Question 1
Question 1: This question requires the establishment of a mathematical model to describe the relationship between the overall thermal conductivity of the fabric and the thermal conductivity of individual fibers. This model needs to consider the structure of the fabric (such as fiber arrangement, void size and distribution, etc.) and the contribution of fibers and air in the voids to heat conduction. In addition, we need to work backwards from the overall thermal conductivity of the fabric to work out the thermal conductivity of individual fibers.
Problem-solving ideas:
Use the basic theory of heat conduction, combined with the thermal conductivity of fibers and air given in the question, and the structural parameters of the fabric (such as fiber diameter, fabric thickness, warp density, weft density, fiber bending angle, etc.) to establish a mathematical model . This model should be able to describe the relationship between the overall thermal conductivity of the plain weave fabric and the thermal conductivity of individual fibers.
The known overall thermal conductivity value of the fabric is substituted into the model, and the thermal conductivity of a single fiber is obtained through reverse calculation.
Question 2
Question 2: The question requires selecting the appropriate diameter of a single A fiber and adjusting the warp density, weft density, and bending angle of the fabric to minimize the overall thermal conductivity of the fabric.
Problem-solving ideas:
Based on the established mathematical model, the problem is transformed into an optimization problem, and the parameter combination (fiber diameter, warp density, weft density, and bending angle) is adjusted to find the parameter combination that minimizes the overall thermal conductivity of the fabric.
Some common optimization algorithms (such as gradient descent, genetic algorithm, simulated annealing, etc.) can be used to solve this optimization problem.
Question 3
Question 3: If the temperature of attachment 1 is actually the temperature of the air on the surface of the fabric on the heat source side, convection heat transfer will occur on that side. This means that our model needs to take into account the effects of convective heat transfer.
Problem-solving ideas:
After considering the impact of convective heat transfer, we need to modify or expand the original mathematical model. This may involve the application of convection heat transfer theory, such as Newton's cooling law.
Based on the new model, re-answer questions one and two. Similarly, problem one requires working backwards to calculate the thermal conductivity of individual fibers, while problem two requires finding a way to minimize the overall thermal conductivity of the fabric.
Note: During all calculations and model building processes, it is necessary to make full use of the experimental data and parameters given in the question.
Question A: Research on structural optimization control of thermal insulation materials.
New thermal insulation material A has excellent thermal insulation properties and is
widely used in high-tech fields such as aerospace, military industry, petrochemicals, construction, and transportation.
At present, the thermal conductivity of a fabric woven from a single fiber of insulation material A can be directly measured; however, the thermal conductivity of a single fiber of 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 (the ratio of length to diameter) is large, making it impossible to measure directly. The thermal conductivity of a single fiber is the basis of the thermal conductivity of fabrics and the basis for establishing various fiber-based thermal conductivity models of fabrics. Establishing a heat transfer mechanism model between the thermal conductivity of a single 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 A of the insulation material, solving the current technical problem that the thermal conductivity of a single fiber A cannot be measured; it can also establish a relationship between the thermal conductivity of a single fiber A of the insulation material and the thermal conductivity of the fabric. On the basis of the relationship model of efficiency, regulating the weaving structure of the fabric and carrying out optimized design can produce
fabrics with excellent thermal insulation properties that better meet the needs of high-tech fields such as aerospace, military industry, petrochemicals, construction, and transportation.
Fabric is a network structure formed by stacking and interweaving a large number of single fibers. This question only studies plain weave fabrics, as shown in Figures 1 and 2. Fabrics made of fibers with different diameters have different basic structural parameters, that is, fiber bending angles, fabric thickness, warp density, weft density, etc., which affect the thermal conductivity of the fabric. For 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
. The bending angle of warp and weft yarns is 10° < θ ≤ 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. The thermal conductivity of static air 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 gaps cannot be ignored.
Figure 1. Schematic cross-section of plain weave fabric
Figure 2. Three-dimensional diagram of plain fabric.
We used a Hotdisk device to heat and measure the fabric in a 25°C laboratory environment. The Hotdisk constant power is 1mW and the action time is 1s. The heat flow is 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 0.1 s with time are shown in Appendix 1.
Please establish a mathematical model to answer the following questions:
Question 1: Assume that the temperature in Appendix 1 is the surface temperature of the fabric on the heat source side, and only consider fiber heat transfer and gas heat transfer between gaps, establish the overall thermal conductivity of the plain fabric and the relationship between the heat transfer of a single fiber Mathematical model of the relationship between conductivities. Under the experimental sample parameter conditions in Appendix 2, the overall thermal conductivity of the plain fabric shown in Figure 2 was measured
to be 0.033W/(mK). Please calculate the thermal conductivity of a single A fiber based on the established mathematical model.
Question 2: Assumptions: 1) The diameter of any single A fiber made into the fabric is 0.3 mm~0.6 mm. 2) For data on the change of surface temperature of the fabric on the side of the heat source with time, please still refer to Appendix 1. 3) The changes in the overall density and specific heat of the fabric due to temperature and fabric structure can be ignored. Please tell me how to select the diameter of a 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 Appendix 1 is actually the temperature of the air on the surface of the fabric on the heat source side, convection heat transfer will occur on that side. Assuming that the convection heat transfer coefficient on the fabric surface is 50 W/(m2 K), please answer the question again.
Question one and question two.