Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm MHD mixed convection in a vertical annulus ﬁlled with Al2O3–water nanoﬂuid considering nanoparticle migration A. Malvandi a,n, M.R. Safaei b, M.H. Kaffash a, D.D. Ganji c a Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran c Mechanical Engineering Department, Babol University of Technology, Babol, Iran b art ic l e i nf o a b s t r a c t Article history: Received 26 November 2014 Received in revised form 16 January 2015 Accepted 23 January 2015 Available online 24 January 2015 In the current study, an MHD mixed convection of alumina/water nanoﬂuid inside a vertical annular pipe is investigated theoretically. The model used for the nanoﬂuid mixture involves Brownian motion and thermophoretic diffusivities in order to take into account the effects of nanoparticle migration. Since the thermophoresis is the main mechanism of the nanoparticle migration, different temperature gradients have been imposed using the asymmetric heating. Considering hydrodynamically and thermally fully developed ﬂow, the governing equations have been reduced to two-point ordinary boundary value differential equations and they have been solved numerically. It is revealed that the imposed thermal asymmetry would change the direction of nanoparticle migration and distorts the velocity, temperature and nanoparticle concentration proﬁles. Moreover, it is shown that the advantage of nanoﬂuids in heat transfer enhancement is reduced in the presence of a magnetic ﬁeld. & 2015 Elsevier B.V. All rights reserved. Keywords: Nanoﬂuid Nanoparticle migration MHD Mixed convection Asymmetric heating 1. Introduction Economic incentives, energy saving and space considerations have increased efforts to construct a more efﬁcient heat exchange equipment. Many techniques have been presented by researchers to improve heat transfer performance, which is referred to as heat transfer enhancement, augmentation, or intensiﬁcation. Bergles [1] was the ﬁrst that classiﬁed the heat transfer enhancement techniques to (a) active techniques which require external forces to maintain the enhancement mechanism such as an electrical ﬁeld or vibrating the surface and (b) passive techniques which do not require external forces, including geometry reﬁnement [2], special surface geometries [3], or ﬂuid additives. Active techniques commonly present a higher augmentation thought they need additional power that increases initial capital and operational costs of the system. In this class, the study of the magnetic ﬁeld has important applications in medicine, physics and engineering. Many industrial types of equipment, such as MHD generators, pumps, bearings and boundary layer control are affected by the interaction between the electrically conducting ﬂuid and a magnetic ﬁeld. The behavior of the ﬂow strongly depends on the orientation and intensity of the applied magnetic ﬁeld. The exerted magnetic ﬁeld manipulates the suspended particles and n Corresponding author. Fax: þ98 21 65436660. E-mail address: amirmalvandi@aut.ac.ir (A. Malvandi). http://dx.doi.org/10.1016/j.jmmm.2015.01.060 0304-8853/& 2015 Elsevier B.V. All rights reserved. rearranges their concentration in the ﬂuid which strongly changes heat transfer characteristics of the ﬂow. The seminal study about MHD ﬂows was conducted by Alfvén who won the Nobel Prize for his works. Later, Hartmann did a unique investigation on this kind of ﬂow in a channel. Effects of MHD on nanoﬂuids have been considered by Sheikholeslami et al. [4–8] on free convection of nanoﬂuids in enclosures, Rashidi et al. [9,10] for entropy generation of nanoﬂuids over a rotating disk, Uddin et al. [11] for hydromagnetic transport of nanoﬂuids over a stretching sheet, and Malvandi et al. [12–15] for considering the effects of magnetic ﬁeld on nanoparticle migration. A good review on this subject is given by Bahiraei and Hangi [16]. Among different passive techniques, particles as additives in the working ﬂuids have burst onto the scene of engineering research which emerged in 1873 [17] and is developing rapidly. The motivation was to improve the thermal conductivity of the most common ﬂuids such as water, oil, and ethylene–glycol mixture, with the solid particles which have intentionally higher thermal conductivity. Then, many researchers studied the inﬂuence of solid–liquid mixtures on potential heat transfer enhancement. But, they were confronted with problems such as abrasion, clogging, fouling and additional pressure loss of the system, which makes these unsuitable for heat transfer systems. In 1995, the word “nanoﬂuid” was proposed by Choi [18] to indicate dilute suspensions formed by functionalized nanoparticles smaller than 100 nm in diameter which had already been created by Masuda et al. [19] as Al2O3–water. These nanoparticles are fairly close in size to the A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 Nomenclature B d cp DB Dh DT g h Ha HTC k k BO Ng Nr NBT Np p qw R T u x, r uniform magnetic ﬁeld strength nanoparticle diameter (m) speciﬁc heat (m2/s2 K) Brownian diffusion coefﬁcient hydraulic diameter (m) thermophoresis diffusion coefﬁcient gravity (m/s2) heat transfer coefﬁcient (W/m2 K) Hartmann number dimensionless heat transfer coefﬁcient thermal conductivity (W/m K) Boltzmann constant (¼ 1.3806488 10 23 m2 kg/s2 K) mixed convective parameter due to temperature gradient mixed convective parameter due to nanoparticle distribution ratio of the Brownian to thermophoretic diffusivities non-dimensional pressure drop pressure (Pa) surface heat ﬂux (W/m2) radius (m) temperature (K) axial velocity (m/s) coordinate system molecules of the base ﬂuid and, thus, can enable extremely stable [20] suspensions with only slight gravitational settling over long periods. Along with the same proposition, theoretical studies emerged to model the nanoﬂuid behaviors. In the beginning, the models were twofold: homogeneous models and dispersion models. In 2006, Buongiorno [21] stated that the homogeneous models are in disagreement with the experimental observations and tend to underpredict the nanoﬂuid heat transfer coefﬁcient. In addition, the dispersion effect is completely negligible due to the nanoparticle size. Thus, Buongiorno developed an alternative model to explain the abnormal convective heat transfer in nanoﬂuids and so eliminate the shortcomings of the homogeneous and dispersion models. He asserted that nanoparticle migration is responsible for the abnormal heat transfer rate in nanoﬂuids. Taking this ﬁnding as a basis, he proposed a two-component four-equation nonhomogeneous equilibrium model for convective transport in nanoﬂuids. Then, a comprehensive survey of convective transport of nanoﬂuids were conducted by Kuznetsov and Nield [22] to study inﬂuence of nanoparticles on natural convection boundary-layer ﬂow past a vertical plate, Goodarzi et al. [23] for two-phase simulation of nanoﬂuids in a shallow cavity, Soleimani et al. [24] for CFD simulation of free convection of nanoﬂuids in a semi-annulus enclosure, Safaei et al. [25] for heat transfer enhancement in a forward-facing contradicting channel, Malvandi et al. [26–30], and Garoosi et al. [31,32]. Recently, Buongiorno's model has been modiﬁed by Yang et al. [33,34] to fully account for effects of the nanoparticle volume fraction. Next, Malvandi et al. [35] considered the modiﬁed model for fully developed mixed convection of nanoﬂuids in a vertical annulus. They indicate that the modiﬁed model is suitable for considering effects of nanoparticle migration in nanoﬂuids. Then, the modiﬁed Buongiorno's model has been applied to different heat transfer concepts, including forced [36– 39], mixed [40–42], and natural convections [43,44]. In the current research, the effects of nanoparticle migration on hydromagnetic (MHD) mixed convection of alumina/water 297 Greek symbols θ ϕ γ η μ ρ σ ζ ε non-dimensional temperature nanoparticle volume fraction ratio of wall and ﬂuid temperature difference to absolute temperature transverse direction dynamic viscosity (kg/m s) density (kg/m3) electric conductivity radius ratio heat ﬂux ratio Subscripts B bf i o p bulk mean base ﬂuid condition at the inner wall condition at the outer wall nanoparticle Superscripts n dimensionless variable nanoﬂuid inside a vertical annular pipe is theoretically investigated. Since the thermophoresis is the main mechanism of the nanoparticle migration, different temperature gradients have been imposed using the assymetric heating. The effects of a uniform magnetic ﬁeld, asymmetric heating effects, the migration of nanoparticles, and how these affect the hydrodynamic and thermal characteristics of the system are of particular interest. 2. Migration of nanoparticles It is well known that nanoparticles do not passively follow the ﬂuid streamlines. Migration of nanoparticles has considerable effects on rheological and thermophysical properties of the nanoﬂuids. For considering the nanoparticle migration, it can be assumed that the suspended nanoparticles can homogeneously be in motion with the ﬂuid, considering a slip velocity relative to the ﬂuid. Because of the very small dimension of the nanoparticles (o100 nm), Brownian and thermophoretic diffusivities are the only signiﬁcant slip mechanisms which are responsible for nanoparticle migration in nanoﬂuids, as Buongiorno [21] stated. Brownian diffusion can be observed due to random drifting of suspended nanoparticles within the base ﬂuid which comes from continuous collisions between nanoparticles and liquid molecules. It is proportional to the concentration gradient and described by the Brownian diffusion coefﬁcient, DB, which is given by the Einstein–Stokes's equation DB = kB T 3πμ bf d p (1) where kB is the Boltzmann's constant, μbf is the dynamic viscosity of the base ﬂuid, T is the local temperature and dp is the nanoparticle diameter. The nanoparticle ﬂux due to Brownian diffusion ( Jp, B ) can be given as 298 A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 J p, B = − ρ pDB ∇ϕ (2) On the other hand, the thermophoresis (“particle” equivalent of the Soret effect), tries to induce the nanoparticles migration in the opposite direction of the temperature gradient (warmer to colder region), causing a non-uniform nanoparticle distribution. The thermophoresis is described by the thermal diffusion, DT, which is given by DT = β μ bf ρ bf ϕ (3) where β = 0.26(kbf /(2k bf + k p )). The nanoparticle ﬂux due to thermophoresis ( Jp, T ) can be calculated as Jp, T = − ρp DT ∇T T (4) Therefore, the total nanoparticle ﬂux consists of two parts as described above ⎛ ∇T ⎞ ⎟ Jp = Jp, B + Jp, T = − ρp ⎜DB ∇ϕ + DT ⎝ T ⎠ (5) Since D B ∼ T and DT ∼ ϕ (depend on the ﬂow ﬁeld), it is advantageous to re-write Eq. (5) as follows [45]: ⎛ ∇T ⎞ ⎟ Jp = − ρp ⎜CBT ∇ϕ + CT ϕ ⎝ T ⎠ (6) where CB = D B /T and CT = DT /ϕ do not depend on the ﬂow ﬁeld. As a result, distribution of nanoparticle can be obtained via ∂t (ϕ) + ∇⋅ (uϕ) = − ⎛ ∇T ⎞ 1 ⎟ ∇ J p = ∇⋅ ⎜CB T ∇ϕ + CT ϕ ⎝ ρp T ⎠ ( ) (7) 3. Problem formulation and governing equations Consider an MHD, laminar and two-dimensional ﬂow of the alumina/water nanoﬂuid inside a vertical annular pipe, which is subjected to different heat ﬂuxes at the inner (qi″) and outer (qo″) qi″/qo″, which characterizes walls. The ratio of the heat ﬂuxes is ε = the degree of the thermal asymmetry. The geometry of the problem is shown in Fig. 1. A two-dimensional coordinate frame has been selected where the x-axis is aligned vertically and the r-axis is normal to the walls. A modiﬁed two-component heterogeneous model is employed for the nanoﬂuid in the hypothesis that the Brownian motion and the thermophoresis are the only signiﬁcant bases of nanoparticle migration. This model involves the following assumptions: incompressible ﬂow, no chemical reactions, dilute mixture, negligible viscous dissipation, negligible radiation, and local thermal equilibrium between the nanoparticles and base ﬂuid. Consequently, the basic incompressible conservation equations of the mass, momentum, thermal energy, and nanoparticle fraction can be expressed in the following manner [21,35]: Continuity equation ∂t (ρ) + ∇⋅ (ρu) = 0 (8) Fig. 1. The geometry of physical model and coordinate system. τ = μ (∇u + (∇u)t ) is the shear stress, β is the nanoﬂuid thermal expansion, g is the gravity, and q is the energy ﬂux relative to the nanoﬂuid velocity, which can be expressed as the sum of the conduction and diffusion heat ﬂux as below q= ) (ϕ − ϕB ) ⎤⎥⎦ g is (11) = ϕρp + (1 − ϕ) ρ bf cp ϕρp c p p + (1 − ϕ) ρ bf c pbf ρ , k = k bf (1 + 7.47ϕ), β ∂t (ρcT ) + ∇⋅ (ρc uT ) = − ∇⋅q + h p ∇⋅J p hp ⏟ ρ (9) Energy equation where h p Jp nanoparticle diffusion heat flux μ = μ bf (1 + 39.11ϕ + 533.9ϕ2), = ( + Further, ρ , μ, k , c are the density, dynamic viscosity, thermal conductivity, and speciﬁc heat capacity of alumina/water nanoﬂuid respectively, depending on the nanoparticle volume fraction as follows: Momentum equation ∂ t (ρu) + ∇. (ρuu) = − ∇p + ∇⋅τ − σB 2u ⎡ + ⎢⎣ (1 − ϕ B ) ρbf β (T − TB ) − ρp − ρbf k∇ T − conduction heat flux the speciﬁc (10) enthalpy of nanoparticles, = ϕρp βp + (1 − ϕ) ρ bf βbf ρ (12) 299 A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 Table 1 Validation of the results with the ones reported by Kays and Crawford [46] when Ha = Nr = Ng = ϕ B = 0 . ζ ε Kays and Crawford [46] Present study HTCo HTCi HTCi HTCo 0.2 0 0.5 1 1.5 0 10.493 89.463 21.426 4.883 5.151 5.450 5.787 0 10.513 88.712 21.397 4.883 5.150 5.449 5.784 0.6 0 0.5 1 1.5 0 109.481 11.218 8.635 5.099 5.812 6.758 8.071 0 115.490 11.248 8.646 5.099 5.813 6.759 8.072 1a 0 0.5 1 1.5 0 17.484 8.234 7.000 5.385 6.511 8.234 11.195 0 17.500 8.235 7.000 5.385 6.512 8.235 11.200 a In order to avoid singularity at ζ ¼1, the results are obtained at ζ ¼0.99999. Table 2 Comparison of NuB with the reported data of Yang et al. [33] when Nr = Ha = γ = Ng = 0, ϕ B = 0.02. NBT Yang et al. Present work Error (%) 0.2 0.4 0.6 0.8 1 2 4 6 8 10 10.359 10.238 10.189 10.163 10.146 10.11 10.09 10.086 10.082 10.08 10.3591 10.239 10.19 10.1635 10.1467 10.111 10.092 10.0861 10.0828 10.081 0.0010 0.0098 0.0098 0.0049 0.0069 0.0099 0.0198 0.0010 0.0079 0.0099 Table 3 Grid independence test for different values of dη when Ha ¼0, NBT ¼ 1, ϕ B = 0.02, ε¼ 0.5, ζ¼ 0.4, and Nr ¼ Ng ¼50. where bf stands for base ﬂuid and p for particle. In addition, the thermophysical properties of Al2O3 nanoparticle and base ﬂuid (water) are also provided as follows: ⎧ ⎪ ⎪ c p p = 773 J /kg K , = 998.2 kg/m3 , ⎪ ⎪ ρp = 3880 kg/m3 , Al 2O 3⎨ = 0.597 W /m K, ⎪ k p = 36 W /m K , ⎪ = 9.93 × 10−4 kg/m s, ⎪ β = 8.4 × 10−6 1/K ⎪ p 4 − = 2.066 × 10 1/K ⎩ k bf μ bf β bf 10−5 25.8497 8.1136 148.0673 5 × 10−6 25.8520 8.1124 148.0666 10−6 25.8573 8.1109 148.0641 HTCo Np dp 1 d ⎛ du ⎞ − σB 2u + ⎡⎣ (1 − ϕ B ) ρβ (T − TB ) − (ρ − ρ bf )(ϕ − ϕ B ) ⎤⎦ g = 0 ⎜ rμ ⎟ − r dr ⎝ dr ⎠ dx (14) ρc p u c pbf = 4182 J /kg K, ρbf HTCi equating ∇h p = c p ∇T , one may simply obtain governing equations for steady, incompressible, hydrodynamically and thermally fully developed ﬂow as follows: Fig. 2. Algorithm of the numerical method. ⎧ ⎪ ⎪ ⎪ ⎪ Water⎨ ⎪ ⎪ ⎪ ⎪ ⎩ dη ⎛ ∂ϕ DT ∂T ⎞ ∂T dT 1 d ⎛ dT ⎞ ⎟ ⎜rk ⎟ + ρp c p p ⎜DB + = ⎝ ∂r dx r dr ⎝ dr ⎠ T ∂r ⎠ ∂r CT ϕ ∂T ⎞ 1 ∂ ⎛ ∂ϕ + ⎟=0 ⎜CB T r ∂r⎝ T ∂ r⎠ ∂r (13) Since the rheological and thermophysical properties ( ρ , μ, k and c ) are dependent to the nanoparticle concentration, the nanoparticle distribution equation, Eq. (7), should be coupled with Eqs. (8)–(10). Thus, substituting Eq. (11) into Eq. (10) and (15) (16) According to Buongiorno [21], heat transfer associated with nanoparticle diffusion (second RHS term of Eq. (15)) can be neglected in comparison with the other terms. In addition, By averaging Eq. (15) from r = Ri to Ro and according to the thermally fully developed condition for the uniform wall heat ﬂux (dT /dx = dTB/dx ) and introducing the following non-dimensional 300 A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 parameters: η= 2r , Dh θ= T − Ti , (qo + qi ) D h u⁎ = Eqs. (14)–(16) can be reduced as u , D h2/ μbf ( − dp/ dx) γ= ⎛1 d2u⁎ 1 dμ dϕ ⎞ du⁎ =−⎜ + ⎟ 2 μ dϕ dη ⎠ dη ⎝η dη μ bf ⎡1 − Nr ϕ − ϕ + Ngρ⁎ β ⁎ (θ − θ ) − Ha2u⁎⎤ − B ( ⎣ ⎦ B) 4μ (qo + qi ) D h TB kbf kbf ⎤ ⎡ βp 1 1 ⎥, β⁎ = ⎢ + ⎢⎣ 1 + ((1 − ϕ) ρ bf / ϕρ p ) β bf 1 + (ϕρ p /(1 − ϕ) ρ bf ) ⎥⎦ q″ ε= i, qo″ k bf d2θ =− 2 k dη ρ⁎ = (1 − ϕ) + ϕρ p / ρ bf (D h (qo + qi )/ kbf ) ρ bf β bf g , Ng = −dp/ dx Nr = (ρ − ρ bf ) g Ha2 = − (dp/ dx) σB 2D h2 μbf , ⎡ ρcu⁎ ⎛ k/k bf ⎞ dθ ⎤ (1 + ζε) dϕ ⎢− ⎟ ⎥ + ⎜7.47 + ⎢⎣ ρcu⁎ (1 + ζ)(1 + ε) ⎝ η ⎠ dη ⎥⎦ dη ∂θ ∂ϕ ϕ =− ∂η NBT [1 + γθ]2 ∂η Ri , ζ= Ro (17) (18) (19) (20) where the average value of parameters can be calculated over the cross-section by Fig. 3. The effects of NBT (ϕ B = 0.06 ) and ϕ B (NBT ¼0.5) on nanoparticle distribution (ϕ/ϕ B ), velocity (u/uB) and temperature (θ/θB) proﬁles when ε ¼0.5, Nr ¼ 50, Ng¼ 50, Ha¼ 5, and ζ ¼0.6. A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 301 Fig. 4. The effects of Ng and Nr (Ha¼ 5) and Ha (Nr ¼Ng ¼ 50) on nanoparticle distribution (ϕ/ϕ B ), velocity (u/uB) and temperature (θ/θB) proﬁles when ε¼ 0.5, ϕ B = 0.06 , NBT ¼ 0.5, and ζ¼ 0.6. Γ ≡ 1 A ∫A Γ dA= 1 π( R o2 − Ri2 ) ∫R Ro Γ (2πr) dr i (21) T B⁎ , Hence, the bulk mean dimensionless temperature and the bulk mean nanoparticle volume fraction ϕ B can be obtained by θB ≡ ρc puθ ρc pu , ϕB = u⁎ϕ u⁎ the solid boundary (no-slip condition). Different heat ﬂuxes are taken into account at the walls, qi″ for the inner wall and the heat ﬂux at outer wall is considered to be qo″. As a result, appropriate boundary conditions for this problem can be expressed as r = Ri : u = 0, (22) 3.1. Boundary conditions The ﬂuid velocity at all ﬂuid–solid boundaries is equal to that of − ki ∂T = qi″, ∂r CT ϕ ∂T ∂ϕ + = 0. r = R o: u = 0, T ∂r ∂r CT ϕ ∂T ∂T ∂ϕ ko = qo″, CB T + = 0. T ∂r ∂r ∂r CB T (23) Substuting Eq. (20) into Eq. (26), the boundary conditions can be expressed as 302 A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 Fig. 5. The effects of ζ (ε¼ 0.5) and ε (ζ ¼ 0.6) on nanoparticle distribution (ϕ/ϕ B ), velocity (u/uB) and temperature (θ/θB) proﬁles when ϕ B = 0.06 , Nr ¼50, Ng ¼50, Ha ¼5, and NBT ¼0.5. η= ζ : u⁎ = 0, 1−ζ ϕ = ϕw = ϕi η = ε ∂θ , =− ∂η 2 (1 + 7.47ϕi ) (1 + ε) 1 : u⁎ = 0, 1−ζ 1 ∂θ = ∂η 2 (1 + 7.47ϕo ) (1 + ε) (24) 4. Numerical method and accuracy Eqs. (18)–(20) are solved in conjunction with the boundary conditions of Eq. (24) by means of the Runge–Kutta–Fehlberg scheme. Convergence criterion is considered to be 10 6 for relative errors of the velocity, temperature, and nanoparticle volume fraction. The numerical procedure involves a reciprocal algorithm in which ϕw , ρcu⁎, and θ B are used to solve the governing equations. The process is repeated until a prescribed value of ϕ B reached, and the relative errors between the assumed values of ρcu⁎ and θ B with the calculated ones after solving Eqs. (18)–(20) are lower than 10 6. In view of helping others to regenerate their own results and provide possible future references, the numerical algorithm is shown graphically in Fig. 2. To check the accuracy of the numerical code, the results obtained for a horizontal annulus with Nr ¼ ϕB ¼Ha¼ Ng¼ 0 and different values of ε and ζ are compared to the reported results of Kays and Crawford [46] in Table 1. Further, a comparison for HTCt (kbf /k B ) with the reported results of Yang et al. [33] is presented in Table 2. Obviously, the maximum percentage difference is less than 0.02%; so, the results are in a desirable accuracy. In A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 Fig. 6. The effects of NBT and ϕ B on the total heat transfer coefﬁcient (a), and the pressure drop (b) when ε¼ 0.5, Nr ¼50, Ng¼ 50, Ha¼ 5, and ζ ¼0.6. addition, the numerical code developed here is run on three different integration steps (dη ) of 10 5, 5 10 6, and 10 6 to verify the results are independent of the grid size. The obtained numerical results are presented in Table 3. The results clearly indicate the grid independent of the code. Accordingly, all the numeric results obtained here are carried out using the integration step dη = 10−6 . 5. Results and discussions Migration of nanoparticles, the viscosity and thermal conductivity distributions are determined by the mutual effects of the Brownian diffusion and the thermophoresis. Here, these effects are considered by means of NBT, which is the ratio of the Brownian diffusion to the thermoporesis. With d p ≅ 20 nm and ϕ B ≅ 0.1, the ratio of Brownian motion to thermophoretic forces NBT ∝ 1/d p can be changed over a wide range of 0.2–10 for alumina/water nanoﬂuid. In addition, the results have been carried out for γ ≅ (Tw − TB )/Tw = 0.1, since its effects on the solution is 303 Fig. 7. The effects of Ha and Nr on the total heat transfer coefﬁcient (a), and the pressure drop (b) when ε ¼0.5, ϕ B = 0.06 , NBT ¼ 0.5, Ng ¼50, and ζ¼ 0.6. insigniﬁcant, see Refs. [33,34]. 5.1. Velocity, temperature and concentration proﬁles The effects of NBT on the nanoparticle volume fraction (ϕ/ϕ B ), velocity (u/uB), and temperature (θ/θB) proﬁles are shown in Fig. 3a. For the lower values of NBT, the nanoparticle volume fraction ejects themselves at the heated walls and accumulate at the core region. In contrast, at the higher values of NBT, a more uniform nanoparticle distribution can be obtained. Migration of the nanoparticles from the walls toward the core region at the lower values of NBT constructs nanoparticle-depleted regions near the walls. This reduces the viscosity, and so reduces the shear stress of the nanoﬂuid on the walls. However, the viscosity and shear stress increase in the core region. Hence, the velocity near the walls increases, especially on the inner wall, which has a lower nanoparticle volume fraction. Furthermore, it can be easily seen that the nanoparticle concentration takes its lowest value at the inner wall (the higher wall heat ﬂux), a slight increase to the maximum in the region far from the wall, but decreased rapidly 304 A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 Fig. 8. The effects of ζ and ε on the total heat transfer coefﬁcient (a), and the pressure drop (b) when ϕ B = 0.06 , Nr ¼50, Ng¼ 50, Ha¼ 5, and NBT ¼ 0.5. towards the outer wall (the lower wall heat ﬂux). This is purely because the heat ﬂux at the inner wall is higher than at the outer wall; so the temperature gradient and the thermophoresis are greater at the inner wall. It is not surprising that the peak of the nanoparticle concentration is closer to the outer wall, having a lower heat ﬂux. The variations in the nanoparticle volume fraction (ϕ/ϕ B ), velocity (u/uB), and temperature (θ/θB) proﬁles for different values of ϕ B are shown in Fig. 3b. There is a downward trend for the volume fraction of nanoparticles in the core region as well as a reversed behavior for that at the walls, as increasing ϕ B . In other words, an increase in ϕ B leads to a more uniform nanoparticle volume fraction distribution. Also, it can be observed that for the lower values of ϕ B , the nanoparticle depletion effects become signiﬁcant and the velocities move further to the outer walls. Fig. 4a shows the effects of mixed convective parameters due to temperature gradient (Ng) and nanoparticle distribution (Nr) on the proﬁles. When Ng and Nr increase, peak of the velocity no longer remains at the core region, but moves toward the heated walls, particularly near the outer wall where the nanoparticle concentration is the lowest. This is because increasing Nr and Ng, intensify the buoyancy forces, which accelerate the momentum near the walls and due to a constant mass ﬂow rate inside the annulus, the velocities in the core region decrease. Accordingly, the nanoparticle concentration and temperature gradients reduce; so, their proﬁles become more uniform, as mixed convective parameters increase. An examination of Fig. 4b reveals a continuing increase in the velocities of the ﬂuid close to the outer wall (lowest nanoparticle concentration), followed by a decrease in the core region, as Ha increases. In essence, the velocities are forced to move slowly close to the outer wall (low viscosity region). This is because the transverse magnetic ﬁeld induces a resistive type force (Lorentz force), which is a retarding force on the velocity ﬁeld. Thus, the velocities reduce at the core region and due to a constant mass ﬂow rate, the velocities near the walls should increase. The increment of the velocities is more likely to take place near the outer wall which has a lower viscosity region. The effects of radii (ζ) and heat ﬂuxes (ε) ratios on the proﬁles have been demonstrated in Fig. 5a and b, respectively. As ζ increases, the effects of inner wall is increased, which strength the effects of viscous forces; so the velocities shift toward the inner wall. As a result, the temperature gradients are reduced. Regarding Fig. 5b, for ε < 1 the nanoparticle rich region is constructed near the inner wall, having a lower heat ﬂux. Increasing ε shifts the peak of the nanoparticle volume fraction toward the outer wall. This is due to the fact that the thermophoresis, which is related to the temperature gradient, is the mechanism of the nanoparticle migration. Any change in ε leads the temperature gradient at the walls to change, so changes the thermophoresis. For ε < 1, the temperature gradient at the outer wall is more than that at the inner wall; so the nanoparticle migration at the outer wall is greater, leading the nanoparticle accumulated region to move toward the inner wall. This phenomena continues until ε = 0, in which there is no temperature gradient at the inner wall; so a nanoparticle accumulation region formed at the inner wall. The effects of ε on nanoparticles distribution have considerable inﬂuence on the velocity and temperature proﬁles. Evidently, a regular symmetry in the velocity proﬁle disappears and the peak of the velocity proﬁle moves toward the outer wall (the lower viscosity region), as ε decreases. However, the dip point of the temperature proﬁle increases and moves toward the inner wall in which the nanoparticles accumulated (the higher thermal conductivity region). In fact, the velocity proﬁle has a tendency to shift toward the nanoparticle depleted region, however, for the temperature proﬁle it is the opposite. 5.2. Heat transfer rate and pressure drop The dimensionless heat transfer coefﬁcient (HTC) at the inner and the outer walls can be deﬁned respectively as HTCi = qi″ Dh hi Dh ε = =− k bf (Ti − TB ) k bf (1 + ε) θ B (25) HTCo = qo″ Dh ho D h 1 = = k bf (To − TB ) k o (1 + ε)(θo − θ B ) (26) The total heat transfer ratio can be expressed as HTCt = HTCi Ri + HTCo R o HTCi ζ + HTCo = Ri + R o ζ+1 (27) and the non-dimensional pressure drop can be deﬁned as ⎛ dp ⎞ ⎛ μ bf uB ⎞ ρ ⎟= B Np = ⎜− ⎟/⎜⎜ ⎝ dx ⎠ ⎝ Dh2 ⎟⎠ ρu⁎ (28) Figs. 6–8 show the effects of the parameters ϕ B , NBT, Nr, Ha, ζ, A. Malvandi et al. / Journal of Magnetism and Magnetic Materials 382 (2015) 296–306 and ε on the total heat transfer rate and the pressure drop, respectively. In addition, the solid line represents the corresponded value for the base ﬂuid. It is shown in Fig. 6a and b that increasing NBT intensiﬁes the heat transfer rate, however, for the pressure drop it is the opposite. Furthermore, increasing ϕ B leads to a rise in both the heat transfer rate and the pressure drop. In fact, the total heat transfer coefﬁcient and the pressure drop for the nanoﬂuid are always greater than that of the base ﬂuid. Fig. 7a and b shows the effects of Nr and Ha on the heat transfer rate and the pressure drop. For nanoﬂuids, increasing Nr reduces the heat transfer rate, whereas the pressure drop is increased. Accordingly, Nr has a negative effects on the performance. When Ha increases, the heat transfer rate of the base ﬂuid is increased, while for the nanoﬂuid it is vice versa. Therefore, it can be concluded that there is no merit in coupling of the magnetic ﬁeld and nanparticles inclusion. In other words, inclusion of nanoparticles would decrease the heat transfer rate of the ﬂuids in the presence of the magnetic ﬁeld. The effects of radii ratio ζ and heat ﬂux ratio ε on heat transfer rate and pressure drop are shown in Fig. 8a and b. It is obvious that the heat transfer rate is very sensitive to ζ for ε o1. In this range, inclusion of nanoparticles enhanced the heat transfer rate, which is more apparent for the lower values of ζ. It can be observed that for the lower values of ε and ζ, where the heat ﬂux and surface at the outer wall is relatively greater than that at the inner wall, the heat transfer rate becomes negative (direction of the heat transfer rate has been changed at the inner wall). Increasing ζ at a constant value of ε, enhances the heat transfer rate. Further rise in ζ, increases the effects of viscous forces and decreases the velocities; so, the heat transfer rate is reduced. In conclusion, there is an optimum value for ζ at ε o1, which results in a greatest heat transfer coefﬁcient. In contrast, for ε 41, the heat transfer rate has an increasing trend with ε and ζ. In addition, the pressure drop is almost reduced with increasing ζ, except for the higher values of ε, as it can be observed in Fig. 8b. 6. Summary and conclusions An MHD mixed convection of Al2O3–water nanoﬂuid inside a vertical annulus is theoretically investigated. Walls are subjected to different heat ﬂuxes; qi″ for the inner wall and qo″ for the right wall, and nanoparticles are assumed to have a slip velocity relative to the base ﬂuid. Assuming a fully developed ﬂow and heat transfer, the basic partial differential equations including continuity, momentum, and energy equations have been reduced to two-point ordinary boundary value differential equations before they are solved numerically. The effects of different parameters including the ratio of Brownian motion to thermophoretic diffusivities NBT, the ratio of heat ﬂuxes at the walls ε, Hartmann number Ha, and bulk mean nanoparticle volume fraction ϕ B on the heat transfer rate and the pressure drop were investigated in detail. The major ﬁndings of this paper are as follows: The imposed thermal asymmetry would change the direction of nanoparticle migration and distorts the velocity, temperature and nanoparticle concentration proﬁles. Accordingly, the heat transfer rate and the pressure drop for the nanoﬂuids are signiﬁcantly sensitive to the imposed thermal boundary condition. The nanoparticles are more likely to accumulate toward the wall with a lower heat ﬂux. As a result, an in-homogeneous distribution of nanoparticles developed which leads to a nonuniform distribution of the viscosity and thermal conductivity of nanoﬂuids. Thus, the velocities shift toward the nanoparticle depleted region, however, the dip point of the temperature 305 proﬁles move toward the nanoparticle accumulated region. Inclusion of nanoparticles in the presence of a magnetic ﬁeld has a negative effect on the performance. There is an optimum value for ζ at ε o1, which results in a greatest heat transfer rate. 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