Investigation of non-stationary thermal conductivity in a multilayer rubber-metal product during post-vulcanization cooling and before vulcanization

. In this article, we considered the issues of obtaining analytical solutions to a non-stationary problem of heat and mass transfer for a multilayer elastomeric material. We determined the criteria influencing the temperature change process in the processed material. We have obtained a dependence that establishes a functional relationship between the primary heat transfer criteria and the temperature of the processed material. The dependence is the initial ratio in developing an engineering method for calculating an industrial installation.


Introduction
Various branches of industry place high demands on the anticorrosion protection of parts of machines and apparatuses operating in conditions of strong, aggressive working environments at elevated temperatures and pressures, with periodic changes in the composition of these environments with their effective mixing.
Of the existing surface protection methods, rubbering occupies a special position due to the specific mechanical properties of rubber.Rubber is characterized by high elasticity, cushioning ability, good wear resistance, high fatigue strength characteristics, heat and frost resistance, resistance to aggressive media, and heat, gas, and water resistance.Rubber is used to protect against aggressive media in the manufacture of chemical equipment, in the automotive, aviation, and engineering industries, as well as for the manufacture of various parts: gaskets, shock absorbers, buffers, bearings, wear rings to protect drill pipes from wear, etc.
Vulcanization of coatings is the final and most responsible process in the entire work cycle on the gumming of any metal object, which is accompanied by high energy costs and especially needs improvement.
The modes of heat treatment of rubber-metal objects are established experimentally, according to the results of laboratory and bench tests, or by measuring the temperature in the product and then determining the duration of vulcanization.
The subject of work is the theoretical study of heat and mass transfer processes [1][2][3][4] with convective heat supply.The authors of the works [5][6][7][8][9][10][11] made a significant contribution to the study of heat and mass transfer.
It should be noted not only the theoretical studies of the authors of works [5][6][7][8] but also the practical results of Russian researchers [12,13], which were continued in the form of the development of thermal regimes for the heat treatment of polymer coatings for enterprises in the real sector of the economy [14][15][16].
Many works are devoted to the problem of studying heat transfer processes for a model of a flat multilayer structure [7][8][9][10][11], which, in turn, affected the mathematical model obtained in this work and became the basis for these studies.
The authors in [17,18], as a result of which the dependencies of heat transfer in nonstationary conditions were obtained, carried out mathematical modeling of heat and mass transfer processes in complex heat engineering systems.
Anyway, the analysis of the works of the listed authors allows us to conclude that heat and mass transfer processes are essential and significant in the heat treatment of rubber coatings and state that internal problems in the heat treatment of elastomer coatings require further study and solution.

Study Description
In cooling a rubber-metal product, internal heat generation is present at the initial moment.Evidence of the internal heat generation associated with the internal vulcanization reaction is that during vulcanization, the elastomeric coatings in the middle were heated above the temperature of the heat carrier.The increase in temperature is due to the content of bound sulfur in the elastomer: the more it is, the higher the temperature.In ebonite mixtures, up to 920 kJ/kg of rubber is released during vulcanization [5,6].
The speed of vulcanization and its flow (degree of vulcanization) depends on the temperature and, consequently, the amount of thermal energy supplied.In turn, the thermal effect of the reaction is a function of the degree of vulcanization.During vulcanization, its speed can be quantified.If a lot of heat is released in the system, and its reactivity is limited, volatile products are released, and the polymer is transformed and destroyed.Comparing the rates of heat release and vulcanization, one can notice the conditions under which the thermal effects of vulcanization have a negative impact on the quality of linings of rubber-metal objects.This is especially noticeable at high vulcanization temperatures (above 428 K).
The main task of the thermal conductivity of a multilayer lining is to find the temperature field inside the elastomer based on its known characteristics [9,10].
Let us consider a model of a cooled sample: a layer of rubber or ebonite is attached to the metal layer with the help of an adhesive seam.The thermophysical properties of steel, determined by thermal conductivity and thermal diffusivity, differ significantly from the same properties of adhesive and rubber layers.Thus, the thermal conductivity of steel is more than two times higher than the thermal conductivity of rubber.
In this regard, we considered a multilayer flat wall separating media whose temperatures change arbitrarily with time.Let us assume that the heat exchange on the outer surfaces of the wall with the media occurs according to Newton's law, and within each of its layers, an internal source of heat that changes in time acts.Then, let us assume that ideal thermal contact occurs between the layers of the wall, and the thermophysical properties of the layers and the intensity of internal heat sources do not depend on temperature.Then, we can reduce the problem of determining the non-stationary temperature field in the multilayer flat wall to integrate the following differential equation of non-stationary heat conduction [7,8,14]: Under initial (2) and boundary (3) conditions: (, 0) = (); (2) [() where x -linear coordinate measured from one of the outer surfaces of the wall;  1 () and  2 () -media temperatures;  1 and  2 -heat transfer coefficients.
The thermophysical characteristics of the multilayer wall as a whole and the internal heat sources acting in it with the intensity   (, ) as a function of the x coordinate are represented as: where   and    -respectively, the thermal conductivity coefficients and the volumetric heat capacity of the i layer of the wall;    -intensity of internal heat sources in the i layer of the wall;   -interface coordinate of the i and i+1 layers of the wall; n -number of layers; _( −   ) -asymmetric identity function [11].
If we pass from variable x to a new independent variable z according to the relation: The solution of equation ( 5) will be sought in the form: We require that the quasi-stationary component (, ) of the general solution (8) satisfies the differential equation ( 9 Then, if we accept then solution (8) will satisfy differential equation ( 5) and boundary conditions (7).Substituting equation ( 8) into the initial condition ( 6), we obtain an expression for determining the coefficients Am: where   2 -squared norm of eigenfunctions   (), From equation ( 13) we find an equation ( 16) for determining the coefficients   (): Determining the quasi-stationary component of the desired temperature field (8) from the solution of equation ( 9) with subsequent satisfaction of the inhomogeneous boundary conditions (10), we obtain (17): Here and in what follows, in order to simplify mathematical notations, it is assumed that   () = ().To solve the special Sturm-Liouville problem (11), (12), we reduce the differential equation (11) to a partially degenerate form: where _( −   ) = _( −   ) -asymmetric impulse function [11].
The general solution of equation ( 18) can be represented as follows [13]: where   and   -integration constants.Solving successively the system of algebraic equations, we find unknown quantities and obtain the solution of equation (11) in closed form: Satisfying the general solution (20) to the boundary conditions (12), we find, up to an arbitrary constant, the form of the eigenfunctions (21) and the characteristic equation for determining the eigenvalues km (22): If now, taking into account the found form of eigenfunctions (21), we calculate the coefficients Am and am(τ) according to expressions ( 14)-( 16) according to expressions ( 14) -( 16) and substitute their values and expression (17) into the desired solution (8), then constructing a general solution to the problem posed will end.

Implementation of the mathematical model
Consider a rubber-metal plate.The initial temperature distribution over the wall thickness is constant and equals 428 K.The heat transfer coefficients at the plate boundaries are α1 = α2 = 200.The ambient temperature is 283 K. Figure 1 shows a graph of the temperature distribution over the thickness of a rubber-metal sample.First comes a layer of steel, then a layer of ebonite.Dashed lines show the interface between the layers.The temperature distribution was observed at different times.The following time points correspond to the numbers on the graph curves: 1 -after 30 s, 2 -after 60 s, 3 -after 120 s, 4 -after 160 s, 5 -after 180 s, 6 -after 240 s, 7 -after 360 s, 8 -after 500 s, 9 -after 700 s, 10 -after 1530 s, 11 -after 1600 s, 12 -after 1650 s.Points are plotted on the graph corresponding to the temperature values obtained empirically.The solid line is the result of analytical solutions.As we can see, the convergence of the results is relatively high.As the cooling time decreases, the discrepancy between the calculated and experimental data does not exceed 1-2%, gradually reducing and tending to zero.

Conclusions and discussion
The development and implementation of computational methods for determining the thermal regimes of vulcanization will make it possible to intensify and optimize the process while maintaining the high quality of gummed products.
The developed methods of calculation and the presented design solutions make it possible to create new and improved existing industrial installations that would reduce the cooling process duration without compromising the quality of the rubber-metal product.The practical value of the results of the work lies in the development and implementation of an engineering method for calculating the cooling of coatings of gummed objects.The proposed method is designed to intensify the heat treatment process and improve the quality of rubber-metal products and the performance of rubber-coating equipment of anticorrosion shops.
The study's results on the intensification of heat transfer during post-vulcanization cooling of rubber coatings with subsequent constructive implementation and evaluation of efficiency became the basis of dissertation research.
The results of the study were tested at the industrial enterprises of Severstal OJSC (Cherepovets), Ammophos OJSC (Cherepovets), and Agrokhim OJSC (Sokol) when creating gummed objects.They formed the basis of their technological process.

E3SFig. 1 .
Fig. 1.Graph of the temperature distribution over the thickness of the rubber-metal sample.
):In this case, the eigenvalues km and the corresponding eigenfunctions   () are determined from the solution of a homogeneous boundary value problem: