Reducing the thickness of the insulation layer of building walls based on the study of their temperature and moisture regime

. The thickness of the building insulation is calculated with the table values of the moisture of building materials, which is included in the regulatory documents with a margin. The study proposes the method for determining the thickness of the insulation layer using the values of operational moisture. To determine the operational moisture of materials, the equation based on the moisture potential is applied, the solution of which is sought using a discrete-continuum approach. The proposed equation makes it possible to determine the distribution of the moisture potential function in the wall enclosing structure. Thereafter, using the moisture potential scale, the distribution of mass moisture over the thickness of the enclosure for the period of maximum moisture accumulation is determined. The equation for determining the thickness of the insulation layer, in which the calculated value of the operational moisture is explicitly substituted, is derived. An algorithm for calculating the thickness of the insulation layer, taking into account the unsteady-state heat and moisture regime of the building envelope, is given. The application of the proposed method on a wall enclosing structure with insulation from expanded polystyrene boards is illustrated. It was found that the thickness of the insulation can be reduced by 26 mm from 120 mm to 94 mm while maintaining the reduced heat transfer resistance of the building wall. It is noted that the proposed method can be applied in the design of new wall enclosing structures.


Introduction
During the operation of the building, building materials accumulate moisture in their thickness. Water in the material conducts heat well, therefore, it negatively affects the thermal protection of buildings [1].
Every building material has a dry thermal conductivity. This thermal conductivity is achieved by drying the building material in a special oven. According to modern regulations in construction industry, building materials can be calculated under operating conditions A or B. The operating conditions of the building material depend on the construction area, humidity zone and the moisture regime of building premises. Humidity conditions of the building premises can be dry, normal, damp, wet. The thermal conductivity of a building material under operating conditions A is higher than the thermal conductivity of a dry building material, but lower than the thermal conductivity under operating conditions B [2][3][4][5][6].
If, under the same boundary conditions, we calculate the thickness of the insulation of the building wall under operating conditions A and B, we obtain that the wall, which is built under operating conditions B, has the longest thickness of the insulation layer [7][8][9][10][11].
Despite the fact that this method takes into account the effect of moisture on the thermal conductivity of building wall materials, the values of the operational moisture of building materials were obtained in Soviet times based on the results of field experiments and included in regulatory documents with a margin [12][13][14][15][16].
Thus, due to the development of discrete-continuous calculation methods, it is possible to calculate the moisture behavior of the building wall, taking into account the climatic impact on it and calculate the value of the operational moisture of the building material. Based on this value of operational mass moisture, it is possible to calculate the operational thermal conductivity of building wall materials and obtain the thickness of the insulation layer, taking into consideration the unsteady-state moisture regime [17][18][19].

The Problem
To develop the method for calculating the thickness of the insulation layer of a building wall using the values of operational mass moisture which are obtained by modeling the unsteady-state heat and moisture regime using a discrete-continuous approach.

Application of the discrete-continuous approach to calculate the operational mass moisture and operational thermal conductivity of the building materials of the building wall
In previous studies, the moisture transfer equation which represents the transfer of moisture under the action of the difference in the moisture potential and takes into account the influence of the temperature field was obtained: where t E -saturated water vapor pressure, Pa; τ -time, s; x -coordinate, m, F material heat-humidity characteristic coefficient, m 2 /(s⋅Pa). The differential equation (1) can be solved using a discrete-continuum approach involving a grid breakdown of the space-time region along the x axis and the search for a solution along the y axis ( Figure 1). The discrete-continuum approach makes it possible to derive the equation for the dependence of the moisture potential on time: where p -the coefficient of the external boundary condition for a building enclosing structure, Pa/s 2 ; B -a column vector, the first and last elements of which describe the boundary conditions on the outer and inner surfaces of the enclosing structure, other elements are equal to 0 for a multi-layer enclosing structure; L -a column vector, the first element of which is equal to one, other elements are equal to 0 for a building enclosing structure.

Derivation of the equation for calculating the thickness of the insulation layer taking into account the unsteady-state moisture regime
The dependence of the thermal conductivity coefficient on mass moisture is determined by the formula: where w  and 0  -thermal conductivity of a building material in wet and dry conditions, W/(m• ͦ C); w -mass moisture content of the building material, kg/kg; w  increase in thermal conductivity with a change in the moisture of the building material, W/(m• ͦ C).
The heat transfer resistance of the building wall is the sum of the thermal conductivity resistance of the material layers and the heat transfer resistance of air and material surfaces: where cond R -conditional resistance to heat transfer, (m 2 • ͦ C )/W.
Equation (4) is a conditional resistance to heat transfer which does not take into consideration the point (dowels, insulation fasteners) and linear (window and door slopes, corners of the building, junction of the wall to the plinth, etc.) thermal inhomogeneities which is located in the wall structure. In order to take them into account, it is necessary to calculate the value of the reduced resistance to heat transfer: expression describing the decrease in resistance to heat transfer due to the influence of point and linear thermal inhomogeneities, W/(m 2 • ͦ C). According to regulatory requirements, the value of reduced heat transfer resistance must be more or equal than the value of required resistance to heat transfer: where red R -required resistance to heat transfer, (m 2 • ͦ C )/W. Let us substitute the conditional resistance to heat transfer from equation (4) We substitute the reduced resistance to heat transfer from equation (7) into inequality (6).
Let us express the thickness of the insulation from inequality (8): In inequality 9, the value  To consider the unsteady-state moisture regime, it is necessary to enter the operational mass moisture into equation (9). To do this, let us substitute the operational thermal conductivity (3) into inequality (9): Formula (10) allows us to calculate the thickness of the insulation layer taking into account the unsteady-state moisture regime by substituting the operational moisture into its structure, which is determined using the discrete-continuum approach.

Algorithm for calculating the thickness of the insulation layer using the unsteady-state heat and moisture regime of the building envelope
1. The enclosing structure of the building wall is divided into sections as shown in Figure 1

Determining the maximum moisture content time moment in the building wall and the value of moisture content of building materials
For mathematical modeling, the building wall structure in Moscow consisting of aerated concrete base and polystyrene foam insulation was chosen (table 1). Polystyrene insulation layer 4 External plaster layer The calculation of this wall according to the regulatory documents showed that with the thickness of the base layer of 0.3 m, the thickness of the insulation layer should be 0.12 m.
Enclosing structure of a residential building is presented (Figure 2). Mathematical modeling of the unsteady-state moisture regime was performed in order to determine the maximum moisture period.
Change in the insulation operational moisture of the building wall during the year is presented (Figure 3). The moisture distribution in the investigated wall of the building at the moment of maximum moisture accumulation is presented. (Figure 4). The achieved values of operational humidity and standard values of moisture content of building wall materials are presented (Table 2).

Determining the thickness of the insulation layer taking into account the unsteady-state moisture regime of the building envelope
The thickness of the insulation layer was determined by the formula (10) using the values of the calculated moisture in accordance with the to table 2. As a result, the thickness of the building wall insulation was obtained as 94 mm, which is 26 mm less than the value of the insulation thickness according to the normative moisture of building materials. The enclosing structure of a residential building after clarifying its heat and moisture behavior is presented ( Figure 5). Insulation savings are obtained due to the modeling of the unsteady-state heat and moisture regime of the building wall as well as consideration of variable climatic influences on the fence, the physical properties of building materials and the inertia of the moistening process are achieved.

Conclusion
The new effective method taking into account point and linear thermal inhomogeneities and the unsteady-state moisture regime of building materials caused by climatic influences and the inertia of the moistening process for determining the thickness of the insulation layer for the building wall was developed. The moisture content of a building material is determined using the discrete-continuum approach. Determination of the value of operational thermal conductivity is calculated according to the well-known formula for the dependence of thermal conductivity on mass moisture. The results of the calculations showed the possibility of achieving the same values of the reduced resistance to heat transfer with the savings of 26 mm of the thickness of the insulation layer due to unsteadystate heat and moisture regime of the fence. This method can be applied in practical engineering work for the design of new enclosing structures.