations of the electric field and is denoted as:

[0040] M1X = Y1,

[0041] where X is a column vector with (N + 1) + 2(M + 2NM) dimensions composed of the unknown coefficients to be solved, which includes N + 1 real constants A 0,0 , A j,0 , j = 1, 2,... N and the real and imaginary parts of M + 2NM complex constants A 0,k , A j,-k , A j,k j = 1, 2,... N; k = 1, 2,... M; M1 is a real coefficient matrix with N + 4NM rows and (N + 1) + 2(M + 2NM) columns;

[0042] Select W collocation points z pi , i = 1, 2,…W on the outer boundary L0, and require that these collocation points be as evenly distributed on L0 as possible; substituting the electric field complex potential function of the matrix into the outer boundary conditions of the electric field, we obtain:

[0043]

[0044] A total of W complex equations are obtained at these W collocation points. Separating the real and imaginary parts of this complex coefficient system of equations, 2W real coefficient equations are obtained, denoted as the outer boundary equations of the electric field:

[0045] M2X = Y2

[0046] Wherein, M2 is a real matrix with 2W rows (N+1)+2(M+2NM) columns;

[0047] The determination of the electric field complex potential function is reduced to solving the equation system M1X=Y1 and M2X=Y2, and the following function L(X) is constructed using the Lagrange multiplier method:

[0048] L(X)=(M2X-Y2) T (M2X-Y2)+Θ T M1X

[0049] Where the superscript T represents the transpose of the matrix, and Θ is the Lagrange multiplier, which is an undetermined real column vector; the variation of the function L(X) with respect to the variable X is written as:

[0050]

[0051] From the above formula, we can get that the extreme point of the function L(X) satisfies:

[0052]

[0053] At the same time, the extreme point of the function L(X) is required to satisfy the formula M1X=Y1, so:

[0054]

[0055] Where [Θ] is the Lagrange multiplier, which is a column vector determined by the above formula. Using the mathematical software Maple to solve the equation, we can obtain a solution that strictly satisfies the internal boundary condition M1X=Y1 and approximately satisfies the external condition M2X=Y2:

[0056]

[0057] Once the electric field complex potential function is determined, it is used as a known quantity to solve the thermal field complex potential function.

[0058] As a further limitation of the present invention, the step 4) specifically comprises: at the inner boundary L of each inclusion and the matrix j On the surface, temperature continuity and energy flow conservation are satisfied:

[0059]

[0060] Add the two equations in the above formula and substitute the truncated form of the thermal field complex potential function g0(z) and g j (z), we get the boundary equation of the thermal field:

[0061]

[0062] Among them, Ω j (σ) is a known function of σ:

[0063]

[0064]

[0065]

[0066]

[0067]

[0068] Substituting the thermal field complex potential function of the substrate into the temperature field outer boundary condition, we get:

[0069]

[0070]

[0071] Similarly, since the complex potential function h0(z) of the electric field in the matrix has been calculated, it is Pi The value h0(z pi ) is the known quantity:

[0072]

[0073] According to the inner and outer boundary conditions of the thermal field, the constrained least squares method is constructed using the Lagrange multiplier method to solve the two sets of boundary conditions, and the solution of the thermal field complex potential function that satisfies the inner boundary conditions of the temperature field and approximately satisfies the outer boundary conditions of the temperature field is obtained.

[0074] As a further limitation of the present invention, the step 5) specifically includes: when the electric field complex potential function h in the inclusion and the matrix j (z), j = 0, 1, 2, ... N and the thermal field complex potential function g j (z), j = 0, 1, 2, ... N is determined, then the thermoelectric field variables at any position in the inclusion and matrix, including temperature, electric potential, current density and heat flux density, are calculated according to the definition formula; after obtaining the temperature field in the thermoelectric plate, the energy conversion efficiency of the thermoelectric plate is calculated by the following formula:

[0075]

[0076] In the formula, η represents the effective energy conversion efficiency of the thermoelectric plate, Q represents the net heat flux density on a boundary, and the max() function represents the maximum of the quantities in the brackets.

[0077] The present invention adopts the above technical scheme, and compared with the prior art, the beneficial effects are as follows: according to the plate shape and the size and position of the inclusions, the series expressions of the electric field complex potential function and the thermal field complex potential function in the inclusion and the matrix material are established; according to the thermal-electric field continuity conditions on the interface between the inclusion and the matrix, the inner boundary conditions that need to be accurately satisfied for the electric field potential function and the thermal field potential function are obtained. On the outer edge of the matrix, the outer boundary equations of the matrix electric field potential function and the thermal field potential function are obtained by the collocation method. Finally, using the Lagrange multiplier method, a linear equation group that is accurately satisfied by the inner boundary and approximately satisfied by the outer boundary is constructed, and the semi-analytical solution of the coupled thermal-electric field in the matrix and the inclusion is obtained by solving. The temperature field calculation results can be directly used for the evaluation of the energy conversion efficiency of thermoelectric semiconductors and the thermal stress analysis of thermoelectric composite materials. When the thermoelectric coupling coefficient of the thermoelectric matrix and the inclusion is set to zero, the established analytical model can simulate the situation where the metal conductor contains multiple conductive inclusions. When the electrical conductivity and thermal conductivity of the inclusion are set to zero, the situation where the inclusion degenerates into an insulating medium (such as air) can also be simulated. The method of the present invention is applicable to the numerical solution method of the effective performance of the thermoelectric plate of the material containing any number of circular inclusions, any position, and any size, and the internal electric-thermal strong coupling temperature field of the finite thermoelectric semiconductor under the combined action of strong current and large temperature difference. The model can be directly used for the effective performance evaluation of thermoelectric composite materials and the optimization design of the microstructure of thermoelectric composite materials, and has great value in promoting the research and development and application of high-efficiency and high-reliability thermoelectric composite materials. The analysis method of the present invention has the advantages of flexibility, reliability, and fast convergence. It is directly used for the evaluation and prediction of the energy conversion efficiency of thermoelectric composite materials, and can promote the research and development and application of high-efficiency thermoelectric devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 Schematic diagram of a thermoelectric plate of finite size containing multiple circular inclusions under the combined action of potential difference and temperature difference in the method of the present invention.

[0079] Figure 2 Embodiment 1 of the present invention contains three types of different loading schematic diagrams of finite-size thermoelectric materials.

[0080] Figure 3 The temperature distribution results at the interface between the inclusion and the matrix in Example 1 of the present invention are compared with the finite element method.

[0081] Figure 4 Schematic diagram of the effect of insulating holes on the temperature gradient at the interface of thermoelectric inclusions in Example 1 of the present invention. Figure 5 Schematic diagram of the effect of metal inclusions on the temperature gradient at the interface of thermoelectric inclusions in Example 1 of the present invention. Figure 6 Schematic diagram of the effect of metal inclusion size on the effective conversion efficiency of a thermoelectric plate in Example 2 of the present invention.

[0082] Figure 7 The effect of the size of the thermoelectric inclusions on the effective conversion efficiency of the thermoelectric plate in the second embodiment of the present invention. DETAILED DESCRIPTION

[0083] like Figure 1 A method for predicting the effective performance of a thermoelectric plate of finite size containing any number of circular inclusions is shown, comprising the following steps:

[0084] 1) According to the plate shape and inclusion size and position, establish the series expressions of the electric field complex potential function and thermal field complex potential function in the inclusion and matrix material;

[0085] Without considering the temperature dependence of material parameters, the fully coupled electric-heat transport equation inside a homogeneous thermoelectric plate on the xy plane can be expressed as:

[0086]

[0087] In the formula, and Represent the current density vector and heat flux density vector in the thermoelectric plate, respectively. The superscript T represents the matrix transposition. φ(x, y) and T(x, y) represent the electric potential and temperature distribution function on the plate, respectively. δ and κ represent the electrical conductivity and thermal conductivity, respectively. ε is the Seebeck coefficient of the thermoelectric plate, also known as the thermoelectric coupling coefficient. At the same time, the electrical-thermal transport process inside the material satisfies the conditions of conservation of matter (carriers) and conservation of energy:

[0088]

[0089] In the formula, is the energy flow density vector. Since only electrical energy and thermal energy are converted to each other during the thermoelectric transport process, the energy flow density vector can be expressed as:

[0090]

[0091] According to the conditions of current conservation and energy conservation in the plate, the thermal-electric field control equation can be expressed as:

[0092]

[0093] Among them, U(x,y)=φ(x,y)+ε·T(x,y) represents the total thermoelectric potential distribution including the Seebeck effect; according to the complex function theory, the thermoelectric potential U(x,y), temperature T(x,y) and electric potential φ(x,y) can be expressed as:

[0094] U=φ+εT=Re[h(z)]

[0095]

[0096]

[0097] Among them, z = x + I y, is an imaginary unit, h(z) and g(z) are the electric field complex potential function and the thermal field complex potential function, respectively, and Re represents the real part of the complex function. With the help of the above two thermoelectric field related potential functions, the expressions of the current, energy flow and heat flux density components in the thermoelectric plate are

[0098] j ex -I·j ey = -δh′(z)

[0099]

[0100]

[0101] It can be seen that the physical quantities related to the electric field and temperature field in the thermoelectric plate can be completely determined by the two introduced analytical functions (the electric field complex potential function h(z) and the thermal field complex potential function g(z)). From the coordinate transformation relationship, it can be obtained that the thermocurrent component in polar coordinates and the component in rectangular coordinates satisfy

[0102] j en -I·j et =e Iθ (j ex -I·j ey )

[0103] j un -I·j ut =e Iθ (j ux -I·j uy )

[0104] q n -I·q t =e Iθ (q x -I·q y )

[0105] In the formula, the subscripts n and t represent the radial component and the tangential component respectively, and θ represents the angle between the radial direction and the positive direction of the horizontal coordinate axis. Let s represent any boundary, and the net current density Q on it is e and the net energy flux density Q u Equal to

[0106] Q e =∫j en (s)ds=∫(j ex dy-j ey dx)=-δIm[h(z)]

[0107]

[0108] Where Im represents the imaginary part of the complex function.

[0109] According to the properties of the analytical function, the area S occupied by the jth inclusion is j The electric field complex potential function and thermal field complex potential function inside the inclusion have the following series form:

[0110]

[0111]

[0112] Where N is the number of inclusions, A j,0 and B j,0 ,j=1,2,…N are 2N real constants, which are related to the equilibrium potential and uniform temperature of each inclusion; A j,k and B j,k ,j=1,2,…,N;k=1,2,…+∞is the unknown complex constant to be determined by the boundary conditions; z 0j is the coordinate of the center of the jth circle inclusion, R j is the radius of the jth circular inclusion; the area S0 occupied by the matrix is ​​the area inside the area surrounded by the outer boundary L0 and the circular inclusion S j The electric field complex potential function and thermal field complex potential function in the matrix material have the following series form:

[0113]

[0114] Among them, A 0,0 and B 0,0 are two real constants, which are related to the equilibrium potential and uniform temperature on the substrate; A 0,k , B 0,k , A j,-k and B j,-k are unknown complex coefficients; constants z0 and R0 are two constants related to the finite matrix size.

[0115] 2) According to the thermal-electric field continuity conditions on the interface between inclusions and matrix, the precise inner boundary conditions and the approximate outer boundary conditions of the inclusion-matrix interface are established;

[0116] Since the inclusion and the matrix are perfectly connected, the matrix material domain S0 and the inclusion domain S j Interface L j The thermoelectric potential, net current, temperature, and net energy flow on both sides are continuous, so the inner boundary conditions are:

[0117]

[0118] Where U (0) (z) is the thermoelectric potential on the substrate side, U (j) (z) is the thermoelectric potential on the inclusion side; is the net current on the substrate side, is the net current on the mixed side; T(0) (z) is the temperature on the substrate side, T (j) (z) is the temperature on the inclusion side; is the net energy flow on the substrate side, is the net energy flow on the inclusion side; the superscripts (0) and (j) indicate that the physical quantity is on the matrix side and the inclusion side, respectively;

[0119] The outer boundary conditions are established. On the outer edge of the matrix material, the electric potential and temperature of the upper and lower boundaries are given constants, so the upper and lower boundary conditions are:

[0120] U (0) (z) = φ 01 +ε0T 01 ,T (0) (z) = T 01 ,z∈CD

[0121] U (0) (z) = φ 02 +ε0T 02 ,T (0) (z) = T 01 ,z∈AB

[0122] Where ε0 is the Seebeck coefficient of the matrix material, φ 01 and T 01 are the potential and temperature of the lower boundary, respectively, 02 and T 02 are the electric potential and temperature of the upper boundary respectively; CD and AB are the upper and lower outer boundaries of the finite plate;

[0123] The left and right boundaries of the matrix material are in contact with the air and are considered to be electrically and thermally insulated; the normal current component and normal heat flow component on the left and right boundaries of the matrix material are zero, that is, the net current density and net energy flow density are both zero, so the left and right boundary conditions are:

[0124]

[0125] Among them, DA and BC are the left and right outer boundaries of the finite plate.

[0126] 3) According to the boundary conditions, a coefficient matrix is ​​established, and the Lagrange multiplier method is used to construct a linear equation system that is exactly satisfied by the inner boundary and approximately satisfied by the outer boundary. The electric field complex potential function h0(z) in the matrix and the electric field complex potential function h0(z) in the inclusion are obtained by solving the equation. j Semi-analytical solution of (z);

[0127] According to the electric field boundary conditions in the boundary conditions, solve the electric field potential function in the matrix and the inclusion. The electric field complex potential function h0(z) in the matrix and the electric field complex potential function h0(z) in the inclusion are solved. j(z), j = 1, 2, …, N; on the inner boundary, the continuity of the thermoelectric potential and the conservation of current on both sides of the matrix and inclusion interface require that:

[0128]

[0129] where the inner boundary L j any point on it satisfies z - z 0j = R j e iθ = R j σ, where σ = e iθ represents any point on the unit circle and satisfies

[0130] Substituting the truncated form of the electric field complex potential function with a finite number of terms in the matrix and inclusion into the electric field inner boundary conditions gives:

[0131]

[0132] where M is the number of series terms intercepted. Using the mathematical software Maple, the constant terms on both sides of the equation are taken out respectively, and the coefficients of σ 0 are made equal to obtain N real coefficient equations:

[0133]

[0134] When j = 1 to N, the coefficients of the same power terms of σ k , k = ±1, ±2,... ±M on both sides of the equation are made equal to obtain a complex equation system with 2MN complex coefficients A 0,k , A j,-k , A j,k j = 1, 2,... N; k = 1, 2,... M; separating the real and imaginary parts of the complex coefficients gives 4MN real coefficient linear equations; the total of N + 4MN real coefficient linear equations are called the electric field inner boundary equation system and denoted as:

[0135] M1X = Y1,

[0136] where X is a column vector of (N + 1) + 2(M + 2NM) dimensions composed of the unknown coefficients to be solved, which includes N + 1 real constants A 0,0 , A j,0 , j = 1, 2,... N and the real and imaginary parts of M + 2NM complex constants A 0,k , A j,-k , A j,k j = 1, 2,... N; k = 1, 2,... M; M1 is a real coefficient matrix with N + 4NM rows and (N + 1) + 2(M + 2NM) columns;

[0137] On the outer boundary L0, W collocation points zpi ,i=1,2,…W, and require these points to be distributed as evenly as possible on L0; Substituting the electric field complex potential function of the matrix into the outer boundary condition of the electric field, we get:

[0138]

[0139]

[0140] At these W points, W complex equations are obtained in total. Separating the real and imaginary parts of the complex coefficient equations, 2W real coefficient equations are obtained, which are recorded as the electric field outer boundary equations:

[0141] M2X=Y2

[0142] Wherein, M2 is a real matrix with 2W rows (N+1)+2(M+2NM) columns;

[0143] The determination of the electric field complex potential function is reduced to solving the equation system M1X=Y1 and M2X=Y2, and the following function L(X) is constructed using the Lagrange multiplier method:

[0144] L(X)=(M2X-Y2) T (M2X-Y2)+Θ T M1X

[0145] Where the superscript T represents the transpose of the matrix, and Θ is the Lagrange multiplier, which is an undetermined real column vector; the variation of the function L(X) with respect to the variable X is written as:

[0146]

[0147] From the above formula, we can get that the extreme point of the function L(X) satisfies:

[0148]

[0149] At the same time, the extreme point of the function L(X) is required to satisfy the formula M1X=Y1, so:

[0150]

[0151] Where [Θ] is the Lagrange multiplier, which is a column vector determined by the above formula. Using the mathematical software Maple to solve the equation, we can obtain a solution that strictly satisfies the internal boundary condition M1X=Y1 and approximately satisfies the external condition M2X=Y2:

[0152]

[0153] Once the electric field complex potential function is determined, it is used as a known quantity to solve the thermal field complex potential function.

[0154] 4) According to the temperature field interface continuity conditions and boundary conditions, establish the inclusion thermal field potential function g that satisfies the inner boundary equation group and the outer boundary equation group j (z) and the matrix thermal field potential function g0(z); the Lagrange multiplier method is used to obtain the least square solution of the temperature field that exactly satisfies the inner boundary and approximately satisfies the outer boundary;

[0155] At the inner boundary L between each inclusion and the matrix j On the surface, temperature continuity and energy flow conservation are satisfied:

[0156]

[0157] Add the two equations in the above formula and substitute the truncated form of the thermal field complex potential function g0(z) and g j (z), we get the boundary equation of the thermal field:

[0158]

[0159] Among them, Ω j (σ) is a known function of σ:

[0160]

[0161]

[0162]

[0163]

[0164]

[0165] Substituting the thermal field complex potential function of the substrate into the temperature field outer boundary condition, we get:

[0166]

[0167]

[0168] Similarly, since the complex potential function h0(z) of the electric field in the matrix has been calculated, it is Pi The value h0(z pi ) is the known quantity:

[0169]

[0170] According to the inner and outer boundary conditions of the thermal field, the constrained least squares method is constructed using the Lagrange multiplier method to solve the two sets of boundary conditions, and the solution of the thermal field complex potential function that satisfies the inner boundary conditions of the temperature field and approximately satisfies the outer boundary conditions of the temperature field is obtained.

[0171] 5) The obtained electric field complex potential function and thermal field complex potential function are used to represent the physical quantities inside the material for evaluating the energy conversion efficiency of thermoelectric semiconductors;

[0172] When the electric field complex potential function h in the inclusion and matrix j (z), j = 0, 1, 2, .N. and the thermal field complex potential function g j (z), j = 0, 1, 2, ... N is determined, then the thermoelectric field variables at any position in the inclusion and matrix, including temperature, electric potential, current density and heat flux density, are calculated according to the definition formula; after obtaining the temperature field in the thermoelectric plate, the energy conversion efficiency of the thermoelectric plate is calculated by the following formula:

[0173]

[0174] In the formula, η represents the effective energy conversion efficiency of the thermoelectric plate, Q represents the net heat flux density on a boundary, and the max() function represents the maximum of the quantities in the brackets.

[0175] Embodiment 1 discloses a method for predicting the electric-thermal coupling temperature field of a finite-size thermoelectric plate considering multiple circular inclusions of arbitrary positions and sizes. The specific method is as follows:

[0176] Step 1) Set the geometric dimensions and material parameters of the thermoelectric plate and the inclusion, and establish Figure 2 The thermoelectric composite system shown (including three circular inclusions of finite-sized thermoelectric plates) is subjected to the combined action of a given potential difference and a temperature difference. In this implementation case, commercial thermoelectric material bismuth telluride (Bi2Te3) is selected as the matrix material, and the inclusions are high-performance thermoelectric semiconductors Ag (Pb 1-y Sn y ) m Sb 2-m , pure metal Sn and holes (air), the specific material parameters are shown in Table 1. The substrate size is 12mm×16mm, the inclusion diameter is 3mm, the potential difference Δφ applied to the upper and lower boundaries of the thermoelectric substrate is 25mV, and the temperature difference ΔT is 50K.

[0177] Table 1 Thermoelectric material parameters

[0178]

[0179] Step 2) according to the size and position of the inclusion, establish the series expression of the electric field complex potential function and the thermal field complex potential function inside the inclusion and the matrix, and establish the precise inner boundary condition and outer boundary approximate condition of the inclusion-matrix interface;

[0180] Step 3) establishing a coefficient matrix according to the boundary conditions, using the constrained least squares method to solve the corresponding electric field boundary value problem, and obtaining the numerical solutions of the electric field complex potential functions h1(z)~h3(z) inside the inclusion and the electric field complex potential function h0(z) inside the matrix;

[0181] Step 4) According to the temperature field interface continuity conditions and boundary conditions, the inclusion thermal field complex potential functions g1(z)~g3(z) and the matrix thermal field complex potential function g0(z) of the inner boundary equation group and the outer boundary equation group that need to be satisfied are established, and the Lagrange multiplier method is also used to obtain the least squares solution of the temperature field that accurately satisfies the inner boundary and approximately satisfies the outer boundary;

[0182] Step 5) The obtained electric field potential function and thermal field potential function are used to represent the physical quantities inside the material, such as temperature T, electric potential φ, etc. According to the definition and Calculate the temperature gradient and analyze the effect of inclusion spacing on the temperature field concentration.

[0183] from Figure 3 It can be seen that the results obtained by the numerical method proposed in this case are highly consistent with the finite element results obtained based on COMSOL Multiphysics.

[0184] Figure 4 It is shown that when the insulating hole approaches the thermoelectric inclusion in the horizontal direction, the horizontal temperature gradient near the thermoelectric inclusion decreases and the vertical temperature gradient increases, indicating that heat conduction in this reverse direction is more hindered, leading to heat accumulation.

[0185] Figure 5 It is shown that when the metal inclusion approaches the thermoelectric inclusion in the horizontal direction, the gradient of the temperature field around the thermoelectric inclusion increases in the horizontal direction and decreases in the vertical direction. This is related to the better thermal conductivity of the metal and the easier it is for heat to penetrate the metal inclusion.

[0186] Embodiment 2

[0187] This implementation case considers that the effective conversion efficiency of a finite-size thermoelectric plate with four circular inclusions is affected by the size of the inclusions. The specific solution process is similar to that of Example 1. First, the solution of the temperature field on the finite-size thermoelectric plate is obtained, then the normal heat flux density at the high-temperature end and the low-temperature end of the thermoelectric plate is calculated, and finally the effective conversion efficiency of the thermoelectric plate is calculated. In this implementation case, the commercial thermoelectric material bismuth telluride (Bi2Te3) is still selected as the matrix material, and the inclusion materials are high-performance thermoelectric semiconductors Ag (Pb 1- y Sn y ) m Sb 2-m and pure metal Sn. The substrate size is 12 mm × 12 mm, and the temperature difference applied between the upper and lower boundaries is ΔT = 50 K. Figure 6and Figure 7 The effects of metal and semiconductor as additions on the effective conversion efficiency of the thermoelectric plate are shown respectively. The thermoelectric performance of metal is relatively low, and as the volume fraction of metal inclusions increases, the effective energy conversion efficiency of the entire thermoelectric plate decreases; while the thermoelectric performance of semiconductors is good, and the effective thermoelectric performance of the plate increases with the increase of the volume fraction of semiconductor inclusions.

[0188] The present invention is not limited to the above-mentioned embodiments. On the basis of the technical solution disclosed in the present invention, technicians in this field can make some substitutions and deformations to some technical features therein according to the disclosed technical content without creative labor, and these substitutions and deformations are all within the protection scope of the present invention.

Claims

1. A method for predicting the effective performance of a finite-size thermoelectric plate containing any number of circular inclusions, characterized in that: The following steps are involved: 1) According to the plate shape and inclusion size and position, establish the series expressions of the electric field complex potential function and thermal field complex potential function in the inclusion and matrix material; 2) According to the electric field continuity conditions on the interface between the inclusion and the matrix and the charge load on the outer edge of the thermoelectric plate, the inner boundary conditions and the outer boundary conditions of the electric field are established; 3) According to the electric field boundary conditions, a coefficient matrix is ​​established. Using the Lagrange multiplier method, a linear equation system is constructed that is exactly satisfied by the inner boundary and approximately satisfied by the outer boundary. The electric field complex potential function h0(z) in the matrix and the electric field complex potential function h0(z) in the inclusion are obtained by solving the equation system. j Semi-analytical solution of (z); 4) According to the temperature field interface continuity condition and the thermal load on the outer edge of the thermoelectric plate, the matrix thermal field potential function g0(z) and the inclusion thermal field potential function g j (z) The inner boundary equations and the outer boundary equations that need to be satisfied; using the Lagrange multiplier method, find the least squares solution of the temperature field that exactly satisfies the inner boundary and approximately satisfies the outer boundary; 5) The obtained electric field complex potential function and thermal field complex potential function are used to represent the physical quantities inside the material, which are used for the calculation of the coupled temperature field inside the thermoelectric semiconductor and the evaluation of the effective thermoelectric performance.

2. The effective performance prediction method of a finite-size thermoelectric plate containing any number of circular inclusions according to claim 1, characterized in that: The step 1) specifically includes: According to the properties of the analytical function, the area S occupied by the jth inclusion j The electric field complex potential function and thermal field complex potential function inside the inclusion have the following series form: Where N is the number of inclusions, A j,0 and B j,0 ,j=1,2,…N are 2N real constants, which are related to the equilibrium potential and uniform temperature of each inclusion; A j,k and B j,k ,j=1,2,…,N;k=1,2,…+∞is the unknown complex constant to be determined by the boundary conditions; z 0j is the coordinate of the center of the jth circle inclusion, R j is the radius of the jth circular inclusion; the area S0 occupied by the matrix is ​​the area inside the area surrounded by the outer boundary L0 and the circular inclusion S j The electric field complex potential function and thermal field complex potential function in the matrix material have the following series form: Among them, A 0,0 and B 0,0 are two real constants, which are related to the equilibrium potential and uniform temperature on the substrate; A 0,k , B 0,k , A j,-k and B j,-k are unknown complex coefficients; constants z0 and R0 are two constants related to the finite matrix size.

3. The effective performance prediction method of a finite-size thermoelectric plate containing any number of circular inclusions according to claim 1, characterized in that: The step 2) specifically includes: Since the inclusion and the matrix are perfectly connected, the matrix material domain S0 and the inclusion domain S j Interface L j The thermoelectric potential, net current, temperature, and net energy flow on both sides are continuous, so the inner boundary conditions are: Where U (0) (z) is the thermoelectric potential on the substrate side, U (j) (z) is the thermoelectric potential on the inclusion side; is the net current on the substrate side, is the net current on the mixed side; T (0) (z) is the temperature on the substrate side, T (j) (z) is the temperature on the inclusion side; is the net energy flow on the substrate side, is the net energy flow on the inclusion side; the superscripts (0) and (j) indicate that the physical quantity is on the matrix side and the inclusion side, respectively; The outer boundary conditions are established. On the outer edge of the matrix material, the electric potential and temperature of the upper and lower boundaries are given constants, so the upper and lower boundary conditions are: U (0) (z)=φ 01 +ε0T 01 ,T (0) (z)=T 01 ,z∈CD U (0) (z)=φ 02 +ε0T 02 ,T (0) (z)=T 01 ,z∈AB Where ε0 is the Seebeck coefficient of the matrix material, φ 01 and T 01 are the potential and temperature of the lower boundary, respectively, 02 and T 02 are the electric potential and temperature of the upper boundary respectively; CD and AB are the upper and lower outer boundaries of the finite plate; The left and right boundaries of the matrix material are in contact with the air and are considered to be electrically and thermally insulated; the normal current component and normal heat flow component on the left and right boundaries of the matrix material are zero, that is, the net current density and net energy flow density are both zero, so the left and right boundary conditions are: Among them, DA and BC are the left and right outer boundaries of the finite plate.

4. The effective performance prediction method of a finite-size thermoelectric plate containing any number of circular inclusions according to claim 1, characterized in that: The step 3) specifically includes: According to the electric field boundary conditions in the boundary conditions, solve the electric field potential function in the matrix and the inclusion. The electric field complex potential function h0(z) in the matrix and the electric field complex potential function h0(z) in the inclusion are solved. j (z), j = 1, 2, ..., N; On the inner boundary, the thermoelectric potential continuity and current conservation conditions on both sides of the matrix and inclusion interface require: Among them, the inner boundary L j Take office a little satisfaction zz 0j =R j e iθ =R j σ, where σ = e iθ Denotes any point on the unit circle that satisfies Substituting the truncated form of the electric field complex potential function in the matrix and inclusions with a finite number of terms into the boundary conditions of the electric field, we obtain: Where M is the number of intercepted series terms. The constant terms on both sides of the equation are extracted using the mathematical software Maple. 0 The coefficients of are equal to obtain a system of N real coefficients: When j = 1 to N, let both sides of the equation be about σ k , k=±1,±2,...±M, the coefficients of the same power terms are equal, and we get the complex coefficient A 0,k , A j,-k , A j,k j=1,2,...N;k=1,2,...M,the number of complex equations is 2MN; separate the real part and imaginary part of the complex coefficients, and get a 4MN real coefficient linear equation system; the total N+4MN real coefficient linear equation system is called the electric field inner boundary equation system and is recorded as: M1X=Y1, Among them, X is a column vector of dimension (N + 1)+2(M + 2NM) composed of unknown coefficients to be solved, which includes N + 1 real constants A 0,0 , A j,0 , j = 1, 2,... N, and M + 2NM complex constants A 0,k , A j,-k , A j,k , j = 1, 2,... N; k = 1, 2,... M, the real and imaginary parts; M1 is a real coefficient matrix with N + 4NM rows and (N + 1)+2(M + 2NM) columns; Select W number of collocation points z on the outer boundary L0 pi ,i=1,2,…W, and require these points to be distributed as evenly as possible on L0; Substituting the electric field complex potential function of the matrix into the outer boundary condition of the electric field, we get: At these W points, W complex equations are obtained in total. Separating the real and imaginary parts of the complex coefficient equations, 2W real coefficient equations are obtained, which are recorded as the electric field outer boundary equations: M2X=Y2 Wherein, M2 is a real matrix with 2W rows (N+1)+2(M+2NM) columns; The determination of the electric field complex potential function is reduced to solving the equation system M1X=Y1 and M2X=Y2, and the following function L(X) is constructed using the Lagrange multiplier method: L(X)=(M2X-Y2) T (M2X-Y2)+Θ T M1X Where the superscript T represents the transpose of the matrix, and Θ is the Lagrange multiplier, which is an undetermined real column vector; the variation of the function L(X) with respect to the variable X is written as: From the above formula, we can get that the extreme point of the function L(X) satisfies: At the same time, the extreme point of the function L(X) is required to satisfy the formula M1X=Y1, so: Where [Θ] is the Lagrange multiplier, which is a column vector determined by the above formula. Using the mathematical software Maple to solve the equation, we can obtain a solution that strictly satisfies the internal boundary condition M1X=Y1 and approximately satisfies the external condition M2X=Y2: Once the electric field complex potential function is determined, it is used as a known quantity to solve the thermal field complex potential function.

5. The effective performance prediction method of a finite-size thermoelectric plate containing any number of circular inclusions according to claim 1, characterized in that: The step 4) specifically includes: j On the surface, temperature continuity and energy flow conservation are satisfied: Add the two equations in the above formula and substitute the truncated form of the thermal field complex potential function g0(z) and g j (z), we get the boundary equation of the thermal field: Among them, Ω j (σ) is a known function of σ: Substituting the thermal field complex potential function of the substrate into the temperature field outer boundary condition, we get: Similarly, since the complex potential function h0(z) of the electric field in the matrix has been calculated, it is Pi The value h0(z pi ) is the known quantity: According to the inner and outer boundary conditions of the thermal field, the constrained least squares method is constructed using the Lagrange multiplier method to solve the two sets of boundary conditions, and the solution of the thermal field complex potential function that satisfies the inner boundary conditions of the temperature field and approximately satisfies the outer boundary conditions of the temperature field is obtained.

6. The effective performance prediction method of a finite-size thermoelectric plate containing any number of circular inclusions according to claim 1, characterized in that: The step 5) specifically includes: when the electric field complex potential function h in the inclusion and the matrix j (z), j = 0, 1, 2, ... N and the thermal field complex potential function g j (z), j = 0, 1, 2, ... N is determined, then the thermoelectric field variables at any position in the inclusion and matrix, including temperature, electric potential, current density and heat flux density, are calculated according to the definition formula; after obtaining the temperature field in the thermoelectric plate, the energy conversion efficiency of the thermoelectric plate is calculated by the following formula: In the formula, η represents the effective energy conversion efficiency of the thermoelectric plate, Q represents the net heat flux density on a boundary, and the max() function represents the maximum of the quantities in the brackets.

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