Mathematical modeling of the mass transfer of the drying process of a multilayer thermal insulation coating

. Polymer thermal insulation materials are widely used in modern industry and technological production of energy carriers. Thermal insulation with polymer coatings is one of the main ways to protect thermal equipment from temperature effects, corrosion, cavitation, erosion, and other influences, reducing the consumption of expensive materials. However, although polymeric materials can significantly reduce the cost of heat losses, their use is kept at a relatively low level. This is due to the low level of culture in the construction industry and the desire to save on projects, even at the expense of quality. The important issue of forming a reliable system “polymer sheet – adhesive film – environment” is given minimal attention, which, as a result, greatly affects the performance and efficiency of the operation of power facilities. In this paper, we studied the problem of mathematical modeling of the mass transfer of the process of drying a multilayer thermal insulation coating on a polymer basis. The proposed method for calculating the concentration and temperature fields allows for optimizing the drying process and improving the quality and reliability of the technological process.


Introduction
Industrial construction uses a wide range of polymer thermal insulation materials.Polymeric thermal insulation materials are highly effective in aggressive operating conditions and where low thermal conductivity is required.Polymer thermal insulation materials contribute to a significant extent to reducing the material consumption of construction, reducing fuel consumption for heat supply of capital construction projects, and transportation of thermal energy [1,2].Insulating the surfaces of industrial equipment and heat pipelines with effective thermal insulation significantly reduces heat loss and helps reduce product costs.At the same time, comfortable conditions are created in the production premises.
At present, a sufficient number of publications are devoted to the issues of modeling heat and mass transfer in porous materials [3,4], including the theoretical study of heat and mass transfer processes with convective heat supply, described in the works [5][6][7][8][9][10][11].
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] mathematically modeled heat and mass transfer processes in complex heat engineering systems.As a result, the authors obtained the dependencies of heat transfer in non-stationary conditions.
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 thin polymer coatings.However, the most frequent question is heat and mass transfer in the layer.At the same time, the issue of forming a multilayer structure due to the reliable joining of thermal insulation with the main surface has yet to be adequately studied.

Physical model of the "polymer substrate -adhesive filmenvironment" system
Polymer thermal insulation materials belong to the group of the most effective.Polymer thermal insulation materials belong to the group of the most effective.This includes materials on an organic (synthetic) basis: foam plastics, honeycomb plastics, polyurethane foams, foamed rubbers, as well as materials based on synthetic fibers.
The physical model of the drying process of the thermal insulation coating is shown in Fig. 1.Practically used adhesive compositions contain the following groups of ingredients:  halogen-containing film-forming polymer (one or more) with a reduced dissociation energy of the C-halogen bond;  adhesive film elasticizers (polar elastomers, various rubbers with a low degree of halogenation, esters, etc.);  adhesion modifiers (aromatic di-and mononitroso compounds, p-quinone dioxime in combination with an oxidizing agent, molecular complexes of resorcinol and urotropin, various adhesive-active synthetic resins, etc.);  vulcanizing agents (amines, nitrogen-containing resins, metal oxides, etc.);  fillers (carbon black, mineral fillers -colloidal silicon dioxide, titanium dioxide, talc, bentonite, etc.);  heat stabilizers (metal salts, epoxy compounds, amines, etc.);  organic volatile solvent or mixture of solvents.The process of drying the glued canvas is a necessary step in the formation of multilayer thermal insulation coatings.Drying of the adhesive coating is a complex of interrelated processes:  solvent diffusion into the polymer array;  solvent diffusion into the adhesive film and evaporation of volatile compounds from the adhesive film into the environment.

Mathematical model of the "polymer substrate -adhesive film -environment" system
In the course of mathematical modeling of the process of drying a glued polymer thermal insulation sheet, we made the following simplifying assumptions:  the physical parameters of the polymer layer and the adhesive film are the same;  the thickness of the polymer layer and the adhesive film remain unchanged;  the influence of evaporation processes in the problem of mass transfer is not taken into account;  the field of solvent concentration along the thickness of the film and the polymeric substrate at each moment of time is constant;  the current concentration of the solvent in the environment is negligible compared to the initial.
The course of diffusion processes is determined by the presence of two phases in the "polymer substrate-adhesive film-environment" system.It consists of the transition of a substance from one phase to another, and the mass flows at the phase boundaries are presumably equal.
The process of solvent diffusion from the adhesive film into the polymer layer and the environment is described by a system of differential equations with initial and boundary conditions.The symbols with index r refer to the polymer layer, and those with index c refer to the adhesive film.
where  ∈ [−ℎ  ; 0]; Cr -mass concentration of the solvent in the polymer layer, kg/m 3 ; kr -coefficient of effective diffusion of the solvent in the polymer layer, m 2 /s.
where  ∈ [−ℎ с ; 0]; Cс -mass concentration of the solvent in the adhesive film, kg/m 3 ; kс -coefficient of effective diffusion of the solvent in the adhesive film, m 2 /s.
The boundary conditions from the definition of the diffusion-permeable layer and the domain of problem ( 1 The conditions for the continuity of diffusion flows at the interface between the adhesive film and the polymer layer are described by the equation ( 4): The density of the diffusion flux from the adhesive film into the environment is equal to the difference between the initial and current mass concentrations.
where  * , Cc -respectively, the initial and current mass concentration of the solvent in the adhesive film, kg/m 3 ; β -mass transfer coefficient, m/s.At the same time, we assume that ℎ → ∞.
The boundary conditions will take the form: Since the numerical values of the diffusion coefficients of the adhesive film and the polymer composition are in the following numerical range: 3•10 -10 ≤kr ≤ 4.5•10 -10 and 3.5•10 - 10 ≤kc ≤ 5•10 -10 m 2 /s [7] and by virtue of the simplifying assumption, in solving the diffusion problem ( 1)-( 2) we take kr/kc=1.
The solution of differential equations ( 6) can be written directly in the form: where А, В, А1, В1 -constants about the z.
×  ( Formulas ( 12) -( 13) represent analytical expressions for calculating the mass concentration of the solvent in the adhesive film and the polymer layer at various times.
The resulting mathematical model qualitatively reflects the distribution pattern of the mass concentration of the solvent in the adhesive film and in the polymer layer, which can be observed in the concentration profiles (Figure 2).
where Src, Sco are the areas of interfaces in the "polymer substrate -adhesive film" and "adhesive film -environment" systems, respectively, m 2 ; the first term of the equation is the amount of solvent diffused into the polymer from the adhesive film, kg/m 2 ; the second term of the equation is the amount of solvent removed from the adhesive film into the environment, kg/m 2 ; M is the initial mass of the solvent in the adhesive layer, kg.
The obtained dependence makes it possible to analyze the temperature changes in the polymer substrate during the drying of the adhesive coating.
Improvement of adhesive formulations increasing requirements for the production technology of multilayer insulated products expands the scope of polymer thermal insulation materials in various industries.Calculating the fields of concentration and temperature makes it possible to optimize the drying process in industrial conditions, reducing the operating costs of the energy carrier.

Fig. 1 .
Fig. 1.Scheme of solvent diffusion from the adhesive film into the polymer with simultaneous evaporation of the solvent into the environment, where hr and hc -thickness of the polymer layer and adhesive film, respectively, m; t -time, s; z -spatial coordinate.

E3SFig. 2 .
Fig. 2. Distribution profiles of the solvent concentration in the adhesive film Cc (a) and in the polymer layer Сr (b) at hc = 2•10 -3 m; hr = 2•10 -2 m; Dc = Dr = 4•10-10 m 2 /s; β = 2.5•10 -7 m/s for different values of time τ: 1 -60 s; 2 -120 s; 3 -180 s.The quality indicators of the adhesive film are characterized by the degree of moisture content, which changes over time.To calculate the drying time τ* of the adhesive coating from condition (4), we compose the equation: