Introduction

Light Emitting Diode or LED lamps have taken the place of discharge lamps such as mercury discharge lamps1,2 and metal halide lamp3,4 due to better energy efficiency, small size, environmental friendliness, low UV radiation, easy control and low maintenance. So, due to this advantages of LED lamps, many researchers focused on its improvements and optimizations. For instance, Ben Salah and Ben Hamida5,6 investigated the heat transfer in LED geometries using air and PCM as alternative cooling materials. They also compared the results based on Ha and Graham7 study for a chip-on-board packing of LED arrays with high power. So, Ben Hamida et al.8 optimized some parameters such as: thickness, material and size of all subcomponents for single-chip LED to reduce the junction temperature and increase light output as well as device reliability. They considered six parts for the case modeling named die, die-attach, metallization, thermal interface material, substrate and heat sink to find the best material and concluded that GaN as LED-Chip material, SiC for Die material, Au as material metallization, 100In for Die-attach material, AIN DBC was the greatest substrate package, Copper for heat sink was the most suitable material and 100In solder as thermal Interface Material for designated parts. Also, other suitable materials are proposed by researchers such as CsPbBr3 as the perovskite light emitting diodes proposed by Jia et al.9. Yu et al.10 investigated the LED performance covered by cesium lead halide perovskite CsPbBr3 through a small organic molecule material named 1, 3, 5-tri (m-pyrid-3-yl-phenyl) benzene (TmPyPB) as a chemical stabilizer. In an experimental study about the LED cooling, Pan et al.11 used the cutting copper fiber oriented sintered heat sinks (CCFOSHS) and found that by decreasing the porosity, heat transfer efficiency of CCFOSHS was improved slightly, but pressure drops were increased, noticeably. Lin et al.12 used the nanofluid-cooled microchannel heat sink for the light-emitting diodes (LEDs) and concluded that nanofluids reduced the thermal resistance more than 42.4% where 0.5% TiO2 nanofluid improved the heat transfer efficiency up to 38.6% compared to pure water. Another study on LED cooling improvements is performed by Lin et al.13 using Taguchi parametric study for the micro-channel. Also, Huang and Wang14 found the optimal fins geometry for the circular micro-channel of LED lighting heat sinks, numerically. They found that thermal resistance of systems for the optimum design were decreased by 16.8% and 11.0% than the “initial” and “Type 2” design heat sinks, respectively.

There are also some researchers focused on the novel designs for LED heat sink applications to improve the thermal managements. Wang et al.15 designed novel tubular oscillating heat pipe with sintered copper particles for this application and reported that the temperature of LED array was inversely related to the illumination intensity. Tang et al.16 developed an integrated heat sink with vapor chamber for LED thermal managements and showed that the high-power LED yields a favorable performance using proposed unit. Kim et al.17 proposed the using cupper-oxide (CuO) composite coating on aluminum-alloy heat sink to enhance the heat dissipation of LED module and observed good results due to the improved thermal radiation property. In a different study, Park et al.18,19 designed a chimney over a circular heat sink in a downlight LED and reported that installing chimney can increase the cooling efficiency of heat sink up to 20%. Microchannels not only is applicable for LED cooling, but also they are very useful instruments for thermoelectric generators20, natural circulation loops21 and etc. which different base fluids such as PCM22 and nanofluids23 are widely used to improve their performance.

Nanofluids due to improvements thermal properties have wide applications and motivated researchers to use them in various aspects. Ben Hamida et al.24 used Ethylene Glycol-Copper Nanofluid under magnetic fields in an enclosure. Ben Jaballah et al.25 applied the hybrid nanofluid for the performance enhancement of bubble absorber for cooling applications. Hatami and Ganji26, Hatami et al.27 and Tang et al.28 used the nanofluid in porous media between two coaxial cylinders, wavy microchannel and wavy cavity, respectively. Also, Hatami et al.29,30,31,32 used the optimization techniques such as Response Surface Methodology (RSM) to find the optimize geometries including nanofluids in heat transfer applications. Massoudi et al.33,34 also investigated the nanofluids application in free convection under the influence of magnetic field.

Many researchers tried to find the correlations of nanofluids properties in different application. Alsarraf et al.35 used a multifunctional optimization for the nanofluid properties to cool the electronic heat sink through the natural convection. Also, Shahsavar et al.36 investigated the variable properties of Fe3O4/CNT/water hybrid nanofluid on the forced convection of mini-channel heat exchanger and found that the error of computed heat transfer rate was not exceeded tha 2.91%. Gheynani et al.37 studied the effect of CuO nanoparticles diameter on heat transfer on non-Newtonian nanofluid in a microtube through the changes in the thermal properties of nanofluid. Kavusi and Toghraie38 tested the various nanofluids (which have different thermal properties) on the performance of a heat pipe and reported that nanoparticle concentration had the greatest effect on the fluid thermal conductivity and thermal resistance. Not only different nanofluids properties were investigated by researches, but also different applications are considered for them. For instance, Gholami et al.39, Barnoon et al.40, Toghraie et al.41 and Arasteh et al.42 utilized the nanofluids and investigated their behavior under different conditions for microchannel, cavity with rotating cylinders, L-shaped porous ribs in microchannel and porous heat sink, respectively.

Based on above discussed literature review, a few studies focused on the nanofluid application for LED cooling using finned micro-channel. So, to fill this gap of study, it is tried to enhance the cooling efficiency of a LED by using the Al2O3-water nanofluid and finned arrangements in a microchannel using numerical Galerkin weighted residual Finite Element Method (GFEM), simultaneously. Also, RSM is used to find the optimized dimensions for the fin numbers, lengths and thicknesses, numerically.

Problem description

A 3D microchannel filled by Al2O3–water nanofluid is considered as shown in Fig. 1 for LED cooling by fin arrangements. Microchannel dimensions are: Height = 100 mm; Width = 80 mm and Depth = 50 mm and the Power of LED = 1, 2 and 3 W as shown in Table 1 in details. The velocity of inlet of nanofluid was considered as 0.001 m/s where the temperature of inlet of nanofluid considered to be 25 °C. The micro-channel is equipped with fin arrangements for better cooling performance. The main objective of current research is finding the optimum values for fin numbers, lengths and thicknesses, respectively as shown in Fig. 1 for the micro-channel geometry. For modeling the problem, a transient flow, incompressible three-dimensional flow from the laws of conservation of mass, momentum, and energy is considered. The governing equations represent the transient flows are8:

Figure 1
figure 1

(a) Structure of the LED package; (b) Sample Dimensions of heat sink, (c) Geometry of LED cooling by finned micro-channel filled by nanofluid (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Table 1 Properties of dimensions and materials of LED package8.
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Mass conservation equation:

$$\frac{\partial \rho }{{\partial t}} + \frac{{\partial (\rho {\kern 1pt} u)}}{\partial x} + \frac{{\partial (\rho {\kern 1pt} v)}}{\partial y} + \frac{{\partial (\rho {\kern 1pt} w)}}{\partial z} = 0$$
(1)

where ρ refers to the mass density and u, v, and w represent the velocities according to x, y, and z, respectively.

Momentum conservation equation according to x:

$$\frac{\partial (\rho \,u)}{{\partial t}} + \frac{{\partial (\rho {\kern 1pt} u{\kern 1pt} u)}}{\partial x} + \frac{{\partial (\rho {\kern 1pt} u{\kern 1pt} v)}}{\partial y} + \frac{{\partial (\rho {\kern 1pt} u{\kern 1pt} w)}}{\partial z} = - \frac{\partial p}{{\partial x}} + \left[ {\frac{\partial }{\partial x}(\eta \frac{\partial u}{{\partial x}}) + \frac{\partial }{\partial y}(\eta \frac{\partial u}{{\partial y}}) + \frac{\partial }{\partial z}(\eta \frac{\partial u}{{\partial z}})} \right]$$
(2)

Momentum conservation equation according to y:

$$\frac{{\partial (\rho {\kern 1pt} v)}}{\partial t} + \frac{{\partial (\rho {\kern 1pt} u{\kern 1pt} v)}}{\partial x} + \frac{{\partial (\rho {\kern 1pt} v{\kern 1pt} v)}}{\partial y} + \frac{{\partial (\rho {\kern 1pt} v{\kern 1pt} w)}}{\partial z} = - \frac{\partial p}{{\partial y}} + \left[ {\frac{\partial }{\partial x}(\eta \frac{\partial v}{{\partial x}}) + \frac{\partial }{\partial y}(\eta \frac{\partial v}{{\partial y}}) + \frac{\partial }{\partial z}(\eta \frac{\partial v}{{\partial z}})} \right] + \rho {\kern 1pt} g$$
(3)

Momentum conservation equation according to z:

$$\frac{{\partial (\rho {\kern 1pt} w)}}{\partial t} + \frac{{\partial (\rho {\kern 1pt} u{\kern 1pt} w)}}{\partial x} + \frac{{\partial (\rho {\kern 1pt} v{\kern 1pt} w)}}{\partial y} + \frac{{\partial (\rho {\kern 1pt} w{\kern 1pt} w)}}{\partial z} = - \frac{\partial p}{{\partial z}} + \left[ {\frac{\partial }{\partial x}(\eta \frac{\partial w}{{\partial x}}) + \frac{\partial }{\partial y}(\eta \frac{\partial w}{{\partial y}}) + \frac{\partial }{\partial z}(\eta \frac{\partial w}{{\partial z}})} \right]$$
(4)

In above equations, p denotes the pressure, g shows the gravity, and η is related to the dynamic viscosity.

Energy conservation equation:

$$\frac{{\partial (\rho {\kern 1pt} C{\kern 1pt} T)}}{\partial t} + \frac{{\partial (\rho {\kern 1pt} u{\kern 1pt} C{\kern 1pt} T)}}{\partial x} + \frac{{\partial (\rho {\kern 1pt} v{\kern 1pt} {\kern 1pt} C{\kern 1pt} T)}}{\partial y} + \frac{{\partial (\rho {\kern 1pt} w{\kern 1pt} \,C{\kern 1pt} T)}}{\partial z} = \left[ {\frac{\partial }{\partial x}(\lambda \frac{\partial T}{{\partial x}}) + \frac{\partial }{\partial y}(\lambda \frac{\partial T}{{\partial y}}) + \frac{\partial }{\partial z}(\lambda \frac{\partial T}{{\partial z}})} \right]$$
(5)

where, T and C are the temperature and the specific heat capacity, respectively and λ is the thermal conductivity.

The practical boundary conditions are assumed as follows:

  • There is a uniform heat flux on top of the die, and other surfaces are adiabatic.

  • The microchannel inlet boundary condition is considered to be the uniform temperature of Tin with a uniform velocity of Uin.

  • At microchannel outlet, Pressure outlet boundary condition with zero gradients is assumed.

  • No slip condition is imposed on the surface of microchannel walls.

  • All the microchannel outer walls are insulated.

In this study, the thermal conductivity of the materials used in the LED package as well as the water properties are considered dependent on temperature8. For water temperature-dependent properties:

(A) Dynamic viscosity:

273.15 K < T < 473.15 K:

$$\eta = { 1}.{37995668}0{4 } - \, 0.0{21224}0{19151}{*} {\text{T }} + { 1}.{36}0{4562827}{*} {1}0^{{ - {4}}}{*} {\text{T}}^{{2}} - { 4}.{6454}0{9}0{319}{*} {1}0^{{ - {7}}}{*} {\text{T}}^{{3}} + { 8}.{9}0{42735735}{*} {1}0^{{ - {1}0*}} {\text{T}}^{{4}} - { 9}.0{79}0{692686}{*} {1}0^{{ - {13}}}{*} {\text{T}}^{{5}} + { 3}.{8457331488}{*} {1}0^{{ - {16}}}{*} {\text{T}}^{{6}}$$
(6)

473.15 < T < 553.15 K:

$$\eta = 0.00{4}0{1235783 } - { 2}.{1}0{746715}*{1}0^{{ - {5}}} *{\text{T }} + { 3}.{85772275}*{1}0^{{ - {8}}} *{\text{T}}^{{2}} - { 2}.{3973}0{284}*{1}0^{{ - {11}}} *{\text{T}}^{{3}}$$
(7)

T in K and dynamic viscosity in Pa.s

(B) Thermal capacity:

273.15 < T < 553.15 K

$${\text{C }} = 12010.1471 - 80.4072879*{\text{T }} + 0.30866854*{\text{T}}^{{2}} - 5.38186884*10^{{- {4}}} *{\text{T}}^{{3}} + 3.62536437*10^{-7} *{\text{T}}^{{4}}$$
(8)

T in K and Thermal Capacity in J/(kg*K).

(C) Mass density

273.15 < T < 293.15 K.

$$\rho = 0.0000{63}0{92789}0{34}*{\text{T}}^{{3}} - 0.0{6}0{367639882855}*{\text{T}}^{{2}} + {18}.{92293824}0{7}0{66}*{\text{T }} - {95}0.{7}0{4}0{55329848}$$
(9)

293.15 < T < 373.15 K.

$$\rho = 0.0000{1}0{335}0{53319}*{\text{T}}^{{3}} - 0.0{13395}0{65634452}*{\text{T}}^{{2}} + {4}.{96928883265516}0*{\text{T}} + { 432}.{257114}00{8512}$$
(10)

T in K and mass density in kg/m3.

(D) Thermal conductivity

273.15 < T < 1000 K

$$\lambda = - 0.{869}0{83936} + 0.00{89488}0{345}{*}{\text{T}} - {1}.{58366345}{*}{1}0^{{ - {5}}}{*}{\text{T}}^{{2}} + {7}.{97543259}{*}{1}0^{{ - {9}}}{*}{\text{T}}^{{3}}$$
(11)

T in K and Thermal conductivity in W/(m*K). For the nanofluid properties, following equations are applied35:

$$\rho_{nf} = \left( {{1} - \phi } \right)\rho_{bf} + \varphi \rho_{p}$$
(12)
$${C}_{{p}_{nf}}=\frac{\left(1-\varphi \right){\left(\rho {C}_{p}\right)}_{bf}+\varphi {(\rho {C}_{p})}_{p}}{{\rho }_{nf}}$$
(13)
$$\begin{gathered} \frac{{k_{nf} }}{{k_{bf} }} = 0.991 + 0.276T\varphi + 77.6\varphi^{2} + 3641.231T\varphi^{2} + \hfill \\ \frac{0.00217}{{\sin \left( {T - \varphi } \right)}} - 6.01 \times 10^{ - 6} T^{2} - 3647.099T\varphi \sin \left( \varphi \right) \hfill \\ \end{gathered}$$
(14)
$$\mu_{nf} = \mu_{static} + \mu_{Brownian}$$
(15)
$$\mu_{nf} = \frac{{\mu_{f} }}{{\left( {1 - \varphi } \right)^{2.5} }} + 5 \times 10^{4} \beta \varphi \rho_{f} \left( {C_{p} } \right)_{f} \frac{{\mu_{f} }}{{k_{f} \Pr }}\sqrt {\frac{{k_{b} T}}{{\rho_{p} d_{s} }}} f\left( {T,\varphi } \right)$$
(16)

where

$$f\left( {T,\varphi } \right) = \left( {2.8217 \times 10^{ - 2} \varphi + 3.917 \times 10^{ - 3} } \right)\left( {\frac{T}{{T_{0} }}} \right) + \left( { - 3.0669 \times 10^{ - 2} \varphi - 3.91123 \times 10^{ - 3} } \right)$$
(17)
$$\beta = 8.4407\left( {100\varphi } \right)^{ - 1.07304}$$
(18)

Methodology of solution

In this study, COMSOL-Multiphysics commercial Galerkin finite element method (GFEM) software is applied for the modeling of problem. GFEM is a numerical method to discretize and solve the coupled partial differential equations governed on any physical phenomenon. GFEM by COMSOL is not only accurate and adaptable, but also it is simple in modeling the complicated physics as well as the time-dependent simulation of physical problems in environment, chemical, energy, electrical, heat transfer and other multi-physic problems. Actually, GFEM with COMSOL can help the user to solve the problems with multi-physics such as electrical–mechanical systems easily, while the solution of these problems by other commercial software (such as FVM based ANSYS-FULENT) or by hand is not possible8. During the modeling process, user must define the boundary conditions after the geometry modeling to show the conditions which must be responded during the solution. Boundary conditions can be defined as the distributed forces, point forces, positional constraints and thermal effects such as temperature changes or applied heat energy. Additionally, the GFEM user can certainly spot any vulnerability in design with the complete visualizations GFEM produces and then use the novel data to have a new design8.

Central Composite Design or CCD is a technique of Design of Experiment (DoE) to achieve the suitable points of each independent variable based on their possible values (level). CCD contains an imbedded factorial or fractional factorial design with center points which is improved by a “star points” group and let’s estimate of curvature. After introducing the minimum number of cases with CCD, RSM must be applied to find the optimum values for the fin numbers, lengths and thicknesses. Actually, RSM find a first or second-order polynomial relation between the response (here temperature) and independent variables (fin numbers, lengths and thicknesses). This polynomial equation consider the interaction between the and surface curvatures and fitted in the shape of31,32

$$y = A_{0} + \sum\limits_{i = 1}^{n} {A_{i} x_{i} } + \sum\limits_{i = 1}^{n} {A_{ii} x_{i}^{2} } + \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left. {A_{ij} x_{i} x_{j} } \right|_{i < j} } }$$
(19)

where xi and xj indicate the geometry independent variables (fin numbers, lengths and thicknesses) and A is the coefficient of polynomial equation. The essential parameter for RSM optimization is “Desirability” which returns the desirable ranges of responses (di). Actually, objective function can be altered from zero outside of the limits to unity at the aim and the point with the maximum desirability function by following equation is the optimized case obtained by numerical solutions:

$$D = \left( {d_{1} \times d_{2} \times ... \times d_{n} } \right)^{\frac{1}{n}} = \left( {\prod\limits_{i = 1}^{n} {d_{i} } } \right)^{\frac{1}{n}}$$
(20)

where n indicates the number of responses. If each of responses or factors drops outside their desirability range, the overall function will be zero. More details about this optimization method can be found in31,32.

Results and discussions

As described in above sections, it is tried to obtain an optimized geometry for the microchannel filled by nanofluid for LED cooling. Dimensions of LED package as well as the materials are presented in Table 1. Based on previous study in8, the best materials for LED-chip is GaN, Die from Si, Die-attach is made by Au-20Sn, substrate is copper and heat sink material is considered to be Al. LED has 1, 2 and 3 W power and the microchannel dimensions was Height = 100 mm; Width = 80 mm and Depth = 50 mm. In order to find a suitable mesh grid, numerous mesh numbers (from extremely coarse to finer mesh) were used for this 3D model as demonstrated in Table 2. These values confirm that fine mesh is a suitable grid type for this study due to its accuracy and acceptable time of calculations. Then, CCD is applied to have the proposed geometries for the considered parameters as shown in Table 3. Fin number, fin diameter and fin thicknesses are three main parameters under the study which their levels were presented in Table 3. CCD proposed 11 cases as the minimum required cases which are shown in Table 4 by details.

Table 2 Mesh independency of the micro-channel.
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Table 3 Geometry parameters of fins arrangement for microchanel and defined levels.
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Table 4 CCD proposed cases for finned geometry with Power of LED = 1, 2 and 3 W Microchannel dimensions: Height = 100 mm; Width = 80 mm and Depth = 50 mm.
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To validate our generated 3D model with the previous model of Ha and Graham7, the temperature distribution of the LED package is simulated with the equivalent settings modeled by Ha and Graham7 as presented in Fig. 1 and generated mesh is depicted in Fig. 2. Figure 3 shows the 3D model simulated by Ha and Graham7 using the commercial ANSYS-FLUENT software where Power of LED lamp was 1 W, heat transfer coefficient was 10 W/m2/K and the temperature of ambient was 25 °C. From the Fig. 3a,b it can be observed that current code which also was used in8 has similar results with7 where the minimum and maximum temperatures for both methods are about 53.22 °C and 60.37 °C, respectively. Actually a minor temperature difference between these two models is existed which does not exceed than 0.001%. Also, the vertical temperature profile along the LED package centerline is plotted in Fig. 3c, which a good agreement between current 3D COMSOL-Multiphysics model and 3D ANSYS-FLUENT model of Ha and Graham7 was observed.

Figure 2
figure 2

Mesh generated in the domain and LED finned section (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figure 3
figure 3

Temperature distribution of the LED package: (a): Our 3D code (b): Ha and Graham code7 (c), Comparison of vertical temperature profile along the center line (z axis)8 (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figures 4, 5 and 6 show the temperatures, velocities and streamlines of nanofluid in both LED and microchannel domains and the results of junction temperature for three LED powers (1, 2 and 3 W) of all 11 designed cases are shown in Table 5 at final time, t = 620 s. It is obvious that when the number of fins increased the cooling process is better occurred and maximum temperature on the LED is decreased. Figures 7 and 8 shows the 2D contours of velocity and streamlines of 11 cases, respectively. It can be observed that cases 6, 8 and 9 due to greater fin numbers (8 fins) has more compact microchannel and the velocity between the fins is greater. But for the cases 2, 3, 5 and 10 nanofluids slowly flow between the fins due to more free spaces. So, this parameter has an important role in heat transfer due to effect on both heat transfer surface and changing the flows regime. Also, the cases which have lengthy fins (Cases 1, 5 and 9) have smaller region of maximum velocity in micro-channel (red area) due to making obstacles against the flow. Because these three parameters has influence on each other, a parametric study is required to find the most important parameter among them. Figure 9 shows the average temperatures during the time for P = 3 W. Cases 3, 10 and 6 have maximum temperatures (worst cooling performance) and cases 9, 1 and 5 have the minimum temperatures or best cooling performances. From Table 4, it can be seen that Cases 3, 10 and 6 are in lowest level of fin length, while cases 9, 1 and 5 are in upper level of fins length. Figures 10 and 11 also present the temperatures during the time when LED power is 2 W and 1 W, respectively. These figures confirm the same results of cooling the LED by finned microchannel. Also, it can be found that number of fins is another important parameter while the fins thickness is not very significant compared to two other parameters due to its small effect of the cooling performance resulted from small effect on both surface and flow regime. From this figures, it can be concluded that by different designs of fins arrangements, it is possible to increase the nanofluid temperature up to 6.5% which enhanced the LED cooling process. Figure 12 shows these descriptions in 2D and 3D contours to find the effect of each parameter on the results. As seen, slope of curves and surfaces for the parameter B is greater than parameter A and greater than parameter C. Also the minimum temperatures (best cooling) is occurred when all the parameters are in upper levels, so desirability function of optimized case in Fig. 13 is depicted which illustrate that better case (desirability = 1) occurs when B = 20 and A = 8. Based on these results, RSM proposed the best cases which is presented in Table 6 as optimized geometries. Figure 14 compares the Nusselt number and pressure drops of all cases. Since the pressure drops are very small, so it can be negligible in optimization and single objective is performed for this case of study. The results of optimized case is depicted in Fig. 15, while the effect of different nanoparticles volume fraction is presented in Fig. 16 which says that larger nanoparticles volume fraction has greater Nusselt number due to temperature dependent properties of base water and greater thermal conductivity of nanofluid as well as the lower pressure drop for smaller nanoparticles concentrations.

Figure 4
figure 4

Temperature contours for the finned micro-channel of different geometries (11 Cases) (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figure 5
figure 5

Velocity contours for the finned micro-channel of different geometries (11 Cases) (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figure 6
figure 6

Streamline contours for the finned micro-channel of different geometries (11 Cases) (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Table 5 Results of final temperature (t = 620 s) for designed geometries.
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Figure 7
figure 7

2D Velocity contours for the finned micro-channel of different geometries (11 Cases) (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figure 8
figure 8

2D Streamline contours for the finned micro-channel of different geometries (11 Cases) (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figure 9
figure 9

Average Temp. when P = 3 W for different 11 cases (a) complete view, (b) zoomed view.

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Figure 10
figure 10

Average Temp. when P = 2 W for different 11 cases (a) complete view, (b) zoomed view.

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Figure 11
figure 11

Average Temp. when P = 1 W for different 11 cases (a) complete view, (b) zoomed view.

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Figure 12
figure 12

2D and 3D contour of geometry parameters on temperature.

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Figure 13
figure 13

Desirability function of optimized geometry.

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Table 6 Optimized geometry proposed by RSM.
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Figure 14
figure 14

(a) Nusselt number (b) Pressure drops graphs for P = 3 W for all designed cases.

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Figure 15
figure 15

Velocity, temperature and streamline for optimized case (COMSOL Multiphysics 5.6, https://www.comsol.com).

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Figure 16
figure 16

Effect of different nanoparticles concentration on Nusselt number of the optimized case.

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Conclusion

In this study, the geometry of a finned microchannel is optimized for application in LED cooling by nanofluids, numerically. COMSOL-Multiphysics commercial code was used and Al2O3-water as the cooling nanofluid flow was selected as the working fluid. Three powers (1, 2 and 3 W) are used for the LED power output and to find the optimized arrangement of fins for reaching the minimum temperature (as cooling efficiency), Central composite design (CCD) was applied to propose minimum possible geometries (11 Cases). Finally, GFEM results of modeling were optimized by the Response Surface Method (RSM) which confirmed that the optimized case include parameters as Fin numbers = 8, Fin length = 20 and Fin thickness = 4 where nanoparticles with φ = 0.05 caused maximum cooling efficiency for this case by greater Nusselt number and lower pressure drop.