*2.2. Determination of Heat Transfer Coe*ffi*cients (h) and Dynamic Thermal Transmittance (U)*

Water absorbs the solar near-infrared radiation and increases the temperature as it flows through the window. The mass flow rate and the thermal properties of glass panes must be studied to allow designers to apply energy-management strategies. Sometimes it might be interesting to increase the water temperature and store that energy for heating purposes. Other times it might be appropriate to reject as much solar radiation as possible without heating the water. The thermal transmittance (*U*) measures the heat transfer through the glazing and the European Standard determines its value [39,40]. Figure 2 illustrates the heat transfer coefficients (*hi*, *hw*, *hg*, and *he*), the temperature distribution in the WFG layers (θ*i*, θ*1*, θ*2*, θ*3*, θ*w*, θ*e*), and the absorptance of layers (*A1, A2, Aw, A3*).

**Figure 2.** Temperature distribution in a triple water flow glazing (WFG) at a specific height, with heat transfer coefficients (*hi*, *hw*, *hg*, *he*), heat fluxes through a triple WFG (*q1, q2, q3, q4, q5, q6*), solar radiation and absorptance of layers (*A1, A2, Aw, A3*). (**a**) Triple WFG with water chamber outdoors. (**b**) Triple WFG with water chamber indoors.

The Equation (1) gives the outdoor heat flux:

$$
\eta\_{\varepsilon} = h\_{\varepsilon} (\theta\_{\varepsilon} - \theta\_{1}),
\tag{1}
$$

where *he* is the outdoor convective coefficient, θ*<sup>1</sup>* is the superficial temperature of the outermost glass pane, and θ*<sup>e</sup>* is the outdoor temperature. The beam solar irradiance, diffuse irradiance, and the angle of incidence should be given. Regarding indoor boundary conditions, the indoor heat flux is given by the Equation (2):

$$q\_i = h\_i(\theta\_3 - \theta\_i)\_\prime \tag{2}$$

where *hi* is the indoor convective coefficient, θ*<sup>3</sup>* is the superficial temperature of the inner glass pane, and θ*<sup>i</sup>* is the indoor temperature that can be a constant value or calculated solving the indoor thermal problem. Newton's law also models the heat transfer inside gas chambers. The heat flux in a gas chamber between two parallel surfaces is expressed by Equation (3):

$$q\_i = h\_{\mathcal{S}} (\theta\_i - \theta\_{i+1})\_\prime \tag{3}$$

where *hg* is the heat transfer coefficient of gas chambers, this coefficient considers the radiative heat transfer between the parallel glass panes and the natural convective transport. This value can be constant or calculated, knowing the emissivity of the two glass planes and an experimental correlation for the natural convective transport. In the water chamber, the situation is different. Heat flux is proportional to the temperature difference between the water temperature θ*<sup>w</sup>* and the glass pane temperatures. This coefficient takes into account the heat transport mechanism forced by the water flow inside the chamber. Since the water is opaque to far-infrared, the radiative heat transfer mechanism is not present in the water chamber. Hence, the heat flux through glass panes in contact with the water

chamber is expressed in Equation (4) for glass panes outside the water chamber, and Equation (5) when the glass pane is after the water chamber:

$$q\_i = h\_w(\theta\_{i-1} - \theta\_w),\tag{4}$$

$$q\_{i+1} = h\_w(\theta\_w - \theta\_i)\_\prime \tag{5}$$

where *hw* is the heat transfer coefficient for the water chamber. By default *hw* = 50.

When it comes to considering the absorptance of layers (*Ai*), the heat flux can be expressed as in Equation (6).

$$q\_{i+1} = q\_i + A\_i \, i\_0 \, . \tag{6}$$

Spectral and thermal properties of WFG have been explained in previous articles [35]. The authors used data from commercial software and developed equations to evaluate the influence of water flowing through glass panes. Equation (7) considers the energy balance at each layer and Newton's definition of heat flux.

$$q\_i = q\_{i-1} + A\_w \ i\_0 + \dot{m} \varepsilon (\theta\_{IN} - \theta\_{OUT}). \tag{7}$$

Equations (8)–(17) show the heat flux of WFG with water chamber indoors.

$$q\_1 \ = h\_\epsilon \ (\theta\_\epsilon - \theta\_1),\tag{8}$$

$$q\_2 = q\_1 + A\_1 \, i\_{0\text{ }\prime} \tag{9}$$

$$q\_2 = h\_{\mathcal{g}} \left(\theta\_1 - \theta\_2\right),\tag{10}$$

$$q\_3 = q\_2 \,, \tag{11}$$

$$
\eta\_4 = h\_w \left(\theta\_2 - \theta\_w\right),
\tag{12}
$$

$$q\_4 = q\_3 + A\_2 \, i\_0 \, . \tag{13}$$

$$q\_5 = h\_w \left(\theta\_w - \theta\_3\right),\tag{14}$$

$$q\_{\mathbb{S}} = q\_4 + A\_w \, i\_0 + \dot{m} \varepsilon (\theta\_{IN} - \theta\_w) \, \tag{15}$$

$$q\_6 := h\_i \left(\theta\_3 - \theta\_i\right),\tag{16}$$

$$q\_6 = q\_5 + A\_3 \, i\_0 \, . \tag{17}$$

where *hi* is the interior heat transfer coefficient; *he*, the exterior heat transfer coefficient; *hg*, the air-cavity heat transfer coefficient; and *hw*, the water heat transfer coefficient. The thermal transmittances (*Ue, Ui*) depend on the heat transfer coefficients. Equations (18) and (19) refer to WFG with the water chamber outdoors (Case 1), and Equations (20) and (21) show the expressions for WFG with a water chamber indoors (Cases 2, 3).

$$\frac{1}{\text{L}I\_c} = \frac{1}{h\_c} + \frac{1}{h\_w} \,\text{\,\,\,}\tag{18}$$

$$\frac{1}{M\_i} = \frac{1}{h\_i} + \frac{1}{h\_{\mathcal{K}}} + \frac{1}{h\_{w'}} \tag{19}$$

$$\frac{1}{\text{L}\_c} = \frac{1}{h\_c} + \frac{1}{h\_\mathcal{g}} + \frac{1}{h\_w} \,\text{\,}\tag{20}$$

$$\frac{1}{M\_i} = \frac{1}{h\_i} + \frac{1}{h\_w}.\tag{21}$$

*U* represents the thermal transmittance between the room and outdoors, and *Uw* represents the thermal transmittance between the water chamber and indoors. 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 air chamber. Equations (22) and (23) show the expressions for *U* and *Uw*, respectively:

$$\mathcal{U}\_{\text{iv}} = \frac{\mathcal{U}\_{i}\dot{m}c}{\dot{m}c + \mathcal{U}\_{\text{c}} + \mathcal{U}\_{i}} \,' \tag{22}$$

$$
\mathcal{U}\mathcal{U} = \frac{\mathcal{U}\_{\text{eff}}\mathcal{U}\_{i}}{\dot{m}c + \mathcal{U}\_{\text{c}} + \mathcal{U}\_{i}}\mathcal{V} \tag{23}
$$

where *m˙* is the mass flow rate and *c* is the specific heat of water.

Table 1 shows a complete description of the different layers and the selected glazing average values. Visual transmittance (*Tv*) is the measurement of visible light (380 to 780 nm) passing through the glazing. The thermal parameters depended on the mass flow rate. If the water was flowing, the *g*-factor became lower. When the water chamber was stopped, the amount of energy entering the building increased. The thermal transmittance (*U*) was almost zero at the design flow rate (2 L/min m2). When *m˙* = 0, *U* depended on the air chamber. The thermal transmittance (*Uw*) as defined in Equation (22) measures the heat losses or gains between the water chamber and the indoor air. *Uw* = 0 when *m˙* = 0. At the working flow rate, its value was high (6.4 W/m2K) when the water chamber was close to indoors, and it was low (0.9 W/m2K) when the water chamber was outdoors. The first case had the water chamber outdoors and a low-emissivity coating in face 4. When *m˙* = 0, the *U* value was 1.041 W/m2K, and the *g*-factor was 0.25. At the operating *m˙* , the *g*-factor became 0.22 and the *U* value, 0.128 W/m2K. The visual transmittance (*Tv* = 0.51) was the lowest of the selected cases. The second case had a water chamber facing indoors. A low-emissivity coating was placed in face 3 and a solar PVB layer in the central pane. The *U* value ranged from 0.066 to 1.041 W/m2K and a variable *g*-factor was 0.24 when the flow was ON, and 0.59 when the flow was OFF, adapting to the outdoor environment in both summer and winter conditions. The visual transmittance was 0.53. The third case had a highly selective coating in face 2. It yielded a *U* value of 0.995 W/m2K when the flow was OFF, and 0.063 W/m2K when the flow was ON. The *g*-value varied between 0.22 and 0.27. The visual transmittance (*Tv* = 0.55) was the highest of the selected cases. The energy transmittance (*T*), did not show significant variations, and it ranged from 0.20 in Case 1 to 0.21 in Cases 2 and 3.


**Table 1.** Spectral and thermal properties of the studied WFGs.

<sup>1</sup> Visual transmittance (*Tv*), energy transmittance (*T*). <sup>2</sup> Thermal transmittance from water chamber to interior (*Uw*), Thermal transmittance of triple glazing (*U*), g-factor (*g*).
