ARTICLE

J. Cent. South Univ. (2019) 26: 1133-1145

DOI: https://doi.org/10.1007/s11771-019-4076-9

Electrical conductivity effect on MHD mixed convection of nanofluid flow over a backward-facing step

SELMEFENDGL Fatih¹, ZCAN OBAN Seda¹, ZTOP Hakan F.²

1. Department of Mechanical Engineering, Celal Bayar University, 45140 Manisa, Turkey;

2. Department of Mechanical Engineering, Technology Faculty, Flrat University, 23119 Elazlg, Turkey

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: In this study, magneto-hydrodynamics (MHD) mixed convection effects of Al₂O₃-water nanofluid flow over a backward-facing step were investigated numerically for various electrical conductivity models of nanofluids. A uniform external magnetic field was applied to the flow and strength of magnetic field was varied with different values of dimensionless parameter Hartmann number (Ha=0, 10, 20, 30, 40). Three different electrical conductivity models were used to see the effects of MHD nanofluid flow. Besides, five different inclination angles between 0°–90°is used for the external magnetic field. The problem geometry is a backward-facing step which is used in many engineering applications where flow separation and reattachment phenomenon occurs. Mixed type convective heat transfer of backward-facing step was examined with various values of Richardson number (Ri=0.01, 0.1, 1, 10) and four different nanoparticle volume fractions (f=0.01, 0.015, 0.020, 0.025) considering different electrical conductivity models. Finite element method via commercial code COMSOL was used for computations. Results indicate that the addition of nanoparticles enhanced heat transfer significantly. Also increasing magnetic field strength and inclination angle increased heat transfer rate. Effects of different electrical conductivity models were also investigated and it was observed that they have significant effects on the fluid flow and heat transfer characteristics in the presence of magnetic field.

Key words: electrical conductivity; nanofluids; backward-facing step; MHD flow; mixed convection

Cite this article as: SELMEFENDGL Fatih, ZCAN OBAN Seda, ZTOP Hakan F. Electrical conductivity effect on MHD mixed convection of nanofluid flow over a backward-facing step [J]. Journal of Central South University, 2019, 26(5): 1133–1145. DOI: https://doi.org/10.1007/s11771-019-4076-9.

1 Introduction

Heat transfer enhancement is a necessity for energy efficiency in many engineering applications which consists of three methods that are described as passive, active and hybrid methods. Using nanofluid (nanosized particles that are suspended in base fluids like water, ethylene glycol. etc.) is an efficient passive method that has been subject to various experimental and numerical studies. Most investigations have the common result that nanofluids enhance heat transfer significantly compared to base fluids with little pressure drop [1–12]. Electrically conductive nanofluids are used with magnetic field applied to the flow and this can be considered a passive heat transfer enhancement method. MHD convection with nanofluids has been performed in various studies with different geometries. AMINOSSADATI et al [13] numerically studied magnetic field effects of forced convection nanofluid flow in partially heated microchannel. Results showed that heat transfer enhancement is better with higher Hartmann and Prandtl numbers and increasing nanofluid volume concentration increased heat transfer for all values of these two numbers. MHD mixed convection of Al₂O₃–water nanofluid flow in a sinusoidal channel was investigated numerically by RASHIDI et al [14]. Heat transfer is increased by increasing Reynolds number, amplitude sine waves and nanofluid volume concentration. SHEIKHOLESLAMI et al [15–18] studied numerically MHD natural convection nanofluid flow with different geometries. Results commonly indicated that Nusselt number increases by increasing Rayleigh number and volume concentration while it decreases by increasing Hartmann number. Mixed convection with Al₂O₃–water nanofluid in a lid driven trapezoidal cavity under the effect of an inclined magnetic field was numerically investigated for various electrical conductivity models by SELIMEFENDIGIL et al [19]. Average Nusselt number increased with increasing Richardson number while decreased with increasing Hartmann number. LARIMI et al [20] studied MHD forced convection ferrofluid (Fe₃O₄- water) flow under the influence of various non- uniform transverse magnetic field arrangements. Results clearly showed that average heat transfer increases by increasing Richardson number and decreasing Hartmann number.

Flow separation and reattachment is an important phenomenon used in many engineering and heat transfer applications and backward-facing step is one of the simplest geometries of this occurrence. This geometry can be seen in many applications of engineering such as flow around buildings, airfoils, nuclear and electronic cooling devices. Numerical studies of convection in backward-facing step have several examples in literature because of its importance. Also using nanofluids for more rapid cooling in this geometry has been studied by several researchers. The earliest numerical study of nanofluid flow in a backward- facing step geometry is the investigation of ABU-NADA [21] with different types of nanofluids and different nanoparticle volume fractions in a channel. Results showed that nanofluids with high thermal conductivity (such as Ag and Cu) have better heat transfer enhancement in the vicinity of top and bottom walls while nanofluids with low thermal conductivity (such as TiO₂) have better enhancement within the recirculation zones. KHERBEET et al [22] studied laminar flow and heat transfer with two types of nanofluid over microscale backward-facing step experimentally and numerically. The measurement results revealed that the water-SiO₂ nanofluid has the highest Nusselt number and it increases with increasing volume fraction. Effects of different step heights on heat transfer were investigated by the same researchers [23] and results revealed that increasing step height has positive effects on heat transfer augment. SELIMEFENDIGIL et al [24], studied mixed convection flow over a backward facing step with a rotating cylinder subjected to nanofluid. Nusselt number plots indicated that there is almost a linear increase in the heat transfer enhancement with increasing Reynolds number and nanoparticle volume fraction. Laminar forced convection of pulsating nanofluid flow over a backward-facing step with a corrugated bottom wall was numerically examined by the same researchers [25]. According to the results, heat transfer increases as the Reynolds number, length and height of the corrugation wave increase. MOHAMMED et al [26–28] studied mixed convection nanofluid flow over different backward-facing step geometries (vertical, horizontal, baffle mounted on top wall). Results showed that heat transfer increases due to increment of nanoparticle volume fraction and Reynolds number, and it decreases with the increment of nanoparticle diameter. AL-ASWADI et al [29] numerically studied laminar forced convection flow of nanofluids over a 2D horizontal backward facing step placed in a duct. Eight types of nanoparticles with 5% volume fraction were dispersed in base fluid and flow, velocity and skin friction properties were studied. HESMATI et al [30] studied numerically mixed convection heat transfer over a 2D backward facing step with an inclined slotted baffle by using nanofluids. Different types of nanofluids with different nanoparticle volume fractions and particle diameters were considered for the problem. It was clearly observed that nanofluids with higher nanofluid volume fraction and smaller nanoparticle diameter affect the heat transfer considerably. Influence of inclination angle of magnetic field on mixed convection of nanofluid flow over a backward facing step and entropy generation was studied numerically by SELIMEFENDIGIL et al [31]. Heat transfer characteristics for different magnetic field orientation angles has been investigated by changing Reynolds number, Hartmann number and solid particle volume fraction.

Results revealed that increasing Reynolds number and solid particle volume fraction increases average heat transfer while increasing Hartmann number makes a decreasing effect on average heat transfer. Also magnetic field inclination angle has considerably effects on local and average heat transfer.

Magnetic field has remarkable effects on heat transfer with electrically conductive fluids. The influence of magnetic field on electrical conductivity and heat transfer with nanofluids is important for engineering applications such as coolers of nuclear reactors, purification of molten metals and many other systems. Numerical studies on electrically conductive nanofluids generally focus on Maxwell’s electrical conductivity model for fluid suspensions [32]. Bruggeman model for non-homogenous macroscobic suspensions is another commonly used model to obtain electrical conductivity of nanofluids [33]. GANGULY et al [34] studied electrical conductivity measurement of water based alumina nanofluid experimentally with different temperatures and particle volume fractions. They composed a correlation for electrical conductivity of nanofluid which depends on nanoparticle volume fraction and temperature. MINEA et al [35] performed measurement of electrical conductivity with 12 nm diameter alumina particles suspended in water. According to their correlation, electrical conductivity is the increasing function of nanoparticle volume fraction and temperature. Different electrical conductivity models developed experimentally in literature depend on volume fraction, shape and size of nanoparticles in different based fluids [36–41].

In this study, electrical conductivity of alumina-water nanofluid over a backward facing step in mixed convective flow under the effects of an inclined magnetic field was numerically examined. Three different existing electrical conductivity models for the alumina-water nanofluid were used for a range of Richardson number, Hartmann number, magnetic inclination angle and solid nanoparticle volume fraction. The results are shown in terms of streamline, isotherm, local and average Nusselt number plots. The results of this study can be used in a variety of thermal engineering problems as mentioned above.

2 Mathematical formulation

2.1 Physical problem

A schematic description of the problem is shown in Figure 1. The geometry is a channel with backward-facing step. The backward step height is H and the height of the channel is 3H. At the inlet of the channel velocity is imposed as parabolic and the temperature is uniform (T=T_c). The walls of the channel are assumed to be adiabatic, except the bottom wall downstream of the channel is heated to a uniform temperature (T=T_h). The working fluid is alumina-water nanofluid under the influence of an oriented magnetic field. The thermophysical properties of the fluid are assumed to be constant. The flow through the channel is considered laminar, 2D, incompressible Newtonian flow. Table 1 presents the thermophysical properties of water and alumina nanoparticles at reference temperature. The Prandtl number of base fluid is 6.1 and the thermophysical properties of base fluid are considered to be constant. The flow is assumed to be incompressible and Newtonian. Thermal equilibrium is considered between fluid phase and nanoparticles and no slip condition between them is assumed.

Figure 1 Schematic diagram of physical model (a) and mesh generation of computational domain (b)

Table 1 Thermophysical properties of pure water and Al₂O₃ nanoparticles

2.2 Governing equations

The governing equations of continuity, momentum and energy for 2D Cartesian coordinates are written as follows [42]:

                              (1)

             (2)

 (3)

               (4)

In energy equation, the viscous dissipation and Joule heating are neglected. Also imposed and induced magnetic field is negligible in comparison with applied magnetic field.

2.3 Thermophysical properties of nanofluids

The effective density, specific heat and thermal expansion coefficient equations are given as follows [43]:

             (5)

             (6)

              (7)

The effective viscosity model of Koo- Kleinstreuer for micro mixing homogeneous suspensions including Brownian motion and nanoparticle volume fraction effects is written as the following equation [44]:

 (8)

The effective thermal conductivity formula is Koo-Kleinstreuer model considering Brownian effects which are the outcome of nanoparticle collision [44]:

k_nf=k_st+k_Brownian                                 (9)

            (10)

     (11)

where k_st is the static thermal conductivity in Ref. [32]. The effective electrical conductivity for alumina-water nanofluid is calculated by using three different models. Model 1 (M1) is the Maxwell’s model for random suspensions including spherical micro-sized particles written as follows [32]:

              (12)

Model 2 (M2) is a developed version of Maxwell’s model which is for relatively small particles (nanoparticles) attained experimentally by GANGULY et al [34]. Electrical conductivity is the basic function of solid particle volume fraction and temperature written as follows:

      (13)

Model 3 (M3) is the correlation of an experimental study of MINEA et al [35] for nanofluids considering nanoparticle volume fraction and temperature written as follows:

                (14)

The variables in the governing equations can be implemented to non-dimensional form using parameters as follows:

,

,

 (15)

Governing equations are converted to dimensionless form as follows [24]:

                           (16)

      (17)

                            (18)

 (19)

2.4 Boundary conditions

The dimensional boundary conditions for the channel with backward-facing step are written as follows:

At the channel inlet, velocity is unidirectional and parabolic: U=1, θ=0,

At the channel exit, fully developed flow conditions are assumed:

On the bottom wall downstream the step, temperature is constant: θ=0.

The channel walls except the bottom wall downstream of the step are assumed to be adiabatic and no-slip conditions occur:

Local Nusselt number is defined as the equation below:

                         (20)

Local Nusselt number is integrated along the bottom wall to obtain spatial average Nusselt number written as follows:

                          (21)

3 Solution methodology

Galerkin weighted residual finite element method is used to solve the governing equations given in Eqs (1)–(4). A commercial CFD code Comsol which uses finite element method was utilized for this purpose. In this method, computational domain was divided into non- overlapping regions and the field variables are approximated by using the Lagrange polynomials of different orders. The convergence of the solution is assumed when the relative error of each variable satisfies the following convergence criteria;

                        (22)

4 Grid independence study and code validation

Grid generation is important to achieve accurate results in minimal time of computation. In this study, computational domain is discretized into triangular elements. Table 2 shows the average Nusselt number on the bottom wall downstream of the step for various grid sizes at the values of parameters Ha=0 and Ha=40 for φ=0.015, Ri=1, γ=45°, M1 model. Considering the physical model of backward-facing step and the time of computation, G5 is determined to be suitable for the problem that has 23385 triangular elements and 2100 rectangular elements. The discretized version of computational element has appropriate interval with totally 25485 elements.

Table 2 Grid independence study

The numerical code is validated with benchmark studies in literature in order to verify accuracy of the present study. The values of reattachment length divided to step height at Reynolds number of 100 for expansion ratio 2 are shown in Table 3. According to the comparison between former studies [45–49] with the present study, a good agreement can be seen in results. Another comparison was made with the results of RUDRAIAH et al [50] for the case in natural convection under the effect of magnetic field. The average Nusselt number values for various values of Hartmann number are given in Table 4 for Grashof number of 2×10⁵.

The average Nusselt number comparisons are given in Table 4 for two Grashof number values and for different Hartmann number values.

Table 3 Comparison of reported reattachment length values (x) with the present study (Re=100, E_R=2)

Table 4 Comparison of average Nusselt number for MHD free convection with different Hartmann numbers

5 Results and discussion

MHD mixed convection of alumina-water nanofluid flow over a backward-facing step was studied numerically with three different electrical conductivity models. The varying parameters were nanoparticle volume fraction of φ=0, 1%, 1.5%, 2% and 2.5%, Hartmann number of Ha=0, 10, 20, 30 and 40, Richardson number of Ri=0.01, 0.1, 1 and 10, magnetic field inclination angle γ=0°, 30°, 45°, 60° and 90°. In addition, numerical study and all simulations were performed for three electrical conductivity models. The effects of all parameters with different electrical conductivity models were simulated as isotherms and streamlines for each condition. In this section, results of each parameter are evaluated and compared with each other.

5.1 Effects of Richardson numbers

Richardson number is the dimensionless parameter which indicates ratio of natural convection to forced convection in a mixed convection flow. For further definition, Richardson number increases, meaning that natural convection becomes important. In the present study, Richardson number was varied between 0.01 to 10 by changing Reynolds number. Figures 2 and 3 show the streamlines and isotherms of varying Ri numbers when other parameters are fixed to the values, f=1.5, γ=45° and Ha=30 for M2 model. The streamlines are colored by the velocity magnitude. Size of recirculation zone behind the step is increasing and the reattachment point occurs further to the step as Richardson number decreases. According to the isotherms in Figure 3, it can be seen that Richardson number has considerably effects on temperature gradient. As Richardson number increases, the region where isotherms are more clustered along the bottom wall is getting closer to the step. This region is the place where the reattachment point occurs.

Effects of varying Richardson numbers on local Nusselt number (Nu_x) along the bottom wall is shown in Figure 4 for Ha=30, f=1.5 and γ=45° in M2 model. As seen on the figure, increment of Richardson number diminishes heat transfer in comparison lower values of Ri. Otherwise, local Nusselt number increases to the peak value near the reattachment point and after that point it decreases slowly to a nearly constant value for the further regions of the wall. The highest value of Nusselt number is attained for Richardson number of 0.01 as Nu_x≈18.5 approximately at 4H distant from the step.

Figure 2 Streamlines for different Richardson numbers on M2 model (φ=0.015, γ=45°, Ha=30):

Figure 3 Isotherms for different Richardson numbers on M2 model (φ=0.015, γ=45°, Ha=30):

Figure 4 Effects of Richardson number on local Nusselt number along hot wall (φ=0.015, γ=45°, Ha=30, M2)

Average Nusselt number values are given in Table 5 for different Richardson numbers for all nanoparticle volume fractions and base fluid for M2 model. As it can be clearly seen from the table,increment of Richardson number due to natural convection effects causes a decrease in the average heat transfer along the bottom wall.

Table 5 Effects of Richardson number on average heat transfer (Nu_m) along bottom wall (φ=0.015, γ=45°, Ha=30, M2)

5.2 Effects of Hartmann numbers

Hartmann number indicates the strength of external magnetic field applied to the flow. In many studies, it is shown that magnetic effect provides flow stability and control. The present study was made for absence of magnetic field and four different in the Hartmann number values. Streamlines and isotherms are illustrated for these five conditions in Figures 5 and 6. Other parameters were fixed at the constant values: nanoparticle volume fraction is 1.5%, Richardson number is 1 and inclination angle of magnetic field is 45° for Model 2. Considering both natural convection and forced convection effects are perceivable for Ri=0 condition, it can be said that the presence of magnetic field suppresses the flow and causes streamlines to take form parallel to the step. Recirculation zone behind the step is getting smaller as Hartmann number increases and streamlines gets closer to the step due to the retarding effect of magnetic field. For the value of Ha=40, no vortex is seen behind the step and reattachment length is approximately 0.9H. Isotherms are illustrated in Figure 6 for varying Hartmann numbers. Steep thermal gradients are seen near the reattachment point and they are more clustered as Hartmann number increases. In addition, reattachment point gets closer to the step in the increment of Hartmann number. Figure 7 graphs local Nusselt number distribution along the hot wall for different Hartmann numbers for volume fraction 1.5%, Richardson number 1 and inclination angle 45° in model 2. As it can be seen from the figure, local heat transfer increases with increasing Hartmann number along the wall. The maximum heat transfer value is approximately Nu_x≈8.4 in Ha=40 and it occurs near the reattachment point at about 1.8H. Table 6 demonstrates effects of Hartmann number on average heat transfer for the values, Ri=1 and γ=45° for all nanoparticle volume fractions in electrical conductivity model 2. Increasing magnetic field causes heat transfer augment for all nanofluid concentrations. In this model the maximum heat transfer enhancement with magnetic field effect is 87.63% for Ha=40 in comparison with the absence of magnetic field. Magnetic field effects were shown to dampen the fluid motion and reduce convection in cavity flow applications. In many cases where flow separation and reattachment occurs magnetic field has the potential to reduce the size of separation and results in heat transfer enhancement as it has been shown in some recent studies.

Figure 5 Streamlines for different Hartmann numbers on M2 model (φ=0.015, γ=45°, Ri=1):

Figure 6 Isotherms for different Hartmann numbers on M2 model (φ=0.015, γ=45°, Ri=1):

Figure 7 Effects of Hartmann number on local Nusselt number along hot wall (φ=0.015, γ=45°, Ri=1, M2)

Table 6 Effects of Hartmann number on average heat transfer (Nu_m) (φ=0.015, γ=45°, Ri=1, M2)

5.3 Effects of inclination angle

In the present study, external magnetic field was applied to the flow. Direction of magnetic field was oriented with five different inclination angles between 0° and 90°. Figure 8 demonstrates streamlines for varying inclination angles while other parameters are fixed at Ha=30, φ=0.015, Ri=1 for model 2. For horizontally oriented inclination angle (γ=0°), there is a similarity with non-magnetic field (Ha=0) condition on streamlines and the reattachment length is just a little closer to the step because of the very weak effect of horizontal magnetic field. As the angle of magnetic field goes inclined to vertical position (between 30°–90°), recirculation zone gets smaller and reattachment point is located closer to the step. For the vertically oriented angle, the suppressing effect is mostly pronounced and streamlines align almost parallel to the step so that there is no vortex occurrence in this condition.

Isotherms are illustrated in Figure 9 for varying angles of magnetic field while other parameters are constant (Ha=30, φ=0.015, Ri=1, M2). More clustering of isotherms is observed as the inclination angle increases and boundary layer thickness decreases behind reattachment point along the hot wall. The maximum local heat transfer value is seen in the reattachment point at location of 1.8H with vertically oriented magnetic field in Figure 10. Average Nusselt number value increases as the inclination angle increases for all models according to Table 7 and the highest Nu_m values are seen in model 3.

Figure 8 Streamlines for different magnetic field inclination angles on M2 model (φ=0.015, γ=45°, Ri=1):

Figure 9 Isotherms for different magnetic field inclination angles on M2 model (φ=0.015, γ=45°, Ri=1):

5.4 Comparison of electrical conductivity models

In the present study, three different correlations which were obtained from experimental investigations are used to estimate electrical conductivity of alumina-water nanofluid. Maxwell’s model is rather for micro-particulate suspensions with sphere shaped particles in the mixture. The other models are developed versions of Maxwell model.

Figure 10 Effects of inclination angle on local Nusselt number along hot wall (φ=0.015, Ha=30, Ri=1, M2)

Table 7 Effects of inclination angle number on average heat transfer (Nu_m) (φ=0.015, Ha=30, Ri=1, M2)

Figures 11–14 demonstrate average Nusselt number values in comparison with three electrical conductivity models with different values of varied parameters. The minimum values of Nu_m are seen on model 1 while the maximum values are on model 3 in almost all graphics. This difference is more pronounced when forced convection effects (Figure 11), nanoparticle volume fraction(Figure 14) and magnetic field effects (Figures 12 and 13) are higher (e.g., Ha=30–40, or γ=60°–90°).

Figure 11 Average Nusselt number values with varying Richardson numbers for all electrical conductivity models (Ha=30, φ=0.015, γ=45°)

Figure 12 Average Nusselt number values with varying Hartmann numbers for all electrical conductivity models (Ri=1, φ=0.015, γ=45°)

Figure 13 Average Nusselt number values with varying inclination angles for all electrical conductivity models (Ha=30, φ=0.015, Ri=1)

Figure 14 Average Nusselt number values with different particle volume fractions for all electrical conductivity models (Ha=30, Ri=1, γ=45°)

6 Conclusions

Numerical study for laminar mixed convection nanofluid flow over a backward-facing step under oriented magnetic field is carried out for three electrical conductivity models. It is expected that the variations of the flow conditions and by change of the thermo-physical properties, the heat transfer and fluid flow characteristics are changed. It is also expected that different electrical conductivity models result in variations of heat transfer rates. Simulations revealed the following results:

1) Increasing forced convection effects by means of decreasing Richardson number cause heat transfer enhancement for all particle volume fractions and electrical conductivity models. The maximum enhancement in average heat transfer is seen when particle volume fraction is 0.025 with 277.94% for model 1 (M1) 479.31% for model 2 (M2) and 496.18% for model 3 (M3).

2) Magnetic field has considerable effects on heat transfer and flow characteristics. Increasing magnetic field strength suppresses the flow and provides flow stability. These phenomena cause an enhancement on heat transfer for backward-facing step geometry in the present study. As a conclusion it can be said that increasing Hartmann number increases heat transfer. The maximum heat transfer enhancement between no-magnetic condition and Ha=40 condition is obtained for 0.025 concentration of nanofluid with 4.22% for model 1, 34.92% for model 2 and 46.98% for model 3.

3) Effects of applied magnetic field changes are due to the inclination angle. For horizontal magnetic field, flow characteristics and heat transfer are similar with non-magnetic condition because the magnetic field effects are very weak in this position. Increasing angle causes an increase in average heat transfer. The maximum values of heat transfer enhancement between horizontal and vertical magnetic field are obtained for 0.025 concentration of nanofluid with 4.96% for model 1, 56.65% for model 2 and 92.83% for model 3.

4) Adding nanoparticles increases heat transfer for all electrical conductivity models and for all values of variable parameters. The maximum enhancement on heat transfer is at the value of varied parameters: Ri=0.01, Ha=40 and γ=90°. Model 3 presents the highest results in heat transfer enhancement.

The results of the present study could be extended to include step ratio effects, different thermal boundary conditions, various nanofluid thermal conductivity models and unsteady effects.

Nomenclature

L

Characteristic length

ρ

Density, kg/m³

μ

Dynamic viscosity, kg/(m·s)

υ

Kinematic viscosity, m²/s

g

Gravity, m/s²

P

Pressure vector

Re

Reynolds number

Pr

Prandtl number

Ha

Hartmann number

Gr

Grashof number

Ra

Rayleigh number

Ri

Richardson number

A_R

Aspect ratio

E_R

Expansion ratio

B

Magnetic induction intensity vector

E

Electric current

H

Step height

J

Current density

A

Surface area

c_p

Specific heat, J/(kg·K^–1)

β

Thermal diffusion coefficient

φ

Volume concentration of nanoparticles

δ

Thermal boundary layer thickness

σ

Electrical conductivity

u, v

Velocity vector

h

Convection heat transfer coefficient, W/(m²·K)

k

Thermal conductivity ratio, W/(m·K)

T

Temperature, K

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(Edited by YANG Hua)

中文导读

导电性对纳米磁流体后向混合对流的影响

摘要：本文对纳米磁流体的几种电导率模型进行了数值模拟，研究了Al₂O₃-水纳米磁流体后向流动的混合对流效应。对流体施加均匀的外部磁场时，通过改变无量纲参数Hartmann数(Ha=0，10，20，30，40)实现磁场强度的变化。应用3种不同的电导率模型监测纳米磁流体的流动。同时，在研究过程中还选取了0°～90°范围内的5中倾角下的外部磁场。后向流动常用于求解流体分流和再合流现象的工程问题。根据不同电导率模型，采用不同的 Richardson数(Ri=0.01，0.1，1，10)和4种不同的纳米粒子体积分数(f=0.01，0.015，0.02，0.025)，对混合型后向流的对流换热进行研究。采用商业代码COMSOL有限元方法进行计算。结果表明，添加纳米颗粒增强了传热效果，增加磁场强度和倾角也增加了热传递速率。在对流体流动施加外部磁场的情况下，不同电导率模型对流体的流动和传热效果也有不同的影响。

关键词：导电性；纳米流体；后向台阶；磁流体流动；混合对流

Received date: 2018-07-20; Accepted date: 2018-10-11

Corresponding author: SELIMEFENDIGIL Fatih, PhD; Tel: +90-236-2012370; E-mail: fthsel@yahoo.com; ORCID: 0000-0002- 5453-2091