## 3 Numerical Model of the Data Center
### Boundary Conditions  
The boundary conditions defined in the numerical models were based on the data collected in the studied real data center. To the values collected was also added the value of pressure loss in the server racks, which are provided in [44]. It is important to point out that, from now on all, coordinate values and parameters are presented with dimensionless quantities to simplify the comparison between case studies. Dimensionless values are identified with a subscript \"adm\" in all equations of this study. The dimensionless quantities range from 0 to 1. It should be noted that, in the case of the temperature, the maximum value corresponds to a temperature value exceeding the operating specifications of the server racks. It is an air temperature value with the capacity to cause malfunctions or to damage to the equipment present in the ICT rooms, while this equipment is intended to operate uninterruptedly. In this study, as previously mentioned, the PRESTO! interpolation method was used mainly due to the presence of porous medium [45]. The porous medium is an agglomerate of particles that are close, leaving small voids between them. Due to the existence of these voids, it is allowed to circulate liquid or gaseous elements that can fill them. Thus, the larger are these voids, the greater is the porosity of this material, that is, a greater amount of another element is required to fill them. In this study, only porosity was considered for the server racks, due to micro perforated doors that allow air to be introduced into the server rack. Besides the porosity present in the doors of the server rack, there are still the components inside that occupy the empty space inside the server rack, thus not allowing free air circulation. Therefore, for this study, the porosity of the server rack was considered as a whole (carcass and elements) and with the value of 80%, provided by the data center. Due to the existence of porous media, it is necessary to calculate the coefficient of inertial loss \"C2\" applied to the server racks [45], which is given by the following expression:  
\[K_{L^{\prime}}=\frac{\Delta p}{\frac{1}{2}\rho v^{2}} \tag{10}\]  
where \(K_{L^{\prime}}\) is the loss factor, \(\Delta p\) is the loss variation in Pa, and \(v\) is the velocity, in m\(\cdot\)s\({}^{-1}\). The coefficient of inertial loss, \(C_{2}\) is expressed as follows:  
\[C_{2}=\frac{K_{L^{\prime}}}{\textit{thickness of the porous medium}} \tag{11}\]  
The coefficient of inertial loss (\(C_{2}\)) is measured in m\({}^{-1}\). For the resolution of Equation (10), the pressure variation value of 7.5 Pa was adjusted to the velocity of m\(\cdot\)s\({}^{-1}\) at the velocities present in [44]. The equations presented in Section 2 are solved in the fluid zone. The fluid zone was considered having an initial dimensionless temperature of 7.09 \(\times\) 10\({}^{-4}\), that is, the average dimensionless temperature measured within the ICT room. The turbulence intensity and the hydraulic diameter are the parameters that allow the iterative calculation of the turbulent kinetic energy and the coefficient of dissipation of the turbulent kinetic energy. Regarding the intensity of turbulence, it indicates the direction of the flow. Generally it is considered 1% for low turbulence intensity and for high turbulence intensity it is considered 10% [45]. In this study, a turbulence intensity of 5% was considered in the extraction zone, as it is located in the containment zone of the hot aisles, which provides a well-defined direction of air flow. Concerning the intensity of turbulence in the air vent area, 5% was also considered since the air vent grids allow directing the air. These values were based on the experimental and numerical results provided in [46; 47; 48; 49]. The boundary conditions are presented in Table 2 and are characterized by dimensionless values. The area of air vents presents values that are always equal, while in the extraction area different values occur, since one of the elements that compose this area is smaller.