J Therm Anal Calorim DOI 10.1007/s10973-016-5560-1

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside a vertical microannulus in the presence of heat generation/absorption S. A. Moshizi1 • A. Malvandi2

Received: 19 December 2015 / Accepted: 14 May 2016  Akade´miai Kiado´, Budapest, Hungary 2016

Abstract Intensity and direction of nanoparticle migration are able to tune thermophysical properties of nanofluids to improve the thermal performance of heat exchange equipments. The intensity strongly depends on nanoparticle diameter and cannot be actively controlled while different directions of nanoparticle migration are achieved with asymmetric heating. In the current study, the mixed convective heat transfer of Al2O3–water nanofluid inside a vertical microannulus is investigated theoretically considering different modes of nanoparticle migration. The model employed for the nanoparticle–fluid mixture is able to fully account for the effects of nanoparticle slip velocity relative to the base fluid originating from the thermophoresis (nanoparticle slip velocity due to temperature gradient) and Brownian motion (nanoparticle slip velocity due to concentration gradient). To consider surface roughness in the microannulus, Navier’s wall slip condition is employed at the solid–fluid interface. It is revealed that the asymmetric heating at the walls alters the orientation of nanoparticle migration and deforms the symmetry of the flow field. In addition, despite temperature-dependent buoyancy forces, concentration-dependent buoyancy forces have considerable effects on the flow fields and nanoparticle migration.

List of symbols cp Specific heat (m2 s-2 K-1) d Nanoparticle diameter (m) DB Brownian diffusion coefficient DT Thermophoresis diffusion coefficient g Gravity (m s-2) h Heat transfer coefficient (kg s-3 K-1) k Thermal conductivity (kg m s-3 K-1) kBO Boltzmann constant (¼ 1:3806488  1023 m2 kg s-2 K-1) L Annulus length (m) N Slip velocity factor (m3 s-1) Nu Nusselt number NBT Ratio of the Brownian to thermophoretic diffusivities Nr Mixed convection parameter p Pressure (kg m-1 s-2) Q0 Dimensional heat generation or absorption coefficient (W m3 K-1) 00 q Surface heat flux (kg s-3) R Radius (m) T Temperature (K) u Axial velocity (m s-1) x, r Coordinate system

Keywords Nanofluid  Nanoparticles migration  Mixed convection  Thermal asymmetry  Microannulus  Modified Buongiorno’s model

Greek symbols / Nanoparticle volume fraction c Ratio of wall and fluid temperature difference to absolute temperature g Transverse direction b Volume expansion coefficient (K-1) h Non-dimensional temperature l Dynamic viscosity (kg m-1s-1) q Density (kg m-3) k Slip parameter

& A. Malvandi [email protected] 1

Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran

2

Young Researchers and Elite Club, Karaj Branch, Islamic Azad University, Karaj, Iran

123

S. A. Moshizi, A. Malvandi

Subscripts B Bulk mean bf Base fluid p Nanoparticle ref Reference value wi Condition at the inner wall wo Condition at the outer wall

Karimipour et al. [8], and Yadav et al. [9, 10]. Theoretical investigation on the effects of nanofluids has been systematically reported and well documented, which can be found in the open literature [11–16]. In addition, rigorous readers can find a comprehensive review of the last experimental studies on nanofluids in the literature such as [17, 18]. Nanoparticle migration effects

Superscripts * Dimensionless variable

Introduction Nanofluids are principally used as thermal enhancing agent and have many purposes in chemical processes, tribological applications, medical applications (drug delivery), pharmaceutical applications, surfactant, and coating. They are also employed extensively for cooling in heat exchangers, solar PV, microelectromechanical systems (MEMs). In order to predict the nanofluids behavior into convective heat transfer, various models are proposed by many researchers over the last years. The models, however, depend on certain inputs from experimentation. Each of them gives close agreement with the experimental data which is referred as the ‘‘best model’’ by those authors. The proposed models are commonly classified according to slip velocity between the base fluid and nanoparticles into two general types: homogeneous models (homogeneous suspensions) and dispersion ones. The homogeneous model represents that no slip velocity exists between the base fluid and nanoparticles [1]. In 2006, Buongiorno [2] theoretically demonstrated that the homogeneous model is in disagreement with the experimental findings and it underestimates the heat transfer coefficient of nanofluids; moreover, because of the nanoscale size of particles, the dispersion effect is negligible. Therefore, nanoparticle fluxes were considered by Buongiorno in accordance with the seven slip mechanisms: inertia, fluid drainage, Brownian diffusion (or Brownian motion), thermophoresis (or thermophoretic diffusion), Magnus effect, diffusiophoresis, and the gravity. Buongiorno claimed that the thermophoresis and Brownian diffusion are the crucial slip mechanisms in nanofluids and the other slip mechanisms can be neglected. Accordingly, a two-component (solid and fluid) four-equation (continuity, momentum, energy, and nanoparticle flux) non-homogeneous equilibrium model was proposed by Buongiorno to model the convective heat transport in nanofluids. Many researchers, then, have studied convective heat transfer in nanofluids after taking Buongiorno’s model into consideration in different geometries, for instance Sheremet et al. [3–5], Sheikholeslami et al. [6], Bahiraei and Mashaei [7],

123

In 2013, Buongiorno’s model was modified by Yang et al. [19, 20] to fully account for the effects of the nanoparticle volume fraction distribution. This model does not ignore the dependency of thermophysical properties of nanofluids (including thermal conductivity and viscosity) to nanoparticles concentration. In other words, non-uniformity of the thermophysical properties has been considered. Malvandi et al. [21], later, used the modified model to examine the mutual impacts of buoyancy and nanoparticle migration on the mixed convection of nanofluids in vertical annuli. Then, Malvandi and Ganji [22] investigated the impacts of nanoparticle migration as well as asymmetric heating at the walls on forced convective heat transfer of MHD alumina/ water nanofluid in microchannels. Bahiraei [23] employed the two-phase Euler–Lagrange method to study heat transfer characteristics of the CuO–water nanofluid in a straight tube under a laminar flow regime and concluded that the thermophoretic and Brownian forces affect the convective heat transfer, especially at greater distances from the tube inlet. Moshizi et al. [24] studied the convective heat transfer and pressure drop characteristics of Al2O3–water nanofluid inside a concentric pipe. Their results indicated that the nanoparticles move from the wall with higher heating energy toward the wall with lower heating energy (along the temperature gradient) due to the thermophoretic force. Hedayati and Domairry [25, 26] investigated the effects of nanoparticle migration on titania/water nanofluids in horizontal and vertical channels. The popularity of nanoparticle migration modeling can be gauged from the numerous published literatures such as [27–29]. Motivation The study of the flow with internal heat generation/absorption in nanofluids is of special interest and has many practical applications in manufacturing processes in industry. The effects of heat generation/absorption on thermal convection are significant where there is high temperature difference between the surface and the ambient fluid. Possible heat generation also alters the temperature distribution; for instance, the most conspicuously practical applications include the particle deposition rate in nuclear reactors, electronic chips, and semiconductor wafers. In this paper, an analysis is presented for the

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside…

laminar fully developed mixed convection of alumina/ water nanofluid inside a concentric microannulus which is subjected to a constant volumetric internal heating. This can be accomplished, for example, by external radiation as well as viscous dissipation, or by a neutron flux in a lithium blanket that occurs in fusion reactors. The nanoparticle volume fraction distribution is obtained considering the nanoparticle fluxes originating from the Brownian diffusion and thermophoresis. As the thermophoresis is the key mechanism of the nanoparticle migration, different temperature gradients are imposed by different wall heat fluxes: q00wi for the inner wall surface and q00wo for the outer one. Because of low-dimensional structures in microtubes, a linear slip condition is considered at the surfaces, which appropriately represents the non-equilibrium region near the fluid/solid interface. The impacts of a heat generation/ absorption, asymmetric heating, and different modes of nanoparticle migration on the thermal performance are of particular interest.

Problem formulation Let us consider a steady and laminar flow of alumina/water nanofluid inside a vertical concentric microannulus in the presence of heat source/sink. The physical model is illustrated in Fig. 1, where the Cartesian coordinates x and r are aligned parallel and normal to the walls, respectively. The walls are heated uniformly by external means at a rate of q00wo and q00wi the outer and inner walls, respectively. The ratio of the heat fluxes is 0 \ e = q00wi =q00wo \ 1, which characterizes

• • • • • • •

Incompressible flow, No chemical reactions, Negligible external forces, Dilute mixture, Negligible viscous dissipation, Negligible radiation, Local thermal equilibrium between the nanoparticles and base fluid.

Accordingly, the basic incompressible conservation equations can be expressed in the following manner [2, 31]:   1d du dp  rl  þ ð1  /wo Þqbf 0 bðT  TB Þ r dr dr dx    ð1Þ  qp  qbf 0 ð/  /wo Þ g ¼ 0   dT 1 d dT  rk qcp u  Q0 ðT  TB Þ dx r dr dr   o/ DT oT oT þ ¼0 ð2Þ  qp cp DB or T o r or   1 o o/ DT oT DB þ ¼0 ð3Þ r or or T or In Eqs. (1)–(3), u, T, and P represent the axial velocity, local temperature, and pressure, respectively. Further, the Brownian diffusion coefficient DB and thermophoretic diffusion coefficient DT are defined by DB ¼

Ri

Ro g ″ qwi

Symmetry Line

the degree of the thermal asymmetry. From the numerical solutions provided by Koo and Kleinstreuer [30] for the most standard nanofluid flows inside a channel of around 50 lm, the viscous dissipation can be neglected. Also, the employed nanofluid model has the following assumption:

heat sink /source

x

″ qwo

kBO T ; 3plbf dp

DT ¼ 0:26

kbf lbf /; 2kbf þ kp qbf

ð4Þ

where kBO is the Boltzmann constant and dp is the nanoparticle diameter. Further, q; l; k and c are the density, dynamic viscosity, thermal conductivity, and specific heat capacity of water/alumina nanofluid, respectively, which depend on the nanoparticle volume fraction. Their correlations are given in Table 1. In addition, the thermophysical properties of alumina nanoparticles and base fluid (water) are also provided in Table 2. Table 1 Relations of thermophysical properties

r, η

Nanofluid Flow

Fig. 1 Geometry of physical model and coordinate system

Thermophysical properties

Relation

Viscosity

l = lbf(1 ? 39.11/ ? 533.9/2)

Thermal conductivity

k = kbf(1 ? 7.47/)

Density Heat capacity

q = /qp ? (1 - /)qbf c¼

/qp cp þð1/Þqbf cbf q

Thermal expansion

b¼

/bp qp þð1/Þqbf bbf q

123

S. A. Moshizi, A. Malvandi Table 2 Thermophysical properties of the base fluid and the nanoparticles Physical properties

Fluid phase (water)

Al2O3

cp /J kg-1 K-1

4182

773

q/kg m

998.2

3880

k/W m-1 K-1

0.597

36

l 9 104/kg m-1 s-1 b/K-1

9.93 2.066 9 10-4

– 8.4 9 10-6

-3

The appropriate boundary conditions can be expressed as lwi du oT o/ DT oT ; k ¼ q00wi ; DB þ ¼ 0: or or qwi dr T or l du oT o/ DT oT r ¼ Ro :u ¼ N wo ; k ¼ q00wo ; DB þ ¼ 0: or or qwo dr T or ð5Þ r ¼ Ri :u ¼ N

and material properties of typical water-based nanofluid with alumina nanoparticles, from which one can calculate the coefficients of the governing equations. Considering Eq. (1), the scale analysis can be written as follows Dp lDU  ; ð1  /Þqbf 0 gbDT ; ðqp  qbf 0 ÞgD/ Lref L2ref |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |ffl{zffl} 101 2:88104

ð6Þ

3:52103

From Eq. (6), it can be deduced that the buoyancy effect due to the temperature gradient (second RHS term) can be neglected with respect to the buoyancy effect due to nanoparticles concentration gradient (third RHS term) and also the shear stress terms (first RHS term). Thus, this term can be removed from Eq. (1). Accordingly, the scale analysis for Eq. (2) can be expressed as

LHS

RHS 8 DT > > > RHS1 ¼ kbf ð1 þ 7:47/B Þ L2 > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflref ffl} > > > > 6:1109 > > > < RHS2 ¼ QDT

U DT P > |ffl{zffl} ref  /B qp cpp þ ð1  /B Þqbf cpbf 1109 > Dx 1 0 > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > 2:08108 > > B DB D/ > DT DT C > C DT > RHS3 ¼ qp cpp B þ > @ > > Lref Tref Lref A Lref |ffl{zffl} |fflffl{zfflffl} > |fflfflffl{zfflfflffl} |{z} : 6 310

where N represents slip velocity factor. It can be seen that in the modified Buongiorno model [Eqs. (1)–(3)], the viscosity and thermal conductivity of nanofluids are both depended on the nanoparticle volume fraction. On the other hand, the nanoparticle volume fraction is non-uniformly distributed by the ratio of Brownian motion and thermophoresis. Thus, it can be stated the modified Buongiorno model is able to consider the effect of nanoparticle volume fraction distribution.

Analysis on the order of magnitudes Equations (1)–(3) can be simplified with the scale analysis. Let us consider our physical conditions, which are typically occurring in the flow of alumina/water nanofluids inside microtubes. Table 3 shows the device

123

2:95108

1:85107

ð7Þ

6:67105

Equation (7) indicates that the LHS term is in the same order of magnitude with the first and second terms of RHS and these are about 105 times more than the third RHS term. In fact, heat transfer associated with nanoparticle diffusion (third RHS term) can be neglected in comparison with the other terms. Therefore, the governing Eqs. (1)–(3) can be written as:    1d du dp  rl ð8Þ   qp  qbf 0 ð/  /wo Þg ¼ 0 r dr dr dx   dT 1 d dT  rk ð9Þ  Q0 ðT  TB Þ ¼ 0 qcu dx r dr dr   1 o o/ DT oT DB þ ¼0 ð10Þ r or or T or By averaging Eq. (9) from r = Ri to Ro and according to the thermally fully developed condition for the uniform

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside… Table 3 Device and material properties of alumina/water nanofluid Reference variables

Values

lref/m s-1

0.01

/ref

0.02

lref/kg m-1 s-1

0.00198

Lref = Dh/m

7.5 9 10-5

-1

b/K

(T-TB)ref/K

50

qp/kg m-3

3880

qbf/kg m-3

998.2

Q/W m-3 K-1

2 9 107 10

Tref/K Dx/m

300 0.01

D/

0.1

hB

qcp u h qcp u

ð16Þ

More explanation on obtaining Eqs. (12)–(14) is included in ‘‘Appendix’’. The Nusselt numbers at the inner and outer walls are defined as Nuwi ¼

hwi Dh q00wi Dh e ¼ ¼ kwi ðTwi  TB Þkwi ð1 þ eÞð1 þ 7:47/wi Þhwi

hwo Dh q00wi Dh ¼ kwo ðTwo  TB Þkwo 1 ¼ ð1 þ eÞð1 þ 7:47/wo Þhwo

Nuwo ¼

ð18Þ

respectively. The Nusselt number based on the thermal conductivity of base fluid can be defined as the non-dimensional heat transfer coefficient by

ð11Þ

Equations (8)–(10) can be reduced to   d2 u 1 1 dl d/ du lbf þ  ¼  ½1  Nr ð/  /wo Þ g l d/ dg dg dg2 4l

hDh k ¼ Nu ¼ Nuð1 þ 7:47/Þ kbf kbf

htotðbfÞ

   d2 h kbf qcu ð1 þ feÞ r r þ  ¼  h þ h 2  dg 4 k qcu ð1 þ fÞð1 þ eÞ 4   d/ k=kbf dh þ 7:47 þ dg dg g

ð13Þ

o/ / oh ¼ 2 og og NBT ½1 þ ch

ð14Þ

with the boundary conditions f 1 þ 39:11/wi þ 533:9/2wi du

:u ¼ k g¼ ; 1f /wi qp qbf þ ð1  /wi Þ dg oh e ¼ og 2ð1 þ 7:47/wi Þð1 þ eÞ 1 1 þ 39:11/wo þ 533:9/2wo du

:u ¼ k ; 1f /wo qp qbf þ ð1  /wo Þ dg ð15Þ

ð19Þ

So, the total heat transfer coefficient enhancement may be evaluated as htot

ð12Þ

oh 1 ¼ ; / ¼ /wo og 2ð1 þ 7:47/wo Þð1 þ eÞ

2lbf ; Dh qbf

and

g¼

g¼

k¼N

ð17Þ

dTB wall heat flux (dT dx ¼ dx ) and introducing the following nondimensional parameters,

2r u T T

; h ¼ 00 00 B ; ; u ¼ D2 q dp ð wo þqwi ÞDh Dh h lbf  dx kbf  00  qwo þ q00wi Dh Ri DBB /B c¼ ; ; f ¼ NBT ¼ TB kbf Ro D TB c   qp  qbf 0 g Q0 D2h q00

; e ¼ 00wi r¼ ; Nr ¼ kbf qwo dp=dx

u / ; u

/B ¼

2 9 10-4

g/m s-2

where the slip parameter k, the bulk mean dimensionless temperature hB, and the bulk mean nanoparticle volume fraction /B can be obtained

Ri hwi þ Ro hwo ðRi hwi þ Ro hwo Þbf fNuwi ð1 þ 7:47/wi Þ þ Nuwo ð1 þ 7:47/wo Þ ¼ ðfNuwi þ Nuwo Þbf ¼

Finally, the pressure gradient can be defined as !  , dp lbf uB q Ndp ¼  ¼ B 2 dx qu ðDh Þ

ð20Þ

ð21Þ

Numerical method Equations (12)–(14) form a system of coupled ordinary differential equations and should be solved in conjunction with the boundary conditions of Eq. (15). The equations are strongly nonlinear and have unknown parameters (/wi, /wo, qcp u , and h) which should be found with a trial–error technique. The numerical algorithm is shown graphically in Fig. 2. Obviously, the numerical procedure involves a reciprocal algorithm in which /wi, /wo, qcp u , h(1), and h are used to calculate the values of /B, qcp u , hB, and h. Firstly, a suitable value for qcp u in the order of 104 has

123

S. A. Moshizi, A. Malvandi

Start

Quantifying the input variables

Initial guess ρ cu*, θ, φ w and θ (1)

Solve the ODE equations

⏐(φ new– φold)/φ old⏐≤10–6

YES

⏐(ρ cu*new– ρ cu*old)/ρ cu*old⏐≤10–6

NO

Modifying φ w with considering φ B

YES

⏐(θ new– θold)/θold⏐≤10–6

NO

NO

Modifying ρ cu*

Modifying θ

YES

θ B ≤10–6

Finish

NO

Modifying θ (1)

Fig. 2 Algorithm of the numerical method

been presumed. Then, the bulk mean temperature h is guessed in the order of 10-2. These two assumptions lead Eq. (13) to be solved. Finally, to determine the boundary conditions, /w and h(1) are guessed. At this stage, the governing equations and boundary conditions are fully defined and can be solved with a numerical method. Here, Runge–Kutta–Fehlberg algorithm and an adaptive method with the order O(h4) with an error estimator of order O(h5) are used. It can adapt the number and position of the grid points during each iteration to improve the accuracy. The process of solving is repeated until the value of /B reaches the predetermined value, and the relative errors between the assumed values of qcp u and h with the calculated ones after solving Eqs. (12)–(14) become lower than 10-6. Furthermore, the convergence criterion is considered to be 10-6 for relative errors of the velocity, temperature, and nanoparticle volume fraction. The corrected values of /wi, /wo, h, and qcp u for a special case are tabulated in Table 4. Table 4 Results of the numerical solution when Nr ¼ 125= r ¼ 5:0= k ¼ 0:01=e ¼ 1:0 /B

NBT

h

qcp u

/wi

0.02

0.3

0.007752

25870.768605

0.017962858

0.016141707

1

0.009184

41273.700312

0.01940046

0.01869293

5

0.009546

47132.31879

0.01988068

0.0197288

123

/wo

Results and discussions In many studies, a nanofluid is considered to be a homogeneous liquid, and its material properties are assumed to be constant in all positions of the system. This assumption is not realistic and causes some misunderstandings in the heat transfer mechanism of a nanofluid. Therefore, the examination of the motion of nanoparticles in a nanofluid is essential for examining nanofluids as a heat transfer medium in heat transfer equipments. Microscopically, because of the very small dimensions of nanoparticles (\100 nm), there are two possible reasons for the inhomogeneity: Brownian and thermophoretic diffusions. The Brownian diffusion is due to random drifting of suspended nanoparticles in the base fluid which originates from continuous collisions among the nanoparticles and liquid molecules. The Thermophoresis induces nanoparticle migration from warmer to colder region (in the opposite direction of the temperature gradient), making a non-uniform nanoparticle volume fraction distribution. According to the 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 Brownian diffusion to the thermophoresis. With dp % 20 nm and /B % 0.1, the ratio of Brownian motion to thermophoretic forces NBT  1/dp can be

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside…

(a)

3.6 ε=0 ε = 0.7 ε=1 ε = 1.5

3.2 2.8 2.4 2

u/uB

changed over a wide range of 0.2–10 for alumina/water B nanofluid. In addition, c ffi TwTT may change from 0 to 0.2; w however, its effects on the solution are insignificant (see Refs. [19]); so the results have been carried for c % 0.2. In this section, the accuracy of the results is firstly demonstrated and then the effects of pertinent parameters on the flow and thermal fields are investigated. Finally, the physical quantities of interest including heat transfer rate and pressure drop are obtained for all the parameters.

1.6 1.2 0.8 0.4 0

Accuracy of the results

–0.4

(b)

Present work

Kays et al. [32] Nuwo 5.036 6.413

Nuwi 0 13.111

Nuwo

0 1

0 13.117

2

8.403

8.846

3

7.504

14.207

4

7.122

36.306

7.122

36.223

(c)

1

1.2

1.4

1.26462

9.42435

1.26463

9.42433

ε=0 ε = 0.7 ε=1 ε = 1.5

0.8

0.6

10-6

2

0.9

8.842

5 9 10

1.8

1

14.215

9.42542

-6

1.6

1.1

8.401

1.26459

10-5

η

1.2

7.503

Ndp =Ndp ðbfÞ

2

–0.02

0.7

htot/htot(bf)

dg

1.8

0

5.036 6.417

Table 6 Grid independence test for different values of dg when NBT ¼ 0:15= e ¼ 1:0= r ¼ 5:0 and Nr = 30

2

0.02

wo

Nuwi

1.8

ε=0 ε = 0.7 ε=1 ε = 1.5

0.04

φ/φB

q00

e ¼ q00wi

1.6

0.06

–0.04

Table 5 Comparison of numerical results for Nuwi and Nuwi with the ones reported by Kays et al. [32] when Nr ¼ r ¼ k ¼ / ¼ 0

1.4

η

Velocity, temperature, and concentration profiles The effects of e; /B ; r; k; and Nr on the velocity (u/uB), temperature (h), and nanoparticle volume fraction (///B) profiles are shown in Figs. 3– 7, respectively. The inner

1.2

θ

To check the accuracy of the numerical code, the results obtained for a concentric annulus with Nr = k = r = 0, and e = 1 are compared to the analytical results of Kays et al. [32] in Table 5. Obviously, the results are in a desirable agreement. In addition, the numerical code is run on three different integration steps (dg) of 10-5, 5 9 10-6, and 10-6 to verify the results whether to be independent of the grid size. The obtained numerical results are presented in Table 6. The results clearly indicate the grid independence of the code. It must be mentioned that all the numerical results obtained here have been carried out using the integration step dg = 10-6.

1

1

1.2

1.4

η

1.6

Fig. 3 Effects of e on a nanoparticle distribution (///B), b velocity (u/uuB.uB), and c temperature (h) profiles when Nr ¼ 125; NBT ¼ 0:15; k ¼ 0:01; r ¼ 5:0 and /B = 0.02

wall is placed at g = 1, whereas g = 2 represents the outer wall. The figures reveal that the nanoparticle volume fraction is lower near the walls and reaches its peak the middle of the microchannel. In other words, nanoparticles

123

S. A. Moshizi, A. Malvandi

(a)

2.8

φB = 0.02

2.4

φB = 0.04

2

φB = 0.08

1.2 1

1.6

u/uB

u/uB

1.4

(a)

3.2

1.2

0.8

σ = –5 σ=0 σ=5

0.6

0.8

0.4 0.4

0.2

0 –0.4

1

1.2

1.4

1.6

1.8

0

2

1

1.2

1.4

(b)

(b)

0.07

0.05

1.8

2

1.8

2

1.8

2

0.07

φB = 0.02

0.05

φB = 0.08

0.03

σ = –5 σ=0 σ=5

φB = 0.04

θ

θ

0.03

1.6

η

η

0.01

0.01

–0.01

–0.01

–0.03

–0.03 1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

(c)

1.2

1.2

1.1

1.1

1

1

φ/φB

φ/φB

(c)

0.9

φB = 0.02

0.8

0.6

0.9

φB = 0.08

0.7 0.6

1

1.2

1.4

1.6

σ = –5 σ=0 σ=5

0.8

φB = 0.04

0.7

1.6

η

η

1.8

2

η

1

1.2

1.4

1.6

η

Fig. 4 Effects of /B on a nanoparticle distribution (///B), b velocity (u/uuB.uB), and c temperature (h) profiles when Nr ¼ 125; NBT ¼ 0:15; k ¼ 0:01; r ¼ 5:0 and e = 1.0

Fig. 5 Effects of r on a nanoparticle distribution (///B), b velocity (u/uuB.uB), and c temperature (h) profiles when Nr ¼ 125; NBT ¼ 0:15; /B ¼ 0:02; k ¼ 0:01 and e = 1.0

migrate from the wall toward the central region of annulus. This is because heating a wall surface imposes a temperature gradient on the nanofluid creating a thermophoretic force. The thermophoretic force then pushes the particles from the heated walls toward the central region—in the

direction of the heat transfer—which leads to a rise in the nanoparticles concentration there and decreases it near the heated walls. The effects of e are shown in Fig. 3. Obviously, for e = 0 where the inner wall is kept adiabatic (no

123

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside… 1.4

(a)

1.4

1.2

1.2

1

1

0.8

0.8

u/uB

u/uB

(a)

0.6

0.6

λ=0 λ = 0.01 λ = 0.04 λ = 0.08

0.4

0.2

0.2 0

Nr = 75 Nr = 100 Nr = 125 Nr = 150

0.4

0

1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

(b)

(b)

0.07

0.05

λ=0 λ = 0.01 λ = 0.04 λ = 0.08

1.6

1.8

2

1.8

2

Nr = 75 Nr = 100 Nr = 125 Nr = 150

θ

θ

0.01

–0.01

–0.01

–0.03

–0.03 1

1.2

1.4

1.6

1.8

2

1

1.2

1.4

η

η

(c)

1.2

1.2 1.1

1.1

1

φ/φB

1

φ/φB

2

0.03

0.01

(c)

1.8

0.07

0.05

0.03

1.6

η

η

0.9

λ=0 λ = 0.01 λ = 0.04 λ = 0.08

0.8 0.7

0.9

0.7 0.6

0.6 1

1.2

1.4

1.6

Nr = 75 Nr = 100 Nr = 125 Nr = 150

0.8

1.8

2

η

Fig. 6 Effects of k on a nanoparticle distribution (///B), b velocity (u/uuB uB), and c temperature (h) profiles when Nr ¼ 125; NBT ¼ 0:15; /B ¼ 0:02; r ¼ 5:0 and e = 1.0

temperature gradient at the inner wall), the nanoparticles only migrate from the outer wall, so a highly poor region of nanoparticles concentration is formed at the outer wall, thereby reducing the viscosity and thermal conductivity of nanofluids. Accordingly, the momentum and temperature

1

1.2

1.4

1.6

η

Fig. 7 Effects of Nr on a nanoparticle distribution (///B), b velocity (u/uuB uB), and c temperature (h) profiles when r ¼ 5:0; NBT ¼ 0:15; /B ¼ 0:02; k ¼ 0:01 and e = 1.0

gradient increase near the outer wall. For 0 \ e \ 1, the heat flux at the inner wall leads the nanoparticles to move from the inner wall toward the central region. Thus, similar to the outer wall, a nanoparticle depleted region is formed at the inner wall. However, the intensity of depleted region

123

S. A. Moshizi, A. Malvandi

(a)

3.7 3.2

ε=0 ε = 0.7 ε=1 ε = 1.5

htot/htot (bf) (ε)

2.7 2.2 1.7 1.2 0.7 0.2 10–1

100

101

NBT

(b)

60

Ndp tot/Ndp tot (bf) (ε)

50

ε=0 ε = 0.7 ε=1 ε = 1.5

40 30 20 10 0 10–1

100

101

NBT Fig. 8 Effects of e on the heat transfer coefficient ratio (a), and the pressure drop ratio (b) when k = 0.01, Nr = 30, /B = 0.02 and r = 5.0

at the inner wall is lower than that at the outer wall which is due to a lower temperature gradient induced by the inner wall heat flux. Increasing e reduces the nanoparticles concentration at the inner wall and shifts the peak of nanoparticles concentration toward the center of microannulus. For e = 1 (thermal symmetry at the walls), the nanoparticles distribution becomes almost symmetric and nanoparticle accumulated region forms in the middle of the microannulus. Notably, as e reaches unity, the nanoparticle distribution becomes almost symmetric but not absolute symmetric, since the imposed heating energy at these two walls is different because the wall with higher area receives more thermal energy. Further increase in e shifts the peak of the nanoparticles concentration toward the outer wall. The effects of e on nanoparticles concentration have considerable influence on the velocity and temperature profiles. Obviously, the velocity and temperature profiles become more symmetric for e = 1 in which the nanoparticle volume fraction is nearly symmetric. For e \ 1, the

123

symmetry in the velocity and temperature profiles disappears and the peak (dip) of the velocity (temperature) profile shifts toward the outer wall (the lower viscosity region). Figure 4 depicts the variations of the nanoparticle volume fraction (///B), velocity (u/uB), and temperature (h) profiles for different values of /B. As it is clear, nanoparticles distribute more uniformly at the higher values of /B. In fact, there is a decreasing trend for the nanoparticles concentration in the core region as well as a reversed behavior for that near the walls, as /B increases. This means that the migration of nanoparticles enhances for the lower nanoparticle volume fraction. However, increasing /B boosts the impacts of the nanoparticle volume fraction on the viscosity; hence, it can be observed that for the higher values of /B, the nanoparticles depletion effect becomes significant and the peak of the velocity profile shifts further toward the walls. The effects of heat generation/absorption (r) are shown in Fig. 5. When there is a heat absorption (r \ 0), more energy will be absorbed by the fluid, so the dip of the temperature profile decreases. In fact, the temperature profile becomes smoothly uniform for the negative values of r. This leads to a slight increase in the temperature gradient at the outer wall, thereby enhancing the nanoparticle migration there. Accordingly, the velocity profile moves slightly back toward the central region. In contrast, heat generation (r [ 0) increases the temperature gradients and the nanoparticle migration at the outer wall. This is the reason that, in the presence of heat generation, the velocity profiles move toward the outer wall. The effects of the slip parameter (k) are shown in Fig. 6. The slip parameter k signifies the amount of slip velocity at the fluid–solid interface. Increasing k leads to a rise in the slip velocity in the vicinity of the walls, and because of the constant mass flow rate in the channel, the velocity in the central region reduces. In fact, the velocities in the central region shift toward the walls leading to a more uniform velocity profile, as k increases. The momentum augmentation at the walls increases the heat removal ability of the flow and reduces the walls’ temperature and its gradient. Accordingly, the nanoparticle migration from the heated walls reduces. Finally, Fig. 7 shows an increasing trend in the velocity profile at the walls (the low viscosity region) followed by a decrease in the core region, as Nr increases. In essence, all of the incoming fluid moves quickly toward the walls. Obviously, the increment of the momentum at the outer wall (which has the lowest viscosity) is greater than that at the inner counterpart, which causes the temperature at the outer wall to reduce significantly. In addition, an increase in Nr shifts the peak of the nanoparticle volume fraction toward the outer wall and reduces the intensity of the nanoparticles depletion there. This is because increasing Nr

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside…

(a)

(a)

5 φB = 0.02

4

σ = –5 σ=0 σ=5

φB = 0.04

φB = 0.08

3

htot/htot (bf) (σ)

htot/htot (bf) (φB)

1.22

2

1.2

1.18

1

0 10–1

100

1.16 10–1

101

103

(b)

101

100 10–1

100

6.5

σ = –5 σ=0 σ=5

5.5

φB = 0.04 φB = 0.08

Ndp tot/Ndp tot (bf) (σ)

Ndp tot/Ndp tot (bf) (φB)

φB = 0.02

102

101

NBT

NBT

(b)

100

101

NBT

4.5

3.5

2.5

1.5 10–1

100

101

NBT

Fig. 9 Effects of /B on the heat transfer coefficient ratio (a) and the pressure drop ratio (b) when Nr ¼ 125; r ¼ 5:0; e ¼ 1:0; and k ¼ 0:01

Fig. 10 Effects of r on the heat transfer coefficient ratio (a) and the pressure drop ratio (b) when e = 0.5, Nr = 30, /B = 0.02 and k = 0.01

intensifies the concentration-dependent buoyancy force, which tries to reduce the nanoparticles concentration gradient. Therefore, nanoparticles move from the higher concentration region toward the lower concentration one and the nanoparticles concentration profile becomes more uniform; consequently, the peak of nanoparticles concentration declines and moves toward the outer wall, as Nr increases.

pressure drop significantly. Moreover, it can be seen that the heat transfer enhancement and pressure drop increment remained almost stable for the higher values of NBT. This is due to the fact that at the higher values of NBT (the Brownian diffusion dominates the thermophoresis), the nanoparticles uniformly distributed leading to a uniform viscosity and thermal conductivity. As a result, the variations of heat transfer rate and pressure drop are reduced. From Fig. 8, it is clear that the heat flux ratio e significantly changes the heat transfer behavior. For the case where the inner wall heat flux is greater that that at the outer wall (e C 1), the heat transfer coefficient enhances about 20 % for all the range of 0.2 \ NBT \ 10. As the value of inner wall heat flux becomes smaller than that of the outer one (e \ 1), the heat transfer coefficient decreases. This trend continues until e = 0 (the adibatic inner wall) which heat transfer coefficient experiences a significant increment of about 350 % at NBT = 0.2. This nonlinear behavior is due to the dependency of heat transfer

Heat transfer rate and pressure drop Figures 8– 12 show the effects of pertinent parameters e, /B, r, k, and Nr on (a) the heat transfer ratio (h[nf]/h[bf]) and (b) the pressure drop ratio (Ndp[nf]/Ndp[bf]) successively. From the figures, the pressure drop ratio has a decreasing trend with increasing NBT. Routinely, NBT is higher for smaller nanoparticles (NBT  DB  1=dp ), so smaller nanoparticles have the merit of a slight increase in the pressure drop. However, larger nanoparticles increase the

123

S. A. Moshizi, A. Malvandi 1.3

(a)

λ=0 λ = 0.01 λ = 0.04 λ = 0.08

1.2

1.25 Nr = 75 Nr = 100 Nr = 125 Nr = 150

1.23

htot/htot (bf) (Nr)

htot/htot (bf) (λ)

1.25

(a)

1.21

1.19

1.15 1.17

1.1 10–1

100

1.15 10–1

101

(b)

100

101

NBT

NBT

(b)

7

9 8

λ=0 λ = 0.01 λ = 0.04 λ = 0.08

5

Ndp tot/Ndp tot (bf) (Nr)

Ndp tot/Ndp tot (bf) (λ)

6

4 3 2 1 10–1

Nr = 75 Nr = 100 Nr = 125 Nr = 150

7 6 5 4 3 2

100

101

NBT

1 10–1

100

101

NBT

Fig. 11 Effects of k on the heat transfer coefficient ratio (a) and the pressure drop ratio (b) when Nr ¼ 125; r ¼ 5:0; /B ¼ 0:02; and e ¼ 1:0

Fig. 12 Effects of Nr on the heat transfer coefficient ratio (a) and the pressure drop ratio (b) when k = 0.01, e = 0.5, /B = 0.02 and r = 5.0

coefficient on momentum as well as thermal conductivity enrichments at the walls, which goes up against each other. These two mechanisms, in fact, act against each other, and the net result of these phenomena determines the overall heat transfer ratio at each wall. As e increases above unity (e C 1), the momentum enrichment at the inner wall (which has a higher heat flux) dominates the decrement of thermal conductivity there; thus, the total heat transfer rate increases. However, for e \ 1, the momentum augmentation at the outer wall cannot overcome the thermal conductivity reduction leading the heat transfer coefficient to decrease. This trend goes on till e = 0 which the momentum enrichment at the outer wall is considerable and its effect on heat transfer rate dominates the thermal conductivity reduction. Consequently, it can be stated that the best heat transfer enhancement is obtained for the adiabatic inner wall and for larger nanoparticles (the lower values of NBT). For the smaller nanoparticles, however, there is no significant migration of nanoparticles and the asymmetric

heating has not considerable impact on the heat transfer and pressure drop coefficients. Regarding Fig. 9, it can be observed that the inclusion of nanoparticles enhances the heat transfer rate and the pressure drop, which is more apparent for the lower values of NBT. Thus, larger nanoparticles signify (dp  1/NBT) a better cooling performance at the higher volume fraction of nanoparticles. This is due to the fact that, in the lower values of NBT, the effect of the nanoparticle volume fraction is considerable since the viscosity and thermal conductivity of nanofluid are the most sensitive to the nanoparticle volume fraction among the other properties. Figure 10a, b shows that the heat absorption r \ 0 has a positive effect on the pressure drop and heat transfer rate in the lower values of NBT. However, for the higher values of NBT, it has a negative effect. For the case of heat generation, the heat transfer rate sharply decreases for the lower values of NBT, while the pressure drop increases. Hence, the performance of the system reduces in the case of heat

123

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside…

generation for the lower values of NBT. This negative behavior vanishes with increasing NBT (for smaller nanoparticles). Therefore, it can be concluded that the smaller nanoparticles are suitable for the system with heat generation, whereas larger nanoparticles are suggested for the heat exchange equipments in the presence of heat absorption. Regarding Fig. 11, the heat transfer ratio enhances with increasing the slip parameter (k). In addition, increasing the slip parameter reduces the pressure drop ratio. Hence, it can be concluded that the slip parameter has a positive effect since it allows a higher heat transfer rate along with a lower pressure loss. Finally, Fig. 12 indicates that the mixed convection due to the non-uniform nanoparticle distribution has a positive effect on the heat transfer rate because increasing Nr leads to a rise in the heat transfer ratio. However, this augmentation is followed by a pressure drop penalty since the pressure drop has an increasing trend with increasing Nr. This effect significantly depends on the nanoparticle size (NBT); so, using smaller nanoparticles (higher values of NBT) prevents the considerable changes in the heat transfer rate and the pressure drop.

Summary and conclusions Nanoparticle migration is a basic phenomenon in nanoparticles–fluid mixtures, which appears due to the slip velocity of nanoparticles relative to the base fluid, thereby significantly affecting the thermo physical properties of nanofluids. In this paper, an analysis is presented for the laminar fully developed mixed convection of alumina/ water nanofluid inside a concentric microannulus, which is subjected to a constant volumetric internal heating/cooling. The Brownian motion and thermophoretic diffusivity are considered by using the modified Buongiorno model to consider the effects of nanoparticle migration. To consider different modes of nanoparticle migration, two different heat fluxes have been considered at the walls: q00wi on the inner wall and q00wo on the outer wall. By examining the order of magnitudes for various terms of the governing equations, it is concluded that despite the temperature-dependent buoyancy forces, concentration-dependent buoyancy forces have considerable effects on the flow fields and nanoparticle migration. The major findings of this paper can be expressed as: •

The maximum enhancement in the heat transfer rate is obtained for the adiabatic inner wall and for the larger nanoparticles (thermophoresis dominant regime). For the smaller nanoparticles, however, there is no

•

•

•

significant migration of nanoparticles and asymmetric heating has not considerable impact on the heat transfer and pressure drop coefficients. Smaller nanoparticles are suitable for the system with heat generation, whereas larger nanoparticles are suggested for the heat exchange equipments in the presence of heat absorption. Larger nanoparticles also signify a better cooling performance at the higher volume fraction of nanoparticles. Slippage at the fluid–solid interface has a positive effect on the cooling performance. Increasing the slip parameter increases the heat transfer rate and decreases the pressure drop. This is a useful attribute, particularly in microchannels since it achieves the higher heat transfer enhancement followed by a significant reduction in the pumping power required to drive the nanofluid flow.

It should be stated that the current study can be developed to model the developing fluid flow and heat transfer of nanofluids in the presence of magnetic field. Furthermore, it is advantageous to consider the effect of nanoparticle migration in the other passive heat transfer enhancement equipments such as curved tubes and helical heat exchangers.

Appendix Momentum equation    1d du dp  rl ð22Þ   qp  qbf 0 ð/  /wo Þg ¼ 0 r dr dr dx      ! 1 du dg dl d/ dg du dg d2 u dg 2 l þr þ rl 2 r dg dr d/ dg dr dg dr dg dr  dp   qp  qbf 0 ð/  /wo Þg ¼ 0 dx 2r dg 2 ¼ g¼ ! Dh dr Dh 

(24) in (23):   4l 4 dl d/ du 4l d2 u dp þ  þ D2h g D2h d/ dg dg D2h dg2 dx    qp  qbf 0 ð/  /wo Þg ¼ 0 lbf l

ð23Þ ð24Þ

ð25Þ

9 (25):

  4lbf d2 u 4lbf 4lbf 1 dl d/ du lbf dp þ ¼ þ 2 l dx D2h dg2 D2h g Dh l d/ dg dg   lbf qp  qbf 0 ð/  /wo Þg þ l

ð26Þ

123

S. A. Moshizi, A. Malvandi

u ¼

D2h lbf

u

 ddpx

ð27Þ

  d2 u 1 1 dl d/ du lbf þ  ¼  g  l d/ dg  dg dg2 4l lbf qp  qbf 0 g

ð/  /wo Þ þ ð28Þ 4l dp= dx   d2 u 1 1 dl d/ du lbf  ¼ þ ½1  Nr ð/  /wo Þ g l d/ dg dg dg2 4l ð29Þ 



qp  qbf 0 g

Nr ¼ dp=dx

ð30Þ

Thermally fully developed relations   dT 1 d dT ¼ rk qcu þ Q0 ðT  TB Þ dx r dr dr qcuT ; qcu

TB

h¼

ð31Þ

T  TB T  TB ¼ ; ðq00wo þq00wi ÞDh ð1 þ eÞ q00wo Dh kbf kbf

Ri f¼ ; Ro

e¼

q00wi q00wo

 For each parameter ðuÞ:u

ð32Þ

Z

1 A

udA A

1  ¼  2 p Ro  R2i

ZRo Ri

1

2ð 1  f Þ 2  2prudr ¼  1  f2

dTB 1 ¼ dx pðR20  R2i Þ

qcu

ZR0 

ð33Þ

Z1f gudg f 1f

   1o oT rk þ Q0 ðT  TB Þ 2prdr r dr dr

Ri

ð34Þ 

qcu

dTB 2 oT ¼ 2 rk 2 dr dx R0 ð1  f Þ 1 1f

Z



Q 0 ð1 þ eÞ

Ro

q00wo Dh Dh g h kbf 2

þ Ri



2 2 Ro ð1  f2 Þ

Dh dg 2



ð35Þ

f 1f

oT k or

 ¼ Ro

123

q00wo ;

oT k or

 ¼ Ri

q00wi

ð36Þ

qcu

  dTB 2 ¼ Ro q00wo þ Ri q00wi 2 2 dx Ro ð1  f Þ 1 Z1f 2D3h Q0 q00wo ð1 þ eÞ þ hgdg 4R2o ð1  f2 Þkbf

ð37Þ

f 1f

qcu

dTB 4q00wo Dh Q0 q00wo ð1 þ eÞ ¼ ð1 þ feÞ þ h kbf dx Dh ð1 þ fÞ

ð38Þ

Energy equation   dT 1 d dT ¼ rk qcu ð39Þ þ Q0 ðT  TB Þ dx r dr dr    dT 1 d q00wo Dh dh ¼ rk ð1 þ eÞ qcu dx r dr kbf dr q00wo Dh h ð40Þ þ Q0 ð1 þ eÞ kbf   dT 4q00wo 1 d dh q00 Dh ¼ ð1 þ eÞ rk qcu þ Q0 ð1 þ eÞ wo h dx dg Dh kbf r dg kbf ð41Þ  dT 4q00 1 Dh dh dk d/ dh d2 h ¼ ð1 þ eÞ wo þ rk 2 k þr qcu dx d/ dg dg dg Dh kbf r 2 dg q00wo Dh h þ Q 0 ð1 þ eÞ kbf ð42Þ   00 dT 4ð1 þ eÞqwo k=kbf ðdk=d/Þ d/ dh ¼ qcu þ dx kbf dg dg Dh g 00 2 4q ðk=kbf Þ dh ð43Þ þ wo ð1 þ eÞ 2 Dh dg q00 Dh þ Q0 ð1 þ eÞ wo h kbf 

dT dTB ¼ dx dx 00 4qwo Dh Q0 q00wo ð1 þ eÞ ð1 þ feÞ þ h ¼ Dh ð1 þ fÞqcu qcukbf

Fully developed condition:

(44) in (43):   4q00wo Dh Q0 q00wo ð1 þ eÞ qcu ð1 þ feÞ þ h ¼ kbf Dh ð1 þ fÞ qcu   4ð1 þ eÞq00wo k=kbf ðdk=d/Þ d/ dh þ kbf dg dg Dh g þ

4q00wo ðk=kbf Þ d2 h q00 Dh ð1 þ eÞ 2 þ Q0 ð1 þ eÞ wo h Dh dg kbf

ð44Þ

ð45Þ

Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside…

Q0 D2h ; kbf

dk ¼ 7:47kbf d/    d2 h kbf qcu ð1 þ feÞ r r þ  ¼  h þ h dg2 4 k qcu ð1 þ fÞð1 þ eÞ 4   d/ k=kbf dh þ 7:47 þ dg dg g

r¼

k ¼ kbf ð1 þ 7:47/Þ !

Nanoparticle fraction conservation equation   1 o o/ DT oT DB þ ¼0 r or or T or   o/ DT oT og þ ¼0 DB og T og o r T ¼ TB þ ð1 þ eÞ

q00wo Dh h kbf

ð46Þ

ð47Þ

ð48Þ ð49Þ ð50Þ

(50) and (24) in (49): o/ q00wo Dh DT ð1 þ eÞ oh i ¼ h q00wo Dh og h TB DB kbf og 1 þ ð1 þ eÞ

ð51Þ

TB kbf

 c¼



q00wo þ q00wi Dh TB kbf

ð52Þ

DB / T

ð53Þ

DT / /

ð54Þ

(52), (53) and (54): DT DTB TB / DT 1 / ¼ ¼ B DB DBB T /B DBB ð1 þ ð1 þ eÞchÞ /B NBT ¼

DBB kbf TB /B   DTB Dh q00wo þ q00wi

o/ q00wo Dh ð1 þ eÞDTB / oh ¼ 2 og ½1 þ ð1 þ eÞch TB kbf DBB /B og /ð1 þ eÞ oh ¼ 2 og NBT ½1 þ ð1 þ eÞch o/ / oh ¼ 2 og NBT ½1 þ ch og

ð55Þ ð56Þ

ð57Þ ð58Þ

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