*2.1. Simplified Model of Triple WFG*

Water flow glazing (WFG) allows the flow of water between two glass panes. Water captures the solar NIR and increases its temperature through the window. The flow of the water enables the homogenization of the building envelope temperature so that designers can apply energy-saving strategies, such as energy storage or solar energy rejection. Table 1 shows the list of symbols that have been used in equations.


**Table 1.** List of symbols.

Figure 1 shows the heat flux and the temperature distribution when the outdoor temperature (θ*e*) is higher than the indoor (θ*i*) through a triple WFG. The thermal transmittance, *U*, is the parameter that explains the heat transfer. The European Standard determines its value and a calculation method [58,59].

**Figure 1.** (**a**) Heat fluxes through a triple water flow glazing (WFG). Solar radiation and absorptance of layers, *A1, A2, Aw,* and *A3*. (**b**) Temperature distribution in a triple WFG at a specific height, with heat transfer coefficients, *hi*, *hw*, *hg*, and *he*.

Previous studies explained the thermal and spectral properties of WFG and its behavior [60,61]. This research used a simplified set of equations from those studies, along with data from commercial software, to assess the performance of the prototype defined in Section 2.2. Equations (1) to (10) show the heat fluxes through the different layers of the glazing. They consider the energy balance at each layer and the Newton's definition of heat flux.

$$q\_1 = h\_c(\theta\_c - \theta\_1),\tag{1}$$

$$q\_2 = q\_1 + A\_1 i\_{0\prime} \tag{2}$$

$$q\_2 = h\_{\mathcal{J}}(\theta\_1 - \theta\_2),\tag{3}$$

$$q\_3 = q\_{2'} \tag{4}$$

$$q\_4 = h\_w(\theta\_2 - \theta\_w),\tag{5}$$

$$q\mathfrak{u} = q\mathfrak{z} + A\mathfrak{z}i\mathfrak{o}\_{\mathfrak{z}}\tag{6}$$

$$q\_5 = h\_w(\theta\_w - \theta\_3),\tag{7}$$

$$q\_5 = q\_4 + A\_w i\_0 + \dot{m} (\theta\_{IN} - \theta\_{\text{uv}})\_\prime \tag{8}$$

$$q\_6 = h\_i(\theta\_3 - \theta\_i),\tag{9}$$

$$q\_6 = q\_5 + A\_3 i\_0. \tag{10}$$

The *U* values depend on the interior heat transfer coefficient, *hi*, the exterior heat transfer coefficient, *he*, the air chamber heat transfer coefficient, *hg*, and the water heat transfer coefficient.

$$\frac{1}{hL\_c} = \frac{1}{h\_c} + \frac{1}{h\_\mathcal{g}} + \frac{1}{h\_{w'}} \tag{11}$$

$$\frac{1}{\mu I\_i} = \frac{1}{h\_i} + \frac{1}{h\_{iw}}.\tag{12}$$

*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 absorptance, *Av*, depends on the energy absorbed by the glass panes and by the water:

$$A\_{\upsilon} = A\_1 \left(\frac{lI\_{\varepsilon}}{h\_{\varepsilon}}\right) + A\_2 \left(\frac{1}{h\_{\mathcal{g}}} + \frac{1}{h\_{\varepsilon}}\right) lI\_{\varepsilon} + A\_3 \left(\frac{lI\_{\varepsilon}}{h\_i}\right) + A\_{\text{i\varepsilon}}.\tag{13}$$

Solving the Equations (1) to (10) and using the values of Equations (11) to (13):

$$
\theta\_{OUT} = \frac{i\_0 A\_\upsilon + \mathcal{U}\_i \theta\_i + \mathcal{U}\_\varepsilon \theta\_\varepsilon + \dot{m}c\theta\_{IN}}{\dot{m}\varepsilon + \mathcal{U}\_\varepsilon + \mathcal{U}\_i}. \tag{14}
$$

Water heat gain is a power magnitude, and it is measured in watts (W). Equation (2) shows the analytical expression.

$$P = \dot{m}c(\theta\_{\text{OUT}} - \theta\_{\text{IN}})\_\prime \tag{15}$$

where *P* is the power absorbed by the water, θ*OUT* and θ*IN* the temperature of water leaving and entering the glazing, respectively, *m˙* is the mass flow rate, and *c* is the specific heat of the water. The mass flow rate is a measurement of the amount of mass passing by a point over time. The goal of absorbing the same power, *P*, can be achieved with a high *m˙* , which results in a low-temperature increase or a low *m˙* , which results in a high-temperature difference between the inlet and outlet. Equation (16) results by combining Equations (14) and (15).

$$P = \frac{\dot{m}c}{\dot{m}c + \mathcal{U}\_c + \mathcal{U}\_i} (\dot{i}\_0 A\_v + \mathcal{U}\_i (\theta\_i - \theta\_{\text{IN}}) + \mathcal{U}\_c (\theta\_c - \theta\_{\text{IN}})),\tag{16}$$

where θ*<sup>e</sup>* and θ*<sup>i</sup>* are, respectively, the temperature outdoors and indoors of the room. Analyzing the Equation (16), the power absorbed by the water varies with θ*IN.* Considering the rest of the equation as a constant, *P0*, *P* linearly decreases with slope *(Ui* + *Ue*). Figure 2 shows that at a specific value of *m˙* , there is a maximum absorbed power, *P*, depending on θ*IN.*

$$P = P\_0 - \theta\_{IN} (\mathcal{U}\_\varepsilon + \mathcal{U}\_i) \,. \tag{17}$$

If boundary conditions do not change with time, the solution becomes constant when the system reaches a steady state. In this set of test cases, sunrays are perpendicular to the glazing. This assumption eliminates uncertainties associated with the dependence of each layer's absorptance with the angle of incidence. Two case studies are considered: (a) and (b). Table 2 defines the outdoor and indoor boundary conditions. Case (a) studies the influence of θ*IN* with a fixed absorptance, and case (b) considers the impact of the absorptance when θ*IN* is fixed.

**Table 2.** Outdoor and indoor boundary conditions for WFG steady thermal performance.


Figure 2 illustrates the power variations with water flow rate, *m˙* . There is a maximum water heat gain when *mc ˙* >> *Ue* + *Ui*. The maximum power absorption occurs when *Av* is high and θ*IN* is low. If the goal is to reject the solar energy, *Av* must be as low as possible, with solar control coatings in the outermost glass panes. In this case (*mc ˙* >> *Ue* + *Ui*), the absorbed power (*P*) is the maximum power (*Pmax*)

$$P\_{\text{max}} = i\_{\text{O}}A\_{\text{F}} + \mathcal{U}\_{\text{i}}(\theta\_{\text{i}} - \theta\_{\text{IN}}) + \mathcal{U}\_{\text{e}}(\theta\_{\text{e}} - \theta\_{\text{IN}}),\tag{18}$$

$$P\_{\text{max}} = i\_0 A\_{\text{\textquotedbl}} + \mathcal{U}\_l \theta\_l + \mathcal{U}\_\mathcal{E} \theta\_\mathcal{E} - \theta \text{IN} \left( \mathcal{U}\_l + \mathcal{U}\_\mathcal{E} \right). \tag{19}$$

**Figure 2.** Power absorbed by the water chamber of the WFG: (**a**) constant *Av* at different inlet temperatures (θ*IN*); (**b**) constant inlet temperature (θ*IN*) with different *Av*.

Combining Equations (14) and (19), the value of θ*OUT* is:

$$
\theta\_{\rm OUT} = \theta\_{\rm IN} + \frac{P\_{\rm max}}{\dot{m}c + \mathcal{U}\_{\rm \varepsilon} + \mathcal{U}\_{i}}.\tag{20}
$$

θ*STAGNATION* is the temperature of the water when the mass flow rate is stopped. Figure 3 illustrates the relationship between θ*OUT* and *m˙* in two cases: (a) and (b). Case (a) shows that θ*STAGNATION* is the same if the water absorption *Av* does not change; case (b) shows the variations of θ*STAGNATION* with different conditions of *Av*. When m is close to zero, then ˙ θ*OUT* gets the maximum value, which is θ*STAGNATION*. When *m˙* is very high, then θ*OUT* = θ*IN*.

**Figure 3.** Outlet temperature (θ*OUT*) of the WFG: (**a**) constant *Av* at different inlet temperatures; (**b**) constant inlet temperature (θ*IN*) with different *Av*.

Equation (21) shows the total heat flux, *q*:

$$q = \text{g}i\_0 + \text{l}I(\theta\_\varepsilon - \theta\_i) - \text{l}I\_w(\theta\_i - \theta\_{\text{IN}}),\tag{21}$$

where *g* describes the proportion of solar energy transmitted indoors, *U(*θ*e*−θ*i)* expresses the heat exchange between the room and outdoors, and *Uw (*θ*i*−θ*IN)* represents the heat exchange between the water chamber and indoors. As per Equations (34) and (35) in [50]:

$$
\mathcal{U}\mathcal{U}\_w = \frac{\mathcal{U}\_l \dot{m} c}{\dot{m}c + \mathcal{U}\_l + \mathcal{U}\_l'},
\tag{22}
$$

$$
\mathcal{U}I = \frac{\mathcal{U}\_{\mathfrak{c}}\mathcal{U}\_{\mathfrak{i}}}{\dot{m}\mathcal{c} + \mathcal{U}\_{\mathfrak{c}} + \mathcal{U}\_{\mathfrak{i}}}.\tag{23}
$$

Since the WFG is a dynamic envelope, the *g* factor depends on the flow rate. When the water flows, the *g* factor decreases, and when the water flow stops, the solar energy enters the building, and the *g* factor increases. The thermal transmittance, *U*, is almost zero when the flow rate is the design flow rate because the water chamber isolates the building from outdoor conditions. When the water flow stops, the thermal transmittance depends mainly on the components that make up an insulated glass unit: the glass panes, coatings, spacer, sealing, and the gas filling the sealed space.
