# Numerical Analysis on the Performance of a Radiant Cooling Panel with Serpentine-Based Design
## 2 Methods
### Computational Fluid Dynamics (CFD)  
This research will be conducted using computational fluid dynamics (CFD) software, known as Ansys Fluent [7; 8]. The Ansys Fluent software is used to simulate the fluid flow and investigate the heat transfer characteristics for different pipe arrangement configurations of the RCP. The Ansys Fluent software uses the Navier Stokes equation in the finite volume method (FVM) to analyze the results of the simulation [9; 10]. The steady-state continuity equation, momentum conservation equation, and energy conservation equation are solved by Ansys Fluent with the basic governing equations described below [11; 12]:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0 \tag{1}\]  
The continuity equation is shown in Equation (1), where \(u\), \(v\) and \(w\) are the velocity component in \(x\), \(y\) and \(z\) directions, respectively. \[\rho\bigg{(}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}\bigg{)}=\rho gx-\frac{\partial p}{\partial x}+ \mu\bigg{[}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{ \partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\bigg{]} \tag{2}\]  
\[\rho\bigg{(}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z}\bigg{)}=\rho gy-\frac{\partial p}{\partial y}+ \mu\bigg{[}\frac{\partial^{2}v}{\partial x^{2}}+\frac{\partial^{2}v}{ \partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}}\bigg{]} \tag{3}\]  
\[\rho\bigg{(}u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}\bigg{)}=\rho gz-\frac{\partial p}{\partial z}+ \mu\bigg{[}\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{ \partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}}\bigg{]} \tag{4}\]  
The momentum conservation equation in three dimensions is shown in Equations (2)-(4). \[\partial\bigg{(}\frac{\rho u_{i}T}{\partial x_{i}}\bigg{)}=\frac{\partial}{ \partial x_{i}}\bigg{(}\frac{k}{C_{p}}\frac{\partial T}{\partial x_{i}}\bigg{)} +S_{T} \tag{5}\]  
The energy conservation equation is shown in Equation (5), where \(T\) is the fluid temperature, \(u_{i}\) is the velocity in different directions in Cartesian coordinates, \(x_{i}\) is the Cartesian coordinates, \(k\) is the thermal conductivity, \(C_{p}\) is the constant-pressure specific heat of fluid, and \(S_{T}\) is the radiative energy source term [11; 13]. The Navier-Stokes equations are then transferred into discretized form for numerical solution. By using the FVM, the general form of the Navier-Stokes equation is integrated over a control volume, and the Gauss divergence theorem is applied. Equation (6) is used as a starting point for computational procedures in the FVM [12]. \(\phi\) is introduced as a general variable for the conservative form of all fluid flow equations [12]:  
\[\int_{cv}\frac{\partial(\rho\phi)}{\partial t}dV+\int_{cv}div(\rho\phi u)dV= \int_{cv}div(\Gamma grad\phi)dV+\int_{cv}S_{\phi}dV \tag{6}\]  
where \(\frac{\partial(\rho\phi)}{\partial t}\) is the rate of increase of the \(\phi\) fluid element, \(div(\rho\phi u)\) is the net rate of flow of \(\phi\) out of the fluid element, \(div(\Gamma grad\phi)\) is the rate of increase of \(\phi\) due to diffusion, and \(S_{\phi}\) is the rate of increase of \(\phi\) due to the source terms. The computational domain is divided into a finite number of close-proximity control volumes as discrete elements. They are located in the centroid of the control volume, and are called variables of interest [14]. FVM uses interpolation to integrate the differential forms of the governing equations, to create discretized equations [14]. The general discretized equation for interior nodes in three dimensions is shown in Equation (7) [12]:  
\[a_{p}\phi_{p}=\sum a_{nb}\phi_{nb}+S_{u} \tag{7}\]  
where \(\Sigma\) is the diffusion summation of all neighboring nodes (\(nb\)) and \(a_{nb}\) is for the neighboring coefficients.