Determination of the heat-retaining capacity of layered materials for shoe lining by the method of mathematical planning of the experiment

. A layered composite material for shoes was obtained by the adhesive bonding method. The middle layer of the material consists of a non-woven material made from a mixture of camel and sheep wool, the top and bottom layers consist of cotton jersey, and a polymer adhesive is located between the layers. The layers are bonded by thermal duplication at a temperature of 150±5°C for 2.0±0.2 minutes. As an optimization parameter, the heat-retaining capacity of the material was chosen depending on the thickness, surface density and percentage of camel wool. The strength and thermophysical properties of the layered material are determined.


Introduction
Laminated composite materials are becoming an important segment of the textile industry. Textile and nonwoven materials are used to reinforce polymer composites [1][2][3], in membrane materials for clothing and footwear [4], etc. A large group is represented by insulated nonwoven materials with heat-retaining and thermoregulation properties [5][6][7][8].
The fibrous raw material is the basic component for nonwovens, which determines the main properties of the final products, not only in terms of chemical properties, but also in terms of physical or other functional properties. Nonwoven materials used for clothing and footwear should have both high sorption and permeability [9]. These requirements are met, first of all, by materials based on natural fibers. Waste wool fibers are a potential source of thermal and acoustic applications due to their natural properties [10,11]. Among natural fibers, wool is one of the oldest textile fibers used by mankind [12]. Wool is much coarser than other fibrous materials, and wool factories accumulate a fair amount of coarse wool fibers that are not spun or knitted. Meanwhile, there is a shortage of warm, comfortable, light, soft lining materials for clothing and footwear based on natural raw materials. Multilayer nonwoven materials with satisfactory physical and mechanical properties have been obtained for use in clothing and footwear parts [13]. Material based on camel wool is used for lining winter insulated shoes.
The purpose of this study is to optimize the heat-retaining capacity of a composite layered material based on wool, depending on the parameters of formation, using the method of mathematical planning of the experiment.

Materials and methods
Non-woven materials from camel and sheep wool fibers were provided by the PE "M. Sayfullayeva" enterprise in Tashkent. To obtain a non-woven material, the waste of camel and sheep wool fibers is laid along the length in an even layer and fastened with a needlepunched method (Fig. 1). Parameters of the needle punching process: web working length -2.2 m, needle punching frequency -280-300 min-1, punching density -6000 m2, web output capacity -80-84 m/s. Cotton knitted fabrics of various types were provided by JV UZTEX LLC, Chirchik, Tashkent region. Dry polymeric film adhesive was purchased at the market of Tashkent city.
The layered material is formed by stacking layers in turn and simultaneously duplicating under pressure when heated. The completed material is passed through heated pressure rollers. The temperature of the roll is 150±5°C, the contact time of the material with the roll is 2.0±0.2 min. Fig. 2 shows a diagram of the construction of the resulting layered material. The physical and mechanical properties of materials were studied at the CENTEXUZ Certification Center TITLP, room temperature 20 ± 3°C, relative air humidity 65 ± 5%. The dependence of the tensile strength on the relative elongation of materials was determined according to the interstate standard "GOST 3813-72 (ISO 5081-77, ISO 5082-82) Textile fabrics and piece-articles. Methods for determination of breaking under tension" using a tensile testing machine AG-1, the maximum breaking force of the device was 1000 N. The resistance of materials to abrasion was determined according to the state standard "GOST 9913-90 Textile materials. Methods for determination of wear resistance" using a device for determining the resistance of fabrics to abrasion M 235/3, the rotation speed of the abrasives was 47.5±2.5 rpm, the sample size was a circle with a diameter of 50 mm.
The heat holding property of the materials was evaluated according to the "ASTM D7984-21 Standard Test Method for Measurement of Thermal Effusivity of Fabrics Using a Modified Transient Plane Source (MTPS) Instrument" using an AW-2 thermal conductivity instrument.

Results and discussion
To study multifactorial systems, the most appropriate is the use of statistical methods for planning an experiment, which provide for determining the number and conditions for conducting experiments that are necessary and sufficient to solve the problem with the required accuracy. These methods make it possible in many cases, with a minimum number of experiments, to obtain mathematical models of multifactorial processes, which are an equation that relates the optimization parameter to factors (the response function).
In general, the response function, which is also the optimization parameter , can be represented by the expression: are independent variable factors. The simplest mathematical model is a polynomial, which is linear with respect to unknown coefficients and therefore simplifies the processing of observations. The polynomial of the first degree in general form is expressed by the equation: = 0 + 1 1 + 2 2 + ⋯ + + 12 1 2 + 13 1 3 ⋯ + 12… 1 2 … For three factors, this equation has the form: = 0 + 1 1 + 2 2 + 3 3 + 12 1 2 + 13 1 3 + 23 2 3 + 123 1 2 3 The choice of the area of the experiment is carried out on the basis of a priori information, and at the same time, the main (zero) levels of factors and the intervals of their variation are established. Moreover, the main levels of factors are chosen in such a way that their combination corresponds to the value of the optimization parameter, which is as close as possible to the optimal one.
For the convenience of recording the conditions of the experiment and processing experimental data, the levels of factors encode: +1 -the upper level; -1 -lower; 0 -main. The coded value of the factor is determined by the expression: =̃−̃0 (4) where, ̃ is the natural value of the i-th factor; ̃0 -natural value of the main level of the i-th factor; -interval of variation of the i-th factor. If the number of levels of each factor is , and the number of factors is , then the number of all combinations of factor levels (the number of experiments in a full factorial experiment) is determined by the expression: The heat-retaining capacity determined experimentally on an AW-2 device was chosen as the optimization parameter y.
The optimization parameter is influenced by the following factors: 1 -thickness, mm; 2 -surface density, g/m2; 3 -percentage of camel wool mixed with sheep wool, %. Selected intervals of variation and levels of factors are given in table 1.  The experiments were not duplicated. To determine the variance of the optimization parameter, three experiments were carried out when the factors were found at the main levels.
The obtained values of the optimization parameter , its average value ̅, the deviations of the optimization parameter values from its average value ( − ̅) and the squares of these deviations are given in Table 3. The processing of the results of the experiment in the absence of duplication of experiments was carried out in the following sequence.
1. Calculation of the variance 2 of the reproducibility of the experiment (Table 3): where, 0 -number of parallel experiments at the zero point; -the value of the optimization parameter in the -th experiment; ̅ -arithmetic mean value of the optimization parameter in 0 parallel experiments. The dispersion of the reproducibility of the experiment is 2 = 0,6583.
The regression coefficients characterizing the effects of interaction are determined by the formula: where, i, l -factor numbers; j -row or experience number in the planning matrix; -the value of the optimization parameter in the -th experiment; , -coded values (±1) of factors i and l in the -th experiment.
3. Checking the static significance of the coefficients of the regression equation. The significance of the coefficients can be checked in two ways: 1) by comparing the absolute value of the regression coefficient with confidence intervals 2) using Student's ttest.
The significance of the coefficients was checked by the first method. To determine the confidence interval, the variances of the regression coefficients are calculated by the expression: where, 2 { } -variance of the i-th regression coefficient; -number of rows or experiences in the planning matrix. The confidence interval ∆ is found by the formula: where, -tabular value of the Student's test at the accepted level of significance and the number of degrees of freedom , with which the variance 2 was determined; 4. Determination of the variance аD 2 of adequacy by the formula: where, -observed value of the optimization parameter in the -th experiment; ̂ -the value of the optimization parameter calculated by the model for the conditions of the -th experiment; − the number of degrees of freedom, which for a linear model is determined by the expression = − ( + 1), where is the number of factors. At a 5% significance level and numbers of degrees of freedom for the numerator 1 = − ( + 1) = 8 − (3 + 1) = 4 and for the denominator 2 = − 1 = 3 − 1 = 2, the tabular value of the criterion =19.3. Since < , then the model expressed by equation (13) is adequate.
According to the obtained model (13), the optimization parameter increases with increasing factors 1 and 3 . The factor 1 has the greatest influence on the optimization parameter. With an increase in the thickness and surface density of the layered material, its heat-retaining capacity increases, but these indicators are limited by the requirements of the standard for insulated shoes. Therefore, 2.4 ± 0.1 mm was taken as the optimal thickness of the material, and the surface density was 704 g/m 2 .
The main physical and mechanical properties of materials obtained according to optimal parameters were determined in comparison with analogues ( Table 5). The layered material "Steppa" of the company "SIRETESSILE" (Italy) was chosen as an analogue. The heatretaining layer of this material consists of 70% wool and 30% polyester, which is fixed on a polyester fabric base. The resulting layered material has higher physical and mechanical properties than the control material. The data sheet for the "Steppa" material does not list the tensile strength and elongation values. According to the EN ISO 20345/A1:2007 Personal protective equipment -Safety footwear standard, the lining material for shoes must meet the following basic requirements: minimum tear force -15 N, resistance to abrasion -not less than 25600 cycles. In general, the composite layered material has satisfactory physical and mechanical properties, including heat-retaining capacity.

Conclusion
Waste fibers of camel and sheep wool are suitable fibrous components in shoe linings. The outer layers of the resulting material are made of cotton jersey, and the inner non-woven layer is made of wool. The layers are bonded together with polyacrylic adhesive. The method of mathematical planning of the experiment determined the optimal thickness, surface density, percentage composition of the non-woven layer of the material for maximum heat-retaining capacity. Composite layered material based on non-woven wool has satisfactory physical and mechanical properties.