Nanofluids are immersions of nanosized particles in a base fluid. The base fluid could be water, ethylene glycol, etc., whereas there are varieties of nanoparticles to choose from, for example, copper (Cu), aluminum oxide ( Al2O3 ), titanium ( TiO2 ), etc. Compared to pure fluids, nanofluids exhibit a few superior properties, including their enhanced thermal conductivity and heat transfer coefficient. Massive research has been conducted by several researchers explaining the flow and thermal behavior of nanofluids. An experimental study by He et al. [1] investigates the heat and flow behavior of TiO2 nanofluids flowing through a vertical pipe. Results show that nanoparticles enhanced the convective heat transfer coefficient for both laminar and turbulent flow regimes, enhancing laminar flow being much smaller than for turbulent flow. A two-dimensional experimental analysis by Shafahi et al. [2] using aluminum oxide ( Al2O3 ), titanium oxide ( TiO2 ) and copper oxide (CuO) in water base fluid, showed that temperature distribution depends on effective thermal conductivity. In another study by Pang et al. [3], thermal enhancement in Al2O3 was found to be less than that in silicon dioxide ( SiO2 ). This anomaly was attributed to the larger cluster size of SiO2 and it was concluded that clustering increases thermal conductivity. A similar study on thermal conductivity by Lee et al. [4] showed that thermal conductivity and viscosity increase with increasing the nanoparticle volume fraction.

In a study by Truong et al. [5] to investigate if nanoparticles increase Critical Heat Flux (CHF) on a passively engineered heating surface, bares and blasted plates were used to establish a baseline for the CHF measurement and the nanofluids used included diamond, alumina and zinc oxide with volume fractions of 1%, 10%, and 20%, respectively. The CHF value for diamond showed an 11% enhancement, and that for both zinc oxide and alumina showed 35% enhancement. Extended research was carried by Wen [6] in further increasing the already enhanced CHF value for nanofluids reviews experiments performed on enhanced CHF and investigated possible mechanisms that increase CHF enhancement. A simplified dry patch model was developed, and the structural disjoining pressure was investigated by calculating interfacial shapes for pure fluid and various concentrations of nanofluids. An experiment by Chun et al. [7] elucidates the effect of nanofluids on boiling heat transfer of silicon, silicon carbide, and water using platinum (Pt) wire as a heat source. The experiment shows that nanoparticle-coated Pt wires are cooled down at a much higher rate compared to the bare Pt wires cooled by water and Si nanofluids.

Among works done on natural convective heat transfer, Wen et al. [8] formulated the transient and steady heat transfer coefficient for different concentrations of nanofluids under natural convective conditions in a range of Rayleigh numbers (Ra) from 106 to 109 . Lin et al. [9] studied the heat and fluid flow of natural convection in a cavity filled with Al2O3 /water nanofluid operating under differentially heated walls. The results show that heat transfer characteristics of the nanofluid can be increased when the ratio of minimum to maximum nanoparticle diameter is increased from 0.001 to 0.007 nm or the mean nanoparticle diameter is decreased from 250 to 5 nm. Corcione [10] conducted a theoretical study on the natural convection heat transfer of nanofluids using three different nanoparticles-Cu, Al2O3 , and TiO2 in two different base fluids-water and ethylene glycol. Ashorynejad et al. [11] showed that the Nusselt number drops off as the Hartmann number increases; however, it rises with the rise of the Rayleigh number and nanoparticle volume fraction. The magnetic field increases or decreases with the influence produced by the existence of nanoparticles concerning the Rayleigh number. Yu et al. [12] illustrated that nanofluid has a comparatively higher heat transfer coefficient for the same Reynolds number, and this coefficient increases with the increasing value of the mass fraction of CuO nanoparticles. They also reported that nanofluid does not have major influence on heat transfer factors at very small volume concentrations and increasing mass fraction causes the pressure of nanofluid to increase. Mutuku [13] clearly showed that CuO-EG nanofluids lead to a swift decline of heat at the boundary layer. Alsoy-Akgün [14] observed that the behaviours of all the variables are subjective to the varying values of parameters (Rayleigh number, Hartmann number, and particle fraction). Zahan et al. [15] found that increasing the value of the Rayleigh number and divider position causes an increase in the heat transfer, but an increase in Hartmann number reduces the heat transfer. Also, they figured that increasing solid volume fraction improves the heat transfer performance. Alsabery et al. [16] showed that the porous layer increment considerably influences the heat transfer. Xiong et al. [17] considered a square cavity with a square thermal column to study the natural convection of the SiO2 -water nanofuid.

Among works done with nanofluids in various geometries, the numerical study by Vajjha et al. [18] investigates heat transfer of nanofluids placed in the flat tubes of an automobile radiator. The results show a considerable increase in average heat transfer coefficient with increasing concentration and Reynolds number. In another numerical study by Maiga et al. [19] to investigate thermal characteristics of the forced convective flow of nanofluids, Al2O3 /Ethylene glycol and Al2O3 Al/water nanofluids flowing in tube and radial geometric configurations were considered. Results show that nanofluids enhanced the heat transfer coefficient, and this enhancement is higher at higher particle concentrations. Abu-Nada et al. [20] studied the effect of inclination angles on a 2D enclosure filled with Cu/water nanofluid. Khanafer et al. [21] numerically investigated the heat transfer characteristics of nanofluids within two-dimensional side heated enclosures for various pertinent parameters. Results show that the presence of nanoparticles enhances the heat transfer characteristics of a fluid and the heat transfer characteristics of nanofluids increase significantly with an increase in volume fraction. Saleem et al. [22] demonstrated that platelet shape geometry has maximum convective flow and a more random pattern of isotherms due to Darcy’s number. Javaherdeh et al. [23] carried out a computational investigation to analyze laminar natural convection heat characteristics in a wavy cavity filled with CuO /water nanofluid. The magnetic field caused the local Nusselt number to decrease in value, at the hot wall, in their work. Moreover, increasing nanoparticle concentration resulted in enhancing the heat transfer performance. Nevertheless, the presence of nanoparticles leads to a noteworthy improvement in heat transfer for all values of the Rayleigh number. Rahimi et al. [24] considered a hollow L-shaped cavity filled with SiO2--TiO2 /water-EG mixture of nanofluid to examine the natural convective heat transfer and fluid flow by lattice Boltzmann method.

Jung et al. [25] conducted an experimental study to investigate the heat transfer coefficient and friction factor of nanofluids in a rectangular-shaped microchannel. Kondaraju et al. [26] conducted a numerical simulation to investigate the effects of coagulation of particles on thermal conductivity and found that heat transfer increases with an increase in volume fraction. Oztop et al. [27] conducted another numerical study to examine the heat and fluid flow due to the natural convection of nanofluids inside a partially heated rectangular-shaped enclosure. Results indicate that the heat transfer enhances with an increase in Rayleigh number as well as with an increase in particle concentration. The numerical investigation on mixed convection heat transfer by Kherbeet et al. [28] shows that the Nusselt number increases with increasing the solid volume fraction and Reynolds number and also reveals that the nanofluid containing SiO2 nanoparticles have the highest Nusselt number. To investigate the forced convection flow behavior of Al2O3 nanofluid in a radial cooling system, Yang et al. [29] conducted a numerical simulation and found that heat transfer enhances with a rise in Reynolds number and particle concentration. A comprehensive review that presents an all-inclusive study on forced convective heat transfer enhancement was presented by Kakac et al. [30].

Yang et al. [31] conducted an experimental study on nanofluids that exhibit both viscosity and elastic properties and found that thermal conductivity increases with increasing particle volume concentration and temperature. Viscosity was found to be higher when using visco-elastic nanofluid rather than the base fluid, and it increases with increasing particle volume concentration but decreases when the temperature was increased. A study performed by Khanafer et al. [32] analyses the synthesis of thermo-physical properties of nanofluids and their contribution in heat transfer enhancement. It was concluded that the types of models that give appropriate values at different temperatures were not clearly defined, suggesting further investigations in the measurement of nanofluid properties.

An interesting numerical study on nanoparticles of various shapes, performed by Fan et al. [33], investigates the effects of thermal conductivity ratio, particle volume fraction, and particle morphology on multiple aspects of nanoparticles such as temperature gradient, phase lags of heat flux, and thermal conductivity. Results reveal that two aspects of nanoparticle geometry affect thermal conductivity, i.e., particle’s radius of gyration and/or non-dimensional interfacial area. In a study performed on transient buoyancy-driven convective heat transfer of bottom-heated water-based nanofluids by Yu et al. [34], results with and without considering Brownian motion were recorded for each volume fraction value. Here a phenomenon called Pitchfork bifurcation was observed for Gr>5.60×104 , the critical Grashof number. Ghalambaz et al. [35] showed that adding a mixture of nanoparticles for a conductive-dominant system (low Rayleigh number) causes an enhancement in the heat transfer. They showed that the Rayleigh number and the thermal conductivity ratio are the escalating factors of the heat transfer rate. Their results showed that the local Nusselt number at the surface of the conjugate wall shrinks significantly for a convective-dominant flow (high Rayleigh number) and an excellent thermally conductive wall.

Comparative analyses on the effects of different nanoparticles on the heat transfer characteristic are available. However, research on the impact of varying base fluid on the nanofluid heat transfer characteristics is rare in the literature. Based on the literature review and to the best of the authors’ knowledge, there is no scholarly work comparing water and ethylene-glycol as the base fluids used for heat transfer medium as nanofluid. To remove the bulk heat or to cool a device in an industry or laboratory is a significant event for enhancing the efficiency of a machine or electrical system, and choosing an appropriate base fluid while using nanofluid plays an important role. The present research is a comparative investigation of the flow and thermal performance between nanofluids and pure-fluids flowing in such a square cavity whose bottom wall is partially heated by a heat source of variable length. The governing non-dimensional Navier Stokes and energy equations are solved numerically using the finite volume method with the staggered grid.

A square cavity has been considered with a heat source placed at its bottom wall, as shown in Fig. 1. The top border is entirely adiabatic, and a part of the bottom wall is kept heated while the rest is adiabatic. The sidewalls are cool. Considering this cavity contains nanofluid, the flow and thermal behavior are investigated when the bottom wall is heated at different lengths and different base fluids and solid particles. There are some predefined underlined assumptions for the simplicity of the problem, and these are: fluid flow is laminar, steady-state, Newtonian, and incompressible. Viscous dissipation, as well as radiation effects, are neglected here. For buoyancy force density variation, Boussinesq approximations have been considered. The flow is governed by the following Navier Stokes and energy equations:

(1) ∂u∂x+∂v∂y=0

(2) ∂u∂t+u∂u∂x+v∂u∂y=−1ρnf∂p∂x+μnfρnf(∂2u∂x2+∂2u∂y2)

(3) ∂v∂t+v∂v∂y+u∂v∂x=−1ρnf∂p∂y+μnfρnf(∂2v∂y2+∂2v∂x2)+g(ρβ)nf(T−Tc)ρnf

(4) ∂T∂t+u∂T∂x+v∂T∂y=αnf(∂2T∂x2+∂2T∂y2)

where ρnf is the density, μnf is the viscosity, βnf is the thermal expansion coefficient, αnf is the thermal diffusivity of nanofluid, and g is the acceleration due to gravity.

To non-dimensionalizing the above mentioned governing equations, the appropriate non-dimensional variables are defined as follows:

X=xL,Y=yL,ϵ=xhL,U=uLαf,V=vLαf,Θ=T−TcTh−Tc,

(5) P=pL2ρfαf2,Pr=νfαf,Ra=gβ(Th−Tc)L3Prνf2,τ=αftL2,

νnf=μnfρnf,αnf=knf(ρcp)nf

where xh is the dimensional heat source length of the bottom wall, Pr is the Prandtl number and Ra is the Rayleigh number.

The effective viscosity for a suspension containing small spherical solid nanoparticles is given by Brinkman [36]:

(6) μnf=μf(1−ϕ)1.5

The effective density and thermal expansion coefficient of a fluid containing solid nanoparticles can be given as [37]:

(7) ρnf=(1−ϕ)ρf+ϕρs

(8) (ρβ)nf=(1−ϕ)(ρβ)f+ϕ(ρβ)s

The heat capacitance of nanofluid is given as:

(9) (ρcp)nf=(1−ϕ)(ρcp)f+ϕ(ρcp)s

The Maxwell–Garnetts model [38] for effective thermal conductivity of a mixture of base fluid along with the particular concentration of nanoparticles states:

(10) knf=ks+2kf−2(kf−ks)ϕks+2kf+(kf−ks)ϕkf

Applying the above non-dimensional variables into Eqs. (1)–(4) yield the following non-dimensional equations:

(11) ∂U∂X+∂V∂Y=0

(12) ∂U∂τ+U∂U∂X+V∂U∂Y=−1(1−ϕ)+ϕρsρf∂P∂X+Pr(1−ϕ)2.5[(1−ϕ)+ϕρsρf](∂2U∂X2+∂2U∂Y2)

(13) ∂V∂τ+U∂V∂X+V∂V∂Y=−1(1−ϕ)+ϕρsρf∂P∂Y+Pr(1−ϕ)2.5[(1−ϕ)+ϕρsρf](∂2V∂X2+∂2V∂Y2)+RaPrΘ[11+(1−ϕ)ρfϕρsβsβf+11+(1−ϕ)ρfϕρs]

(14) ∂Θ∂τ+U∂Θ∂X+V∂Θ∂Y=knfkf[(1−ϕ)+ϕ(ρcp)s(ρcp)f](∂2Θ∂X2+∂2Θ∂Y2)

The physical properties of the particles and fluid taken into consideration are tabulated in Tab. 1.

To solve the non-dimensional governing Eqs. (11)–(14) the following boundary conditions are used:

(15) Θ=0,U=V=0,atX=0,1,0<Y<1

Θ=1,U=V=0,atY=0,1−ϵ2≤X≤1+ϵ2

∂Θ∂Y=0,U=V=0,atY=0,0≤X≤1−ϵ2,1+ϵ2≤X≤1

∂Θ∂Y=0,U=V=0, at Y=1,0≤X≤1

where ϵ is the non-dimensional heat source length which can vary.

The rate of heat transfer is measured in terms of the Nusselt number. The local Nusselt number is defined by

Nu(X)=−knfkf∂Θ∂YY=0

and the average Nusselt number on the heated bottom wall is defined by

Nu¯=1ϵ∫1−ϵ21+ϵ2Nu(X)dx

A SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm, also known as a pressure-corrector method, is used to integrate the problem (11)–(14) numerically in the given cavity. The SIMPLE algorithm involves the nodal momentum contributions and cell-face dissipation coefficients in the pressure-correction equation on a collocated or staggered grid arrangement. For the pressure and temperature, computational results are computed and stored at the center of the node. A three-point backward difference formula is used for the time derivative of the velocity and temperature, whereas the central difference quotient is used for the convective and diffusion terms. After discretizing the governing Eqs. (11)–(14), the resulting quasi-linear algebraic equations are solved using the line Gauss–Siedel method. In this study, the local mesh refinement is performed with a tangential hyperbolic stretching function. Further, a constant time step Δτ=104 is used to ensure the CFL number lies between 0 and 1.

In order to validate our code, the present results have been compared to the benchmark solution of Davis [42], in which side heated cavity flow was considered (Tab. 2). In this comparison, the results are presented in terms of the average Nusselt number and maximum U and V velocities for three different Rayleigh numbers. From this table, it is seen that results are in good agreement with the benchmark solutions of Davis [42]. Khanafer et al. [21] conducted a numerical study on the natural convection of nanofluid in a side heated square cavity. A comparison has been made with these numerical results for Gr 105 and 0.1(10) . The results are shown in Fig. 2a, which clearly illustrate a good agreement of our solutions with Khanafer et al. [21]. In addition, one comparison is also performed with the experimental study conducted by Calcagni et al. [43] in which the present geometric configuration is utilized, that is, localized heating from the bottom and symmetrically cooling from the sides. For comparison, solutions are obtained for pure fluid with Pr = 0.71 Ra = 1.205 × 10⁵ and heat source length ɛ = 4/5, and the agreement is good indeed, as depicted in Fig. 2b.

For the present problem, a grid independence test has been conducted for Pr = 16.6, Ra=105 , ϵ=3/5 and ϕ=0.1 . Three grid arrangements are taken into consideration as 100 × 100, 140 × 140, and 180 × 180, and the results are shown in Fig. 2c. For these three different grid arrangements, the results are almost independent for temperature distribution Θ . Therefore, it is reasonable to select the 140 × 140 control volumes, and the rest of the simulations are performed with this grid size.

Simulations have been carried out for several Rayleigh numbers, Ra, from 103 to 105 and nanoparticle volume fraction between 0 . Five separate cases are considered here as follows.

The objective of the work is to investigate the flow and thermal behavior of nanofluid in a square cavity compared to ordinary pure fluid with localized heating at the bottom wall. At the end of the research, it is found that nanofluids are superior to pure fluid in transferring heat away from a cavity where the bottom border is heated. The length of heat source ϵ has a significant impact on flow behavior and the rate of heat transfer from the heat source. Lower the ϵ higher is the heating effect taking place, and hence the temperature distribution is almost the same at high ϵ values whereas it is very much different for low ϵ values. It is also found that the Rayleigh number affects the flow and thermal behavior of nanofluids. As far as the average Nusselt number is concerned, it decreased with an increase in ϵ . Changing base fluid to water instead of ethylene glycol reduces the average Nusselt number for any particular Ra indicating that ethylene glycol is a better transporter of heat energy away from a system rather than water. Choosing the appropriate nanoparticle in the nanofluid has also been found to be a key factor since it is observed that when the solid particle was Cu, the value of the average Nusselt number was higher than it was when the solid particle was Al2O3 . In a nutshell, it can be said that if there is a large laboratory or a system where there are plenty of devices generating a huge amount of bulk heat, then cooling with the help of nanofluids, instead of cooling via fans or pure fluids, will be a very effective and wise method since it will drive heat away faster and will increase the overall efficiency of the heat-generating components involved.

The results obtained from this research can be of good use to manage the bulk heat in telecom field data centers. Since nanofluids are proven to have a faster heat transfer rate than pure fluids, their use as a coolant will more likely reduce the load on chillers and air conditioners, bringing down the electricity cost. A comparative analysis between two different base fluids (water and ethylene glycol) has been carried out. The study would help engineers utilize an optimum nanofluid to maximize the heat transfer rate. Moreover, the impact of different nanoparticles (Al₂O₃ and Cu) on the flow circulation and heat transfer rate was also analyzed. As a consequence, this article could be a guideline for thermal engineers to design more efficient heat transfer equipment.