# Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature
###### Abstract
For conduction heat transfer, the most widely used discretization methods in topology optimization are the Finite Volume Method and the Finite Element Method [26; 27; 28; 29]. Nevertheless, most of the studies available in the specific literature related to cooling systems are limited to steady state conditions since topology solutions involve many design variables and specific optimization algorithms, resulting in an increased complexity of the technique that leads to a high numerical cost [25]. Another optimization method proposed to maximize heat dissipation while keeping a low pressure drop for the cooling fluid is the so-called Constructal law. [PERSON] and [PERSON] [30] defined the Constructal law as \"For a finite-size flow system to persist in time (to live), its configuration must change in time such that it provides easier and easier access to its currents (fluid, energy, species, etc.)\". The Constructal law takes advantage of the optimized configurations already present in nature to implement them into more efficient refrigeration systems. In this sense, various research works have focused on optimizing the cooling capacity of systems based on extended surfaces with shapes inspired by nature [31; 32; 33; 34]. The Constructal law has also been used in cooling systems relying on forced convection by flowing liquids. There are some works in the literature that focus on optimizing compact heat exchangers, increasing the exchange surface while minimizing the volume. [PERSON] [35] studied forced convection in a microchannel heat exchanger by an evolutionary design. The author varied the shape and length of the microchannel to find an optimal number of channels for a fixed heat loss and volume. The study was carried out for different shapes of the channel, including rectangles, ellipses, and regular polygons. An expression was derived to determine the ideal shape of the channel and the results were compared to those available in the literature. The work carried out by [PERSON] et al. [36], who optimized the inner diameter for different Prandtl numbers and various channel shapes, was used for this comparison. Additional works studied the performance of heat exchangers with complex tree-shaped configurations based on the Constructal law [37; 38; 39]. Recently, the Constructal law has been applied to numerical simulations of cooling systems based on flat cooling plates with channels of different shapes to reduce the maximum temperature of operation and to homogenize the temperature distribution across the plate. [PERSON] et al. [40] focused on optimizing the distribution of serpentine channels in radiant cooling panels to minimize the fluid pumping cost. [PERSON] et al. [41] developed a simplified numerical approach to evaluate the effect of the shape of the channels on the heat dissipation of a squared metallic plate subjected to a uniform heat flux, assuming a constant temperature for the wall of the channel. In this line, [PERSON] et al. [42] performed numerical simulations to determine the temperature distribution of cooling plates with various channel shapes, also under a uniform heat flux, considering, in this case, heat transfer by forced convection at the channel surface. In this work, the thermal performance of metallic cooling plates based on channels through which a cooling fluid circulates is analyzed. The effect of the shape of channels on heat dissipation is quantified using a finite differences approach that considers forced convection heat transfer from the plate to the cooling fluid circulating through the channel. The numerical model allowed the comparison of the temperature distribution across the plate for commonly used configurations, with square and fork alike channels, and for novel configurations based on shapes inspired by nature, specifically resembling the outline of flowers. A simple geometrical evaluation of the configurations is proposed, analyzing the distribution of minimum distance of all points across the plate to the channel.