Modeling the Process of Moisture Diffusion by a Flame-Retardant Coating for Wood

Description of performance of fire-protective coatings during operation of a wooden construction structure is a separate and complex task that covers both stages of the protection process: both moisture protection and further heat transfer that occurs when the coating swells. It has been proven that they imply creating a layer at the surface of the material, which prevents the penetration of moisture to wood when the swelling of a wooden structure and the destruction of the coating begins. Due to this, it becomes possible to determine the effect of flame retardants and the properties of protective formulations on the process of decelerating the rate of moisture absorption of wood. When using fire-retardant coatings for wood, as it is indicated by the research results, typical processes imply the formation of a protective layer under the impact of temperature and a decrease in humidity, which slow down the moisture diffusion processes. It seems likely that such a mechanism of a fire-retardant coating is a factor in regulating the degree of formation of a weather-resistant protective layer and the effectiveness of heat and moisture insulation of the material. We have modeled the process of moisture transfer by a fire-retardant coating; the diffusion coefficient was determined and the estimation dependences were derived, which made it possible to obtain a change in the dynamics of moisture when drying a fire-retardant coating. Based on the derived dependences, the moisture diffusion coefficient of a fire-retardant coating was calculated, which amounts to 0.163·10-9 m2/s. The results from determining the mass loss of the coating sample during drying indicate the ambiguous effect of the nature of a protective agent on the change in humidity. In particular, this implies the availability of data sufficient for the qualitative implementation of the process of inhibition of moisture diffusion and the identification, on its basis, of the point in time that gives rise to a drop in the coating efficiency. The features of inhibiting the process of moisture transfer to wood, which was treated with a fire-retardant coating, include several aspects. Specifically, the use of water-insoluble flame retardants and other components, as well as a polymer binder, characterized by the formation of a heat-insulated layer at the wood surface

standardized efficiency can only be used taking into consideration the character of change in its flame-retardant properties during operation.
Therefore, it is necessary to study the conditions of moisture diffusion through a coating to wood and to substantiate the conditions for protecting a material against the effect of water by the coating.

Literature review and problem statement
Modern fire protection methods include the use of swelling coatings. They represent complex systems of organic and inorganic components [5], characterized by high intumescent capacity. However, it is not obvious for which classes of operation they belong. The effectiveness of using flame retardants based on organic substances was shown in [6]. Due to the effect of fire retardants based on polyphosphoric acids and foaming agents, it is possible to significantly influence the formation of a porous layer of foam coke. Works [5,6] show the component formulations, their efficiency, as well as the thermal characteristics of coatings, but the conditions of using coatings and their performance characteristics were not indicated.
A description of the performance of intumescent coatings, one of the tasks of which is to relate experimental data to existing theoretical models, is given in work [7]. This makes it possible, at least in principle, to evaluate the simplifications accepted, only with respect to thermal stability, which consider a thermophysical model whose solution is given by polynomials that are not related to the physical content.
A mathematical model was considered for a change in the temperature and humidity fields of a fire-retardant coating based on the laws of conservation of matter and energy. The models immediately imply a specific type of functional dependences with a set of undefined coefficients, and the task is reduced to determining the numerical value for these coefficients, which is associated with high inaccuracy [8].
The effectiveness of application of the coating's components based on organic substances is shown in paper [9], where, due to the action of flame retardants based on polyphosphoric acids and foamers, one can significantly affect the formation of a protective layer of foam coke. However, it is necessary to study the conditions for barrier formation for thermal conductivity and moisture capacity and to establish the effective action of a coating with the protective layer formation.
A significant increase in the stability, density, and strength of a protective layer is achieved due to the targeted formation of certain additives that form high-temperature compounds [10]. However, there are no relevant physical and chemical data to confirm this process.
The effect of inorganic fillers on a water-based flame-retardant coating proved its effectiveness; however, the mechanism for swelling a coating was not specified and the operating conditions of the coating were not identified in [11]. The authors present an analytical model of the fire resistance and thermal degradation of the porous structure of foam coke in a fire-retardant coating, which takes into consideration the pore shapes, but a given model does not consider which phase transformations a coating undergoes when operating in a humid environment [12].
Paper [13] proposed a mathematical model and a procedure for numerical study into the kinetics of the state of heat and humidity of a capillary-porous body, built on the simultaneous equation for solving thermal conductivity and moisture transfer. However, the cited study is characteristic of inorganic material and cannot be attributed to wood. Therefore, modeling the process of moisture diffusion through a fire-retardant coating for wood, the impact of the components that are part of them on this process, is an unresolved component in ensuring the fire resistance of building structures. This has necessitated our research in this area.

The aim and objectives of the study
The aim of this study is to identify patterns of moisture diffusion through a flame-retardant coating. This would make it possible to justify the application of a flame-retardant coating at sites with high humidity.
To achieve the goal, the following tasks were solved: -to model parameters for moisture diffusion through a fire-retardant coating for wood; to establish patterns in moisture transport through a flame-retardant coating at a change in temperature and moisture content.

1. Examined materials used in the experiment
The research was performed using a system consisting of ammonium polyphosphate (APP), melamine, pentaerythritol (PER) and a dispersed PVA-based binder. The fillers such as titanium dioxide (8 %), talc (2 %) were added to the above formulation.
Experimental coating samples were prepared based on a system containing 18÷20 % of APP, 12÷14 % of melamine, 10÷12 % of PER, 16 % of dispersed PVA, and water. The resulting mass was agitated, fillers were injected in the amount of 10 % and disks with a diameter of 36 mm with a thickness of 3÷4 mm were formed: 3 pieces (Table 1, Fig. 1). The disks were sanded tightly together. Sample No. 3 was immersed in distilled water, excess water was removed using filter paper. The disks were assembled in one (14.22 mm thick), a thermocouple was inserted in the middle at a distance of 8.8 mm, the side surface of a sample was wet-isolated with paraffin, that is a measuring cell was created (Fig. 2).

Fig. 2. Measuring cell
The mass of a wet middle sample (sample 2) was 3.74 g and the total sample mass was 12.62 g.
At the same time, the moisture content of the middle part of the sample and the entire sample was determined, which was 17.03 % and 5.05 %, respectively.

2. Procedure for determining the indicators of samples' properties
The research into modeling the process of moisture diffusion through a coating was conducted using the basic provisions of mathematical physics.
To find values for a coefficient of moisture diffusion through a coating, special equipment was designed and manufactured (Fig. 3). A sample together with the thermocouple and a flat heater was put on scales to control the temperature and moisture loss by the sample.
We experimentally determined the process of moisture diffusion by a flame-retardant coating according to the method given in [14]. The method implies that a sample of the flame-retardant coating is placed on a flat heater, which in turn is placed on the scales. The heating was switched on, the mass loss of the sample was measured and the temperature in the middle of the sample was monitored. Based on the results of mass loss, the rate of moisture evaporation was determined and the coefficient of moisture diffusion from the sample of the flame-retardant coating was calculated. 3 -sample, 4 -thermocouple, 5 -analog-to-digital converter, 6 -computer, 7 -temperature controller A criterion for the complete loss of moisture by a flame-retardant coating exposed to thermal action is the stability of coating mass over 3 days.

Modeling parameters of moisture diffusion through a flame-retardant coating for wood
As a result of treating wood with flame-retardant coatings, a capillary-porous layer is formed at the surface, which can protect the material from moisture penetration during operation. Wood itself is capable of absorbing moisture and swelling, which leads to loss of adhesion by the coating and its shedding.
Given the above, there is a task to study the mass transfer of moisture through the layer of a flame-retardant coating to wood at a change in heat and moisture content in the environment.
In order to define the process of moisture diffusion through the flame-retardant layer of a coating, the method for resolving a problem on mass transfer for a three-layer disk with different moisture capacity is suggested. At the initial time, the lower surface of the disk is heated to a temperature of drying (30 °С), which is maintained constant over the entire process of heating, with temperature distribution occurring through the coating. In this case, moisture is evaporated from the upper end of the disk (Fig. 4).
The differential equations describing the process of thermal diffusion of moisture through a flame-retardant coating are composed of the heat and mass transfer equations [15]: where Т is the temperature, °С; а is the coefficient of temperature diffusivity, m 2 /s; t is the time of sample drying, s; u is the moisture of a sample, %; d m is the moisture diffusion coefficient, m 2 /s; δ is the thermogradient coefficient, 1/degree; ε is the phase transition coefficient; r is the specific heat of water evaporation, J/kg. At a constant drying speed, the change in humidity is constant: After substituting (3) in (1) and appropriate transformations, the equation takes the following form: where λ are the coefficients of thermal conductivity of a material, W/(m•°С); с is the thermal capacity of a wet material, J/(kg•K); γ 0 is the density of a coating, kg/m 3 . Temperature distribution in a sample is described by equation: Considering the initial and boundary conditions: We obtain: After integrating (2) for x in the range from 0 to R and appropriate transformations we obtain: We accept the parabolic law of moisture distribution in a body: and initial and boundary conditions: 11) or -at x=0 Then, from (6), (10), we obtain: After integrating (13) for x in the range from 0 to R and appropriate transformations we obtain:  (16) Using boundary conditions (11), (12), we obtain: Substituting (17) into (16), we obtain the thermo-gradient coefficient: For isothermal conditions, since A*=0, we obtain: where u ∂ / ∂τ is the rate of moisture mass loss by a sample, %; R is the distance to a thermocouple, m; R 1 is the thickness of the middle part of a sample, m; ū is the moisture capacity of a sample, %; ū 1 is the moisture capacity of the middle part of a sample, %.
Thus, a dependence has been derived from the system of heat and mass transfer, initial and boundary conditions, to determine the coefficient of moisture diffusion through a coating.

Experimental study into moisture mass loss by a flameretardant coating for wood and its results
To determine the coefficient of moisture diffusion, a study was conducted into its evaporation under the action of a heating device. The results from determining the temperature and a change in moisture through a coating under the action of a heater are shown in Fig. 5, 6.
One can see from Fig. 5 that under the action of a heater on coating samples, intensive heat transfer started and a slight increase in the middle of the sample occurred over 3 days. The temperature did not change thereafter. The intensive mass loss by the sample continued to fall over 8 days, after that, it leveled to a certain extent (Fig. 6).
The moisture loss by the sample was calculated from equation: where m 0 is the sample mass before tests, g; m 1 is the sample mass after drying, g. The rate of sample mass loss was calculated based on where u is the sample moisture, %; t is the time of tests, s. , the diffusion coefficient was calculated, which for a given sample was 0.163•10 -9 m 2 /s, which attributes the product to those materials that slowly conduct moisture.

Discussion of results from studying the process of the progress of phase transformations
When using flame-retardant coatings for wood, as it is indicated by the results of our study (Fig. 6), it is a natural process to form a protective layer under the action of temperature and a decrease in moisture, which slow down the processes of moisture diffusion. It seems likely that such a mechanism of the flame-retardant coating is a factor in regulating the degree of formation of a weatherproof protective layer and the efficiency of thermal and moisture insulation of the material. This agrees with the data reported in [6,16], where the authors also relate the effectiveness of protection against moisture to using insoluble flame retardants. In contrast to the results from studies [8,17], the data obtained on the impact of moisture diffusion process by a flame-retardant coating and changes in moisture-insulating properties allow us to argue about the following: the main regulator of the process is not only the formation of a coating layer, but also the use of water-insoluble flame retardants and a polymeric binder, which, under the action of temperature-humidity fields, provide for the coating adhesion; -significant effect on the process of wood protection from fluctuations in temperature and moisture when applying a flame-retardant coating is produced towards the formation of water-insoluble capillary porous coatings at the surface of a natural combustible material.
Such conclusions may be considered appropriate from a practical point of view, since they allow a reasonable approach to determining the required formulation for a fire retardant. From a theoretical point of view, they suggest the determination of the mechanism of fire protection processes, which are certain advantages of this study. The results of determining the mass loss by a coating sample during drying (Fig. 6) indicate the ambiguous impact of the nature of a protective agent on the change in moisture. In particular, this implies the availability of data sufficient for the qualitative conducting of the process of inhibition of moisture diffusion and the detection, on its basis, of the moment that gives rise to a fall in coating efficiency. Such a detection will allow us to investigate the transformation of a coating surface, which moves in the direction of elevated temperature, and to identify those variables that significantly affect the onset of the transformation of this process.

Conclusions
1. We have modeled the process of moisture transfer by a flame-retardant coating, determined the diffusion coefficient and derived estimation dependences, which make it possible to establish a change in the moisture dynamics during drying of a flame-retardant coating. Based on the derived dependences, a diffusion coefficient at fire protection was calculated, which reaches 0.163•10 -9 m 2 /s for a coating.
2. Features of slowing down the process of moisture transport inside wood, which is treated with a flame-retardant coating, include several aspects. Namely, the use of water-insoluble flame retardants and other components, as well as a polymeric binder, which are characterized by the formation of a heat and moisture-proof layer at wood surface. This indicates the possibility of targeted control over the processes of moisture transfer to wood through the use of a flame-retardant coating.

Acknowledgement
Authors are grateful for the financial support of this work, conducted within the framework of the financing