**1. Introduction**

The issue of heat transfer dynamics is closely related to the subject of the energy efficiency of buildings, which is important at the design and construction stages, as well as during the operation of buildings or their parts. This is visible in many legal regulations and policies aimed at improving the energy efficiency of buildings. This is due to the fulfillment of the provisions of Art. 7 of Directive 2002/91/EC [1] and Art. 20 of Directive 2010/31/EU on the energy performance of buildings [2], according to which European Union Member States must take measures to provide all participants in the construction process with a wide range of information on different methods and practices for improving the energy performance of buildings. Moreover, Art. 12 of Directive 2012/27/EU on energy efficiency [3] obliges EU Member States to take appropriate measures to promote and enable the efficient use of energy by consumers. The provisions of the above directives have been implemented in the Polish legal system through Art. 11 sec. 1 of the Act of 15 April 2011, amended on 20 May 2016 on energy efficiency, [4] and Art. 40 of the Act of 29 August 2014 on the energy performance of buildings [5]. According to Directive (EU) 2018/844 of the European Parliament and of the Council of 30 May 2018 [6], amending Directive 2010/31/EU on the energy performance of buildings and Directive 2012/27/EU on energy efficiency, clear and ambitious targets for the renovation of the existing building

stock have a great significance. Therefore, efforts to improve the energy performance of buildings would actively contribute to enhance the energy independence of the Union and would also have enormous potential to create jobs in the Union. In this context, Member States should take into account the need to clearly link their long-term renovation strategies to relevant initiatives to support skills development and training in the construction and energy efficiency sectors. These provisions oblige the minister responsible for construction, spatial planning, and development and housing to conduct information, education, and training activities regarding available energy efficiency improvement measures, as well as to conduct a campaign of information to improve the energy performance of buildings. The purposefulness and methodology of determining the energy performance of a building result from the regulations [7,8]. Activities in the field of improving the energy efficiency of buildings, which are the subject of many scientific publications, such as [9], are part of shaping the climate and energy policy, ensuring, inter alia, the reduction of greenhouse gas emissions and constitute one of the most important challenges resulting from membership in the European Union. The Union is committed to efforts to develop a sustainable, competitive, safe and low-carbon energy system while maintaining the security of heat and energy supplies. In the context of infrastructure responsible for ensuring the security of heat supply, an important issue is its assessment in the aspect of supply security, taking into account economic and environmental conditions, presented in the paper [10]. The results of energy saving calculations may be of interest for the investors, engineers, and policy makers who intend to minimize the difference between the planned and real energy savings analyzed in the paper [11]. Aspects of energy savings and energy supply management in buildings have been analyzed in many publications, e.g., papers [12,13], with elements of heat supply safety simulation presented in the paper [14]. These issues play an important role in social, technical, and political terms. These aspects are related also with heat losses in the buildings and district heating systems, which was also underlined in the work [15]. The analysis of sensitivity of energy distribution for residential buildings is presented in the paper [16]. The energy reduction effects of the thermal labyrinth system were analyzed in the paper [17].

In changing climatic conditions, phenomena occurring in buildings are influenced by a number of parameters, such as: air temperature and humidity, wind direction and speed, cloudiness, azimuth and height of the sun, solar radiation, or even the management of the surroundings. The analysis of external climate parameters, such as temperature, air humidity and wind conditions, for the needs of outdoor thermal comfort have been included in paper [18].

The heat balance in buildings results from the analysis of heat losses and gains. It is made for a building, assuming appropriate parameters in order to select appropriate devices to meet the requirements of thermal comfort. Example of analysis of indoor air parameters contain the work [19,20]. Thermal comfort optimization in microgrids equipped with renewable energy sources and energy storage units was analyzed in the paper [21].

Issues of heat transfer in heat exchangers was emphasized in the work [22]. There is a correlation between some parameters of the isolation of buildings and the wind free stream velocity and wind-to-surface angle. In the work [23], it has been shown that the convective heat transfer coefficient value strongly depends on the wind velocity. The influence of the thermal insulation thicknesses of external walls on heating cost from the ecological and economic assessment is analyzed in [24].

The way to achieve high energy efficiency of buildings along with the required quality of the internal environment are advanced technologies in both control [25] and construction where, for example, phase change materials can be used [26]. Examples of building energy management analyzes using increased thermal capacitance and thermal storage management are shown in work [27]. Tools for increasing energy efficiency in the examples [28] and in the integration of HVAC systems are presented in the works [29,30].

In order to analyze the modelling and simulation of heat transfer in buildings, the theory and application of this type of tool are collected and characterized by Clarke [31]. The methods to analyzing building energy and control systems are often used, such as using the equation-based, object-oriented Modelica in the paper [32]. Methods based on the coupling of three different types of simulation models, namely spectral optical model, computational fluid dynamics model, and building energy simulation, are presented in [33]. Physical phenomena, notably optical, thermodynamic, and fluid dynamic processes, have been analyzed for commercial buildings with double-skin façades. The modelling of heat transfer taking advantage of heat energy accumulation in building walls is the goal of the work [34]. The paper is focused on the future optimization of a control strategy. The issue of simulating heat transfer through point thermal bridges is the subject of the paper [35].

On the basis of the literature analysis of the subject of this work, it can be stated that there is a lack of ordering and development of methods for analyzing heat transfer dynamics using changeable external conditions. Existing works in this field mainly deal with the issues of energy control and control of the HVAC system's parameters, while there are no studies taking into account the changeability of atmospheric conditions and their impact on the dynamics of heat transfer. We realize that this is important in a changing climate, where the heating season and the summer season stand out. The issue presented in the article may be helpful in the analysis of the thermal inertia of a building in order to optimize the operation of HVAC systems.

The literature review presented in the article confirms that the approach to dynamics of heat transfer in buildings, proposed in the manuscript, is innovative with regard to the analysis of the impact of the variability of external conditions on energy efficiency and it was not previously applied in this way. This issue was the subject of this work.

#### **2. Methodology**

The room can be treated as a closed object, limited by building partitions, located in the space, treated as the surroundings. Due to the lack of a thermodynamic equilibrium between them, there are energy interactions between the room and the surroundings. To explain it simply, there is a heat transfer, considered as heat losses or gains.

Heat gains can originate from:


Heat losses can be caused:


The thermal balance includes heat fluxes presented schematically in Figure 1.

Heat gains from people Ql, from devices Qu, and from lighting Qe can be taken together as internal heat gains Qw.

$$\mathbf{Q\_w} = \mathbf{Q\_l} + \mathbf{Q\_u} + \mathbf{Q\_e} \tag{1}$$

Due to variable environmental parameters, both heat gains and losses change over time. For a room to be kept at a constant temperature, the heat gains and heat losses must be balanced, and this balance is ensured through appropriate heating systems. At the same time, the above means that the demand for heat for heating facilities is not constant but depends on external weather conditions. The weather conditions that determine heat exchange are the temperature of the outside air and solar radiation.

**Figure 1.** Balance sheet diagram.

Due to the thermal inertia, changes in partition temperature are slower than changes in air temperature. Therefore, the amplitude of changes in heating power Qg is not proportional to the amplitude of outside air temperature change Te.

The equation for the heat balance of the room is as follows:

$$\mathbf{Q\_{g}} + \mathbf{Q\_{os}} + \mathbf{Q\_{w}} = \mathbf{Q\_{0}} + \mathbf{Q\_{si}} + \mathbf{Q\_{V}} + \mathbf{Q\_{SW}}\tag{2}$$

The individual components of Equation (1) have been explained below. Heat transferred from indoor air to the wall:

$$\mathbf{Q}\_{\rm si} = \alpha\_{\rm i} \cdot \mathbf{A}\_{\rm si} \cdot (\mathbf{T}\_{\rm i} - \mathbf{T}\_{\rm wi}) \tag{3}$$

Heat transferred from indoor air to interior walls:

$$\mathbf{Q\_{sw}} = \mathbf{a\_i} \cdot \mathbf{A\_{sw}} \cdot (\mathbf{T\_i} - \mathbf{T\_{sw}}) \tag{4}$$

The heat of solar radiation penetrating the window [8]:

$$\mathbf{Q\_{\alpha\circ}} = \mathbf{z} \cdot \mathbf{w\_{\circ}} \cdot \mathbf{l\_{\circ}} \mathbf{A\_{\circ}} \tag{5}$$

Heat from internal sources:

$$\mathbf{Q\_w} = \mathbf{q\_A} \cdot \mathbf{A\_f} \tag{6}$$

Heat to prepare the ventilation air:

$$\mathbf{Q}\_{\mathbf{V}} = \mathbf{V} \cdot \mathbf{p} \cdot \mathbf{c}\_{\mathbf{P}^\*} (\mathbf{T}\_{\mathbf{i}} - \mathbf{T}\_{\mathbf{e}}) \tag{7}$$

or

$$\mathbf{Q\_w} = \mathbf{V\_{Ve}} \cdot \mathbf{A\_f} \tag{8}$$

Heat loss through the window:

$$\mathbf{Q\_o} = \mathbf{U\_o} \cdot \mathbf{A\_o} \cdot \left(\mathbf{T\_i} - \mathbf{T\_e}\right) \tag{9}$$

The temperature of the internal surface of an outer wall is a result of the influence of the inner environment and the heat conduction in this wall.

The heat conduction in the wall is caused by the temperature distribution, which is unsteady due to changing external climatic conditions.

The temperature of the outer surface of the outer wall is influenced by the transfer of heat to the outside air, the absorption of solar radiation, and the emission of radiation to the sky.

Heat transferred from the outer wall surface to the outside air:

$$\mathbf{Q\_{sc}} = \alpha\_{\mathbf{c}} \cdot \mathbf{A\_{sc}} \cdot \left(\mathbf{T\_{we}} - \mathbf{T\_{e}}\right) \tag{10}$$

Radiant heat losses from the outer wall to the skyfall:

$$\mathbf{Q\_{sen}} = \boldsymbol{\sigma} \cdot \boldsymbol{\varepsilon\_{sn}} \cdot \mathbf{A\_{se}} \cdot \left(\mathbf{T\_n^4} - \mathbf{T\_{we}^4}\right) \tag{11}$$

Solar radiation of heat absorbed by the outer surface of the outer wall [8]:

$$\mathbf{Q\_{re}} = \varepsilon\_{\text{s}} \mathbf{A\_{sa}} \mathbf{I} \tag{12}$$

Signs:

αi—coefficient of heat transfer from the wall surface to the internal air,

αe—heat transfer coefficient from the external wall surface to the outside air,

Asi—the surface of the outer wall inside the room,

Asw— the surface of the internal walls of the room,

Ase—the outer surface of the outer wall,

Af—reference surface (floors),

Ao—window area,

V—ventilation air stream,

qA—indicator of internal heat sources,

VVe—ventilation rate,

z—shading coefficient,

ws—radiation transmittance coefficient,

<sup>σ</sup><sup>s</sup> <sup>=</sup> 5.67 <sup>×</sup> <sup>10</sup>−<sup>8</sup> [W/m2K4]—radiation constant,

εsn—radiation absorption coefficient of the outer surface of the outer wall,

εs—radiation emission coefficient of the outer surface of the outer wall,

I—solar radiation intensity,

Ti—internal air temperature,

Te—outside air temperature,

Twi—temperature of the inner surface of the outer wall,

Tsw—surface temperature of internal walls,

Twe—external wall surface temperature,

Tn—skyfall temperature.

Equation (2) shows the required heat output to heat the room during the heating season.

$$\mathbf{Q\_g} = (\mathbf{Q\_o} + \mathbf{Q\_{si}} + \mathbf{Q\_V} + \mathbf{Q\_{sw}}) - (\mathbf{Q\_{o6}} + \mathbf{Q\_w}) \tag{13}$$

*Energies* **2020**, *13*, 6469

The use of a heating device with the required thermal power and automatic temperature control results in maintaining the room temperature in accordance with the regulations [7].

In most cases, this temperature is taken as a constant value (Ti = const).

Outside the heating season, the heating devices are turned off (Qg = 0), and the internal air temperature is determined based on the thermal balance.

After using the thermal balance equations and transformations, the internal air temperature is described by the equation:

$$\mathbf{T\_i} = \frac{\mathbf{z} \cdot \mathbf{w\_s} \cdot \mathbf{I} \cdot \mathbf{A\_o} + \mathbf{q\_A} \cdot \mathbf{A\_f} + \left(\mathbf{U\_o} \cdot \mathbf{A\_o} + \mathbf{V} \cdot \mathbf{p} \cdot \mathbf{c\_p}\right) \mathbf{T\_e} + \mathbf{a\_i} \cdot \mathbf{A\_{si}} \ T\_{\rm wi} + \mathbf{a\_i} \cdot \mathbf{A\_{sw}} \ T\_{\rm sw}}{\left(\mathbf{U\_o} \cdot \mathbf{A\_o} + \mathbf{V} \cdot \mathbf{p} \cdot \mathbf{c\_p}\right) + \mathbf{a\_i} \cdot \mathbf{A\_{si}} + \mathbf{a\_i} \cdot \mathbf{A\_{sw}}} \tag{14}$$

Due to the variable temperature of the external air, outside the heating season also the temperature of the inside air is variable. The temperature of the inner surface of the walls is also variable.

This temperature is the result of heat transfer through the external wall between the room and the outside air.

With the simplifying assumption that the outside air temperature changes periodically according to the cosine function, the solution to the problem of heat conduction in a semi-infinite medium is described by the equations, according to [36,37]:

It was assumed that the air temperature changes according to the equation:

$$\mathbf{T\_f} = \mathbf{T\_{fo}} \cdot \cos(\omega \mathbf{\cdot} \mathbf{t}) \tag{15}$$

in which

$$
\omega = \frac{2 \cdot \pi}{\text{t}\_o} \tag{16}
$$

$$\mathbf{v} = \frac{\boldsymbol{w}}{2 \cdot \pi} \tag{17}$$

where:

Tfo—amplitude of air temperature changes,

t—time,

ω—period of temperature changes,

to—change period time,

ν—frequency of changes,

The temperature T at depth x, below the surface, is described by the following dependencies:

$$\mathbf{T} = \mathbf{C}\_2 \cdot \mathbf{T}\_{\text{fo}} \cdot \mathbf{e}^{-\sqrt{\frac{\mathbf{D}}{2} \cdot \mathbf{x}} \cdot \mathbf{x}} \cdot \cos(\omega \cdot \mathbf{t} + \sqrt{\frac{\mathbf{D}}{2 \cdot \mathbf{a}}} \cdot \mathbf{x} + \mathbf{C}\_2) \tag{18}$$

$$\mathcal{C}\_1 = \frac{1}{\sqrt{1 + 2 \cdot \frac{\lambda}{\alpha} \cdot \sqrt{\frac{\alpha \overline{\alpha}}{2 \cdot \mathbf{a}}} + 2 \cdot \left(\frac{\lambda}{\alpha} \cdot \sqrt{\frac{\alpha \overline{\alpha}}{2 \cdot \mathbf{a}}}\right)^2}}\tag{19}$$

$$\mathbf{C}\_2 = -\text{arctg}\frac{1}{1 + \frac{\mathbf{g}}{\lambda}\cdot\sqrt{\frac{2\mathbf{g}}{\omega}}}\tag{20}$$

where:

λ—thermal conductivity,

a—thermal diffusivity of the area,

α—coefficient of heat transfer from air to the surface.

The constant C1 determines the degree of air temperature reduction resulting from the transfer of heat, while the constant C2 means the delay in wave propagation due to the transfer of heat to the surface.
