CN113962130A - Calculation method for three-dimensional contact heat transfer - Google Patents

Calculation method for three-dimensional contact heat transfer Download PDF

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CN113962130A
CN113962130A CN202111256136.5A CN202111256136A CN113962130A CN 113962130 A CN113962130 A CN 113962130A CN 202111256136 A CN202111256136 A CN 202111256136A CN 113962130 A CN113962130 A CN 113962130A
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严成增
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Abstract

The invention provides a three-dimensional contact heat transfer calculation method, which divides a granular medium or a continuous-discontinuous medium into a plurality of polyhedral solid units through grid division; determining an overlapping area of two polyhedral entity units according to the node coordinates of the two polyhedral entity units which are in contact with each other; calculating the contact heat flow between the two polyhedral solid units; according to the contact heat flow between the two polyhedral solid units, utilizing an interpolation shape function to obtain the heat flow of the node of each polyhedral solid unit; combining a calculation method of heat conduction inside a particle medium or inside a continuous medium to obtain the temperature of each node in the polyhedral solid unit; and the simulation of three-dimensional contact heat transfer of the contact particle medium or the continuous-discontinuous medium can be realized by circulating the steps. The technical scheme provided by the invention has the beneficial effects that: the heat transfer by contact between continuous-discontinuous media or particulate media can be calculated.

Description

Calculation method for three-dimensional contact heat transfer
Technical Field
The invention relates to the technical field of three-dimensional heat conduction, in particular to a three-dimensional contact heat transfer calculation method.
Background
The continuous-discontinuous medium or particle medium problem relates to engineering application problems of civil engineering, geotechnical engineering, geological engineering, mining, pharmacy, chemical engineering, agriculture and the like. With the development of computer hardware and the development of computational simulation technology, the numerical simulation method is gradually becoming a powerful tool for particle medium or continuous-discontinuous medium contact heat transfer analysis and calculation. Currently, the primary computational method for solving the contact heat transfer problem is the discrete element method. When solving the three-dimensional contact heat transfer, the discrete element method mainly adopts the round balls to represent the granular medium, and even for the granules with complex shapes, the granular medium is represented by the round balls or the combination of the round balls, so that the three-dimensional contact heat transfer of the granular medium is finally converted into the contact heat transfer before the two round balls. Although this method has the advantage of high computational efficiency, the shape of the particles cannot be well characterized, and the computational accuracy is relatively low. Therefore, the existing methods still have many problems in calculating the three-dimensional contact heat transfer.
Disclosure of Invention
In view of the above, the embodiments of the present invention provide a method for calculating three-dimensional contact heat transfer, which can calculate the contact heat transfer between continuous-discontinuous media or granular media.
The embodiment of the invention provides a method for calculating three-dimensional contact heat transfer, which comprises the following steps:
s1, dividing the particle medium or continuous-discontinuous medium into a plurality of polyhedral solid units through grid division, and converting the contact heat transfer between the particle medium or continuous-discontinuous medium which are mutually contacted into the contact heat transfer between the polyhedral solid units which are mutually contacted;
s2, determining the overlapping area of two polyhedral solid units according to the node coordinates of the two polyhedral solid units which are in contact with each other;
s3, calculating the contact heat flow between two polyhedral solid units, wherein the contact heat flow between the two polyhedral solid units is equal to the sum of the heat flows passing through the outer boundary surfaces of the overlapped region;
s4, according to the contact heat flow between two polyhedral solid units, utilizing an interpolation shape function to obtain the heat flow of the node of each polyhedral solid unit;
s5, combining a calculation method of heat conduction inside the particle medium or inside the continuous medium to obtain the temperature of each node in the polyhedral solid unit;
s6 the steps S1-S5 are repeated, and the simulation of three-dimensional contact heat transfer of the contact particle medium or the continuous-discontinuous medium can be realized.
Further, step S3 includes:
s31, recording two polyhedron solid units which are contacted as an active unit and a passive unit, recording the part of the external boundary surface of the active unit, which is intersected with the interior of the passive unit, as an active external boundary surface Sc, and recording the part of the external boundary surface of the passive unit, which is intersected with the interior of the active unit, as a passive external boundary surface St;
s32 the expression of the heat flux flowing from the passive unit to the active unit through the active outer boundary surface is:
Figure BDA0003323923810000021
in the formula, PscAt any point within the active outer boundary, Tt (Ps)c) Is a point PscTemperature in the passive cell, Tc (Ps)c) Is a point PscTemperature in the active cell, hcIs the contact heat transfer coefficient;
the expression of the heat flux flowing into the passive unit from the active unit through the passive outer boundary surface is as follows:
Figure BDA0003323923810000022
in the formula, PstAt any point within the passive outer boundary, Tc (Ps)t) Is a point PstTemperature in the active cell, Tt (Ps)t) Is a point PstTemperature in the passive unit, hcIs the contact heat transfer coefficient;
s33 expression of the total contact heat flow of the passive unit flowing into the active unit is:
Figure BDA0003323923810000023
in the formula (I), the compound is shown in the specification,
Figure BDA0003323923810000031
the outer boundary surface of the overlapping region of the two polyhedral solid units;
then the process of the first step is carried out,
Figure BDA0003323923810000032
in the formula, hcFor contact heat transfer coefficient, TtIs the temperature, T, of any point on the outer boundary surface of the overlap region in the passive cellcIs the temperature at any point on the outer boundary surface of the overlap region within the active cell.
Further, when the active outer boundary surface is denoted as a polygon B, the expression of the heat transferred from the passive unit to the active unit through the polygon B is:
Figure BDA0003323923810000033
in the formula, hcIs a contact heat transfer coefficient, PsBIs any point within the polygon B, SBIs the area of polygon B;
according to the fact that the temperature in the polyhedral solid unit follows linear distribution, the expression of the heat transferred from the passive unit to the active unit through the polygon B is as follows:
Figure BDA0003323923810000034
in the formula, SBIs the area of polygon B, nBΔ T (Bi) is the temperature difference between the passive unit and the active unit at the vertex Bi of the polygon B.
Further, step S4 includes:
s41, dividing the polygon B into a plurality of triangles, and obtaining the action points of the heat flow of each triangle according to the area of the polygon B, thereby obtaining the action points of the heat flow in the area of the polygon B;
s42, obtaining an interpolation shape function Ni of the action point in the active unit and the passive unit according to the action point of the heat flow in the polygon B area;
the heat flow distributed by each node of the S43 active unit is Qci＝NiQs，NiIs at the active point of actionInterpolation shape function in the unit, Qs is the heat flow on the active outer boundary surface;
each node of the passive unit is distributed with heat flow rate Qcj＝-NjQs，NjIs an interpolation shape function of an action point in the passive unit, and Qs is the heat flow on the active outer boundary surface;
s44, calculating the heat flows on the contact surfaces of the other outer boundary surfaces of the active unit and the other passive units by adopting the method, distributing the heat flows to the corresponding nodes of the active unit and the passive units, and finally obtaining the total contact heat transfer of each node.
Further, step S43 includes:
the active unit and the passive unit are tetrahedron ABCD and tetrahedron EFGH respectively, the active outer boundary surface is positioned on the triangular surface ABC, and the interpolation shape function of the action point in the active unit is NA,NB,NCInterpolation shape function N with action points in passive unitsE,NF,NG,NH；
The heat flow distributed by each node of the active unit is as follows:
Figure BDA0003323923810000041
the heat flow distributed by each node of the passive unit is as follows:
Figure BDA0003323923810000042
further, the polyhedral solid element in step S1 is gridded for one or more of tetrahedron, pentahedron, hexahedron, and three-dimensional voronoi.
Further, in step S5, the method of calculating the heat conduction inside the granular medium or inside the continuous medium is one of a finite element method, a finite volume method, a finite difference method, a boundary element method, a finite-discrete element method, a discontinuous deformation analysis method, and a numerical prevalence method.
The technical scheme provided by the embodiment of the invention has the following beneficial effects: the particle medium or the continuous-discontinuous medium is divided into a plurality of polyhedral solid units, so that the contact heat transfer between the particle medium or the continuous-discontinuous medium which are mutually contacted is converted into the contact heat transfer between the polyhedral solid units which are mutually contacted, the contact heat transfer of the continuous-discontinuous medium or the particle medium with a complex shape can be very conveniently simulated, and the temperature evolution of the continuous-discontinuous medium or the particle medium can be obtained by combining with other methods for solving the heat conduction in the continuous-discontinuous medium or the particle medium.
Drawings
FIG. 1 is a schematic structural diagram of a contacted granular medium divided into a plurality of polyhedral solid elements;
FIG. 2 is a schematic view of a contact between two polyhedral solid elements;
FIG. 3 is a schematic diagram of the contact heat transfer of a passive cell to an active cell through a triangular face ABC.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be further described with reference to the accompanying drawings.
The embodiment of the invention provides a method for calculating three-dimensional contact heat transfer, which comprises the following steps:
s1, dividing the particle medium or continuous-discontinuous medium into a plurality of polyhedral solid units through grid division, and converting the contact heat transfer between the particle medium or continuous-discontinuous medium which are mutually contacted into the contact heat transfer between the polyhedral solid units which are mutually contacted; the polyhedral solid element may be one or more of a tetrahedron, a pentahedron, a hexahedron, a three-dimensional voronoi, or any other polyhedron, and in this embodiment, the divided polyhedral solid element is a tetrahedron (see fig. 1).
S2, determining the overlapping area of two polyhedral solid units according to the node coordinates of the two polyhedral solid units which are in contact with each other;
s3, calculating a contact heat flux between two polyhedral solid elements, wherein the contact heat flux between the two polyhedral solid elements is equal to the sum of heat fluxes passing through outer boundary surfaces of the overlapping region.
Specifically, S31 designates two polyhedral solid cells in contact as an active cell and a passive cell, a portion where an active cell outer boundary surface intersects with the inside of the passive cell is designated as an active outer boundary surface Sc, and a portion where a passive cell outer boundary surface intersects with the inside of the active cell is designated as a passive outer boundary surface St.
In this embodiment, the contact heat transfer of two discrete particulate media of fig. 1 can be converted to contact heat transfer between two tetrahedral units; FIG. 2 shows the contact between tetrahedral unit ABCD and tetrahedral unit EFGH, where ABCD is the active unit β c, EFGH is the passive unit β t, and the intersection of the active and passive unit outer boundary surfaces is the active outer boundary surface Sc, which is the polygon B in FIG. 20B1B2B3The intersection of the passive cell boundary surface and the active cell is the passive boundary surface St, which is the polygon B in FIG. 20B1FG. Polygonal surface B2B3GF. Triangle FB1B2And triangle B0B3G。
S32 the expression for the heat flow from the passive cell β t through the active outer boundary surface Sc into the active cell β c is:
Figure BDA0003323923810000061
in the formula, PscAt any point in the active outer boundary surface Sc, Tt (Ps)c) Is a point PscTemperature, Tc (Ps) in the passive cell beta tc) Is a point PscTemperature in the active cell β c, hcIs the contact heat transfer coefficient;
the expression of the heat flow of the active unit β c flowing into the passive unit β t through the passive outer boundary surface St is:
Figure BDA0003323923810000062
in the formula, PstAt any point within the passive outer boundary surface St, Tc (Ps)t) Is a point PstTemperature in the active cell β c, Tt (Ps)t) Is a point PstTemperature in the passive unit beta t, hcIs the contact heat transfer coefficient;
the heat flux of the passive unit β t flowing into the active unit β c through the passive outer boundary surface St is equal in magnitude and opposite in direction to the heat flux of the active unit β c flowing into the passive unit β t through the passive outer boundary surface St, and the expression of the heat flux of the passive unit β t flowing into the active unit β c through the passive outer boundary surface St is:
Figure BDA0003323923810000063
s33 expression of the total contact heat flow of the passive unit flowing into the active unit is:
Figure BDA0003323923810000071
in the formula (I), the compound is shown in the specification,
Figure BDA0003323923810000072
the outer boundary surface of the overlapping region of the two polyhedral solid units;
the contact heat flow of the whole passive tetrahedron into the active tetrahedron is:
Figure BDA0003323923810000073
in the formula, hcFor contact heat transfer coefficient, TtIs the temperature, T, of any point on the outer boundary surface of the overlap region in the passive cellcIs the temperature at any point on the outer boundary surface of the overlap region within the active cell.
Specifically, as shown in fig. 3, the active outer boundary surface is denoted as polygon B (B)0B1B2B3) The heat transferred from the passive unit to the active unit through the polygon BThe expression for the quantity is:
Figure BDA0003323923810000074
in the formula, hcIs a contact heat transfer coefficient, PsBIs any point within the polygon B, SBIs the area of the polygon B.
Since the temperature in the polyhedron follows a linear distribution, the temperature distribution is controlled by the temperature control system
Figure BDA0003323923810000075
Are also subject to a linear distribution,
Figure BDA0003323923810000076
difference of (2)
Figure BDA0003323923810000077
Also following a linear distribution, the expression for the heat transferred by the passive unit to the active unit through the polygon B is:
Figure BDA0003323923810000078
in the formula, SBIs the area of polygon B, nBΔ T (Bi) is the temperature difference between the passive unit and the active unit at the vertex Bi of the polygon B.
S4, distributing the contact heat flow between the two polyhedral solid units to the nodes of the two polyhedral solid units, and obtaining the heat flow of the nodes of each polyhedral solid unit by utilizing an interpolation shape function.
Specifically, since the heat flows are linearly distributed on the polygon B, it is necessary to find the action points of the distributed heat flows so as to distribute the heat flows to the active cells and the respective nodes of the cells at S41. Dividing the polygon B into a plurality of triangles, and obtaining the action points of the heat flow of each triangle according to the area of the polygon B, thereby obtaining the action points of the heat flow in the area of the polygon B;
s42 is according toThe action points of the heat flow in the polygonal B area are used for obtaining interpolation shape functions N of the action points in the active unit and the passive unit respectivelyi；
The heat flow distributed by each node of the S43 active unit is Qci＝NiQs，NiIs an interpolation shape function of an action point in the active unit, and Qs is the heat flow on an active outer boundary surface;
each node of the passive unit is distributed with heat flow rate Qcj＝-NjQs，NjQs is the heat flux on the active outer boundary surface as an interpolated shape function of the point of action within the passive cell.
In this embodiment, the active unit and the passive unit are a tetrahedron ABCD and a tetrahedron EFGH, respectively, the polygon B of the active outer boundary surface is located on a triangle ABC, and the interpolation shape functions of the action points in the triangle ABC are NA,NB,NCInterpolation shape function N of action point in passive tetrahedral EFGHE,NF,NG,NH；
The heat flow allocated to each node of the active unit is:
Figure BDA0003323923810000081
the heat flow distributed by each node of the passive unit is as follows:
Figure BDA0003323923810000082
NA,NB,NCas an interpolated shape function of the total heat flux action point within the triangular plane Δ ABC, NE,NF,NG,NHIs an interpolated shape function of the total heat flow action point within the passive cell EFGH. QsABCThe heat flux on the triangular face ABC is equal to the heat flux Qc on the polygon B.
S44 calculating the heat flow of each node of the active unit and the passive unit after the contact heat transfer through the active outer boundary surface, and adopting the methodCalculating the heat flows on the contact surfaces of other outer boundary surfaces of the active unit and other passive units, distributing the heat flows to the corresponding nodes of the active unit and the passive units, and finally obtaining the total contact heat transfer Q of each nodec。
S5, combining a calculation method of heat conduction inside the particle medium or inside the continuous medium to obtain the temperature of each node in each polyhedral solid unit; the method of calculating the heat conduction inside the granular medium or inside the continuous medium is a Finite Element Method (FEM), a Finite Volume Method (FVM), a Finite Difference Method (FDM), a boundary element method, a finite-discrete element method (FDEM), a Discrete Element Method (DEM), a discontinuous deformation analysis method (DDA), a numerical prevalence method (NMM), or other types of methods.
S6 the steps S1-S5 are repeated, and the simulation of the three-dimensional contact heat transfer of the granular medium or the continuous-discontinuous medium can be realized.
The invention provides a three-dimensional contact heat transfer calculation method for a granular medium or a continuous-discontinuous medium, which can be used for conveniently simulating the contact heat transfer of the continuous-discontinuous medium or the granular medium with a complex shape by dividing the granular medium or the continuous-discontinuous medium into a plurality of polyhedral solid units and converting the contact heat transfer between the mutually contacted granular medium or the continuous-discontinuous medium into the contact heat transfer between the mutually contacted polyhedral solid units, and can be combined with other methods for solving the internal heat transfer of the continuous-discontinuous medium or the granular medium to obtain the temperature evolution of the continuous-discontinuous medium or the granular medium.
In this document, the terms front, back, upper and lower are used to define the components in the drawings and the positions of the components relative to each other, and are used for clarity and convenience of the technical solution. It is to be understood that the use of the directional terms should not be taken to limit the scope of the claims.
The features of the embodiments and embodiments described herein above may be combined with each other without conflict.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (7)

1. A method for calculating three-dimensional contact heat transfer, comprising:
s1, dividing the particle medium or continuous-discontinuous medium into a plurality of polyhedral solid units through grid division, and converting the contact heat transfer between the particle medium or continuous-discontinuous medium which are mutually contacted into the contact heat transfer between the polyhedral solid units which are mutually contacted;
s2, determining the overlapping area of two polyhedral solid units according to the node coordinates of the two polyhedral solid units which are in contact with each other;
s3, calculating the contact heat flow between two polyhedral solid units, wherein the contact heat flow between the two polyhedral solid units is equal to the sum of the heat flows passing through the outer boundary surfaces of the overlapped region;
s4, according to the contact heat flow between two polyhedral solid units, utilizing an interpolation shape function to obtain the heat flow of the node of each polyhedral solid unit;
s5, combining a calculation method of heat conduction inside the particle medium or inside the continuous medium to obtain the temperature of each node in the polyhedral solid unit;
s6 the steps S1-S5 are repeated, and the simulation of three-dimensional contact heat transfer of the contact particle medium or the continuous-discontinuous medium can be realized.
2. The method for calculating three-dimensional contact heat transfer according to claim 1, wherein step S3 includes:
s31, recording two polyhedron solid units which are contacted as an active unit and a passive unit, recording the part of the external boundary surface of the active unit, which is intersected with the interior of the passive unit, as an active external boundary surface Sc, and recording the part of the external boundary surface of the passive unit, which is intersected with the interior of the active unit, as a passive external boundary surface St;
s32 the expression of the heat flux flowing from the passive unit to the active unit through the active outer boundary surface is:
Figure FDA0003323923800000011
in the formula, PscAt any point within the active outer boundary, Tt (Ps)c) Is a point PscTemperature in the passive cell, Tc (Ps)c) Is a point PscTemperature in the active cell, hcIs the contact heat transfer coefficient;
the expression of the heat flux flowing into the passive unit from the active unit through the passive outer boundary surface is as follows:
Figure FDA0003323923800000021
in the formula, PstAt any point within the passive outer boundary, Tc (Ps)t) Is a point PstTemperature in the active cell, Tt (Ps)t) Is a point PstTemperature in the passive unit, hcIs the contact heat transfer coefficient;
s33 expression of the total contact heat flow of the passive unit flowing into the active unit is:
Figure FDA0003323923800000022
in the formula (I), the compound is shown in the specification,
Figure FDA0003323923800000023
the outer boundary surface of the overlapping region of the two polyhedral solid units;
then the process of the first step is carried out,
Figure FDA0003323923800000024
in the formula, hcFor contact heat transfer coefficient, TtIs the temperature, T, of any point on the outer boundary surface of the overlap region in the passive cellcIs the temperature at any point on the outer boundary surface of the overlap region within the active cell.
3. The method for calculating three-dimensional contact heat transfer according to claim 2, wherein the expression of the heat quantity transferred from the passive unit to the active unit through the polygon B is given by taking the active outer boundary surface as the polygon B:
Figure FDA0003323923800000025
in the formula, hcIs a contact heat transfer coefficient, PsBIs any point within the polygon B, SBIs the area of polygon B;
according to the fact that the temperature in the polyhedral solid unit follows linear distribution, the expression of the heat transferred from the passive unit to the active unit through the polygon B is as follows:
Figure FDA0003323923800000026
in the formula, SBIs the area of polygon B, nBΔ T (Bi) is the temperature difference between the passive unit and the active unit at the vertex Bi of the polygon B.
4. The method for calculating three-dimensional contact heat transfer according to claim 3, wherein the step S4 includes:
s41, dividing the polygon B into a plurality of triangles, and obtaining the action points of the heat flow of each triangle according to the area of the polygon B, thereby obtaining the action points of the heat flow in the area of the polygon B;
s42, obtaining an interpolation shape function Ni of the action point in the active unit and the passive unit according to the action point of the heat flow in the polygon B area;
the heat flow distributed by each node of the S43 active unit is Qci＝NiQs，NiIs an interpolation shape function of an action point in the active unit, and Qs is the heat flow on an active outer boundary surface;
each node of the passive unit is distributed with heat flow rate Qcj＝-NjQs，NjIs an interpolation shape function of an action point in the passive unit, and Qs is the heat flow on the active outer boundary surface;
s44, calculating the heat flows on the contact surfaces of the other outer boundary surfaces of the active unit and the other passive units by adopting the method, distributing the heat flows to the corresponding nodes of the active unit and the passive units, and finally obtaining the total contact heat transfer of each node.
5. The method for calculating three-dimensional contact heat transfer according to claim 4, wherein the step S43 includes:
the active unit and the passive unit are tetrahedron ABCD and tetrahedron EFGH respectively, the active outer boundary surface is positioned on the triangular surface ABC, and the interpolation shape function of the action point in the active unit is NA,NB,NCInterpolation shape function N with action points in passive unitsE,NF,NG,NH；
The heat flow distributed by each node of the active unit is as follows:
Figure FDA0003323923800000031
the heat flow distributed by each node of the passive unit is as follows:
Figure FDA0003323923800000041
6. the method for calculating three-dimensional contact heat transfer according to claim 1, wherein the polyhedral solid elements in the step S1 are gridded for one or more of tetrahedrons, pentahedrons, hexahedrons, and three-dimensional voronoi.
7. The method for calculating three-dimensional contact heat transfer according to claim 1, wherein in step S5, the method for calculating the heat transfer inside the granular medium or inside the continuous medium is one of a finite element method, a finite volume method, a finite difference method, a boundary element method, a finite-discrete element method, a discontinuous deformation analysis method, and a numerical prevalence method.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117057210A (en) * 2023-08-28 2023-11-14 天津大学 Heat conduction simulation method based on particle discrete elements

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117057210A (en) * 2023-08-28 2023-11-14 天津大学 Heat conduction simulation method based on particle discrete elements
CN117057210B (en) * 2023-08-28 2024-01-23 天津大学 Heat conduction simulation method based on particle discrete elements

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