A numerical simulation for magnetohydrodynamic nanofluid flow and heat transfer in rotating horizontal annulus with thermal radiation.

The impact of an axial magnetic field on the heat transfer and nanofluid flow among two horizontal coaxial tubes in the presence of thermal radiation was considered in this study. The impact of viscous dissipation was also considered. The well-known KKL (Koo-Kleinsteuer-Li) model was applied to approximate the viscosity of the nanofluid and the effective thermal conductivity. Furthermore, proper transformations for the velocity and temperature were applied in this study to obtain a set of ODEs (ordinary differential equations) for basic equations governing the flow, heat and mass transfer. In addition, the 4th order Runge-Kutta (RK) numerical scheme was applied to solve the differential equations along with the associated boundary conditions. The impacts of different parameters, including Hartmann number, Reynolds number, radiation parameter and aspect ratio, on the heat transfer and flow features were studied. According to the results, the value of the Nusselt number increases with an increase in the radiation parameter, Hartmann number and aspect ratio and a decrease in the Reynolds number and Eckert number. This journal is © The Royal Society of Chemistry.

Introduction

Effective heat transfer in liquids has become a challenging issue in industries and academia because of its importance in the improvement of the efficiency of different devices. Accordingly, different approaches have been proposed by researchers to improve the ability of heat transfer in fluids; one of the best methods to advance the heat transfer rate in liquids is the use of nanofluids, which are materials suspended with nanoparticles (size ≤ 50 nm). These nanoparticles significantly improve the convective heat transfer in liquids by altering the basic features of the base fluid. Carbides or carbon nanotubes (CNTs), metals and oxides are usually used as particles. Because of the superior features of nanofluids, extensive studies have been performed to evaluate the effectiveness of this heat transfer approach. The results obtained from different studies indicate that the effect of addition of nanoparticles to the base fluid becomes more significant under the influence of a magnetic field. To achieve the full potential of nanofluids, it is important to investigate the impact of different impressive parameters that affect the behavior of nanofluids.

Computational approaches have been broadly applied for the simulation of engineering-related issues;[1-15] in the field of nanofluids, researchers have applied these approaches to investigate the behavior of these kinds of liquids to achieve the full potential of nanofluids. The influences of the 2nd order temperature jump and velocity slip B.Cs for 3rd-grade nanofluids over a coaxial tube have been investigated by Zhu et al.[16] Their findings show that thermophoresis movement and Brownian motion cause the temperature to increase. For power-law nanofluids, Lin et al. have examined the Marangoni convection flow and heat transfer driven by the temperature gradient.[17] Furthermore, using a model that contains the influences of thermophoresis and Brownian motion, the boundary layer flow of a nanofluid over an extending plate was studied by Khan and Pop.[18] They have considered the Prandtl number, Brownian motion, Lewis number and thermophoresis in their study and demonstrated that the reduced Nusselt number is a declining function of each nondimensional number. Khan et al.[19] have also scrutinized the laminar 3D nanofluid flow by a bi-directional stretching sheet. According to this study, the convective heat transfer in different fluids improves because of the presence of nanoparticles in the host liquid.

When the value of the convection heat transfer factor is insignificant, thermal radiation significantly affects the total surface heat transfer. Bakier[20] investigated the influence of thermal radiation on mixed convection from a vertical surface in a porous material. The corresponding equations were solved by a 4th-order RK scheme in the abovementioned study. The influence of magnetic field and radiation on the mixed convection stagnation point flow over a vertical stretching plate in a porous structure was investigated by Hayat et al.[21] According to their findings, for both types of flows, i.e. assisting and opposing, the values of the local Nusselt number and coefficient of skin friction were tabulated. Hayat et al.[22] studied the impact of thermal radiation and Joule heating on the MHD flow of a Maxwell fluid in the presence of thermophoresis. The impact of thermal radiation on heat transfer and magnetohydrodynamics nanofluid flow was scrutinized by Sheikholeslami et al.[23] Their findings show that an increase in the radiation parameter causes a decrease in the thickness of the concentration boundary layer.

The investigation of the electrically conductive fluid flow, called magnetohydrodynamic (MHD), is a topic of numerous studies reported in the literature because of its various applications. The study of solar plasma, terrestrial cores and stellar structures are examples of its applications in astrophysics and geophysics. Moreover, MHD has numerous industrial applications such as in the extraction of geothermal energy, nuclear reactors, heat and mass transfer and the stability of convective flows. Sheikholeslami et al.[24] have studied the heat transfer and nanofluid flow characteristics among two horizontal parallel sheets in a rotary system and proved that an increase in the Reynolds number and volume fraction of the nanoparticle leads to an increase in the Nusselt number. However, the Nusselt number reduces as the Eckert number, rotation and magnetic parameters increase. Sheikholeslami and Ganji[25] have investigated the magnetohydrodynamic and ferrohydrodynamic influences on convective heat transfer and ferrofluid flow and found that depending on the value of the Rayleigh number, magnetic number has a different influence on the Nusselt number. The influence of magnetic field on the CuO–water heat transfer and nanofluid flow in a chamber that is heated from beneath was scrutinized by Sheikholeslami et al.[26] According to their findings, at high Rayleigh numbers, the influence of heat source length and Hartmann number is more obvious. Reddy et al.[27] scrutinized the MHD flow, heat and mass transfer features of water combined with Cu and Ag nanoparticles above a rotary disk via a porous media by chemical reactions, thermal radiation and partial slip. They studied the impact of significant parameters, including chemical reactions, temperature slip, thermal radiation, velocity slip, nanoparticle volume fraction and magnetic and porous parameters, on concentration, temperature, azimuthal velocity and radial velocity evaluations in the boundary layer zone. The influence of the abovementioned parameters on local Sherwood number, local skin friction coefficient and local Nusselt number was further examined. According to their findings, an increase in the volume fraction parameter of the nanoparticles results in the elevation of the temperature.

Reddy and Chamkha[28] have further scrutinized the MHD flow and heat and mass transfer of a viscous incompressible nanofluid over a uniform sheet through a porous medium considering the chemical reaction, thermal radiation, heat generation/absorption, thermo-diffusion and diffusion-thermo effect. They examined the impact of important parameters, such as Dufour parameter, Soret parameter, magnetic parameter, Prandtl number, the volume fraction of nanoparticles, space-dependent and temperature-dependent heat source/sink parameters, on the Sherwood number, Nusselt number, skin-friction coefficient, nanoparticle concentration fields, velocity and temperature. Al-Mudhaf and Chamkha[29] used a numerical approach to scrutinize the Marangoni convection flow over a smooth surface because of the existence of a temperature and concentration gradient. They studied the impact of the thermo-solutal surface tension ratio, heat generation or absorption coefficient, Hartmann number, the chemical reaction coefficient and the suction or injection parameter on the quantities belonging to the Sherwood and Nusselt numbers, boundary-layer mass flow rate, wall velocity, concentration profiles, temperature and velocity. Based on their results, it can be concluded that Sherwood and Nusselt numbers and the wall velocity increase by a first-order chemical reaction. However, it has a regressive effect on the mass flow rate in the boundary layer. Furthermore, it is predicted that the Sherwood and Nusselt numbers, the wall velocity and the boundary-layer mass flow rate are augmented with an increase in the thermo-solutal surface tension ratio. The KKL correlation was used by Sheikholeslami Kandelousi for the simulation of heat transfer and nanofluid flow in a permeable channel.[30] According to the findings, the augmentation of heat transfer is directly related to the Reynolds number when the power law index vanishes. However, for other amounts of power law index, a reverse trend was observed. Recently, different researchers have scrutinized heat transfer improvement in nanofluids.[31-41]

In this study, the influence of an exterior axial magnetic field used for the obliging convection nanofluid flow among two horizontal concentric tubes was numerically investigated considering the impacts of thermal radiation and viscous heat dissipation. A rotary internal tube with a low constant angular velocity value induced the forced flow, and the external tube was fixed. The impacts of numerous parameters, namely, aspect ratio, radiation parameter, Hartmann number and Reynolds number, on the characteristics of heat transfer and flow were studied.

Problem definition

To resolve engineering-related problems, the main governing equation should be initially derived, and effective terms must be determined. Then, the applied boundary conditions are defined according to the physics of the problem.[42-51] Since most of the engineering-related problems contain nonlinear terms in the main governing equations, various techniques[52-73] have been developed to solve these equations.

A steady, laminar and unidirectional flow is considered. Thus, the components of the velocity in axial and radial directions as well as the derivatives of the velocity with respect to θ and z vanish. In this situation, the equations governing the flow of the nanofluid and heat transfer in cylindrical coordinates becomewhere q represents the radiation heat flux that can be specified according to the Rosseland estimation as . Furthermore, βR and σ represent the mean absorption factor and Stefan–Boltzmann coefficient, respectively. With adequately small alterations in the fluid temperature inside the flow, a linear function of T can be used to express T4.[72] We can use the well-known Taylor's series expansion to expand T4 around T2. By ignoring higher order terms, T4 ≅ 4T23T − 3T24 was obtained. Hence, the effective density, electrical conductivity and heat capacity of the nanofluid can be stated as follows:

Brownian motion significantly affects the effective thermal conductivity. According to Koo and Kleinstreuer,[30] two different components, i.e. a Brownian motion part and the particle's conventional static part, compose the effective thermal conductivity. The influences of base fluid combinations, types of particle, temperature, particle volume fraction and particle size are considered in this double-component thermal conductivity model.where kstatic denotes the Maxwell-based static thermal conductivity. The improved thermal conductivity constituent created by the micro-sized convective heat transfer of a particle's Brownian motion and influenced by the surrounding fluid motion was acquired by analyzing the Stokes' flow around a spherical domain (i.e. nanoparticles). Koo and Kleinstreuer[30] presented two experimental functions (β and f) and combined the interaction among nanoparticles in addition to the temperature influence in the model, which resulted into

Various base fluids and nanoparticles have various functions. In this study, only nanofluids based on water were considered. The following format is valid for CuO–water nanofluids:where the constants b (i = 0 to 10) are based on the types of nanoparticles. Since the CuO nanoparticles are widely applied in different applications, such as in electronic and optoelectronic devices, field-effect transistors, gas actuators and solar cells, we used this type of nanoparticles in this study. Moreover, based on the values defined in Tables 1 and 2 for the coefficients of the CuO–water nanofluids, the values of R2 for the CuO–water nanofluids were found to be 96% and 98%, respectively.[30] Consequently, the KKL correlation is represented as follows:

Thermophysical properties of water and nanoparticles[14]

	ρ (kg m−3)	C p (J kg−1 K−1)	k (W m−1 K−1)	d p (nm)	σ (Ω m)−1
Pure water	997.1	4179	0.613	—	0.05
CuO	6500	540	18	29	10−10

The coefficient values of CuO–water nanofluids[14]

Coefficient values	CuO–water
a 1	−26.593310846
a 2	−0.403818333
a 3	−33.3516805
a 4	−1.915825591
a 5	6.42185846658 × 10−2
a 6	48.40336955
a 7	−9.787756683
a 8	190.245610009
a 9	10.9285386565
a 10	−0.72009983664

The laminar nanofluid flow in micro heat sinks was further studied by Koo and Kleinstreuer[30] using the effective nanofluid thermal conductivity model introduced by them. The authors suggest the following relation for the effective viscosity because of micromixing in suspensions:where denotes the viscosity of the nanofluid. Hence, we can write the governing relation and the corresponding B.Cs, i.e. from eqn (1)–(3), in a dimensionless format as follows:whereas

Note that for the readers' convenience, the star signs have been dropped from the equations. The Nusselt number Nu and the skin friction coefficient Cf along the internal wall can be expressed as follows:

Various researchers have introduced different goal parameters.[74-95]

Numerical method

The corresponding equations in this study were solved by the 4th order RK scheme. To use this numerical method, initially, the governing differential relations must be reduce into a set of 1st order ODEs. Let . Various numerical techniques can help researchers in modeling.[96-130] The following system of equations can be achieved:and the related initial conditions are as follows:

Next, eqn (17) along with the initial conditions (18) can be solved using the 4th order RK integration scheme. Note that to approximate the proper values for the unknown initial conditions u1 and u2, Newton's scheme was used, and the process was repeated till the B.Cs at v*(1) = 0, θ(1) = 0 were satisfied. MAPLE was used to perform the computations. Furthermore, the supreme value of r* = 1 for each set of factors was recognized when the values of the anonymous B.Cs at r* = η did not alter to an effective loop with error less than 10−6. Optimized models can be used for better accuracy.[131-149]

Results and discussion

In the current study, the heat transfer and nanofluid flow among two horizontal coaxial tubes was studied. Note that the fluid flow influenced by an axial magnetic field and the impact of thermal radiation were also considered. The impacts of aspect ratio, Hartmann number, radiation parameter and Reynolds number on the heat transfer and flow features were studied (Fig. 1).

Fig. 1

Geometry of the problem.

The results of the current study for the velocity profile were compared with those of Aberkane et al.[73] in Fig. 2. According to this figure, a good agreement can be observed between these findings, which shows the reliability of the method used in this study. The flow velocity and temperature changes at different Hartmann numbers are shown in Fig. 3. According to this figure, an increase in the value of the Hartmann number causes a decrease in both the temperature and the velocity. Furthermore, Fig. 4 demonstrates the impact of Reynolds number on the velocity of the fluid flow and temperature. The results show that as the Reynolds number increases, the thermal boundary layer thicknesses and velocity increases. The changes in the Nusselt number and skin friction coefficient at different Hartmann and Reynolds numbers are demonstrated in Fig. 5. According to this plot, an increase in the Hartmann number and a decrease in the Reynolds number result in the augmentation of the Nusselt number. Moreover, in the case of skin friction constant, similar trend can be detected in this figure.

Fig. 2

Comparison of the results obtained by Aberkane et al.[73] and present results of the velocity profile for η = 0.5 and Ha = 4.

Fig. 3

Effect of Hartmann number on the velocity and temperature profiles when Pr = 6.8, η = 0.5, Ec = 0.01, Rd = 0.1, Re = 1, ϕ = 0.04.

Fig. 4

Effect of Reynolds number on the velocity and temperature profiles when Pr = 6.8, η = 0.5, Ha = 1, Ec = 0.01, Rd = 0.1, ϕ = 0.04.

Fig. 5

Effect of Reynolds number and Hartmann number on the Nusselt number and skin friction coefficient when Pr = 6.8, η = 0.5, Ec = 0.01, Rd = 0.1, ϕ = 0.04.

The impact of the Eckert number on the Nusselt number and temperature profile is presented in Fig. 6. This graph shows that the augmentation of the Eckert number causes the thickness of the thermal boundary layer to reduce. Hence, the Nusselt number is affected by the Eckert number and reduced upon its augmentation. The change in temperature and Nusselt number at different radiation parameters is revealed in Fig. 7. According to this figure, the value of temperature reduces when the radiation parameter is taken into account, and an increase in the radiation parameter results in the augmentation of the Nusselt number. The influences of the aspect ratio on the skin friction constant and Nusselt number are presented in Fig. 8. This plot displays that an increase in the amount of the aspect ratio leads to a reduction in the space between the hot and the cold walls. Hence, the value of the Nusselt number increases with an increase in the aspect ratio.

Fig. 6

Effect of the Eckert number on the temperature profile and Nusselt number when Pr = 6.8, η = 0.5, Rd = 0.1, Re = 1, ϕ = 0.04.

Fig. 7

Effect of the radiation parameter on the temperature profile and Nusselt number when Pr = 6.8, η = 0.5, Ec = 0.01, Re = 1, ϕ = 0.04.

Fig. 8

Effect of aspect ratio on the Nusselt number and skin friction coefficient when Pr = 6.8, η = 0.5, Ec = 0.01, Rd = 0.1, Re = 1, ϕ = 0.04.

Conclusions

In the current study, the impact of axial magnetic field on the heat transfer and nanofluid flow among two horizontal coaxial tubes was examined. It was supposed that the magnetic field considered in this study was uniform and constant. Moreover, the influence of thermal radiation and viscous dissipation was considered. The corresponding PDEs were initially converted to a set of ODEs and then solved by the 4th-order RK scheme. The impact of active parameters on the heat transfer and flow was considered. The results indicate that an increase in the radiation parameters and Hartmann number causes the velocity and the temperature boundary layer thicknesses to decrease. However, an enhancement in the Eckert number and Reynolds number may lead to an increase in the velocity and temperature boundary layer thicknesses.

Conflicts of interest

There is no conflict of interest related to this paper.