Hybrid Nano-Jet Impingement Cooling of Double Rotating Cylinders Immersed in Porous Medium

Abstract

A cooling system with impinging jets is used extensively in diverse engineering applications, such as solar panels, electronic equipments, battery thermal management, textiles and drying applications. Over the years many methods have been offered to increase the effectiveness of the cooling system design by different techniques. In one of the available methods, nano-jets are used to achieve a higher local and average heat transfer coefficient. In this study, convective cooling of double rotating cylinders embedded in a porous medium is analyzed by using hybrid nano-jets. A finite element formulation of the thermo-fluid system is considered, while impacts of Reynolds number, rotational speed of the double cylinders, permeability of the porous medium and distance between the cylinders on the cooling performance are numerically assessed. Hybrid and pure fluid performances in the jet cooling system are compared. It is observed that the cooling performance improves when the rotating speed of the cylinder, permeability of the medium and jet Reynolds number are increased. The heat transfer behavior when varying the distance between the cylinders is different for the first and second cylinder. Higher thermal performances are achieved when hybrid nanofluid with higher nanoparticle loading is used. An optimization algorithm is used for finding the optimum distance and rotational speeds of the cylinders for obtaining an improved cooling performance, while results show higher effectiveness as compared to a parametric study. The outcomes of the present work are useful for the thermal design and optimization of the cooling system design for configurations encountered in electronic cooling, energy extraction and waste heat recovery.

1. Introduction

Jet impingement (J-I) cooling provides improved thermal performance, while many applications are encountered for J-I such as in thermal management, electronic cooling, photovoltaic (PV) systems, textiles, drying and many more [1,2,3,4,5,6]. In PV systems, higher cell temperature can be reduced by using impinging jets and it is noted among the most effective active cooling systems [7,8]. The basic flow features and heat transfer (HT) mechanism for slot J-I systems have been studied extensively. In cooling applications of hot surfaces, the surface geometry effects, jet to target surface distance along with the other parameters such as fluid type and flow rate are among the important parameters for the overall effectiveness of the cooing system with impinging jets.

The J-I system has been considered for flat and curved surfaces [9,10,11]. However, in some cases, rotational effects should be considered in J-I cooling, such as in metal processing, turbine blades cooling and many more. Metzger et al. [12] developed an experimental procedure for measuring the local HT on rotating surfaces using liquid crystal surface coatings. They used a single circular jet, which was mounted to the disk surface. They obtained local Nu in terms of disk and Re. Carper Jr et al. [13] performed an experimental work on the HT from a rotating disk by using a single oil J-I system. The uniform surface temperature was considered for the disk as well as the impacts of flow rate, jet nozzle, disk diameters and angular velocities of disk on the HT. Correlations were developed for average Nu in terms of the rotational Re, jet Re, jet Pr and radial location of jet. Lallave et al. [14] analyzed the conjugate HT for confined slot J-I on a rotating disk which was uniformly heated. The study was conducted by considering different values of Re, rotational rate, Ekman number, nozzle to target space and disk thickness. It was observed that in most conditions, the local HT coefficient rises by activating the rotations. Ayadi et al. [15] performed a numerical study for a convective HT from rotating the cylinder under a magnetic field. They considered different strengths of the magnetic field for different domains, while the size of the cylinder was found to be influential when the rotations were activated.

The jet cooling system may be operated in various regions including porous zones. The porous layer-coupled HT systems are used in convection for flow and HT control [16,17,18,19]. The porous layer or obstacles can be considered in thermal systems for HT intensification. The cooling system with confined J-I system may be considered for the porous region or part of the medium may be porous. Many studies have been performed for assessing the effectiveness of J-I cooing when the porous medium is considered [20,21,22,23,24]. Wong and Saeid [25] numerically analyzed the J-I cooling for a hot surface in a porous channel for the mixed convection flow regime. HT enhancements were found by increasing the porosity, while deterioration of HT was profound for higher values of Rayleigh numbers. Zahmatkesh and Ali Naghedifar [26] performed oscillatory convection of J-I cooling for a partially heated surface which was immersed in a porous medium. When the medium porosity was increased, HT rates were intensified, while the oscillatory response was found to be dependent upon the Re, Grashof number and Darcy number. Hussain et al. [27] performed numerical work on the confined slot J-I in an open cavity with a porous layer. The cavity had a trapezoidal shape when the Darcy-Forchheimer model was adopted. It was observed that when the porous layer thickness was increased, the average Nu deteriorated.

Nanofluids (NF) technology has been considered in the J-I cooling system for further performance enhancements. Over the years, many advancements have been achieved when NFs are integrated in various thermal systems for refrigeration, thermal management, energy storage, solar power and many more [28,29,30]. Many different NFs have been produced and sophisticated correlations have been developed for accurately describing the thermophysical properties [31,32,33,34,35,36,37,38,39,40]. The hybrid NFs have been preferred in HT applications for many reasons, including cost, better thermal performance features due to the synergistic effects of individual nanoparticles (NPs) and stability [41,42,43]. In J-I applications, the utilization of NFs provided an additional increase in the overall thermal performance [44,45,46,47,48,49]. The amount of the increment depends upon the loading amount of NPs, shape and size of the NPs, base fluid type and other operating parameters. NFs have been used when rotations are considered or with porous media applications. In those systems, the effectiveness of the NFs have been shown [50,51,52,53,54,55,56,57].

In the present study, slot J-I cooling of hot double rotating cylinders (RCs) which are embedded in a porous medium is numerically assessed. Even though some studies exist that consider the rotational effects of surfaces for J-I applications, convective HT from double cylinders mounted in a porous region has never been considered. Hybrid NF using water with alumina + copper NPs is used for further intensification of the cooling performance. An optimization study is also used for getting the highest cooling rate from the hot cylinders in terms of the rotational speed of the RCs and the distance between them. The outcomes of the present work are useful for the thermal design and optimization of the cooling system design for configurations encountered in electronic cooling, energy extraction, waste heat recovery and many different systems.

2. Modeling Approach

J-I cooling of double RCs embedded in the porous region is explored numerically, while a schematic view is given in Figure 1. Two identical hot cylinders which are kept temperature of Th are embedded in the porous region with a porosity of and permeability of K. The J-I cooling system includes three identical slots of width w = 0.01 m which are separated by the distance sj = 5 w. The distance between the inlet and the bottom plate is H = 10 w, while the porous layer height is Hp = 0.6 H and the plate length is L = 150 w. The cylinder radius is Rc = 0.3 Hp and the distance between them is sc. They are rotating with a speed of $\Omega$ in a clockwise direction. The cylinder center locations are (xc1, yc1) for the first cylinder and (xc2, yc2) for the second cylinder. The horizontal locations are xc1 = 0.48L, xc2 = xc1 + sc + 2Rc, while the vertical locations are yc1 = yc2 = 0.5 Hp.

Just as the HT fluid is considered, so too are the PF and NF considered. The NF is a hybrid type which includes alumina and copper NPs in water, which is the PF of the cooling system. The NP and PF properties are given in

Table 1. Thermophysical properties of PF and NPs [58].

Properties	Water	Cu	Al2O3
cp—J /kg K	4179	385	765
k—W /m K	0.613	401	40
$\rho$—kg / m${}^3$	997.1	8933	3970
$\mu$—kg / m s	$8.9\times {10}^{-4}$	-	-

. Experimental data were used for obtaining the thermophysical properties in the work of Ref. [58]. NP volume fractions of 0.01 and 0.02 are used in the PF to obtain the HNF.

Table 2. Viscosity and thermal conductivity of NF at two different solid volume fractions of NPs [58].

$\mathit{\varphi }$ (%)	$\mathit{\varphi }$—Cu (%)	$\mathit{\varphi }$—Al2O3 (%)	k (W/mK)	$\mathit{\mu }\times {10}^3$ (kg/ms)
1	0.038	0.962	0.657008	1.602
2	0.0759	1.9241	0.684992	1.935

shows the thermal conductivity and viscosity for 1 and 2 percent of the solid volume fraction.

A 2D, laminar flow model is considered, while impacts of thermal radiation, free convection and viscous dissipation are not taken into account. Above the porous zone, conservation equations of mass, momentum and energy are stated as [59,60]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$\begin{array}{cc}\hfill & \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial x}+{\nu }_{nf}\left({\nabla }^{2}u\right)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial y}+{\nu }_{nf}\left({\nabla }^{2}v\right)\hfill \end{array}$$

(3)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_{nf}{\nabla }^{2}T.$$

(4)

In the porous region where hot cylinders are embedded, the generalized Darcy–Brinkmann–Forchheimer extended model is used [59,60]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(5)

$$\begin{array}{cc}\hfill & \frac{1}{{\epsilon }^2}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial x}+\frac{{\nu }_{nf}}{\epsilon }\left({\nabla }^{2}u\right)\hfill \\ \hfill & -{\nu }_{nf}\frac{u}{K}-\frac{F_c}{\sqrt{K}}u\sqrt{u^2+v^2}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \frac{1}{{\epsilon }^2}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{1}{{\rho }_{nf}}\frac{\partial p}{\partial y}+\frac{{\nu }_{nf}}{\epsilon }\left({\nabla }^{2}v\right)\hfill \\ \hfill & -{\nu }_{nf}\frac{v}{K}-\frac{F_c}{\sqrt{K}}v\sqrt{u^2+v^2}\hfill \end{array}$$

(7)

where $\epsilon$ and K denote the porosity and permeability, respectively.

In the model, the non-dimensional parameters are given as:

$$\begin{array}{cc}\hfill & {\mathrm{F}}_c=\frac{1.75}{\sqrt{150{\epsilon }^2}},\mathrm{Pr}=\frac{\nu }{\alpha },\mathrm{Re}=\frac{u_j{D}_h}{\nu },\mathrm{Rew}=\frac{\Omega D_h^2}{\nu },\mathrm{Da}=\frac{K}{D_h^2},\hfill \end{array}$$

(8)

where $D_h$ is the characteristic length ($D_h=2w$), while Re, Pr and Da denote the Reynolds number, Prandtl number and Darcy number, respectively.

Boundary conditions are stated as:

• slot jet inlets:
  $u=0,v=uc,T=Tc$.
• At the exit of slot-I system:
  $\frac{\partial u}{\partial x}=0,\frac{\partial v}{\partial x}=0,\frac{\partial T}{\partial x}=0$
• On the first hot cylinder (C1) surface:
  $u=-\Omega (y-yc1),v=\Omega (x-xc1),T=T_h$.
• On the second hot cylinder (C2) surface:
  $u=-\Omega (y-yc2),v=\Omega (x-xc2),T=T_h$.
• At the plates (top plate wall and walls between the slots):
  $u=v=0,\frac{\partial T}{\partial y}=0$
• At the interfaces between the porous layer and NF region:
  ${\Phi }_f={\Phi }_p,J{(\Phi )}_f=J{(\Phi )}_p$
• At the walls of the plates (top and bottom):
  $u=v=0,\frac{\partial T}{\partial n}=0$.

where $\Phi$ can be any of the $p,u,v,T$, while J is either the heat flux or the shear stress [61,62].

As the solution method, GWR-FEM is considered. Main steps in the modeling with FEM flow and HT problems can be found in many sources [63,64]. Field variable approximation is performed by using different ordered Lagrange FEMs as in [65]:

$$\begin{array}{cc}\hfill & u=\sum _{k=1}^{N^u}{\Psi }_k^{u,v}{U}_k,v=\sum _{k=1}^{N^v}{\Psi }_k^{u,v}{V}_k,\hfill \\ \hfill & p=\sum _{k=1}^{N^p}{\Psi }_k^p{P}_k,T=\sum _{k=1}^{N^u}{\Psi }_k^T{T}_k,\hfill \end{array}$$

(9)

where ${\Psi }^{u,v},{\Psi }^p$ and ${\Psi }^T$ are the shape functions. The related nodal values of the elements are given by the terms $U,V,P$ and T. When they are used in the relevant equations, residual R is established.

The weighted average of R is set to be zero as in the following equation:

$${\int }_V{W}_{k}RdV=0,$$

(10)

where $W_k$ denotes the weight function.

In the numerical code, the SUPG (streamline upwind Petrov–Galerkin) is used for handling the instability, while BICGStab (BiConjugate Gradient Stabilized) is used for flow and HT modules. As the converge criteria, a value of 10${}^{-7}$ is used where it is stated as in the following [66]:

$$\left|\frac{{\Gamma }^{(i+1)}-{\Gamma }^i}{{\Gamma }^{(}i+1)}\right|\le {10}^{-7}.$$

(11)

Here, $\Gamma$ denotes any of the field variables, while i is the iteration number. A commercial computational fluid dynamics (CFD) code Comsol [67] is used.

Cooling performance of the hot rotating cylinders C1 and C2 is evaluated by using Nusselt numbers (Nu), which are stated as follows:

$${\mathrm{Nu}}_s=\frac{h_s{D}_h}{k_{nf}}=-\frac{D_h}{T_h-T_c}\frac{\partial T}{\partial s},{\mathrm{Nu}}_m=\frac{1}{S_{}}{\int }_0^{L_{}}{\mathrm{Nu}}_{s}ds.$$

(12)

where $h_s$, $k_{nf}$, $D_h$ and S are the local HT coefficient, thermal conductivity, characteristic length and total length of the hot parts.

A mesh independence test (MIT) is conducted, considering that various grid systems have different numbers of mixed type (triangular + quadrilateral) elements. Figure 2a shows the average Nu variations with various grid sizes at two different rotations of the cylinders. The grid system with 317,943 elements is selected. The mesh distribution is shown in Figure 2b where it is refined towards the interfaces and near the walls.

Two different validation works are used to check the capability of the code in solving the convective HT for slot J-I and for convection with a porous medium. In the first study, numerical results of Ref. [68] is used where slot J-I on an isothermal surface is analyzed in a laminar case. Comparisons of local Nu variations at several spatial locations are shown in

Table 3. Code validation 1: Local Nusselt numbers for slot J-I HT considering different locations along a hot surface, (Re = 100).

X	Local Nu—Ref. [68]	Local Nu—Current
1.31	0.49	0.50
1.94	1.27	1.22
2.64	2.39	2.23
3.87	2.79	2.89
7.55	1.01	1.02
18.22	0.296	0.309

for a Reynolds number of 100. In another study, convection in a porous enclosure which is differentially heated is explored and results in Refs. [69,70].

Table 4. Code validation 2: Comparisons of average Nu for convective HT in a porous enclosure which is differentially heated.

Case in Study	Ra = 100	Ra = 1000
Current	3.13	13.78
Ref. [69]	3.02	13.72
Ref. [70]	3.16	14.06

shows the comparison results of average Nu at two different Rayleigh numbers (Ra). The maximum difference between the results is obtained below $2\%$.

3. Results and Discussion

J-I cooling from two identical hot C-RCs which are embedded in porous media is explored by using hybrid NLs. The permeability of the porous regions where cylinders are placed is varied between Darcy number 0.00001 to 0.1. The RCs rotational speeds are taken between Rew = 0 and Rew = −500, while Re based on slot size is considered between 100 and 500. Hybrid NF is considered by using water with alumina and copper NPs with different loading amounts. The distance between the C-RCs is varied between sc = Rc and Sc = 5Rc. Cooling performance of both hot RCs is explored.

Impacts of Re on streamline distributions are shown in Figure 3 for stationary and rotating cylinder cases. The cylinders are rotating in a clockwise direction. In the absence of rotations, recirculation regions (RRs) are formed near the inlet due to confinement and entrainment. Multiple RRs are formed between the slots while their sizes become larger with higher Re. When rotations become active, RRs are established near the C-RCs. At lower Re, the impact of rotations is more pronounced and a large vortex near the left cylinder is formed at Rew = −500. Figure 4 shows the impacts of the rotation of cylinders on the flow pattern variations at the highest and lowest permeability of the embedded zone. When Da is low, the resistance of the medium to the flow is higher, while impacts of the rotations of the cylinders have slight impacts on the streamline variations near the slots and above the non-porous zone. For higher permeability of the medium, the flow field near the upper plates and in between the slots are highly affected by the variation of Rew. At the highest Rew, in the vicinity of the left RC, a vortex is seen, while the vortex near the mid-slot extends toward the region between the cylinders due to the rotation of the cylinders. A two-layer vortex form is seen near the right plate, while the lower one is established within the porous region.

Average Nu versus Re shows the increment for both cylinders, while the increment amount depends on the rotational speed, as shown in Figure 5. When cylinders are stationary (Rew = 0), average HT from the cylinder on the right (C2) is higher as compared to the HT of the cylinder on the left (C1). However, when rotations are activated, at Re = 300 and Re = 400, the average Nu is higher for C1 as compared to C2. This could be attributed to the fact that at Rew = −400 and Rew = −500, a vortex is formed near cylinder C1 which acts as an additional obstacle, which results in higher local thermal transport. Average Nu becomes 2.8 and 2.9 times higher at the highest Re when compared to the case at Re = 100 for stationary cylinders C1 and C2. When rotations are active, these values become 5.5 and 4.6. The impact of Re on the average Nu is significant when rotations are used. For lower permeability of the medium, average Nu rises with higher rotations for both cylinders. This is due to the low impact of rotation of the flow field variations above the porous zone, while thermal transport is enhanced due to the rotating of the cylinders. The average Nu rises by about $30\%$ and $14\%$ for cylinders C1 and C2, respectively, at the highest speed with lower permeability of the zone. When permeability of the medium is increased, average Nu increases with higher Rew for cylinder C1 while it deteriorates for cylinder C2 from Rew = −200 to Rew = −500. The average HT is increased by about $64\%$ with rotations at the highest speed for cylinder C1 while it deteriorates by about $29\%$ for cylinder C2. The performance of the J-I system is increased by using NF and increasing the NP loading amount in the base fluid, as shown in Figure 6b,c. For cylinders C1 and C2, PF and NF behavior is different when varying the rotational speed. The HT rates are higher when NF is used. For higher NP loading in the PF, HT amounts are further enhanced for cylinders C1 and C2. Average Nu generally rises with Rew for cylinder C1 when the NF has a solid volume fraction of $1\%$ and $2\%$. However, for cylinder C2, the highest HT values are seen at Rew = −200 when using HNF, but the behavior is different when using PF. At the highest rotational speed, HT performance is deteriorated for cylinder C2 when all fluid types are used.

Increasing the permeability of the embedded region results in better cooling of the hot cylinders, as shown in Figure 7. The resistance of the domain where hot cylinders are mounted to the cold fluid flow is higher with lower Da, while RRs are formed only in the zone above the porous region. When permeability is increased, the impacts of rotations become important. In this case, flow recirculations are formed between the cylinders, while RRs between the slots are also affected. A saturation type curve is obtained when average Nu versus permeability of the embedded zone is inspected. The discrepancy between the average Nu for cylinder C1 and C2 becomes higher at the highest Da when the rotational effects become important. The average HT becomes 5.2 times higher when values at the lowest and highest permeabilities are compared for cylinder C1, while there is $10\%$ discrepancy between C1 and C2 at Da = 0.1. Another factor that can be used for control of convection is the distance between the hot cylinders. Without rotations, when the distance between the RCs is increased, the vortex size increases (Figure 8). At the highest Sc, four vortex centers are seen between the first two slots and RCs. Behind the cylinder C2, two large vortices are formed, while near the upper plate above the cylinder C2, an additional vortex is established. The characteristics of the flow field change when varying the distance between the cylinder in the case of rotations as compared to the motionless cylinder case. At the smallest distance (sc = Rc), a large vortex is seen near the bottom wall toward the cylinder C1 and its size is reduced with greater distances. At Sc = 3Rc, one large vortex is formed between the cylinders, while at Sc = 5Rc, the vortices are broken into three parts. Above the porous region, near the plates, the RRs are also affected by the variation of the distance between the hot cylinders. The HT behavior shows different characteristics from C1 and C2 for varying distance depending upon the rotations of the cylinders. At Rew = 0, the optimum distance when the highest HT rates are achieved is seen at sc = Rc for cylinder C1 and Sc = 4Rc for cylinder C2, while the lowest HT values are obtained at distances Sc = 3Rc and = Rc. When rotations are active, increasing the distance generally has a positive impact on the overall HT for cylinder C1; but for cylinder C2, average Nu first reduces until Sc = 3Rc and then increases up to Sc = 5Rc (Figure 9). In the absence of rotations, average Nu variations become $17\%$ and $23.5\%$ for cylinder C1 and C2, respectively, when varying the distance between the RCs, while these values become $97.8\%$ and $35.9\%$ when rotations are active at the highest speed.

Due to the non-monotonic behavior of HT with varying Rew and distance between the cylinders, an optimization routine has been used to maximize the HT rates from the cylinders. As the optimization function, the average Nu from the hot cylinders is considered, while the distance between the RCs and rotational speeds are varied. As the cylinder impact is more influential for higher permeability of the embedded region, Da value is taken as 0.1.

When the J-I system with hot RCs is parametrized, control variables are considered, while a function of the PDE solution is achieved, which is the average Nu from the hot cylinders in the PDE constraint optimization. The PDE problem is considered as an equality constraint in the general problem of optimization as [71,72]:

$$\begin{array}{cccc}\hfill & \underset{\psi }{\mathrm{minimize}}\hfill & \hfill & \mathrm{f}(\Phi (\psi ),\psi )\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill & \hfill & \zeta (\Phi (\psi ),\psi ))=0,\hfill \\ \hfill & \hfill & & lb\le G(\Phi (\psi ),\psi ))\le ub.\hfill \end{array}$$

(13)

where $\psi$ repents the control variables. The $\Phi$ and G denote the PDE solutions and constraints, respectively. In the current work, the COBYLA (Constrained Optimization BY Linear Approximations) algorithm is utilized. It is a gradient-free method. Interpolation at the vertices of a simplex is conducted. Cylinder rotations (Rew) and distance between them (sc) are considered as the parameters of interest. Lower ($lb$) and upper bounds ($ub$) are given as: $-500\le Rew\le 0$ and $rc\le sc\le 5rc$.

The search for the maximum of the objective function in the range of relevant parameters is shown in Figure 10a. The optimum values of Rew and sc are found as (Rew, sc) = (−483,4.425Rc). At this operating point, the average Nu from cylinders C1 and C2 are found as 36.13 and 33.35, while the overall Nu is found as 34.74. When parametric CFD is used, the highest overall Nu is obtained as 32.67 for the operating point of (Rew, sc) = (−500, 5Rc). In this case, average Nu from cylinders C1 and C2 become 34.6 and 30.75, respectively. Comparison of streamlines between the parametric CFD and optimum operating point is shown in Figure 10b,c. The obtained average Nu at the optimum is $4.5\%$ higher for C1 and $8.5\%$ higher for C2 when compared to the best case of parametric CFD.

4. Conclusions

The convective HT performance for the slot J-I cooling system of double rotating hot cylinders embedded in porous region is analyzed. The following conclusions were achieved:

• When rotations are activated, the RRs are affected in between the slots while additional vortices are formed near the cylinders which have significant impacts on the convective HT from the cylinders.
• When rotation is considered, the impact of Re on HT becomes significant. For the stationary cylinder case, the average HT at Re = 500 becomes 2.8 and 2.9 times higher when compared to values at Re = 500 for the first and second cylinder, while they become 5.5 and 5.6 higher when rotations are used.
• The impact of rotation on the behavior of the HT for cylinder C1 and C2 depends greatly upon the permeablity of the medium in which the cylinders are immersed. For the lowest permeability, the average Nu rises by about $30\%$ and $14\%$ for C1 and C2, respectively. However, at higher permeability, the average Nu rises by $64\%$ for C1 and deteriorates by $29\%$ for C2 at the highest rotational speed.
• The distance between the cylinders is influential on the overall cooling performance of the J-I cooling system. The variations in the average Nu when varying the distance between the RCs are $17\%$ and $23.5\%$ for C1 and C2 without rotations, respectively, while they are $97.8\%$ and $35.9\%$ with the highest rotational speed.
• An optimization routine based on COBYLA is used to find the optimum parameters of interest for achieving the best cooling performance.
• The optimum value of (Rew, sc) at Re = 300 is obtained as (−483,4.425Rc). When compared to the best case of parametric CFD, the average Nu becomes $4.5\%$ and $8.5\%$ higher for cylinders C1 and C2, respectively.

This study can be extended to include the pressure drop, different thermal boundary conditions on the cylinders and different NF types. Different aspects of NFs such as nanoparticle shapes and two-phase modeling can be considered. These will increase the applicability of the current results.