CN110362935B - Heat conduction modeling and calculating method for plastic-steel door and window frame - Google Patents

Heat conduction modeling and calculating method for plastic-steel door and window frame Download PDF

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CN110362935B
CN110362935B CN201910648426.0A CN201910648426A CN110362935B CN 110362935 B CN110362935 B CN 110362935B CN 201910648426 A CN201910648426 A CN 201910648426A CN 110362935 B CN110362935 B CN 110362935B
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邹涛
曲一兵
昃向博
李祥忠
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Abstract

The invention discloses a modeling and calculating method for heat conduction of a plastic-steel door and window frame. Constructing a temperature function of the frame and a closed cavity inside the frame by adopting a finite difference method based on a two-dimensional steady-state heat conduction differential equation without an internal heat source; and performing boundary processing on the boundary of the cavity and the frame by adopting a harmonic-mean method, and combining with a difference method to construct a heat conduction function at the boundary. And under the third type of boundary conditions, performing simulation calculation on the constructed mathematical model by using MATLAB to obtain the equivalent heat transfer coefficient of the frame. The method can effectively analyze the mathematical characteristics of the heat transfer of the frame, better process the boundaries of different materials, greatly reduce the workload of simulation calculation on the basis of ensuring the model precision and is beneficial to the realization of high-precision simulation calculation.

Description

Heat conduction modeling and calculating method for plastic-steel door and window frame
Technical Field
The invention belongs to the technical field of heat conduction calculation of plastic-steel doors and windows, and particularly relates to a heat conduction modeling and calculating method of a plastic-steel door and window frame.
Background
The plastic-steel door and window frame is composed of a plurality of cavities with different sizes and PVC plastics, the structure is complex, and the heat transfer process is very complex. In the heat conduction calculation process of the plastic-steel door and window frame, the boundary part of the frame and the closed cavity exists, and special calculation needs to be carried out on heat conduction at the boundary part. And under the third type of boundary conditions, the closed cavity is equivalent to a solid material, the solid material is converted into equivalent thermal conductivity, and the heat conduction calculation is carried out.
For the heat transfer calculation of the plastic-steel door and window frame, a one-dimensional thermal resistance principle or a two-dimensional finite element method is generally adopted for heat transfer calculation, and the method has the problems of large calculation error, relatively complex calculation and incapability of well describing a physical model. And for junctions of different materials, the processing is rough, and the calculation precision is not high.
Disclosure of Invention
Aiming at the defects or improvement requirements in the prior art, the invention provides the heat conduction modeling and calculating method for the plastic-steel door and window frame, which is beneficial to the realization of high-precision simulation calculation on the basis of ensuring higher calculation precision and has important significance on whole window heat transfer analysis.
To achieve the above object, according to one aspect of the present invention, there is provided a method for modeling and calculating heat conduction of a plastic-steel jamb, comprising the steps of:
(1) based on a two-dimensional steady-state heat conduction equation, a difference method is selected for heat transfer calculation, and the two-dimensional heat conduction differential control equation without an internal heat source is as follows:
Figure RE-GDA0002166609770000011
the solution conditions of the control equation, i.e. the boundary conditions, are of the third type, i.e. the surface heat transfer coefficient h and the ambient temperature t between the object on the boundary and the surrounding fluid are specified:
Figure RE-GDA0002166609770000021
at the adiabatic surfaces, the boundary conditions are satisfied:
Figure RE-GDA0002166609770000022
(2) the method comprises the following steps of (1) discretization, node equation establishment, wherein the main methods for establishing the discrete equation comprise a Taylor series expansion method and a thermal equilibrium method, assuming that (i, j) is an inner node, the inner node can be obtained by an outer node method in the thermal equilibrium method, and when dx is dy:
Figure RE-GDA0002166609770000023
(3) establishing a node physical quantity algebraic equation, and coding the grid nodes of the heat-conducting object for the convenience of research; selecting coordinate axis x-axis and coordinate axis y, and establishing a mathematical model by a physical model of a frame and an actual external environment; a plurality of cavities are arranged in the middle of the frame, and coding is carried out by calculating equivalent thermal conductivity coefficients of the cavities;
(4) when the numerical simulation is carried out on the heat transfer process, the condition that the equivalent heat conductivity coefficient at the interface of different materials, namely the air heat conductivity coefficient in the frame and the cavity, is suddenly changed exists, and the heat conductivity function at the boundary is constructed by calculating the equivalent heat conductivity coefficient at the boundary;
(5) substituting the heat-conducting function formula into MATLAB for coding, and adopting a Gauss-Seidel iteration method for iterative solution to determine the minimum error of the iterative convergence condition; and (3) obtaining a temperature field of the physical model by using MATLAB calculation, and completing equivalent heat transfer coefficient calculation according to a Newton cooling formula and a Fourier law:
Q＝hA(T-T 0 )
wherein h is the heat exchange coefficient of the surface of the object, A is the contact surface area with a heat source, and T 0 Is the initial temperature, i.e. the ambient temperature, T is the object surface temperature;
K＝Q/(A×t)
where K is the heat transfer coefficient, Q is the average heat transfer, A is the projected area, and t is the heat transfer temperature difference (the heat transfer temperature difference is herein 40 ℃).
Preferably, the steps (2) and (3) are specifically: respectively establishing temperature functions of the frame, the external environment, the internal nodes of the frame and the internal nodes of the closed cavity according to the following equations;
for the convection straight boundary node of the frame:
Figure RE-GDA0002166609770000031
Figure RE-GDA0002166609770000032
for corner points where adiabatic boundary and convection boundary of the frame meet:
Figure RE-GDA0002166609770000033
for adiabatic straight boundary nodes of the bezel:
Figure RE-GDA0002166609770000034
for adiabatic boundary nodes of the bezel:
Figure RE-GDA0002166609770000035
for common interior nodes of the frame and closed cavity:
Figure RE-GDA0002166609770000036
wherein h is 1 、h 2 Is the heat transfer coefficient, lambda, of the surface of the object F The heat conductivity of the object is dx which is the step length.
Preferably, the step (4) is specifically: the calculated cavity equivalent thermal conductivity coefficient has a larger difference with the thermal conductivity coefficient of PVC plastics, and the heat flux passing through the interface mainly depend on a restrictive link, namely a material with small thermal conductivity; calculating the equivalent heat conductivity coefficient of the PVC plastic and the junction of the closed cavity by adopting a harmonic mean method, and combining the equivalent heat conductivity coefficient with an external node method; harmonic mean method, i.e. satisfying the equation:
Figure RE-GDA0002166609770000041
the following can be obtained:
Figure RE-GDA0002166609770000042
namely, it is
Figure RE-GDA0002166609770000043
When lambda is E ＜＜λ F When it is ready to use
Figure RE-GDA0002166609770000044
The internal node formula of the difference method is as follows:
Figure RE-GDA0002166609770000045
combining the internal node formula with the harmonic mean method, the temperature equation at the boundary of different media can be obtained as follows:
Figure RE-GDA0002166609770000046
preferably, a frame convection straight boundary node formula, an adiabatic straight boundary node formula, an intersection angle point formula of an adiabatic boundary and a convection boundary, an adiabatic boundary node formula, and a frame and closed cavity internal node formula are respectively established according to the actual boundary condition of the frame by a two-dimensional heat conduction difference method. And combining the formula with a physical model, establishing a frame and cavity temperature function, and performing frame heat conduction modeling.
Preferably, the formula (2) of the nodes inside the frame is established through an external node method, and the formula (3) of the heat transfer temperature function at the boundary of different media is established by combining the formula (1) obtained through a harmonic mean method. Calculating the equivalent thermal conductivity of each closed cavity and the PVC plastic profile, substituting the equivalent thermal conductivity into a formula (3), constructing a thermal conductivity function at each boundary, and substituting the thermal conductivity function, the PVC plastic profile and the closed cavity temperature function into MATLAB for simulation calculation.
Generally, compared with the prior art, the technical scheme conceived by the invention has the following beneficial effects: (1) based on the multilayer flat wall two-dimensional steady-state heat conduction differential equation without the internal heat source, the frame is subjected to mathematical modeling by adopting a finite difference method, and heat transfer calculation is performed by MATLAB, so that the calculation precision is more accurate on the basis of ensuring the model precision. (2) The method of combining the harmonic mean method and the difference method is adopted to solve the problem of heat transfer boundary of different materials, and the method has simple steps and higher calculation speed.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a schematic view of modeling the heat conduction of the frame of the plastic-steel door/window according to the embodiment of the present invention; in the figure: 1. PVC plastic profiles; 2. sealing the cavity; 3. the junction of the plastic section and the closed cavity.
FIG. 2 is a schematic diagram illustrating a modeling and calculation method for heat conduction of a frame of a plastic-steel door/window according to an embodiment of the present invention; in the figure: 1. PVC plastic profiles; 2. the cavity 1 is closed; 4. the cavity 2 is closed; 5. A closed cavity 3; 6. the cavity 4 is closed; 7. a closed cavity 5; 8. closing the cavity 6.
FIG. 3 is a schematic diagram illustrating the principle of the heat conduction blending and averaging method for the plastic-steel door/window jamb according to the embodiment of the invention; in the figure, E, F represents two media, e represents the junction of E, F media and lambda represents E Is the thermal conductivity of the E medium, λ F Is the thermal conductivity of the medium F, (deltax) e Is the width of the division grid.
Fig. 4 is a schematic diagram illustrating calculation of equivalent heat transfer coefficients of a plastic-steel door and window frame according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features designed in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The method is based on a two-dimensional steady-state heat conduction equation, selects a difference method to carry out heat transfer calculation, respectively establishes heat transfer equations of a frame and a closed cavity, establishes a boundary mathematical model by using a harmonic mean method, and carries out solving.
The heat conduction modeling and calculating method for the plastic-steel door and window comprises the following steps:
(1) based on a two-dimensional steady-state heat conduction equation, a difference method is selected for heat transfer calculation, and the two-dimensional heat conduction differential control equation without an internal heat source:
Figure RE-GDA0002166609770000061
the solution conditions of the control equation, i.e. the boundary conditions, are of the third type, i.e. the surface heat transfer coefficient h and the ambient temperature t between the object on the boundary and the surrounding fluid are specified:
Figure RE-GDA0002166609770000062
at the adiabatic surfaces, the boundary conditions are satisfied:
Figure RE-GDA0002166609770000063
(2) the method comprises the following steps of carrying out regional discretization, establishing a node equation, wherein the main methods for establishing the discrete equation comprise a Taylor series expansion method and a thermal equilibrium method, assuming that (i, j) is an inner node, and obtaining the node equation by an outer node method in the thermal equilibrium method, and when dx is dy:
Figure RE-GDA0002166609770000064
(3) establishing a node physical quantity algebraic equation, and coding the grid nodes of the heat-conducting object for convenient research; selecting coordinate axis x-axis and coordinate axis y, establishing a mathematical model by a physical model of a frame and an actual external environment, wherein a plurality of cavities exist in the middle of the frame, and coding is carried out by calculating equivalent thermal conductivity coefficients of the cavities;
(4) when the numerical simulation is carried out on the heat transfer process, the condition that the equivalent heat conductivity coefficient at the interface of different materials, namely the air heat conductivity coefficient in the frame and the cavity, is suddenly changed exists, and the heat conductivity function at the boundary is constructed by calculating the equivalent heat conductivity coefficient at the boundary;
(5) substituting the heat conduction function formula into MATLAB for coding through the steps, carrying out iterative solution by adopting a Gauss-Seidel iterative method, and determining the minimum error of the iterative convergence condition; and (3) obtaining a temperature field of the physical model by using MATLAB calculation, and completing equivalent heat transfer coefficient calculation according to a Newton cooling formula and a Fourier law:
Q＝hA(T-T 0 )
wherein h is the heat exchange coefficient of the surface of the object, A is heatSource contact surface area, T 0 Is the initial temperature, i.e. the ambient temperature, T is the object surface temperature;
K＝Q/(A×t)
where K is the heat transfer coefficient, Q is the average heat transfer, A is the projected area, and t is the heat transfer temperature difference (the heat transfer temperature difference is herein 40 ℃).
Preferably, the steps (2) and (3) are specifically: respectively establishing temperature functions of the frame, the external environment, the internal nodes of the frame and the internal nodes of the closed cavity according to the following equations;
for a convective straight boundary node of the bounding box:
Figure RE-GDA0002166609770000071
Figure RE-GDA0002166609770000072
for corner points where adiabatic boundary and convection boundary of the frame meet:
Figure RE-GDA0002166609770000073
for adiabatic straight boundary nodes of the bezel:
Figure RE-GDA0002166609770000074
for adiabatic boundary nodes of the bezel:
Figure RE-GDA0002166609770000081
for common interior nodes of the frame and closed cavity:
Figure RE-GDA0002166609770000082
wherein h is 1 、h 2 Is the heat transfer coefficient, lambda, of the surface of the object F Is the object thermal conductivity, and dx is the step size.
Preferably, the step (4) is specifically: the difference between the equivalent thermal conductivity of the cavity and the thermal conductivity of PVC plastic is large, and the heat flux passing through the interface mainly depend on a restrictive link, namely a material with small thermal conductivity; calculating the equivalent thermal conductivity coefficient of the PVC plastic at the junction of the PVC plastic and the closed cavity by adopting a harmonic averaging method, and combining the equivalent thermal conductivity coefficient with an external node method; harmonic mean, i.e. satisfying the equation:
Figure RE-GDA0002166609770000083
the following can be obtained:
Figure RE-GDA0002166609770000084
namely, it is
Figure RE-GDA0002166609770000085
When lambda is E ＜＜λ F When is at time
Figure RE-GDA0002166609770000086
The internal node formula of the difference method is as follows:
Figure RE-GDA0002166609770000087
combining the internal node formula with the harmonic mean method, the temperature equation at the boundary of different media can be obtained as follows:
Figure RE-GDA0002166609770000088
preferably, a frame convection straight boundary node formula, an adiabatic boundary and convection boundary intersection corner point formula, an adiabatic boundary node formula and a frame and closed cavity internal node formula are respectively established according to the actual boundary condition of the frame by a two-dimensional heat conduction difference method. And combining the formula with a physical model, establishing a frame and cavity temperature function, and performing frame heat conduction modeling.
Preferably, the formula (2) of the nodes inside the frame is established through an external node method, and the formula (3) of the heat transfer temperature function at the boundary of different media is established by combining the formula (1) obtained through a harmonic mean method. Calculating the equivalent thermal conductivity of each closed cavity and the PVC plastic profile, substituting the equivalent thermal conductivity into a formula (3), constructing a thermal conductivity function at each boundary, and substituting the thermal conductivity function, the PVC plastic profile and the closed cavity temperature function into MATLAB for simulation calculation.
Taking a frame with the sectional material BR58NC as an example, the length of the frame is 6cm, the width of the frame is 5.2cm, and the total projection area of the cavity is 11.04cm 2 . Under the third type of boundary condition of the external environment, indoor and outdoor are: t is t 1 ＝20℃,h 1 ＝8W/m 2 ·℃，t 2 ＝-20℃,h 2 ＝23W/m 2 Temperature. By referring to the data, the thermal conductivity of PVC plastic is 0.17W/(m × K), and the step size is selected to be 0.001. The closed cavities 1 to 6 are equivalent to solid materials and have thermal conductivities of 0.048W/(m × K), 0.045W/(m × K), 0.0421W/(m × K), 0.117W/(m × K), 0.0515W/(m × K) and 0.052W/(m × K), respectively. As shown in fig. 2, the outer wall edges af and fe and the inner wall surface bc are convection straight boundary nodes; boundary nodes a, f, b and c are corner points at the intersection of the heat insulation boundary and the convection boundary; the adiabatic straight boundary nodes are ab, cd and de; the adiabatic boundary node is d. Selecting an x axis and a y axis of a coordinate axis, wherein the unit length of the x axis and the y axis is 1mm, and taking a left lower angular point as a (1,1) point.
And establishing a calculation model through a formula, putting the calculation model into MATLAB for calculation, and obtaining a calculation result graph as shown in figure 4.
From FIG. 4 we can see that the process of the invention gives an equivalent heat transfer coefficient K of 1.7W/(m) 2 XK). By looking up data, the cross section of the PVC cavity is of a three-cavity structure from indoor to outdoor, no metal reinforcing rib exists, and the K is approximate to 2W/(m) 2 xK) with a relative error of 15%, which is relatively close to the value obtained by the method of the invention. The calculation time of the method is 3s through MATLAB, and the calculation time is 3sThe speed is high.
It will be readily understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention and is not intended to limit the invention, and any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included within the scope of the present invention.

Claims (5)

1. A modeling and calculating method for heat conduction of a plastic-steel door and window frame is characterized by comprising the following steps:
(1) based on a two-dimensional steady-state heat conduction equation, selecting a difference method for heat transfer calculation, and selecting a two-dimensional heat conduction differential control equation without an internal heat source:
Figure RE-FDA0002166609760000011
the solution conditions of the control equation, i.e. the boundary conditions, are of the third type, i.e. the surface heat transfer coefficient h and the ambient temperature t between the object on the boundary and the surrounding fluid are specified:
Figure RE-FDA0002166609760000012
at the adiabatic face, the boundary condition is satisfied:
Figure RE-FDA0002166609760000013
(2) the method comprises the following steps of (1) discretization, node equation establishment, wherein the main methods for establishing the discrete equation comprise a Taylor series expansion method and a thermal equilibrium method, assuming that (i, j) is an inner node, the inner node can be obtained by an outer node method in the thermal equilibrium method, and when dx is dy:
Figure RE-FDA0002166609760000014
(3) establishing a node physical quantity algebraic equation, and coding the grid nodes of the heat-conducting object for the convenience of research; selecting coordinate axis x-axis and coordinate axis y, and establishing a mathematical model by a physical model of a frame and an actual external environment; a plurality of cavities are arranged in the middle of the frame, and coding is carried out by calculating equivalent thermal conductivity coefficients of the cavities;
(4) when the numerical simulation is carried out on the heat transfer process, the condition that the equivalent heat conductivity coefficient at the interface of different materials, namely the air heat conductivity coefficient in the frame and the cavity, is suddenly changed exists, and the heat conductivity function at the boundary is constructed by calculating the equivalent heat conductivity coefficient at the boundary;
(5) substituting the heat-conducting function formula into MATLAB for coding, and adopting a Gauss-Seidel iteration method for iterative solution to determine the minimum error of the iterative convergence condition; and (3) obtaining a temperature field of the physical model by using MATLAB calculation, and completing equivalent heat transfer coefficient calculation according to a Newton cooling formula and a Fourier law:
Q＝hA(T-T 0 )
wherein h is the heat exchange coefficient of the surface of the object, A is the contact surface area with a heat source, and T 0 Is the initial temperature, i.e. the ambient temperature, T is the object surface temperature;
K＝Q/(A×t)
where K is the heat transfer coefficient, Q is the average heat transfer, A is the projected area, and t is the heat transfer temperature difference (the heat transfer temperature difference is herein 40 ℃).
2. The modeling and calculating method for heat conduction of plastic-steel doors and windows according to claim 1, wherein the steps (2) and (3) are specifically as follows:
for the convection straight boundary node of the frame:
Figure RE-FDA0002166609760000021
Figure RE-FDA0002166609760000022
for corner points where adiabatic boundary and convection boundary of the frame meet:
Figure RE-FDA0002166609760000023
for adiabatic straight boundary nodes of the bezel:
Figure RE-FDA0002166609760000024
for adiabatic boundary nodes of the bezel:
Figure RE-FDA0002166609760000025
for common internal nodes of the frame and closed cavity:
Figure RE-FDA0002166609760000026
wherein h is 1 、h 2 Is the heat transfer coefficient, lambda, of the surface of the object F The heat conductivity of the object is dx which is the step length.
3. The modeling and calculating method for heat conduction of plastic-steel doors and windows according to claim 1 or 2, wherein the step (4) is specifically: the difference between the equivalent thermal conductivity of the cavity and the thermal conductivity of PVC plastic is large, and the heat flux passing through the interface mainly depend on a restrictive link, namely a material with small thermal conductivity; calculating the equivalent thermal conductivity coefficient of the PVC plastic at the junction of the PVC plastic and the closed cavity by adopting a harmonic averaging method, and combining the equivalent thermal conductivity coefficient with an external node method; harmonic mean, i.e. satisfying the equation:
Figure RE-FDA0002166609760000031
the following can be obtained:
Figure RE-FDA0002166609760000032
namely, it is
Figure RE-FDA0002166609760000033
When lambda is E ＜＜λ F When is at time
Figure RE-FDA0002166609760000034
The internal node formula of the difference method is as follows:
Figure RE-FDA0002166609760000035
combining the internal node formula with the harmonic mean method, the temperature equation at the boundary of different media can be obtained as follows:
Figure RE-FDA0002166609760000036
4. the modeling and calculating method for heat conduction of plastic-steel door and window frame as claimed in claim 2, wherein the frame convection straight boundary node formula, the adiabatic straight boundary node formula, the corner point formula at the intersection of the adiabatic boundary and the convection boundary, the adiabatic boundary node formula, and the node formula inside the frame and the closed cavity are respectively established according to the actual boundary condition of the frame by a two-dimensional heat conduction difference method. And combining the formula with a physical model, establishing a frame and cavity temperature function, and performing frame heat conduction modeling.
5. The modeling and calculation method for heat conduction of plastic-steel door and window frames as claimed in claim 3, wherein the formula (2) for the nodes inside the frame is established by an external node method, and the formula (3) for the heat transfer temperature function at the boundary of different media is constructed by combining the formula (2) obtained by the harmonic mean method with the formula (1). Calculating the equivalent thermal conductivity of each closed cavity and the PVC plastic profile, substituting the equivalent thermal conductivity into a formula (3), constructing a thermal conductivity function at each boundary, and substituting the thermal conductivity function, the PVC plastic profile and the closed cavity temperature function into MATLAB for simulation calculation.
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