【2023 Huashu Cup National Undergraduate Mathematical Contest in Modeling】A problem analysis and complete paper on structural optimization control of thermal insulation materials

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1 topic

Research on Structural Optimal Control of Thermal Insulation Materials for Problem A

The new heat insulation material A has excellent heat insulation properties and is widely used in high-tech fields such as aerospace, military industry, petrochemical, construction, and transportation.

At present, the thermal conductivity of a fabric woven from a single fiber of thermal insulation material A can be directly measured; however, the thermal conductivity of a single fiber of thermal insulation material A (which can be assumed to be a constant value in the experimental environment of this question), because Its diameter is too small and its aspect ratio (ratio of length to diameter) is too large to be measured directly. The thermal conductivity of a single fiber is the basis of the thermal conductivity of fabrics, and it is also the basis for establishing various fiber-based fabric thermal conductivity models. Establishing a heat transfer mechanism model between the thermal conductivity of a single thermal insulation material A fiber and the overall thermal conductivity of the fabric has become a research focus. This model can not only obtain the thermal conductivity of a single fiber of thermal insulation material A, and solve the current technical problem that the thermal conductivity of a single fiber of thermal insulation material A cannot be measured; On the basis of the relational model of the efficiency, control the weaving structure of the fabric and optimize the design to produce fabrics with excellent thermal insulation properties that better meet the needs of high-tech fields such as aerospace, military industry, petrochemical, construction, and transportation.

The fabric is a network structure formed by stacking and interweaving a large number of single fibers. This topic only studies the plain weave fabric, as shown in Figure 1 and Figure 2. Fabrics made of fibers with different diameters have different basic structural parameters, that is, fiber bending angle, fabric thickness, warp density, weft density, etc., which affect the thermal conductivity of the fabric. In this question, assume that the vertical section of any single fiber A is circular, and each fiber in the fabric is always a curved cylinder. Warp and weft bending angle 10° < $\theta$ ≤ 26.565°。

Thermal conductivity is one of the most important indicators of the physical properties of fibers and fabrics. There are gaps between the fibers of the fabric, and the air in the gaps is static air, and the static air thermal conductivity is 0.0296 W/(mK). When calculating the thermal conductivity of fabrics, both the heat transfer between fibers and the heat transfer of air in the voids cannot be ignored.

Figure 1. Schematic diagram of plain weave fabric section
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Figure 2. Three-dimensional image of plain weave fabric
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Under the laboratory environment of 25 ℃, the fabric is heated and measured with the Hotdisk device. The constant power of the Hotdisk is 1mW, and the action time is 1s. The heat flow is just transferred to the other side of the fabric at 0.1s. The experimentally measured data on the temperature change of the fabric on the side of the heat source between 0 and 0.1s with time is shown in Appendix 1, as follows.

Appendix 1 Data of Temperature Variation with Time

moment(s)	temperature (°C)
0	25.000
0.02	25.575
0.04	25.693
0.06	25.807
0.08	25.896
0.10	25.971

Experimental sample parameters:

The diameter of a single A fiber is d=0.6mm, the thickness of the fabric is h=2d, and the warp density is $\rho_s = 60 fibers/10cm$ , the weft density is $\rho_w =80 threads/$ $10cm$ $r_{w} = 80 threads /10 cm$ , warp bending angle $degrees\theta_s = 19.8$ $degrees$ $i_{s} = 19.8 degrees$ , weft bending angle $degrees\theta_w = 25.64 degrees$ , the overall specific heat of the fabric is $0.05MJ/m^3K$ , the overall thermal diffusivity of the fabric is $0.663mm^2/s$ 。

Please build a mathematical model and answer the following questions:

Question 1 : Assuming that the temperature in Attachment 1 is the surface temperature of the fabric on the heat source side, and only considering the heat transfer of fibers and the gas in the gaps, establish a mathematical model of the relationship between the overall thermal conductivity of plain weave fabrics and the thermal conductivity of a single fiber. Under the conditions of the experimental sample parameters in Appendix 2, the overall thermal conductivity of the plain weave fabric shown in Figure 2 is measured to be 0.033W/(mK). Please calculate the thermal conductivity of a single A fiber according to the established mathematical model.

Question 2 : Assumptions: 1) The diameter of any single A fiber made of fabric is 0.3 ~0.6. 2) The data of the surface temperature change with time of the fabric on the side of the heat source still refer to Table 1. 3) The changes in the overall density and specific heat of the fabric due to temperature and fabric structure can be ignored. May I ask how to choose the diameter of single A fiber and adjust the warp density, weft density and bending angle of the fabric so that the overall thermal conductivity of the fabric is the lowest.

Question 3 : If the temperature in Attachment 1 is actually the temperature of the air on the surface of the fabric on the heat source side, then convective heat transfer will occur on this side, assuming that the convective heat transfer coefficient on the fabric surface is 50 W/(m2K), please answer question 1 again and question two.

2 Problem Analysis

2.1 Question 1

This problem is to solve the thermal conductivity of a single fiber. First, for the overall heat transfer mechanism of the plain weave fabric, considering the two parts of fiber heat transfer and gas heat transfer in the void, the thermal conductivity of a single fiber can be solved by establishing a relationship model between the macroscopic thermal conductivity of the fabric and the thermal conductivity of the fiber. Secondly, the shape of the fiber is cylindrical, and the flow field, temperature field distribution inside the fiber and the thermal resistance of heat transfer between the fiber surface and the air medium are obtained from the Stokes equation and the energy equation, and then the thermal conductivity of a single fiber is obtained. Conductivity. Finally, the thermal conductivity of the fabric as a whole can be calculated from the experimental data and the heat conduction equation, and the results can be used to deduce the thermal conductivity of individual fibers.

The mathematical models to be used are:

Stokes equation and energy equation.
Heat conduction equation and boundary conditions in cylindrical coordinate system.
Nonlinear least squares model fitting analysis.
The one-dimensional heat conduction problem of the overall thermal conductivity of the fabric is solved from the heat conduction equation and experimental data.

2.2 Question 2

This question is to analyze how to choose the diameter of a single A fiber and how to adjust the warp density, weft density, and bending angle of the fabric to make the overall thermal conductivity of the fabric the lowest. First, it is necessary to understand the heat transfer process of the fabric. The heat conduction of fabrics can generally be regarded as the realization of heat transfer between fibers. Therefore, it is necessary to consider the influence of the thermal conductivity of the fiber on the overall thermal conductivity of the fabric. Second, the diameter has an effect on the thermal conductivity of the fiber. In general, the larger the diameter, the poorer the ability to conduct heat between fibers, resulting in lower overall thermal conductivity. Conversely, the smaller the diameter, the better the thermal conductivity between the fibers, resulting in an increase in the overall thermal conductivity. In addition, warp density, weft density, and bending angle also affect the overall thermal conductivity of the fabric. By adjusting the warp density, weft density and bending angle, the contact area and contact length between fibers can be changed, thereby affecting the effect of heat conduction.

The mathematical models to be used are:

The heat conduction model is used to describe the heat conduction process between fibers. An example is the heat conduction equation (Fourier's law of heat conduction), which takes into account parameters such as the diameter and thermal conductivity of the fibers.
The effects of changes in density and specific heat on heat transfer are simplified.
Finally, the finite difference or finite element method is used to solve the heat conduction model, and finally the overall thermal conductivity of the fabric is obtained.

2.3 Question 3

Considering the convective heat transfer, the temperature of the fabric surface is no longer simply the sum of the thermal resistance of the fiber and the air, but also considers the effect of the convective heat transfer. At this time, the relationship model among the overall thermal conductivity of the fabric, the thermal conductivity of the fiber, and the convective heat transfer coefficient on the surface of the fabric can be established. First, the fluid mechanics equation and the heat conduction equation can obtain the surface temperature and the distribution of the flow field and the temperature field. Secondly, the obtained surface temperature is added to the thermal resistance of the fiber, the thermal resistance of the air medium, and the convective heat transfer coefficient to obtain the overall thermal resistance of the fabric, and then the overall thermal conductivity is calculated. Finally, the thermal conductivity of a single fiber is deduced using the overall thermal conductivity obtained from the experiment and known parameters such as fabrics and fibers.

2.4 Question 4

Considering the convective heat transfer of air on the surface of the heat source side, it is necessary to redefine the heat conduction model of the fabric and consider the influence of convective heat transfer through a mathematical model. At this time, the heat conduction of the fabric mainly has two parts: the heat conduction between the fibers and the convective heat transfer of the air on the surface of the fabric. Since the influence of various factors on the overall thermal conductivity is involved in problem 2, multivariate function optimization can be used to find the optimal solution. If the multivariate function optimization model is established, the overall thermal conductivity of the fabric is used as the objective function, and then the optimal solution is found by adjusting the diameter, warp density, weft density, and bending angle of a single A fiber. In the objective function, the heat conduction of the fabric can be described by the heat conduction equation; while the convective heat transfer can be described by Newton's cooling law, where the convective heat transfer coefficient is 50 W/(m2K).

3 Introduction to the paper

Title: Research on structural optimization control of thermal insulation materials

Key words: heat conduction convection heat insulation material structure optimization control

Summary:

This paper mainly studies the heat conduction of flat fabrics. By optimizing various parameters of the fabric, fabrics with stronger heat insulation capabilities are obtained. Fabrics with strong heat insulation capabilities are widely used in aerospace. At the same time, fabrics with strong thermal insulation ability also have relatively weak electrical conductivity, which can be used in various aspects such as insulating clothing, insulating gloves, etc., and has high applicability.

For problem 1: We use the Fourier heat conduction equation to establish a mathematical model, simplify the cylindrical single fiber into a square column with the same volume, and combine the measured air thermal conductivity and the overall thermal conductivity of the fabric through the thermal resistance model. The thermal conductivity of a single fiber was obtained to be 0.03627 W/(mK).

For problem 2: the thermal conductivity of a single fiber obtained from problem 1, we then optimize the diameter, warp density, weft density, and bending angle of the model, we can get the optimal solution as the most densely woven, fiber diameter It is 0.3752mm, and the fabric with the best thermal insulation performance is obtained.

For question 3: Since convection has a great influence on the transfer of heat, it is necessary for us to add convection to our model. By establishing a new model, we find that the thermal conductivity of a single fiber is 0.0245 W/( mK), at the same time we can optimize the parameters of the fiber, we can get the fabric parameters with the best thermal insulation coefficient. The introduction of convection makes the model of fabric heat transfer more perfect, and more accurate fabric parameters can be calculated, which plays a more important role in the optimization of fabric parameters.

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