CN110941882A - Thermal performance analysis method of composite material with curved interface - Google Patents
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Abstract
The invention discloses a thermal performance analysis method of a composite material with a curved interface, and provides a rapid and accurate numerical analysis method for the thermal performance of a multilayer composite material with a two-dimensional problem and a curved interface type; the method provides interface connection conditions for describing physical quantity jumping 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 transfer equation; meanwhile, a control equation and interface conditions are dispersed by using an interface immersion method, and a stable and convergent numerical value format with second-order precision is obtained; the invention establishes a mathematical model and a numerical simulation method by solving the problem that the two-dimensional composite material has the curve interface type, and the method can be used for carrying out high-precision rapid analysis on the heat conduction process and the thermal resistance performance of a multi-layer composite material such as sandwich steel, a sleeve module 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 curved interface.
Background
With the development of scientific technology and industrial production, people put higher demands on heat conducting materials. Compared with the traditional heat conduction materials (metal, metal oxide, ceramic and the like), the high polymer heat conduction composite material is concerned by the scientific community because of the advantages of low price, small relative density, easy processing, short molding period and the like. Heat-conducting polymer composites have been successfully used in a variety of fields and are increasingly in demand, such as the fields of illumination, aerospace, automotive, electronics, and batteries. The thermal conductivity of composites having thermal interfaces has received much attention from researchers. The thermal interface material is a composite material which is prepared by filling heat-conducting particles with a high polymer material serving as a matrix. The thermal interface material can effectively improve the heat conduction between two solid interfaces and plays an important role in the performance, the service life and the stability of an electronic device. Particularly, when two solid interfaces are in direct contact, the actual contact area between the interfaces is very small due to the roughness of the interfaces, and is only 1% -2% of the apparent contact area, and the interface contact thermal resistance is very large at the moment. Therefore, the thermal conductivity of the thermal interface composite material is rapidly and quantitatively analyzed, and the research significance and the application value of the performance research, development and effective utilization of the thermal interface material are very good.
At present, most of the thermal performance research of the composite material adopts an experimental research method, the experimental method is limited by the complexity of experimental equipment and the composite material and the arbitrary constraint of an interface, and a comprehensive and systematic analysis on the thermal performance of the tested material is difficult to realize. On the other hand, many researchers have proposed theoretical models, such as equivalent thermal conductivity models of two-phase composites, based on experiments and data for some simple two-phase composites. In addition, researchers adopt a Monte Carlo numerical simulation method to perform statistical analysis on the phenomenon that the composite material has a wide distribution of random unequal diameter pore canals by adopting ANSYS software. The invention provides a method based on the combination of a physical property mechanism and mathematical modeling, which carries out comprehensive, accurate and detailed numerical analysis on the heat conduction process of a composite material by solving a control equation of the heat conduction process of the composite material. The method is not limited by the interface geometry of the composite material, and the composite material with the interface of any shape can be analyzed.
Disclosure of Invention
The invention aims to provide a thermal performance analysis method of a composite material with a curved interface, and a mathematical model and a numerical simulation method are established by solving the problem that a two-dimensional composite material has the curved interface type. The method can conveniently and quickly 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 thermal analysis of a composite material having a curvilinear interface, comprising the steps of:
s 1: analyzing the actual physical process of the multilayer heat transfer problem, and carrying out analysis and reasonable assumption by multiple models:
firstly, a material model with two layers of different media is considered, and the different materials have clear interfaces; for convenience, the problem of constant heat transfer is not considered, namely the temperature reaches a stable state in the transfer process of the composite material, and the absorption of the material to heat is neglected;
assuming that the material in each layer is isotropic and uniformly distributed, the three-dimensional model can be considered by simplifying the three-dimensional model into a two-dimensional problem along the thickness direction (vertical direction) of the material;
based on the above analysis and assumptions, the problem under study can be reduced to a two-dimensional thermal conduction problem, which is characterized as follows:
(a) consists of two substances, each two layers having distinct interfaces;
(b) the physical quantity at the material interface is interrupted;
(c) the temperature is transferred from outside to inside and from high to low;
carrying out numerical simulation on the steady-state heat conduction process of the composite material with the curve interface; interlayers of different materials are regarded as non-ideal interfaces through reasonable mathematical assumptions, and relevant interface connection conditions are proposed aiming at the characteristic that the physical properties on the interfaces are discontinuous; and converting the actual physical problem into a mathematical problem to solve.
: a mathematical description and control equation describing the heat transfer process of the multilayer material is given:
(a) two-dimensional problem the steady-state heat conduction process inside different media can be described by the following diffusion equation:
wherein
And is
Is a given function;
(b) interface between different media
The jump and the discontinuity can occur, and the following connection conditions are adopted for depiction:
wherein
Is an interface
In a region
The unit of upper outer normal direction,
representing the jump value of the variable at the interface,
and
respectively represent temperature
At the interface
Limits on both sides, i.e.
the equation (3) establishes the relationship between the temperatures of the left and right sides of the interface, and the temperature jumps of the two sides of the interface can be seen to be the sumThe heat flux across the interface is proportional with a proportionality coefficient of
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.
: according to the parameters, thickness and heat conduction coefficient characteristics of the interlayer material, providing interface boundary conditions for describing discontinuous conditions of physical quantities on the interlayer;
dividing jump condition on interface into two parts along coordinate axis
And
memory for recording
Is unit normal vector along the interface
Point of direction
And is and
the tangent vector on the interface can be expressed as
Then the jump condition on the interface can be re-expressed as
Jump of physical quantity on interface
In the tangential direction can be expressed as,
wherein
The formula (6) can be represented as
Similarly, the formula (7) can be represented as
Multiplying equation (10) by
To obtain
Solving the following system of linear equations
Then the process of the first step is carried out,
and
can be expressed as
And
to pair
And
about
Can obtain
According to the control equation
Therefore, the temperature of the molten metal is controlled,
: dispersing the heat conduction model of the multi-layer heat insulation material by adopting a mathematical method to obtain a dispersion linear equation set;
for the two-dimensional problem, a dynamic template mode is selected for format construction; dividing all grid nodes into two types according to the position relation between the interface position and the grid nodes: (1) regular grid points, namely selecting a five-point template, wherein all the grid points in the template are positioned on one side of a material interface; (2) selecting 7-point templates with grid points distributed on two sides of a material interface;
performing orthogonal mesh generation on the calculation area to obtain a calculation mesh; it is assumed that the finite difference format of equation (1) can be expressed as
For regular grid points
For regular grid points, if the 5-point format is adopted for direct dispersion, the corresponding coefficient is
And is
Truncation error of the format of
：
For irregular grid points
For irregular grid points, 7-point template is adopted for constructing format
Then, adaptively finding 7 points in the template according to the geometric relationship of the interface, and unfolding the non-central point variable to the central point by a Taylor unfolding method; by utilizing the relationship between the decomposition and the derivative of the physical quantity on the interface, a relational expression that the unknown quantity on the grid points is respectively described by the single-side node of the interface can be obtained; finally, the expression of the truncation error is utilized and is required to satisfy
(ii) a Further, the coefficient in the formula (12) can be found by adopting a undetermined coefficient method, and the construction process of the numerical value format is completed.
: solving the discrete linear equation set and analyzing the result;
the model and value formats are validated by a problem with an accurate solution.
The invention has the following beneficial effects:
firstly, the method can be used for carrying out high-precision rapid analysis on the heat conduction process and the heat resistance performance of a multi-layer composite material such as sandwich steel, a heat-resistant firewall and the like;
secondly, the method provided by the invention has no any limitation on the geometric shape of the interface, so that the numerical simulation can be carried out on the heat conduction process of the combined fitting with the curved edge interface with any shape;
thirdly, the invention of the patent adopts a computer numerical simulation method to research the thermal performance of the multilayer material, so that 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 the composite multilayer material made of any material;
fourthly, the method is simple and reliable, and when the material quality changes, only corresponding physical parameters need to be adjusted in a computer, so that the method is an economic and money-saving research method and can help research, develop and analyze projects to save cost and time.
Drawings
FIG. 1 is a schematic view of a composite model I with a curved outer interface according to the present invention.
FIG. 2 is a schematic representation II of the composite model of the present invention having an interface within a curve.
FIG. 3 is a schematic diagram of the three-dimensional isotropic problem simplified into a two-dimensional problem model I according to the present invention.
Fig. 4 is a schematic diagram of the three-dimensional isotropic problem simplified into a two-dimensional problem model II according to the present invention.
FIG. 5 is a template for the two-dimensional curve interface problem of the present invention.
FIG. 6 is a computational grid of the present invention.
FIG. 7 is a comparison of the numerical solution and the exact solution of the present invention.
FIG. 8 is a diagram illustrating an error of a numerical solution according to the present invention.
The number of the grids is 80 x 80,
the calculation error of time.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments.
Problem analysis and model assumptions
First, consider a material model with two layers of different media with clear interfaces between the different materials, as shown in fig. 1 and 2:
for convenience, the problem of constant heat transfer is considered, namely the temperature reaches a stable state in the transfer process inside the composite material, and the absorption of the material to heat is neglected; assuming that the material within each layer is isotropic and uniformly distributed, the three-dimensional model can be considered by simplifying it into a two-dimensional problem along the thickness direction (vertical direction) of the material, as shown in fig. 3 and 4:
based on the above analysis and assumptions, the problem under study can be reduced to a two-dimensional thermal conduction problem, which is characterized as follows:
(a) consists of two substances, each two layers having distinct interfaces;
(b) the physical quantity at the material interface is interrupted;
(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. The physical model of the sandwich steel material is shown in figure 1. Interlayers of different materials can be regarded as non-ideal interfaces through reasonable mathematical assumptions, and relevant interface connection conditions are provided aiming at the characteristic that physical properties on the interfaces are discontinuous. And converting the actual physical problem into a mathematical problem to solve. Since the steady-state heat conduction process is described by the second order partial differential equation (poisson equation). Therefore, the control equation of the physical problems of the heat transfer process and the heat resistance performance of the sandwich steel plate can be shown by the formulas (1) to (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) Two-dimensional problem the steady-state heat conduction process inside different media is described by the following diffusion equation:
wherein
And is
Is a given function;
(b) interface between different media
The jump and the discontinuity can occur, and the following connection conditions are adopted for depiction:
wherein
Is an interface
In a region
Upper unit outer normal direction.
Representing the jump value of the variable at the interface.
And
respectively represent temperature
At the interface
Limits on both sides, i.e.
equation (3) establishes the relationship between the temperatures of the left and right sides of the interface, and it can be seen that the temperature jumps of the two sides of the interface are proportional to the heat flux passing through the interface, and the proportionality coefficient is
As can be seen from equation (4), heat flow throughThe sides of the interface are equal, i.e., no heat is absorbed or generated at the interface.
Decomposition and derivative relationships at curvilinear interfaces
First, a jump condition on an interface is divided into two parts along a coordinate axis
And
memory for recording
Is unit normal vector along the interface
Point of direction
And is and
(ii) a The tangent vector on the interface can be expressed as
(ii) a Then the jump condition on the interface may be re-expressed as
Jump of physical quantity on interface
In the tangential direction can be expressed as,
wherein
The formula (6) can be represented as
Similarly, the formula (7) can be represented as
Multiplying equation (10) by
To obtain
Solving the following system of linear equations
Then the process of the first step is carried out,
and
can be expressed as
And
to pair
And
about
Can obtain
According to the control equation
Therefore, the temperature of the molten metal is controlled,
discrete equations and constructs numerical formats
For the two-dimensional problem, a dynamic template mode is selected for format construction. Dividing all grid nodes into two types according to the position relation between the interface position and the grid nodes, (1) regular grid points, selecting a five-point template, wherein all the grid points in the template are positioned at one side of a material interface; (2) and (4) selecting 7-point templates with irregular points, wherein grid points in the templates are distributed on two sides of the material interface, as shown in figure 5.
The calculation grid can be obtained by performing orthogonal grid division on the calculation region as shown in fig. 6. It is assumed that the finite difference format of equation (1) can be expressed as
4.1 regular grid points
For regular grid points, if the 5-point format is adopted for direct dispersion, the corresponding coefficient is
And is
Truncation error of the format of
：
4.2 irregular grid points
For irregular grid points, 7-point template is adopted for constructing format
Then, 7 points in the template are found in a self-adaptive mode according to the geometric relation of the interface, and the non-central point variable is expanded to the central point through a Taylor expansion method. And by using the relationship between the decomposition and the derivative of the physical quantity on the interface, a relational expression that the unknown quantity on the grid points is respectively described by the single-side node of the interface can be obtained. Finally, substituting the expression of 7 points in the template into the expression (12) and requiring that the truncation thereof meets the requirement of error
. Further, the coefficient in the equation (12) can be found by the undetermined coefficient method. Thus, the construction process of the numerical value format is completed.
(5) Solving the system of equations and analyzing the results
The model and value formats are first validated by a problem with an accurate solution. As can be seen from Table 2, the algorithm proposed in the present study can accurately simulate the temperature discontinuity and jump conditions on 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. The computational grid when the interface is circular is given from fig. 6. Figure 7 shows that the numerical solution matches the exact solution very well figure 8 gives the error of the numerical solution, and it can be seen that the error decreases as the number of grids increases.
Example 2 consider a calculation region of
The interface position satisfies the condition
By dividing the area into two parts, an accurate solution to the problem can be given
Wherein the diffusion coefficient is
The heat flow on the interface satisfies the conservation condition
Example 1 at different grid numbers
The error and the convergence order of the signal,
TABLE 2
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.
Claims (1)
1. A method for thermal analysis of a composite material having a curvilinear interface, comprising the steps of:
s 1: analyzing the actual physical process of the multi-layer heat transfer problem, and performing analysis and reasonable assumption by multiple models
Firstly, a material model with two layers of different media is considered, and the different materials have clear interfaces; the problem of constant heat transfer is considered, namely the temperature reaches a stable state in the transfer process inside the composite material, and the absorption of the material to heat is neglected; 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 a two-dimensional problem along the thickness direction of the materials;
based on the above analysis and assumptions, the problem under study can be reduced to a two-dimensional thermal conduction problem, which is characterized as follows:
(a) consists of two substances, each two layers having distinct interfaces;
(b) the physical quantity at the material interface is interrupted;
(c) the temperature is transferred from outside to inside and from high to low;
carrying out numerical simulation on the steady-state heat conduction process of the composite material with the curve interface; the interlayer between different materials is regarded as a non-ideal interface through mathematical assumption, and provides related interface connection conditions aiming at the characteristic of discontinuous physical properties on the interface; converting the actual physical problem into a mathematical problem to solve;
s 2: a mathematical description and control equation describing the heat transfer process of the multilayer material is given:
(a) two-dimensional problem the steady-state heat conduction process inside different media is described by the following diffusion equation:
wherein
And is
Is a given function;
(b) interface between different media
The jump and the discontinuity can occur, and the following connection conditions are adopted for depiction:
wherein
Is an interface
In a region
Upper unit outer normal direction;
representing a jump value of the variable at the interface;
and
respectively represent temperature
At the interface
Limits on both sides, i.e.
equation (3) establishes the relationship between the temperatures of the left and right sides of the interface, and it can be seen that the temperature jumps of the two sides of the interface are proportional to the heat flux passing through the interface, and the proportionality coefficient is
(ii) a 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;
s 3: according to the parameters, thickness and heat conduction coefficient characteristics of the interlayer material, providing interface boundary conditions for describing discontinuous conditions of physical quantities on the interlayer;
dividing jump condition on interface into two parts along coordinate axis
And
memory for recording
Is unit normal vector along the interface
Point of direction
And is and
(ii) a The tangent vector on the interface can be expressed as
(ii) a Then the jump condition on the interface may be re-expressed as
Jump of physical quantity on interface
In the tangential direction can be expressed as,
wherein
The formula (6) can be represented as
Similarly, the formula (7) can be represented as
Multiplying equation (10) by
To obtain
Solving the following system of linear equations
Then the process of the first step is carried out,
and
can be expressed as
And
to pair
And
about
Can obtain
According to the control equation
Therefore, the temperature of the molten metal is controlled,
s 4: dispersing the heat conduction model of the multi-layer heat insulation material by adopting a mathematical method to obtain a dispersion linear equation set;
for the two-dimensional problem, a dynamic template mode is selected for format construction; dividing all grid nodes into two types according to the position relation between the interface position and the grid nodes: (1) regular grid points, namely selecting a five-point template, wherein all the grid points in the template are positioned on one side of a material interface; (2) selecting 7-point templates with grid points distributed on two sides of a material interface;
performing orthogonal mesh generation on the calculation area to obtain a calculation mesh; it is assumed that the finite difference format of equation (1) can be expressed as
For regular grid points
For regular grid points, if the 5-point format is adopted for direct dispersion, the corresponding coefficient is
And is
Truncation error of the format of
：
For irregular grid points
For irregular grid points, 7-point template is adopted for constructing format
Then, adaptively finding 7 points in the template according to the geometric relationship of the interface, and unfolding the non-central point variable to the central point by a Taylor unfolding method; by utilizing the relationship between the decomposition and the derivative of the physical quantity on the interface, the relational expression that the unknown quantity on the grid points is respectively described by the single-side nodes of the interface can be obtained, and the corresponding relational expression is obtained; finally, the expression of the truncation error is utilized and is required to satisfy
(ii) a Further, the coefficient in the formula (12) can be found by adopting an undetermined coefficient method, and the construction process of the numerical value format is completed;
s 5: solving the discrete linear equation set and analyzing the result;
the model and value formats are validated by a problem with an accurate solution.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN112036039A (en) * | 2020-09-01 | 2020-12-04 | 内蒙古科技大学 | High-precision numerical analysis method for thermal performance of conjugated heat transfer material |
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Cited By (2)
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| CN112036039A (en) * | 2020-09-01 | 2020-12-04 | 内蒙古科技大学 | High-precision numerical analysis method for thermal performance of conjugated heat transfer material |
| CN112036039B (en) * | 2020-09-01 | 2022-12-23 | 内蒙古科技大学 | High-precision numerical analysis method for thermal performance of conjugated heat transfer material |
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