CN110941882A - A method for thermal analysis of composite materials with curvilinear interface - Google Patents
A method for thermal analysis of composite materials with curvilinear interface Download PDFInfo
- Publication number
- CN110941882A CN110941882A CN201910988709.XA CN201910988709A CN110941882A CN 110941882 A CN110941882 A CN 110941882A CN 201910988709 A CN201910988709 A CN 201910988709A CN 110941882 A CN110941882 A CN 110941882A
- Authority
- CN
- China
- Prior art keywords
- interface
- equation
- composite material
- grid points
- expressed
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
- 238000000034 method Methods 0.000 title claims abstract description 52
- 239000002131 composite material Substances 0.000 title claims abstract description 34
- 238000002076 thermal analysis method Methods 0.000 title description 2
- 230000008569 process Effects 0.000 claims abstract description 27
- 238000004458 analytical method Methods 0.000 claims abstract description 17
- 238000012546 transfer Methods 0.000 claims abstract description 12
- 239000000463 material Substances 0.000 claims description 41
- 239000010410 layer Substances 0.000 claims description 12
- 238000004364 calculation method Methods 0.000 claims description 8
- 230000014509 gene expression Effects 0.000 claims description 8
- 230000001788 irregular Effects 0.000 claims description 8
- 238000011160 research Methods 0.000 claims description 8
- 238000010276 construction Methods 0.000 claims description 7
- 238000009792 diffusion process Methods 0.000 claims description 7
- 239000011229 interlayer Substances 0.000 claims description 7
- 238000000354 decomposition reaction Methods 0.000 claims description 4
- 230000000704 physical effect Effects 0.000 claims description 4
- 238000010521 absorption reaction Methods 0.000 claims description 3
- 239000000126 substance Substances 0.000 claims description 3
- 239000012774 insulation material Substances 0.000 claims description 2
- 238000012067 mathematical method Methods 0.000 claims description 2
- 238000004088 simulation Methods 0.000 abstract description 7
- 229910000831 Steel Inorganic materials 0.000 abstract description 6
- 239000010959 steel Substances 0.000 abstract description 6
- 239000011185 multilayer composite material Substances 0.000 abstract description 4
- 238000013178 mathematical model Methods 0.000 abstract description 3
- 238000007654 immersion Methods 0.000 abstract 1
- 238000010586 diagram Methods 0.000 description 3
- 239000006185 dispersion Substances 0.000 description 3
- 239000002184 metal Substances 0.000 description 3
- 241000764238 Isis Species 0.000 description 2
- 238000011161 development Methods 0.000 description 2
- 238000002474 experimental method Methods 0.000 description 2
- 230000004907 flux Effects 0.000 description 2
- 239000007787 solid Substances 0.000 description 2
- 241000287196 Asthenes Species 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 238000004422 calculation algorithm Methods 0.000 description 1
- 239000000919 ceramic Substances 0.000 description 1
- 230000008859 change Effects 0.000 description 1
- 239000002322 conducting polymer Substances 0.000 description 1
- 229920001940 conductive polymer Polymers 0.000 description 1
- 239000004020 conductor Substances 0.000 description 1
- 230000007423 decrease Effects 0.000 description 1
- 238000009826 distribution Methods 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 238000005286 illumination Methods 0.000 description 1
- 238000009776 industrial production Methods 0.000 description 1
- 239000011159 matrix material Substances 0.000 description 1
- 230000007246 mechanism Effects 0.000 description 1
- 239000007769 metal material Substances 0.000 description 1
- 229910044991 metal oxide Inorganic materials 0.000 description 1
- 150000004706 metal oxides Chemical class 0.000 description 1
- 238000000465 moulding Methods 0.000 description 1
- 239000002245 particle Substances 0.000 description 1
- 229920000642 polymer Polymers 0.000 description 1
- 239000002861 polymer material Substances 0.000 description 1
- 239000011148 porous material Substances 0.000 description 1
- 238000012545 processing Methods 0.000 description 1
- 238000007619 statistical method Methods 0.000 description 1
- 230000009897 systematic effect Effects 0.000 description 1
Images
Landscapes
- Investigating Or Analyzing Materials Using Thermal Means (AREA)
Abstract
本发明公开了一种具有曲线界面的复合材料的热性能分析方法，本发明为二维问题具有曲线界面类型的多层复合材料的热性能提供一种快速，精确的数值分析方法；本方法通过对物理过程的合理近似，提出了描述界面上物理量跳跃的界面连接条件；并结合二维定常热传导方程对多层复合材料的传热过程进行数学描述；同时，利用浸入界面方法对控制方程和界面条件进行离散，并得到了一个稳定、收敛且具有二阶精度的数值格式；本发明通过对二维复合材料具有曲线界面类型的问题，建立了数学模型及数值模拟的方法，本方法可以用来对夹芯钢材，套筒模件等多层复合材料的热传导过程和热阻性能进行高精度的快速分析方法。
The invention discloses a thermal property analysis method of a composite material with a curved interface. The invention provides a fast and accurate numerical analysis method for the thermal properties of a two-dimensional problem of a multilayer composite material with a curved interface type; the method adopts For a reasonable approximation of the physical process, the interface connection conditions that describe the jump of physical quantities on the interface are proposed; the heat transfer process of the multilayer composite material is mathematically described by combining with the two-dimensional steady heat conduction equation; at the same time, the control equation and the interface are analyzed by the immersion interface method. The conditions are discretized, and a stable, convergent numerical format with second-order precision is obtained; the present invention establishes a mathematical model and a numerical simulation method through the problem that the two-dimensional composite material has a curve interface type, and the method can be used for A high-precision rapid analysis method for the thermal conduction process and thermal resistance performance of sandwich steel, sleeve modules and other multi-layer composite materials.
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)
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910988709.XA CN110941882B (en) | 2019-10-17 | 2019-10-17 | Thermal performance analysis method for composite material with curve interface |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910988709.XA CN110941882B (en) | 2019-10-17 | 2019-10-17 | Thermal performance analysis method for composite material with curve interface |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN110941882A true CN110941882A (en) | 2020-03-31 |
| CN110941882B CN110941882B (en) | 2023-09-29 |
Family
ID=69906057
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201910988709.XA Active CN110941882B (en) | 2019-10-17 | 2019-10-17 | Thermal performance analysis method for composite material with curve interface |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN110941882B (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112036039A (en) * | 2020-09-01 | 2020-12-04 | 内蒙古科技大学 | A high-precision numerical analysis method for thermal properties of conjugated heat transfer materials |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| FR2769167A1 (en) * | 1997-09-29 | 1999-04-02 | Centre Nat Rech Scient | REINFORCED SUPERCONDUCTING MATERIAL, SUPERCONDUCTIVE CAVITY, AND METHODS OF MAKING |
| WO2008112013A1 (en) * | 2006-10-04 | 2008-09-18 | The Regents Of The University Of California | Carbon nanotube arrays as thermal interface materials |
| CN109564786A (en) * | 2016-08-08 | 2019-04-02 | 通用原子公司 | Engineering SIC-SIC composite material and entirety SIC layer structure |
| CN109926675A (en) * | 2019-03-29 | 2019-06-25 | 安徽双桦热交换系统有限公司 | A kind of NB continous way soldering core heating means of Soldering Technology of Automobile Radiators |
-
2019
- 2019-10-17 CN CN201910988709.XA patent/CN110941882B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| FR2769167A1 (en) * | 1997-09-29 | 1999-04-02 | Centre Nat Rech Scient | REINFORCED SUPERCONDUCTING MATERIAL, SUPERCONDUCTIVE CAVITY, AND METHODS OF MAKING |
| WO2008112013A1 (en) * | 2006-10-04 | 2008-09-18 | The Regents Of The University Of California | Carbon nanotube arrays as thermal interface materials |
| CN109564786A (en) * | 2016-08-08 | 2019-04-02 | 通用原子公司 | Engineering SIC-SIC composite material and entirety SIC layer structure |
| CN109926675A (en) * | 2019-03-29 | 2019-06-25 | 安徽双桦热交换系统有限公司 | A kind of NB continous way soldering core heating means of Soldering Technology of Automobile Radiators |
Non-Patent Citations (3)
| Title |
|---|
| 万长风;林继;洪永兴;: "边界节点法模拟瞬态热传导及Matlab工具箱开发", 能源与环保, no. 02 * |
| 张程宾;韩群;陈永平;: "基于MATLAB的传热学课程虚拟仿真实验平台设计", 实验技术与管理, no. 01 * |
| 石琳;程素森;阮新伟;许良友;: "高炉铸铜冷却壁的热性能分析", 钢铁, no. 06 * |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112036039A (en) * | 2020-09-01 | 2020-12-04 | 内蒙古科技大学 | A high-precision numerical analysis method for thermal properties of conjugated heat transfer materials |
| CN112036039B (en) * | 2020-09-01 | 2022-12-23 | 内蒙古科技大学 | A high-precision numerical analysis method for thermal properties of conjugated heat transfer materials |
Also Published As
| Publication number | Publication date |
|---|---|
| CN110941882B (en) | 2023-09-29 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN111709171B (en) | Isogeometric solving and heat dissipation topology generation method for heat flow strong coupling problem | |
| Gao et al. | Element differential method for solving general heat conduction problems | |
| CN110728087B (en) | A Numerical Analysis Method for Thermal Properties of Sandwich Multilayer Composites | |
| Mandal et al. | High resolution finite volume computations on unstructured grids using solution dependent weighted least squares gradients | |
| Zhang et al. | An improved meshless method with almost interpolation property for isotropic heat conduction problems | |
| WO2024041642A1 (en) | Time-domain heat conduction simulation method and storage medium | |
| Xie et al. | Mathematical modeling of heat transfer in mold copper coupled with cooling water during the slab continuous casting process | |
| CN112036039B (en) | A high-precision numerical analysis method for thermal properties of conjugated heat transfer materials | |
| CN115758841A (en) | Finite element reduced modeling method for digital twinning application-oriented transformer temperature field | |
| CN115659908B (en) | Three-unit unbalanced porous medium method of printed circuit board heat exchanger | |
| CN107563038A (en) | A kind of new thermal contact resistance finite element solving method | |
| Navah et al. | Thermofluid topology optimization for cooling channel design | |
| CN110941882A (en) | A method for thermal analysis of composite materials with curvilinear interface | |
| Röver et al. | Design and numerical assessment of an additively manufactured Schwarz diamond triply periodic minimal surface fluid-fluid heat exchanger | |
| Kapadia et al. | Sensitivity analysis for solid oxide fuel cells using a three-dimensional numerical model | |
| CN105808508B (en) | It is a kind of to solve the random orthogonal method of deploying for not knowing heat conduction problem | |
| Mercier et al. | Heat transfer associated to natural convection flow in a partly porous cavity | |
| Kumar et al. | A mathematical model and numerical solution of a one dimensional steady state heat conduction problem by using high order immersed interface method on non-uniform mesh | |
| CN116882253A (en) | Modelica-based thermal member modeling method, device and medium | |
| CN110196983B (en) | High-dimensional random heat conduction problem spectrum analysis method based on distribution point theory | |
| CN111310391A (en) | Simulation method of plate-fin heat exchanger | |
| Nguyen et al. | High‐order implicit time‐stepping with high‐order central essentially‐non‐oscillatory methods for unsteady three‐dimensional computational fluid dynamics simulations | |
| CN118797958B (en) | Intelligent design method of floor heating insulation module based on swarm optimization algorithm | |
| Yang et al. | Zonal free element method for solving nonlinear transient heat conduction problems | |
| Ramšak et al. | Heat diffusion in fractal geometry cooling surface |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |