*3.1. Steady Boundary Conditions*

If boundary conditions do not vary with time, the steady state does not depend on thermal mass and specific heat of components. Hence, a benchmark test case based on constant boundary conditions was the easiest way to start with validation. The study comprised three different cases of WFG. When the system was circulating, the design flow rate was 2 L/min m2, and the inlet temperature was set at a constant value θ*IN*. When the flow rate was stopped, the outlet temperature

of the water chamber θ*OUT* was called the stagnation temperature. The outputs of these test cases were the thermal power transported by the flow rate and the water heat gain. Figure 4 shows the dynamic *U* and *Uw* values defined in Equations (22) and (23). The red line represents the design flow rate (2 L/min m2 = 0.029 Kg/s m2). If the flow rate were above the design value, it would not affect the transmittances.

**Figure 4.** Thermal transmittances of three WFG case studies depending on the mass flow rate, *m˙* . (**a**) Thermal transmittance of the WFG modular units (*U*). (**b**) Thermal transmittance between the water chamber and indoors (*Uw*).

Equation (24) shows the absorptance, *Av*, that depends on the energy absorbed by the glass panes and by the water:

$$A\_{\upsilon} = A\_1 \Bigl(\frac{lL\_{\varepsilon}}{h\_{\varepsilon}}\Bigr) + A\_2 \Bigl(\frac{1}{h\_{\mathcal{J}}} + \frac{1}{h\_{\varepsilon}}\Bigr) lI\_{\varepsilon} + A\_3 \Bigl(\frac{lL\_{\varepsilon}}{h\_i}\Bigr) + A\_{\upsilon}.\tag{24}$$

Equation (25) shows that the *g*-factor for WFG also depends on the mass flow rate:

$$\mathcal{g} = \left(\frac{\mathcal{U}\_i}{\dot{m}c + \mathcal{U}\_\varepsilon + \mathcal{U}\_i}\right) A\_v + A\_i + T\_\prime \tag{25}$$

where *Ai* is the secondary internal heat transfer factor. The direct solar energy transmittance (*T*) is related to the visible and NIR wavelengths. *Ai* is negligible when the water chamber is facing indoors, but if it is facing outdoors and with high values of *hw*, *Ai* can be calculated With Equation (26).

$$A\_i = A\_3 \left( 1 - \frac{lI\_i}{h\_i} \right) \tag{26}$$

Table 2 shows the thermal transmittance of WFG at the design flow rate. The interior or exterior convective heat transfer mechanism can be modeled by constant values or by more elaborate models given by the norm ISO 15099:2003. By default, constant values of *hi* = 8 and *he* = 23 were used. However, by selecting the ISO model, *hi* and *he* could be determined precisely using the European Standard [40]. A typical value for the heat transfer coefficient of the water chamber, *hw*, was 50 W/m2K. The heat transfer coefficient of an argon chamber was *hc* = 1.16 W/m2K. The air chamber emissivity was very low, so the heat transfer coefficient due to radiation, *hr*, could be neglected. Therefore, the heat transfer coefficient of the argon chamber, *hg* = *hc*+ *hr*, was 1.16 W/m2K. The specific heat capacity of the fluid was *c* = 3600 J/kg K. *A1* is the absorptance of the exterior glass pane, *A2*, is the absorptance of the middle glass pane, and *A3* is the absorptance of the interior one. *Aw* is the absorptance of the water chamber. The highest *Ai* (0.01) is shown in Case 1, when the water chamber was placed outdoors.


**Table 2.** Absorptances and thermal transmittances of WFG (*m˙* = 2 L/min m2).

Absorptances of glass panes (*Aj*), Absorptances of water layer (*Aw*), Total absorptance of water flow glazing (*Av*), Interior thermal transmittance (*Ui*), Exterior thermal transmittance (*Ue*), Thermal transmittance of triple glazing (*U*), Thermal transmittance from water chamber to interior (*Uw*).

Equation (27) results from solving the Equations (9)–(28):

$$
\theta\_{OUT} = \frac{i\_{\rm O}A\_{\rm v} + \ell I\_{\rm i}\theta\_{\rm i} + \ell I\_{\rm c}\theta\_{\rm c} + \dot{m}c\theta\_{\rm IN}}{\dot{m}c + \ell I\_{\rm c} + \ell I\_{\rm i}}.\tag{27}
$$

Equation (28) shows the analytical expression of water heat gain (*P*).

$$P = \dot{m}c(\theta\_{\text{OlIT}} - \theta\_{\text{IN}})\_\prime \tag{28}$$

where θ*OUT* and θ*IN* are the temperatures of water leaving and entering the glazing, respectively; *m˙* is the mass flow rate, and *c* is the specific heat of the water. By combining Equations (27) and (28), Equation (29) shows the power as a function of absorptance (*Av*) and thermal transmittances (*Ui, Ue*).

$$P = \frac{\dot{m}c}{\dot{m}c + \mathcal{U}\_c + \mathcal{U}\_i} (i\_0 A\_v + \mathcal{U}\_i (\theta\_i - \theta\_{\text{IN}}) + \mathcal{U}\_c (\theta\_\varepsilon - \theta\_{\text{IN}}))\_\prime \tag{29}$$

where *Av* comes from Equation (24). When the system reaches a steady state, the boundary conditions do not change with time. The assumption that solar radiation is perpendicular to the interfaces makes the absorptance of each layer non-dependent of the angle of incidence. When the mass flow rate is high enough (*mc ˙* >> *Ue* + *Ui*), the absorbed power (*P*) reaches its peak value (*Pmax*). Table 3 shows constant input values in winter and summer conditions. Indoor air temperature (θ*i*), outdoor air temperature (θ*e*), interior and exterior heat transfer coefficients (*hi, he*), and solar irradiance (*I*). Equations (29)–(32) were used to calculate *Ue* and *Ui*.


**Table 3.** Constant input parameters in winter and summer.

Once the glazing reaches the steady state in winter and in summer, thermal performances are determined. Using Equations (27) and (29), and considering the inputs in Table 3, Table 4 shows the following outputs in winter and summer, respectively. *P* is the water heat gain, *T* is the transmittance of the glazing, θ*OUT* is the outlet temperature when the flow rate is *m˙* = 2 L/min m2, and θ*<sup>s</sup>* is the stagnation temperature when *m˙* = 0.

**Table 4.** Simulation outputs. Steady state in winter and summer conditions.


If energy management in winter is based on energy harvesting, Case 2 showed the best performance. Its water heat gain in winter was ten times as high as in Case 1. On the other hand, if energy management in summer is based on energy rejection, Case 3 was the best choice. In summer, the water heat gain of Case 1 was 1.5 times as much as Case 2. When it came to cooling capacity, Case 3 showed the best performance. It had to dissipate around 131.9 W/m2, whereas Case 2 had to dissipate 418.7 W/m2. Case 3 showed excellent properties for energy rejection strategies in warm climates because it showed the least absorbed power in summer, whereas the transmittance (*T*) was not much higher than in other cases.

Considering the simulation results in a steady state, Case 2 showed the best performance for water heat absorption throughout the year. It was selected for the south elevation. Case 3 showed the best performance for energy rejection, and it was selected for east and west facades. The next subsection studies the simulation results of the selected cases in transient conditions.
