**3. Dynamics of Heat Transfer through an External Wall—Case Study**

The external wall of the room is subjected to variable outside air temperature. Moreover, the external surface is influenced by the variable intensity of solar radiation. Climatic conditions cause periodic temperature variability in daily and annual cycles. Climate changes also cause changes in the temperature of external surfaces such as external building envelopes. Surface temperature changes are transferred deep into the material according to the principles of heat transfer. With a sufficient wall thickness, it can be treated as a semi-infinite medium for which the heat conduction problem has been analytically solved.

For example data:

Tfo = 20 ◦C, <sup>a</sup> <sup>=</sup> 0.485 <sup>×</sup> <sup>10</sup>−<sup>6</sup> m2/s, to = 24 h, <sup>α</sup> <sup>=</sup> 12 W/(m2·K), <sup>λ</sup> <sup>=</sup> 0.82 W/(m2·K).

The tendencies of temperature changes in the semi-infinite medium is shown in Figure 2. A negative coordinate x indicates temperature changes in the adjacent fluid.

**Figure 2.** Temperature changes along the depth in a semi-infinite medium [38].

The graphically presented temperature changes in the medium and equations [36,37] allow for the following conclusions:


The variability of temperature over time at different depths is shown in Figure 3. The presented considerations apply to a semi-infinite medium. However, they may be the basis for the analysis of temperature distribution in media with a finite, sufficiently large thickness. An example of such a medium are the external walls of a building, which is influenced by external air of variable temperature. There is a daily and annual periodicity of changes. Daily changes do not significantly affect the temperature distribution inside the wall under the surface, especially on the inner wall surface (from the room side). Annual changes, on the other hand, affect heat losses and, consequently, seasonal heat demand for heating.

**Figure 3.** An example of temperature changes with time at depth x in a semi-infinite medium [38].

In relation to the semi-infinite medium model, the following differences should be taken into account:


The periodic variability of outdoor air temperature and solar radiation intensity can be described by the Fourier series of the form [39]:

$$\mathbf{y} = \frac{1}{2}\mathbf{a}\_0 + \sum\_{\mathbf{i}=1}^{\mathbf{k}} \mathbf{i} \cdot \mathbf{a}\_{\mathbf{i}} \cdot \cos(\omega \cdot \mathbf{t}) + \sum\_{\mathbf{i}=1}^{\mathbf{k}} \mathbf{i} \cdot \mathbf{b}\_{\mathbf{i}} \cdot \sin(\omega \cdot \mathbf{t}) \tag{21}$$

The coefficients of the Fourier series should be determined on the basis of real data obtained, e.g., from measurements.

The temperature of the outside air and the intensity of solar radiation are climatic parameters and they are measured at meteorological stations. The results of long-term measurements are available on the website of the Ministry of Infrastructure [40].

For the selected weather station, we can read the average monthly and hourly average outside temperature values determined from multi-year measurements.

Using these values, data can be approximated by a Fourier series. For the Rzeszow-Jasionka meteorological station, the annual changes in the values of average daily outside air temperatures are shown in Figure 4, where the line resulting from the approximation of the data by the Fourier series (Te (apr)) is also plotted.

**Figure 4.** Annual variability of outside air temperature for Rzeszow city.

The parameters of the Fourier series for approximation calculations of the outside air temperature are presented in Table 1.



As can be seen from the diagram, there is a high agreement of the measurement data with the results obtained from the approximation equation. Significant differences occur in the summer, outside the heating season.

A similar procedure can be applied to record the variability of solar radiation, but due to the consideration of the external wall, only the radiation to the vertical plane will be important.

The Fourier series parameters for the approximation calculations of the radiation intensity per vertical surface (Wh/m2d) are summarized in Table 2.

Exemplary results for the Rzeszow-Jasionka actinometrical station are shown in Figure 5.

The comparison of the measurement results and those calculated from the Fourier series equation for the direction of the southern and northern radiation is presented in Figure 5.

Marking N in Figure 5 depicts the radiation changes from the North. Labeling S denotes imaging of changes in radiation from the southern side. Marking N-a represents the results of approximation of radiation changes from the North using the Fourier series. Then, by analogy, S-a refers to the results of the approximation of radiation changes from the south side.


**Table 2.** Fourier series parameters for approximation calculations of the radiation intensity.

Remarks on the compliance of the approximation with the data are the same as in the case of the approximation of the outside air temperature.

Increasing the accuracy of the approximation is possible by increasing the amount of Fourier series components. For the assumed purpose of the analysis of the temperature distribution in the external wall, the assumed accuracy is sufficient.

#### **4. Heat Losses through the External Wall—Case Study**

The external wall separates the outside environment and the interior of the room. Such a wall is subjected to an externally variable air temperature as well as solar radiation. Climatic conditions cause periodic temperature variability as well as the variability of radiation intensity, and these changes occur in daily and annual periods. The resulting wall temperature distribution makes it difficult to provide a strict analytical solution, especially in the case of a multi-layer partition.

Simplified methods can be used to solve special cases, e.g., the finite difference method (MRS).

The analysis of heat loss through the external partition is made below, with the following assumptions.

On the side of the inner wall (in the room), the air temperature remains constant. This is the case during the heating season with automatic temperature control (thermostatic valves).

The heat transfer from the inside air to the surface takes place by transfer, with a transfer factor taking into account the radiation.

On the outer side of the outer wall, heat is transferred to the outside air by taking over.

On the outer side of the outer wall, heat transfer also takes place through radiation to the outer space (the sky).

The outside air temperature changes throughout the year according to the climatic conditions. Solar radiation falls on the outer surface of the wall with periodic, annual variability.

The values of external temperature and radiation intensity were taken as daily averages.

The image of the partition in question, with significant values marked, is shown in Figure 6.

**Figure 6.** Cross-section through the outer wall.

The variable temperature distribution in the wall was determined by the finite difference method based on heat transfer equations, taking into account the thermal balance of the room.

Due to the variable temperature of the outside air, the temperature of the inside surface of the outside walls is also variable.

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

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

When applying the finite difference method (FDM) to solve the transient heat conduction in the outer wall, formulas based on the calculation scheme are obtained (Figure 7).

**Figure 7.** Calculation scheme.

The formulas resulting from the discretization of areas were used for the calculations. Inner surface temperature of outer wall:

$$\mathbf{T\_{wi}} = \frac{\mathbf{T\_i} + \frac{\lambda\_i}{\alpha\_i \Delta \mathbf{x\_l}} \cdot \mathbf{T\_1}}{\mathbf{1} + \frac{\lambda\_i}{\alpha\_i \Delta \mathbf{x\_l}}} \tag{22}$$

Surface temperature at the boundary of the layers:

$$\mathbf{T\_w} = \frac{\frac{\lambda\_1}{\Delta \mathbf{x\_1}} \mathbf{\cdot} \mathbf{T\_{j-1}} + \frac{\lambda\_2}{\Delta \mathbf{x\_2}} \mathbf{\cdot} \mathbf{T\_{j+1}}}{\frac{\lambda\_1}{\Delta \mathbf{x\_1}} + \frac{\lambda\_2}{\Delta \mathbf{x\_2}}} \tag{23}$$

The temperature of the outer surface of the outer wall was determined from the balance sheet (Figure 8):

$$\mathbf{Q\_p} + \mathbf{Q\_{ne}} = \mathbf{Q\_{se}} + \mathbf{Q\_{sen}} \tag{24}$$

**Figure 8.** Balance diagram of the external surface.

Taking into account the Equations (10)–(12) and the heat conduction equation under the surface of the partition (24), it can be concluded that:

$$\mathbf{Q\_{P}} = \frac{\lambda\_{2}}{\Delta \mathbf{x\_{2}}} \cdot \mathbf{A\_{\\$6} \cdot (T\_{\mathbf{k}-1} - T\_{\text{We}})} \tag{25}$$

After substitution and transformations, the temperature of the outer surface of the outer wall is described by the relationship:

$$\mathbf{T\_{we}} = \frac{\varepsilon\_{\rm 8} \cdot \mathbf{I} + \frac{\lambda\_{\rm 2}}{\Delta \mathbf{x}\_{2}} \cdot \mathbf{T\_{k}} - 1 + \mathbf{a}\_{\rm 6} \cdot \mathbf{T\_{e}} - \varepsilon\_{\rm n} \cdot \left(\mathbf{T\_{we}^{4}} - \mathbf{T\_{n}^{4}}\right)}{\alpha\_{\rm 6} + \frac{\lambda\_{\rm 2}}{\Delta \mathbf{x}\_{2}}} \tag{26}$$

Due to the presence of Twe, the equation is solved by the iteration method. Signs:

Tn—skyfall temperature

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

εs—surface radiation absorption coefficient

εsn—surface radiation emission coefficient

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

λ2—thermal conductivity coefficient of the insulating layer

Δx2—step of discretization of the insulating layer

The above equations are completed with boundary conditions.

