Thermal enhancement of microchannel heat sink using pin-fin configurations and geometric optimization

The rectangular cross-section MCHS has been chosen for this study as it exhibits the least thermal resistance[32]. The computational domain and geometry of the microchannel heat sink with three pin-fin configurations used are shown in figures 1 and 2. The diameter of the pin is kept constant in all configurations. The geometrical details of the computational domain and pin fins are described in table 1. The dimensions used are consistent with the experimental data of [1]. The thermophysical properties of the solid (copper) and liquid (water) are given in table 2. Constant heat flux has been applied to the base of the MCHS. Cold water enters the MCHS inlet at 293.15K with different velocities corresponding to the Reynolds Number range of 100–1000. All the walls of the MCHS that are exposed to the surroundings are perfectly insulated, except the base wall where the heat flux has been applied.

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Table 1. Physical dimensions [1].

S.no	Symbol and description	Values (mm)
1	${P}_{x}$(Transverse Pitch)	1
2	${P}_{y}$ (Longitudinal Pitch)	0.25
3	$D$ (Diameter)	0.035
4	${W}_{{ch}}\,$(Width of Channel)	0.35
5	${H}_{{ch}}$ (Height of Channel)	0.5
6	${L}_{{ch}}$ (Length of Channel)	10
7	$W$ (Width)	0.6
8	$H$ (Height)	1.5

Table 2. Material properties [1].

Material properties	Copper	Water
Density $\left({\boldsymbol{kg}}{\boldsymbol{/}}{{\boldsymbol{m}}}^{{\boldsymbol{3}}}\right)$	8978	998.2
Dynamic Viscosity $\left({\boldsymbol{kg}}{\boldsymbol{/}}{\boldsymbol{m}}{\boldsymbol{/}}{\boldsymbol{\sec }}\right)$	—	0.001003
Specific Heat ${\boldsymbol{(}}{\boldsymbol{J}}{\boldsymbol{/}}{\boldsymbol{kg}}{\boldsymbol{/}}{\boldsymbol{K}}{\boldsymbol{)}}$	386	4184
Thermal Conductivity $\left({\boldsymbol{W}}{\boldsymbol{/}}{\boldsymbol{m}}{\boldsymbol{/}}{\boldsymbol{K}}\right)$	387.6	0.6

The following assumptions are made to simplify the numerical modeling. The fluid is considered Newtonian and incompressible with a no-slip boundary condition at the wall. Due to the low Reynold's number, the flow is laminar'[33]. The thermal and physical properties of the solid and fluid domains are considered constant. Heat transfer and flow rate are analyzed under steady-state conditions. Additionally, natural convection and radiation modes of heat transmission are disregarded, and the impact of body force, gravity, and viscous heat dissipation are neglected. Walls are assumed to be smooth to ignore the effects of surface roughness and thermal slip length.

The following governing equations have been used for the numerical modeling based on the assumptions discussed earlier.

The continuity equation for steady, incompressible flow [34] is given as

$$\begin{eqnarray}&&\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0\end{eqnarray} \tag{ 1 }$$

$u,$ $v$ and $w$ are components of fluid velocity in x, y, and z directions, respectively

The momentum equation for steady, incompressible flow [34] is given as

$$\begin{eqnarray}&&\rho \left(V.{\rm{\nabla }}\right)V=-{\rm{\nabla }}P+\mu {{\rm{\nabla }}}^{2}V\end{eqnarray} \tag{ 2 }$$

The energy equation for steady, incompressible flow[34] is as follows;

$$\begin{eqnarray}&&u.{\rm{\nabla }}T=\frac{\lambda }{\rho {{\rm{C}}}_{p}}{{\rm{\nabla }}}^{2}T\end{eqnarray} \tag{ 3 }$$

The energy equation in the solid domain is

$$\begin{eqnarray}&&\lambda {{\rm{\nabla }}}^{2}T=0\end{eqnarray} \tag{ 4 }$$

Where $u$ is the velocity at the inlet of the microchannel, $\rho$ is the density of water, $P$ is the pressure and ${C}_{p}$ Is the specific heat at constant pressure.

The boundary conditions to solve the governing equations for three configurations of MCHS are given in table 3.

Table 3. Boundary condition for MC-BW, MC-SW, MC-Mixed.

Inlet Flow Velocity	$u={u}_{{in}}$
Inlet Temperature	${T}_{{in}}$ = 293.15K
Outlet Pressure	$P={P}_{{out}}=1{\rm{atm}}$
Constant Heat Flux at Bottom Wall	$-{k}_{s}\frac{\partial {T}_{s}}{\partial y}={q}_{w}$ = 1,000,000 W m−2
At Solid–liquid Interface	$u=v=w=0$
	$-{k}_{s}\frac{\partial {T}_{s}}{\partial n}=-{k}_{f}\frac{\partial {T}_{f}}{\partial n}$
Adiabatic wall Boundary	$\frac{\partial {T}_{f}}{\partial x}=\frac{\partial {T}_{s}}{\partial x}=0$

2.3.1. Calculation parameters

The relevant expressions are provided in this subsection to calculate the heat transfer properties and fluid flow in MCHS. The Reynolds number (Re) can be defined as:

$$\begin{eqnarray}&&{Re}=\frac{{\rho }_{f}{u}_{m}{D}_{h}}{{\mu }_{f}}\end{eqnarray} \tag{ 5 }$$

${{\rm{u}}}_{{\rm{m}}}$ is the average or mean velocity of the fluid and ${{\mu }}_{{\rm{f}}}$ is the dynamic viscosity of the fluid. Where the hydraulic diameter is given as:

$$\begin{eqnarray}&&{\,D}_{h}=\frac{2{H}_{{ch}}{W}_{{ch}}}{{H}_{{ch}}+{W}_{{ch}}}\end{eqnarray} \tag{ 6 }$$

The friction factor represents the fluid's pressure loss when interacting with the surface and is used to find the thermal enhancement factor.

$$\begin{eqnarray}&&f=\frac{2{D}_{h}\unicode{x02206}p}{{L}_{{ch}}{\rho }_{f}{u}_{{m}^{2}}}\end{eqnarray} \tag{ 7 }$$

Where ${L}_{{ch}}$ denotes the whole length of the channel, and ∆p denotes the pressure decrease throughout its length.

The average heat transfer coefficient is calculated as follows:

$$\begin{eqnarray}&&h=\frac{{q}_{w}{A}_{b}}{2\left({W}_{{ch}}+{H}_{{ch}}\right){L}_{{ch}}\unicode{x02206}T}\end{eqnarray} \tag{ 8 }$$

${{\rm{q}}}_{{\rm{w}}}$ is heat transfer through the base, ${{\rm{A}}}_{{\rm{b}}}$ is the area of the base

$$\begin{eqnarray}&&\unicode{x02206}T={T}_{w}-{T}_{f}\end{eqnarray} \tag{ 9 }$$

${{\rm{T}}}_{{\rm{w}}}$ Average channel wall temperature while ${{\rm{T}}}_{{\rm{f}}}$ the average temperature of the fluid.

Nusselt number is the ratio of convective heat transfer to conduction at the fluid boundary.

$$\begin{eqnarray}&&{Nu}=\frac{h{D}_{h}}{{k}_{f}}\end{eqnarray} \tag{ 10 }$$

The overall performance of the microchannel is calculated by finding the thermal enhancement factor that defines the ratio of the increase of Nusselt number against an increase in friction factor.

$$\begin{eqnarray}&&{\eta }=\frac{{Nu}/N{u}_{0}}{\sqrt[3]{f/{f}_{0}}}\end{eqnarray} \tag{ 11 }$$

The total thermal resistance is written as,

$$\begin{eqnarray}&&{\,R}_{{tot}}={R}_{{cond}}+{R}_{{conv}}+{R}_{{cap}}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{R}_{{tot}}=\frac{{T}_{b}-{T}_{w}}{{q}_{w}{A}_{b}}+\frac{{T}_{w}-{T}_{f}}{{q}_{w}{A}_{b}}+\frac{{T}_{f}-{T}_{{in}}}{{q}_{w}{A}_{b}}=\frac{{T}_{b}-{T}_{{in}}}{{q}_{w}{A}_{b}}\end{eqnarray} \tag{ 13 }$$

The average wall temperature is given as follows:

$$\begin{eqnarray}&&{T}_{w}=\frac{\int {T}_{w-x,y}{dydx}}{\int {dydx}}\end{eqnarray} \tag{ 14 }$$

The average base temperature is calculated by:

$$\begin{eqnarray}&&{\,T}_{b}=\frac{\int {T}_{b-x,y}{dydx}}{\int {dydx}}\end{eqnarray} \tag{ 15 }$$

Average bulk fluid temperature is given by:

$$\begin{eqnarray}&&{T}_{f}=\frac{\int {T}_{f-i,x}{\rho }_{f-i,x\,}\left|\vec{v}.\vec{{dA}}\right|{dx}}{\int {\rho }_{f-i,x\,}\left|\vec{v}.\vec{{dA}}\right|{dx}}\,\end{eqnarray} \tag{ 16 }$$

Net Total rate of entropy generation:

$$\begin{eqnarray}&&{\dot{S}}_{{gen}}={\dot{S}}_{{gen},\unicode{x02206}T}+{\dot{S}}_{{gen},\unicode{x02206}P}\end{eqnarray} \tag{ 17 }$$

Entropy generation due to heat transfer:

$$\begin{eqnarray}&&{\dot{S}}_{{gen},\unicode{x02206}T}={\iiint }_{{\rm{\Omega }}}\,{\dot{S}}_{{gen},\unicode{x02206}T}^{{\rm{\text{'}\text{'}\text{'}}}}{dV}=\frac{{q}_{w}{A}_{{base}}\left({T}_{W}-{T}_{f}\right)}{{T}_{f}{T}_{W}}\end{eqnarray} \tag{ 18 }$$

Entropy generation due to pressure losses:

$$\begin{eqnarray}&&{\,\dot{S}}_{{gen},\unicode{x02206}p}={\iiint }_{{\rm{\Omega }}}\,{\dot{S}}_{{gen},\unicode{x02206}p}^{{\rm{\text{'}\text{'}\text{'}}}}{dV}=\frac{\dot{m}}{{\rho }_{f}{T}_{f}}\unicode{x02206}p\end{eqnarray} \tag{ 19 }$$

${N}_{s}$ denotes the total augmented entropy generation rate. This is the ratio of the MCHS's overall volumetric generation rates with various configurations of trefoil ribs.

$$\begin{eqnarray}&&{N}_{s}=\frac{{\dot{S}}_{{gen}}}{{\dot{S}}_{{gen},0}}\end{eqnarray} \tag{ 20 }$$

The ANSYS FLUENT 2021 R2 software, based on the finite volume method, is used to conduct three dimensional steady laminar flow numerical simulations. The QUICK and second upwind interpolation schemes solve the diffusive and convective components of the momentum, energy, and continuity equations. Additionally, the SIMPLE algorithm is utilized for the pressure-velocity coupling. The numerical solution is truncated when the residual of momentum, energy, and continuity equations are reduced to 10⁻⁶ level.

A tetrahedral mesh is created to divide the solid and fluid computational domain into finite elements. Four alternative element sizes are evaluated to determine whether the results depend on the number of elements. For the computational domain with four different element sizes, the temperature distribution profile is plotted along the height of the channel. The number of elements in coarse, medium, fine, and very fine grids is 696673, 760310,948704, and 1008215, as illustrated in table 4. It can be seen that the results using a fine grid (984704 elements) and a very fine grid (1008215 elements) overlap, as shown in figure 3. Since results do not change beyond a fine grid, hence mesh size of 98470 elements is used throughout this study.

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The Nusselt Number and friction factor obtained from numerical results are compared with the experimental data from Wang et al [35] for validation purposes, as shown in figure 4. The operating conditions, geometrical parameters, and materials used in the simulation are the same as the experimental data. The maximum error between the current numerical simulation and experimental data for the Nusselt number and friction factor is 0.85% and 0.95% respectively. It can be seen clearly that numerical results are in good agreement with the experimental results.

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When pin fins are added to the walls of MCHS, they act as barriers to the flow, which results in flow separation. The pressure difference between the high-pressure region upstream of the fin and the low-pressure region downstream causes wake formation. This wake formation creates regions of low velocity and increases turbulence, as shown in figure 5. Vortices are shed periodically in the wake region when the fluid flows past the pin. In the wake zone, vortices increase mixing, generate turbulence, and improve heat transmission in the microchannel. Vortices and flow recirculation regions increase the convective heat transfer characteristics, causing more heat dissipation from the heated surface.

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Figure 6 displays the velocity contours of various pin-fin configurations. When the MC-SC is used, a fully developed boundary layer forms. When pin-fins are introduced in the microchannel, the boundary layer is disrupted before it is fully developed. The pin-fins disturb the flow dynamics, preventing the boundary layer from reaching its fully developed state. This disruption can affect heat transfer and flow behavior within the microchannel.

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Figure 7 illustrates the pressure distribution within the channel for different MCHS configurations. A stagnation zone is visible upstream of the pin fins, and a low-pressure zone is evident downstream, forming a recirculation zone. The inclusion of pin-fins configuration results in a higher pressure drop as the fluid encounters a flow restriction.

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Mixed pins increase the turbulence in the MCHS, increasing the energy dissipation in the fluid, thus leading to non-uniform flow distribution across the microchannel heat sink surface. The non-uniform flow distribution forms stagnant zones where the fluid velocity is lower and has a higher resistance to flow, leading to a higher pressure drop across the microchannel heat sink. Figure 8 displays the pressure drop for different MCHS configurations at varying Reynolds numbers. As the velocity increases, the wall shear stress increases, requiring more pressure to overcome it, leading to an increasing trend in pressure drop with the Reynolds number. Figure 8 demonstrates that the MCHS with pin-fin configurations has a higher pressure drop than MC-SC because of additional irreversible pressure losses due to wake formation downstream of the pin-fins and frictional losses. MC-MIXED has the highest pressure drop at all Reynolds numbers because the presence of pin-fins on both sides and the bottom of the channel wall creates more disturbance in flow.

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In an MCHS with pin fins, wall shear stress increases due to increased flow obstruction. Because of pin-fins, more interaction occurs between the fluid and the channel surface, which increases wall shear stress on the MCHS surface. Figure 9 shows the correlation between the Reynolds number range and wall shear for different microchannel pin-fin arrangements and a smooth microchannel (MC-SC). The results indicate that wall shear increases with an increase in Reynolds number. The smooth microchannel exhibits the lowest wall shear since there are no pins present to obstruct fluid movement. On the other hand, the MC-MIXED configuration has the highest wall shear due to pins on both the side and base walls of the microchannel, which restrict fluid passage and reduce its velocity along the flow direction. Among the pin-fin configurations, MC-SW exhibits the lowest wall shear, whereas MC-BW has a lower wall shear than MC-MIXED but a higher wall shear than the MC-SW configuration.

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Figure 10 shows the relationship between the Reynolds number and Darcy friction coefficient [36] for various pin configurations and smooth channels. The graph indicates that the friction coefficient slightly increases in smooth channels as the Reynolds number decreases. This occurs because the velocity of the fluid is increasing, leading to a decrease in friction factor. Conversely, the highest increase in friction coefficient with the lowest Reynolds number occurs when pins are configured with their side walls. The friction factor is highest for all pin-fin configurations at Re = 200. Among different configurations, the MC-MIXED configuration exhibits the highest increase in friction factor with Re due to the presence of more pins contributing to the friction in the microchannel compared to other configurations. In contrast, the MC-SW has the lowest increase in friction factor with Re compared to other pin-fin configurations.

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This study aims to propose an efficient microchannel heat sink (MCHS) that can effectively dissipate heat from the channel. Figure 11 demonstrates that the MC-SC has a high temperature and non-uniform temperature distribution compared to pin-fin configurations due to its low Nu and high thermal resistance. On the other hand, MCHS with pin-fin configurations shows a lower temperature and a more uniform temperature distribution, as it dissipates a large amount of heat to the fluid due to low thermal resistance for convection. Non-uniform temperature distribution induces thermal stresses in both the MCHS and the host chip, harming electronic chip safety and integrity. By employing MCHS with pin-fin configurations, not only can a superior cooling performance be achieved by dissipating a large amount of heat from the electronic chip, but it also promotes the safe operation of these electronic chips by eliminating thermal stresses and high-temperature hot spots.

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Heat transfer effectiveness between the channel walls and fluid through convection is commonly measured using the Nusselt Number, a dimensionless parameter. Figure 12 depicts the relationship between the Nusselt Number (Nu) and the Reynolds number (Re). As the graph shows, the pin-fin configuration MCHS employs on its channel walls leads to a higher Nusselt number due to improved fluid mixing and flow disturbance. Furthermore, the convection heat transfer coefficient improves at higher flow velocities, increasing Nu with Re. Among the various pin-fin configurations, MC-SC exhibits the smallest increase in Nusselt number with Reynolds number. This is because of boundary layer development, which causes a decrease in local Nusselt numbers.

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On the other hand, the highest increase in Nusselt number is observed in the MC-MIXED configuration. As a result, MC-MIXED offers the highest convective heat transfer and can remove the most heat from the channel compared to the other configurations discussed. Meanwhile, MC-SW shows the lowest increase in Nusselt number with an increase in Re among the different pin-fin configurations.

Figure 13 illustrates the correlation between thermal resistance and Reynolds number. As thermal resistance declines, the rate of heat transfer increases. The mixed pins in MCHS increase the fluid turbulence, which breaks down the boundary layer and increases the mixing between the fluid and the channel walls, leading to more effective heat convection. As a result, the thermal resistance of the microchannel heat sink decreases. The trend in the graph suggests that thermal resistance decreases with increasing Reynolds number, primarily due to the increased velocity of the fluid, leading to an improvement in the convective heat transfer coefficient. Among the different configurations, MC-Mixed exhibits the lowest thermal resistance and, thus, the highest Nu due to its increased surface area for heat dissipation. In contrast, MC-SC has the highest thermal resistance due to its low Nusselt number and less wall surface area for heat dissipation, resulting in a lower heat transfer rate.

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Although the pin-fins improve the thermal performance of MCHS, they also decrease hydraulic performance, owing to an increase in the pressure drop within the channel. To assess the overall performance enhancement of MCHS, the thermal enhancement factor is computed by comparing the Nusselt number improvement resulting from the pin-fins with the increase in pressure drop. The addition of pin fins increases the surface area available for heat transfer. It thus enhances the convective heat transfer coefficient, which increases the thermal enhancement factor. Figure 14 illustrates the variation of the thermal enhancement factor with Re. The thermal enhancement factor increases until Re = 600, indicating that the Nusselt number improvement outweighs the pressure losses. However, after Re = 600, the thermal enhancement factor declines because pressure losses surpass the Nusselt number improvement. Among the different configurations, the Mixed wall configuration has the highest thermal enhancement factor of η = 1.4 at Re = 600, and the MC-SW configuration has the lowest thermal enhancement factor.

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The presence of the pin fins causes the fluid flow in MCHS to encounter greater resistance. As the Reynolds number increases, the frictional losses and pressure drop also increase. The MCHS with pin-fins has a greater surface area-to-volume ratio than the MCHS without pins, which leads to faster heat transfer rates. Hence, a higher surface area to volume ratio and greater resistance to fluid flow results in higher pressure losses, which in turn cause a more significant generation of entropy. Figure 15 illustrates the correlation between the Reynolds number and the entropy generated due to pressure drop within MCHS. Pressure losses inside MCHS lead to entropy generation [34]. Smooth channels have the lowest pressure drops, resulting in lower entropy generation. On the other hand, the presence of pin fins leads to more irreversibilities and greater pressure drop, resulting in higher entropy generation. MC-MIXED exhibits the highest entropy generation among the different configurations due to the maximum number of pins. At the same time, MC-SW shows the lowest entropy generation among the pin-fin configurations.

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Entropy generation due to temperature occurs due to irreversibilities in the heat transfer process, resulting in the dissipation of energy in the form of heat and an increase in the overall entropy of the system. Figure 16 illustrates the correlation between the Reynolds number and the entropy generation resulting from heat transfer. As the Reynolds number increases, the entropy generation due to heat transfer decreases. MC-MIXED exhibits the lowest entropy generation among the pin-fin configurations, while MC-SC has the highest entropy generation due to heat transfer among all pin-fin configurations.

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Figure 17 shows that the total entropy generation decreases with an increase in the Reynolds number. The entropy generation due to temperature has a more significant impact on the overall entropy generation than the entropy generation due to pressure drop. MC-SC exhibits the highest rate of entropy formation and the slowest rate of heat transmission. Among the pin fin configurations, MC-SW displays the highest rate of entropy formation. In contrast, the MIXED configuration has the lowest entropy generation due to the highest heat transfer rate in MC-MIXED.

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Figure 18 displays the relationship between the Reynolds Number and the augmentation entropy generation number (N_s ) for various pin-fins configurations. When ${N}_{s}\,$ is less than 1, the pin-fin configuration performs better regarding heat transfer [34]. Thus, different pin fin configurations should have ${N}_{s}$ less than one to achieve better cooling performance than a smooth channel, which has a ${N}_{s}$ value of 1. The addition of pin-fins in MCHS reduces the total entropy augmentation number by reducing the contribution of entropy generation caused by fluid friction. The entropy augmentation number decreases due to enhanced mixing, increased heat transmission, and less friction from pin fins. With an increase in Reynolds number, almost all pin fin configurations show a declining trend, reaching their maximum at Re = 200 for side wall pin fin configuration. After Re = 600, the ${N}_{s}$ increases due to the rise in pressure drop. Among different configurations, the MC-Mixed pin fins arrangement exhibits the lowest ${N}_{s}$ and gives the best performance in terms of heat transfer, while MC-BW has the lowest increase in ${N}_{s}$ with Re.

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Figure 19 compares the Reynolds number and transport efficiency, indicating that the transport efficiency increases with an increase in the Reynolds number. Among the different configurations, the MC-MIXED configuration exhibits the maximum transport efficiency due to the least thermal resistance, resulting in higher heat transfer than MC-SC, which has the lowest transport efficiency.

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The RSM is a collection of statistical methods for creating experimental models with responses and independent variables. These techniques aim to evaluate and enhance a response that depends on several independent variables. In each experiment, the independent variables are changed to determine the reason for these variations in the results. Using this method, the most important variable may be found, and its interactions with other process factors on the examined responses are also conceivable. Before being broadened to mimic numerical trials, Box and Draper's RSM [37] was initially developed to simulate experimental reactions. The second-order polynomial has been demonstrated to be the best equation for fitting the studied data. Using this technique, experimental models are fitted.

$$\begin{eqnarray*}&&Y={\beta }_{o}+\displaystyle \sum _{i=1}^{k}{\beta }_{i}{X}_{i}+\sum \displaystyle \sum _{i\lt j}{\beta }_{{ij}}{X}_{i}{X}_{j}\end{eqnarray*}$$

$$\begin{eqnarray}&&+\,\displaystyle \sum _{i=1}^{\,k}\,{\beta }_{{ii}}{X}_{{i}^{2}}+\varepsilon \end{eqnarray} \tag{ 21 }$$

Where the polynomial coefficients are ${\beta }_{o},$ ${\beta }_{i},$ ${\beta }_{{ij}},$ and ${\beta }_{{ii}}.$ The inputs of the model are given the values. ${X}_{i}$ and ${X}_{j}.$ The number of model inputs and the examined response are indicated, respectively, by the letters k and $Y.$ [36]

With the addition of the pin-fins, thermal performance increases; to find the optimized values of transverse pitch (${P}_{x}$), longitudinal pitch (${P}_{y}$), and the diameter of the pin (${D}_{{pin}}$) for base wall pins, RSM is used. The range of three independent variables and the goal of optimization are shown in table 5. To find optimal geometry, RSM optimization is used at Re 300, which resulted in the geometry of transverse pitch (${P}_{x}$) = 2.50 mm, longitudinal pitch $({P}_{y})$= 0.25 mm, and the diameter of the pin (${D}_{{pin}}$) = 0.045 mm, with optimal values of Nusselt number and pressure drop with the desirability of 85.1%, as shown in table 6. To analyze the performance of the pin fin design, the Design Expert provided twenty comparisons using six center points. Table 7 demonstrates the Nusselt number and pressure drop values, calculated using numerical calculations and RSM model predictions. It is observed that the values predicted by the RSM models and numerical calculations are approximately identical, indicating that the values predicted by the RSM model are correct. The most significant Nusselt number is at row 1 (${P}_{X}$ = 1.00 mm, ${P}_{y}\,$= 0.15 mm, and ${D}_{{pin}}$ = 0.045 mm) while the most significant pressure drop is at the 4th row (${P}_{x}\,$= 2.50 mm, ${P}_{y}\,$= 0.25 mm, and ${D}_{{pin}}\,$= 0.045 mm), as shown in table 8. In the first row, the number of pins is more due to a low transverse pitch, which improves the convection heat transfer coefficient by providing a greater exposed area resulting in a higher Nu. In contrast, in the 4th row, the transverse pitch is large; hence, the pins are less, resulting in a lower pressure drop. To examine the optimal performance of a microchannel, we must find a geometry that gives the optimum values of Nusselt number (Nu) and pressure drop (ΔP).

Table 5. Range of independent variables.

Name	Goal	Lower limit	Upper limit
A-${P}_{x}$	Is in range	1	2.5
B- ${P}_{y}$	Is in range	0.15	0.25
C: ${D}_{{pin}}$	Is in range	0.035	0.045
Nu	Maximize	7.45037	10.4
P drop	Minimize	1495.75	3489.74

Table 6. Solutions.

S.no	${{\rm{P}}}_{{\rm{x}}}$	${{\rm{P}}}_{{\rm{y}}}$	${{\rm{D}}}_{{\rm{pin}}}$	Nu	P drop	Desirability
1	2.50	0.25	0.045	9.77974	1661.36	0.851(Selected)
2	2.49	0.25	0.045	9.77623	1662.19	0.850
3	2.50	0.25	0.045	9.76345	1659.68	0.848
4	2.46	0.25	0.045	9.75823	1667.08	0.846
5	2.50	0.25	0.045	9.75839	1669.57	0.845
6	2.50	0.25	0.045	9.73851	1657.11	0.844
7	2.45	0.25	0.045	9.7501	1669.62	0.844
8	2.50	0.25	0.045	9.74004	1676.54	0.840
9	2.50	0.25	0.045	9.69844	1652.97	0.838
10	2.50	0.25	0.044	9.66369	1649.36	0.832

Table 8. Set of runs given by RSM for MC-BW.

Run		Variables
	${P}_{x}$ (mm)	${P}_{y}$ (mm)	${D}_{{pin}}$ (mm)
1	1.00	0.15	0.045
2	1.66	0.25	0.040
3	1.00	0.15	0.035
4	2.50	0.25	0.045
5	1.66	0.15	0.040
6	2.50	0.20	0.040
7&8	1.66	0.25	0.035
9	1.00	0.25	0.045
10	2.50	0.15	0.035
11	1.66	0.20	0.040
12	1.66	0.20	0.040
13	1.66	0.20	0.045
14	1.66	0.20	0.040
15	1.66	0.20	0.035
16	1.66	0.20	0.040
17	1.00	0.20	0.040
18	1.66	0.20	0.040
19	1.66	0.20	0.040
20	2.50	0.15	0.045

The ANOVA provides the relationship between the independent variables and responses under study. It also determines the significance of variables among each other. Additionally, it is used to find the reliability and relevance of the regression models. Tables 9 and 10 give Nusselt number and Pressure drop results generated using ANOVA. The 'F-value' shows the significance of the model's terms, and the 'P-value' is the probability of the F-test. The lower the P-value, the more effective the term. According to the ANOVA results, the term is statistically significant when the P-value is less than 0.05. In the case of MC-BW, Nusselt number (Nu), ${A}({Px}),{C}({Dpin}),{and\; AB}({Px}\times {Py})$ are independent variables that could significantly affect the value of Nusselt number due to 'P-value' less than 0.05. The 'P-value' for Pressure drop (ΔP) is less than 0.05, indicating that A (Px), B (Py), C (Dpin), AB, AC, BC, A², and B² can all have a significant impact on the ΔP. Within the case of MC-Mixed, for Nusselt number (Nu), ${A}({Px}),{B}({Py}),{C}({Dpin}),{D}({Pside}-{wall}),$ and AB have a 'P-value' less than 0.05 suggesting that these independent variables in the model could significantly influence Nu. Furthermore, the 'P-value' becoming less than 0.05 for Pressure drop (P), ${A}({Px}),{B}({Py}),{C}({Dpin}),$ ${and\; D}({Pside}-{wall})$ have a more significant effect on the dependent variable. Table 11 provides the four functions to determine Nusselt number (Nu) and pressure drop (ΔP) data accuracy. The coefficient of determination (${R}^{2}$) describes how well the regression model fits the derived data. The R-squared and adjusted R-squared values indicate a better fit for the model. The coefficient of determination of 71.34% and 99.72% is acquired for Nusselt number (Nu) and pressure drop (ΔP), respectively. This means 71.34% of Nu and 99.72% of the ΔP data fit the regression model.

Table 9. ANOVA'S results for Nu for MC-BW.

Sources	Sum of squares	DF	Mean square	F Value	P-Value
Model	9.98	6	1.66	5.39	0.0054(significant)
A(${P}_{x}$)	2.05	1	2.05	6.65	0.0229
B (${P}_{y}$)	0.42	1	0.42	1.37	0.2632
C(${D}_{{pin}}$)	1.55	1	1.55	5.02	0.0432
AB	2.80	1	2.80	9.09	0.0099
AC	0.32	1	0.32	1.03	0.3289
BC	0.69	1	0.69	2.22	0.1598
Residual	4.01	13	0.31
Correlation	13.99	19

Table 10. ANOVA'S Data for ∆P for MC-BW.

Source	Sum of squares	DF	Mean Square	F Value	p-value Prob > F
Model	4.838E+006	9	5.376E+005	392.53	< 0.0001
A-${P}_{x}$	3.266E+006	1	3.266E+006	2384.54	< 0.0001
B-${P}_{y}$	2.385E+005	1	2.385E+005	174.17	< 0.0001
C-${D}_{{pin}}$	3.111E+005	1	3.111E+005	227.14	< 0.0001
AB	34796.89	1	34796.89	25.41	0.0005
AC	8066.46	1	8066.46	5.89	0.0356
BC	15343.89	1	15343.89	11.20	0.0074
A^2	1.972E+005	1	1.972E+005	144.00	< 0.0001
B^2	21519.70	1	21519.70	15.71	0.0027
C^2	27.78	1	27.78	0.020	0.8896
Residual	13694.70	10	1369.47
Lack of Fit	13694.70	5	2738.94
Pure Error	0.000	5	0.0
Correlation	4.852E+006	19

RSM method is used to give the best polynomial equations for Nusselt number (Nu) and pressure drop (ΔP) as

$$\begin{eqnarray*}&&{\left({Nu}\right)}^{1}=28.46453-6.94528\times {P}_{x}-86.51897\end{eqnarray*}$$

$$\begin{eqnarray*}&&\times \,{P}_{v}-274.65393\times {D}_{{pin}}+18.67703\times {P}_{x}\times {P}_{v}\end{eqnarray*}$$

$$\begin{eqnarray}&&+\,62.84530\times {P}_{x}\times {D}_{{vin}}+1237.71787\times {P}_{v}\times {D}_{{Pin}}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray*}&&{\left(\unicode{x02206}P\right)}^{1}=1182.34429-2549.7684{\rm{* }}\,{P}_{x}\end{eqnarray*}$$

$$\begin{eqnarray*}&&+\,14354.21633\times {P}_{v}+1.04777\times {E}^{5}\times {D}_{{Pin}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&+\,2144.74537\times {P}_{x}\times {P}_{v}10326.35\times {P}_{x}\times {D}_{{Pin}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&-\,1.92240\times {E}^{5}\times {P}_{v}\times {D}_{{Pin}}+469.93409\times {P}_{{x}^{2}}\end{eqnarray*}$$

$$\begin{eqnarray}&&-\,34442.01368\times {P}_{{v}^{2}}-1.2374\times {E}^{5}\times {D}_{{{Pin}}^{2}}\end{eqnarray} \tag{ 23 }$$

Figure 20 of the RSM perturbation graphs illustrates the effects of independent factors on the responses, including transverse pitch, longitudinal pitch, pin diameter, and side wall pitch. The impact of the variables on the response increases with the slope of the variable. According to the graph, the Nusselt number (Nu) decreases by 10.6% when the transverse pitch $({Px})$ is changed from 1 mm to 2.5 mm while keeping ${Py}$ = 0.2 mm and ${Dpin}$ = 0.04 mm constant, 3.3% when the longitudinal pitch $({Py})$ is changed from 0.15 mm to 0.25 mm by keeping ${Dpin}$ = 0.04 and ${Px}$ = 1.75 constant, and the Nusselt number (Nu) increases by 8.23% when the ${Dpin}$ is changed from 0.035 mm to 0.045 mm by keeping the ${Px}$ = 1.75 and ${Py}$ =0.2 constant. In addition, the pressure drop (ΔP) decreases by 44.6% when the transverse pitch $({Px})$ is changed from 1 mm to 2.5 mm by keeping ${Py}$ = 0.2 mm and ${Dpin}$ = 0.04 mm constant, 20.8% when the longitudinal pitch $({Py})$ is changed from 0.15 mm to 0.25 mm have keeping ${Dpin}$ = 0.04 and ${Px}$ = 1.75 constant, and increases 35.4% when the ${Dpin}$ is changed from 0.035 mm to 0.045 mm by keeping the ${Px}$ = 1.75 and ${Py}$ =0.2 constant. According to the above numerical data, the transverse pitch $({Px})$ and ${Dpin}$ have a more significant impact on the Nusselt number (Nu) and pressure drop (ΔP) than the longitudinal pitch $\left({Py}\right).$

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Figure 21 indicates that the transverse pitch $({P}_{x})$ is more effective on the Nusselt number (Nu) and pressure drop (ΔP) compared to other factors. This means that it is possible to optimize the Nusselt number (Nu) and pressure drop (ΔP) by changing the value of the transverse pitch $({P}_{x}).$ The three-dimensional contours of Nusselt number (Nu) and pressure drop (ΔP), shown in figure 21, demonstrate the two-by-two response surface relationship between all three independent variables while keeping one constant. From figure 20, it is clear that the increase in transverse pitch increases Nu and pressure drop. At the same time, the longitudinal pitch and diameter of the pin have minor effects on Nusselt number (Nu) and pressure drop (ΔP). When the transverse pitch is low, the number of pin-fins will increase, increasing the surface area for convective heat transfer and Nu. Increasing the number of pin-fins also causes flow hindrances and pressure drop. Similarly, an increase in diameter increases Nusselt number (Nu) and pressure drop (ΔP), while an increase in longitudinal pitch decreases the pressure drop. To achieve the optimum pressure drop and Nu, we can use the geometry predicted by RSM, as shown in table 12.

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Table 12. Optimized parameters for MC-BW.

${P}_{x}$ (mm)	${P}_{y}$ (mm)	${D}_{{pin}}$ (mm)	∆P	Nu	Desirability
2.50	0.25	0.045	1661.36	9.77974	0.851

For the optimization of mixed pins configurations, the range of four independent variables is given in table 13. The RSM optimization gives the geometry (${P}_{x}$= 1.0 mm, ${P}_{y}$= 0.150 mm, ${D}_{{pin}}$= 0.035 mm and ${{P}}_{{side\; wall}}=1.250{mm}$) with optimum values of Nusselt number (Nu) and pressure drop (ΔP) with the desirability of 60.9%, shown in table 14. For the Nusselt number 2FI model is selected, and for the pressure drop linear model is selected by the RSM. For the Nusselt number from table 15 model F-value of 15.10 implies that the model is significant. There is only a 0.01% chance that an F-value of 15.10 could occur due to noise. P-values less than 0.0500 indicate that model terms are significant. A, B, C, D, and AB are significant model terms. Values greater than 0.10 indicate that the model terms are not significant. For the pressure drop from table 16 model F-value of 24.24 implies that the model is significant. This value indicates that there is only a 0.01% chance of noise. A P-value less than 0.05 indicates that model terms are significant. A, C, and D are significant model terms in this case. To examine the optimal performance of a mixed pin-fins configuration to find a geometry that gives the optimum Nusselt number (Nu) and pressure drop (ΔP) at Re 300. RSM provided twenty-seven runs for MC-MIXED, as given in table 17, with one central point used to find the optimized parameters. The predicted values of the Nusselt number and pressure drop by the RSM can be found using these equations.

$$\begin{eqnarray*}&&{\rm{Sqrt}}\left({\rm{\Delta }}P\right)=67.94-7.15{P}_{x}-1.67{P}_{y}\,\end{eqnarray*}$$

$$\begin{eqnarray}&&+\,3.46{D}_{{pin}}-3.57{\,P}_{{side}\,{wall}\,}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray*}&&{Sqrt}\left({Nu}\right)=4.39561+\left(-0.507432\right)\times {P}_{x}\end{eqnarray*}$$

$$\begin{eqnarray*}&&+\left(-6.07121\right)\times {P}_{y}+2.70806\times {D}_{{pin}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&+\,0.47424\times {\,P}_{{side}\,{wall}\,}+1.43058\times {P}_{x}{P}_{y}\end{eqnarray*}$$

$$\begin{eqnarray*}&&+\,2.95461\times {P}_{x}{D}_{{pin}}+\left(-0.0483753\right)\end{eqnarray*}$$

$$\begin{eqnarray*}&&\times {P}_{x}{\,P}_{{side}\,{wall}\,}+76.199\times {P}_{y}{D}_{{pin}}+\left(-0.956223\right)\end{eqnarray*}$$

$$\begin{eqnarray}&&\times \,{P}_{y}{\,P}_{{side}\,{wall}\,}+(-9.49737)\times {D}_{{pin}}{\,P}_{{side}\,{wall}\,}\end{eqnarray} \tag{ 25 }$$