# Numerical Analysis on the Performance of a Radiant Cooling Panel with Serpentine-Based Design
## 2 Methods
### Mathematical Model  
Based on the sketch in Figure 1, chilled water flows from the inlet of the copper tube and goes through the channels of the tube. The top and sides of the radiant panel are insulated, and heat flux is applied from beneath the radiant panel. Therefore, heat exchange occurs between the aluminum plate and the chilled water flowing through the copper tube. The fluid inlet temperature and ambient temperature are kept constant throughout the simulation. The plate area and flow volume are defined as Equations (8) and (9).  
\[A_{p}=W\times L \tag{8}\]  
\[V_{flow}=\frac{\pi}{4}\;D^{2}L_{tube} \tag{9}\]  
where \(W\) and \(L\) is the width (m) and length of the plate (m), respectively. \(D\) is the tube diameter (m) and \(L_{tube}\) is the flow path length (m). The governing equations for the model are shown in Equations (1)-(5).  
The total heat flux received by the panel is expressed by Equation (10). The total heat flux is the summation of radiation and convection heat transfer between the surrounding temperature and the panel's surface.  
\[q_{rc}^{\prime\prime}=h_{r}\big{(}AUST-T_{p}\big{)}\;+\;h_{c}\big{(}T_{amb}-T_{ p}\big{)} \tag{10}\]  
where \(h_{r}\) is the radiation heat transfer coefficient, \(AUST\) is the area-weighted average temperature of all the uncontrolled surfaces in the conditioned space, \(T_{p}\) is the effective radiant panel surface temperature and \(h_{c}\) is the convection heat transfer coefficient. The  
Figure 1: Flow configurations: (**a**) one-inlet-one-outlet serpentine, (**b**) counterflow two-inlet-two-outlet serpentine, (**c**) three-inlet-three-outlet serpentine configurations. Note: inlet (blue arrow) and outlet (green arrow).  
equation for radiation and convection heat transfer coefficient is expressed as shown in Equations (11) and (12), respectively [23]:  
\[h_{r}=\frac{\sigma\sum_{j=1}^{n}F_{\epsilon_{s-j}(\frac{T_{j}^{n}-T_{j}^{n}}{T_{j }})}}{\left(AUST-T_{p}\right)} \tag{11}\]  
where \(\sigma\) is the Stephan-Boltzmann constant, and \(F_{\epsilon_{s-j}}\) is the radiation interchange factor between the panel surface temperature and \(j\)th surface temperature, \(T_{j}\).  
\[h_{c}=\frac{q^{\prime\prime}{}_{c}}{\left(T_{amb}-T_{p}\right)} \tag{12}\]  
where \(T_{amb}\) is the ambient air temperature.  
However, based on data from the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE), the radiation and convective heat fluxes for a suspended RCP may also be estimated by using Equations (13) and (14) [24]:  
\[q_{r}^{\prime\prime}=5\times 10^{-8}[\left(AUST+273\right)^{4}-\left(T_{p}+273 \right)^{4}] \tag{13}\]  
\[q_{c}^{\prime\prime}=2.31(T_{amb}-T_{p})^{0.31}\left(T_{amb}-T_{p}\right) \tag{14}\]  
For the calculation of heat flux, an initial value of plate temperature, \(T_{p}\), is needed to be assumed and adjusted accordingly throughout the simulation. _AUST_ is assumed to be the same as \(T_{amb}\) for ease of calculation, normally when the cases and design adhere to ASHRAE 1999a [16; 25]. Table 1 below shows the heat transfer characteristics, based on the mean panel temperature determined by [PERSON] et al. [25].