Evaluation of indoor temperature for various building envelopes damaged

The purpose of this paper is to represent inner air temperature calculation algorithm provided that interior space is formed by several building envelopes which transmit thermal losses. The chosen external wall includes an area with damaged structural layers. We examine one-dimensional thermal transfer between hot (inner) and cold (outside) space in steady-state conditions. The derived equation allows calculating the estimated inner temperature of the air taking into consideration the various combination of building envelopes on condition that one of them is damaged.


Introduction
The problem is to analyze the process of microclimate formation considering variable properties of building envelopes with damaged zone.
Analysis of previous studies shows the engineering solutions based on one-dimensional thermal transfer in steady-state conditions [1,2]. However, there are no investigations devoted to the estimation of indoor air temperature under the assumption that building envelope contains the damaged area. Some investigations [3,4,5,6,7,8] are focused on the problem of microclimate parameters evaluation in old buildings with visibly damaged zones on building envelopes. Others refer to the problem of buildings envelope modernization but generally, they oriented to the choice of convenient technical solution for building reconstruction and analysis of thermal insulation impact on microclimate [9,10,11].
The problem of indoor temperature determination for various building envelopes on condition that one of them (external wall) contains damaged zone with fractured structural layers leads us to the following tasks: -to specify the initial data of the problem concerned; -to derive a mathematical expression for inner air temperature determination provided that one of considered building envelopes (the external wall) is damaged by a certain area of erosion; -to calculate and analyze the results of estimation for a particular case.
-to generalize an analytic expression; -to carry out a numerical modeling using ELCUT software.

Initial data
Previous investigations [12,13] allowed us to define initial data and boundary conditions for inner air temperature determination in case of the only heat transfer through an external wall with damaged zones. Analytic determination of indoor temperature for various building envelopes on condition that one of them (external wall) contains damaged zone with fractured structural layers requires the following assumptions: -we examine plane wall, window, floor slab or roof slab containing several construction layers with different thicknesses 1,2...n  , mm , this solid is infinite in y and z According to the assumptions given in [13], we assume that the plane multi-layer wall contains damaged area caused by erosion. The thickness of this area . . It is plausible to assume that the presence of damaged area on the external side of the plane wall results in the temperature change of the both inner and exterior surfaces ( Let us consider two variants of indoors with several types of building envelopes. For the first variant, we consider a room with external wall wich contains damaged zone and installed window (Fig. 1a). As for the second variant, we study a room enclosed by the floor slab, the roof slab and the external wall with damaged zone and installed window (Fig. 1b).
If we take in consideration mentioned above for the first case the heat flows through different building envelopes can be given as the following mathematical expression: -heat flux through the plane multi-layer wall with the surface F , 2 m , can be written as:  t -estimated temperature of the inner air, о С .
After the transformation we can obtain the following formula: -according to [14], heat flux through the window with a surface w F , 2 m , which is installed in the external wall without damages, can be defined as: -the heat flux through the window with a surface w F , 2 m , which is installed in an external wall with damaged zone can be given as: x -the part of damaged area in the whole wall and which (according to [13])can be expressed as: The value of dam x is ranging from 0 to 1.
Let us suppose that the heating equipement produces the equal quantity of thermal energy whether the wall (with window) contains damaged area or not. The The estimated temperature of the interior air can be derivated from (8) Taking into consideration (7) an expression (9) can be written as: Concerning the second variant, heat flux through the building envelopes including installed window in a plane external wall with or without damaged zone can be given in the same way as for (1) Similarly to the 1 st variant let us assume that , o C (15) wall n , w n -coefficient which depends on the positioning of the building envelope (wall, window) against the external air.
Expressions (10) and (15) are quite lengthy. Its obviously if we consider more building envelopes we will obtain larger expression. Therefore the previous formula can be written in general form as:

Analysis
The widespead in Dnipropetrovsk for the eighties of XXth century standard composition of multi-layer wall was chosen as an exemple for calculation. Figure 2 shows the cross-section of the wall.  Table 1. According to the obtained data, the coefficient of heat transfer and thermal resistance of the wall varies linearly [13] for 1,3 % per each 10 mm of damaged area thickness.
Estimated inner air temperature in the room enclosed by an external wall with damaged zone and installed window was calculated using the expression (10). The results are given in Table 2 and shown on Fig. 3a. Estimated inner air temperature in the room enclosed by floor slab, roof slab, an external wall with damaged zone and installed window was calculated using the expression (15). The results are given in Table 3 and shown on Fig. 3b.  Fig. 3. Evaluation of indoor airspace temperature depending on the damaged area change on condition that the room is enclosed by the following building envelopes: а) external wall with damaged zone and installed window; b) floor slab, roof slab, external wall with damaged zone and installed window Analysis of the data given in Table 2, 3 and previous results presented in [4] leads to the statement that in the case of increasing of building envelopes number (taking into account only damages of the external wall) estimated parameters are decreasing, depending on calculated (basic) temperature, by: -2,45% provided that the room is enclosed by external wall with an installed window; -0,35% on condition that the room is enclosed by floor slab, roof slab, an external wall with an installed window.
The numerical analysis was carried out by ELCUT 6.3 workbench, a program which allows calculating a three-dimensional or a two-dimensional steady-state temperature distribution and heat transfer [15,16]. The software is based on the finite element analysis using an appropriate grid. Properties of used materials are taken into consideration. Two basic models of rooms with various building envelopes (including damaged walls) were considered for simulations. Mentioned software allows evaluating the inner air temperature in the case of the basic variant without the damaged zone. Results are represented on the Fig. 3 a, c. Also, the variant which contains the damaged zone was investigated and the results are shown on Fig. 3 According to the obtained results of the simulation for the both basic variants (with and without the damaged zone), we can state that there is a decreasing of the airspace temperature and the same effect was detected between the layers of building envelope. Table  4 represents temperatures on the external and inner surfaces of the building envelope and also it shows the temperature between the structural layers of the building envelope under the computer simulation. Since the data selection for the modelling was carried out in the place of the thickest damage, so we compare these data with the results of the calculations for the analogical thickness of the damaged zone from the Table 5.   The results represented in Table 4 and 5 are compared in Table 6. The calculated temperature values on the inner surface of the wall differ from the results of modelling by 2,4 %. This result doesn't exceed the tolerance.
The divergence of calculated results and computer modelling data (ELCUT 6.3) can be explained by the following: -in case of two-dimensional simulation we don't take into account the part of damaged zone on the surface of plane wall dam x ; -in the case of mathematical modelling we take into account equal thickness for the building envelope in damaged zone, whereas in the case of computer modelling mentioned parameter is varying.

Conclusions
This paper is devoted to the analytical investigation (for climite conditions of Dnipropetrovsk) of indoor airspace temperature variation provided that the room is enclosed by several multi-type building envelopes and one of them contains damaged zones. The obtained data reveals that the presence of damaged zones leads to decrease of temperature values at the border of each wall layer, therefore we observe the drop of indoor airspace temperature.
Taking into consideration the results presented in [4] we can state that in the case of increasing of building envelopes number (taking into account only damages of the external wall) estimated parameters are decreasing, depending on calculated (basic) temperature, by: -2,45% provided that the room is enclosed by external wall with an installed window; -0,35% on condition that the room is enclosed by floor slab, roof slab, an external wall with an installed window.
Results of mathematical and computer modelling (ELCUT 6.3) let us assume that its possible to evaluate the temperature change on external and inner surfaces of the building envelope and the temperature between layers on condition that the part of damaged zone on the surface of the plane wall dam x is taken into account.