CN110941882B - Thermal performance analysis method for composite material with curve interface - Google Patents
Thermal performance analysis method for composite material with curve interface Download PDFInfo
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Abstract
The invention discloses a thermal performance analysis method of a composite material with a curve interface, which provides a rapid and accurate numerical analysis method for the thermal performance of a multilayer composite material with a curve interface type for a two-dimensional problem; the method provides interface connection conditions for describing physical quantity jump on an interface through reasonable approximation of a physical process; the heat transfer process of the multilayer composite material is mathematically described by combining a two-dimensional steady heat conduction equation; meanwhile, a control equation and interface conditions are discretized by using an immersion interface method, and a stable, convergent and second-order-precision numerical format is obtained; the invention establishes a mathematical model and a numerical simulation method by solving the problem that the two-dimensional composite material has a curve interface type, and the method can be used for carrying out a high-precision rapid analysis method on the heat conduction process and the thermal resistance performance of the multi-layer composite material such as sandwich steel, sleeve modules and the like.
Description
Technical Field
The invention relates to the application field of multi-medium composite materials, in particular to a thermal performance analysis method of a composite material with a curve interface.
Background
With the development of scientific technology and industrial production, higher requirements are put on heat conducting materials. Compared with the traditional heat conduction materials (metal, metal oxide, ceramic and the like), the high-molecular heat conduction composite material has the advantages of low price, small relative density, easiness in processing, short molding period and the like, and is paid attention to the scientific community. The heat conductive polymer composite material has been successfully applied in various fields and has been increasingly demanded, such as the lighting field, the aerospace field, the automotive field, the electronic field, the battery field and the like. The thermal conductivity of composites with thermal interfaces is of great interest to researchers. The thermal interface material is a composite material which is formed by taking a high polymer material as a matrix and filling heat conducting particles. The thermal interface material can effectively improve the heat conduction between two solid interfaces and plays a vital role in the performance, the service life and the stability of electronic devices. In particular, when two solid interfaces are in direct contact, the roughness of the interfaces can lead to a very small actual contact area between the interfaces, and only 1% -2% of the apparent contact area is needed, so that the interface contact thermal resistance is very large. Therefore, the thermal conductivity of the thermal interface composite material is rapidly and quantitatively analyzed in numerical value, and the thermal interface composite material has very good research significance and application value for the performance research, development and effective utilization of the thermal interface material.
At present, most of thermal performance researches of composite materials adopt experimental research methods, the experimental methods are limited by experimental equipment and complexity of the composite materials, and any constraint of interfaces, so that the thermal performance of the tested materials is difficult to be comprehensively and systematically analyzed. On the other hand, many researchers have proposed some theoretical models, such as equivalent thermal conductivity models of two-phase composites, based on experiments and data, for some simple two-phase composites. Also, researchers have adopted Monte Carlo numerical simulation to statistically analyze the phenomenon of random non-uniform diameter channels in composite materials using ANSYS software. The invention provides a method based on combination of physical mechanism and mathematical modeling, which is used for comprehensively, accurately and detailed numerical analysis of the heat conduction process of a composite material by solving a control equation of the heat conduction process of the composite material. Moreover, the method is not limited by the interface geometry of the composite material, and the composite material with any interface shape can be analyzed.
Disclosure of Invention
The invention aims to provide a thermal performance analysis method of a composite material with a curve interface, which establishes a mathematical model and a numerical simulation method by solving the problem that a two-dimensional composite material has a curve interface type. The method can conveniently and rapidly predict the temperature change of the inner side and the outer side of the composite material, so that the heat conduction performance of the composite material can be evaluated.
In order to solve the technical problems, the following technical means are adopted:
a method for analyzing thermal properties of a composite material having a curvilinear interface, comprising the steps of:
s1: analyzing the actual physical process of the multi-layer heat transfer problem, and analyzing and reasonably supposing the multi-model:
firstly, considering a material model with two layers of different media, wherein a clear interface is formed between different materials; for convenience, the problem of stable heat transfer, namely that the temperature reaches a stable state in the process of transferring the inside of the composite material, is not considered, and the absorption of the material to heat is ignored;
assuming that the materials in each layer are isotropic and uniformly distributed, the three-dimensional model can be considered by simplifying the three-dimensional model into two-dimensional problems along the thickness direction (vertical direction) of the materials;
based on the above analysis and assumptions, the studied problem can be reduced to a two-dimensional heat conduction problem, which is characterized as follows:
(a) Consists of two substances, and a distinct interface is formed between every two layers;
(b) A break in a physical quantity at the material interface;
(c) The temperature is transferred from outside to inside and from high to low;
performing numerical simulation on the steady-state heat conduction process of the composite material of the curve interface; the interlayers among different materials are regarded as non-ideal interfaces through reasonable mathematical assumption, and related interface connection conditions are provided according to the characteristic of discontinuous physical properties on the interfaces; and converting the actual physical problem into a mathematical problem to solve.
: a mathematical description describing the heat transfer process of a multilayer material and a control equation are given:
(a) The steady state heat conduction process inside different media of a two-dimensional problem can be described by the following diffusion equation:
(1)
(2)
wherein the method comprises the steps ofAnd->Is a given function;
(b) Interface between different mediaJumping and interruption can occur, and the following connection conditions are adopted for describing:
(3)
(4)
wherein the method comprises the steps ofFor interface->In area->Unit external normal direction on ∈10->Representing the jump value of the variable at the interface,and->Respectively indicate temperature +.>At interface->Limit values on both sides, i.e.
And->Respectively representing diffusion coefficients of media at two sides of the interface;
the relation between the temperatures at the left and right sides of the interface is established in the formula (3), and the jump of the temperatures at the two sides of the interface is proportional to the heat flow passing through the interface, and the proportionality coefficient isIt can be seen from equation (4) that the heat flow is equal across the interface, i.e. no heat is absorbed or generated at the interface.
: according to the characteristics of parameters, thickness and heat conduction coefficient of the interlayer material, providing interface boundary conditions for describing the discontinuous condition of physical quantity on the interlayer;
dividing the jump condition on the interface into two parts along the coordinate axisAnd->Record->Along +.>Point to->And->The tangent vector on the interface can be expressed as +.>Then the jump condition on the interface can be re-expressed as
(5)
(6)
Jump of physical quantity on interfaceIn the tangential direction can be expressed as,
(7)
wherein the method comprises the steps of
（8)
Formula (6) can be expressed as
(9)
Similarly, formula (7) can be expressed as
(10)
Multiplying formula (10)Obtaining
Solving a system of linear equations as follows
Then the first time period of the first time period,and->Can be expressed as
And
for a pair ofAnd->About->Can obtain
From the control equation
Thus, the first and second substrates are bonded together,
。
: dispersing the heat conduction model of the multilayer heat insulation material by adopting a mathematical method to obtain a discrete linear equation set;
for the two-dimensional problem, selecting a dynamic template mode to perform format construction; dividing all grid nodes into two types according to the position relation between the interface position and the grid nodes: (1) A five-point template is selected by the regular grid points, and all the grid points in the template are on one side of the material interface; (2) Irregular points, selecting a 7-point template, wherein grid points in the template are distributed on two sides of a material interface;
performing orthogonal grid subdivision on the calculation region to obtain a calculation grid; it is assumed that the finite difference format of equation (1) can be expressed as
(11)
For regular grid points
For regular grid points, the 5-point format is adopted for direct dispersion, and the corresponding coefficient is
And is also provided with
The truncation error of the format is：
For irregular grid points
For irregular grid points, 7-point templates are adopted for construction format
(12)
Then, 7 points in the template are found in a self-adaptive mode according to the geometric relation of the interface, and non-central point variables are unfolded to the central point through a Taylor unfolding method; the relation between the decomposition and the derivative of the physical quantity on the interface is utilized to obtain a relation expression that the unknown quantity on the grid point is respectively described by the nodes on the single side of the interface; finally, the expression of truncation error is utilized and is required to satisfyThe method comprises the steps of carrying out a first treatment on the surface of the Further, the coefficient in the formula (12) can be found by adopting a pending coefficient method, and the construction process of the numerical format is completed.
: solving a discrete linear equation set, and analyzing a result;
the model and numerical format are verified by questions with exact solutions.
The beneficial effects of the invention are as follows:
firstly, the invention can be used for high-precision rapid analysis of the heat conduction process and the heat resistance of the sandwich steel, the heat-resistant firewall and other multi-layer composite materials;
secondly, the method provided by the invention has no any limitation on the geometric shape of the interface, so that the thermal conduction process of the combined fitting with any curved boundary surface can be numerically simulated;
thirdly, the invention of the patent adopts a computer numerical simulation method to study the thermal performance of the multilayer material, so the invention is not limited by physical factors such as the type, the thickness and the like of the material, and can be applied to composite multilayer materials of any material;
fourth, the method of this patent is simple, reliable, only need adjust corresponding physical parameter in the computer when the material changes, therefore, it is an economic and saving of money research method, can help research and development and analysis project save cost and time.
Drawings
FIG. 1 is a schematic diagram of a composite material model I with an external interface curve according to the present invention.
FIG. 2 is a schematic diagram of a composite model with an interface within the curve of the present invention, illustrating II.
FIG. 3 is a schematic diagram of the three-dimensional isotropic problem simplified to two-dimensional problem model I of the present invention.
FIG. 4 is a schematic diagram of the three-dimensional isotropic problem simplified to two-dimensional problem model II of the present invention.
FIG. 5 is a template of the two-dimensional curve interface problem of the present invention.
FIG. 6 is a computational grid of the present invention.
Fig. 7 is a numerical solution and exact solution comparison of the present invention.
FIG. 8 is a diagram illustrating the error of the numerical solution of the present invention.
The number of grids is 80 x 80,calculation errors at that time.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments.
Problem analysis and model assumptions
First, consider a material model with two layers of different media, with a clear interface between the different materials, as shown in fig. 1, 2:
for convenience, the problem of steady heat transfer is considered, namely, the temperature reaches a stable state in the process of transferring the inside of the composite material, and the absorption of the material to heat is ignored; assuming isotropic and uniform distribution of material within each layer, the three-dimensional model can be considered as a matter of simplifying into two dimensions along the thickness direction (vertical direction) of the material, as shown in fig. 3 and 4:
based on the above analysis and assumptions, the studied problem can be reduced to a two-dimensional heat conduction problem, which is characterized as follows:
(a) Consists of two substances, and a distinct interface is formed between every two layers;
(b) A break in a physical quantity at the material interface;
(c) The temperature is transferred from outside to inside and from high to low;
the method provided by the invention carries out numerical simulation on the steady-state heat conduction process of the sandwich steel. A schematic diagram of the physical model of the sandwich steel is shown in FIG. 1. The interlayer between different materials can be regarded as a non-ideal interface through reasonable mathematical assumption, and related interface connection conditions are provided according to the characteristic of discontinuous physical properties on the interface. And converting the actual physical problem into a mathematical problem to solve. Since the steady state heat transfer process is described by a second order partial differential equation (a solution equation). Therefore, the control equation of the physical problems of the heat transfer process and the thermal resistance performance of the sandwich steel sheet can be shown by the formulas (1) - (4). The dimensions and thermal conductivity of the sandwich steel plates are shown in table 1 below:
TABLE 1 parameters of different metallic materials
。
Establishing a mathematical model
(a) The steady state heat conduction process inside different media of two-dimensional problem is described by the following diffusion equation:
(1)
(2)
wherein the method comprises the steps ofAnd->Is a given function;
(b) Interface between different mediaJumping and interruption can occur, and the following connection conditions are adopted for describing:
(3)
(4)
wherein the method comprises the steps ofFor interface->In area->The unit external normal direction. />Representing the jump value of the variable at the interface.And->Respectively indicate temperature +.>At interface->Limit values on both sides, i.e.
And->Respectively representing diffusion coefficients of media at two sides of the interface;
the relation between the temperatures at the left and right sides of the interface is established in the formula (3), and the jump of the temperatures at the two sides of the interface is proportional to the heat flow passing through the interface, and the proportionality coefficient isIt can be seen from equation (4) that the heat flow is equal across the interface, i.e. no heat is absorbed or generated at the interface.
Decomposition and derivative relationships at curve interfaces
First, the jump condition on the interface is divided into two parts along the coordinate axisAnd->Record->Along +.>Point to->And->The method comprises the steps of carrying out a first treatment on the surface of the The tangent vector on the interface can be expressed as +.>The method comprises the steps of carrying out a first treatment on the surface of the The jump condition on the interface can be re-expressed as
(5)
(6)
Jump of physical quantity on interfaceIn the tangential direction can be expressed as,
(7)
wherein the method comprises the steps of
(8)
Formula (6) can be expressed as
(9)
Similarly, formula (7) can be expressed as
(10)
Multiplying formula (10)Obtaining
Solving a system of linear equations as follows
Then the first time period of the first time period,and->Can be expressed as
And
for a pair ofAnd->About->Can obtain
From the control equation
Thus, the first and second substrates are bonded together,
。
discrete equations and constructing numerical formats
For two-dimensional problems, the format construction is performed by selecting a dynamic template mode. Dividing all grid nodes into two types according to the position relation between the interface position and the grid nodes, wherein (1) rule grid points are selected, five-point templates are selected, and all grid points in the templates are positioned on one side of a material interface; (2) Irregular points, a 7-point template is selected, and grid points in the template are distributed on both sides of the material interface, as shown in fig. 5.
Orthogonal meshing of the computational region may result in a computational mesh as shown in fig. 6. It is assumed that the finite difference format of equation (1) can be expressed as
(11)
4.1 regular grid points
For regular grid points, the 5-point format is adopted for direct dispersion, and the corresponding coefficient is
And is also provided with
The truncation error of the format is：
4.2 irregular grid points
For irregular grid points, 7-point templates are adopted for construction format
(12)
And then, 7 points in the template are adaptively found according to the geometric relation of the interface, and non-central point variables are unfolded to the central point through a Taylor unfolding method. And the relation between the decomposition and the derivative of the physical quantity on the interface is utilized to obtain a relation expression that the unknown quantity on the grid point is respectively described by the nodes on the single side of the interface. Finally, substituting the expression of 7 points in the template into the expression (12) and requiring that the truncation thereof is erroneously satisfied. Further, the coefficient in expression (12) can be found by using the predetermined coefficient method. Thus (2)The construction process of the numerical format is completed.
(5) Solving the equation set and analyzing the result
The model and numerical format are first validated by the problem with exact solutions. As can be seen from table 2, the algorithm proposed by the present study can accurately simulate the intermittent and jump conditions of the temperatures at both sides of the interface, and the error is continuously reduced with the increase of the grid number, and the approximate second-order precision is maintained. A computational grid when the interface is circular is given from fig. 6. Fig. 7 shows that the numerical solution and the exact solution agree very well fig. 8 shows the error of the numerical solution, and it can be seen that the error decreases as the number of meshes increases.
Example 2 consider the calculation region to beInterface position satisfies the condition->Dividing the region into two parts gives an accurate solution to the problem
Wherein the diffusion coefficient is
The heat flow on the interface meets the conservation condition
Example 1 under different grid numbersError and convergence order, ++>
TABLE 2
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (1)
1. A method for analyzing thermal properties of a composite material having a curvilinear interface, comprising the steps of:
s1: analysis of the actual physical process of the multilayer heat transfer problem, multimode analysis and rational assumptions
Firstly, considering a material model with two layers of different media, wherein a clear interface is formed between different materials; considering the steady heat transfer problem, namely that the temperature reaches a stable state in the process of transferring the inside of the composite material, and neglecting the heat absorption of the material; assuming that the materials in each layer are isotropic and uniformly distributed, the three-dimensional model can be considered by simplifying the three-dimensional model into two-dimensional problems along the thickness direction of the materials;
based on the above analysis and assumptions, the studied problem can be reduced to a two-dimensional heat conduction problem, which is characterized as follows:
(a) Consists of two substances, and a distinct interface is formed between every two layers;
(b) A break in a physical quantity at the material interface;
(c) The temperature is transferred from outside to inside and from high to low;
performing numerical simulation on the steady-state heat conduction process of the composite material of the curve interface; the interlayers among different materials are regarded as non-ideal interfaces through mathematical assumption, and related interface connection conditions are provided according to the characteristic of discontinuous physical properties on the interfaces; converting the actual physical problem into a mathematical problem for solving;
s2: a mathematical description describing the heat transfer process of a multilayer material and a control equation are given:
(a) The steady state heat conduction process inside different media of two-dimensional problem is described by the following diffusion equation:
(1)
(2)
wherein the method comprises the steps ofAnd->Is a given function;
(b) Interface between different mediaJumping and interruption can occur, and the following connection conditions are adopted for describing:
(3)
(4)
wherein the method comprises the steps ofFor interface->In area->Unit external normal direction onOrientation; />Representing jump values of the variables at the interface; />Andrespectively indicate temperature +.>At interface->Limit values on both sides, i.e.
And->Respectively representing diffusion coefficients of media at two sides of the interface;
the relation between the temperatures at the left and right sides of the interface is established in the formula (3), and the jump of the temperatures at the two sides of the interface is proportional to the heat flow passing through the interface, and the proportionality coefficient isThe method comprises the steps of carrying out a first treatment on the surface of the As can be seen from equation (4), the heat flow is equal across the interface, i.e., no heat is absorbed or generated at the interface;
s3: according to the characteristics of parameters, thickness and heat conduction coefficient of the interlayer material, providing interface boundary conditions for describing the discontinuous condition of physical quantity on the interlayer;
dividing the jump condition on the interface into two parts along the coordinate axisAnd->Record->Along +.>Point to->And->The method comprises the steps of carrying out a first treatment on the surface of the The tangent vector on the interface can be expressed as +.>The method comprises the steps of carrying out a first treatment on the surface of the The jump condition on the interface can be re-expressed as
(5)
(6)
Jump of physical quantity on interfaceIn the tangential direction can be expressed as,
(7)
wherein the method comprises the steps of
(8)
Formula (6) can be expressed as
(9)
Similarly, formula (7) can be expressed as
(10)
Multiplying formula (10)Obtaining
Solving a system of linear equations as follows
Then the first time period of the first time period,and->Can be expressed as
And
for a pair ofAnd->About->Can obtain
From the control equation
Thus, the first and second substrates are bonded together,
s4: dispersing the heat conduction model of the multilayer heat insulation material by adopting a mathematical method to obtain a discrete linear equation set;
for the two-dimensional problem, selecting a dynamic template mode to perform format construction; dividing all grid nodes into two types according to the position relation between the interface position and the grid nodes: (1) A five-point template is selected by the regular grid points, and all the grid points in the template are on one side of the material interface; (2) Irregular points, selecting a 7-point template, wherein grid points in the template are distributed on two sides of a material interface;
performing orthogonal grid subdivision on the calculation region to obtain a calculation grid; it is assumed that the finite difference format of equation (1) can be expressed as
(11)
For regular grid points
For regular grid points, the 5-point format is adopted for direct dispersion, and the corresponding coefficient is
And is also provided with
The truncation error of the format is：
For irregular grid points
For irregular grid points, 7-point templates are adopted for construction format
(12)
Then, 7 points in the template are found in a self-adaptive mode according to the geometric relation of the interface, and non-central point variables are unfolded to the central point through a Taylor unfolding method; the relation between the decomposition and the derivative of the physical quantity on the interface is utilized to obtain a relation expression of the unknown quantity on the grid point, which is described by the nodes on the single side of the interface, so as to obtain a corresponding relation expression; finally, the expression of truncation error is utilized and is required to satisfyThe method comprises the steps of carrying out a first treatment on the surface of the Further, by using the coefficient method to be determinedCoefficients in the formula (12) can be found to complete the construction process of the numerical format;
s5: solving a discrete linear equation set, and analyzing a result;
the model and numerical format are verified by questions with exact solutions.
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| WO2008112013A1 (en) * | 2006-10-04 | 2008-09-18 | The Regents Of The University Of California | Carbon nanotube arrays as thermal interface materials |
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