Accordingly, the analysis can be decoupled into two distinct phases。

The first phase consists of modeling the thermo-fluid dynamic phenomena occurring in the under-slating ventilation ducts。 From a general point of view, the ducts can be seen as channels heated on the upper side (the same as in bare plate concept)。 The air is heated inside the channel and drives internal natural convection。

The duct is modeled as an inclined channel between two parallel surfaces with geometry and sizes illustrated in Figure 2。

The analytical model is one-dimensional and based on the following assumptions:

-Tile as a homogeneous and isotropic body;

-Negligible thickness of the tile (lumped parameters);

-Air as ideal gas with constant properties。

-Adiabatic lower duct surface;

-Air flow in quasi-steady-state conditions

In order to determine the flowrate and temperature of the air at the outlet of the ventilation channel, adopting a similar approach as in [18,19], the generalized Bernoulli’s equation inside the channel (Eq。 1), continuity equation (Eq。 2), air energy balance (Eq。 3) and the tile energy balance (Eq。 4) must be combined:

(1)

(2)

(3)

Where the indices in and out represent the inlet and outlet sections respectively。 Eq。 (1) is rewritten as: (5) with (6) where is the isobaric thermal expansion coefficient [K-1] evaluated at the film temperature 。 The energy loss term R is defined as (7) where is the friction factor, usually calculated from empirical correlations。 For the case of a buoyancy driven flow in a rectangular channel with a ribbed surface, no correlations have been found by the authors in the literature。 Hence, the friction factor has been experimentally determined as shown in the next section。 The various terms of Eq。 (3) and Eq。 (4) are expressed as follows: (8) (9) (10) (11) (12) The solar irradiance on the roof (Gtile) is calculated summing the direct normal irradiance (GDNI) and the diffuse irradiance (GDIFF) as expressed in the following equation。 (13) The incidence angle ( ) is calculated using standard approach for solar angle calculation [20,21] The radiative heat losses are estimated by defining conveniently the apparent sky temperature。 In this study it has been calculated by [22] correlation: (14) where is the water vapor partial pressure and is called serenity index, i。e。 the ratio between irradiation at ground level and irradiation outside the earth’s atmosphere, both calculated on a flat surface。 The integral emissivity (ε) and absorptivity (α) in Eq。 (11) and Eq。 (12) have been experimentally tuned on the tile as discussed in the next section。 The convective heat transfer coefficients and in Eq。(9) and Eq。(10) have been calculated

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6 from available correlations。 In particular, the convective coefficient , at the outer surface, considers both natural convection ( ) and forced convection due to wind action ( , according to the following relation [23]: (15) where and are taken from [24]and [25]: (16) Where L is the length of the pitch and wwind is the wind speed。 (17) Where k is the air thermal conductivity, the Prandtl number and is defined as (18) The convective coefficient , at the inner surface, is determined by the correlation of [26]: (19) extended to the case of inclined channels setting: (20) The Biot number for the tile always resulted sufficiently low to apply the lumped parameter analysis。 The solution of the thermal model requires an iterative procedure summarized in Figure 4。 Since the roof has two pitches with different orientation, i。e。 different irradiation patterns, the calculation will be performed for both pitches (leading to subscripts 1 and 2)。 Once the convergence is reached, the air flowrate (Eq。 21) and temperature (Eq。 22) at the heat pump evaporator on top of the roof are calculated as follows: (21) (22) The overall heating power recovered and transferred to the evaporator is determined with reference to the air ambient temperature: (23) The higher temperature compared to ambient air allows increasing the COP of the heat pump, therefore reducing the primary energy consumption for a given thermal output。 simplified heat pump model was developed to determine the performances of an air-to-water heat