# Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature
## 2 Theory
### Numerical Model
\begin{array}{l}\frac{k_{\mathrm{a}}}{L}0.54\mathrm{ R}a_{Lc}^{1/4}\ \mathrm{if}\ 10^{4}\leq\mathrm{R}a_{Lc}\leq 10^{7}\\ \frac{k_{\mathrm{a}}}{L}0.18\mathrm{R}a_{Lc}^{1/3}\ \mathrm{if}\ 10^{7}\leq \mathrm{R}a_{Lc}\leq 10^{11}\end{array} \right.\ \mathrm{if\ air\ is\ over\ the\ plate} \tag{4}\]\[h_{\rm h,ab}=\frac{k_{\rm a}}{L_{\rm c}}0.27\rm Ra_{Lc}^{1/4}\ \rm if\ 10^{5}\leq Ra_{Lc}\leq 1 0^{10}\ \ \ \rm if\ air\ is\ under\ the\ plate \tag{5}\]  
where \(L_{c}\) is the characteristic length of the horizontal surface, i.e., area over perimeter \(L_{c}=L/4\), \(k_{\rm a}\) is the thermal conductivity of air, and \(\rm Ra_{Lc}\) is the Rayleigh number referred to this characteristic length, defined as follows:  
\[\rm Ra_{Lc}=\frac{g\cdot\beta_{a}(T-T_{a})L_{c}^{3}}{\alpha_{a}\cdot v_{a}} \tag{6}\]  
where \(\beta_{a}\) is the thermal expansion coefficient associated with air, \(g\) is the gravity acceleration, and \(\alpha_{a}\) and \(v_{a}\) are the air thermal diffusivity and kinematic viscosity, respectively. The contribution of the heat dissipated by natural convection at the lateral surfaces of the cooling plate was evaluated based on the estimation of the convection heat transfer coefficient for the lateral surfaces, \(h_{\rm h,al}\), obtained from the [PERSON] and [PERSON] [44] correlation for vertical plates, which reads:  
\[h_{\rm h,al}=\frac{k_{\rm a}}{L_{c}}\left(0.825+\frac{0.387\cdot\rm Ra_{Lc}^{ 1/6}}{\left(1+\left(\frac{0.492}{\rm Pr_{\rm a}}\right)^{9/16}\right)^{8/27}} \right)\ \rm if\ Ra_{Lc}>1 \tag{7}\]  
where \(L_{\rm c}\) is the characteristic length of the vertical surface, which, in this case, corresponds to the plate thickness \(L_{\rm c}=\rm s\), and \(\rm Pr_{\rm a}\) is the Prandtl number for air, defined as \(\rm Pr_{\rm a}=\mu_{\rm a}\ c_{pa}/k_{\rm a}\). Considering the low temperature difference between the plate and ambient air, and the reduced dimensions of the plate, low values of the Ra are expected, and thus, heat transfer between the surface of the plate and ambient air will occur by laminar free convection. The properties of air required by the correlations were evaluated at the film temperature, i.e., the average temperature between the plate and the ambient air. A variable plate temperature ranging from its initial temperature \(T_{0}=55\ ^{\circ}\)C and the constant ambient temperature of \(T_{\rm a}=25\ ^{\circ}\)C was considered for the calculation. The temperature-dependent natural convection coefficients obtained from the correlations were thus averaged in the whole temperature range from \(T_{0}\) to \(T_{\rm a}\), obtaining average values of \(h_{\rm h,at}=14\ \rm W/(m^{2}K)\) for the top surface, \(h_{\rm h,ab}=7\ \rm W/(m^{2}K)\) for the bottom surface, and \(h_{\rm h,al}=19\ \rm W/(m^{2}K)\) for the lateral vertical surfaces. The heat dissipated from the plate to the water flowing in the channel was estimated based on typical correlations for forced convection coefficients. Considering the diameter of the channel \(d=6\ \rm mm\), the velocity of water \(v_{\rm w}=1\ \rm m/s\), and the properties of water at \(T_{\rm w}=25\ ^{\circ}\)C, the Reynolds number for flowing water, defined as \(\rm Re=\rho_{\rm f}\ v_{w}\ d/\mu_{\rm f}\), is \(\rm Re=6699\). For such a high value of the Reynolds number, a turbulent regime is expected.