Effects of radiation and chemical reaction due to graphene oxide nanofluid flow in concentric cylind

Abstract

Keywords: Buongiorno Model; Entropy Generation; Graphene Nanofluids

Introduction

Nanofluids are used to overcome the drawbacks of microfluids and viscous fluids, such as the erosion of flow channels, clogging, and deposition. While modeling their flow, thermophoretic diffusion and Brownian motion are considered for their significance in heat and mass transfer [1]. Besides the advantages of nanofluids on the whole, the

type of nanoparticles dispersed in the fluids greatly influence the behavior of the nanofluids. In view of their thermophysical properties, graphene nanoparticles are the best choice for enhanced thermal performance. They significantly improve the performance of fluids in devices such as radiators and heat exchangers when dispersed either as nanoparticles or hybrid nanoparticles [2, 3, 4, 5].

*Corresponding author. *E-mail address: jagadeeshwarp@iiitt.ac.in, jagadeeshwar.pashikanti@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University Press, İstanbul, Turkey Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

The commonly used graphene nanofluids in literature are graphene oxide nanofluids and composite graphene nanofluids with ethylene glycol and water as basefluids [1]. Azimi et al.[6] evaluated the heat transfer analytically when graphene oxide nanofluid flows unsteadily within moving plates. In their study, a comparative analysis is conducted by dispersing GO, alumina, titania, and silver nanoparticles and it is interpreted that silver nanofluid was effected in the highest Nusselt number and it increased with volume fraction. Gul et al. [7] examined the 2D flow in an upright channel. In this study, the authors concluded that GO with ethylene glycol has stronger thermal efficiency than waterbased GO. While Ullah et al. [8] examined the 3D flow where the fluid is pressed between a vertical channel. Both investigations emphasize the impacts of strong magnetic effects on nanofluid flow when the medium is considered to be porous and concluded that magnetic field parameter controlled the fluid flow. The slip flow of titanium dioxide and graphene oxide nanofluids using the LevenbergMarquardt scheme in neural networks is studied by Khan et al. [9]. The influence of flow variables on velocity, temperature, and concentration profiles are discussed and the absolute error values are found to range between 10−1 to 10−8. Extensive applications of thermal radiation and chemical reaction effects in different propulsion systems, atomic plants, turbines, the manufacturing of polymers, fertilizers, and dyes, etc. signify their importance in Computational Fluid Dynamics. Noteworthy investigations considering the impacts of chemical reaction and thermal radiation, such as the study by Srinivasacharya and Ayano [10], investigated the impacts of chemical reaction on a micropolar fluid with MHD and cross-diffusion effects, with the conclusion that the microrotation, velocity, and concentration profiles decayed with chemical reaction parameter. Srinivasacharya and Shafeeurrahman [11] investigated the flow of a chemically reacting nanofluid for entropy generation analysis with magnetic field and Joule heating effect. From the results obtained, it is concluded that the entropy, velocity, and temperature of the nanofluid are enhanced by the Joule heating parameter, whereas, the velocity is enhanced and the temperature and entropy generation are depleted by the chemical reaction parameter. Masood and Farooq [12] conducted an analysis on thermal stratification and thermal radiation effects due to the MHD flow of graphene oxide and silver nanoparticles in water. The inferences of this investigation are that convective heat transfer decreases with both the stratification parameter and radiation parameter. Jha and Samaila [13] studied the impacts of thermal radiation and nonlinear density variation on the MHD flow over an inclined plate. The results show an elevation in heat transfer with angle of inclination and thermal radiation. Oyedepo et al. [14] modeled the thermal performance of alumina nanofluids as coolants in car radiators. Results signified a 14% elevation in heat transfer coefficient for a 4% increase in volume fraction. Rathore and Sandeep [15] modelled and investigated the blood-based silver and gold nanofluid flow in a stenosed artery with variable heat

source and magnetic field effects. This study helps with localized photothermal therapy (PTT) to treat cancer. Algehyne et al. [16] computationally examined the hybrid nanofluid flow due to partially ionized graphene oxide and silver over a stretching sheet. The investigation signifies a decrease in velocity with Hartmann number and an increase with Hall current and ion-slip parameter. Moreover, many classical and novel methods for numerical and analytical investigations in cylindrical geometries have been widely used. For example, Zeeshan et al. topologically approached the flow of nanofluids using a combined Genetic algorithm and Nelder-Mead method [17] with the conclusion that the Dufour number and the Dufour-solutal Lewis number enhance the Sherwood number while depleting the Nusselt number. Kumar et al. [18] explored the study on Powell-Eyring fluid flow in concentric rotating cylinders and found from the results obtained that the velocity profiles increase with the Powell-Eyring parameter and rotation parameter. [19] investigated fractional Oldroyd-B nanofluid for 1-D flow with ethylene glycol-based molybdenum disulfide, copper, alumina, and silver nanoparticles. The conclusions suggest that molybdenum disulfide nanofluid has higher velocity profiles in comparison to the other nanofluids. Studies such as the analysis of the squeezing flow of an ionic liquid by Shah et al. [20] with a conclusion that copper nanoparticles among copper, alumina, and titania are the best to enhance heat transfer is worth mentioning. Also, an enhancement in the velocity boundary layer is found to be unique to copper nanofluids. A numerical examination of nanofluid flow in a porous annulus is conducted by Miles and Bessaih [21] and entropy and convective heat transfer are observed to increase with the addition of nanoparticles. Limited applied and computational studies on graphene nanofluids demand further exploration of their flow. Also, thermal radiation and chemical reaction effects have extensive industrial applications, such as the manufacturing of dyes, fertilizers, and polymers, spacecraft, turbines, atomic plants, missiles, satellites, etc. Despite all the computational investigations on concentric cylinders, there is a lack of literature on the convective study due to graphene oxide nanofluid flow with chemical reaction effects and thermal radiation impacts in concentric cylinders. The investigation aims to bridge the gap by studying the flow of graphene oxide nanoparticles in water between concentric cylinders. The modeled equations are numerically solved, and the values are graphically represented.

Governing Equations

The geometry of the flow consists of an inner vertical cylinder placed concentrically with an outer cylinder of radii a and b, respectively. The following assumptions are made to model the flow between the vertical cylinders. • Graphene oxide nanofluid is assumed to flow with constant pressure gradient ∂P/∂Θ.

The flow is driven by the rotation of the outer cylinder which has an angular velocity Ω. • u is the tangential velocity with which the fluid flows. • The flow is steady, incompressible, axisymmetric, and fully developed meaning the fluid has constant density, the equations describing the flow are the same at any point throughout the channel and are independent of time, and angular coordinate. • The body forces due to gravity, radiative heat flux, and chemical reaction effects, the effects caused by characteristic nanoparticle phenomena namely thermophoresis and Brownian motion are considered. The outline of the flow model is shown in Figure

1. Implementing the above-mentioned assumptions,

Boussinesq approximations, and boundary layer approximations, the variations with respect to axial coordinate disappear. Thus the governing partial differential equations of the problem adapting Buongiorno nanofluid model [22] are given by, (1)

Figure 1. Outline of the flow in concentric cylinders (created by the author). outer and inner cylinders respectively, km and h are mass and heat transfer coefficients, and Dm denotes mass diffusivity and κ thermal conductivity. Let σ* be Stefan’s constant and k* be Roseland’s mean absorption coefficient, then radiative heat flux is represented by qr = (−4σ*/3k*)(∂T4/∂r). T4 is expanded in Taylor series about Ta to obtain T4a ≊ 4T3a T −3T4a by ignoring the higher order derivatives. Properties of nanofluids are defined as [7, 29] Table 1. Thermophysical properties [25, 26, 27, 28] Property (Units)

(5) along with the following no-slip convective boundary conditions [23]

(6) Here, the equations (1) - (5) respectively represent the nanofluid continuity equation, r−momentum equation, Θ−momentum equation, energy equation, nanoparticle continuity equation [24]. The notations Tb, Ta, Cb, and Ca indicate the fluid temperature and concentration at the

The suffixes bf, sp and nf refer to base fluid, nanofluid, and solid particle and the quantities defined above are respectively viscosity, thermal diffusivity, density, thermal expansion coefficient, specific heat capacity, and thermal conductivity. Table 1 presents the values of thermophysical properties. The idea of similarity transformation broadly refers to the transformation of the given problem to a similar

and simpler problem using the variables called similarity variables. In this study, it displays the physical similarities within the given problem such as the similarities of velocity, temperature, and concentration profiles. These variables are used to reduce the number of independent variables of the problem under consideration. The transformation of the partial differential equations is achieved through the similarity variables,

(7) The derived practically important quantities such as local Sherwood number Shz and Nusselt number Nuz and skin friction Cf are

(12) where A5 = 4(1 − Φ)−2.5 ((1−Φ)+Φρsp/ ρbf )−1. The entropy generation number is derived and assessed in the next section. Analysis on Entropy Generation The entropy generation rate, [30]

The constants Ai, i = 1 to 4 and the coefficients in the above equations namely the Grashof number Gr, Reynolds number Re, buoyancy ratio Nr, radiation parameter Rd, Brownian motion parameter Nb, chemical reaction parameter CR, thermophoresis parameter Nt, constant pressure gradient P1, Schmidt number Sc, thermal diffusivity αnf, thermal and concentration Biot number Bit and Bic and heat capacity ratio τ are defined as

is composed of entropy due to temperature, viscous dissipation, and concentration. From the characteristic entropy generation rate SG0 = κnf(Tb-Ta)2/(TaL)2 and SG, entropy generation number can be written as (14) On nondimensionalization,

(16) where the total entropy number is composed of three parts, the entropy number due to combined mass and heat transfer NSm, fluid friction NSf and heat transfer NSh. The nondimensional parameters in (15) are defined as Eckert number Ec = Ω2/(Cpbf (Tb − Ta)), Prandtl number Pr = μbfCpbf / κbf, constant parameter χ = d2/L2, temperature parameter

ΩT = (Tb − Ta)/Ta, concentration parameter ΩC = (Cb − Ca)/ Ca and the combined mass and heat transfer parameter Mm = RDBCa/κbf. By estimating Bejan number Be = NSh/NS [31], the cause for generated entropy is determined. Bejan number, if > 0.5, heat transfer is influenced, if < 0.5, combined heat and mass transfer and fluid friction is the major cause, and if = 0.5, all the three irreversibilities contribute equally [32].

Chebyshev polynomials Tk(ξ) = cos(k cos−1(ξ)) are used and f, θ and ϕ are iterated at ξj = cos (πj /N), j = 0, 1, 2, ...N, the Gauss Lobatto collocation points and the domain [η0, 1] is mapped to these points by η = {(1−η0)ξ+(1+η0)}/2. Thus, the unknowns and their derivatives are given by

Numerical Solution

The Spectral quasilinearization method (SQLM) [23, 33, 34] is used to solve the equations (8) - (11). This technique has been successfully used in a limited number of studies and is shown to give better accuracy at lower orders than the spectral homotopy analysis method converges rapidly, and is thus superior to some semi-analytical methods such as the Adomian decomposition method, the Laplace transform decomposition technique, the variational iteration method, and the homotopy perturbation method. The SLM and SQLM are non-perturbation methods requiring neither the presence of an embedded perturbation parameter nor the addition of an artificial parameter. These methods are therefore free of the major limitations associated with other perturbation methods. Nonlinear terms are linearised by ignoring the higher derivatives in Taylor series expansion about the solution. Let fr, θr and ϕr and fr+1, θr+1 and ϕr+1 be the solutions and improved solutions of the differential equations. Thus, the linearised equations with the boundary conditions are

Here, the Chebyshev differentiation matrix is given by D = D/2. Substitution of the above approximations in (17) - (20) yields, (23) associated with the boundary conditions

We choose the initial functions as f0 = (η−η0)/(1−η0) , θ0 = {A3Bit(η−η0)+2√η0}/ {A3Bit(1−η0)+2(1+√η0)} , ϕ0 = {Bic(η−η0)+2√η0} / {Bic(1−η0)+2(1+ √ η0)} to satisfy the boundary conditions (20). These initial conditions are iterated to obtain the numerical solution.

Results And Discussion

The numerical results of temperature, velocity, and concentration are estimated in the section for varied values in the practical range [35, 36]. Parameter values are taken as Bit = 0.8, Bic = 0.3, Gr = 2 × 105, Nb = 2×10−4, Re = 300, Nt = 3×10−4, Nr = 2, Sc = 170, Rd = 5 and CR = 0.08 unless mentioned otherwise and Pr = 6.5 and Φ = 0.01. SLM approximation of 100th order is taken to obtain converging results at the third iteration. The results for the case of Gr = P1 = Rd

= CR = 0, from the present study are compared to the results from Sinha and Chaudhary [37] in Table 3 and the values are observed to be in good agreement. Figures 2 - 4 depict the influence of Re on f, NS and Be. As the Reynolds number increases, viscous forces increase and hence the flow velocity is supposed to increase. But in this case, the velocity is observed to decrease due to the presence of thermal radiation (Fig. 2). Similarly, the entropy number also decreases (Fig. 3) increasing the Bejan number and suggesting that the entropy comes from mass transfer and fluid friction (Fig. 4). Figures 5 - 8 represent the effects of Bit on f, θ, NS and Be. When Bit increases, velocity slightly decreases (Fig. 5). Whereas, increasing the Bit improves the heat transfer coefficient thus increasing the heat transfer from the inner to outer cylinder. Hence, there is a rise in the temperature of the nanofluid at the outer cylinder, whereas the temperature falls at the inner cylinder (Fig. 6). From Fig. 7, it is

observed that NS increases with the values of Bit. This is because the increasing heat transfer coefficient improves the nanofluid temperature. An increase in temperature means that the particles of the substance have greater kinetic energy. The faster-moving particles have more disorder than particles that are moving slowly at a lower temperature, and hence NS increases. This results in an increased Be, thus emphasizing the dominance of heat transfer on the increased entropy (Fig. 8). The impacts of Bic on f, ϕ, NS, and Be are shown in Figures 9 - 12. An increase in Bic boosts the mass transfer by enhancing the coefficient of convective mass transfer. This results in an enhanced nanofluid velocity (Fig. 9). The enhanced mass transfer aids in the movement of nanoparticles from the inner to outer cylinder, thus causing increased concentration values at the outer cylinder and decreased values at the inner cylinder (Fig. 10). Whereas, NS increases for η < 0.386, again increases for 0.386 < η <

Table 2. Nusselt number, Sherwood number, and skin friction values Re

0.906 and decreases for η > 0.906 (Fig. 11). This results in a decreasing Be for η < 0.386, decreasing for 0.386 < η < 0.906 and increasing for η > 0.906 (Fig. 12). Hence, fluid friction and mass transfer influence the entropy all over the flow channel. The effects of Rd and CR on NS and Be and ϕ are respectively presented in Figures 13 - 15. When the radiation parameter is increased, NS increases (Fig. 13). But the impact of Rd on the temperature profile is ineffective. Thus, from this, we infer that the heat transfer effected by radiative heat flux completely vanishes in the generated entropy. Considering the Bejan number, increasing NS causes Be to increase (Fig. 14), thus affirming the influence of heat transfer on the increasing entropy when the radiation parameter is involved. Likewise, when the values of CR are raised, the enhancing rate of chemical reaction propagates the

nanoparticle concentration. Hence, when CR is increased, there is an inflation ϕ values (Fig. 15). Table 2 shows Nu, Sh and Cf values at the outer cylindrical surface. Nusselt number is the ratio between convection to conduction at the surface, and at the surface, Nu increases with increasing Bic. This implies that the convection at the outer surface increases with diffusion and convective mass transfer, because, as the concentration Biot number is increased, convective mass transfer improves and the nanoparticles move with higher kinetic energy, thus improving the surficial heat transfer by convection. Whereas, Nu decreases with Bit and Rd, thus enhancing the conductive heat transfer. The impact of Rd on Nu also explains its ineffectiveness on the temperature profile. Whereas the impact of Bit on Nu emphasizes that the increased θ profiles were due to conduction and

not convection, as expected. Similarly, when Bic and CR increase, the values of Sh decreases and increases respectively. Thus, the concentration Biot number enhances the concentration profiles in view of diffusion and not convection, similar to the case of Bit. While CR enhances the rate of chemical reaction consequently enhancing convective mass transfer at the surface. Similarly, the skin friction drag, even if it is very small is impacted by the embedded parameters. Moreover, Bit reduces the skin friction by enhancing the nanoparticle characteristics, while Re, Bic, Rd and CR elevate the skin friction drag at the surface.

Conclusion

Flow of graphene oxide nanofluid between concentric cylinders with the effects of thermal radiation and chemical reaction parameter is studied and the following are inferred: • Heat transfer is enhanced by thermal Biot number. • Mass transfer is increased by chemical reaction parameter and concentration Biot number. • Skin friction is preferably reduced by enhancing the thermophoresis parameter, Brownian motion parameter, and thermal Biot number. • Bejan number is calculated to be Be > 0.5 for η > 0.625, and hence the generated entropy is due to heat transfer close to the outer cylinder. • Enhancing the radiation parameter depletes the Nusselt number and increases skin friction and enhancing the chemical reaction parameter enhances the Sherwood number and skin friction. The problem is useful widely ranging from turbine manufacturing, aircraft propulsion systems, coolants, automobile radiators, and heat exchangers to pharmacotherapy. The study can be extended by analyzing the non-Newtonian behavior of the fluid.

Nomenclature

Bic Bit C Ca Cb Cf Cp CR DB DT Ec f g Gr h k KC Mm Nb Nr NS Nt NSm Nu P p Pr qr R r Rd Re SG Sc Sh T Ta Tb u

concentration Biot number thermal Biot number concentration of the nanofluid concentration of the nanofluid at the inner cylinder concentration of the nanofluid at the outer cylinder skin friction specific heat capacity (JK−1kg−1) chemical reaction parameter Brownian diffusivity (m2s−1) thermophoretic diffusivity (m2s−1) Eckert number dimensionless velocity acceleration due to gravity (ms−2) Grashof number heat transfer coefficient (Wm−2K−1) mass transfer coefficient (ms−1) rate of chemical reaction (s−1) heat parameter Brownian diffusion coefficient number buoyancy ratio dimensionless entropy generation number thermophoretic diffusion coefficient number entropy due to combined heat and mass transfer Nusselt number dimensionless pressure pressure (Pa) Prandtl number radiative heat flux (Wm−2) universal gas constant radial co-ordinate radiation parameter Reynolds number entropy generation rate Schmidt number Sherwood number Nanofluid temperature (K) Nanofluid temperature at the inner cylinder (K) Nanofluid temperature at the outer cylinder (K) tangential velocity component (ms-1)

Greek symbols αnf thermal diffusivity (m2s−1) β thermal expansion coefficient (K−1) χ constant parameter η similarity variable κ thermal conductivity (Wm−1K−1) λ mixed convection parameter μ dynamic viscosity (Nsm−2) Ω angular velocity (rad s−1) concentration parameter ΩC temperature parameter ΩT Φ nanoparticle volume fraction ϕ dimensionless concentration ψ stream function

density (kgm−3) heat capacity ratio tangential coordinate dimensionless temperature variable viscosity parameter

Acknowledgement

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

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

Ethics

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Statement On The Use Of Artificial Intelligence

Artificial intelligence was not used in the preparation of the article.

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