## 2 Methodology  
In this research, CFD was used to develop an AHU mixing temperature prediction model utilizing return airflow, outdoor airflow and outdoor temperature. CFD is a tool for analyzing and predicting complex thermal fluid flow phenomena and is based on the governing equations of a continuous fluid that basically satisfy the laws of the conservation of mass, momentum and energy [10]. These governing equations are often represented by the Navier-Stokes equations, which are partial differential equations that describe the behavior of a fluid and the physical phenomena associated with it. In CFD, such partial differential equations are converted into differentiated algebraic equations that are to be solved numerically [11]. By setting the initial and boundary conditions to solve the equations, one can accurately predict the behavior of a fluid over time and space [12].  
For our fluid analysis, we used the continuity equation, the Navier-Stokes equations and the energy equation to determine the heat transfer and temperature distribution. Of the aforementioned equations, the energy equation is a differential equation that represents the law of energy conservation, which refers to the time and space variation of energy in a fluid, and it is represented by Equation (1) [9].  
\[\frac{\partial}{\partial t}\bigg{(}\rho\bigg{(}\mathrm{e}+\frac{\mathrm{v}^{ 2}}{2}\bigg{)}\bigg{)}+\
abla\cdot\bigg{(}\rho\mathrm{v}\bigg{(}\mathrm{h}+ \frac{\mathrm{v}^{2}}{2}\bigg{)}\bigg{)}=\
abla\cdot\bigg{(}\mathrm{k}_{\mathrm{ eff}}\
abla\mathrm{T}-\sum_{j}\mathrm{h}_{j}\overrightarrow{\mathrm{J}}_{j}+ \overline{\pi}_{\mathrm{eff}}\cdot\overrightarrow{\mathrm{v}}\bigg{)}+S_{h}+ \
abla\cdot\overrightarrow{\mathrm{q}}_{r} \tag{1}\]  
Here, \(\rho\) is the density (kg/m\({}^{3}\)), \(\mathrm{e}\) is the internal energy (J/kg), \(\mathrm{v}_{2}^{2}\) is the kinetic energy term (J/kg), \(\mathrm{v}\) is the flow velocity vector (m/s), \(\mathrm{h}\) is the enthalpy (J/kg), \(\mathrm{k}_{\mathrm{eff}}\) is the effective thermal conductivity (W/m\(\cdot\)K), T is the temperature (K), \(\
abla\)T is the temperature gradient (K/m), \(\mathrm{h}_{j}\) is the enthalpy of component i (J/kg), \(\mathrm{J}_{j}\) is the component diffusive flux (kg/m\({}^{2}\cdot\)s), \(\overline{\pi}_{\mathrm{eff}}\) is the effective viscous stress tensor, \(S_{h}\) is the volumetric heat source (W/m\({}^{3}\)), and \(\overrightarrow{\mathrm{q}}_{r}\) is the radiant energy flux (W/m\({}^{2}\)).  
Based on the assumption that heat transfer within a fluid or solid does not occur by conduction, this equation transform as follows:  
\[\mathrm{k}_{\mathrm{eff}}=0 \tag{2}\]  
Equation (2), and only the part of Equation (1) that corresponds to enthalpy, is an expression for temperature.  
In this case, the enthalpy per component is defined by Equation (3).  
\[\mathrm{h}_{j}=\int_{T_{\mathrm{ref}}}^{T}\mathrm{c}_{pj}\mathrm{dT} \tag{3}\]  
Here, \(T_{\mathrm{ref}}\) is the reference temperature (K), T is the current temperature (K), and \(\mathrm{c}_{pj}\) is the specific heat of the static pressure of component j (J/kg\(\cdot\)K).  
For ideal gasses, the enthalpy h can be expressed according to the relationship given in Equation (4).  
\[\mathrm{h}=\mathrm{c}_{v}\mathrm{T}+\frac{\mathrm{P}_{\mathrm{abs}}}{\rho} \tag{4}\]  
Here, \(\mathrm{c}_{v}\) is the static specific heat (J/kg\(\cdot\)K) and \(\mathrm{p}_{\mathrm{abs}}\) is the absolute pressure (Pa).  
The temperature is derived using Equation (5).  
\[\mathrm{T}=\frac{1}{\mathrm{c}_{v}}(\mathrm{h}-\frac{\mathrm{P}_{\mathrm{abs} }}{\rho}) \tag{5}\]Based on Equation (5), the mixing temperature can be calculated by setting T\({}_{\rm i}\) as the temperature at a specific coordinate in the CFD simulation and T\({}_{\rm mix}\) as the average temperature at N points. This can be expressed as Equation (6).  
\[\rm T_{mix}=\frac{1}{N}\sum_{i=1}^{N}T_{i}=\frac{1}{N}\sum_{i=1}^{N}(\frac{1}{c_ {v}}(h_{i}-\frac{P_{abs}}{\rho})) \tag{6}\]