2.2.2. Boundary Conditions

The momentum boundary conditions in the walls were considered no-slip stationary with a constant rugosity of 0.5. The thermal boundary conditions modeled the incident solar radiation on the cover and the absorber plate as a heat generation source, calculated using Equations (13) and (14).

$$q\_{\mathcal{C}}(t) = \left( I(t) \left[ \overline{\mathfrak{a}}\_{\mathfrak{c}} + \overline{\mathfrak{a}}\_{\mathfrak{c}} \overline{\tau}\_{\mathfrak{c}} \overline{\mathfrak{q}}\_{p} \varrho\_{r}^{2\langle n \rangle} \right] \frac{A\_{\mathfrak{c}}}{A\_{p}} \right) / w\_{\mathfrak{c}} \tag{13}$$

$$q\_p(t) = \left( I(t) \, \overline{\pi}\_c \boldsymbol{q}\_r^n P\_\mathcal{S} \left[ \overline{\pi}\_p + \overline{\pi}\_p \overline{\boldsymbol{q}}\_p \overline{\boldsymbol{q}}\_c \overline{\boldsymbol{q}}\_r^{2/n} \frac{A\_p}{A\_c} \right] \frac{A\_c}{A\_p} \right) / w\_p \tag{14}$$

where *q*(*t*) is the heat generation, *I*(*t*) is the solar irradiance, *Pg* is the gap loss factor (0.96), *α* is the absorptivity, *τ* is the transmissivity, ρ is the reflectivity, *Ac* is the cover area, *Ap* is the absorber plate area, and *wc* and *wg*, are the thickness of the cover and plate, respectively. The optical properties of the cover are *α<sup>c</sup>* = 0.05, *τ<sup>c</sup>* = 0.89 y ρ*<sup>c</sup>* = 0.05, the receiver are *α<sup>p</sup>* = 0.95 and ρ*<sup>p</sup>* = 0.05, and from the reflector ρ*<sup>r</sup>* = 0.91.

The heat transfer coefficient from the cover to the environment (HTCc-a) was obtained by applying the flow around finite flat-plates methodology reported in [30], while the convection losses of the external walls were calculated using the heat transfer coefficient (HTCb) correlation proposed by [31]. Additionally, in laminar flow, the heat transfer coefficient from the flat-plate receiver to the fluid in the cavity (HTCp-cav) was estimated using the discretized Fourier's law, considering the local temperature normal to the wall [26]. Moreover, in turbulent flow, the HTCp-cav was determined using the law of the wall for estimating the local temperature of the fluid by applying the Reynolds analogy [26].

The heat conduction in the exterior walls of the CPC was modeled as shell conduction, whereas the radiation losses in the cover were calculated with an emissivity value of *ε<sup>c</sup>* = 0.81, and the sky temperature (*Ts*) was calculated with the correlation proposed by Swinbank (Equation (15)), reported in [32], where *Ta* refers to the ambient temperature.

$$T\_s = 0.0552 T\_a^{1.5} \tag{15}$$

Regarding the turbulence parameters, a turbulence intensity of 5% and a turbulent viscosity ratio of 10 were applied. Table 4 summarizes the parameters of the boundary conditions of the CFD modeling.


**Table 4.** Boundary conditions considered in the CFD modeling of the U-shape double-pass CPC.

The pressure-based solver with the Coupled scheme was selected. In addition, a Second-Order scheme for spatial discretization was implemented because of numerical simulation stability. The formal truncation errors of individual terms in the governing equations were calculated; the error for the HTCp-cav was 1.27%, 0.051% for the shear stress, and 0.05% for the Nusselt number (with a security factor of Fs = 3) [32]. In addition, values of 1 × <sup>10</sup>−<sup>4</sup> for the mass residual and a mass imbalance of 5.3 × <sup>10</sup>−8% and <sup>2</sup> × <sup>10</sup>−<sup>7</sup> energy residual were accomplished. The verification of the results was carried out by quantifying the uncertainty of the numerical calculations. For Nu, the spatial error of 0.51% was obtained, while for HTCp-cav and shear stress were 0.66% and 0.37%, respectively. The grid convergence index was also verified, finding out 0.05% for Nu, 0.19% for shear stress, and 0.09% for HTCp-cav.
