**3. Simulation Results**

The window frame thermal transmittance was calculated from the following equation [10]:

$$\mathbf{U}\_{\rm f} = \frac{\mathbf{L}\_{\rm f}^{2\rm D} - \mathbf{U}\_{\rm P} \mathbf{b}\_{\rm P}}{\mathbf{b}\_{\rm f}},\tag{1}$$

where:

$$\mathcal{L}\_{\text{f}}^{\text{2D}} = \frac{\Phi}{\Theta\_{\text{i}} - \Theta\_{\text{e}}}. \tag{2}$$

The results of the numerical calculations of five models are given in Table 2.

Results show that each simulation result differs less than 3% from the values given in the Standard [10]. Thus, it can be stated that the software Ansys Fluent [14] could be applied as a tool for window frame thermal transmittance calculations. The lower row of Table 2 is supplemented with the results of the simulations of the fifth model, whichh is a real PVC window frame hosting a triple glazing.

#### **4. Window Frames Thermal Transmittance Measurements**

The second stage of the work was focused on the measurements of window frame thermal transmittance and comparison of the experimental results with the simulations for validation.

The frame measurements should meet the demands of the Standard EN 12412-2 [11] and should be performed in a calorimetric chamber (Figure 7). This device consists of two spaces separated with a surround panel, which consists of a cold chamber and a warm chamber. The specimen to be tested was mounted into the surround panel, which was made of thick insulation material with known thermal conductivity. In the case of window frame thermal transmittance measurements, the glazing was substituted by an insulation panel with (known) low thermal conductivity, and the same thickness as the glazing.

**Figure 7.** The view of the Laboratory of Thermal Engineering calorimetric chamber situated in Cracow University of Technology, Poland.

In the cold chamber, the air was kept in steady-state thermal conditions (0 ◦C), through a control and data acquisition system. The same approach was used in the warm chamber, where the air temperature set point was fixed to 20 ◦C. The test specimen cold side faced a baffle. A set of ventilators was installed on the top of the baffle to push the air upwards in controlled forced convection conditions on the space between the baffle and the specimen. The heart of the measurement stand is the hot box system. It is a smaller guarded chamber embracing the specimen on its warm side. In the lower hot box part, ventilators were installed in order to move the air downwards along the tested specimen, to create steady-state natural convection conditions. The hot box was equipped with internal radiators. The idea of the measurement is that the radiators provide the amount of heat demanded to ensure the temperature difference between the internal and the external hot box walls is 0 K, i.e., no heat is transferred through the hot box walls. That means that all heat power from the radiators is supplied to the specimen, to the surround panel and to the flanking losses between them [16,17].

After more than three hours of steady-state conditions in the whole apparatus, the temperatures of air, surface temperatures of baffle, reveal, insulation filling and surround surfaces were measured on both cold and warm sides, along with the heat power supplied by the radiators. Then, the calculations given by the Standard EN 12412-2 [11] were performed to obtain the value of the window frame thermal transmittance.

The view of the tested window frame filled with the insulation panel and mounted into the surrounding panel in the calorimetric chamber of the Laboratory of Thermal Engineering at Cracow University of Technology is shown in Figure 8.

The uncertainty of the frame section was calculated using the error propagation rule from the equation:

$$
\Delta \mathbf{U}\_{\mathbf{f}} = \sqrt{\left(\frac{\Delta \mathbf{U}\_{\mathrm{m},\mathrm{t}} \partial \mathbf{U}\_{\mathrm{f}}}{\partial \mathbf{U}\_{\mathrm{m},\mathrm{t}}}\right)^{2} + \left(\frac{\Delta \mathbf{A}\_{\mathrm{t}} \partial \mathbf{U}\_{\mathrm{f}}}{\partial \mathbf{A}\_{\mathrm{t}}}\right)^{2} + \dots + \left(\frac{\Delta \mathbf{A}\_{\mathrm{n}} \partial \mathbf{U}\_{\mathrm{f}}}{\partial \boldsymbol{\Theta}\_{\mathrm{n}}}\right)^{2}}\tag{3}
$$

where the uncertainties of all the quantities of Equation (3) had to be calculated according to the error propagation rule. The measured frame thermal transmittance results equaled to 1.063 W/(m2K), with an uncertainty of about 8%.

The discrepancy between the measured and simulated Ansys Fluent values of the frames thermal transmittance was about 3%. The good agreemen<sup>t</sup> of measurements with the simulations enforces the reliability of the finite volume analysis for the evaluation of window frames thermal performance.

**Figure 8.** View of the tested window frame in the surround panel on the warm side.

The measurement results as well as the calculation results based on the Standard EN 12412-2 [11] are given in Table 3.


**Table 3.** Measurement and calculation results of real frame thermal transmittance.

#### **5. Sensitivity Analysis of the Influence of Input Data on the Value of Window Frames Thermal Transmittance**

In this section, after the simulation code validation, analysis of the influence of input data on the results of window frames thermal transmittance value was performed. The geometry presented in Figure 2 (frame D3 from the Standard [10]) was chosen for the analysis. This is a PVC frame with steel reinforcements, EPDM seals and 11 air gaps within its geometry.

The analysis was performed using Ansys DesignXplorer [14]. This tool enabled us to parameterize the desired input data in Fluent in the assumed range of variations in order to achieve the response surface and to analyze the optimized outputs. The response surface can be used to predict the performance of the data without needing to run the actual simulation. This model was applied for the sensitivity analysis. In the present study, the sensitivity analysis method called "screening" was applied. By means of this approach, it was determined how the change of the input values of a certain parameter influenced the output. The sensitivity is a quotient where the nominator is the di fference between the output values Y(X + D) and Y(X), where D is a percentage variation of the input X and it is the term on the denominator. The higher the sensitivity, the higher the impact of the analyzed variable.

In order to prepare the optimization process, it was necessary to create variable input parameters in a given range instead of constant values. The parametric calculations took into account thermal conductivities of the solid materials as well as the surface heat transfer coe fficients both on cold and warm frame sides. Moreover, the thermal conductivities of air gaps in the frame were treated as changeable parameters as well. The Standard [10] introduces the method of calculating the equivalent thermal conductivities of air gaps within the frame geometries, considering one-dimensional convective and radiative heat transfer through the gas gaps. As a result, the gap thermal equivalent conductivity treats the gas as a solid material and these values constituted the input data in Ansys Fluent [14] calculations for the frame thermal transmittance. In order to perform the sensitivity calculations, it was necessary to create user defined functions (UDF) in C++ to calculate equivalent conductivities of air gaps in Fluent, based on pre-run established gap temperatures and emissivity, which were imported to the UDF by a get-parameter function. These values were then parameterized in Design Explorer. The output parameters were the heat fluxes on the boundary surfaces and a frame U-value that was set as the optimized parameter. The ranges of the variable parameters used for the sensitivity analysis are listed in Table 4.


**Table 4.** Ranges of parameters variation in the sensitivity analysis.

The preliminary sensitivity calculations indicated that thermal conductivities of steel and EPDM have no influence on the results, mainly because of the low amount of these materials in the whole geometry. For this reason, among all the solids, only PVC conductivity was varied in the sensitivity calculation process.

The sensitivities of four output parameters described in Table 4, as calculated in Ansys Fluent [14], are given in Figure 9. A higher parameter sensitivity value means that it has a bigger impact on the output results.

**Figure 9.** Sensitivities of parameters P1–P4, described in Table 4.

The results indicated that the values of heat transfer coe fficients on cold and warm frame sides have dominant impact on the results of frame thermal conductivities together with the PVC thermal conductivity, while the air gaps emissivity alone has a negligible e ffect on the whole performance.

The surface heat transfer coe fficients depend mainly on the air velocity and, at a lower level, on the inner and outer surfaces emissivity. Therefore, their prominent modifications are linked to the convection heat transfer, an e ffect not tunable by the window frame construction.

The rationing is not to include parameters not directly connected to the building component on the evaluation of its performance. Thus, in the last part of the sensitivity analysis, they were kept constant.

For the same reason, the temperature variations on both cold and warm sides were not investigated, despite the fact that inner and outer temperatures change significantly depending on building location and weather conditions. These temperatures vary quickly in time, especially in the hot season.

On the other hand, the e ffect of the air gaps equivalent thermal conductivity variation had a substantial impact on the window frame thermal transmittance. It was varied between the worst condition (air in the gap and surface emissivity equal to 0.90) and the most e fficient one (which represents the situation of air gaps filled with a super insulation material). Intermediate values may represent filling air gaps with di fferent insulation materials like wool, aerogel or di fferent gases like argon, and giving the surfaces di fferent emissivity [18].

The results of this part of the optimization process indicated that the values of equivalent thermal conductivities of gaps (P5) have the strongest influence on thermal transmittance of window frames with respect to the PVC thermal conductivity (Figure 10).

**Figure 10.** Sensitivities of PVC thermal conductivity (P3) and multiplier of air gap equivalent conductivity (P5).

Summing up the sensitivity analysis results, both window frame construction and external environmental boundaries considerably influence its thermal performance, with the gaps playing the most important role among the parameters where a designer could intervene.
