# Heat and Mass Transfer Enhancement of MHD Hybrid Nanofluid Flow in the Presence of Activation Energy

Authors: M. Shanmugapriya; R. Sundareswaran; P. Senthil Kumar
Journal: International Journal of Chemical Engineering
 (2021)

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## Abstract

In this study, water is apprehended as conventional fluid with the suspension of two types of hybrid nanoparticles, namely, single-walled CNTs (SWCNTs) and multiwalled CNTs (MWCNTs). The influence of a magnetic field, thermal radiation, and activation energy with binary chemical reaction has been added to better examine the fine point of hybrid nanofluid flow. The mathematical structure regarding the physical model for hybrid nanofluid is established and then the similarity variables are induced to transmute the leading PDEs into nonlinear ODEs. These equations were solved using the shooting technique together with RKF 4-5th order for various values of the governing parameters numerically. The results of prominent parameters were manifested through graphs and tables. The results indicate that the hybrid nanofluid SWCNT−MWCNT/water is fully adequate in cooling and heating compared to other hybrid nanofluids. In addition, the rise in the value of activation energy E upsurges the nanoparticle transfer rate of hybrid nanofluid.


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## Body

## 1. Introduction

Hybrid nanofluids are a new kind of working fluid that is made by suspension of two different types of nanoparticles with sizes (under 100 nm) into the conventional fluid (water, oils, ethylene glycol, biological fluids, etc.). This new form of nanofiluid is having higher thermal conductivity and thermo-physical properties than conventional fluids. In recent years, these hybrid nanofluids were used in various heat transfer applications such as heat pipe, solar energy, refrigeration as well as heating, heat exchanger, ventilation, air conditioning system, coolant in machining and manufacturing, biomedical, space, ships, defense, etc. A comprehensive review on hybrid nanofluids in heat transfer applications is given in Sarkar et al. [1]; Sidik et al. [2]; Sundar et al. [3]; and Sajid and Ali [4]. Khashi’ie et al. [5] presented analytical and numerical solutions of MHD flow of Cu−Al2O3/water hybrid nanofluid towards a shrinking cylinder. The effect of velocity and thermal slips and chemical and thermal radiation on TiO2/Al2O3-water-based hybrid nanoliquid over a stretching sheet in both steady and unsteady was examined by Santhi et al. [6]. Khan et al. [7] discussed the impact of two different nanoparticles Cu and Al2O3 with two different base fluids H2O and C2H6O2 along a stretched surface under the impact of nonlinear thermal radiation, inclined magnetic field, and mass suction. Their results showed that the Cu−Al2O3/H2O−C2H6O2 hybrid nanofluid is more effective in cooling and heating compared to Cu−Al2O3/H2O hybrid nanofluid, and a simple nanofluid Al2O3/H2O. Saeed et al. [8] established the analytical solution of Darcy-Forchheimer MHD hybrid nanofluid flow and heat transfer using HAM. They found that increasing values of Brownian motion increases the temperature profile of the hybrid nanofluid. Sheikholeslami et al. [9] investigated the entropy generation and free convection of MHD hybrid nanofluid flow within a porous tank. They observed that the growth of permeability reduces the exergy loss while Nuave augments with rise of Da.Carbon nanotubes (CNTs) are the cylindrical structures of carbons atoms (graphene). CNTs are classified into three groups depending on the number of graphene layers: single-walled CNTs (SWCNT) consisting of one graphene layer with a diameter of 0.5–1.5 nm, double-walled carbon nanotube (DWCNT), and multiwalled CNTs (MWCNTs) dwelling of many graphene layers interlinked nanotubes, with diameters of ranges between 10 and 100 nm. Due to the unique properties such as high mechanical strength, rigidity, hardness, adhesion, high dimensional ratio, chemical stability and high thermal and electrical conductivity, they provide many applications in nanotechnology, chemical and biochemical sensors, catalytic, conductive plastics, structural composite materials, etc. Prabhavathi et al. [10] analyzed the slip effects of SWCNTs and MWCNTs based Maxwell nanofluid flow and identified that the temperature of both nanofluids intensifies as the values of Maxwell parameter rises. Sreedevi and Sudarsana Reddy [11] deliberated the impact of radiation on MWCNT-kerosene-dependent nanofluid over a wedge. Recently, several authors (Tassaddiq et al. [12]; Esfe et al. [13]; Al-Hanaya et al. [14]; Upreti et al. [15, 16]) studied heat and mass transfer analysis of CNT-based hybrid nanofluid over different geometries.Activation energy is a term introduced in 1889 by Svante Arrhenius that is defined the minimum energy attained through the atoms or molecules required to start a chemical reaction. Activation energy along with binary chemical reaction existing in mass transfer has numerous applications in chemical and geothermal engineering, oil reservoirs, water and oil emulsions, food processing, etc. Initially, this concept was introduced by Bestman [17]. Some researchers reported the impact of activation energy in binary chemically reactive flow of hybrid nanofluids under numerous features (Ijaz Khan et al. [18]; Ahmad and Nadeem [19]; Zaib et al. [20]; Khan et al. [21]).The research regarding the boundary layer flow with heat and mass transfer, involving hybrid nanofluid, has revealed to be realistically significant in engineering processes. However, a cautious review of the current literature reveals that the flow over a moving wedge embedded in the hybrid nanofluid was not taken into account. Due to broad fascinating in energy sector, the present work investigate the impact of activation energy with binary chemical reaction on mass and heat transport characteristics of magnetohydrodynamic (MHD) flow of aSWCNT−MWCNT/water hybrid nanofluid induced by moving wedge. The resultant equations were solved using shooting technique along with RKF 4-5th order. Results for velocity, temperature, concentration profiles, skin friction coefficient, heat transfer rate, and nanoparticle transfer rate are graphically displayed and debated briefly.


## 2. Mathematical Formulation

### 2.1. Physical Description

The steady two-dimensional MHD boundary layer flow of aSWCNT−MWCNT/water hybrid nanofluid induced by moving wedge was investigated. It is assumed that the wedge is moving with the velocity u˜w∗x=U˜w∗x˜m and the free stream velocity u˜e∗x,t=U˜fs∗x˜m. Here, m=β/2−β is the Hartree pressure gradient in which β=2m/1+m is the wedge angle parameter which can be symbolized as Ω˜∗=πβ1 (see Figure 1). Temperature and concentration hybrid nanofluids at the wall are T˜w and C˜w, respectively, while T˜⟶T˜fs and C˜⟶C˜fs are ambient temperature and concentration.Figure 1
Physical interpretation of flow model.


### 2.2. Flow Analysis

The modeled equations based on the above assumptions are stated as follows [6]:(1)∂u˜∂x+∂v˜∂y=0,(2)u˜∂u˜∂x+v˜∂v˜∂y−u˜e∂u˜e∂x=μhnfρhnf∂2u˜∂y2−σB˜02ρhnfu˜e−u¯,(3)u˜∂T˜∂x+v˜∂T˜∂y−1ρCphnf∂qr∂y=khnfρCphnf∂2T˜∂y2,(4)T˜4≅4T˜∞3T˜−3T˜∞4,qr=−4σ∗3k∗∂T˜4∂y=−16σ∗T˜∞33k∗∂T˜∂y,u˜∂C˜∂x+v˜∂C˜∂y+kr2C˜−C˜fsT˜T˜fsn0e−Ea/κT˜fs=DB∂2C˜∂y2+DTT˜fs∂2T˜∂y2,with(5)u˜|y=0=u˜wx=U˜wxm,v˜|y=0=0,T˜|y=0=T˜w,C˜|y=0=C˜w,u˜|y=∞=u˜ex=U˜fsxm,T˜|y=∞=T˜fs,C˜|y=∞=C˜fs.In (4), kr2T˜/T˜fsn0e−Ea/κT˜fs exemplify the modifies Arrhenius equation in which kr is the rate of chemical reaction rate, n0 is fitted rate constant, and Ea is the modified Arrhenius function.


### 2.3. Transformation

In order to transform the modeled equations, the following transformations are established [22]:(6)ψ=2ϑfU˜fsm+10.5x0.5m+1fζ,ζ=m+1U˜fs2ϑf0.5x0.5m−1y,θζ=T˜−T˜fsT˜w−T˜fs,φζ=C˜−C˜fsC˜w−C˜fs,with the components of velocity u˜,v˜ and stream function ψ as(7)u˜=∂ψ∂y=U˜fsxmf′ζ,v˜=−∂ψ∂x=m+1ϑfU˜fs20.5x0.5m−1fζ+ζm−1m+1f′ζ.


### 2.4. Thermophysical Properties ofSWCNT−MWCNT/water-Based Hybrid Nanofluid

The dynamic viscosityμhnf, the effective dynamic density ρhnf, the specific heat or heat capacitanceρCphnf, and the thermal conductivity khnf of the hybrid nanofluid are specified as(8)μhnf=μf1−φ12.51−φ22.5,ρhnf=φ2ρSWCNT+1−φ21−φ1ρf+φ1ρMWCNT,ρCphnf=φ2ρCpSWCNT+1−φ21−φ1ρCpf+φ1ρCpMWCNT,khnf=knf∗kMWCNT+2knf−2φ2knf−kMWCNTkMWCNT+2knf+2φ2knf−kMWCNT,where knf=kf∗kSWCNT+2kf−2φ1kf−kSWCNT/kSWCNT+2kf+2φ1kf−kSWCNT.φ1 and φ2 are used for the solid volume fraction of MWCNT and SWCNT respectively, ρf,ρCpf and kf are the respective densities, specific heat, and thermal conductivity of the conventional fluid. The thermophysical properties of the present flow modeled are listed in Table 1.Table 1
Physical properties of base fluidsH2O and nanoparticles CNTs [19].
PropertyConventional fluidNanoparticlesH2OMWCNTSWCNTCpJkg−1k−14179796425kWm−1k−10.61330006600ρkgm−3997.116002600


### 2.5. Transformed Problems

Plugging the above-mentioned transformation and thermophysical properties,SWCNT−MWCNT/water-based hybrid nanofluid, the momentum, and thermal and concentration boundary layers became(9)f‴−A2A12mm+1f′2−1−f′f″+A1M2m+11−f′=0,(10)1+RA4θ″−PrA3A44mm+1f′θ−fθ′=0,(11)φ″+NtNbθ″−Le4mm+1f′φ−fφ′−Leσ1+δθn0exp−E1+δθφ=0,subject to(12)f|ζ=0=0,f′|ζ=0=γ,θ|ζ=0=1,φ|ζ=0=1,f′|ζ⟶∞=1,θ|ζ⟶∞=0,φ|ζ⟶∞=0,where(13)A1=1−φ1−2.51−φ2−2.5,A2=1−φ21−φ1+φ1ρSWCNTρf+φ2ρMWCNTρf,A3=1−φ21−φ1+φ1ρCpSWCNTρCpf+φ2ρCpMWCNTρCpf,A4=khnfknf.Equations (9)–(12) include the nondimensional magnetic parameter M, radiation parameter R, Prandtl number Pr, Brownian movement parameter Nt, thermophoresis parameter Nb, Lewis number Le, moving wedge parameter γ, temperature relative parameter δ, chemical reaction parameter σ, and activation energy parameter E which are defined as follows:(14)M=σB02ρfU∞,R=16σ∗T˜∞33k∗kf,Pr=ϑfαf,Nt=τDTT˜w−T˜fsT˜fsϑf,Nb=τDBC˜w−C˜fsϑf,Le=ϑfDB,γ=U˜wU˜fs,δ=T˜w−T˜fsT˜fs,σ=kr2c0x2m,E=EaκT˜fs.


### 2.6. Physical Quantities

The dimensionless forms of skin friction coefficientS˜fx=μhnf/ρfu˜e2∂u˜/∂y|ζ=0, heat transfer rate H˜tx=x˜−khnf∂T˜/∂y|ζ=0−16σ∗/3k∗T˜3∂T˜/∂y|ζ=0/kfT˜w−T˜fs, and nanoparticle transfer rate C˜tx=−x˜∂C˜/∂y|ζ=0/C˜w−C˜fs are defined as(15)S˜fxRex0.5=11−φ12.51−φ22.5m+12f″0,H˜txRex−0.5=−khnfknfm+121+Rθ′0,C˜txRex0.5=−m+12φ′0,where Rex=u˜exx˜/ϑf is the Reynolds number.


## 2.1. Physical Description

The steady two-dimensional MHD boundary layer flow of aSWCNT−MWCNT/water hybrid nanofluid induced by moving wedge was investigated. It is assumed that the wedge is moving with the velocity u˜w∗x=U˜w∗x˜m and the free stream velocity u˜e∗x,t=U˜fs∗x˜m. Here, m=β/2−β is the Hartree pressure gradient in which β=2m/1+m is the wedge angle parameter which can be symbolized as Ω˜∗=πβ1 (see Figure 1). Temperature and concentration hybrid nanofluids at the wall are T˜w and C˜w, respectively, while T˜⟶T˜fs and C˜⟶C˜fs are ambient temperature and concentration.Figure 1
Physical interpretation of flow model.


## 2.2. Flow Analysis

The modeled equations based on the above assumptions are stated as follows [6]:(1)∂u˜∂x+∂v˜∂y=0,(2)u˜∂u˜∂x+v˜∂v˜∂y−u˜e∂u˜e∂x=μhnfρhnf∂2u˜∂y2−σB˜02ρhnfu˜e−u¯,(3)u˜∂T˜∂x+v˜∂T˜∂y−1ρCphnf∂qr∂y=khnfρCphnf∂2T˜∂y2,(4)T˜4≅4T˜∞3T˜−3T˜∞4,qr=−4σ∗3k∗∂T˜4∂y=−16σ∗T˜∞33k∗∂T˜∂y,u˜∂C˜∂x+v˜∂C˜∂y+kr2C˜−C˜fsT˜T˜fsn0e−Ea/κT˜fs=DB∂2C˜∂y2+DTT˜fs∂2T˜∂y2,with(5)u˜|y=0=u˜wx=U˜wxm,v˜|y=0=0,T˜|y=0=T˜w,C˜|y=0=C˜w,u˜|y=∞=u˜ex=U˜fsxm,T˜|y=∞=T˜fs,C˜|y=∞=C˜fs.In (4), kr2T˜/T˜fsn0e−Ea/κT˜fs exemplify the modifies Arrhenius equation in which kr is the rate of chemical reaction rate, n0 is fitted rate constant, and Ea is the modified Arrhenius function.


## 2.3. Transformation

In order to transform the modeled equations, the following transformations are established [22]:(6)ψ=2ϑfU˜fsm+10.5x0.5m+1fζ,ζ=m+1U˜fs2ϑf0.5x0.5m−1y,θζ=T˜−T˜fsT˜w−T˜fs,φζ=C˜−C˜fsC˜w−C˜fs,with the components of velocity u˜,v˜ and stream function ψ as(7)u˜=∂ψ∂y=U˜fsxmf′ζ,v˜=−∂ψ∂x=m+1ϑfU˜fs20.5x0.5m−1fζ+ζm−1m+1f′ζ.


## 2.4. Thermophysical Properties ofSWCNT−MWCNT/water-Based Hybrid Nanofluid

The dynamic viscosityμhnf, the effective dynamic density ρhnf, the specific heat or heat capacitanceρCphnf, and the thermal conductivity khnf of the hybrid nanofluid are specified as(8)μhnf=μf1−φ12.51−φ22.5,ρhnf=φ2ρSWCNT+1−φ21−φ1ρf+φ1ρMWCNT,ρCphnf=φ2ρCpSWCNT+1−φ21−φ1ρCpf+φ1ρCpMWCNT,khnf=knf∗kMWCNT+2knf−2φ2knf−kMWCNTkMWCNT+2knf+2φ2knf−kMWCNT,where knf=kf∗kSWCNT+2kf−2φ1kf−kSWCNT/kSWCNT+2kf+2φ1kf−kSWCNT.φ1 and φ2 are used for the solid volume fraction of MWCNT and SWCNT respectively, ρf,ρCpf and kf are the respective densities, specific heat, and thermal conductivity of the conventional fluid. The thermophysical properties of the present flow modeled are listed in Table 1.Table 1
Physical properties of base fluidsH2O and nanoparticles CNTs [19].
PropertyConventional fluidNanoparticlesH2OMWCNTSWCNTCpJkg−1k−14179796425kWm−1k−10.61330006600ρkgm−3997.116002600


## 2.5. Transformed Problems

Plugging the above-mentioned transformation and thermophysical properties,SWCNT−MWCNT/water-based hybrid nanofluid, the momentum, and thermal and concentration boundary layers became(9)f‴−A2A12mm+1f′2−1−f′f″+A1M2m+11−f′=0,(10)1+RA4θ″−PrA3A44mm+1f′θ−fθ′=0,(11)φ″+NtNbθ″−Le4mm+1f′φ−fφ′−Leσ1+δθn0exp−E1+δθφ=0,subject to(12)f|ζ=0=0,f′|ζ=0=γ,θ|ζ=0=1,φ|ζ=0=1,f′|ζ⟶∞=1,θ|ζ⟶∞=0,φ|ζ⟶∞=0,where(13)A1=1−φ1−2.51−φ2−2.5,A2=1−φ21−φ1+φ1ρSWCNTρf+φ2ρMWCNTρf,A3=1−φ21−φ1+φ1ρCpSWCNTρCpf+φ2ρCpMWCNTρCpf,A4=khnfknf.Equations (9)–(12) include the nondimensional magnetic parameter M, radiation parameter R, Prandtl number Pr, Brownian movement parameter Nt, thermophoresis parameter Nb, Lewis number Le, moving wedge parameter γ, temperature relative parameter δ, chemical reaction parameter σ, and activation energy parameter E which are defined as follows:(14)M=σB02ρfU∞,R=16σ∗T˜∞33k∗kf,Pr=ϑfαf,Nt=τDTT˜w−T˜fsT˜fsϑf,Nb=τDBC˜w−C˜fsϑf,Le=ϑfDB,γ=U˜wU˜fs,δ=T˜w−T˜fsT˜fs,σ=kr2c0x2m,E=EaκT˜fs.


## 2.6. Physical Quantities

The dimensionless forms of skin friction coefficientS˜fx=μhnf/ρfu˜e2∂u˜/∂y|ζ=0, heat transfer rate H˜tx=x˜−khnf∂T˜/∂y|ζ=0−16σ∗/3k∗T˜3∂T˜/∂y|ζ=0/kfT˜w−T˜fs, and nanoparticle transfer rate C˜tx=−x˜∂C˜/∂y|ζ=0/C˜w−C˜fs are defined as(15)S˜fxRex0.5=11−φ12.51−φ22.5m+12f″0,H˜txRex−0.5=−khnfknfm+121+Rθ′0,C˜txRex0.5=−m+12φ′0,where Rex=u˜exx˜/ϑf is the Reynolds number.


## 3. Computational Scheme

A shooting technique together with RKF 4-5th order integration scheme is adopted to establish the computational results of nonlinear differential equations (9)–(11) with the boundary restrictions represented in equation (12). Following the basic methodology of shooting technique, we reduce this dimensionless boundary value flow system into first-order system by using the pursuing procedure:(16)f=u1,f′=u2,f″=u3,f‴=u3′=A2A12mm+1u22−1−u1u3−A1M2m+11−u2,θ=v1,θ′=v2,v2′=Pr1+R/A4A3A44mm+1u2v1−u1v2,φ=w1,φ′=w2,w2′=−NtNbv2′+Le4mm+1u2w1−u1w2+Leσ1+δv1n0exp−E1+δv1w1.with(17)u1|ζ=0=0,u2|ζ=0=γ,v1|ζ=0=1,w1|ζ=0=1,u3|ζ⟶∞=α1,v2|ζ⟶∞=α2,w2|ζ⟶∞=α3.


## 4. Outcomes and Discussions

We graphically analyze the behavior of various sundry parameters on fluid velocity, temperature, and concentration disseminationf′ζ,θζandφζ in Figures 2–21. The ranges of constraints in this research are considered as 0.1≤m≤0.4,0.1≤M≤3,0.5≤R≤2,0.1≤Pr≤2,0.2≤Nt≤0.8,0.5≤Nb≤2,1≤Le≤2.5,0.1≤σ≤0.7,0≤δ≤3,1≤E≤4,0.5≤γ≤5,0.01≤φ1,φ2≤0.1. Figures 2–7, given the impact of φ1 and φ2 on f′ζ,θζandφζ. The intensification of φ1 and φ2 reduces the thickness of the boundary layer flow, which arises in an increase of fluid velocity. We boost the quantity of both CNTs in H2O, the heat absorbing capacity of the fluid intensifies. As a result, the temperature profile is enhanced. Additionally, the higher values of φ1 and φ2 decay the mass transfer rate. Figures 8 and 9 demonstrate the effects of m (Hartree pressure gradient parameter) on f′ζandθζ. The augmented values of maugmented the surface drag force and f′ζ increases. A reverse configuration is grasped for growing values of minθζ.Figure 2
φ1 variation on f′ζ.Figure 3
φ2 variation on f′ζ.Figure 4
φ1 variation on θζ.Figure 5
φ2 variation on θζ.Figure 6
φ1 variation on φζ.Figure 7
φ2 variation on φζ.Figure 8
m variation on f′ζ.Figure 9
m variation on θζ.Figure 10
M variation on f′ζ.Figure 11
M variation on θζ.Figure 12
γ variation on f′ζ.Figure 13
γvariation on θζ.Figure 14
R variation on θζ.Figure 15
Pr variation on θζ.Figure 16
Pr variation on φζ.Figure 17
Nt variation on φζ.Figure 18
Nb variation on φζ.Figure 19
Le variation on φζ.Figure 20
σ variation on φζ.Figure 21
δ variation on φζ.The variation off′ζ on M (magnetic parameter) is shown in Figures 10 and 11. An increment in magnetic strength strengthens the velocity and weakens the temperature. This is due to the fact that with growing values of M, the Lorentz forces enhances, which raises the resistive force to the hybrid nanofluid motion. The response of f′ζandθζ to the variation of γ (moving wedge parameter) are illustrated in Figures 12 and 13. Rising of γ increases the velocity distribution and diminishes the temperature distribution. Furthermore, the thickness of the hydrodynamic boundary layer depreciation and thermal boundary layer are upgraded.Figure14 demonstrates the effect of RRadiation parameter on θζ. It explains that, due to the increment of R, depreciate k∗ (the mean absorption coefficient), consequently increases ∂qr/∂y (heat flux radiation) and the radiative heat transfer rate into the hybrid nanofluid that caused an increase in θζ. Figures 15 and 16 show the disparities of θζ and φζ on up surging values of Pr is the proportion of the molecular diffusivity over thermal diffusivity, increase in Pr in the hybrid nanofluid has reduce the thermal diffusivity. Therefore, the temperature of SWCNT−MWCNT/water-based hybrid nanofluid worsens as Pr intensifies, while the reverse phenomena occurs in φζ. The variation of φζ via Nt (thermophoresis parameter) and Nb (Brownian motion) is displayed in Figures 17 and 18. With increase in the value of Nt, thickening the concentration boundary layer, φζ increases. Nb portrays a declining performance of φζ. Figure 19 reveals that an elevation in Le (Lewis number) diminishes the mass diffusion and φζ; this is because that Le inversely related to the mass diffusion.Figure20 illustrates the φζ profiles decline due to the intensification of σ (dimensionless reaction rate) in the entire flow region. Physically, growing values of σ causes an increment in the term σ1+δθn0exp−E/1+δθ, it helps to calamitous kr (chemical reaction rate) that decreases φζ. Figure 21 shows the abating phenomena of hybrid nanoparticle concentration φζ due to the fluctuation of δ. The portrayal of E (activation energy) in distribution of φζ is shown in Figure 22. It is found that φζ attained a maximum level when E is assigned the maximum value. Physically, due to involvement of E afford some extra energy to system, which enhances the chemical reaction rate and hence φζ increases with higher values of E.Figure 22
E variation on φζ.The current results of the present problem are compared with the available literature and divulged in Table2. Tables 3–5 show the computational results of skin friction, heat transfer rate, and nanoparticle transfer rate against certain physical parameters. These tables shows that S˜fxRex0.5 enhances for larger values of m,M,φ1 and φ2, while it decays for γ. H˜txRex−0.5 upsurges for increasing values of R,φ1, and φ2, while decreases for m,M,Pr and γ also the larger values of Pr,Nt and E intensifies C˜txRex−0.5. Thus, hybrid nanofluid evinced an eminent performance in S˜fxRex0.5, H˜txRex−0.5, and C˜txRex−0.5 compared to other nanofluid.Table 2
Comparison ofS˜fxRex0.5 at selected values m when M=φ1=φ2=0.
mUllah et al. [23]Gopikrishna and Shanmugapriya [22]Present results00.46960.46960.46961/110.65500.65500.65501/50.80210.80210.80211/30.92770.92770.92771/21.03891.03891.038911.23261.23261.2326Table 3
Numerical values ofS˜fxRex0.5.
mMγφ1φ21/1−φ12.51−φ22.5m+1/2f″00.10.10.50.10.10.44500.20.48460.30.51600.40.54150.40.65161.00.94022.01.28553.01.55630.50.44501.5−0.53543.0−2.58895.0−6.16930.010.44550.030.44510.050.44490.100.44500.010.46100.030.45700.050.45320.100.4450Table 4
Numerical values ofH˜txRex−0.5.
mMRPrγφ1φ2−khnf/knfm+1/21+Rθ′00.10.11.06.20.50.10.1−0.86980.2−0.98650.3−1.07650.4−1.14860.4−0.89351.0−0.91832.0−0.94003.0−0.95290.5−0.91731.0−0.86981.5−0.82922.0−0.79450.1−0.17410.72−0.32651.0−0.37822.0−0.51880.5−0.86981.5−1.23113.0−1.64445.0−2.07800.01−1.01760.03−0.98420.05−0.95120.10−0.86980.01−1.02040.03−0.98650.05−0.95290.10−0.8698Table 5
Numerical values ofC˜txRex−0.5.
mMRPrNtNbLeσδEγφ1φ2−m+1/2φ′00.10.11.06.20.10.11.00.50.50.50.50.10.1−0.40390.2−0.45490.3−0.49440.4−0.52610.4−0.41421.0−0.42352.0−0.43023.0−0.43360.5−0.39771.0−0.40391.5−0.43412.0−0.45980.1−0.80840.72−0.74431.0−0.71982.0−0.64350.20.00950.40.83640.61.66340.82.19030.5−0.73471.0−0.77601.5−0.78982.0−0.79671.0−0.40391.5−0.61482.0−0.78492.5−0.93020.1−0.39670.3−0.40030.5−0.40390.7−0.40750.0−0.39731.0−0.41602.0−0.45203.0−0.49631.0−0.57042.0−0.47923.0−0.43494.0−0.41390.5−0.40391.5−0.47253.0−0.56485.0−0.67120.01−0.29080.03−0.31690.05−0.34240.10−0.40390.01−0.29250.03−0.31800.05−0.34310.10−0.4039


## 5. Conclusions

Effect of thermal radiation and activation energy with binary chemical reaction ofSWCNT+MWCNT/H2O hybrid nanofluid is studied. The key findings were as follows:Widening ofMcaused skin friction S˜fxRex0.5 increases and heat transfer H˜txRex−0.5 and mass transfer C˜txRex−0.5 rates to be decreasedHeat transferH˜txRex−0.5 is intensified subject to large radiation RUpsurge ofγ leads to enhance f′ζ and decline θζandφζφζ upgrades via Pr and Nt and it degrades Nbφζ enhances by larger activation energy E and it decays through σ,δAn increase inφ1 and φ2 diminish f′ζ and φζ and enhances θζFrom the above analysis, we can conclude that the impact of heat transfer inSWCNT−MWCNT/water-based hybrid nanofluid is more powerful compared to the common fluid



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*Source: 9473226-2021-08-25.xml*