*Article* **Activation Energy Performance through Magnetized Hybrid** *Fe***3***O***4–***PP* **Nanofluids Flow with Impact of the Cluster Interfacial Nanolayer**

**M. Zubair Akbar Qureshi 1,†, Qadeer Raza 1, Aroosa Ramzan 1, M. Faisal 1, Bagh Ali 2, Nehad Ali Shah 3,† and Wajaree Weera 4,\***


**Abstract:** The current work investigated the mass and heat transfer of the MHD hybrid nanofluid flow subject to the impact of activation energy and cluster interfacial nanolayer. The heat transport processes related to the interfacial nanolayer between nanoparticles and base fluids enhanced the base fluid's thermal conductivity. The tiny particles of *Fe*3*O*<sup>4</sup> and *PPy* were considered due to the extraordinary thermal conductivity which is of remarkable significance in nanotechnology, electronic devices, and modern shaped heat exchangers. Using the similarity approach, the governing higher-order nonlinear coupled partial differential equation was reduced to a system of ordinary differential equations (ODEs). *Fe*3*O*4*–PPy* hybrid nanoparticles have a considerable influence on thermal performance, and when compared to non-interfacial nanolayer thermal conductivity, the interfacial nanolayer thermal conductivity model produced substantial findings. The increase in nanolayer thickness from level 1 to level 5 had a significant influence on thermal performance improvement. Further, the heat and mass transfer rate was enhanced with higher input values of interfacial nanolayer thickness.

**Keywords:** hybrid nanofluid; heat and mass transfer flow; MHD; *Fe*3*O*4*–PPy* hybrid nanoparticles; interfacial nanolayer; activation energy

**MSC:** 00-01; 99-00

#### **1. Introduction**

The activation energy is concomitant with chemical reaction and has a noteworthy role in heat and mass transfer, free convective boundary layer flows in the fields of oil container engineering and geothermal reservoirs. The exploration of thermal transportation in fluid flows is an attractive topic for researchers due to its wide applications. Moreover, thermal stability and instability are in high demand in the current era. Buongiorno et al. [1] proposed the relationship in nanofluids by incorporating Brownian diffusion and thermophoresis. Gurel [2] investigated the melting heat transport of phase change materials subject to the melting boundary condition. The finite volume approach was used to obtain numerical solutions. Dhlamini et al. [3] studied the binary chemical reactions in the mixed convective flow of nanofluids with activation energy. The effects of bioconvection, changing thermal conductivity, and activation energy past an extended sheet were investigated by Chu et al. [4]. Water was regarded as a conventional fluid by Wakif et al. [5] and in this investigation the dynamics of radiative-reactive Walters-b fluid due to mixed convection conveying gyrotactic microorganisms, tiny particles experience haphazard motion, thermomigration, and Lorentz force. To further explore the precise point of hybrid nanofluid

**Citation:** Qureshi, M.Z.A.; Raza, Q.; Ramzan, A.; Faisal, M.; Ali, B.; Shah, N.A.; Weera, W. Activation Energy Performance through Magnetized Hybrid *Fe*3*O*4–*PP* Nanofluids Flow with Impact of the Cluster Interfacial Nanolayer. *Mathematics* **2022**, *10*, 3277. https://doi.org/10.3390/ math10183277

Academic Editor: Camelia Petrescu

Received: 15 August 2022 Accepted: 3 September 2022 Published: 9 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

flow, the impact of the magnetic field, heat radiation, and activation energy with a binary chemical reaction was introduced. Sreenivasulu et al. [6] investigated the influence of activation energy on the hybrid nanofluid flow via a flat plate with viscous dissipation and a magnetic field. Ramesh et al. [7] investigated the influence of thermal performance on the usual heat source/sink effect. Chemical reaction and activation energy effects are accounted for in the mass equation. Wasif et al. [8] prepared to extend cavity construction for a broader variety of purposes, beginning with the determination of cavity shape owing to the increase in heat transmission inside that selected cavity for diverse boundary conditions. Kumar et al. [9] investigated the flow of a tangent hyperbolic fluid through a transferring stretched surface. Nonlinear radiation was used to offer warm shipping properties. Activation energy indicated different elements of mass transfer. As a result of an unstable flow over a stretched surface in an incompressible rotating viscous fluid with the appearance of a binary chemical reaction and Arrhenius activation energy, Awad et al. [10] theorized that the spectral relaxation method (SRM) could be employed to evaluate the linked highly nonlinear problem of partial differential equations. Rekha et al. [11] explored the influence of a heat source/sink on nanofluid flow through a cone, wedge, and plate while utilizing a dispersion of aluminum alloys (AA7072 and AA7075) as a base nanoparticles fluid water. The activation energy and porous material are also taken into account in the simulation.

*Fe*3*O*4*–PPy* core–shell nanoparticles were created using polymer polypyrrole (PPy), which could be used for cancer combination therapy that is image-guided and controlled remotely after being functionalized with polyethylene glycol. The *Fe*3*O*<sup>4</sup> core, which breaks down slowly in physiological conditions, is used in this system as a magnetically controlled device for cure administration as well as a magnet. Magnetic iron oxide nanoclusters were coated for a lightabsorbing near-infrared resonance imaging comparison to create a multifunctional nanocomposite. Due to their remarkable function in separation technology, *Fe*3*O*<sup>4</sup> NPs have been garnering a lot of attention. In addition to having exceptional catalytic and good magnetic characteristics, the nanocomposite developed using the straightforward and easily produced *Fe*3*O*<sup>4</sup> NPs also exhibits good dispensability and biocompatibility [12,13]. Suri et al. [14] studied iron oxide–PPy nanocomposites as gas and moisture sensors. Sun et al. [15] reported nanoparticles of Fe3O4–PANI with a very thin PANI covering of the core–shell with badly improved microwave absorption properties. Zhao et al. [16] described a method for making *Fe*3*O*4*/PPy* nanocomposites.

Combined mass and heat transfer flows associated with chemical reactions play a crucial function in a wide range of applications, such as transport phenomena, cooling, and heating processes in electronics, binary diffusion systems, absorption reactors, polymer processing, solar energy systems, and the plastics industry. Salmi et al. [17] presented the similarity analysis through the finite element method (FEM) to investigate the non-Fourier behavior of the heat and mass transfer. Roy et al. [18] introduced a binary chemical reaction with an activation energy effect on the heat and mass transfer of a hybrid nanofluid flow over a permeable stretching surface. Oke et al. [19] presented the combined heat and mass transfer effects in the presence of magnetohydrodynamic ternary ethylene glycol-based hybrid nanofluids over a rotating three-dimensional surface with the impact of suction velocity. The unstable MHD convective flow with the heat and mass transfer characteristics for a noncompressible gelatinous electrical system was studied by Babu et al. [20]. The effects of the magnetic field on fluid flow and heat transfer rate have been documented in numerous earlier investigations [21–25]. Sreedevi et al. [26] examined heat and mass transport through a nanofluid. Salmi et al. [27] presented a unique analysis of the combined effects of Hall and ion slip currents, the Darcy–Forchheimer porous medium, and nonuniform magnetic field under the suspension of hybrid nanoparticles with heat and mass transfer aspects. Santhi et al. [28] discussed the unsteady magnetohydrodynamics heat and mass transfer analysis of a hybrid nanofluid flow over a stretching surface with chemical reaction, suction, and slip effects. Raja et al. [29] presented a novel idea of a radiative heat and mass flux 3D hybrid nanofluid, RHF. A Bayesian regularization technique based on backpropagated neural networks (BRT-BNNs) was employed to estimate the solution of the proposed model. Shah et al. [30] investigated the Numerical simulation

of a thermally enhanced EMHD flow of a heterogeneous micropolar mixture comprising (60%)-ethylene glycol (EG), (40%)-water (W), and copper oxide nanomaterials (CuO). Bidyasagar et al. [31] analyzed the unsteady laminar flow with heat and mass transfer of an incompressible and hydromagnetic Cu–Al2O3/H2O hybrid nanofluid near a nonlinearly permeable stretching sheet in the presence of nonlinear thermal radiation, viscous–ohmic dissipation, and velocity slip. Further, the impacts of heat generation and absorption, chemical reactions, convective heat, and mass conditions at the boundary were also considered by Farooq et al. [32]. Oke [33] investigated the flow of gold water nanofluids across the revolving upper horizontal surface of a paraboloid of revolution. The development of nanofluids has been groundbreaking in terms of improving fluid thermal and electrical conductivity. A real convective magnetohydrodynamic flow of a Cu–engine oil nanofluid along a vertical plate that has been convectively heated was taken into consideration by Kigio et al. [34].

Cluster interfacial nanolayer is a term that refers to a vast variety of extended quasitwo-dimensional nanoobjects that have unique physical and chemical properties, ranging from liposomes and cell membranes to graphene and layered double hydroxide flakes. Tso et al. [35] examined the effects of the interfacial nanolayer on the efficient thermal conductivity of nanofluids. The suggested model gives an equation to predict the nanolayer thickness for various kinds of nanofluids. On the other hand, in the mathematical model of Murshed et al. [36], a similar issue arises. They picked a ratio of nanolayer thermal conductivity to base fluid thermal conductivity of 1.1–2.5. Acharya et al. [37] examined the hydrothermal variations of radiative nanofluid flow by the influence of nanoparticle diameter and nanolayer. Zhao et al. [38] examined molecular dynamics simulation to investigate the effect of the interfacial nanolayer structure on enhancing the viscosity and thermal conductivity of nanofluids.

Having studied the abovementioned literature, to the best of our knowledge, no study has been conducted on the estimation of heat and mass transfer for hybrid nanofluids flowing across orthogonal porous discs subject to the effects of MHD, activation energy, and cluster interfacial nanolayers on the thermal conductivity model for hybrid nanofluid flows. Motivated by the abovementioned literature's wide scope of hybrid nanoparticles and cluster interfacial nanolayers, we considered studying the present elaborate problem. To enhance the nanoparticles stability, we also considered the activation energy. Using suitable similarity transformation quantities, the governing PDEs were transformed into dimensionless ODEs. To solve the system of ODEs, the Runge–Kutta shooting technique was used to draw numerical and graphical results.

#### **2. Formulation of Governing Equations**

The momentum, energy, and concentration equations of the incompressible flow for the two-dimensional velocity field *U* = [*u*(*x*, *y*, *t*), *v*(*x*, *y*, *t*)] of the single-phase model in the presence of chemical reactions, heat source, and activation energy were formulated. In this problem, we assumed a viscous, laminar, incompressible, time-dependent, twodimensional flow of a hybrid nanofluid containing *Fe*3*O*<sup>4</sup> − *PPy*/*H*2*O* through a permeable channel of breadth 2a(t). The induced magnetic field was ignored based on the presumption of a low Reynolds number. Further, the thermophysical properties of the base fluid, the single and hybrid nanofluids are expressed in Tables 1 and 2. Both walls of the channel were absorbent and could move above and below with a time-dependent rate (a (t)). Using these assumptions, the fluid flow-governing equations for conservation of mass, linear momentums, energy, and concentration in vector form are as follows:

$$(\nabla \mathcal{L}I = 0),$$

$$
\rho\_{\rm Inf} \frac{D \mathcal{U}}{D \mathfrak{t}} + \nabla\_{\mathcal{P}} - \nabla.\mathfrak{r} = A \times \mathbb{R} \tag{2}
$$

$$\left(\rho c\_p\right)\_{\ln f} \left(\frac{DT}{Dt}\right) + \nabla.q\_c = \beta k\_r^2 \left(\frac{T}{T\_2}\right)^n \times \exp\left(\frac{-E\_d}{k^\*T}\right)(c - c\_2) \tag{3}$$

$$\frac{D\mathbb{C}}{Dt} - D\nabla^2 \mathbb{C} = -k\_r^2 \left(\frac{T}{T\_2}\right)^n \times \exp\left(\frac{-E\_a}{k^\*T}\right) (c - c\_2) \tag{4}$$

where *<sup>D</sup> Dt* <sup>=</sup> *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* + *U*.∇, *ρ* denotes density, *t* represents time, *τ* shows the extra shear stress of the fluid, *p* is pressure, *T* stands for temperature, *C* represents concentration, *D* stands for the diffusion coefficient, *<sup>A</sup>* <sup>×</sup> *<sup>R</sup>* <sup>=</sup> *<sup>σ</sup>eB*<sup>2</sup> 0 *ρhnf U* signifies magnetic hydrodynamics, *Ea* denotes activation energy, *k*<sup>∗</sup> represents the Boltzmann constant, *k*<sup>2</sup> *<sup>r</sup>* is the chemical reaction constant, and *cp* is the specific heat capacity. The heat flux (*qc*) was defined by Fourier's law of conduction, *qc* = −*khnf* ∇*T*, and *khnf* is the thermal conductivity of a hybrid nanofluid. *T*<sup>1</sup> denotes the temperature of the lower channel wall and *T*<sup>2</sup> denotes the temperature of the upper channel wall shown in Figure 1. According to assumptions, the governing Equations (1)–(4) are written as [39]:

$$\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{5}$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{p\_x}{\rho\_{\ln f}} + \mathbf{v}\_{\ln f} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) - \frac{\sigma\_\varepsilon B\_0^2}{\rho\_{\ln f}} u \tag{6}$$

$$\frac{\partial v}{\partial t} + \mu \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{p\_y}{\rho\_{\text{lnf}}} + \mathbf{v}\_{\text{lnf}} \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) \tag{7}$$

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial y} = a\_{\text{lnf}} \left( \frac{\partial^2 T}{\partial y^2} \right) + \frac{1}{(\rho\_{\text{cp}})\_{\text{lnf}}} \left( \beta k\_r^2 \left( \frac{T}{T\_2} \right)^n \* \exp\left( \frac{-E\_a}{k^\* T} \right) (c - c\_2) \right) \tag{8}$$

$$\frac{\partial \mathbb{C}}{\partial t} + u \frac{\partial \mathbb{C}}{\partial x} + v \frac{\partial \mathbb{C}}{\partial y} = D \left( \frac{\partial^2 \mathbb{C}}{\partial y^2} \right) - k\_r^2 \left( \frac{T}{T\_2} \right)^n \ast \exp\left( \frac{-E\_a}{k^\* T} \right) (c - c\_2) \tag{9}$$

where *ρhnf* denotes the density of the hybrid nanofluid and *σ<sup>e</sup>* represents electrical conductivity. The physical model was used in the Cartesian coordinate system (x, y), and the u and v components of velocity were plotted on the *x*- and *y*-axes, respectively. *B*<sup>0</sup> is the magnetic field strength, *αhnf* is the coefficient of thermal diffusivity of the hybrid nanofluid, ν*hnf* is the kinematic viscosity of the hybrid nanofluid, *T*1, *C*<sup>1</sup> and *T*2, *C*<sup>2</sup> represent the temperature and concentration of the lower and upper plates with *T*<sup>1</sup> > *T*<sup>2</sup> and *C*<sup>1</sup> > *C*2. The mathematical expression for the kinematic viscosity of the hybrid nanofluid and the thermal diffusivity of the hybrid nanofluid are given below:

$$w\_{\ln f} = \frac{\mu\_{\ln nf}}{\rho\_{\ln f}} \; \; \alpha\_{\ln f} = \frac{k\_{\ln nf}}{\left(\rho c\_p\right)\_{\ln f}} \tag{10}$$

where *ρcp hnf* is the hybrid nanofluid's specific capacitance and *khnf* is the hybrid nanofluid's thermal conductivity. The heat variations inside the fluid flow are minimal, so function *T<sup>n</sup>* may be represented linearly. By excluding higher-order components, *T<sup>n</sup>* may be enlarged using Taylor's series concerning temperature *T*2, giving the following approximation:

$$T^n = (1 - n)T\_2^n + nT\_2^{n-1}T \left(\frac{T}{T\_2}\right)^n = (1 - n) + n\frac{T}{T\_2} \tag{11}$$

**Figure 1.** Physical model.

The boundary conditions of the present problem are as follows:

$$\begin{cases} y = -a(t), \ u = 0, \ v = -A\_1 a'(t), \ T = T\_1, & \mathbb{C} = \mathbb{C}\_1 \\ y = a(t), \ u = 0, \ v = A\_1 a'(t), \ T = T\_2, & \mathbb{C} = \mathbb{C}\_2. \end{cases} \tag{12}$$

The time t derivative is represented by the dash, and A is the wall permeability factor. The suitable similarity transformations are as follows:

$$\eta = \frac{y}{a}, \quad u = -\frac{\mathbf{x}\upsilon\_f}{a^2} F\_\eta(\eta, t), \text{ } v = \frac{\upsilon\_f}{a} F(\eta, t), \text{ } \theta = \frac{T - T\_2}{T\_1 - T\_2}, \text{ } \chi = \frac{\mathbb{C} - \mathbb{C}\_2}{\mathbb{C}\_1 - \mathbb{C}\_2} \tag{13}$$

In view of Equation (13), the continuity Equation (5) is satisfied, and Equations (6)–(9) can be written as follows:

$$\frac{\upsilon\_{\text{Info}}}{\upsilon\_f} F\_{\eta\eta\eta\eta} + \alpha \left( 3F\_{\eta\eta} + \eta F\_{\eta\eta\eta} \right) + F\_{\eta} F\_{\eta\eta} - \frac{\alpha^2}{\upsilon\_f} F\_{\eta\eta\dagger} - F F\_{\eta\eta\eta} - \frac{\rho\_f}{\rho\_{\text{Info}}} M F\_{\eta\eta} = 0,\tag{14}$$

$$\begin{split} \theta\_{\overline{\eta}\eta} + \frac{k\_f}{k\_{\inf}} \text{Pr}((1-\varrho\_1-\varrho\_2)+\varrho\_1 \frac{(\rho c\_{\overline{\rho}})\_{\overline{\eta}1}}{(\rho c\_{\overline{\rho}})\_{\inf}} + \varrho\_2 \frac{(\rho c\_{\overline{\rho}})\_{\overline{\eta}2}}{(\rho c\_{\overline{\rho}})\_{\inf}}) (\eta u - F) \theta\_{\overline{\eta}} - \frac{\mu^2}{a\_{\inf}} \theta\_l + \frac{k\_f}{k\_{\inf}} ((1-(\varrho\_1+\varrho\_2)) \\ + (\varrho\_1)(\frac{\rho c\_{\overline{\rho}}}{\rho c\_{\overline{\rho}}}) + (\varrho\_2)(\frac{\rho c\_{\overline{\rho}}}{\rho c\_{\overline{\rho}}}) (1+(n\*\gamma)\theta[\eta]) (1-E+(E\*\gamma)\theta[\eta]) \chi[\eta] = 0 \end{split} \tag{15}$$

$$\chi\_{\eta\eta} + \mathcal{S}\varepsilon (\eta\alpha - F)\chi\_{\eta} - \frac{a^2}{D}\chi\_t + (\mathcal{S}\varepsilon \ast \sigma)(1 + (n \ast \gamma)\theta[\eta])(1 - E + (E \ast \gamma)\theta[\eta])\chi[\eta] = 0 \tag{16}$$

Boundary conditions Equation (12):

$$\begin{array}{ccccc} F = -\text{Re} f\_{\prime\prime}, & F\_{\eta} = 0, & \theta = 1, \text{ and } \begin{array}{c} \chi = 1, \text{ and } \chi = 1, \text{ at } \eta = -1, \\ F = \text{Re} f, & F\_{\eta} = 0, \quad \theta = 0, \text{ and } \begin{array}{c} \chi = 0, \text{ at } \eta = 1. \end{array} \end{array} \tag{17}$$

The Prandtl number is equal to *Pr* <sup>=</sup> (*μcp*)*<sup>f</sup> <sup>k</sup> <sup>f</sup>* , <sup>α</sup><sup>=</sup> *aa* (*t*) *<sup>υ</sup><sup>f</sup>* is the wall expansion ratio, Re = *Aaa* (*t*) *<sup>υ</sup><sup>f</sup>* is the permeability Reynolds number, *<sup>γ</sup>* <sup>=</sup> *<sup>T</sup>*1−*T*<sup>2</sup> *<sup>T</sup>*<sup>2</sup> is the temperature difference parameter, *<sup>σ</sup>* <sup>=</sup> *<sup>k</sup>*<sup>2</sup> *<sup>r</sup>* (1−*γ*) *<sup>a</sup>* is the dimensionless reaction rate, *<sup>M</sup>* <sup>=</sup> *<sup>σ</sup>eB*<sup>2</sup> 0 *a*2 *<sup>μ</sup><sup>f</sup>* is the magnetic parameter, *E* = *Ea <sup>k</sup>*∗*<sup>T</sup>* is the dimensional activation energy parameter, *Sc* <sup>=</sup> *<sup>υ</sup><sup>f</sup> <sup>D</sup>* is the Schmidt number. Finally, we set *F* = *f Re* and considered the case following Majdalani et al. [40], where *α* is a constant, *f* = *f*(*η*), *θ* = *θ*(*η*), and *X*(*η*), which leads to *θ<sup>t</sup>* = 0, *Xt* = 0, and *fηη*<sup>t</sup> = 0; thus, we obtained the following equation:

$$\left(G\_1 f\_{\eta\eta\eta\eta} + f\_{\eta\eta\eta} \left(\mathrm{a}\eta - \mathrm{R}ef\right) + f\_{\eta\eta} \left(\mathrm{3a} + \mathrm{R}ef\_{\eta}\right) - G\_2 M \mathrm{Re} f\_{\eta\eta} = 0\tag{18}$$

$$\left(\theta\_{\eta\eta} + G\_3 G\_4(Pr)((a\eta - Ref)\,\theta\_{\eta} + \frac{k\_f}{k\_{\rm inf}}(Pr\*\sigma\*\lambda)((1 + (\gamma\*n)\theta[\eta])(1 - E + (\gamma\*E)\theta[\eta]))\chi[\eta] = 0\tag{19}$$

$$(\chi\_{\eta\eta} + \mathrm{Sc}(\eta\mathfrak{a} - \mathrm{Ref})\chi\_{\eta} + (\mathrm{Sc} \ast \sigma)(1 + (\gamma \ast \mathfrak{n})\theta[\eta])(1 - E + (\gamma \ast E)\theta[\eta])\chi[\eta] = 0 \tag{20}$$

and

$$\begin{array}{ccccc} f = -1 & & f\_{\eta} = 0, & \quad \theta = 1, & \text{and } \chi = 1, \text{at } \eta = -1 \\ f = 1, & & f\_{\eta} = 0, & \quad \theta = 0, & \text{and } \chi = 0, \text{ at } \eta = 1 \end{array} \tag{21}$$

$$\begin{split} \text{where } G\_{1} &= \left( \frac{1}{\left(1 - (q\_{1} + q\_{2})\right)^{2.5} \left( (1 - (q\_{1} + q\_{2})) + (q\_{1}) \left( \frac{\rho\_{1}}{\rho\_{bf}} \right) + (q\_{2}) \left( \frac{\rho\_{2}}{\rho\_{bf}} \right) \right)} \right), G\_{2} = \\ \left( \frac{1}{\left( (1 - q\_{1} - q\_{2}) + q\_{1} \left( \frac{\rho\_{1}}{\rho\_{bf}} \right) + q\_{2} \left( \frac{\rho\_{2}}{\rho\_{bf}} \right) \right)} \right), G\_{3} &= \left( \left( 1 - (q\_{1} + q\_{2}) \right) + (q\_{1}) \left( \frac{\rho\_{1} \mu\_{1}}{\rho\_{bf} \mu\_{f}} \right) + (q\_{2}) \left( \frac{\rho\_{1} \mu\_{2}}{\rho\_{bf} \mu\_{f}} \right) \right), G\_{4} = \left( \frac{k\_{\rm{int}} \cdot k\_{\rm{int}} \cdot \rho}{k\_{\rm{int}} \cdot k\_{f}} \right)^{-1}. \end{split}$$

**Table 1.** Thermophysical features of a hybrid nanofluid and NPs [41].


**Table 2.** Thermophysical properties of a hybrid nanofluid proposed in [42,43].


Variables *ϕs*<sup>1</sup> and *ϕs*<sup>2</sup> show volume fraction from the first and second NPs, *ρ<sup>f</sup>* is the base fluid density, *ρs*<sup>1</sup> and *ρs*<sup>2</sup> are the density of the first and second solid NPs, *ρCp s*1 and *ρCp <sup>s</sup>*<sup>2</sup> is the thermal capacitance of the first and second solid NPs, the thermal capacitance for the base fluid is represented as *ρCp f* , *khnfl* is the effective nanolayer thermal conductivity of the hybrid nanofluid, *k <sup>f</sup>* and *kb f* represent thermal conductivity of the base fluid, *λ*<sup>1</sup> = 1 + *<sup>h</sup> <sup>r</sup>* , *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>h</sup>* <sup>2</sup>*r*, h is the nanolayer thickness, r is the radius of the particle, *ks*<sup>1</sup> and *ks*<sup>2</sup> are the thermal conductivities of solid nanoparticles, and *knlr* is the thermal conductivity nanolayer.
