Performance Enhancement of Double-Layer Microchannel Heat Sink by Employing Dimples and Protrusions on Channel Sidewalls

Abstract

High-temperature gradient causes thermal stress and is indirectly responsible for other demerits in nonstacked microchannels. High-temperature gradient is overcome by employing double layers in the heat sink. Utilization of dimple and protrusion in such double-layered sink to enhance overall performance is done in the present numerical study. Before placement of dimples and protrusions on sidewalls of the sink, optimum width and depth of channel have been assessed. Microsinks with the protruded-dimpled bottom layer and microsinks with protruded-dimpled layers are investigated. The parameters such as maximum bottom wall temperature difference (ΔT_b), Nusselt number ratio (Nu/Nu_o), and thermal performance factor (η) have been evaluated. The impact of aligned and staggered arrangements of dimple and protrusion is also compared. Deionized water as a coolant for the range of Reynolds number of 89–924 is examined. It has been realized that aligned models offer higher heat transfer coefficient, maximum Nu, and minimum ΔT_b, but in terms of overall performance, staggered sinks are superior. The heat sink with both layers protruded and dimpled, showing Nu/Nu_o and η of 1.36 and 1.32, respectively, is observed as one of the optimum sinks which offers an excellent lowest ΔT_b of 1.48 K.

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1. Introduction

The wide range of applications of the microchannel heat sink has emerged in recent years and is possible due to its good thermal management capacity. Microsizing of channels helps in heat dissipation for the miniaturized electronics devices subjected to high heat flux. It was the novel experimental research work of Tuckerman and Pease [1] that introduced the concept of microchannel heat sink (MCHS) in 1981, which played a path-breaking role in the cooling of IC products. Utilizing their proven fundamentals, such as the large surface area to volume ratio and augmented convective heat transfer coefficient, researchers performed numerous investigations on microchannels to counter the recent challenges. As per the prediction of Karayiannis and Mahmoud [2], the heat flux of future computers could reach 2 MW/m² by 2026, implying a big challenge. So several investigations are done with improved techniques. Attachment of rib, protrusion, porous section into flow-channel and channel surface subtraction by employing cavity, dimple, groove on channel wall are amongst the significant passive enhancement techniques.

Implicating cavities into the flow channel is one of the potential concepts. Xia et al. [3, 4] applied cavities of different shapes in a single layered microchannel and investigated numerically the effects of cavity parameters on heat transfer and fluid flow. Enlarging interface area and subsequent enhancement in heat transfer by the introduction of a different cavity, grooves are predicted in similar studies [5–7]. The researchers [8–12] have articulated that a microchannel with a combination of cavities and ribs creates chaotic advection and flow mixing into the channel in addition to the breaking-up of the thermal, hydraulic boundary layer. Dimpled and protrusions if mounted on the wall of the microchannel, noticeable addition on performance is observed. Wei et al. [13] performed a simulation of the rectangular microchannel with the row of dimples at its bottom surface with air as coolant. Lan et al. [14] investigated the influences of dimples and the protrusions on performance, keeping them in a staggered and aligned position, and revealed that staggered design had better enhancement. Li et al. [15] studied the effects of the staggered and aligned pattern. Li et al. [16] showed that the thermal performance was largely affected by the depth of dimple and protrusion.

Choudhary et al. [17] experimented pin fins with and without wings in the heat sinks. The Nusselt number, friction factor, and thermohydraulic performance were determined for the working fluid of air at Re ranging from 6800 to 15100. The impacts of inline and staggered positioning of pin fin were studied. Staggered fins with wings showed better performance parameters in the experimental results. Bahiraei et al. [18] concluded that the nanofluid, together with the elliptical pin-fins, offers remarkable cooling performance in the heat sink. The larger number of pin-fins demonstrates superior heat transfer by convection which enables the heat sink unit to lower the substrate temperature. The heat transfer coefficient was found with the hike of 50% for the optimum fin density. An extensive experimental investigation was carried out by Patil et al. [19] to study the conventional heat sink channel configured with a pin fin and plate fin. Effects of different geometrical parameters of pin fin and plate fin were determined. The performance of such heat sinks was evaluated through the representation of Nusselt number, friction factor, and thermohydraulic performance. The employment of pin fin and plate fin showed variation in the performance parameters. The study of aligned and staggered arrangements of dimples on the surface wall of plate fin was also an important part of the investigation. In the observation, although the pin fin heat sink yielded better Nusselt number and heat transfer, the plate fin heat sink with dimple arrangement exhibited the maximum thermohydraulic performance. Gupta et al. [20] analyzed the influences of dimples, positions of dimples, and dimensions of dimples on heat transfer experimentally and flow friction. A bigger diameter of dimple and a larger number of dimples promote more heat dissipation. The dimpled heat sink with a bigger diameter and staggered pattern of dimples leads to higher Nu at the cost of a larger friction factor. The fin performance as calculated in the study is also regulated by the configuration of the sink in such a way that a bigger diameter and staggered pattern of dimples bring out higher fin performance. Turkyilmazoglu [21] predicted the temperature distribution and thermal efficiency of the parabolic pin fin. They determined the variations of these thermal parameters with Peclet number and Biot number. They identified the optimum pin fin of the concave-parabolic shape by adjusting a design variable called the fin shape parameter.

Unveiling of the concept of double-layer microchannel heat sink by Vafai and Zhu [22] in 1997 was a new achievement in the microchannel domain. More reliability and durability can be achieved when a single layer or channel is substituted by a double channel in MCHS. Substantial progress in double-layer sink has been reported by the investigators. Vafai and Khaled [23] compared the single- and double-layered microsink numerically and analytically. Chong et al. [24] applied the analytical method to study single- and double-layered microchannels with available correlations. They ascertained that the double layer helped in decreasing pressure drop, which was the key finding of Ref. [22, 23]. Xie et al. [25], like other investigators, had reported that the significantly larger flow area in the case of double-layer sink is the advantage. Xie et al. [26] compared wavy single- and double-layer channels and estimated larger wave amplitude for better results. In a similar investigation, Xie et al. [27] studied the influence of counter and parallel flow for a wavy channel. They revealed that Nu for parallel flow was higher than that of counterflow at low Re and it was opposite at larger Re. Shen et al. [28, 29] took novel structures having staggered flow alternation. Performance fluctuation with different width ratios, aspect ratios, and flow rate was studied by Wu et al. [30] and Wei et al. [31]. Levac et al. [32] assessed single-layer and double-layer channels. Ahmed et al. [33] performed experiments on rectangular and triangular heat sink of aluminium substrate using nanofluids.

Extensive analyses were carried out on double-layer heat sinks through the investigational works of Zhai et al. [34–36]. Wong et al. [37] conducted a numerical study to evaluate parallel flow and counterflow. Wong et al. [38], in another study, selected tapered channel and highlighted that taper channel had better thermal performance in comparison to simple straight channel at higher pumping power. Leng et al. [39] employed an optimization method and noted that the maximum bottom wall temperature change is 3.23 K for an optimal channel. Lin et al. [40], in their analysis, noted the lowest thermal resistance of 0.089 K/W. Transverse flow was studied by Ansari and Kim [41]. Heat sink with a large heat flux of 556 W/cm² was simulated and optimized by Shao et al. [42]. Debbarma et al. [43, 44] increased the performance of double-layer sinks by modifying fluid channel structures. Employment of porous materials in the channel is found beneficial for some specific designs. Ghahremannezhad et al. [45] got the porous double-layer heat sink improved for a certain porous material. Li et al. [46] reported that the suitable combination of lower pressure drop of porous ribs and better conductivity of solid lead to improvement in performance. They recommended an improved model comprising a lower channel of solid copper ribs and an upper layer of porous ribs. Their analysis showed that thermal resistance was reduced by 11%, whereas temperature uniformity was enhanced by 56%. Wang et al. [47] opined that pumping power was reduced remarkably for the porous heat sink. Wang et al. [48] optimized the design parameters of stacked channel. Kulkarni et al. [49] realized that pumping power was inversely proportional to the increase of heat flux. It occurred because fluid viscosity decreases with an increase in heat flux. Rajabifar [50] introduced a new concept of applying phase change material (PCM) slurries into double-layer channels to improve cooling performance.

Upgradation of performance of double-layer microchannel heat sink (DL-MCHS) by implicating dimple and protrusions on the channel wall is the main objective of the present study. Protrusions or any other constrictors employed in channel flow lead to significant heat exchange by virtue of fluid mixing and breaking up of boundary layers. Likewise, the cavity of different shapes, including dimple, enhances performance by producing vortices and transverse fluid movement. In addition to that, the positional pattern of such channel surface subtractors and constrictors inside the flow channel are crucially important in terms of the cooling effect. However, in the study, the prediction of performance enhancement considering all the relevant losses is very much significant. As per the findings reported in the investigations [13–16], the application of dimples and protrusions is found to be beneficial in single layer microchannel heat sink (SL-MCHS). But the influence of dimples and protrusions in the DL-MCHS is not studied by the researchers so far, as per the knowledge of the present authors. Therefore, this work has focused on utilizing dimples and protrusions in the DL-MCHS. It is interesting to know the impacts of dimples and protrusions when these are mounted on only the bottom channel or both bottom and top channels. The variation in flow and thermal characteristics for different patterns like aligned and staggered are also the parts of this investigation so that optimum design with an appropriate combination of dimples and protrusions can be predicted after a thorough study.

2. Descriptions of Microchannel Heat Sink and Numerical Method

2.1. Geometry of Heat Sink

Double-layer microchannel heat sinks (DL-MCHS) of total width, W = 10 mm, and total length, L = 10 mm, are considered in present study. Figure 1 depicts a schematic diagram of double-layer microchannel heat sink (DL-MCHS). A total of fourteen cases (Case 1 to Case 14) of DL-MCHS as described in Tables 1–5 are investigated in four different groups. The fourteen (14) cases are classified into four (4) groups (A, B, C, D) in accordance with the following objectives of the groups. So, Table 2 states the respective objectives of the four groups.

[figure(s) omitted; refer to PDF]

Table 1

Descriptions of microchannel heat sinks (Case 12 to Case 14) having bottom (lower) layer and top (upper) layer with protruded and dimpled channel walls.

	Case	Channel wall feature	Pattern	Pitch (P)/mm
Group D	Case 12	Protruded bottom and top channel	Staggered	1.0
	Case 13	Dimpled and protruded bottom and top channel	Staggered	1.0
	Case 14	Dimpled and protruded bottom and top channel	Staggered	0.5

Table 2

Descriptions of the groups: Group A, Group B, Group C, and Group D.

Groups	Study	Sink channel descriptions	Objectives
Group A	Channel parametric analysis	Top and bottom channel walls are plain	To find optimum channel height (Hc), channel width (Wc), and base height (Hb)
Group B	Protrusion parametric analysis	Only the bottom channel wall is protruded	To find the optimum diameter of the protrusion (Dp)
Group C	Comparison between aligned and staggered pattern	Only the bottom channel wall is protruded and dimpled	To assess the optimum position/pattern of dimple and protrusion
Group D	Single and double layers with dimple and protrusion	Top and bottom channel walls are protruded and dimpled	To examine the impact of dimple and protrusion when placed in two layers

Table 3

Descriptions of double-layer microchannels heat sinks (Case 1 to Case 5) with smooth channel walls.

	Cases	Hc/μm	Wc/μm	Hb/μm
Group A (channel parametric analysis)	Case 1	200	100	150
	Case 2	300	100	150
	Case 3 (case 0)	400	100	150
	Case 4	400	150	150
	Case 5	400	100	200

Table 4

Descriptions of microchannel heat sinks (Case 6 and Case 7) with protruded bottom (lower) channel wall.

	Cases	Dp/μm	dp/μm	Pitch (P)/mm
Group B (protrusion parametric analysis)	Case 6	200	40	2.5
	Case 7	300	40	2.5

Table 5

Descriptions of microchannel heat sinks (Case 8 to Case 11) having protruded and dimpled bottom (lower) channel walls.

	Case	Bottom channel feature	Pattern	Pitch (P)/mm
Group C	Case 8	Protruded bottom channel	Aligned	1.0
	Case 9	Dimpled and protruded bottom channel	Aligned	1.0
	Case 10	Dimpled and protruded bottom channel	Staggered	1.0
	Case 11	Protruded bottom channel	Staggered	1.0

Descriptions of microchannel heat sinks such as Case 1, Case 2, Case 3, Case 4, and Case 5 with smooth channel walls in the upper layer (top channel) and lower layer (bottom channel) both are shown in Table 3. These five cases comprising different channel height (H_c), channel width (W_c), and base height (H_b) are studied in Group A. To find the optimum channel parameters, different values of channel height (H_c), channel width (W_c), and base height (H_b) are evaluated in this parametric analysis. Case 3 is found optimum in terms of better performance and considered as the base channel, i.e., Case 0 in other groups. The channel height (H_c), channel width (W_c), and base height (H_b) of the optimum channel Case 3 (or Case 0) are applied in all other nine Cases of Group B, Group C, and Group D.

Case 6 and Case 7, as studied in Group B, are the microchannel heat sinks with protruded bottom (lower) channel walls. The upper layer has remained a plain channel with smooth sidewalls. Table 4 shows the dimensions of protrusion as mounted in Case 6 and Case 7 at an aligned position. The diameters of the protrusion (D_p) are 200 μm in Case 6 and 300 μm in Case 7, respectively, whereas the depth of protrusion (d_p) is taken as a constant value of 40 μm for both cases, as per Ref. [51]. Optimum D_p of 200 μm along with d_p of 40 μm is used as the protrusion parameters in Group C and D.

Table 5 depicts the descriptions of microchannel heat sinks: Case 8, Case 9, Case 10, and Case 11 having protruded, dimpled bottom (lower) channel walls. The objective of the study of group C is mainly to assess the impacts of the Dimpled and Protruded bottom channel (lower layer), keeping the upper layer as a plain surfaced wall. The aligned and staggered position of dimple and protrusion, as defined in Figure 2, is also studied. The diameter of dimple (D_d) of 200 μm and depth of dimple (d_d) of 40 μm in this Group C for all four cases are considered.

[figure(s) omitted; refer to PDF]

The Group D is analyzed to investigate the effects of both bottom (lower) layer and top (upper) layer with protruded, dimpled channel walls. Here, Case 12, Case 13, and Case 14, having both layers with protruded, dimpled channel sidewalls, are examined. Table 1 shows descriptions of such microchannel heat sinks. Only the staggered position of dimple and protrusion due to its merits is considered in this group.

Water as coolant is flown through the silicon channel substrate in the simulation. Parallel flow in analysis A and counter flow in other analyses are arranged for the study. The thickness of the wall between the bottom and top channel (t_m) is 50 μm, kept unchanged for all cases. Channel wall thickness (W_w) of 200 μm and total channel length (L) of 10 mm are also constant for all cases. Same channel height (H_c) is considered for both bottom (lower) layer and top (upper) layer in all cases.

The optimum channel dimensions, i.e., channel height (H_c) of 400 μm, channel width (W_c) of 100 μm and base height (H_b) of 150 μm are taken for Case 6 and Case 7.

The optimum channel dimensions, i.e., channel height (H_c) of 400 μm, channel width (W_c) of 100 μm, and silicon base height (H_b) of 150 μm are taken for Case 8, Case 9, Case 10, and Case 11. Protrusion dimensions: D_p = 200 μm, d_p = 40 μm; dimple dimensions: D_d = 200 μm and d_d = 40 μm.

The optimum channel dimensions, i.e., channel height (H_c) of 400 μm, channel width (W_c) of 100 μm, and base height (H_b) of 150 μm are taken for Case 12 to Case 14. Protrusion dimensions: D_p = 200 μm, d_p = 40 μm; Dimple dimensions: D_d = 200 μm and (d_d) = 40 μm.

Figure 2 shows the views of the section of DL-MCHS where dimples and protrusions are positioned on the bottom channel. As depicted in Figure 2(b), dimple pairs are vertically aligned at opposite sidewalls, and protrusion pairs are also vertically aligned at opposite sidewalls. Similarly, dimples are vertically staggered at opposite sidewalls and protrusions are vertically staggered at opposite sidewalls, as drawn in Figure 2(c), which in other words implies that dimples and protrusions together are in the aligned position. The pitch (P) is defined as the distance between two consecutive protrusions or dimples.

2.2. Computational Model

A single computational domain, as shown in Figure 3, represents the conjugate unit of symmetrical geometry of the heat sink. This single domain has channel height H_c, channel width W_c and base height or bottom substrate thickness H_b. Half of substrate wall thickness, W_w/2, is selected as the domain wall thickness. The t_m denotes a thin solid layer between two fluid channel layers. The bottom wall is subjected to uniform heat flux. Two outer sidewalls and top wall are assumed adiabatic surfaces indicating no transfer of heat through these walls.

[figure(s) omitted; refer to PDF]

2.3. Mathematical Equation and Computation

The coolant fluid flow in the microchannel heat sink is generally assumed as incompressible, laminar, and steady in order to make the problem formulation simplified and thus, the governing equations are expressed as follows.

Continuity equation:$$\begin{array}{}\text{(1)}& \nabla .\text{V}=0.\end{array}$$

Navier–Stokes equation:$$\begin{array}{}\text{(2)}& \rho \frac{DV}{Dt}=-\nabla p+\mu {\nabla }^2\text{V}.\end{array}$$

Energy equation for fluid: $$\begin{array}{}\text{(3)}& \rho c_v\frac{DT}{Dt}=k{\nabla }^{2}T.\end{array}$$

Energy equation for silicon substrate:$$\begin{array}{}\text{(4)}& k_s{\nabla }^2{T}_s=0.\end{array}$$

At the inlet of lower layer (x = 0): u = u_in, $v$ = 0, ω = 0, T_in = 293 K; At upper layer (x = 0 for parallel flow, x = L for counterflow): u = u_in (parallel), u = -u_in (counter), $v$  = 0, ω = 0, T_in = 293 K; At the outlet of the lower layer and upper layer: p_out = 1 atmospheric; for the bottom wall: q = constant [28, 29, 38, 39].

The radiation effect, gravity, and body forces are assumed as negligible. The interfacial wall is a no-slip boundary. The SIMPLEC method is adopted for numerical solution in Fluent solver. Fluid viscosity is taken as piecewise-linearly variable with temperature. The inlet fluid velocity (u_in) ranges from 0.5 m/s to 4 m/s and the uniform heat flux (q) applied at the bottom surface is 10⁶ W/m². Residual of 10⁻⁶ is the convergence criteria set for continuity and momentum equations whereas 10⁻⁸ is for energy equation.

The following are some other associated equations and performance parameters for the microchannel domain.

The Reynolds number (Re) and hydraulic diameter (D_h) are given as follows:$$\begin{array}{c}\text{(5)}& \mathrm{Re}=\rho u\frac{D_h}{\mu },D_h=\frac{2H_c{W}_c}{H_c+W_c}.\end{array}$$Here, ρ, u, µ, H_c,W_c are fluid density, flow velocity, fluid viscosity, channel height, and channel width, respectively.

Fanning’s friction factor (f) and Nusselt number (Nu), heat transfer coefficient (h) are computed as follows:$$\begin{array}{c}\text{(6)}& f=\frac{\Delta p{D}_h}{2\rho u^{2}L},Nu=\frac{h{D}_h}{k},\\ h=\frac{Q_{b,\text{total}}}{A_{\text{con}}{T}_{b,av}-T_{av}}.\end{array}$$Here, $\mathrm{\Delta }p$, L, and k represent pressure drop, channel length, and thermal conductivity of the fluid. ${\text{\hspace{0.17em}Q}}_{\text{b},\text{total}}$, ${\text{A}}_{\text{con}}$, ${\text{T}}_{\text{b},\text{av}},{\text{\hspace{0.17em}and\hspace{0.17em}T}}_{\text{av}}$ stand for total heat applied at the bottom wall, convection heat removal area, average temperature of the bottom wall, and average fluid temperature of the channel.

The thermal performance factor or heat transfer enhancement factor ($\eta$) [26, 28, 29, 35] is defined as follows:$$\begin{array}{}\text{(7)}& \eta =\frac{Nu}{N{u}_0}{\frac{f}{f_0}}^{1/3},\end{array}$$where $N{u}_0$ and $f_0$ stand for Nusselt number and friction factor of the smooth or base microchannel, respectively.

Thermal resistance (R) is expressed as follows:$$\begin{array}{}\text{(8)}& \text{R}=\frac{\Delta {\text{T}}_{\text{b}}}{{\text{Q}}_{\text{b},\text{total}}}.\text{(9)}& \Delta T_b=T_{b,\max }-T_{b,\min }.\end{array}$$

The pumping power in the bottom and top layers can be written as follows:$$\begin{array}{c}\text{(10)}& P{P}_1=u{\text{A}}_{\text{c}1}\Delta {\text{p}}_{1}N,P{P}_2=u{\text{A}}_{\text{c}2}\Delta {\text{p}}_{2}N.\end{array}$$

Total pumping power (PP) is given as$$\begin{array}{}\text{(11)}& PP=P{P}_1+P{P}_2.\end{array}$$

ΔT_b, T_b,max, T_b,min,${\text{A}}_{\text{c}1},\mathrm{\Delta }{\text{p}}_1,{\text{\hspace{0.17em}A}}_{\text{c}2,}\mathrm{\Delta }{\text{p}}_2$ and N are maximum bottom wall temperature difference, maximum bottom wall temperature, minimum bottom wall temperature, cross-sectional area of the bottom channel, pressure drop in the bottom channel, the cross-sectional area of the top channel, pressure drop in the top channel, and the number of channels at each layer.

2.4. Validation

2.4.1. First Validation

The reliability of the computational work has been verified before the beginning of the present investigation. The present study is compared with the experiential results of Wei et al. [31]. For the same dimensional geometry of double-layer microchannel and uniform heat flux of 7 × 10⁵ W/m^2, the verification is carried out. A lower channel length of 18 mm and upper channel length of 10 mm are considered with the total heating bottom surface area of 10 cm × 10 cm. Results for different coolant flow rates, i.e., 83, 116, 147, and 177 ml/minute, are compared. Silicon and deionized water are, respectively, the heat sink substrate and coolant fluid. The comparison is represented in Figure 4, where 3.66% is observed as the maximum variation in thermal resistance, justifying the agreement of the present work with experiential results.

[figure(s) omitted; refer to PDF]

Values of thermal resistances (K/W) of the present study and Wei et al. [31] are given in Table 6.

Table 6

Values of thermal resistances (K/W) of the present study and study by Wei et al. [31].

Sl	V/(ml/min)	Thermal resistances (K/W)			Percentage error
		Present study	Wei et al. (2007) [13]
1	83	0.266765		0.257	3.66068
2	116	0.193143		0.191	1.109687
3	147	0.159687		0.158	1.056265
4	177	0.140557		0.137	2.530646

2.4.2. Second Validation

In the second validation, a comparison between the average bottom wall temperatures of the present study and Zhai et al.’s [34] is made. Heat sink is the double-layer sink and is studied with counterflow arrangement. The geometrical parameters of the microchannel heat sink are as follows: length of the channel (L) = 10 mm, width of the channel (W) = 3 mm, number of channels = 10. The heat flux at the bottom wall (q) = 100 W/cm².

Tables 7 and 8 represent the dimensions of channels of the heat sink and a comparison of results of the present study and Zhai et al.’s [34], respectively.

Table 7

Dimensions of channels of the heat sink.

Channel height (μm)	Channel width (μm)	Base height (μm)	Middle rib thickness (μm)
100	100	150	50

Table 8

Comparison of results of the present study and study by Zhai et al.’s [34].

Sl	V/(ml/min)	Average bottom wall temperature			Percentage error
		Present study	Zhai et al. [34]
1	6	101.6835		102	-0.31123
2	12	67.27218		66	1.891094
3	18	56.83177		56	1.463565

2.4.3. More Validations with the Experimental Works

The following comparisons are also made to assure the accuracy of the numerical method adopted in the present study. A similar design of microchannel experimented in Tuckerman and Pease [1] has been simulated by applying the present method. The coolant flow rate of 4.7 cm³/s and heat flux of 181 W/cm² at the bottom wall are considered for investigation of microchannel of single layer model having channel height (H_c) of 320 μm, channel width (W_c) of 56 μm, and channel wall thickness (W_w) of 44 μm. Table 9 depicts the comparison of the results of the present study and Tuckerman and Pease [1].

Table 9

Comparison of results of the present study and the study by Tuckerman and Pease [1].

V/(cm3/s)	Thermal resistance (0 C/W)			Percentage error
	Present study	Tuckerman and Pease [1]
4.7	0.106765		0.11	3.030046

In another validation, the experimental work of Chai et al. [6] has been considered. The length of the microchannels is 10 mm, the total width covering the ten microchannels is 3 mm: channel height (H_c), channel width (W_c), and bottom substrate thickness (H_b) are as follows:

  W_c = 0.1 mm = 100 μm

  H_c = 0.2 mm = 200 μm

  H_b = 0.15 mm = 150 μm

Inlet water coolant temperature of 293 K and heat flux of 0.6 MW/m² at the bottom wall are taken for the silicon heat sink. The variations in Nu with respect to different Re as obtained in the study are shown in Table 10.

Table 10

Comparison of results of the present study and the study by Chai et al. [6].

Chai et al. [6]		Present work		% error
Re	Nu	Re	Nu	Nu
149	5.22	149	5.41	3.64
201	6.12	201	6.33	4.41
259	7.11	259	7.41	4.22
311	8.01	311	8.35	4.24
366	8.86	366	9.14	3.16
423	9.52	423	9.87	3.68
478	9.95	478	10.18	2.31
528	10.56	528	10.86	2.84
588	11.36	588	11.80	3.87
643	11.88	643	12.12	2.02
693	12.16	693	12.47	2.55
753	12.59	753	12.91	2.54
805	13.53	805	13.83	2.22

2.5. Grid Independence Test

The computational domain, including the fluid region, has been discretized into hexahedral elements. Figure 5 shows an unmeshed section, the mesh formed at the dimple and protrusion region of the computational model. The independence of the grid is checked to confirm the accuracy of the numerical solution. As such, three different meshed models of Case 6 comprising Mesh (a) 269280, Mesh (b) 404277, and Mesh (c) 792480 elements give negligible differences in thermal resistance and bottom wall temperature readings. Likewise, Mesh (a) and Mesh (b) show a deviation of only 0.66% and 0.57%, respectively, from Mesh (c) in terms of thermal resistance, whereas these deviation figures are 0.040% and 0.051% in terms of bottom wall temperature at u_in = 0.5 m/s. Therefore, a similar mesh of Mesh (b) is utilized for all other cases to avoid large meshed elements like Mesh (c).

[figure(s) omitted; refer to PDF]

Figure 6(a) represents the variations of thermal resistance of Case 6 for different meshed models. Similarly, Figure 6(b) depicts the variations of bottom wall temperature of Case 6 for different meshed models.

[figure(s) omitted; refer to PDF]

3. Results and Discussion

Total of fourteen cases (Case 1 to Case 14) of DL-MCHS are investigated in four different Groups. The fourteen cases are classified into four groups (A, B, C, D) in accordance with their respective objectives. The objectives of the studies performed in groups A, B, C, and D are, respectively, (a) to find optimum channel height (H_c), channel width (W_c), and base height (H_b), (b) to find the optimum diameter of the protrusion (D_p), (c) to assess optimum position/pattern of dimple and protrusion, (d) to examine the impact of dimple and protrusion when placed in two layers.

3.1. Group A

The parametric effect of channel geometry of double-layered heat sink is evaluated in this group. Different channel heights and widths are studied. Thermal resistance (R) and maximum bottom wall temperature difference (ΔT_b) are the performance parameters compared for the five cases. Channel height (H_c) of 200 μm, 300 μm, 400 μm, and channel width (W_c) of 100 μm, 150 μm are the design variables in this analysis. H_c = 400 μm and W_c = 100 μm are observed as the optimum channel height and channel width, respectively. The effect of base height (H_b) has also been tested by increasing its value to 200 μm from 150 μm. The temperature distribution of the channel segment at the middle of streamwise length for these cases is depicted in Figure 7. As per the depiction, it can be illustrated that all five cases show different contours; Case 3 with good uniformity, whereas Case 1 with worst temperature distribution. The average bottom wall temperature (T_b,av) for Case 3 is calculated as 313 K, whereas Case 1 exhibits a maximum T_b,av of 326 K at the same inlet velocity.

[figure(s) omitted; refer to PDF]

Heat distribution between two layers of channels varies along with the flow directions due to the varied value of the thermal resistance for each layer. Figure 8(a) shows the values of thermal resistance R for five cases at different Re. The R of Case 1 is higher for all ranges of Re in comparison to other cases. Case 3 and Case 5 perform better than other cases due to lower R. Case 3 exhibits low thermal resistances in all ranges of Re with the lowest R = 0.119 K/W at Re = 678, which is 48%, 15%, 18%, and 2.5% lower than that of Case 1, Case 2, Case 4, and Case 5, respectively, at the corresponding fluid velocities.

[figure(s) omitted; refer to PDF]

The maximum bottom wall temperature difference is an important parameter evaluated in MCHS to know the temperature distribution subjected to the silicon substrate. For five cases, the relation of maximum bottom wall temperature difference (ΔT_b) with Re is drawn in Figure 8(b). Case 3 gives lesser ΔT_b for all Re, which describes better temperature uniformity across the bottom wall upon the heat application on the bottom surface. Thus, parameters H_c = 400 μm, W_c = 150 μm, H_b = 150 μm of Case 3 are found optimum, and this case is considered as Case 0 in Group B, C, and D. Nu of these channels are not compared since their hydraulic diameters are not same. Hence, comparison of these heat sinks in terms of Nusselt number ratio and thermal enhancement factor is not justified.

3.2. Group B

Elevation of performance of MCHS is possible by implicating protrusion on the channel. This is happening by virtue of an increase of interfacial area and suitable fluid mixing [14, 16]. This analysis is to find the influence of protrusion and estimate the suitable diameter of the protrusion (D_p). The channel of Case 3, as found optimum in Group A, is studied with the application of protrusions on its sidewalls and the diameter of the protrusion (D_p) is varied from 200 μm to 400 μm in this study so that the optimum diameter can be calculated. The depth of protrusion d_p = 40 μm is kept constant since this value has been calculated as the optimum depth of protrusion by the author, Ref. [51]. Impact of protrusion on Nu/Nu_o and η in addition to maximum bottom wall temperature difference is evaluated. Figure 9 evidently shows how Case 6 with D_d = 200 μm is superior. It exhibits well uniformity in bottom wall temperature at all Re than Case 7.

[figure(s) omitted; refer to PDF]

Interestingly, it is determined that larger protrusion produces a larger heat transfer coefficient h. A similar trend is observed in the study of the Nusselt number ratio. Case 7 provides better h and larger Nu/Nu_o at higher Re than Case 6. Case 6 shows better Nu/Nu_o at low Re as per the depiction of Figure 10(a). These results indicate that more surface area helps in enhancement of Nu/Nu_o. The maximum Nu/Nu_o as calculated in this analysis is 1.17 at Re = 674 for Case 7. It is obvious that when water flow impinges on the protrusion, it enables heat to transfer directly to fluid, and thus, temperature in the protruded region is low [16].

[figure(s) omitted; refer to PDF]

As far as pressure drop is concerned, the variation of heat transfer enhancement factor η with Reynold number Re has to be analyzed. Because this parameter consider both the heat transfer augmentation and the pressure drop increase, Figure 10(b) represents this compared data. Although Nu/Nu_o is significantly enhanced in Case 7 at large Re, its higher friction causes deterioration of the η. As such, a large gap between the thermal performances of Case 6 and Case 7 is predicted. Therefore, Case 6 is found to be superior to Case 7 with a maximum η of 1.066 at Re of 264.

3.3. Group C

This part includes Case 8, Case 9, Case 10, and Case 11. The objectives are to assess superiority between aligned and staggered sets and to compare protruded-dimpled sinks with mere protruded sinks. Case 8 and Case 11 consist of protrusions mounted on sidewalls, whereas Case 9 and Case 10 comprise both dimple and protrusion set on sidewalls. Case 8 and Case 9 are aligned sets, Case 10 and Case 11 are staggered sets. The aligned pattern indicates that two protrusions or two dimples are at the same line transversely and arranged on opposite walls. In this analysis, dimples and protrusions are set on layer 1 (bottom channel), keeping layer 2 (top channel) a smooth one. Such dimpled and protruded sink with a vertically aligned and staggered design is found in Figure 2. Aligned heat sinks perform better in terms of thermal parameters such as ΔT_b, Nu, Nu/Nu_o than staggered heat sinks. But Staggered heat sink model, i.e., Case 10 and Case 11 exhibit superiority in terms of hydraulic characteristics and overall performance.

Figure 11 visualizes velocity profiles at the span of middle along x direction for Case 8, Case 9, Case 0, Case 10, and Case 11. All contours shown are taken for inlet velocity (u_in) = 4 m/s. It is evident from the figure that the velocity of the fluid is dominantly increasing at and near the constricted passage of two protrusions due to the reduced flow area. Contrary to that, at the flow passage area between two aligned dimples, the diverging space causes a reduction of fluid velocity. The pressure profiles are portrayed in Figure 12. For Case 9, pressure contour is found to be brighter between two aligned dimples since a major part of dimple is subject to sudden pressure rise. But pressure has been reduced to a certain extent in the staggered model where one dimple is replaced by a protrusion in Case 10. This is well agreed with the finding of Lan et al. [14], which describes that the association of dimples into the channel is an attractive means because dimples enhance heat transfer with low-pressure loss. Figure 13 illustrates the velocity contour, streamlines, and formation of vortices at the ending regions of protrusions in the bottom channel of Case 8 at Re = 673. These vortices improve the intermixing of fluid layers flowing through the constricted region and base region of protrusions.

[figure(s) omitted; refer to PDF]

For all cases, the maximum bottom wall temperature difference (ΔT_b) reduces rapidly with an increase of Reynold’s number. It can be explained that a higher velocity of liquid coolant leads to faster conduction heat transfer in the bottom wall and a higher convection rate in the fluid. This phenomenon is well illustrated in Figure 14. It is evident that vertically aligned sinks, i.e., Case 8 and Case 9, predict better bottom wall temperature distribution than staggered heat sinks or Case 10 and Case 11. The remarkable low ΔT_b as obtained is 1.7175 K for Case 8. Heat sinks with an aligned protrusion, aligned protrusion, and dimple contribute larger augmentation in heat transfer coefficient than the staggered sinks. The underlying reason is that aligned protrusion and aligned dimple cause rigorous mixing of fluid layers, which in turn reduces maximum bottom wall temperature difference (ΔT_b) considerably. Thus, h for aligned and staggered models are 42205.31 and 40192.6 W/(m² K), respectively, at the same fluid velocity of 4 m/s. In the study of pumping power, it is apparently seen that the aligned heat sinks absorb more pumping power than staggered models as per their larger fiction factor for almost all ranges of Reynolds number. The constricted passage between two protrusions in the aligned model is responsible for it.

[figure(s) omitted; refer to PDF]

One key performance parameter of the microchannel is Nu/Nu_o. The aligned protruded sink, Case 8, is providing impressive results in terms of Nu/Nu_o, as shown in Figure 15(a). The protrusions contribute excellently to enhancing heat transfer because the constriction position of protrusions brings in significant mixing phenomena into the channel. Front and downstream portions of protrusion affect differently on heat dissipation. The front part of the protrusion creates sudden obstruction in the coolant flow, and fluid impinges on it. Thus the high amount of heat dissipates at the frontal position, thereby increases Nu locally at the frontal face of protrusion. In contrary, the downstream portion of protrusion is subject to lower heat transfer and a place of fluid separation. Nu/Nu_o of Case 8 reaches a maximum of 1.22 at Re = 346, whereas the largest Nu is 11.25 for this case at Re = 673. Figure 15(b) expresses an interesting relationship between the heat transfer enhancement factor and Re. The apparent trend of η is that it increases with an increase of Re and reaches a peak value at a certain Re, but then it starts falling with an increase of Re and reaches the lowest value at the maximum Re considered in the study. It reveals that the volume flow rate with corresponding Re should be customized to obtain the optimum performance parameters.

[figure(s) omitted; refer to PDF]

This evaluation based on heat transfer and friction loss both concludes that staggered heat sinks are superior to aligned, although aligned models exhibit better ΔT_b, h, and Nu/Nu_o. This happens due to the fact that vertically aligned heat sinks result in higher pressure drop in comparison to staggered heat sinks owing to the rigorous mixing of fluid layers. As per the study output, for Case 8, pressure drop ranges from 6.33 KPa to 91.98 KPa and for Case 10, it ranges from 5.12 KPa to 66.18 KPa for the same range of fluid velocities. The findings indicate that employment of dimple or protrusion, irrespective of the number and positional pattern, always helps in Nusselt number improvement, signifying higher heat release. This improvement is associated with pressure penalty, but even though overall performance, i.e., η remains augmented. In contrary, for Case 8 at Re = 700, the pressure drop becomes very high, which reduces η.

3.4. Group D

In this part, only the staggered sets are analyzed. MCHS with staggered protrusion and MCHS with staggered protrusion combined with dimple are compared. Thermal characteristics along with the hydraulic characteristics of these heat sinks are studied. Case 12 consists of protrusion only. Case 13 is fitted with protrusion and dimple. The staggered pattern indicates that two protrusions or two dimples are not at the same streamwise location. Both the cases have a pitch distance of 1 mm, whereas Case 14 has a pitch distance of 0.5 mm combined with dimple and protrusion. In this analysis, both layer 1 and layer 2 are fitted with dimple and protrusions. To compare the cases, velocity contour, pressure contour, temperature contour, and variation of ΔT_b, Nu/Nu_o, and η are studied. The heat sinks studied in Group D are performing well in all aspects compared to the sinks of Group A, B, and C.

Figure 16 illustrates velocity, pressure, temperature contours in y-z plane at the middle span of the channel along x direction for Case 0, Case 12, Case 13, and Case 14. The channel section in the streamwise direction for the profile plotting is taken in such a way that the presence of dimple and protrusions are visualized. Average bottom wall temperature and maximum interior temperature get reduced for these three cases. Furthermore, better uniformity in temperature distribution in the bottom wall, as well as liquid coolant flow, is clearly observable for Case 14 among the three cases. All contours shown are taken for u_in = 4 m/s.

[figure(s) omitted; refer to PDF]

Counterflow in microchannel heat sink always leads to very low ΔT_b and the addition of protrusion and dimple in the channel structure implicates more reduction in ΔT_b. Figure 17 depicts the comparison of ΔT_b of Case 12, Case 13, and Case 14 at different Re. Case 14, where staggered protrusion and dimple are used, shows an excellent reduction in maximum bottom wall temperature difference, signifying the low thermal stress. ΔT_b as low as 1.48 K is determined for Case 14 at Re = 672. Case 12 and Case 13 also bring a suitable decrease in ΔT_b. In addition to a better uniform temperature, Case 14 also provides augmented h. The short pitch of protrusion and dimple in the channel causes more interception of the boundary layer, frequent intermixing of fluids which in turn enhances convection heat transfer. In the case of the smooth base channel, the boundary layer develops over a longer distance. The introduction of more number of protrusion and dimples in the channel helps in enlarging the surface area too. Increase of protrusion and dimple in Case 14 influences pumping loss to be increased at all Re.

[figure(s) omitted; refer to PDF]

Figure 18 compares the property profiles of the bottom channel for Case 12, Case 13, and Case 14. Momentum transfer takes place strongly in the region where protrusion and dimple are placed. As seen in Figure 18(a), change in velocity in streamwise and in normal direction occurs differently at different locations. For Case 14, the core flow is significantly disturbed by the frequent position of staggered protrusion and dimples which in turn helps to mix the hotter and cooler fluids. And due to this, the average bottom wall temperature and maximum bottom wall temperature difference for Case 14 decreases to a minimum in comparison to Case 12 and Case 13. Fluids get accelerated and do intensify inertia at the constricted space of protrusions for all three cases, whereas it reduces its velocity at a diverging span of dimple as merely observed in Figure 18(a). As per the illustration referring to Figure 18(b), the pressure gradient is not only found in flow direction; in fact, fluid pressure fluctuates in normal or radial direction too. The pressure reduction at the downstream face of protrusions leads to fluid separation and wake formation. On the contrary, a negative pressure gradient at the end region of dimples creates recirculation. How the temperature of flowing fluids is regulated by the position of protrusion and dimples can be apparently elaborated by Figure 18(c). The channel average wall temperature is observed to be quite lesser in Case 12, Case 13, and Case 14 than in the base channel. Fluid temperature across the channel for Case 14 is decreased substantially than that of Case 12 and Case 13 and offers a more uniform temperature all over the channel section. For example, the average interior fluid temperature of Case 14 has been determined as 294.20 K at the highest inlet velocity. Along the smooth constant section of the channel, the fluid tends to cling to the wall, but protrusion and dimples together break this mechanism and result in substantial convection heat exchange. It is agreeable that upstream portion of protrusion provides high inertia and high heat transfer [14]. It is found that wall temperature near the inlet of a dimple is higher than other regions. The increase in the wall temperature near the inlet of a dimple and the trailing edge of a protrusion is attributed to the diverging passage of the flow in each of these regions.

[figure(s) omitted; refer to PDF]

A pictorial view of the variation of Nu/Nu_o with the change of Re for Case 12, Case 13, and Case 14 is illustrated in Figure 19(a). Combination of dimple and protrusion is proved to be worthy of the investigation. The presence of dimple and protrusion at the same transverse position of flow brings heat transfer enhancement. The frontal region of protrusion and downstream of dimple contribute locally in better heat dissipation and improved Nusselt number. Larger number of protrusion and dimple results in pressure gradient in transverse direction in more sections of water flow introducing vortices inside dimples and behind the protrusions. This is the physics behind the improvement in the average Nusselt number for Case 14. For this case, Nu are 7.59 and 11.33 at Re = 99.48 and Re = 672, respectively. Renold’s number has a notable impact on the Nu/Nu_o. At low point Re = 100, Nu/Nu_o is the largest and with an increase of Re, the ratio starts falling rapidly for Case 14. The Nu/Nu_o of Case 14 is 22% and 2.5% bigger than that of Case 12 at the lowest and highest Re, respectively.

[figure(s) omitted; refer to PDF]

The pressure drop plays a vital role in the improvement of η. Heat transfer enhancement factor(η) will not increase with the augmented heat transfer if the pressure drop rises. The maximum η for Case 12 is 1.114 and for Case 13 is 1.109. In Case 14, cumulative friction reduces overall performances at higher Re. So for Case 14, the highest η is 1.32 at the lowest velocity. The pitch length of 0.5 mm brings the best effect at a low flow rate into the channel. The relationship between the heat transfer enhancement factor and Re for Case 12, Case 13, and Case 14 is portrayed in Figure 19(b). Below Re = 500, Case 14 produces highest η followed by Case 12 and Case 13. At the largest Re, this sequence becomes the opposite.

Figures 20 and 21 visualize how property contours differ in layer 1 and layer 2. As observed in Figure 20, pressure at the inlet differs significantly from pressure at the outlet for both the channel layers. The total pressure drops are 72.47 KPa at layer 1 and 73.40 KPa at layer 2, respectively, for the heat sink represented by Case 14 at an inlet velocity of 4 m/s. The lower channel remains hotter than the upper channel and the temperature of the coolant is hotter at the outlet than that of the inlet at each layer, as visible in Figure 21. The corresponding average interior temperature of layer 2 is 293.91 K, lesser than that of layer 1. The average interface temperature and maximum interface temperature for layer 2 are also lower. The top channel receives a maximum interface temperature of 297.8 K, which is about 4 K lesser than that of the lower channel.

[figure(s) omitted; refer to PDF]

The pumping power may also be studied to judge the performances of the present improved version of the double-layer channel (DL-MCHS), especially Case 12, Case 13, and Case 14. The addition of dimple and protrusions apparently compel the pressure drop to rise, thereby increasing the pumping power of channels. Therefore, the augmented Nusselt number must be justified by the rise of the pumping power. As such, Case 12, Case 13, and Case 14 are associated with an accountable rise of the pumping power. To justify this phenomenon, the second overall performance parameter as defined below can be determined.

The second overall performance factor or thermal-hydraulic performance factor (η₁) [26] is defined as follows:$$\begin{array}{}\text{(12)}& {\eta }_1={\frac{Nu/N{u}_0}{PP/P{P}_0}}^{1/3}.\end{array}$$Here, PP = total pumping power for the present cases considering pumping powers of both upper layer and lower layer and PP₀ = total pumping power for Case 0 considering pumping powers of both upper layer and lower layer.

Figure 22(a) shows the variation of PP/PP₀ with Re for Case 12, Case 13, and Case 14. Pumping power hikes are 19%, 20%, and 30% for Case 12, Case 13, and Case 14, respectively, than the Case 0 at inlet velocity of 4 m/s.

[figure(s) omitted; refer to PDF]

Figure 22(b) depicts the variation of η₁ with Re for Case 12, Case 13, and Case 14. The graphical representation of the thermal-hydraulic performance factor (η₁₎ in terms of pumping power, expresses the results close to the results of η evaluation. Although pumping power for Case 14 rises remarkably, the significant enhancement of Nu promotes the hike of thermal-hydraulic performance factor (η_1). The 33% increase in η₁ for Case 14 suggests that its structural design is a suitable one due to heat transfer augmentation as well as higher overall performance irrespective of pumping power loss.

4. Conclusions

(i) In parametric study, it has been noted that although a high channel performs well, a channel with bigger width impede the performance, because of its higher thermal resistance. It suggests that the thick fluid entity in the heat sink is not capable of better convections since the core flow and core fluid layers remain unaffected. The dimension of Case 3 is found optimum since it exhibits low thermal resistances in all ranges of Re with the lowest R = 0.119 K/W at Re = 678, which is 48%, 15%, 18%, and 2.5% lower than that of Case 1, Case 2, Case 4, and Case 5, respectively, at the corresponding fluid velocities.

(ii) The findings indicate that the presence of dimple or protrusion, irrespective of the number and positional pattern, always helps in Nusselt number improvement, signifying higher heat release. This improvement is associated with pressure drop as a penalty. However, overall performance; i.e., η remains augmented. Heat transfer enhancement factor η is greater than 1 for all cases at almost all ranges of Re. It is understood through the plotted temperature profiles and calculation of maximum temperature differences of the fourteen cases that the temperature gradient which causes thermal stress can be controlled by virtue of faster heat transfer, larger coolant volume flow rate. The employment of dimple and protrusion separately or combined way promotes well temperature distribution across the sink substrate and liquid.

(iii) Volume flow rate of coolant plays a crucial role in the performance. It is a notable observation that low volume flow rate, as well as high volume flow rate, do not result in better overall performance in most of the cases. In those cases, the volume flow rate with corresponding Re should be customized to obtain the optimum overall performance parameter.

(iv) It has been realized that aligned models offer higher heat transfer coefficient, maximum Nu, and minimum ΔT_b, but in terms of overall performance, staggered sinks are superior. The aligned protruded sink, Case 8 is, providing the highest h of 42.20 KJ/m²k and the largest Nu of 11.25 at Re = 673. However, staggered sinks offer a better heat transfer enhancement factor (η). Heat sinks with an aligned protrusion, heat sinks with aligned protrusion, and dimple contribute larger augmentation in heat transfer coefficient than the heat sinks with staggered design. The underlying reason is that aligned protrusion and aligned dimple cause rigorous mixing of fluid layers, which in turn reduces maximum bottom wall temperature difference considerably. In the study of pumping power, it is apparently seen that aligned heat sinks absorb more pumping power than staggered models due to their larger fiction factor. Hence, staggered models perform better in the overall term.

(v) The heat sink with both layers protruded and dimpled, showing Nu/Nu_o and η of 1.36 and 1.32, respectively, is observed as one of the best sinks. It offers an excellent lowest ΔT_b of 1.48 K, revealing 63% decrease from that of a smooth base channel.

Glossary

Nomenclature

$A_{c1}$:Cross sectional area of bottom channel, m²

$A_{c2}$:Cross sectional area of top channel, m²

$A_{con}$:Convection heat removal area, m²

D_d:Diameter of dimple, μm

$D_h$:Hydraulic diameter, m

D_p:Diameter of protrusion, μm

H_b:Base height, μm

H_c:Channel height, μm

L:Total length of the microchannel, mm

N:Number of channels at each layer

Nu:Nusselt number

P:Pitch distance between two consecutive dimple of protrusion, mm

$Q_{b,t}$:Total heat applied at bottom wall, W

R:Thermal resistance, K/W

R:Reynolds number

T:Temperature variable of fluid, K

$T_{av}$:Average fluid temperature of channel, K

$T_{b,av}$:Average temperature of bottom wall, K

T_b,max:Maximum bottom wall temperature, K

T_b,min:Minimum bottom wall temperature, K

T_in:Inlet Temperature of fluid, K

$T_s$:Temperature variable of silicon, K

V:Velocity vector

W:Total width of heat sink, mm

W_c:Channel width, μm

W_w:Channel wall thickness, μm

$c_v$:Specific heat constant of fluid, J/(kgK)

d_d:Depth of dimple, μm

$d_h$:Hydraulic diameter, μm

d_p:Depth of protrusion, μm

f:Fanning’s friction factor

h:Heat transfer coefficient, W/(m² K)

k:Thermal conductivity of fluid, W/(mK)

${\text{k}}_{\text{s}}$:Thermal conductivity of silicon, W/(mK)

p_out:Outlet fluid pressure, Pa

q:Heat flux at the bottom wall, W/m²

t_m:Thickness of wall between bottom and top channel, μm

u:Average flow velocity, m/s

u_in:Inlet fluid velocity, m/s

ρ:Fluid density, Kg/m³

$\eta$:Thermal performance factor

Δp:Pressure difference across channel, Pa

µ:Fluid viscosity, Kg/(ms).

0:Base Heat sink

1:Bottom channel

2:Top channel

in:Inlet

out:Outlet.

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