【2023 Huashu Cup National Undergraduate Mathematical Contest in Modeling】A problem analysis, model establishment and references

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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 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 Annex 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 gap, 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. It can be built at this time
. . . slightly

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.
. . . slightly

3 Mathematical models

3.1 Question 1

First, the relationship model between the macroscopic thermal conductivity of the fabric and the thermal conductivity of the fiber is established. The macroscopic heat conduction equation of the fabric is:
$\lambda\nabla T$

Among them, $q$ is the heat flux passing through the unit area per unit time, $\lambda$ is the macroscopic thermal conductivity of the fabric, $\nabla T$ is the gradient of the temperature field.

Assuming that the fabric is only composed of fibers and air, without considering other factors in the fabric, that is, the fabric is a homogeneous isotropic material, and all the fiber directions in the fabric are parallel to the fabric surface. Decompose the fabric into a fiber grid and air gap, assuming that the fiber and the air gap are in point contact, and the thermal resistance at the contact point is negligible, then the overall thermal conductivity of the plain weave fabric and the thermal conductivity of a single fiber can be obtained The relationship between:

$\lambda = \phi\lambda_f + (1-\phi)\lambda_a$

Among them, $\lambda_f$ is the thermal conductivity of a single fiber, $\lambda_a$ is the thermal conductivity of air, $\phi$ is the volume fraction of fibers.

Considering the heat transfer of air in the gap, the thermal conductivity of air can be obtained according to the gas heat conduction equation:

$\lambda_a = {\frac{K}{3}C_p}$

Among them, $K$ is the thermal conductivity coefficient of the gas, $C_p$ is the specific heat capacity of the gas.

The volume fraction of fibers in the fabric can be calculated from information such as the geometric parameters of the fabric and the radius of the fibers:

$\phi = {\frac{\rho_s\pi(d/2)^2}{\rho_s\pi(d/2)^2 + \rho_w\pi(d/2)^ 2 + (h-\pi d^2/4)(1-\pi/4)}} = \frac{4\rho_s}{3\rho_s+4\rho_w}$

Among them, $\rho_s$ is classic and dense, $\rho_w$ is the weft density, $d$ is the diameter of a single fiber, $h$ is the fabric thickness.

Combining the above formulas, the relationship between the macroscopic thermal conductivity of the fabric and the thermal conductivity of a single fiber is obtained:

$\lambda = \frac{4\rho_s}{3\rho_s+4\ rho_w}\lambda_f + \frac{K}{3}C_p\left(1 - \frac{4\rho_s}{3\rho_s+4\rho_w}\right)$

. . . slightly

Finally, the overall thermal conductivity of the fabric is solved according to the experimental data in Appendix 1 and the heat conduction equation.

Let the plane of the fabric be $x - y$ 平面， $The z-$ axis is the direction perpendicular to the plane, $x$ 和 $The y-$ axis is along the warp and weft directions of the fabric respectively. According to the one-dimensional heat conduction equation, it can be obtained:

$\frac{\partial T}{\partial t} = \alpha\frac{\partial^2 T}{\partial z^2}$

Among them, $\alpha$ is the thermal diffusivity.

According to the experimental data, the temperature distribution at time 0s, 0.02s, etc. are obtained respectively, and then according to the heat conduction equation, the heat conduction equation of the fabric at each time point can be obtained by using the difference method. According to the least square fitting method and the heat conduction equation, the overall thermal conductivity of the fabric can be solved, and then the thermal conductivity of a single fiber can be deduced by the formula in question 1.

3.2 Question 2

The overall thermal conductivity of the fabric is first expressed as the sum of the thermal conduction contribution between the fibers and the thermal conduction contribution in the void, namely:

$\lambda=\lambda_{fiber}+\lambda_{air}$

Among them, $\lambda_{fiber}$ Indicates the heat conduction contribution between fibers, $\lambda_{air}$ Indicates the heat conduction contribution in the void.

Next, deduce the two parts separately.

(1) Heat conduction between fibers

It is assumed that the distribution of each fiber in the fabric is a geometric figure parallel to the surface of the fabric, with a columnar distribution in the longitudinal direction. Thus for each fiber, it can be regarded as a length $L$ , the cross-sectional area is $The column of A$ , the center line of the column is parallel to the surface of the fabric.

Let the heat conduction between fibers be $q$ , then:

$q=\frac{\lambda_{fiber}A}{L} \Delta T$

Among them, $\Delta T$ of the fabric. Since the fiber material is a thermal insulation material, the thermal conductivity of the fiber $\lambda_{fiber}=\lambda_{A}$ 。

With the heat conduction formula in hand, it is necessary to consider how to calculate the overall thermal conductivity of the fabric. For plain weave fabrics, the bending angle $\theta of warp and weft yarns$ generally takes a value of $\theta \leq 26.565°$ , by density $\rho_s$ And weft density $\rho_w$ It can be determined according to the dimensional accuracy of the fabric, the thickness of the fabric $h$ of each fiber $d$ are known.

Based on this, the number of fibers per unit area of the fabric can be deduced $n$ ：

$n=\frac{\rho_s+\rho_w}{d^2}$

Therefore, the total contribution of heat conduction between fibers in the fabric can be expressed as:

$\lambda_{fiber}=n\frac{\lambda_{A}}{L}kd$

For example, $k=\frac{\sqrt{\pi}}{4sin\theta}$ , that is, the contact coefficient between fibers.

(2) Heat conduction in the void

. . . slightly

Next, the model needs to be solved to determine the best solution. First, under the assumption that the diameter of a single fiber A is given, the warp density, weft density and bending angle of the fabric need to be adjusted to minimize the total thermal conductivity of the fabric. This can be translated into finding such that the overall thermal conductivity of the fabric $\lambda$ The combination of warp density, weft density and bending angle with the smallest $λ .$

For such extreme value problems, a common method is to use numerical optimization algorithms. In this problem, the numerical solution can be achieved using the minimize function in the SciPy library in Python.

Among them, the objective function that needs to be optimized numerically is:

$hf(x)=n\frac{\lambda_{A}}{L}kd + \frac{\lambda_{air}S_h}{\delta_h}$

Among them, $x$ represents a parameter vector including warp density, weft density and bending angle.

At the same time, constraints need to be considered:

$\theta_s \leq 26.565°$

$\theta_w \leq 26.565°$

$\leq d \leq 0.6mm$

Finally, the optimization method is used to solve the problem. The total thermal conductivity of the fabric is expressed as an objective function, and the bending angle, diameter of each fiber, warp density and weft density are used as optimization variables, and the above three constraints are expressed as constraint functions. There are MATLAB toolbox and optimize toolbox in Scipy in Python for optimization.

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【2023 Huashu Cup National Undergraduate Mathematical Contest in Modeling】Problem analysis, model establishment and references of research on structural optimization control of thermal insulation materials