Numerical Simulation and Sensitivity Analysis Using RSM on Natural Convective Heat Exchanger Containing Hybrid Nano ﬂ uids

This work presents a numerical analysis for exploring heat transfer phenomena in an enclosed cavity using magnetohydrodynam-ics natural convection. Because of the numerous real-world applications of nano ﬂ uids and hybrid nano ﬂ uids in industrial and thermal engineering developments, hybrid nano ﬂ uids are used as ﬂ uid mediums in the ﬂ uid ﬁ eld. A hexagonal-shaped heat exchanger is taken with two circular surfaces along the middle part. The upright circular surface acts as a homogeneous heat source, while the lower circular surface functions as a heat sink. The remaining portions of the adjacent walls are thermally insulated. The copper (Cu) and titanium dioxide (TiO 2 ) nanoparticles are suspended into water to make a hybrid nano ﬂ uid. For solving the corresponding governing equations, the weighted-residual ﬁ nite element method is applied. To explain the major outcomes, isotherms, streamlines, and many others 2D and 3D contour plots are involved graphically with a physical explanation for different magnitudes of signi ﬁ cant parameters: Rayleigh number ð 10 3 ≤ Ra ≤ 10 6 Þ : , Hartmann number ð 0 ≤ Ha ≤ 100 Þ : , and nanoparticle volume fraction ð 0 ≤ ϕ ≤ 0 : 06 Þ : . The novelty of this work is to apply response surface methodology on the natural convective hybrid nano ﬂ uid model, to visualize 2D and 3D effects, and to study the sensitivity of independent parameters on response function. Due to the outstanding thermal properties of the hybrid nano ﬂ uid, the addition of Cu and TiO 2 nanoparticles into H 2 O develops the heat transfer rate to 35.85 % rather than base ﬂ uid. Moreover, a larger magnitude of Ra and the accumulation of mixture nanoparticles result in the thermal actuation of a hybrid nano ﬂ uid. With greater magnetic impact, an opposite response is exhibited.


Introduction
Natural convective heat transfer has drawn consideration from a lot of today's researchers due to its plentiful straight industrial and technical uses in heat exchangers, house cooling and heating systems, nuclear reactors, solar collectors, electrical equipment, fire engineering, petrochemical industries, and so on.The key factors driving the attractiveness of the natural convective heat-transfer process are the easy and truncated price of building geometrical domains, even if forced and natural convection are primarily two methods of convection for heat transfer in a fluid medium.Natural convection has typically happened because of the temperature variance and the buoyancy forces.Due to the simplicity and vast applications of natural convection heat transfer, at different times, frequent studies have previously been done [1][2][3][4][5][6].To increase the efficiency of heat transmission in both natural and force convective thermal engineering processes, nowadays, nanofluid is imposed with common fluid into the fluid domain.Actually, nanofluid is a comparatively new class of thermal engineering fluid combined by suspending nanoparticles (size smaller than 100 nm), such as Cu, Ag, Al 2 O 3 , or TiO 2 , in common liquids.These common liquids (for instance, engine oil, water, pump oil, ethanol, etc.) are known as base fluids.First, Choi and Eastman [7] introduced nanofluid at Argonne National Laboratory, USA, in 1995.Nanofluid has numerous practical applications in industry and health science, such as in semiconductors, solar engineering, thermal storage systems, microelectronics, electronic devices, computer processors, nuclear reactor cooling, biomedical engineering, cancer therapy, etc.To heighten the thermal performance of base fluid, at different times, already there have been extensive studies about Cu, CuO, Ag, Al 2 O 3 , Fe 3 O 4 , and TiO 2 suspension into a base fluid in numerous fluid domains [8][9][10][11][12][13][14][15][16].To explain the significance of nanofluid, Rostami et al. [17] described natural convective heat transfer models with and without the existence of nanoparticles into the base fluids.Ellahi et al. [18] investigated carbon nanotubes (CNTs)-H 2 O nanofluid-based heat transfer performance.Park et al. [19] explored free convection as impacted by a temperature variation between a heated interior cylinder and a cold external slanted square hollow.Waqas et al. [20] explored natural convective heat transfer in a horizontal annulus with a fin using Cu-H 2 O nanofluid.They arrived at the conclusion that raising the value of ϕ lowered the velocity field while raising the value of ϕ and Ra increased the thermal performance.
Furthermore, to get better heat transfer performance nowadays, two different nanoparticles are intermixed into a base fluid, and this suspension is known as a hybrid nanofluid.By choosing the materials of nanoparticles accurately, they would play a significant essence of each other.Actually, a singlecomponent nanofluid does not have all the desirable characteristics needed for a given application.For example, metal oxide nanoparticles (CuO, MgO, TiO 2 , Al 2 O 3 , etc.) have lower heat conductivity and greater chemical stability.On the contrary, nanoparticles that are metallic (Cu, Al, Ag, Au, etc.,) are very reactive and unstable, and they also have good thermal conductivity.As a result, by combining these two types of nanoparticles, improved physical and chemical properties can achieve while also improving the host fluid's capacity for heat transfer.Due to its synergism, hybrid nanofluid exhibits higher thermal conductivity than traditional nanofluids.Recently, numerous researchers have done both numerically and experimentally to study the novel technological concept of hybrid nanofluids [21][22][23][24][25][26].Chamkha et al. [27] studied how well a hybrid nanofluid transferred heat over time in a semicircular chamber.Khadim et al. [28] analyzed Cu-Al 2 O 3 -H 2 O nanofluid in a porous enclosure, making a wavy surface where the bottom surface was taken as a heat source.Biswas et al. [29] explored natural convective magnetohydrodynamics (MHD) nanofluid by taking Cu and Al 2 O 3 into water in a porous oblique wavy vertical wall enclosure that was heated.In another work, Rashad et al. [30] analyzed again the consequences of Cu-Al 2 O 3 -H 2 O nanofluid enclosed in a triangle with a uniform magnetic and thermal field.The simulation was resolved by employing the finite difference technique, where results showed a significant effect for increasing the nanoparticle volume fraction.Takabi and Salehi [31] explained natural convective establishment in a corrugated cavity with a nonuniform heat lowermost wall occupied with an Al 2 O 3 -Cu-H 2 O mixture.Devi and Devi [32] modeled a 3D boundary layer hybrid nanofluid flow across an extending sheet using Newtonian heating and Lorentz force.
Furthermore, the scientific discipline known as MHD is responsible for describing how magnetic force affected an electrically conducting liquid.MHD is predisposed by a variety of factors, including the radiation in X-rays, the magnetic field on earth, cooling of fission reactors, solar wind, therapy for tumor treatment, etc. Due to these practical applications, many researchers examined MHD heat transport in a number of distinct geometries over time [33][34][35][36][37]. Sheikholeslami et al. [38] analyzed natural convection with thermal radiation impact using Al 2 O 3 -water in a closed cavity.Prakash et al. [39] did another numerical analysis on MHD natural convection in a partially ventilated curved permeable aperture containing a nanofluid.They came to the conclusion that as the Hartmann number developed, the overall convection decreased.Mourad et al. [40] also explored natural convective MHD using Fe 3 O 4 -MWCNT-water nanofluid in a porous cavity with consistent magnetic field, where entropy generation was calculated.Over a porous sheet, Mahesh et al. [41] completed a different investigation on MHD hybrid nanofluid to investigate the impacts of radiation.Ahmed et al. [42] explored the consequences of employing hybrid nanofluid on MHD radiative natural convection in porous configurations.
From the aforementioned review of the literature, it is apparent that the MHD natural convection has a great interest to researchers since it has revolutionary implications, across a wide range of engineering fields.Despite several studies on various closed cavities were carried out over time to explore the behavior of the MHD natural convective heat transfer process, only a few studies were carried out on hexagonal enclosures using different nanoparticles in more recent years [43][44][45][46].Again, relatively little research on heat exchangers, mechanical devices that transport heat between two or more liquids, was done in order to develop a quick heat transfer system [47][48][49].Moreover, to visualize the 2D and 3D effects of response function and study the sensitivity analysis of independent factors, the response surface methodology (RSM) is involved in this natural convective hexagonalshaped heat exchanger that contains a hybrid nanofluid.That is, we investigate the sensitivity analysis of the natural convective heat transfer procedure for a hexagonal heat exchanger loaded with Cu-TiO 2 -H 2 O hybrid nanofluid considering the direct implication of magnetic field.According to the highest level of the writers' knowledge, no research has yet been done on this issue.The regulatory equations are simulated using a form of computation known as the finite element method (FEM) [50].So, the leading purpose of this simulation is to investigate natural convection in a closed cavity covered with hybrid nanofluid statistically and numerically and examine how much the overall heat transfer rate is exaggerated by the Ra, Ha, and ϕ.This numerical investigation is designed to provide responses to the subsequent research questions.Such as: (i) What is the behavior of fluid flow and heat transfer rate on this heat exchanger?(ii) What is the importance of adding nanofluid and hybrid nanoparticles into base fluid?(iii) What is the best-fitted correlation between the response function and independent factors?(iv) How much response function is sensitive for the independent factors?

Mathematical and Physical Model
In this work, a hexagonal-shaped heat exchanger is chosen as a fluid domain that is covered by TiO 2 and Cu solid nanoparticles into water (H 2 O).This fluid is steady, Newtonian, and incompressible that has a direct bearing on the magnetic field.The horizontal length of this cavity is L. Two cylindrical pipes, diameter 0.10 L, are taken into the cavity along the middle part of the cavity, where the right side has a uniform heat source (T h ) and the left one acted as a heat sink (T c ).A physical diagram of this proposed hybrid nanofluid model is characterized in Figure 1.However, all of the bordering walls remain insulated, and the gravitational force due to acceleration (g) is acted downward.Moreover, a uniform magnetic field (B 0 ) is acted from the right to left.The shape of the Cu and TiO 2 nanoparticles are estimated alike in shape and size, and all outer walls are considered as no-slip.The taken thermo-physical appearances of base fluid and solid particles are described in Table 1.

Governing Equations.
To formulate this MHD 2D heat exchanger using hybrid nanofluid, the involved governing equations are continuity, momentum, and energy equations that are termed as follows [13]: where the velocity field v ¼ ðu; vÞ: acted along X and Y axes, respectively.The buoyancy forces and the external magnetic field are treated as body forces (F) acting across the Y-axis in the momentum equation.As a consequence, F ¼ The corresponding boundary conditions are defined as follows: where n is the perpendicular vector acted on the heated surface.
2.2.Nanofluid's Properties.The thermo-physical belongings of base fluid and nanoparticles are prearranged earlier.Actually, the belongings of nanofluid lie on base fluid and nanoparticle's characteristics.So, to compute the hybrid nanofluid belongings, the associations described in Table 2 are involved.

Dimensionless Mathematical Model.
The subsequent dimensionless quantities [44] are taken to make dimensionfree equations for governing Equations ( 1)-( 3): Employing the abovementioned dimensionless variables listed above, Equations ( 1)-( 3) turn into the following: where Rayleigh number (Ra), Prandtl number (Pr), and Hartmann number (Ha) are three significant parameters generated from the above equation, which are defined, respectively, as: Ra ¼ . Moreover, the dimension-free boundary conditions reformed as follows: Furthermore, the average heat transfer rate from the surface that was heated is determined by the following:

Mathematical Problems in Engineering
where N is the perpendicular vector that acts on the heated surface.Additionally, the stream function ψ is defined as: Here, the plus sign signifies the anticlockwise  [16,24].
Base fluid and nanoparticle c p (J.kg Density of nanofluid : Specific heat capacity : Thermal conductivity : Thermal diffusivity : Thermal expansion coefficient : Electrical conductivity : where  Mathematical Problems in Engineering circulation, while the minus sign implies the clockwise circulation of streamlines.Moreover, equation of vorticity, and the vorticity vector is Ω.

Numerical Analysis
3.1.Solution Methodology.For this hybrid nanofluid-based hexagonal heat exchanger model, the dimensionless governing Equations ( 5)-( 8) are simulated by utilizing the Galerkin weighted residual FEM.The entire domain is split into distinct triangular form elements for calculating velocity and temperature, taking into account a quadratic interpolation function.Alternatively, the linear interpolation procedure is also engaged to compute the pressure gradient.The Newton-Raphson iteration procedure using MATLAB programing is employed to construct a group of nonlinear algebraic equations to formulate those governing equations.For this approach, the criteria for convergence is derived for all variables that are unable to have a significant impact in the sense that jΠ iþ1 − Π i j: <δ ¼ 10 −5 where ΠðU; V; θÞ: represents the value of the iteration, and i + 1 and i are two sequential steps describes the whole procedure.A complete process for this FEM was elaborately described in [48].A comprehensive flowchart is described in Figure 2.

Grid Sensitivity Analysis.
To obtain the most elements possible using this finite element technique, a grid test is explained by taking the corresponding parameters: Ra = 10 4 , ϕ = 0.02, and Ha = 10.Also, Pr = 6.9 is considered for base fluid.Furthermore, to implement this sensitivity test for making proper meshing, the optimal value of the Nu av is chosen.This fluid model's entire domain has been divided into five separate numbers (2,256, 3,338, 7,250, 20,110, and 26,662) of triangular-shaped elements.Figure 3(a) represents a sample of triangular-type meshing.Also, Table 3 and Figure 3(b) show the calculated values of Nu av for various numbers of triangle elements in this fluid domain.It is obvious that the value of Nu av for the 20,110 number of elements is virtually identical to the value found for the next larger number of components.As a result, the 20,110 triangular components are recommended for meshing and completing this fluid model.

Code Validation.
The scientific reliability of this simulation is tested utilizing streamline and isotherm contours using Park et al. [19], where Ra = 10 3 and Pr = 0.7 are used.Also, the length of the square cavity was one, and the inner circle was 0.2 that is represented in Figure 4.The outermost sides served as heat sinks, while the inner circular surface in the center served as a uniform heat source.The streamline and isotherm contours are very alike to the current result.The Nu av is also compared, in Table 4, to the results of Park et al. [19], which reveal a similar consequence.That is, these results demonstrate good agreement with the present study, which increases trust in the current analysis.

Results and Discussion
The achieved outcomes are reported using streamlines, isotherms, and Nu av to investigate this hexagonal natural convective fluid model by using a hybrid nanofluid.The influence of Ra, Ha, and ϕ are illustrated with physical interpretation using Ra = 10 4 , Pr = 6.9, Ha = 10, and ϕ = 0.02 as standards.The importance of adding different nanoparticles into a base fluid is explained graphically.Moreover, a sensitivity analysis is conducted to demonstrate how Ra, Ha, and ϕ impact the Nu av by involving another statistical method called RSM.This numerical simulation on a natural convective hexagonal heat exchanger can predict a best-fitted regression equation for independent factor and response function.Also, the importance of adding hybrid nanoparticles into base fluid can be explained with 2D and 3D visualization for response function.5 illustrates streamline and isotherm contours that reveal the impact of Ra from 10 3 to 10 6 on fluid motion and heat transfer when Ha = 10, Pr = 6.9, and ϕ = 0.02 are keeping fixed.Here, Figure 5(a) displays the control of Ra on streamlines, showing that at a low value of Ra, the streamlines are almost perfectly uniform (dumbbell) along the vertical mid-point of the cavity.It is clearly observed that when the Ra shifted from 10 3 to 10 5 , the pattern of the streamline is almost the same, but the velocity field increased sufficiently.But at the higher Ra (10 6 ), the streamlining pattern varies dramatically.
The convective mode of heat transfers around the right hot cylinder gradually became stronger due to a significantly higher buoyancy effect.The streamlines have now extended out from the right hot cylinder to the left one.Once more, fluid from the nearby left cylinder that is considerably colder flows into the hot right cylinder.As a result, two small vortices inside the main vortex and the other two rotating rolls on both sides of the hot cylinder and cool cylinder are generated.These vortices show that enormous convection occurs at high Ra values.On the other hand, the isotherm contours demonstrate the temperature transmission mode (conduction or convection) as well as the practical benefits of temperature.Figure 5(b) illustrates how Rayleigh's number affects the contours of the isotherm.It is evident that convection is less within the cavity when the Ra is low (10 3 ), as evidenced by the fact that the isotherm contours are practically parallel along the vertical axis at the central part of the cavity.The cavity center's low isotherm compactness indicates weakly convective temperature flow.With rising Ra, the isotherm contours become excessively deformed and begin to flatten from hot to cool cylinders.Moreover, at high Ra (10 6 ), they almost completely flatten.The real cause of this is an increase in fluid velocity driven by a rise in Ra.As a consequence, the naturally occurring convective heat transmission from the right's hot circular exterior to the left cooler circular exterior.At low Ra (10 3 −10 5 ) value, the rate of heat transmission is relatively smaller but at higher Ra (10 6 ) value, the change of heat is very significant.6 illustrates the effect of Ha, which indicates the influence of the magnetic field, using isotherms and streamlines for keeping ϕ = 0.02, Pr = 6.9, and Ra = 10 4 .The fluctuation of streamlines for diverse values of Ha is depicted in Figure 6(a).

Effect of Hartmann Number. Figure
For each value of the Ha taken into consideration, the figures display a comparable symmetric pattern (dumbbell) along a vertical line at the center of the streamlines.It is also clearly detected that at Ha = 0, in nonattendance of the external magnetic field, the streamlines are at maximum state.
However, with the development of Ha (25, 50, and 100), that means when an exterior magnetic field is acted on the system, the streamlines gradually disappear, which indicates that flow strength drops with a growing magnetic field.The physical significance of this consequence is that when an exterior magnetic field is pragmatic, a greater field interacts with moving fluid, which possesses magnetic impressionability and reduces flow movement inside the cavity.Additionally, the Lorentz force produced by put on a magnetic field has the tendency to resist the fluid movement, which weakens the streamlines inside the cavity.Additionally, the isothermal lines in Figure 6(b) suggest that there is a change, but it is not very noticeable for larger Ha (50 and 100).This has a   7(a).When TiO 2 nanoparticles are added into water, the heat transfer rate is rapidly developed rather than the pure base fluid.Again, if Cu nanoparticles are added into the water, then the heat transfer rate is also developed rather than the pure base fluid.Moreover, if hybrid nanoparticles are added into the base fluid, the Nu av is developed remarkably.This time, the Nu av is the greatest than the mono-nanofluid or only base fluid.Furthermore, Figure 7 and ϕ).RSM provides one of the most successful approaches for modeling multidimensional situations in which the input factors influence the interest-generating responses simultaneously [51].Though other RSM algorithms exist, the second-order (quadratic) RSM approach generally proves adequate for approximating the response.First, a quadratic-type RSM model can be considered as follows: Mathematical Problems in Engineering where y; d 0 ; d i , and d ii denote the response function, terms of intercept, coefficient of linear and quadratic term of ith factor, respectively.Also, d ij expresses the coefficient of interacting term of ith and jth factors.Furthermore, the factors Ra, Ha, and ϕ are used as input factors, while Nu av is occupied as response term (y).The foremost aim of this RSM is to find the best-fitted connection among independent factors related to this proposed heat exchanger model and response function.The central composite design (CCD) based secondorder RSM is involved due to the perfectness of the secondorder regression model in both experimental and numerical simulation-type datasets [52].Here, 0 ≤ ϕ ≤ 0:06; 10 3 ≤ Ra ≤ 10 6 , and 0 ≤ Ha ≤ 100 are considered the lower and upper limits of independent factors.This algorithm takes in a total of 20 runs, 8 cubes, 6 centers, and 6 axial points for 3 independent factors.
In Table 5, the codded levels for input factors of this CCDbased RSM model are described.Also, Table 6 expresses the values of the response function, received from FEM calculation, for different runs.Furthermore, Table 7 shows the results for this RSM-based simulation, where the quantity DOF (degrees of freedom) specifies the maximum number of standalone terms.Also, the score of the sum of squares (SS) is quite satisfactory.Furthermore, the p-value, the probability value for null hypothesis becoming true, is a critical indicator of this statistical research.Since a small p-value implies that the null hypothesis is rejected, a very small p-value (generally less than 5%) recommends the model is statistically significant for the dataset given from numerical simulation.The adjusted R 2 value is 98.91% that demonstrate the framework is appropriate for conniving the Nu av .Another key statistic that needs to be extremely low for a model to be judged appropriate is the lackof-fit.Equation ( 12) expresses a general quadratic RSM model for investigating the association among the parameters Ra, Ha, ϕ, and the Nu av .Mathematical Problems in Engineering Table 8 also shows the estimated coefficients of Equation ( 12) for Nu av with the corresponding p-value.Because of the significance of the p-value, only significant terms based on p-values are employed to build an acceptable regression equation.Nonsignificant terms have been ignored (bold emphasized).
That is, the terms Ha 2 and Ha.ϕ are completely unimportant for the regression Equation ( 12) of Nu av .As a result, the connection between Nu av and the input factors can be mathematically described as follows: −1:16ϕ 2 − 0:9929Ra:Ha þ 0:8413Ra:ϕ : show two and 3D plots of the response surface derived with RSM to investigate the influence of independent factors on Nu av .The impact of Ra and Ha on Nu av is seen in Figure 8(a).This 2D contour map makes it evident that as Ra enlarged but Ha lessened, the response function increased.In a definite value of the magnetic field, the value of the Nu av is developed while ϕ remains constant.For example, when the value of Ra is raised to 500,500 (codded value 0) from 10 3 (codded value-1), the rate of heat transfer is augmented about 300%.Again, when the value of Ra is raised to 10 6 (codded value 1) from 500,500 (codded value 0), the Nu av is also increased about 18.3%.Additionally, Figure 8(b) illustrates a 3D surface view for observing the controls of Ra and Ha on Nu av .The physical explanation of this occurrence is that as the magnitude of Ra grows, the natural convective propensity strengthens simultaneously.On the other hand, in Figures 9 and 10, due to the rise of Ha, the Nu av reduces gradually.The physical enlightenment for this finding is that when an outer magnetic field is provided, a larger field interacts with stirring fluid, which reduces the flow circulation of the entire cavity.Furthermore, Figure 9 expresses another 2D and 3D graphical representations to represent the impact of Ra and ϕ on Nu av for this hybrid nanofluid model.Also, the rate of Nu av is boosted up due to the rise of Ra and ϕ, while Ha remains fixed.At the maximum value of Ra and ϕ (coded value 1), this changing rate of Nu av is maximum, which is clear from the 2D contour Figure 9(a).Similarly, Figure 10 symbolizes the impact of Ha and ϕ on Nu av .In this case, the changing rate of Nu av is developed by raising the size of ϕ with diminishing the magnetic field (Ha).But this changing rate of Nu av is smaller than the previous two cases.Mathematical Problems in Engineering Now, using Equations ( 14) and ( 16), we can determine the rate of sensitivity of the response function (Nu av ), and the outcomes are exhibited in Table 9.Here, to see the sensitivity rate the codded value of Ra is used 0 and 1; codded value of Ha is used at −1, 0, and 1; and the codded value of ϕ is taken as 0, −1, and 1.It is also important to remember that a positive sensitivity indicates that the input factors amplify the reaction.This demonstrates that the Ra and ϕ have a positive influence on the Nu av .A negative sensitivity, on the other hand, demonstrates the polar opposite trend, where growing the input factors gives the reaction to go down.This shows that the input factor Ha has a detrimental impact on Nu av .Furthermore, the straight bar in Figure 11 represents positive sensitivity to Nu av , whereas the flipped bar represents negative sensitivity to Nu av .The sensitivity grade is also shown by the overall length of the vertical bar.

Conclusion
In this work, heat transfer consequences based on natural convective hexagonal heat exchanger consuming hybrid (Cu-TiO 2 -H 2 O) nanofluid are investigated.Considering the relationship of heat transport mechanisms under the impact of the magnetic effect in particular.A numerical solution is built to encompass the complicated phenomena using the Galerkin infinite element method.The statistical methodology known as RSM is used to investigate the sensitivity study of the output function for an in-depth evaluation of the heat conveyance mechanism.The effect of hybrid nanofluid on streamlines, isotherms, and Nu av is studied.
Additionally, utilizing RSM, several 2D and 3D surface plots are provided for clear visualization of the heat transportation method with implicated critical components.A best fitted regression equation is also developed by statistical the RSM technique.Because of the exceptional thermal properties of the hybrid nanofluid, the Cu-TiO 2 -H 2 O exceeds the heat transfer efficiency of the base fluid by 35.85%.Also, the Cu-H 2 O and TiO 2 -H 2 O nanofluids express 34.3% and 29.38% greater heat transmission liken to base fluid, respectively.Furthermore, the Ha has an immediate negative impact on the velocity distribution of this hybrid nanofluid.Finally, the sensitivity analysis shows that the parameters Ra and ϕ have a favorable effect on Nu av , but Ha has a negative effect on heat transmission from heated surfaces.By using RSM, the discussion about sensitivity analysis and explanation of the 3D surface plot, for natural convective hexagonal heat exchanger containing hybrid nanofluid, is the main novelty of this work.

FIGURE 1 :
FIGURE 1: Schematic view of the proposed hexagonal model.

FIGURE 2 :
FIGURE 2: A comprehensive flowchart of the computing technique.

FIGURE 3 :
FIGURE 3: (a) Type of meshing entire domain and (b) the grid test by Nu av .
(b) shows an enhancement of Nu av with adding off mono-nanoparticles or hybrid nanoparticles.The results show that the Nu av of Cu-TiO 2 -H 2 O hybrid nanofluid is more sophisticated than the Cu-H 2 O nanofluids, though the rate of Nu av reduces with rising Ha factor for all types of fluid combination.That is, Figure 7 shows that the Cu-TiO 2 -H 2 O hybrid nanofluid has superior heat transport ability than TiO 2 -H 2 O nanofluid or Cu-H 2 O nanofluid because of its outstanding thermal properties.This is one of the primary motives for involving hybrid nanofluid instead of base fluid or mono-nanofluid.4.4.RSM.The statistical technique RSM is used to analyze the impact of response function (Nu av ) caused by the relevant factors (Ra, Ha,

FIGURE 8 :
FIGURE 8: Effect on Nu av for Ra and Ha: (a) 2D sight and (b) 3D sight.

TABLE 1 :
Thermo-physical individual of base fluid and solid particles

TABLE 3 :
Numerical values of Nu av for different elements.

TABLE 4 :
Numerical values of Nu av with Park et al. [19]..3.Effect of Hybrid Nanofluid.The Nu av is employed with varying values of Ra and Ha to describe and display the heat transfer rate for various types of fluids such as base fluid, Cu-H 2 O, TiO 2 -H 2 O, and Cu-TiO 2 -H 2 O.For changing the value of Ra, the Nu av growths gradually for pure fluid (H 2 O), TiO 2 -H 2 O nanofluid, Cu-H 2 O nanofluid, and hybrid nanofluid (Cu-TiO 2 -H 2 O), which is depicted by Figure physical meaning in that fluid movement is restricted by the acted magnetic field.Thus, the isothermal lines change a very little, as is seen from Figure6(b), and the heat convection caused by fluid movement is minimal.4

TABLE 5 :
Comparison of thermal enactment by Nu av for (a) Ra and (b) Ha.Codded levels for input factors.

TABLE 6 :
Levels of input factors and response function.

TABLE 7 :
ANOVA for Nu av using RSM.

TABLE 8 :
Regression coefficients for Nu av that are predicted based on RSM.
Values indicated in the bold font denote insignificant terms (p>0:05).

TABLE 9 :
Sensitivity analysis for Nu av .