Design of Novel Cooling Systems Based on Metal Plates with Channels of Shapes Inspired by Nature

Abstract

The effect of the channel shape of aluminum plates on cooling capacity was evaluated by studying different configurations. Common shapes of the channel, such as square and fork shapes, were compared with novel configurations inspired by shapes found in nature, specifically the shape of the outline of flowers, inspired these new configurations, consisting of channels with crateriform, salverform, and cruciform shapes. The aim of the study is to evaluate the effect of the channel shape on the cooling capacity of the metal plate. To that end, all the configurations were analyzed from a geometrical point of view, determining the minimum distance of each point across the plate to the channel. A finite difference method was implemented to study both transient and steady state heat dissipation across the plates for each configuration. Even though the effect of the channel shape on the average temperature of the plate is slight, the maximum temperature, the size and location of hot spots, and the temperature homogeneity of the plate are strongly affected by the shape of the channel through which the cooling fluid is circulated. A reduction in the maximum temperature of the plate during transient cooling of around 2 °C for the crateriform and salverform channels and approximately 4.5 °C for the cruciform channel can be attained, compared to the standard configurations. The steady state heat dissipation analysis concluded that the crateriform and salverform configurations reduced the maximum variation in temperature of the common configurations by roughly 15%, whereas a reduction of approximately 28% could be reached by the cruciform configuration. Regarding the homogeneity of temperature across the plate, a reduction up to 34.5% of the index of uniform temperature can be attained using the novel configurations during the steady state refrigeration of the plate. The cruciform channel is the optimal configuration for both transient and steady state cooling processes, reducing the size and temperature of hot spots and improving the temperature homogeneity of the plate, a result already anticipated by the geometrical analysis. In fact, the main conclusions attained from the cooling study are in good agreement with the results of the geometrical analysis. Therefore, the geometrical analysis was found to be a simple and reliable method to design the shape of channels of a cooling system.

1. Introduction

Cooling plays a critical role in many applications, contributing to improving the performance and efficiency of devices generating heat during operation. The presence of hot spots and/or non-uniform distributions of temperature may lead to malfunctioning or inefficient operation of electronic equipment and a reduced longevity [1]. In fact, more than 50% of all integrated circuit failures are related to thermal issues [2]. The incessant development of electronics in search of more compact units in terms of power density requires dissipating more heat on a lower surface to maintain a high efficiency operation, constituting a challenge for cooling systems. Generally, the heat dissipated from the device to the heat sink occurs by conduction and is transferred to the environment by natural, mixed or forced convection by means of extended surfaces or fins with various geometrical designs. The tendency to minimize the size of electronic devices implies a greater difficulty for the fluid to circulate between the fins of the heat sink. Thus, this configuration may be inefficient for dissipating all the necessary heat, which in many cases can cause irreparable damage to the devices generating heat during operation or an efficiency reduction of these elements that may not accomplish the requirements of miniaturization of electronic components [3]. As an alternative, cooling systems based on forced convection by circulation of a cooling liquid through a channel are widely used due to an improved heat dissipation capacity. Depending on the size of the conduits through which the cooling fluid circulates, the channels of the cooling systems are categorized as microchannels, minichannels, or simply channels. For the former, the hydraulic diameter of the channels ranges between 10 and 200 μm [4], the second accounts for channels with hydraulic diameters of 200–3000 μm [5], whereas the latter presents hydraulic diameters above 3 mm. Microchannels provide efficient cooling and their applications typically operate with low fluid velocities, corresponding to a laminar flow, to avoid an undesired high pressure drop [6]. Consequently, the heat transfer improvement by utilization of small channels should be evaluated together with a pressure drop analysis for the fluid, to account for the pumping cost of the cooling fluid [7]. Therefore, channel-based cooling systems should find a compromise between a high heat dissipation across the solid and a low pressure drop for the fluid. Some techniques are reported in the literature to enhance thermal dissipation, focused on diminishing the thermal resistance and the pressure drop of the fluid flowing along the channel [8]. These techniques are generally devoted to studying the influence of the geometry of the heat sink microchannels [9] and the cooling fluid’s properties, including the development of new fluids with better thermal properties. In this sense, some works related to microchannels technology are focused on increasing the thermal conductivity of the cooling fluid by introducing nanoparticles with higher thermal conductivity in the fluid, generating the so-called nanofluids. This enhancement of the heat transfer properties of the nanofluids implies an increment on the convective heat transfer coefficient, improving the heat dissipation [10,11]. Nevertheless, the use of these fluids also produces an increment in the pressure drop due to the higher density and viscosity of nanofluids, penalizing the ratio of heat transfer dissipation to pumping power supply. Another problem associated with nanofluids is the sedimentation caused by a deficient mixing of the nanoparticles, which may generate agglomerates that could end up blocking the channels [12]. Recent studies show numerical simulations and analyses of the impact of employing hybrid nanofluids generated from mixtures of two different nanoparticles on heat transfer and hydrodynamics [13,14].

Several research works have focused on studying the performance of cooling systems based on metal plates with channels through which cooling fluids circulate. For instance, in photovoltaic (PV) technology, the unused solar radiation induces an increment of the PV cell temperature, which results in a reduction in the photovoltaic efficiency. A reliable solution is the refrigeration of the PV cell with channels or serpentines that enhance the heat dissipation and homogenize the cell temperature, promoting an increase in the conversion efficiency [15,16]. There exist several alternative applications of cooling plates with relevance in the last years, such as the necessity of controlling the operation temperature of lithium-ion batteries of electric vehicles and in fuel cells. In these systems, a common refrigeration method is based on indirect liquid cooling, such as a refrigeration plate with mini/micro channels to circulate a cold fluid [17,18,19]. The main advantage of these compact liquid-based refrigeration systems is that they can maintain a proper temperature of the battery, avoiding hot spots under extreme conditions, such as rapid charging and discharging at high atmospheric temperatures [20]. In most of these cooling applications, the reduction in the maximum temperature of the system is a relevant parameter, but also the homogeneity of temperature of the system plays a critical role, e.g., to improve the mismatch of PV cells [21]. The most widely studied configurations for the channels comprise parallel conduits, serpentines, and combinations of both [22,23,24]. However, the shapes of the employed configurations are rarely optimized in these studies. Regarding the optimization of the shape of the channels, Dbouk [25] reviewed the numerical approaches to optimize the design of industrial cooling applications, known as topology optimization. This technique is founded in complex mathematical expressions that minimize or maximize objective functions under several constraints. 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. Bejan and Lorente [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. Muzychka [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 Yilmaz 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. Mosa et al. [40] focused on optimizing the distribution of serpentine channels in radiant cooling panels to minimize the fluid pumping cost. Almerbati 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, Samal 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. The finite differences approach was used to study the capability of each configuration to dissipate heat under both transient and steady state cooling processes. The conclusions attained by the geometrical analysis and the heat transfer study were compared to determine the potential of the simple geometrical analysis, here proposed as a first approach to design proper channel shapes for cooling plates.

2. Theory

2.1. 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:

$$\rho V{c}_p\frac{\partial \mathrm{T}}{\partial \mathrm{t}}=q_{\mathrm{k}}+q_{\mathrm{h},\mathrm{a}}+q_{\mathrm{h},\mathrm{w}}+q_{\mathrm{in}}$$

(1)

where T is the temperature of the plate at each spatial coordinate, V represents the volume where the diffusion equation is defined, ρ and c_p represent the solid material density and specific heat, respectively, q_k is the heat rate by conduction inside the solid body, q_h,a depicts the heat rate dissipated through the external walls of the plate by natural convection to ambient air, q_h,w represents the forced convection heat rate to the cold water circulating along the channel, and q_in denotes the heat rate entering the system.

The cooling plate is assumed to be initially at T₀ = 55 °C and surrounded by ambient air at a room temperature of T_a = 25 °C. Water flows inside the channel at a constant temperature of T_w = 25 °C and a velocity of v_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_w = 25 °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_in. The boundary conditions used in this work were previously employed by Almerbati et al. [41] and Samal 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:

$$\mathrm{IC}: T\left(t=0\right)=T_0$$

(2)

$$\begin{gathered} \mathrm{BC} \mathrm{at} \mathrm{external} \mathrm{walls}: q_{\mathrm{h},\mathrm{a}}^{″}=h_{\mathrm{a}}·\left(T-T_{\mathrm{a}}\right) \\ \mathrm{BC} \mathrm{at} \mathrm{channel} \mathrm{walls}: q_{\mathrm{h},\mathrm{w}}^{″}=h_{\mathrm{w}}·\left(T-T_{\mathrm{w}}\right) \end{gathered}$$

(3)

In which q″ represents the heat flux, h_w is the forced convection coefficient from the channel wall to cold water, and h_a refers to the natural convection coefficient from the external surfaces to ambient air, which can be divided into three components: (i) h_h,ab for the natural convection coefficient of the external horizontal bottom wall of the cooling plate, (ii) h_h,at for the natural convection coefficient of the external horizontal top wall of the cooling plate, and (iii) h_h,al for the natural convection coefficient of the external lateral walls of the cooling plate. The correlation proposed by McAdams [43] for the horizontal hot surfaces in contact with ambient air in upwards and downwards directions was used to determine the convection coefficients h_h,at and h_h,ab, respectively:

$$h_{\mathrm{h},\mathrm{at}}=\left\{\begin{array}{c}\frac{k_{\mathrm{a}}}{L_{\mathrm{c}}}0.54{\mathrm{Ra}}_{L\mathrm{c}}^{1/4} \mathrm{if} {10}^4\le {\mathrm{Ra}}_{L\mathrm{c}}\le {10}^7\\ \frac{k_{\mathrm{a}}}{L_{\mathrm{c}}}0.15{\mathrm{Ra}}_{L\mathrm{c}}^{1/3} \mathrm{if} {10}^7\le {\mathrm{Ra}}_{L\mathrm{c}}\le {10}^{11}\end{array}\right. \mathrm{if} \mathrm{air} \mathrm{is} \mathrm{over} \mathrm{the} \mathrm{plate}$$

(4)

$$h_{\mathrm{h},\mathrm{ab}}=\frac{k_{\mathrm{a}}}{L_{\mathrm{c}}}0.27{\mathrm{Ra}}_{L\mathrm{c}}^{1/4} \mathrm{if} {10}^5\le {\mathrm{Ra}}_{L\mathrm{c}}\le {10}^{10} \mathrm{if} \mathrm{air} \mathrm{is} \mathrm{under} \mathrm{the} \mathrm{plate}$$

(5)

where L_c is the characteristic length of the horizontal surface, i.e., area over perimeter L_c = L/4, k_a is the thermal conductivity of air, and Ra_Lc is the Rayleigh number referred to this characteristic length, defined as follows:

$${\mathrm{Ra}}_{Lc}=\frac{g·{\beta }_a\left(T-T_a\right)L_c^3}{{\alpha }_a·{\upsilon }_a}$$

(6)

where β_a is the thermal expansion coefficient associated with air, $g$ is the gravity acceleration, and ${\alpha }_a$ and ${\upsilon }_a$ 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_h,al, obtained from the Churchill and Chu [44] correlation for vertical plates, which reads:

$$h_{\mathrm{h},\mathrm{al}}=\frac{k_{\mathrm{a}}}{L_c}\left(0.825+\frac{0.387·{\mathrm{Ra}}_{L\mathrm{c}}^{1/6}}{{\left(1+{\left(\frac{0.492}{{\mathrm{Pr}}_{\mathrm{a}}}\right)}^{9/16}\right)}^{8/27}}\right){ \mathrm{if} \mathrm{Ra}}_{L\mathrm{c}}>1$$

(7)

where L_c is the characteristic length of the vertical surface, which, in this case, corresponds to the plate thickness L_c = s, and Pr_a is the Prandtl number for air, defined as Pr_a = μ_a c_pa/k_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 T₀ = 55 °C and the constant ambient temperature of T_a = 25 °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 T₀ to T_a, obtaining average values of h_h,at = 14 W/(m²K) for the top surface, h_h,ab = 7 W/(m²K) for the bottom surface, and h_h,al = 19 W/(m²K) 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 mm, the velocity of water v_w = 1 m/s, and the properties of water at T_w = 25 °C, the Reynolds number for flowing water, defined as Re = ρ_f v_w d/μ_f, is Re = 6699. For such a high value of the Reynolds number, a turbulent regime is expected. Thus, for turbulent flow of water in the channel, the most widely used correlation is that of Gnielinski [45], valid for Reynolds number from 4·10³ to 10⁶, which reads:

$$h_{\mathrm{h},\mathrm{w}}=\frac{k_{\mathrm{f}}}{d}\frac{\left(\xi /8\right)\left(\mathrm{Re}-1000\right){\mathrm{Pr}}_{\mathrm{f}}}{1+12.7{\left(\xi /8\right)}^{0.5}\left({\mathrm{Pr}}_{\mathrm{f}}^{2/3}-1\right)}\left[+{\left(\frac{d}{L_{\mathrm{ch}}}\right)}^{2/3}\right] \mathrm{if} 4\times {10}^3 \mathrm{Re} {10}^6, 0.5 {\mathrm{Pr}}_{\mathrm{f}}20$$

(8)

where d and L_ch are the diameter and length of the channel, respectively, k_f is the thermal conductivity of the fluid, Pr_f is the Prandtl number of the fluid, and ξ is the friction factor, which can be estimated for turbulent flows with the expression proposed by Petukhov [46]:

$$\xi ={\left(0.079 \ln \left(\mathrm{Re}\right)-1.64\right)}^{-2}$$

(9)

Taler [47] proposed a power-type correlation to estimate the Nusselt number of turbulent flows, valid for Reynolds numbers between 3·10³ and 10⁶, from which the convection coefficient can be calculated as follows:

$$h_{\mathrm{w}}=\frac{k_{\mathrm{f}}}{d}0.00881{\mathrm{Re}}^{0.8991}{\mathrm{Pr}}_{\mathrm{f}}^{0.3911} \mathrm{if} 3\times {10}^3\le \mathrm{Re}\le {10}^6, 2\le {\mathrm{Pr}}_{\mathrm{f}}\le 1000$$

(10)

The values of the convection coefficient predicted by the correlation of Taler [47] are similar, although slightly lower, to those obtained from the correlation of Gnielinski [45]. In this work, an average value for the forced convection coefficient to the fluid circulating through the channel of the cooling system will be used, calculated as the mean of the values derived from the correlations of Taler [47] and Gnielinski [45]. Assuming an average velocity of water of v_w = 1 m/s and evaluating the properties of water at T_w = 25 °C, the forced internal convection coefficient results in a value of h_w = 5263.5 W/(m²K).

2.2. Finite Differences Discretization

The diffusion equation, Equation (1), was solved using a transient finite difference implicit numerical scheme. For this purpose, the cooling plate was divided in cells of size Δx in the horizontal direction, and Δy in the vertical direction, so that the temperature can be evaluated at N nodes on the surface of the cooling plate. In view of the square shape of the cooling plate and for the sake of simplicity of the numerical model, Δx and Δy were chosen of the same size, Δx = Δy. Hence, the total number of nodes is distributed across the plate in a square mesh N = N_r·N_r. As indicated in Section 2.1, the temperature distribution is considered uniform in the direction perpendicular to the cooling plate. Nevertheless, the size Δz of the plate thickness direction shall be accounted for in the numerical model. Therefore, the diffusion equation, Equation (1), is solved using a transient finite difference scheme in a total of N nodes along the surface of the cooling plate following Equation (11):

$$\rho ∆x∆y∆z{c}_p\frac{T_{m,n}^{\left(p+1\right)}-T_{m,n}^{\left(p\right)}}{∆t}=q_{\mathrm{k}}^{\left(p+1\right)}+q_{\mathrm{h},\mathrm{a}}^{\left(p+1\right)}+q_{\mathrm{h},\mathrm{w}}^{\left(p+1\right)}+q_{\mathrm{in}}$$

(11)

where the superscript p indicates the time instant and the subscripts m and n refer, respectively, to the nodes in the horizontal and vertical directions of the cooling plate. Consequently, node (1, 1) corresponds to the upper left corner of the plate and node (N_r, N_r) denotes its lower right corner. In Equation (11), the heat transferred by conduction, convection, and the heat entering the system are discretized for each node.

For each node in the system, the heat transferred by conduction to the node can be expressed as the conduction contribution from any adjacent node as follows:

$$q_k^{\left(p+1\right)}=q_{k_{m-1,n}}^{\left(p+1\right)}+q_{k_{m+1,n}}^{\left(p+1\right)}+q_{k_{m,n+1}}^{\left(p+1\right)}+q_{k_{m,n-1}}^{\left(p+1\right)}$$

(12)

where conduction can be evaluated considering Fourier’s law and the lateral area through which conduction occurs. For instance, for the node situated in the right-hand-side of node (m, n), i.e., node (m + 1, n), the contribution to Equation (12) would read:

$$q_{k_{m+1,n}}^{\left(p+1\right)}=k∆y∆z\frac{T_{m+1,n}^{\left(p+1\right)}-T_{m,n}^{\left(p+1\right)}}{∆x}$$

(13)

Convection through the top and bottom surfaces of the cooling plate can be also expressed in a finite difference form as:

$$q_{\mathrm{h},\mathrm{a}}^{\left(p+1\right)}=h_{\mathrm{h},\mathrm{a}}∆x∆y\left(T_{\mathrm{a}}-T_{m,n}^{\left(p+1\right)}\right)$$

(14)

where $h_{\mathrm{h},\mathrm{a}}=h_{\mathrm{h},\mathrm{at}}$ is at the top surface of the plate and $h_{\mathrm{h},\mathrm{a}}=h_{\mathrm{h},\mathrm{ab}}$ is at its bottom surface.

In cases in which the (m, n) node is located in the lateral or corner cells of the cooling plate, natural convection through the top and bottom faces would occur only in half or one quarter of the surface of the node, respectively, and lateral vertical convection appears. This additional term can be expressed, for example, for a side lateral node as:

$$q_{\mathrm{h},\mathrm{a}}^{\left(p+1\right)}=h_{\mathrm{h},\mathrm{al}}∆y∆z\left(T_{\mathrm{a}}-T_{m,n}^{\left(p+1\right)}\right)$$

(15)

where ΔyΔz is the surface through which heat is dissipated. Finally, nodes in which the water channel is present have an additional forced convection term due to the water cooling, $q_{\mathrm{h},\mathrm{w}}^{\left(p+1\right)}$. In this case, the heat transfer surface between the node and the channel is 2ΔyΔz, neglecting the curvature of the channel, since the channel is in the middle plane of the plate.

$$q_{\mathrm{h},\mathrm{w}}^{\left(p+1\right)}=h_{\mathrm{w}}2∆x∆y\left(T_{\mathrm{w}}-T_{m,n}^{\left(p+1\right)}\right)$$

(16)

All the above equations can be combined for each cell, accounting for the different heat rates occurring in the cells, and rewritten in a dimensionless form using the Biot, Fourier, and mesh size dimensionless numbers:

$$\mathrm{Bi}=\frac{h∆z}{k}$$

(17)

$$\mathrm{Fo}=\frac{\alpha ∆t}{∆x^2}$$

(18)

$$\mathsf{\delta }=\frac{∆x}{∆z}$$

(19)

where the conduction length in the vertical direction of the plate, Δz, can vary, accounting for the presence or not of the cooling channel. If no cooling channel is present in node (m, n), Δz coincides with the plate thickness, Δz = s, and heat is transferred by conduction through the whole thickness of the plate. However, the presence of the cooling channel of diameter d diminishes the length through which the cooling plate conducts heat in the direction perpendicular to the thickness, obtaining a thickness of those nodes where the channel is located of Δz = s − d.

With all the above, for an internal node with heat dissipated by natural convection through the top and bottom surfaces and to the water-cooling channel, always considering that Δx = Δy, Equation (11) can be rewritten as:

$$\begin{gathered} T_{m,n}^{\left(p+1\right)}\left[1+4\mathrm{Fo}+{\mathrm{Fo}\mathsf{\delta }}^2\left({\mathrm{Bi}}_{\mathrm{a}}+{\mathrm{Bi}}_{\mathrm{w}}\right)\right]-\mathrm{Fo}\left(T_{m-1,n}^{\left(p+1\right)}+T_{m+1,n}^{\left(p+1\right)}+T_{m,n+1}^{\left(p+1\right)}+T_{m,n-1}^{\left(p+1\right)}\right) \\ -{\mathrm{FoBi}}_{\mathrm{a}}{\mathsf{\delta }}^2{T}_{\mathrm{a}}-2{\mathrm{FoBi}}_{\mathrm{w}}{\mathsf{\delta }}^2{T}_{\mathrm{w}}-\frac{q_{\mathrm{in}}}{\rho c_p}\frac{∆t}{∆x∆y∆z}=T_{m,n}^{\left(p\right)} \end{gathered}$$

(20)

where Bi_a and Bi_w are, respectively, the Biot numbers for air and water, and are referred to with the corresponding conductivities, convection coefficients, and the conduction characteristic length in the vertical direction of the cooling plate, Δz, at the corresponding node. Hence, the fundamental equations of the finite difference model would be only affected by the time-independent specific dimensionless numbers at each node. With this, note that, for instance, in case no water cooling channel is present in the (m, n) node of Equation (20), h_w = 0 W/(m²K) implies Bi_w = 0 and heat would be only dissipated by conduction and natural convection through the top and bottom surfaces of the plate for internal nodes, with an additional natural convection term through the lateral surface for cases in which the nodes are located on the lateral or corners of the cooling plate.

A finite difference expression of the diffusion equation presented in Equation (1) can be constructed in the form of Equation (20) for each node, accounting for the heat conduction transfer with the neighboring nodes and the specificities of lateral nodes, corner nodes, and nodes situated at the channel locations. With this, a matrix of the finite element coefficients, $\mathit{A}$, of size N × N can be constructed, in such a manner that, for each time step:

$$\mathit{A}{\mathit{T}}^{\left(p+1\right)}=T^{\left(p\right)}+\mathit{a}{T}_{\mathrm{a}}+\mathit{b}{T}_{\mathrm{w}}+\mathit{c}{q}_{\mathrm{in}}$$

(21)

where T is a vector of N terms with the temperature at each node of the cooling plate and $\mathit{a}$, $\mathit{b},$ and $\mathit{c}$ are, respectively, vectors of length N representing, at each node of the plate, the finite element coefficients for the heat transfer by natural convection to the air, by forced convection to the water and the heat rate entering the system, respectively. Subsequently, matrix $\mathit{A}$ can be inverted to obtain the transient evolution of the temperature at each node, knowing the temperature distribution at time p and the dimensionless numbers at each node of the system.

The numerical model was solved with a time step of Δt = 10⁻³ s and a mesh size of Δx = Δy = 2.2 mm, i.e., considering N_r = 68 nodes in each direction of the cooling plate and a total number of nodes of N = 4624 equally distributed across the plate, guaranteeing the time-step and mesh independency of the results. A physical time of around 60 s was calculated for the evaluation of the transient cooling of the plate when no heat was introduced in the system, q_in = 0 W. For cases in which q_in ≠ 0 W, the simulations lasted a sufficient time so that steady state conditions were reached, with a tolerance of the maximum change in the temperature of all nodes between successive time steps below 10⁻⁵ K.

Figure 1 shows a flow chart indicating the numerical procedure used to solve the finite difference scheme both in transient (blue lines in the figure) and in steady state (red lines in the figure) conditions. Firstly, the plate and channel geometries are generated in a CAD software and a top view is imported into the numerical scheme. Secondly, the domain is meshed and the channel indexes are identified. Then, the physical properties of the plate and fluids, the initial conditions, and the boundary conditions are introduced in the scheme. This allows for evaluation of the dimensionless parameters of the problem used to construct the matrix and vectors with the finite element coefficients, $\mathit{A},\mathit{a},\mathit{b}$, and $\mathit{c}$. Inversion of matrix $\mathit{A}$ allows for computing the temperature distribution at the subsequent time step. For the transient cooling calculations, the temperature of all nodes of the plate is calculated at time step p + 1 from their temperatures in the previous time step p until the maximum physical time of the simulation t_max is attained. In this case, the time evolution of the temperature of each node of the plate is stored to analyze the transient refrigeration process. In the case of steady state cooling, the temperature of all nodes is determined at time step p + 1 from the previous time step p until the temperature variation between consecutive iterations is below a tolerance of 10⁻⁵ K for all points. For the steady state refrigeration calculations, the temperature distribution of the plate once steady state conditions are reached is stored.

3. Heat Transfer Configurations

The cooling systems analyzed consist of an aluminum plate with a channel drilled inside. The properties considered for the aluminum plate for the heat transfer computations are density ρ = 2650 kg/m³, specific heat c_p = 890 J/(kg·K), and thermal conductivity k = 113 W/(m·K). All the plates are squared and have the same dimensions, with a length of L = 15 cm and a thickness of s = 1 cm. The cross section of the channel is circular, with a diameter of d = 6 mm, and the length of the channel is L_ch = 43.9 cm for all configurations. The shape of the channel along the plate differs for the different configurations analyzed. However, the diameter and length of the channel are the same for all configurations to allow the direct comparison of different cases. The channel is located at the middle plane of the plate thickness, with its inlet and outlet centered in a side of the plate and separated 3.75 cm for all cases studied. Therefore, the external heat transfer surfaces of the plate in contact with ambient air and the internal heat transfer surface of the channel in contact to the cooling fluid are constant for all the configurations considered.

A standard configuration for a channel of a cooling system over a surface is a squared shape, keeping a constant distance to all sides of the surface. This square shaped channel can be observed in Figure 2, where the channel is placed 1.75 cm apart from the plate sides. Another usual configuration for the channel is that typically used for electric resistors, with a fork shape. A smooth fork shape channel configuration, keeping the same inlet, outlet, and length of the channel as the square configuration, is also included in Figure 2.

The performance of the traditional configurations presented in Figure 2 was compared to that of the novel designs inspired by the shapes of the outline of flowers. The novel configurations of the cooling system channel were named after the shape of the flowers that inspired their design, using the botanic terminology proposed in Hickey and King [48]. Figure 3 shows the three novel configurations proposed, which include specifically crateriform, salverform, and cruciform shapes for the channel of the cooling system. All the configurations based on flowers’ outlines kept the same inlet, outlet, diameter, and length of channel as the common configurations depicted in Figure 2 so that the surface of the plate in contact with the cooling fluid is the same for all configurations. Therefore, the thermal results obtained from the analysis of the different configurations here proposed can be directly compared to evaluate the effect of the channel shape on the performance of each cooling device.

4. Results and Discussion

The results of the analysis of the different configurations of the cooling system are compared and discussed in this section. First, all configurations were analyzed from a geometrical point of view. Then, the finite differences numerical method was used to evaluate the capacity of each configuration to cool the plate under transient and steady state heat transfer conditions.

4.1. Geometrical Analysis

All configurations were analyzed from a purely geometrical point of view by calculating the minimum distance of each point of the plate to the channel considering a 2D domain. A grid of 338 × 338 points was built on the 2D domain of the plate and the minimum distance from each point to the cooling channel was computed. Contour plots of the distance from each point across the plate to the channel are depicted in Figure 4 for all the configurations studied. The square shape configuration presents the maximum distance to the channel for points located at the center of the plate, whereas for the fork shape configuration the maximum distances are found at the corners placed at the side where the inlet and outlet of the channel are situated. The maximum distance obtained for both standard configurations is similar; however, using fork shape channels results in a more homogeneous distribution of the distance across the plate compared to a square shape configuration. All the novel configurations proposed, based on the outline of flowers, result in lower maximum distances to the channel. The crateriform configuration shows maximum distances to the channel at the four corners and the center of the plate, while the maximum distances appear at the center of the plate for the salverform channel. Among all configurations, the cruciform channel minimizes the distances for all points of the plate to the channel, reducing the maximum value and inducing a more homogeneous distance to the channel across the whole plate compared to the rest of configurations considered.

The main characteristic parameters of the distribution of distances from all points across the plate to the channel for each configuration, including the maximum distance r_max, the average distance r_m, and the standard deviation of the distance ${\sigma }_r$, are reported in

Table 1. Characteristic values of the geometrical analysis of the different configurations.

Configuration	rmax (cm)	rm (cm)	σr (cm)
Square	5.34	1.26	1.17
Fork	5.34	1.14	0.98
Crateriform	4.93	1.22	1.07
Salverform	4.29	1.13	0.94
Cruciform	3.68	1.11	0.88

. The maximum distance of the common configurations, i.e., square and fork shape channels, coincides, obtaining a significant reduction of this parameter for all the novel configurations proposed. This substantial reduction in the maximum distance to the channel is very relevant for a heat transfer application since the maximum temperature of the plate will be proportional to this distance due to the proportionality of the heat rate by conduction through the solid with the distance. Regarding the average distance of all points of the plate to the channel, the highest value was obtained for the square configuration, whereas in the case of the fork shape channel, the average distance is lower. Thus, even though the maximum distances of the square and fork configurations are similar, the average distance is higher for the square shape channel due to the larger region of points with high distance to the channel close to the center of the plate in this case. The average distance for the crateriform configuration is even larger than that of the fork shape channel, although the maximum distance is lower for the former configuration. The salverform channel leads to a similar value of the average distance to the fork shape configuration, whereas this value is lower for the cruciform channel. The homogeneity of the distribution of distances to the channel across the whole plate can be analyzed based on the standard deviation of the distance ${\sigma }_r$. The less uniform distribution of distances to the channel corresponds to the square configuration, for which the standard deviation of the distance is maximum. In contrast, the more homogenous distribution of distances to the channel was obtained for the cruciform channel, which resulted in the minimum value of the standard deviation of the distances from all points of the plate to the channel.

In view of the characteristic values of the distribution of distances from all points across the plate to the channel, included in Table 1, the novel configurations proposed, or at least some of them, could reduce the maximum, average, and standard deviation of the distances of all points of the plate to the channel through which the cooling fluid circulates. However, the reduction attained for these three parameters with regard to the square configuration differs, obtaining a maximum reduction of 31.1% for the maximum distance, 11.9% for the average distance, and 24.8% for the standard deviation of the distance. Hence, during the operation of the cooling system, the maximum temperature of the plate is expected to be reduced more significantly than the average plate temperature when using the novel configurations of the channel compared to the standard configurations.

4.2. Heat Transfer

The finite differences method presented in Section 2.2 was applied to determine the performance of all configurations of the cooling plate when flowing cold water through the cooling channel under both transient and steady state conditions.

4.2.1. Transient Heat Transfer

The cooling capacity of the different configurations was evaluated by solving a transient heat transfer problem with the finite differences numerical model. For the transient cooling study, no heat flux on the plate surface was considered, thus q_in in Equation (1) is zero. The initial temperature of the whole plate was fixed to 55 °C and heat was dissipated to ambient air and to water flowing through the channel. The temperature of both ambient air and water in the channel was set at 25 °C. Therefore, for a sufficiently long time, the whole plate would reach a final temperature of 25 °C.

A total physical time of 60 s of the cooling process was studied for each configuration. Figure 5 shows the evolution of temperature across the plate for all configurations as time progresses during the first 15 s of the transient cooling process, showing snapshots of the temperature distribution in time intervals of 2 s. After 1 s of cooling, the temperature distribution is similar in all cases, with most of the plate at the initial temperature of 55 °C and a clearly visible channel shape at a lower temperature. However, as time progresses, heat is principally dissipated by forced convection to the water-cooling channel, and the regions close to the channel cool down by conduction. Remarkable differences can be observed in the temperature distribution of the plate for the different configurations studied as transient cooling evolves. The heat dissipation occurs from a large region of high temperature at the center of the plate for the square configuration, whereas heat dissipation mainly takes place from the corners closest to the inlet and outlet of the channel for the fork configuration, which are the points with the largest distance to the channel according to the geometrical analysis. In the case of the crateriform configuration, heat dissipation happens from the center and the four corners of the channel, while the dissipation behavior of the salverform channel plate is similar to that of the square configuration, showing a higher temperature zone at the plate center. Heat dissipation seems to be faster and more homogeneous for the cruciform channel, which is apparently the best option to refrigerate a hot plate by circulating a cold fluid. Noticeably, the temperature distribution maps depicted in Figure 5 are similar to the contour plots of the distance to the channel included in Figure 4, which was an expected result since heat dissipation is affected by the distance to a sink point, i.e., the cooling channel. This confirms the usefulness of the simple geometrical analysis discussed in Section 4.1.

The time evolution of the average and maximum temperature of each configuration can be found in Figure 6 for a time period of 60 s. As anticipated from the results of the geometrical analysis of the configurations, the evolution of the average temperature of all configurations is similar, obtaining slightly lower values for the cruciform shape of the channel, in agreement with the results of the geometrical study. In all cases, an inverse exponential approximation of the average temperature of the plate from the initial temperature to the ambient and cooling fluid temperature was obtained. Higher variations between different configurations were observed for the evolution of the maximum temperature of the plate, as predicted by the geometrical analysis of the configurations. The evolution of the maximum temperature for square and fork shape channel almost collapse, obtaining the highest values among all configurations. This is also in accordance with the geometrical analysis, in which both configurations presented the same maximum distance to the channel, which is higher compared to that of the novel configurations proposed, as reported in Table 1. Lower values for the maximum temperature of the plate were obtained for the crateriform and salverform channels compared to the standard configurations with square and fork shape channels. Even though the maximum distance to the channel was higher for the crateriform than for the salverform configuration, the time evolution of the maximum temperature obtained for crateriform and salverform channels is very similar. This may be attributed to the larger extension of the zone with large distance to the channel at the center of the salverform configuration, from which the dissipation of heat to the channel could be slower than from the smaller regions of high distance to the channel of the crateriform configuration located at the center and corners of the plate. The lowest maximum temperature of the plate was found for the cruciform channel, which is the best configuration among those studied in this work, as previously concluded from the geometrical analysis.

Considering the time evolution of the maximum temperature of the plate during the cooling process shown in Figure 6, the novel configurations proposed, i.e., crateriform, salverform, and cruciform channels, improves the performance of the standard configurations, i.e., square and fork shape channels, during transient heat transfer. To quantify the speed of the cooling process, the time required to attain the mean temperature between the initial and final temperature of the plate, i.e., a temperature of 40 °C, was determined. A similar value was required for the square and fork configurations, with values of 21.8 and 21.5 s, respectively. The time required to reduce the maximum temperature of the plate to 40 °C was decreased to 19.3 and 19.0 s when using the salverform and crateriform configurations, respectively. Among all the configurations studied, the fastest reduction in the maximum temperature of the plate was found for the cruciform configuration, requiring a time of only 16.3 s. Thus, a maximum reduction in the time required to reduce the maximum temperature of the plate from 55 °C to 40 °C of 25.4% was attained using the cruciform configuration.

The maximum temperature of the plate can also be reduced using the novel configurations proposed compared to the standard configurations. This reduction in the maximum temperature was quantified by calculating the reduction in maximum temperature attained from each of the novel configurations with respect to the average of the usual configurations. Figure 7 shows the time evolution of the maximum temperature reduction attained by the novel configurations. The maximum temperature obtained during transient cooling of the plates can be reduced up to 2 °C by the crateriform and salverform configurations and up to 4.5 °C by the cruciform configuration. This temperature reductions are remarkable considering that the plate is initially heated up to only 55 °C, that is, only 30 °C above ambient temperature. Therefore, the maximum temperature reduction reached by the crateriform and salverform channels is around 6.6%, while for the cruciform channel it is approximately 15%. It should be noticed that this reduction is attributed only to the difference in shape of the channel since the length and diameter of the channel is the same for all configurations.

4.2.2. Steady State Heat Transfer

Heat transfer was also evaluated for all configurations under steady state conditions, considering a total power input q_in uniformly distributed across the plate. The total heat rate input was varied from 0 to 200 W and the steady state temperature distribution for each shape of the channel was determined. The plate was assumed to be initially at ambient temperature, i.e., 25 °C, and the temperature of all points of the plates was calculated as time progressed until a steady state temperature distribution was reached. The temperature distribution of the plate for various thermal power inputs, from 25 to 200 W in intervals of 25 W, is depicted in Figure 8 for each configuration. Obviously, the temperature of the plate is proportional to the thermal power input for all configurations, as can be anticipated in view of Equation (1). However, both the temperature distributions and their values vary substantially among the different configurations of the channel considered. The square configuration shows a great zone of high temperature at the center of the plate, whereas the fork shape channel induces the appearance of the maximum temperature at the corner of the plate at the side where the inlet and outlet of the channel are located. Regarding the novel configurations, the crateriform configuration is characterized by the presence of small zones of high temperature at the center and the four corners of the plate, while the salverform configuration shows a maximum temperature zone at the center of the plate, similar, although smaller, in extension compared to the high temperature regions of the square configuration. In contrast to the rest of configurations, the cruciform shape shows a more homogeneous distribution of temperature across the plate, avoiding the presence of hot spots in the plate. The temperature distribution maps obtained for steady state cooling conditions resemble again the contour maps of distance to the channel reported in Figure 4. Thus, the geometrical analysis can be used as a first design tool to determine optimal configurations of channels for heat transfer applications.

The evolution of the mean and maximum temperatures of the plate with the heat rate input can be observed in Figure 9. Both the average and the maximum temperature of the plate present a linear increase with the thermal power input for all configurations, as expected in view of Equation (1). The highest average temperature was obtained for the square shape channel due to the large zone of high temperature at the center of the plate. Among the novel configurations proposed, the highest average temperature was obtained for the crateriform configuration, whereas the lowest values were attained for the cruciform configuration, in agreement with the results obtained from the geometrical analysis of the different channel shapes. In fact, the increase in the average temperature of the plate over the ambient temperature can be reduced by 6.5% using the cruciform configuration compared to the squared channel plate. Concerning the maximum temperature of the plate, even though the maximum distance to the channel was the same for the square and the fork shape configurations, the maximum temperature attained in the plate under steady state conditions was slightly higher for the square shape channel than for the fork configuration, confirming again the importance of the extension of the region of maximum distance to the channel. Similar values of the maximum temperature of the plate under steady state conditions were reached for the crateriform and the salverform configurations due to the higher concentration of points, with higher distance to the channel for the salverform configuration, close to the plate center, than for the crateriform configurations, for which these distant points to the channel are distributed at the center and the four corners of the plate. Once more, the lowest values of the maximum temperature of the plate attained for steady state cooling were obtained for the cruciform configuration, confirming again the higher capability of this channel shape to cool down the plate and the applicability of the simple geometrical analysis proposed to predict the optimal shape for heat transfer purposes.

Considering the linear increase in the maximum temperature of the plate with the heat flux supplied, i.e., q_in″ = q_in/L², the dimensionless temperature proposed by Samal et al. [42], $\overline{T}$, was used to quantify the effect of the shape of the channel on the performance of the cooling plate. The dimensionless temperature $\overline{T}$ is defined as follows:

$$\overline{T}=\frac{T_{\max }-T_{\mathrm{w}}}{q_{\mathrm{in}}^{″}s/k}$$

(22)

where T_max is the maximum temperature of the plate, T_w is the temperature of the cooling water, q_in″ is the input heat flux, s is the plate thickness, and k is the thermal conductivity of the solid plate. The results for the dimensionless temperature can be found in

Table 2. Dimensionless temperature for each configuration.

Configuration	$\overline{\mathit{T}}$ (−)
Square	13.4
Fork	13.2
Crateriform	11.8
Salverform	12.0
Cruciform	10.5

. A similar value of $\overline{T}$ was obtained for the standard configurations, i.e., square and fork shapes of the channel. However, the value of the dimensionless temperature was lower for all the novel configurations proposed. The average dimensionless temperature obtained for the common configurations can be reduced by 11.3, 9.8, and 21.1% using the crateriform, salverform, and cruciform configurations, respectively. Therefore, a significant reduction in the dimensionless temperature can be attained by using the novel configurations, especially for a cruciform shape of the cooling channel.

From a practical point of view, it is also important to evaluate the difference between the maximum and the minimum temperature of the plate as a measure of the uniformity of the temperature distribution in the plate. The maximum variation in temperature of the plate, defined as the difference between the maximum temperature T_max and the minimum temperature T_min, is plotted versus the thermal power input in Figure 10a for all the configurations studied. The maximum temperature variation obtained for the square and fork shape channels collapse, increasing linearly with the thermal power input. A substantial reduction in the maximum variation in temperature can be reached using the proposed novel shapes for the channel. Slightly higher values for the maximum variation in temperature in the plate were obtained for the crateriform compared to the salverform configuration, while the cruciform channel resulted again in the lowest temperature variations among all configurations analyzed. The crateriform and salverform configurations reduced the maximum temperature variation of the standard configurations, i.e., the square and fork configuration, by roughly 15%, whereas a reduction of approximately 28% could be attained by the cruciform configuration. These results agree with the maximum distances obtained from the geometrical analysis of the configurations and the main tendencies could be again directly inferred from the simple geometrical study carried out in Section 4.1. In fact, the slope of the linear increase of the maximum variation in temperature with the thermal power input under steady state conditions for each configuration is related to the maximum distance to the channel.

Depending on the application, it is possible that not only the high temperature but also the extension of the hot spots in which this high temperature occurs is relevant. To account for both the extension of the region of the plate with high temperature and the values of the high temperature, the Index of Uniform Temperature (IUT) proposed by Rahgoshay et al. [49] was determined. The IUT is a measure of the average deviation of the temperature from the mean values, and is defined as follows:

$$\mathrm{IUT}=\frac{\iint \left|T-T_{\mathrm{m}}\right|d{A}_p}{\iint d{A}_p}$$

(23)

The variation of IUT with the heat rate input can be found in Figure 10b. The maximum values of IUT were obtained for the square configuration as a combined effect of the high temperatures and large extension of high temperature region in the plate. However, the values of IUT for the fork, crateriform, and salverform channels are comparable, even though different values for the highest temperatures and extensions of the zone of high temperature were found for these three configurations. The fork configuration is characterized by two hot spots of high temperature at the corners close to the inlet and outlet of the channel. In contrast, the crateriform configuration shows five zones of high temperature, characterized by lower values compared to the fork channel, whereas the salverform configuration presents a larger zone of high temperature at the plate center, with values also lower than the fork configuration. Regarding the cruciform configuration, the values obtained for the IUT were also lower than for the rest of cases studied, confirming again the potential of the cruciform channel to improve the cooling capability of the plate. In fact, the value of the IUT can be reduced by 34.5% using the cruciform configuration compared to the square configuration.

5. Conclusions

The performance of a cooling system consisting of an aluminum plate with an interior channel through which cold water flows was numerically evaluated. Various shapes of the channel were tested, comparing commonly used shapes such as square or fork-alike channels with novel configurations inspired by nature, specifically by the outline of flowers: crateriform, salverform, and cruciform channels. The inlet and outlet location, diameter, and length of the channel were the same for all configurations, so that the effect of the channel shape from a thermal point of view could be quantified by direct comparison of the results obtained for each case. A geometrical analysis of the configurations, based on calculating the minimum distance of all points across the plate to the channel, was carried out. The geometrical analysis concluded that the maximum distance to the channel of the standard configurations, i.e., square and fork channels, coincided and was substantially reduced for the novel configurations. Among all configurations tested, the cruciform channel constitutes the optimum configuration in terms of the minimum distance of the points of the plate to the channel.

The cooling capacity of each configuration was quantified based on a finite differences approach that solves the diffusion equation in the plate, considered as a 2D domain. Transient cooling was studied, assuming a high initial temperature of the plate and heat was dissipated to both ambient air at the external surfaces of the plate and flowing cold water at the surface of the channel. The time evolution of the maximum temperature of the plate showed that the cooling capacity was improved by the proposed novel configurations with regard to the standard shapes of the channel. Even though the time evolution of the average plate temperature was only slightly affected by the channel shape, the maximum temperature of the plates can be reduced up to 2 °C by the crateriform and salverform configurations and up to 4.5 °C by the cruciform configuration during transient cooling. This maximum temperature reduction reached by the crateriform and salverform channels corresponds to roughly 6.6%, while for the cruciform channel a reduction of around 15% was attained. In addition, the sizes of the hot spots located across the plate during transient refrigeration were substantially reduced when using novel configurations with channel shapes based on the outline of flowers, especially for the cruciform configuration. Therefore, the novel configurations tested showed a significant effect on the hot spots and the homogeneity of temperature across the cooling plate.

Steady state heat dissipation was also analyzed, considering a uniform thermal input across the plates. In this case, both the maximum temperature attained by each configuration and the extension of the high temperature regions were considered by an index of uniform temperature. Again, the effect of the channel shape on the average temperature of the plate during steady state refrigeration was slight. However, the plate temperature homogeneity and the presence of hot spots is strongly affected by the shape of the channels through which the cooling fluid is circulated. These effects were quantified by calculating the average dimensionless temperature and the index of uniform temperature, obtaining reductions in the average dimensionless temperature of 9.8, 11.3, and 21.1% using the salverform, crateriform, and cruciform channels compared to the common configurations, respectively. Regarding the index of uniform temperature, a maximum reduction of 34.5% can be attained by the cruciform configuration compared to the square channel plate, confirming the increased homogeneity of the plate temperature for this novel configuration. Both for transient and steady state cooling processes, the cruciform configuration was proven to be the optimal design. In fact, the main conclusions attained from the heat transfer results are in excellent agreement with the results obtained from the simple geometrical analysis of the configurations. Therefore, the geometrical analysis was found to be a straightforward and useful tool to design optimal shapes for the heat transfer characteristics of channels in cooling systems.