**1. Introduction**

The massive use of insulating materials on opaque walls makes windows the weakest part of the building envelope in terms of heat loss [1,2]. The transparent components performance have been strongly improved in recent years, both from the glazing and frame points of view [3–6]. The researchers also examined the edge-of-the-glass region to create better joint solutions [7,8] but the efforts from researchers and manufacturers to lower the windows thermal transmittance are still needed [9].

Window frame thermal transmittance ( *Uf* values) can be obtained both in numerical simulations and in measurements performed in the calorimetric chamber. Both methods are described in standards ISO 10077-2 [10] and EN 12412-2 [11], respectively. The computational approach could be implemented by software such as Frame Simulator [12], Therm [13], etc.

The standard ISO 10077-2, gives some examples of geometries of window frames with all the necessary input data and solutions (calculated *Uf* values). The Standard obliges the users of the method to validate the applied calculation programs and to perform the calculations given in the examples. The users should achieve a level of discrepancy lower than 3% between the results of the calculations and the values given in the Standard. In the first part of the present work, the software Ansys Fluent [14] was validated with the ISO 10077-2 approach to calculate *Uf* values. Fluent was chosen due to the capability of the code to implement optimization routines that allow a relatively quick calculation of a high number of frame design cases under investigation.

The robustness of the software results was further confirmed by measurements performed in standardized conditions in the calorimetric chamber, assessing the di fferences between the simulation and measurement results of an actual window frame thermal transmittance. After these steps, a sensitivity analysis [15] of the di fferent parameters playing a role in the frame thermal transmittance was conducted. The investigation covered the boundary conditions (surface heat transfer coe fficients on cold and warm sides), as well as material thermal conductivities and surface emissivity. The software allows automatic variation of the above-mentioned parameters within a user-defined range, in order to define the most influential ones on the window global performance. As far as the authors' knowledge, the sensitivity analysis approach has not been reported in previous studies and it could constitute a novel tool in the window frames design phase, covering a wide range of cases through simulations with a reasonable computational time.

Both calculations and experiments carried out in this research are valid only for steady-state conditions. The numerical models and the hot box measurements do not take into account the transient effect of boundary conditions (heat transfer coe fficients, boundary temperatures, heat accumulation, overheating due to direct exposure to sunlight, etc.), which could play a significant role when the holistic performance of a window is analyzed.

#### **2. Analyzed Frame Models and Simulation Input Data**

The validation of the window frame thermal transmittance calculations took into account four chosen example models given in Standard ISO 10077-2 [10]: aluminum frame (example I1 from the Standard), PVC frame (example I3), a wooden frame (example I4) and a roof window wooden frame (example I5). The views of the models are given in standard [10]. An additional frame model was analyzed, which was a window frame made of PVC hosting a triple glazing [16] (Figure 1), which was also tested experimentally in a hot box apparatus.

All five models contain air gaps in the frames. The gaps were replaced by a solid with an equivalent thermal conductivity calculated according to Standard ISO 10077-2 [10] indications. Both convective and radiative heat transfer in the gaps were taken into account. The Standard [10] introduces the method of calculation including conductivity of air gaps when they are rectangular as well as when they are non-rectangular. In the latter case, the equivalent width, thickness and area of non-rectangular air gaps are calculated, and they are treated as rectangular, following the standard. The knowledge of surface temperatures of the gaps is required which implies an iterative process. In the first step, the gap surface temperatures were assumed and after heat exchange calculations they were corrected from the temperature distribution and the calculations were repeated again. After calculations of the equivalent values of a gap conductivity, the simulations treat all materials as solids. The material thermal conductivities are shown in Table 1. Surface emissivity was assumed as equal to 0.90. When calculating window frame thermal transmittance, glazing has to be replaced with an insulation panel of the same thickness as glazing and a visible height of 190 mm.

**Figure 1.** Model of real PVC frame with the following materials: 1, PVC; 2, steel; 3, ethylene-propylene diene monomer (EPDM); 4, butyl rubber; 5, insulation panel. Colored lines define the boundary conditions.


**Table 1.** Material thermal conductivities.

Both the end of the insulation panel and the frame part adjacent to the wall were treated as adiabatic. On the internal (warm) side of the frame, the temperature of air θ*i* was fixed at 20 ◦C, while on the external (cold) frame side, the air temperature θ*e* was considered equal to 0 ◦C, as suggested by the Standard [10]. As the Standard [10] assumes different values of heat transfer coefficients on boundaries, the choices taken in the domain are shown in Figure 1. The yellow boundary line shows adiabatic conditions on the end of the frame (left hand side). Blue lines show external boundary conditions with heat transfer coefficient equal to 25 W/(m2K), while red and orange lines represent internal boundary conditions with the values 7.7 W/(m2K) and 5 W/(m2K), respectively.

Ansys Fluent [14] program uses the finite volume method. The calculations were performed in steady-state conditions, with second order up-wind discretization scheme used for the energy equation. Thetemperaturedistributionsforfivegiven geometriesarepresentedinFigures2–6.

**Figure 2.** Temperature distribution on aluminum frame, example I1 from ISO 10077-2.

**Figure 3.** Temperature distribution on PVC frame, example I3 from ISO 10077-2.

**Figure 4.** Temperature distribution on wooden frame, example I4 from ISO 10077-2.

**Figure 5.** Temperature distribution on roof window frame, example I5 from ISO 10077-2.

**Figure 6.** Temperature distribution on real PVC frame.

As can be seen in Figures 2–6, the temperature distribution in the part of glazing is one-dimensional (parallel isotherms), while in the frame the heat flux changes its direction. The warmer the frame on its internal side, the lower the value of frame heat losses. It can be seen that the aluminum one (example I1) is the coldest on its internal side, indicating the worst performance in terms of thermal transmittance (see the results in Table 2).


**Table 2.** Comparison of the simulation results and the results from the standard ISO 10077-2.
