Thermal-hydraulic performance of flat-plate microchannel with fractal tree-like structure and self-affine rough wall

ABSTRACT Inspired by the natural bifurcating structures, tree-like microchannels have been applied for microelectronics cooling. In order to understand the thermal-hydraulic performance of a flat-plat tree-like microchannel, successive branching ratios of tree-like structure are optimized based on minimization of flow resistance. It is shown that the optimal successive diameter ratio of symmetrical and dichotomous structures under volume constraint follows Murray’s law, while the optimal successive length ratio under the constraint of fixed channel area follows the power law 2−2/3. A mathematical model of convection in disc-shaped heat sink composed of a tree-like microchannel with self-affine rough surface is developed by the fractal geometry and finite element method. The flat-plate tree-like micro-channel with optimal successive diameter and length ratio shows enhanced thermal-hydraulic performance. The Nusselt number of the flat-plat tree-like micro-channel increases with the inlet Reynolds number and the self-affine fractal dimension of the rough wall. The present optimization method and mathematical model for the flat-plate tree-like microchannel shed light on the design of flat-plate micro-channel heat sinks and flow channel in fuel cell among other potential cooling applications.


Introduction
With the development of miniaturization and integration of micro-electrical systems (MEMS), high-efficiency cooling technology has become one of the key design issues because of sharp rise of power density of modern integrated circuits (Craighead, 2000;Fan et al., 2022;Le et al., 2022;Sisó et al., 2021).As one of the most attractive cooling technologies, microchannel cooling has attracted extensive attention (Liu, Bahrami, et al., 2022;Tuckerman & Pease, 1981;Xu, Sasmito, et al., 2016;Zhai et al., 2016).Inspired by the efficient natural bifurcation systems with evolutionary advantage such as the trunk veins of plants, the vasculature of biological tissues, the river basins, the porous seepage networks of underground oil reservoirs, etc., the tree-like microchannel has been proposed to improve the thermal-hydraulic performance of traditional parallel and serpentine microchannels (Pence, 2003;Senn & Poulikakos, 2004;Zhang et al., 2019).It has been shown by numerous studies that the tree-like microchannel has reduced the flow resistance, displays improved uniformity of temperature distribution and reduced the risk of channel blockage (Chen et al., 2010;Ho et al., 2021;Liu, Zhu, et al., 2022;Guo et al., 2019;Rubio-Jimenez et al., 2016;Song et al., 2022).
Topology optimization of tree-like networks is important for the understanding of natural bifurcation systems, which is also a prerequisite for its application in industrial transport processes (Alston & Barber, 2016;Liu, Gao, et al., 2022;Yu & Li, 2006).The optimal bifurcation ratio is generally deduced according to the biological bionics principle associating the function and morphology.A structural theory developed by Bejan argues that natural structures are generated from the tendency to achieve optimal performance, and the optimized path from a single point to a finite volume shows the fractal tree-like topology (Bejan, 2005(Bejan, , 2017;;Chen, 2012).Xu et al. optimized the bifurcation structure for different transport processes (Xu et al., 2006;Xu & Yu, 2006).Gosselin (2007) and Gosselin and da Silva (2007) demonstrated the requirement to minimize the pumping power of the network under different size constraints.Jing and Song (2019) obtained the optimal diameter ratio by minimizing flow resistance of tree-like networks with volume and area constraints, respectively.
It is generally accepted that the optimal successive diameter ratio of bifurcation structure obeys Murray's law, it states that the cube of the diameter of a parent vessel equals the sum of the cubes of the diameter of the daughter vessels.Although some scholars have proposed existence of an optimal successive length ratio for the bifurcation structure, there is no accepted rule on the successive length ratio so far (Jing & Song, 2019;West et al., 1997).West et al. (1997) reported the optimal successive length ratio of 2 −1/3 with the principle of maximizing space filling.Adrian Bejan et al. (2000) obtained an optimal successive length ratio of 2 −1/7 for a T-shaped bifurcation with the assumptions of area conservation and volume constraint.While, Chen et al. (2021) proposed that the optimal length ratio between two successive branching vessel is 2 −3/5 by relying on the limitation on the surface area of the channel side.In the applications of tree-like structures, the successive length ratio is generally artificially selected.For example, Adrian Bejan et al. (2000) took the successive length ratio as the same with that of successive diameter ratio (2 −1/3 ), Pence (2003) and Xia et al. (2015) chose the successive diameter ratio of 2 −1/2 to prevent overlapping of channels at high-branching level.Luo et al. (2019) set the successive length ratio of the fractal tree-like network to be to 0.70.Therefore, it is necessary to add appropriate constraints in combination with the reality to obtain the optimal successive length ratio of the tree-like network.
On the other hand, it is worth noting that the rough surface of microchannel has significant effect on the heat and mass transfer properties, which has become one of the hot research topics in thermal science and engineering applications (Kandlikar et al., 2005;Lilly et al., 2007;Rostami et al., 2015;Rostami et al., 2020).Though surfaces of microchannels are usually modeled as a regular or random arrangement of different roughness elements such as sinusoidal, triangular, rectangular etc (Croce & D'Agaro, 2005;He et al., 2021;Kumar, 2019;Rovenskaya & Croce, 2016).However, the multi-scale and random characteristics of surface roughness are difficult to model with Euclidean geometry.While, the fractal geometry has been found to be a powerful mathematical tool to characterize the rough surface topography.Chen et al. (2009) introduced the Weierstrass-Mandelbrot (W-M) function with multi-scale and self-affine properties to characterize the rough surface of microchannel.Their numerical results show that the eddy currents caused by roughness have an effect on microchannel thermal-hydraulic performance, which becomes increasingly significant with increase of the Reynolds number.The pressure loss increases with increase of the self-affine fractal dimension.Yang et al. (2015) used the rough elements (conic peaks) with statistical self-similar fractal characteristics to characterize the rough surface of the microchannel, and deduced the pressure gradient, friction coefficient and Poiseuille number in the laminar flow state, revealing that the flow resistance in the microchannel was related to the fractal dimension, the size of the channel, and the ratio of the maximum diameter to the minimum diameter.It can noteworthy that most previous studies focused on the conjugate fluid flow and heat transfer in the traditional parallel microchannel, but few studies have been carried out to examine the influence of multi-scale rough surface on the thermal-hydraulic performance of treelike microchannel.
Flat-plate heat sink is most commonly used in microchannel cooling system, in order to further improve the heat dissipation efficiency, this study proposes a new optimization method for the successive branching ratios of tree-like network.Combined with the actual application scenario, the influence of rough surface on the thermal-hydraulic performance of the microchannel is considered.This research aims to provide theoretical basis for the design of flat-plate microchannel heat sinks and fuel cell flow channels etc.In this paper, the successive length ratio of tree-like network is optimized first by taking appropriate constraints into account.Then a mathematical model of disc-shaped heat sink composed of tree-like microchannel with rough surface is developed by the finite element method.The topological structure and rough surface of tree-like microchannel are modeled with self-similar and self-affine fractal, respectively.Finally, the influence of successive ratio and rough surface on the thermal-hydraulic performance of fractal tree-like microchannel is discussed accordingly.This paper is organized as 5 sections.In the following section 2, resistance minimization is performed on a bifurcation structure with different constraints to optimize the successive ratio.Then, a mathematical model of discshaped heat sink composed of tree-like microchannel with rough surface is proposed in section 3. The thermalhydraulic performance of fractal tree-like microchannel is discussed in section 4. The concluding remarks are made in the final section 5.

Structural optimization
The following scale factors are introduced to characterize the topological structure of tree-like network (West et al., 1997;Xu et al., 2006) where α and β are the successive length (l) ratio and diameter (d) ratio of the network respectively.As shown in Figure 1, the subscript k ( = 0,1,2 . . . ) denotes the branching level, l 0 and d 0 are the length and diameter of the initial parent channel.In view of the self-similar characteristics of the tree-like network, the scale factors are independent of branching level k.The self-similar fractal scaling law between branching number (n) and scale factors is Under the fully developed assumption of incompressible laminar flow and ignoring the effect of bifurcation, the flow resistance in the channel can be expressed by the Hagen-Poiseuille Equation R = 128μl/(π d 4 ), where μ is the dynamic viscosity of the fluid.The total flow resistance of a single bifurcation structure can be obtained by the series-parallel relationship of the parent and daughter branches.
It has been shown that minimizing the flow resistance under fixed volume 2 )/4 results in Murray's law for successive diameter ratio.For a symmetrical and dichotomous system, this leads to β = 2 −1/3 and the self-similar fractal dimension for diameter distribution is It is necessary to add appropriate constraints in combination with the reality to seek the best structure of the network (Gosselin, 2007).For a two-dimensional fractal tree-like microchannel, the constraint condition of surface area S = π(l 2 0 + l 2 1 + l 2 2 )/4 can be applied to characterize the function area of tree-like network.Therefore, the flow resistance of the bifurcation structure can be minimized with Lagrangian number multiplication.
where the Lagrangian factor η is an arbitrary nonzero constant.According to the extreme value condition, the function F is differentiated and simplified for l 0 , l 1 and l 2 to obtain: Substituting the successive diameter ratio result of Murray's law (β = 2 −1/3 ) into above equation results in: Following the principle of maximizing space filling of a 3D tree-like network (West et al., 1997), the total volume of the parent level and the daughter level of the network is approximately equal.That is, 4/3π where N is the total number of branches.Finally, the optimized successive length ratio α = 2 −1/3 ≈ 0.794 is obtained, and it can be seen that the result is independent of the number of branches k.

Self-affine rough wall
It has been proven that the rough wall contours show multi-scale and self-affine characteristics, which can be modeled by the Weierstrass-Mandelbrot (W-M) function (Majumdar & Tien, 1990) where z is the height value of the wall contour, and x represents the position coordinate of the contour.The fractal dimension of the self-affine wall D s is used to describe the irregularity of the function z at all scales.The scale factor G reflects the magnitude of z and determines its specific size.The spatial frequency of the contour γ is usually taken as 1.5, and the factor i corresponds to the cutoff frequency of the contour structure.Based on W-M function, three self-affine rough surface contour curves were generated by MATLAB and showed in Figure 2. It can be clearly seen that the self-affine fractal dimension shows significant influence on the roughness of curves.The irregularity of the rough surface contour increases with the increment of self-affine fractal dimension.

Numerical simulation
The physical model for a disc-shaped heat sink with tree-like microchannels is shown in Figure 3.The model consists of three parts: the silicon chip as the heat source, the package cover at the top, and the fractal tree-like microchannels embedded in the blue-shaded part of the package cover.The thickness of the package cover is 0.5 mm, and the thickness of the chip and the fractal bifurcated microchannel is 0.25 mm.Considering the size of the model and the actual application scenario of the microchannels, the fluid domain is composed of 10 treelike microchannels, which emanate from the center of the heat sink.
The microchannel size decreases regularly with selfsimilar fractal scaling law.By taking the self-similarity and symmetry of tree-like network as well as the reduction of computational cost, a continuous fractal tree-like microchannel with three branching levels (k = 0,1,2) was modelled.And rough wall inside of branching microchannel was generated by W-M function.As shown in Figure 1, the bifurcation angle θ is defined as the angle between two branches of the same level.In a similar model, the bifurcation angles θ 1 = 44°a nd θ 2 = 40°were set for the first and second branching level, respectively, taking into account the actual machining and optimization performance (Xu, Li, et al., 2016).The Murray's law was applied on the successive diameter ratio (β = 2 −1/3 ≈ 0.794), and three successive length ratios (α = 2 −2/3 ≈ 0.630, α = 2 −1/2 ≈ 0.707 and α = 2 −1/3 ≈ 0.794) were studied.The structural parameters of tree-like microchannel are summarized in Table 1.
In order to simplify the numerical analysis, the following assumptions are applied: (1) steady heat and mass transfer; (2) incompressible laminar flow in a fully developed state; (3) gravity and other forms of physical strength can be ignored.Based on these assumptions, the thermal-hydraulic characteristics of the flat-plate microchannel can be obtained by solving the following equation: where u is the velocity vector, ρ is the fluid density, p is pressure, I is the moment of inertia, K is the viscous stress tensor, M is the volume force vector, d z is the solid thickness, c p is the fluid heat capacity at constant pressure, T s is the solid temperature, q is the solid  onduction heat flux, Q is the heat source, and q 0 is the inward heat flux.It is assumed that the silicon of the channel wall material has constant parameters at T s = 310 K.
Water is used as a fluid working medium with an initial temperature of 293 K.The channel inlet flow velocity varies from 0.10 m/s to 0.50 m/s, and the corresponding Reynolds number (Re = ρv in d 0 /μ) is 44-220.The physical properties of liquid fluid and solid are listed in Table 2.
The following boundary conditions were applied on the flat-plate tree-like microchannel.(1) Inlet: the velocity was taken as 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50 m/s, and the coolant temperature was set as 293.15K. (2) Outlet: a zero outlet pressure was used as the reference pressure.(3) Interface: the interfacial condition at solid/liquid interface requires temperature continuity and relates to the temperature gradients.(4) Wall: a constant heat flux of Q = 10W/cm 2 was applied on the channel wall.
The fluid flow and heat transfer were computed with the finite element code COMSOL, and the PARDISO parallel sparse direct solver was used.Grid-independence test with 9 meshing strategies were carried out on a typical 2D tree-like network (Group3 in Table 1).Figure 4 shows the pressure distribution on one side of the channel wall with different mesh densities.The sharp pressure increase at the end of the first two channel stages is ascribed to the flow stagnation at the bifurcation angle.It can be seen that the pressure difference is lower than 1.1% with the grid encryption.
In order to validate the present numerical method, same microchannel heat sinks as that by Chen et al. (2009) and Xu, Li, et al. (2016) were simulated.Chen et al. (2009) carried out numerical simulation on a straight rectangular microchannel heat sink (50mm × 1mm × 0.1 mm), where the mass flow rate on inlet is 1 g/s and the heat flux on the channel wall is 10W/cm 2 .The rough contours generated by W-M function are shown in Figure 2.While Xu, Li, et al. (2016) studied the flow and thermal fields of a fractal-like multilayer silicon microchannel heat sink with a diameter of 40 mm by CFD, where the mass flow rate on inlet is 200 ml/min and the heat flux on the channel wall is 40W/cm 2 .It can be seen in Figure 5(a) that the predicted pressure drop distribution along the flow direction by the present method is very close to that by Chen et al. (2009). Figure 5(b) indicates that the predicted temperature by the proposed model is in good agreement with the single-layer heat sink model of Xu, Li, et al. (2016), and the average deviation is only 1.3%.Also, the temperature peaks at bifurcations are very close to that by Xu, Li, et al. (2016).
Except the pressure drop and flow resistance, the friction coefficient is used to characterize the flow field in  tree-like microchannel (Liu & Zhu, 2021).
where p m is the pressure drop between the inlet and outlet of the microchannel, L is the channel wall length, and u is the average velocity in the microchannel, ρ is the fluid density.The parent channel diameter d 0 is selected as the characteristic length of the whole system.To analysis the convective heat transfer performance of fractal tree-like microchannels with the rough wall, the average heat transfer coefficient is introduced (Jing et al., 2019) where T is the temperature difference between the fluid-solid interface, T w is the average wall temperature, T in and T out are the average temperatures at the inlet and outlet of the fluid.The Nusselt number of the water flowing through the microchannel system is defined as: where k f is the thermal conductivity at the average fluid temperature.

Pressure drop and resistance
Figure 6 shows the influence of the different rough surfaces on the pressure drop p m and friction coefficient f of the fractal tree-like microchannel model under different length ratios.It can be found that the pressure drop increases and the resistance decreases with the increase of the flow rate.Compared with the smooth surface of the fractal tree-like microchannel, the rough surface can evidently enhance pressure drop and friction coefficient.And both pressure drop and friction coefficient are increased by enhanced fractal dimension.In addition, when the circular area is limited, the fractal tree-like microchannel with α = 0.630 has the lowest pressure drop and friction coefficient.And its advantage in flow characteristics increases with the increase of inlet flow rate compared with that of α = 0.794 and 0.707.It can be also seen that the hydraulic performance of tree-like microchannel with α = 0.707 is better than that of α = 0.794.It can be concluded that the space limiting constraint is not suitable for the optimization of the length ratio of a 2D tree-like microchannel.

Velocity distribution
The velocity distribution can provide basic information to understand the pressure-driven fluid workflow in the microchannel.As shown in Figure 7, the inlet Reynolds number is Re = 110, the tree-like microchannel presents the flow line mixing and separation at two bifurcation angles, and the flow pattern (white line) changes dramatically due to the existence of bifurcation angles.Around the crest of the rough wall, the fluid velocity becomes larger due to the squeezing effect of the smaller channel cross-section.After the fluid flows through the crest of the rough wall, the vortex zone is generated in the trough of the rough wall under the action of two connected crests.It is clearly seen in Figure 7 that the velocity magnitude of tree-like microchannel with self-affine rough wall decreases with the increase of fractal dimension, which can be attributed to enhanced frequency of self-affine rough surface with increased fractal dimension (Figure 2).Also, the larger the fractal dimension, the larger the reflux zone at the rough wall trough, and the greater the effect on the flow pattern.At the same time, the vortex zone exerts a reverse pressure effect on the nearby fluid, which leads to a low-velocity zone near the rough wall.From Table 3, the rough wall of tree-like microchannel inevitably consumes energy and increases the pressure drop while decreases the flow velocity.However, the pressure drop between inlet and outlet of the tree-like microchannel with α = 0.630 is still lower than that of α = 0.794 and 0.707.

Temperature distribution
Compared with the traditional parallel and serpentine microchannels, the fractal tree-like microchannel has long been proven to have the inherent advantages of temperature uniformity and pressure reduction.Figure 8 shows the temperature distribution of the channel when the inlet Reynolds number is Re = 110.The distribution of the maximum and minimum temperature can be seen in this figure.It is found in Figure 8 and Table 3 that the successive length ratio shows important influence on the temperature distribution of tree-like microchannel even with optimal diameter ratio.It is clear that both the maximum temperature and temperature difference of tree-like microchannel with α = 0.630 are lowest compared with that of α = 0.794 and 0.707 under the same inlet velocity and rough surface.It can also be seen from Figure 7 that the extrusion action of the rough wave peak is greater than the reverse pressure action of the eddy current region, which makes the flow rate increase with the increase of fractal dimension.Moreover, the increased wall surface by increasing fractal dimension can enhance heat conduction between working fluid and solid wall.Thus, the temperature uniformity can be further improved by increasing the fractal dimension of channel wall.
The distribution of wall temperature on one side of a fractal tree-like microchannel with smooth wall surface and α = 0.630 at different inlet Reynolds numbers is shown in Figure 9.The bifurcation is an important geometric feature of the tree-like microchannel, and slight fluctuations of local wall temperature occur at the bifurcations.While the temperature fluctuation phenomenon is not obvious (Figure 10).And the wall temperature along the flow direction can be lowered by enhancing inlet Reynolds number, however, the magnitude of reduction decreases sharply.And Figure 10(c) shows that the wall temperature of the fractal tree-like microchannel with α = 0.630 is lower than that with α = 0.794 and 0.707.

Nusselt number
Figure 11 shows that the Nusselt number increases as the inlet Reynolds number increases.Re increases, that is, the fluid velocity increases, the heat transfer performance improves, so the Nu number also increases.Furthermore, the bifurcations of the tree-like microchannel can interrupt the laminar boundary, which also leads to the increase of convective heat transfer rate.Because the extrusion effect of rough surface increases with the increase of fractal dimension, the convective heat transfer rate increases accordingly.Further examination on Nusselt number of fractal tree-like microchannel with different successive length ratios clearly indicates that the convective heat transfer rate of a 2D fractal tree-like microchannel with α = 0.630 is highest.It should be noted that the Nusselt number for low fractal dimension is close to each other, which may be ascribed to the de sharpening processing in image imports.And the blockage of fluid caused by rough surface in tree-like microchannel should be avoided in practical applications.

Comprehensive performance
In order to evaluate the thermal-hydraulic performance of the fractal tree-like microchannel, the ratio of the average heat transfer coefficient to the inlet and outlet pressure drop ξ = h/Δp m is used.Since the fractal tree-like microchannel with α = 0.630 indicates better thermal-hydraulic performance compared with that of α = 0.794 and 0.707, the optimal branching ratios α = 0.630 and β = 0.794 were applied on the 2D tree-like network in this section.Figure 12 shows the influence of inlet Reynolds number and rough surface on the comprehensive performance of the treelike microchannel.The pressure loss inevitably increases with the increase of inlet flow rate, and the thermalhydraulic performance gradually decreases and tends to be stable.Although the inner rough surface of microchannel can enhance heat conduction and convection near the wall, it meanwhile induces sharp increase of the pressure drop.Thus, the comprehensive performance of the tree-like microchannel with rough surface is much lower than that of smooth surface.The competitive effect of rough surface on heat transfer and fluid flow make the relationship between the self-affine fractal dimension and comprehensive performance is non-monotonic.

Discussion
It is found that the effect of successive length ratio on the performance of tree-like microchannel is significant even when the optimal successive diameter ratio is used.Compared with the successive length ratios of α = 0.794 and 0.707, the advantages of thermal-hydraulic characteristics of tree-like microchannel with α = 0.630 increase with the increase of inlet flow, which reflects the effectiveness of the proposed optimization method.It should be pointed out that Murray's law cannot be directly applied to the successive length ratio of two-dimensional flat-plate microchannels.The present theoretical and numerical results indicate that the flat-plate tree-like micro-channel with successive diameter and length ratios of 2 −1/3 ( ≈ 0.794) and 2 −2/3 ( ≈ 0.630) shows enhanced thermal-hydraulic performance.Bifurcations of the tree-like network make the flow pattern and temperature fluctuate, but also improve the overall temperature uniformity.The inner rough surface of microchannel leads to sharp increase in the pressure drop along the flow direction and also increases the heat conduction.The self-affine fractal dimension is positively correlated with the convective heat transfer rate due to the extrusion effect of the rough surface.

Conclusion
In view of wide applications of flat-plate microchannel heat sinks, the thermal-hydraulic performance of fractal tree-like microchannel was evaluated numerically.A new optimization method based on flow resistance minimization for 2D tree-like microchannels is proposed.The effects of successive length ratio and surface roughness as well as inlet Reynolds number on the thermal-hydraulic performance of fractal treelike microchannel are explored.The optimal successive diameter ratio of symmetrical and dichotomous structures under volume constraint follows Murray's law (2 −1/3 ), while the optimal successive length ratio under the constraint of fixed channel area follows the power law 2 −2/3 .The fractal tree-like microchannel with α = 0.630 ( ≈ 2 −2/3 ) indicates better thermal-hydraulic performance compared with that of α = 0.794 ( ≈ 2 −1/3 ) and 0.707 ( ≈ 2 −1/2 ), which proves validity of the proposed optimizing method for a 2D tree-like network.

Declaration of competing interest
I declare on behalf of all authors of this manuscript that this paper has been neither copyrighted, classified, published, nor is being considered for publication elsewhere.There is no any conflict of interest with any financial, personal or other relationships with other people or organizations.
where D l and D d are self-similar fractal dimensions for length and diameter distribution, respectively.A symmetrical dichotomous structure (n = 2, l 1 = l 2 , d 1 = d 2 ) is used, and then the length and diameter of the k th -level branch are l k = l 0 α k and d k = d 0 β k .

Figure 1 .
Figure 1.Schematic diagram of the fractal tree-like microchannel with self-affine rough wall.

Figure 3 .
Figure 3.The physical model of disc-shaped heat sink with tree-like microchannels.

Figure 4 .
Figure 4. Pressure distribution along the channel wall on one side of the fractal tree-like microchannel (k = 2) with different meshing strategies.

Figure 5 .
Figure 5. Model validation: (a) pressure distribution in the straight rectangular microchannel; (b) wall temperature.

Figure 7 .
Figure 7.The flow field at bifurcation of the fractal tree-like microchannels with rough and smooth wall: (a) first level; (b) second level.

Figure 8 .
Figure 8.The temperature distribution of the fractal tree-like microchannel with rough and smooth wall.

Figure 9 .
Figure 9.The wall temperature distribution of the fractal tree-like microchannel with smooth wall surface and α = 0.630 at different inlet Reynolds numbers.

Figure 11 .
Figure 11.The relationship between Reynolds number and Nusselt number of the fractal tree-like microchannel with different length ratios and surface roughness.

Table 1 .
Structural parameters of tree-like microchannel (mm in unit).

Table 3 .
Inlet and outlet pressure drop and temperature distribution of fractal tree-like microchannel with different length ratios (Re = 110).