A thermal energy analysis of binary ( Go-Co/H 2 O ) and ternary ( Go-Co-Zro 2 /H 2 O ) nanofluids based on characterization and thermal performance

Abstract

Hybrid or binary nanofluids have superior mechanical and thermal characteristics but the tri-hybrid nanofluids comprise of more embellished thermal properties, better physical strength, and enhanced stability. The present work characterizes the thermal and physical aspects of the hybrid and tri-hybrid nanofluids. The nano-composition of graphene oxide (Go) and cobalt (Co) is used in the amalgamation of hybrid nanofluid Go-Co/H₂O, whereas the addition of zirconium oxide (ZrO₂) in this mixture gives rise to the ternary Go-Co-ZrO₂/H₂O hybrid nanofluid. The activation energy and viscous dissipation terms are also amended in the governing equations. The mathematical framework consists of a complex natured dynamical system. However, a numerical algorithm based on finite-difference discretization is developed which can solve the system numerically via the MATLAB software. A comparison with the existing literature is provided in order to validate the numerical procedure. From the outcomes, it is noticed that the temperature of hybrid as well as tri-hybrid nanofuid increases rapidly with change in concentration of zirconium oxide and cobalt. Temperature increases up to 20% by taking 0.1 volume fraction of both zirconium oxide and cobalt. Porous medium and activation energy resist the flow and concentration respectively. A comparative judgment evidently reveals that tri-hybrid Go-Co-ZrO₂/H₂O nanofluid has a substantial effect on temperature as equated to hybrid or pure nanofluid.

Keywords

Ternary hybrid nanofluids graphene oxide cobalt zirconium oxide activation energy

Introduction

The nanofluids are engineered in a systematic way having solid fragments of metals or metallic oxides. The metallic or non-metallic materials gives rise to nanofluids when mix in a base liquid.

These fluids not only increase the thermal conductivity but also embellish the thermal characteristics of the base liquids. A substantial increase in the physical and thermal characteristics of the host liquids owe to nanofluids.^1–4 Even a lot of struggle has been made to explore the thermal characteristics of the nanofluids but still efforts are continued. As a result of this queue, the investigations have been extended from nano to hybrid and tri-hybrid nanofluids. The composition of two distinct nanoparticles together with base fluids generates the hybrid nanofluids but the ternary or tri-hybrid nanofluids are composed of three different nano-materials which are immersed in the host liquid. The host liquid may be the engine oil, water, sodium alginate, kerosene oil, ethylene glycol, paraffin, alcohol, and so on. In the same way, tri-hybrid nano-composition may be composed of metals, nitrides, carbides, or metallic oxides materials. The tri-hybrid nanofluids can provide new opportunities to improve mechanical and thermal characteristics in a variety of applications.^5–7

The ternary case of nanofluids has been examined recently by various researchers. Shao et al.⁸ examined the ternary nanofluid considering the impact of natural convection in the flow. The geometry used in this analysis was the porous enclosure. Two hot movable baffles in the enclosure caused the motion of fluid. The prominent effect of Darcy number was studied, in this research, using a simulation method. A review analysis was presented by Adun et al.,⁹ in which, they discussed the stability and thermophysical properties of mono as well as tri-hybrid case of nanofluids. Algehyne et al.¹⁰ investigated the ternary nanoparticles MgO, CoFe₂O₄, and TiO₂ using the Non-Fourier's concept. Their work was based on numerical analysis that designated the fact that tri-hybrid case of nanofluids would contain better efficiency than usual fluids subject to heat transfer enhancement. Elnaqeeb et al.¹¹ interpreted the dynamics of automobiles cooling systems using three different nanomaterial (tri-hybrid case). The importance of heat transport characteristics was also elaborated. The water base flow of three particles Ag-TiO₂-Cu was investigated by Li et al.¹² subject to Thomson slip and temperature jump conditions. The RKF-45 technique was adopted to perform simulations. The results of this study spotted that the velocity slip would be the factor for which the flow speed got reduced.

The present analysis incorporates the nanoparticles such as graphene oxide, cobalt, and zirconium oxide for the composition of ternary mixture Go-Co-ZrO₂/H₂. Each type of nanoparticles has its own special thermal characteristics. Cobalt is a metal with silvery-blue color. The preparation of magnets owe to the cobalt. Some materials like aluminum and nickel when alloyed with the cobalt produce the power magnets. Nowadays, the treatment of cancer has also been carried out with the use of radioactive cobalt. The cobalt oxide (CO₃) has been synthesized in water by Salari et al.¹³ to prepare the water-based nanofluid. They used some experimental methods like SEM and XRD to illustrate the mechanisms of nanomaterial. Sekhar et al.¹⁴ measured the relative viscosity of cobalt-based nanofluid at high temperature. The temperature range limit varied from 30 °C to 60 °C. Cobalt ferrite COFe₂O₄ and silver Ag were comparatively studied by Farooq et al.¹⁵ The purpose of this work was to notice that which nanoparticle (out of COFe₂O₄ and Ag) would exhibit better efficiency. Santhosh and Sivaraj¹⁶ and Murtaza et al.¹⁷ proposed the cobalt particles in the preparation of nanofluid. Both of these studies consisted of numerical analysis of the respective problems. The first one considered the effect of magnetohydrodynamics and the other one taken the second-order slip effect in the model equations.

Graphene oxide and the zirconium oxide also possess better features. The oxidation process of graphite gives rise to the graphene oxide (Go) which is single-atomic-layered material. It is readily available cheap material. The oxide form of graphene (Go) is easily dissolvable in H₂O and other solvents. It is not conductive as the oxygen exists in the lattice graphene. The usages of graphene oxide involve water purification, catalysts, bio-sensors, energy storage devices, and so forth. The zirconium di-oxide ZrO₂ or the zirconia (other name) is often obtained by the fusion of zirconium silicate which is also known as zircon sand. The potential uses of ZrO₂ have been widely found in the ceramic industry. This metal usually exhibits an inorganic behavior. It has strong chemical inertness but low solubility. Some recent studies investigating the features of graphene oxide and zirconium oxide are referred to the references.^18–20

The aim to present this work is the interpretation of three distinct nanoparticles such as graphene oxide, cobalt, and zirconium oxide which make amalgamation Go-Co-ZrO₂ in the water (base fluid). The nano-composition of graphene oxide, cobalt, and zirconium oxide has not been analyzed yet, specifically, taking the combined impacts of activation energy and the viscous dissipation. The new features of water based Go-Co-ZrO₂ ternary hybrid nanofluids are investigated. The analysis also interprets the thermal aspects of hybrid as well as pure nanofluids using Go/H₂O and Go-Co/H₂O respectively.

Mathematical description of the model

Ternary hybrid nanofluid flow is being assumed in a porous medium. Different types of nanofluids such as graphene-oxide-based mono nanofluid Go/H₂O and cobalt-graphene-oxide-based hybrid nanofluid Go-Co/H₂O have been taken into account. A tri-hybrid nano-composition is obtained when the zirconium oxide is amalgamated in the water which is base fluid in the present case. In this way, a tri-hybrid nanoparticles Go-Co-ZrO₂-H₂O nano-composition is prepared. The concentrations and temperatures are specified at the surface and away from surface and are respectively denoted by $C_{W}$ , $C_{\infty}$ , $T_{W}$ , and $T_{\infty}$ . The extending surface possesses the velocity $U_{W} (x)$ . The suction phenomenon occurs when suction velocity $v_{o}$ is greater than 1 ( $v_{o} > 1$ ). The model equations also involve the activation energy as well as viscous dissipation term. The structure of the problem is shown in Figure 1(a).

Figure 1.

(a) Physical model of the problem. (b) Mesh sensitivity analysis.

Flow assumptions

The prominent impacts of porosity (porous medium), viscous dissipation, and activation energy have been assumed in the current analysis.

Porous medium

The pores through which a fluid flows involve the interconnected void spaces that can be referred to as porous medium. The Darcy's law typically expresses the flow through a porous medium and according to this law, the characteristic length scale (L) and the dynamic viscosity (μ) are inversely proportional to the volumetric flow rate (q). On the other hand, q is in direct proportion to the pressure gradient (ΔP). Mathematically, this relation can be written as:

q = - k \frac{Δ P}{μ L}

where,

k is proportionality constant and expresses the permeability of the porous medium. The negative sign indicates that flow occurs in the direction of decreasing pressure.

Viscous dissipation

The mechanical energy conversion, due to the internal friction within a fluid flow, into heat is known as viscous dissipation. It can be formulated using the Navier-Stokes as well as the energy equations and is typically expressed in terms of the rate of thermal energy generation per unit volume $Q_{v}$ within the fluid. The mathematical representation of viscous dissipation is as follows:

Q_{v} = μ (\nabla u)^{2}

Where

$Q_{v}$ is the rate of viscous dissipation per unit volume,

$\nabla u$ is the velocity field gradient,

$μ$ is the fluid's dynamic viscosity.

The above equation portrays that the rate of viscous dissipation is proportional to the dynamic viscosity of fluid and square of the velocity gradient.

Activation energy

The minimum amount of energy (e.g. the fundamental concept in chemical kinetics) relates to activation energy which is required for a chemical reaction to occur. This energy creates the energy barrier for which the molecules must overcome for a reaction to proceed. Arrhenius equation correlates a mathematical relation between the temperature (T), the rate constant (k), and the activation energy Ea at which the reaction occurs:

k = A e^{- \frac{E a}{R T}}

where,

e is the base of the natural logarithm,

A is the pre-exponential factor or frequency factor,

R is the gas constant.

The Arrhenius equation indicates that a lower rate constant will be due to a higher activation energy at a given temperature. This equation can be used to estimate activation energies from experimental data.

Governing equations

The model equations relating the tri-hybrid or ternary hybrid nanofluids have the following form^21–23:

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

(1)

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = υ_{T n f} \frac{\partial^{2} u}{\partial y^{2}} - \frac{μ_{T n f}}{ρ_{T n f} ℜ *} u - \frac{σ_{T n f}}{ρ_{T n f}} B_{0}^{2} u,

(2)

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{T n f}}{{(ρ C_{p})}_{T n f}} \frac{\partial^{2} T}{\partial y^{2}} + \frac{μ_{T n f}}{{(ρ C_{p})}_{T n f}} {(\frac{\partial u}{\partial y})}^{2},

(3)

u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D_{B} \frac{\partial^{2} C}{\partial y^{2}} - K_{r}^{2} (C - C_{\infty}) {(\frac{T}{T_{\infty}})}^{m_{0}} \exp (\frac{- E *}{k * T}) .

(4)

The proposed boundary conditions (BCs) are composed as:

\begin{matrix} y = 0 : u (x, 0) = U_{w} (x) = c x, T (x, 0) = T_{w}, v (x, 0) = - v_{0}, C (x, 0) = C_{w}, \\ y \to \infty : u (x, \infty) = 0, T (x, \infty) = T_{\infty}, C (x, \infty) = C_{\infty} . \end{matrix}}

The model equations (1)–(4) and the BCs (A) evolve the following terms:

$ρ_{T n f}$	density of the tri-hybrid nanofluid	$U_{W} (x)$	stretching velocity
$u$	horizontal velocity component	$(ρ C_{p})_{T n f}$	heat capacity of the tri-hybrid nanofluid
$μ_{T n f}$	viscosity of the tri-hybrid nanofluid	$K_{r}$	rate constant of chemical reaction
$m_{o}$	exponent fixed rate	$K_{T n f}$	thermal conductivity of the tri-hybrid nanofluid
$ℜ *$	permeability of the porous medium	$v_{T n f}$	kinematic viscosity of the tri-hybrid nanofluid
$v$	vertical component of velocity	$E *$	activation energy
$σ_{T n f}$	electrical conductivity of the tri-hybrid nanofluid	$k *$	mean absorption coefficient

The transformed equations

The similarity variables, using boundary layer theory, have been taken in the form²⁴:

\begin{aligned} η = \sqrt{\frac{c}{υ_{f}}} y, ψ = \sqrt{c υ_{f}} x F (η), u = c x F^{'} (η), θ (η) = \frac{T - T \infty}{T_{w} - T \infty}, \\ v = - \sqrt{c υ_{f}} F (η), ϕ (η) = \frac{C - C_{\infty}}{C_{w} - C_{\infty}} . \end{aligned}

(5)

Incorporating the variables (5) to the Equations (1)–(4), we achieve the following dimensionless form:

d_{1} ({f^{'}}^{2} - f f^{″}) = d_{2} {f^{″'} - ε_{p} f^{'}}

(6)

- d_{3} f θ^{'} = \frac{1}{Pr} d_{4} θ^{″} + E_{c k} d_{5} f {^{″}}^{2}

(7)

\frac{1}{S_{c}} ϕ^{″} + f ϕ^{'} - C h_{R} (1 + δ θ)^{m 0} \exp (\frac{- A_{E}}{1 + δ θ}) ϕ = 0

(8)

Where the parameter representing the porosity of the medium is

ε_{p} = \frac{υ_{f}}{c ℜ *}

and

E_{c k} = \frac{U^{2}}{{(c_{p})}_{f} (T_{w} - T_{\infty})}

expresses the Eckert number. The activation energy is denoted by

A_{E} = \frac{E *}{k * T_{\infty}}

and the chemical reaction parameter is described by

C h_{R} = \frac{K_{r} x}{U_{w}}

. The other parameters are Prandtl number, temperature difference parameter, and Schmidt number represented by

P r

δ

, and

S_{c}

respectively. The notations such as

d_{1}, d_{2}, d_{3}, d_{4}

, and

d_{5}

are mentioned below:

d_{1} = (1 - ϕ_{3}) {(1 - ϕ_{2}) {(1 - ϕ_{1}) ρ_{f} + ϕ_{1} (\frac{ρ_{s_{1}}}{ρ_{f}})} + ϕ_{2} (\frac{ρ_{s_{2}}}{ρ_{f}})} + ϕ_{3} (\frac{ρ_{s_{3}}}{ρ_{f}})

(9)

d_{2} = (1 - ϕ_{1})^{2.5} (1 - ϕ_{2})^{2.5} (1 - ϕ_{3})^{2.5}

(10)

d_{3} = (ρ C_{p})_{T h f} = [(1 - ϕ_{3}) {(1 - ϕ_{2}) [(1 - ϕ_{1}) {(ρ C_{p})}_{f} + ϕ_{1} {(ρ C_{p})}_{s_{1}}] + ϕ_{2} {(ρ C_{p})}_{s_{2}}} + ϕ_{3} {(ρ C_{p})}_{s_{3}}] (ρ C_{p})_{f}

(11)

d_{4} = k_{Tnf} = \frac{k_{s_{3}} + 2 k_{hnf} - 2 ϕ_{3} (k_{hnf} - k_{s_{3}})}{k_{s_{3}} + 2 k_{hnf} + ϕ_{3} (k_{hnf} - k_{s_{3}})} k_{hnf},

(12)

where k_{hnf} = \frac{k_{s_{2}} + 2 k_{nf} - 2 ϕ_{2} (k_{nf} - k_{s_{2}})}{k_{s_{2}} + 2 k_{nf} + ϕ_{2} (k_{nf} - k_{s_{2}})} k_{nf}^{'}

(13)

and k_{nf} = \frac{k_{s_{1}} + 2 k_{f} - 2 ϕ_{1} (k_{f} - k_{s_{1}})}{k_{s 1} + 2 k_{f} + ϕ_{1} (k_{f} - k_{s_{1}})} k_{f} .

(14)

Δ_{5} = μ_{T n f} = {{(1 - ϕ_{1})}^{2.5} {(1 - ϕ_{2})}^{2.5} {(1 - ϕ_{3})}^{2.5}}^{- 1} μ_{f}

(15)

In above equations, the terms $ϕ_{1}$ , $ϕ_{2}$ , and $ϕ_{3}$ represent the concentration of graphene oxide, cobalt, and zirconium oxide respectively.

Using equation (9), BCs take the form as:

\begin{matrix} η = 0 : f = S_{n}, f^{'} = 1, θ = 1, ϕ = 1, \\ η \to \infty : f^{'} \to 0, θ \to 0, ϕ \to 0. \end{matrix}}

Where

S_{n} = \frac{v_{0}}{\sqrt{c υ_{f}}}

represents the suction with

v_{o} > 1

Physical quantities and thermal properties of nanoparticles

The relations for the physical quantities that have been frequently used in the engineering applications are deliberated as:

{Re}_{x}^{\frac{1}{2}} C_{f x} = \frac{f^{″} (0)}{{(1 - ϕ_{1})}^{2.5} {(1 - ϕ_{2})}^{2.5} {(1 - ϕ_{3})}^{2.5}}, N u_{x} {Re}_{x}^{\frac{1}{2}} = - \frac{k_{T n f}}{k_{f}} θ^{'} (0), S h_{x} {Re}_{x}^{\frac{1}{2}} = - ϕ^{'} (0)}

Where

{Re}_{x} = \frac{U_{w} x}{υ_{f}}

can be used for the Reynolds number. The characteristics of the proposed nanoparticles together with the base fluid are specified in Table 1. The fundamental characteristics of the tri-hybrid nanofluids are provided in Table 2.

Table 1.

Thermal properties of paraffin cobalt, gold, and zirconium oxide.

Properties	Water (H₂O)( $f$ )	Graphene oxide( $S_{1}$ )	Cobalt( $S_{2}$ )	Zirconium oxide( $S_{3}$ )
$k (W / mK)$	0.6071	5000	100	1.7
$ρ (kg / m^{3})$	997	1800	8900	5680
$C_{p} (J / kgK)$	4180	717	420	502
$σ (s / m)$	5.5 $\times$ 10⁻⁶	1.1 $\times$ 10⁻⁵	1.602 $\times 10^{7}$	2.4 × 10⁻⁶

Table 2.

Thermal characteristics of ternary hybrid nanofluids.^25,26

Electrical conductivity	$\frac{σ_{Tnf}}{σ_{hnf}} = \frac{(1 + 2 ϕ_{3}) σ_{s_{3}} + (1 - 2 ϕ_{3}) σ_{hnf}}{(1 - ϕ_{3}) σ_{s_{3}} + (1 + ϕ_{3}) σ_{hnf}},$ $\frac{σ_{hnf}}{σ_{nf}} = \frac{(1 + 2 ϕ_{2}) σ_{s_{2}} + (1 - 2 ϕ_{2}) σ_{nf}}{(1 - ϕ_{2}) σ_{s_{2}} + (1 + ϕ_{2}) σ_{nf}},$ $\frac{σ_{nf}}{σ_{f}} = \frac{(1 + 2 ϕ_{1}) σ_{s_{1}} + (1 - 2 ϕ_{1}) σ_{f}}{(1 - ϕ_{1}) σ_{s_{1}} + (1 + ϕ_{1}) σ_{f}} .$
Viscosity	$μ_{T n f} = \frac{μ_{f}}{{(1 - ϕ_{1})}^{2.5} {(1 - ϕ_{2})}^{2.5} {(1 - ϕ_{3})}^{2.5}},$
Density	$ρ_{T h f} = (1 - ϕ_{3}) {(1 - ϕ_{2}) [(1 - ϕ_{1}) ρ_{f} + ϕ_{1} ρ_{s_{1}}] + ϕ_{2} ρ_{s_{2}}} + ϕ_{3} ρ_{s_{3}},$
Thermal conductivity	$\frac{k_{Tnf}}{k_{hnf}} = {\frac{k_{s_{3}} + 2 k_{hnf} - 2 ϕ_{3} (k_{hnf} - k_{s_{3}})}{k_{s_{3}} + 2 k_{hnf} + ϕ_{3} (k_{hnf} - k_{s_{3}})}}^{'}$ $\frac{k_{hnf}}{k_{nf}} = {\frac{k_{s_{2}} + 2 k_{nf} - 2 ϕ_{2} (k_{nf} - k_{s_{2}})}{k_{s_{2}} + 2 k_{nf} + ϕ_{2} (k_{nf} - k_{s_{2}})}}^{'}$ $\frac{k_{nf}}{k_{f}} = \frac{k_{s_{1}} + 2 k_{f} - 2 ϕ_{1} (k_{f} - k_{s_{1}})}{k_{s 1} + 2 k_{f} + ϕ_{1} (k_{f} - k_{s_{1}})} .$

Numerical solution

Equations (6) and (8) not only comprise of nonlinear terms but also they are coupled. In this situation, the analytical solution may not be possible. If so, it will be so much time consuming. However, one of the choices to solve such type of equations is the numerical approach. We determine the approximate solutions of the problem by incorporating finite-difference discretization. A complete methodology procedure can be further studied in the articles by Ahmad et al.^27–29 The system (6)–(8) is discretized using following relations:

\begin{matrix} W^{'} (i) = \frac{W_{i + 1} - W_{i - 1}}{2 * h}, Θ^{'} (i) = \frac{Θ_{i + 1} - Θ_{i - 1}}{2 * h}, Φ^{'} (i) = \frac{Φ_{i + 1} - Φ_{i - 1}}{2 * h}, \\ W^{″} (i) = \frac{W_{i + 1} - 2 W_{i} + W_{i - 1}}{h^{2}}, Θ^{″} (i) = \frac{Θ_{i + 1} - 2 Θ_{i} + Θ_{i - 1}}{h^{2}}, Φ^{″} (i) = \frac{Φ_{i + 1} - 2 Φ_{i} + Φ_{i - 1}}{h^{2}} . \end{matrix}}

Figure 1(b) describes the sensitivity analysis. The optimal mesh size that minimizes error while retaining computational efficiency can be found with the help of mesh sensitivity analysis. Higher discretization error results from a coarser level of discretization as the mesh size increases.

The comparison in Table 3 leads towards the precision of algorithmic procedure and accuracy of the code as well.

Table 3.

A comparison with the Prandtl number ( $ϕ_{1} = ϕ_{2} = ϕ_{3} = 0$ ).

$Pr$	Ali et al.³⁰	Khan and Pop³¹	Present results
2	0.91045	0.9113	0.91045
7	1.89083	1.8954	1.89083
20	3.35271	3.3539	3.35271

Results and discussion

This section comprises of the consequences of the physical parameters on the ternary Go-Co-ZrO₂/H₂O, hybrid Go-Co/H₂O, and usual nanofluids Go/H₂O. The volume concentrations of grapheme oxide, cobalt, and zirconium oxide are respectively represented by $ϕ_{1}, ϕ_{2}$ , and $ϕ_{3}$ . In the analysis of three types of nanofluids, the combination of the nanoparticles is taken in such a way that $ϕ_{1} = 0.1$ and $ϕ_{2} = ϕ_{3} = 0$ for the pure nanofluid whereas $ϕ_{1} = ϕ_{2} = 0.1$ and $ϕ_{3} = 0$ for the hybrid nanofluids. In the same way, the nanoparticles volume concentration of three particles is same in the tri-hybrid case of nanofluids, for example, $ϕ_{1} = ϕ_{2} = ϕ_{3} = 0.1$ . The particular values that have been used in approximations are: $S c = 1.5, m_{1} = 2, ϕ_{1} = 0.1, Pr = 6.2$ unless otherwise specified in tables. The velocity as well as temperature profile seems to be deteriorating in the flow regime under the influence of the suction parameter $S_{n}$ (see Figures 2 and 3). The phenomenon for which the suction occurs is the generation of pressure difference. In fact, the fluid is pushed by the high pressure towards the low pressure area that resists the fluid's speed and reduces the temperature as well. On the other side, the optimal solution can be attained by taking the small values of $S_{n}$ .

Figure 2.

Velocity $F (η)$ with change in suction parameter $S_{n}$ .

Figure 3.

Temperature $Θ (η)$ with change in suction parameter $S_{n}$ .

Figures 4 and 5 provide the pictorial evidences for the velocity $F^{'} (η)$ and temperature $θ (η)$ of the ternary nanofluids. Velocity reduces, in either case of nanofluids, with an increase in the porosity of the medium. On the other side, temperature profile uplifts with the effect of the same parameter $ε_{p}$ . The reduction in the velocity was expected as the porous medium usually resists the flow. The small pores size will generate the more resistance. However, the small will be the size of pores, the higher will be the resistive force. It has been come into notice that the porosity parameter can’t attain so small values in the current analysis. When we assign smaller values to this parameter we face interruption in the solution of the problem. It may be due to the fact that the base fluid possesses three distinct nanoparticles which require some space or large pores size so that the particles may pass through these pores easily. The analysis of Table 4 portrays that both parameters, porosity and suction, have increasing effect on the Shear stresses in all cases of nanofluids. But the diminishing values of the heat transfer rate are mainly caused by the dominant effect of porosity, whereas an increase in the Nusselt number is due to the suction phenomenon.

Figure 4.

Velocity $F^{'} (η)$ with change in porosity parameter $ε_{p}$ .

Figure 5.

Temperature $Θ (η)$ with change in porosity parameter $ε_{p}$ .

Table 4.

Shear stress and heat transfer rate with change in $ε_{p}$ and $S_{n}$ .

Parameters		Shear stress
$ε_{p}$	$S_{n}$	Pure nano	Hybrid nano	Tri-hybrid nano
05	0.6	−4.54343070	−5.85648666	−7.21953419
10		−5.38606984	−7.10446661	−8.90194056
15		−6.13443248	−8.14543539	−10.28595846
03	0.3	−3.95894106	−4.71142701	−5.81928864
	0.6	−4.19546514	−5.26435081	−6.40308955
	0.9	−4.44033851	−5.85645074	−7.02348669
Parameters		Heat transfer rate
$ε_{p}$	$S_{n}$	Pure nano	Hybrid nano	Tri-hybrid nano
05	0.6	0.81353743	0.53014437	−0.28542250
10		0.48241943	−0.26198238	−1.36435679
15		0.08556033	−0.92458633	−2.24458966
03	0.3	−0.52247402	−0.08394656	−0.66601912
	0.6	0.82284819	0.90306698	0.24477628
	0.9	2.23782517	1.99898865	1.27624492

It is manifested from Figures 6 and 7 that nanoparticles volume concentration of cobalt and zirconium oxide increase the temperature in the flow regime in either case of nanofluids. Temperature is more prominent in case of ternary hybrid nanofluids (Go-Co-ZrO₂/H₂O) as compared to hybrid (Go-Co/H₂O) or mono (Go/H₂O) nanofluids. The temperature varies with the effect of nanoparticles depending on various factors like concentration and size of nanoparticles, type of nanoparticles, and the surrounding environment. The combination of three distinct nanoparticles can further enhance the heat transfer and modify the phase transition such as melting/boiling points. It can also generate thermoelectric, photo-thermal, and catalytic effects. The nano-composition of three different particles like graphene oxide, cobalt, and zirconium oxide (when dispersed in the base liquid) can provide improved thermal characteristics, efficient heat dissipation, and it facilitates to reduce the temperature gradients. It can be observed from Table 5 that heat transferal rate is got demoted by the Eckert number and the volume fraction of zirconium oxide but an opposite trend is noticed in case of volume fraction of cobalt which increases the heat transfer rate marginally.

Figure 6.

Temperature $Θ (η)$ with change in volume concentration of cobalt $ϕ_{2}$ .

Figure 7.

Temperature $Θ (η)$ with change in volume concentration of zirconium oxide $ϕ_{3}$ .

Table 5.

Heat transfer rate with change in $E_{c k}$ , $ϕ_{2}$ , and $ϕ_{3}$ .

Heat transfer rate
$E_{c k}$	$ϕ_{2}$	$ϕ_{3}$	Pure nano	Hybrid nano	Tri-hybrid nano
0.10	0.1	0.1	2.52481944	2.59596693	2.27259723
0.20			0.82284819	0.90306698	0.24477628
0.30			−0.87912306	−0.78983296	−1.78304466
0.2	0.3	0.1	−0.75612214	−1.45798060	−2.30200963
	0.4		−2.39820994	−3.23562251	−4.35742792
	0.5		−4.74149471	−5.91598162	−7.63615301
0.2	0.1	0.1	1.41860023	0.97709354	0.24477628
		0.3	−0.04348163	−0.91689201	−2.05348105
		0.5	−4.00306632	−5.76979531	−8.18198813

The results of Figure 8 evidently disclose that the Eckert number causes an enhancement in the temperature of the fluid. The kinetic as well as thermal energies can be characterized, in fluid dynamics, by the Eckert number $E_{c k}$ . This dimensionless parameter interlinks the fluid motion (kinetic energy) with the change in temperature (thermal energy) in a flow. The magnitude of the Eckert number further clarifies its characteristics. For instance, if its magnitude is less than one then kinetic energy will dominate and the flow will possess low energy. For the values of $E_{c k} >> 1$ , thermal energy will dominate and the flow will be considered to have high energy. The Eckert number together with the Nusselt number interprets the heat transfer phenomenon.

Figure 8.

Temperature $Θ (η)$ with change in Eckert number $E_{c k}$ .

The mixture of water based tri-hybrid nanoparticles Go-Co-ZrO₂ is assumed to have chemically reactive species. The chemical reaction phenomenon will definitely occur because of the species involvement in the flow. However, the homogeneous mixture will contain the concentration C with unit $mol / m^{3}$ . The profile $Φ (η)$ decelerates with change in the chemical reaction parameter $C h_{R}$ (see Figure 9). A certain minimum amount of energy for which the chemical reaction takes place is known as activation energy. According to the collision theory, a sufficient energy is required to reactant molecules for the collision, and it will initiate the activation energy. The activation energy parameter tends to accelerate the concentration in all three cases of nanofluids, as depicted in Figure 10. It is evident from Table 6 that the chemical reaction elevates the mass transfer but contrarily, activation energy devaluates the mass transfer rate in either case of pure, hybrid, and tri-hybrid nanofluids.

Figure 9.

Concentration $Φ (η)$ with change in chemical reaction parameter $C h_{R}$ .

Figure 10.

Concentration $Φ (η)$ with change in activation energy parameter $A_{E}$ .

Table 6.

Mass transfer rate with change in $C h_{R}$ and $A_{E}$ .

Parameters		Mass transfer rate
$C h_{R}$	$A_{E}$	Pure nano	Hybrid nano	Tri-hybrid nano
0.3	1.0	3.55808526	3.64547092	3.79529250
0.5		4.62960636	4.71680096	4.88324047
0.7		5.51593868	5.60042792	5.77249246
1.5	1	8.15037430	8.22509687	8.40083891
	3	6.76431407	6.86208451	7.07250998
	5	5.54382520	5.66442083	5.90690244

Conclusions

A suitable combination of tri-hybrid nanoparticles might provide the desired results in the factual applications. The current work aims to interpret the characteristics of three distinct types of nanoparticles such as graphene oxide, cobalt, and zirconium oxide. The analysis also covers the features of the problem under the prominent effects of activation energy and viscous dissipation. An algorithmic approach is taken into account to find the numerical solutions of the problem. Some major culminations of the study are as follows:

The suction phenomenon tends to deteriorate the velocity and temperature in the flow regime.

Velocity reduces but temperature enhances, in either case of nanofluids, with an increase in the porosity of the medium.

Both parameters, porosity and suction, have increasing effect on the shear stresses in all cases of nanofluids (pure, hybrid, and ternary hybrid case).

Nanoparticles volume concentration of cobalt (Co) and zirconium oxide (ZrO₂) increase the temperature.

The activation energy parameter tends to accelerate the concentration in nanofluids like Go/H₂O, Go-Co/H₂O and Go-Co- ZrO₂/H₂O.

The temperature to be increase with the Eckert number.

Footnotes

Acknowledgements

Authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/273/45.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Wasim Jamshed

References

Zhang

, et al. MHD Stagnation point flow of nanofluid over a curved stretching/shrinking surface subject to the influence of Joule heating and convective condition. Case Stud Therm Eng 2021; 26: 101184.

Ali

Jamshed

Ahmad

, et al. A self-similar approach to study nanofluid flow driven by a stretching curved sheet. Symmetry 2022; 14: 1991.

Ali

Faridi

Ahmad

, et al. Quasi-linearization analysis for heat and mass transfer of magnetically driven 3rd-grade (Cu-TiO₂/engine oil) nanofluid via a convectively heated surface. Int Commun Heat Mass Trans 2022; 135: 106060.

Gupta

Sharma

. A brief review of nanofluids utilization in heat transfer devices for energy saving. Mater Today Proc 2023. DOI: https://doi.org/10.1016/j.matpr.2023.03.364

Sun

Yang

Zhao

, et al. A review of multifunctional applications of nanofluids in solar energy. Powder Tech 2022; 411: 117932.

Panduro

CEA

Finotti

. A review of the use of nanofluids as heat-transfer fluids in parabolic-trough collectors. Appl Therm Eng 2022; 211: 118346.

Maatoug

Abbasi

Farooq

, et al. Bioconvective Homann flow of tangent hyperbolic nanofluid due to spiraling disk with convective and zero mas flux constraints. J Indian Chem Soc 2023; 100: 100819.

Shao

Nayak

Dogonchi

. Ternary hybrid nanofluid natural convection within a porous prismatic enclosure with two movable hot baffles: an approach to effective cooling. Case Stud Therm Eng 2022; 40: 102507.

Adun

Kavaz

Dagbasi

. Review of ternary hybrid nanofluid: synthesis, stability, thermophysical properties, heat transfer applications, and environmental effects. J Cleaner Prod 2021; 328: 129525.

10.

Algehyne

Alrihieli

Bilal

. Numerical approach toward ternary hybrid nanofluid flow using variable diffusion and non-Fourier’s concept. ACS Omega 2022; 7: 29380–29390.

11.

Elnaqeeb

Animasaun , et al. Ternary-hybrid nanofluids: significance of suction and dual-stretching on three-dimensional flow of water conveying nanoparticles with various shapes and densities. Zeitschrift für Naturforschung A 2021; 76: 231–243.

12.

Puneeth

Saeed

, et al. Analysis of the Thomson and Troian velocity slip for the flow of ternary nanofluid past a stretching sheet. Sci Rep 2023; 13: 2340.

13.

Salari

Khavarpour

Masoumi

, et al. Preparation of cobalt oxide and tin dioxide nanofluids and investigation of their thermophysical properties. Microfluid Nanofluid 2022; 26: 79.

14.

Sekhar

TVR

Nandan

Prakash

. Marisamy muthuraman, investigations on viscosity and thermal conductivity of cobalt oxide- water nano fluid. Mater Today Proc 2018; 5: 6176–6182.

15.

Farooq

Waqas

Fatima

, et al. Computational framework of cobalt ferrite and silver-based hybrid nanofluid over a rotating disk and cone: a comparative study. Sci Rep 2023; 13: 5369.

16.

Santhosh

Sivaraj

. Comparative heat transfer performance of hydromagnetic mixed convective flow of cobalt-water and cobalt-kerosene ferro-nanofluids in a porous rectangular cavity with shape effects. Eur Phys J Plus 2023; 138: 40.

17.

Murtaza

Kumam

Bilal

, et al. Parametric simulation of hybrid nanofluid flow consisting of cobalt ferrite nanoparticles with second-order slip and variable viscosity over an extending surface. Nanotechnol Rev 2023; 12: 20220533.

18.

Ahmad

Ali

Ashraf

, et al. Analysis of pure nanofluid (GO/engine oil) and hybrid nanofluid (GO–Fe3O4/engine oil): novel thermal and magnetic features. Nanotechnol Rev 2022; 11: 2903–2915.

19.

Alktranee

Shehab

Németh

, et al. Effect of zirconium oxide nanofluid on the behaviour of photovoltaic–thermal system: an experimental study. Energy Rep 2023; 9: 1265–1277.

20.

Souayeh

. Simultaneous features of cc heat flux on dusty ternary nanofluid (Graphene + Tungsten Oxide + Zirconium Oxide) through a magnetic field with slippery condition. Mathematics 2023; 11: 54.

21.

Mahmood

Khan

Saleem

, et al. Numerical analysis of ternary hybrid nanofluid flow over a stagnation region of stretching/shrinking curved surface with suction and Lorentz force. J Magn Magn Mater 2023; 573: 170654.

22.

Ali

Ahmad

Nisar

, et al. Simulation analysisof MHD hybrid Cu-Al₂O₃/H₂O nanofluid flowwith heat generation through a porous media. Int J Energy Res 2021; 45: 19165–19179.

23.

Ahmad

Ali

Faridi

, et al. Novel thermal aspects of hybrid nanoparticles Cu-TiO₂ in the flow of ethylene glycol. Int Commun Heat Mass Transf 2021; 129: 105708.

24.

Ahmad

Ali

Rizwan

, et al. Heat and mass transfer attributes of Copper-Aluminum oxide hybrid nanoparticles flow through a porous medium. Case Stud Therm Eng 2021; 25: 100932.

25.

Michael

I,K

Yanovsky

Anusha

, et al. MHD Flow and heat transfer of a ternary hybrid ferrofluid over a stretching/shrinking porous sheet with the effects of Brownian diffusion and thermophoresis. East Eur J Phy 2023; 1: 7–18.

26.

Yashkun

Zaimi

Abu Bakar

, et al. MHD hybrid nanofluid flow over a permeable stretching/shrinking sheet with thermal radiation effect, 2020. Int J Numer Method H https://doi.org/10.1108/HFF-02-2020-0083

27.

Ahmad

Ali

Ashraf

. MHD Flow of Cu-Al₂O₃/water hybrid nanofluid through a porous media. J Porous MEdia 2021; 24: 61–73.

28.

Ahmad

Ashraf

Ali

. Simulation of thermal radiation in a micropolar fluid flow through a porous medium between channel walls. J Therm Anal Calorim 2021; 144: 941–953.

29.

Ahmad

Ashraf

Ali

. Numerical simulation of viscous dissipation in a micropolar fluid flow through a porous medium. J Appl Mech Tech Phys 2019; 60: 996–1004.

30.

Ali

Ahmad

, et al. Insights into the thermal attributes of sodium alginate (NaC6H7O6) based nanofluids in a three-dimensional rotating frame: a comparative case study. Case Stud Therm Eng 2023; 49: 103211.

31.

Khan

Pop

. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf 2010; 53: 2477–2483.