# Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature
## 2 Theory  
### Numerical Model  
The numerical model proposed in this work considers a two-dimensional approximation of the cooling system consisting of a solid plate with a channel. The finite difference numerical model is based in a two-dimensional domain, provided that the temperature distribution across the thickness of the cooling plate is considered uniform. This assumption is supported by the low Biot numbers obtained for both the heat transfer by natural convection from the external surfaces of the plate to ambient air, with values of Bi of order 0.001 and for the heat transfer by forced convection from the surface of the channel to the cold water flowing through it, with a value of Bi below 0.1. Therefore, heat is transferred in the plane of the cooling plate following the diffusion equation:  
\[\rho Vc_{p}\frac{\partial\mathrm{T}}{\partial\mathrm{t}}=q_{\mathrm{k}}+q_{ \mathrm{h,a}}+q_{\mathrm{h,w}}+q_{\mathrm{in}} \tag{1}\]  
where \(T\) is the temperature of the plate at each spatial coordinate, \(V\) represents the volume where the diffusion equation is defined, \(\rho\) and \(c_{p}\) represent the solid material density and specific heat, respectively, \(q_{\mathrm{k}}\) is the heat rate by conduction inside the solid body, \(q_{\mathrm{h,a}}\) depicts the heat rate dissipated through the external walls of the plate by natural convection to ambient air, \(q_{\mathrm{h,w}}\) represents the forced convection heat rate to the cold water circulating along the channel, and \(q_{\mathrm{in}}\) denotes the heat rate entering the system. The cooling plate is assumed to be initially at \(T_{0}\) = 55 \({}^{\circ}\)C and surrounded by ambient air at a room temperature of \(T_{\mathrm{a}}\) = 25 \({}^{\circ}\)C. Water flows inside the channel at a constant temperature of \(T_{\mathrm{w}}\) = 25 \({}^{\circ}\)C and a velocity of \(v_{\mathrm{w}}\) = 1 m/s. Under these conditions, the process can be simplified, considering that the temperature of water along the whole channel is uniform and equal to \(T_{\mathrm{w}}\) = 25 \({}^{\circ}\)C [41]. Therefore, the plate is progressively cooled down by both natural convection to ambient air and forced convection to cold water flowing through the channel until it reaches a steady state temperature, which depends on the heat rate entering the system, \(q_{\mathrm{in}}\). The boundary conditions used in this work were previously employed by [PERSON] et al. [41] and [PERSON] et al. [42] to model the cooling of solid plates refrigerated by a cold fluid circulating through an interior channel. The initial (IC) and boundary (BC) conditions can be thus expressed as follows:  
\[\mathrm{IC:}\ T(t=0)=T_{0} \tag{2}\]  
\[\mathrm{BC\ at\ external\ walls:}\ q_{\mathrm{h,a}}^{\prime\prime}=h_{\mathrm{a}} \cdot(T-T_{\mathrm{a}}) \tag{3}\]  
\[\mathrm{BC\ at\ channel\ walls:}\ q_{\mathrm{h,w}}^{\prime\prime}=h_{\mathrm{w}} \cdot(T-T_{\mathrm{w}})\]  
In which \(q^{\prime\prime}\) represents the heat flux, \(h_{\mathrm{w}}\) is the forced convection coefficient from the channel wall to cold water, and \(h_{\mathrm{a}}\) refers to the natural convection coefficient from the external surfaces to ambient air, which can be divided into three components: (i) \(h_{\mathrm{h,ab}}\) for the natural convection coefficient of the external horizontal bottom wall of the cooling plate, (ii) \(h_{\mathrm{h,at}}\) for the natural convection coefficient of the external horizontal top wall of the cooling plate, and (iii) \(h_{\mathrm{h,al}}\) for the natural convection coefficient of the external lateral walls of the cooling plate. The correlation proposed by [PERSON] [43] for the horizontal hot surfaces in contact with ambient air in upwards and downwards directions was used to determine the convection coefficients \(h_{\mathrm{h,at}}\) and \(h_{\mathrm{h,ab}}\), respectively:  
\[h_{\mathrm{h,at}}=\left\{