MATEMATICAL MODELING OF THE MICROCLIMATE IN BUILDINGS WITH GREEN ROOFS

In the present study, the basic energy flows, forming the microclimate of buildings with green roofs were been presented. Based on the finite differences method, a mathematical model has been developed, describing the building energy exchange processes. For this purpose each element of the building (walls, floor, roof) has been divided into multiple bordering layers and the temperature curve of each one of them was estimated using the heat fluxes entering from the bordering layers. The indoor air temperature was then estimated using the building element’s internal surface. The model’s algorithm is presented in a block-scheme, describing the sequence of evaluations. The developed model allows the microclimate inside buildings with green roofs to be modeled and simulated for different climate zones and insulation parameters.


INTRODUCTION
The green roofs are known to offer very good insulation, compared to the conventional roofs.During the winter season it is a consequence of the high soil substrate density and heat capacity, as well as the insulation offered by the vegetation and during the summer season -mainly by the evapotranspiration process [1,2].
Until now, the green roof insulation cannot be simply expressed as thermal resistance (R-value) because it varies quite significantly for the different seasons, climates and weather conditions [1,3].Existing studies show that during the summer season it is possible to reduce the indoor air temperature by up to a couple of degrees Celsius, if the building parameters are sized in accordance with the local climate [1].
In order to evaluate the influence of the building's parameters (insulation width, insulation parameters etc.) on the indoor air temperature, it is necessary to develop models and means for simulation of its microclimate.There are many known models, which describe the heat exchange through a green roof [4][5][6], but there are no known models, summarizing all the energy flows of such buildings, and allowing the indoor air temperature to be simulated in the time domain.
The goal of this study is to describe all the energy flows in a building with a green roof and to develop a mathematical model, which can be used for simulation of the temperature curve inside buildings with green roofs.

Major energy flows and approximations
In order to properly size the parameters of buildings with a green roof, it is necessary to model all the major energy exchange processes that occur inside it.The major energy flows in a single-storey building are (Figure 1): gr conv.
-the convective heat flow from the green roof (vegetation) surface to the environment, -the long-wave radiation heat flow from the green roof surface to the environment, -the heat flow of the reflected solar energy, -the convective heat flow between the indoor air and the roof's interior surface, -the convective heat flow between the indoor air and the exterior walls, -the convective heat flow between the exterior walls and the outdoor air, -the convective heat flow between the indoor air and the floor's internal surface, -the conductive heat flow between the floor and the soil under the building, -the convective heat flow accumulated in the interior walls, is replaced with convective heat flow between the ceiling's surface and the indoor air of the lower storey.For residential multi-storey buildings the temperature gradient between the indoor airs of the two storeys is usually low and the flow in f .could be neglected.
The one dimensional heat transfer at any point of a homogenous element (wall, etc.) can be described with [7,8]  where is the thermal conductivity, -density and C -the specific heat capacity.However the modern buildings are characterized with multiple insulation layers.Considering the temperature regimes inside and outside the building are non-stationary, the heat transfer through them cannot be evaluated as a simple conduction.In order to model the heat exchange processes in buildings with green roof, in the present study have been adopted the following approximations: 1.Each element of the building (wall, floor, ceiling) is divided into multiple bordering layers; 2. Each layer is characterized with width, specific heat capacity, density and thermal conductivity; 3. The temperature and the thermal characteristics of a certain layer are approximated to be the same in its whole volume; 4. The indoor air temperature and thermal parameters are the same in its whole volume; 5.The soil layer under the building's slab has the same temperature and thermal parameters in its whole volume.
Using the finite difference method in its one dimensional form, the heat flows through the building elements are modeled between each two bordering layers.

Heat flows through the roof
The roof is divided into n bordering layers, with the 1 st one being in contact with the indoor air and the n th (topmost) one -in contact with the environment.The temperatures of these layers are respectively .The energy flows between the different layers and the indoor/outdoor air are presented in Figure 2.
The conductive heat flow between two bordering layers, in the moment of time t, is calculated with [7,8], where is directed from the 1 i th towards i th layer: where The energy flux entering the topmost layer through the vegetation surface is a function of multiple processes, including the solar radiation, evapotranspiration, long-wave radiation and convection.A detailed model for estimation of these energy flows, as well as the resulting total energy entering the topmost layer from the environment, has already been presented in [9].The energy flow in r. , between the bottommost layer (the ceiling surface) and the indoor air, is estimated with [7,8]: where in r h .
is the heat transfer coefficient of the ceiling, T .respectively.The energy flows between the layers and the environment are presented on Figure 3.The heat flows between two bordering floor layers are estimated similarly to equation (1).The heat flows between the floor's internal surface (1 st layer) and the indoor air as well as the floor's external surface (n th layer) and the soil under the slab, are evaluated with [7,8] where is the heat transfer coefficient of the indoor floor surface, -the temperature of the soil under the building's slab, K ; -the thermal conductivity of the soil under the building, -the thermal conductivity of the n th layer of the floor, -the width of the n th layer of the floor, m ; -the width of the underbuilding soil layer, m .T .respectively.The 1 st layer of the exterior walls is in contact with the indoor air and the n th layer -in contact with the outdoor air.For the interior walls both their surfaces are in contact with the indoor air.The energy flows between the bordering interior/exterior wall layers are evaluated according to equation (1).The heat exchange between the exterior walls and the indoor/outdoor air are estimated with: Heat flows through the windows and doors.The windows and doors thermal properties are usually specified by their producers as thermal resistance.Considering they have relatively low surface, width and worse thermal parameters compared to insulation materials, the energy flow through them could be simplified with: where win R is the thermal resistance of the window (or the door),

Estimation of the temperature variation of the building elements
Considering the building elements are divided into multiple bordering layers, the total energy flow entering a certain layer, is estimated as an algebraic sum of the energy flows coming from the two bordering layers.The total energy flow, entering the i th layer, is: where the heat flows The temperatures change of the i th layer for a time interval t is then estimated with [7,8]: where is the specific heat capacity of the i th layer of the building element, 1 1

K kg J
; -the mass of the i th layer of the building element, kg ; i layer.
-the width of the i th layer, m ; i layer.
-the density of the i th layer, Considering all the heat flows between the indoor air and the surrounding surfaces are known, the new indoor temperature t t room T , t seconds later, could be estimated with [7,8]: where: is the temperature of the indoor air in the moment of time t, K ;

ALGORITHM OF THE MODEL
The operation algorithm of the model, describing the microclimate in a building with green roof, is presented on Figure 6.In block 1 is initialized the time variable.In block 2 is called an algorithm for estimation of the heat flows between the green roof layers, the indoor air and the environment according to equations (1), ( 2) and the model presented in [9].The algorithm then estimates the new roof layer temperatures in the moment of time t, according to equation (10).
In blocks 3, 4 and 5 are called the algorithms for estimation of the layer temperatures of the exterior walls, the floor, and the interior walls respectively.Using equations ( 1) to (10) they estimate the new temperatures of each layer in the moment of time t.In block 6 are estimated all the heat flows influencing the indoor air temperature, and then in block 7 is evaluated the new indoor air temperature t seconds later, according to equation ( 12).The blocks 2-8 are iterated for each moment of time until the desired simulation period ends ( MAX t t ).

CONCLUSIONS
In the present study, the major heat exchange processes, influencing the energy balance of buildings with green roofs have been described.Using the finite difference method in its one dimensional form, all the building elements (walls, roof, and floor) have been divided into multiple bordering layers.The conductive heat flow between each two bordering layers has been described, as well as the convection between the elements surface and the indoor/outdoor air.They are used to estimate the temperature curves of all the building elements and their layers.Using the internal surface temperatures of the building elements, the indoor air temperature in the time domain has been described.The used dependencies have been summarized in an algorithm, presented in a block-scheme.
The developed model of the microclimate inside buildings with green roofs allows the indoor air temperature curve inside such buildings to be simulated.It could be used for sizing the building insulation parameters as well as for sizing the green roof's soil substrate depth and vegetation type.
The model also can be used to estimate the efficiency of such buildings for different climate zones by using meteorological data arrays including the air temperature, air humidity, wind speed and cloudiness.
A software application for computer simulation of the microclimate of buildings with green roofs, implementing the presented algorithm, is currently under development and is the object of future investigations.
flow used in the process evapotranspiration from the vegetation surface,

Fig. 1 .
Fig. 1.Major energy flows in a building with green roof.In multi-storey buildings energy flow : air temperature, K .Heat flows through the floor.Then floor is divided into n bordering layers, with the 1 st one being in contact with the indoor air and the n th one -in contact with the underbuilding soil.The temperatures of these layers are

Fig. 2 . 3 . 4 and Figure 5 .
Fig. 2. Heat flows through the green roof.Fig. 3. Heat flows through the floor of the building.Heat flows through the walls.The exterior and the interior walls are also divided into n layers, presented on Figure 4 and Figure 5. Their temperatures are 1 .w T , 2 .w T … n w T .and 1 .i T , 2 .i T … n i

rA
-the surface of the ceiling, 2 m ; f A -the surface of the floor, 2 m ; w A -the surface of the exterior walls, 2 m ; i A -the surface of the interior walls, 2 m ; win A -the surface of the windows/doors, 2 m ; of the indoor air, 3 m .

Fig. 6 .
Fig. 6.Operation algorithm of the model, describing the microclimate of buildings with green roofs.