International Journal of Thermal Sciences

New correlation for Nusselt number of nano fl uid with Ag / Al ₂ O ₃ / Cu nanoparticles in a microchannel considering slip velocity and

temperature jump by using lattice Boltzmann method

Arash Karimipour

Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran

a r t i c l e i n f o

Article history:

Received 20 April 2014 Received in revised form 10 January 2015 Accepted 12 January 2015 Available online

Keywords:

Different nanoparticles Slip velocity Temperature jump LBM

Microflow

a b s t r a c t

Forced convection heat transfer of water-Ag, water-Cu and water-Al2O3nanofluid in a microchannel is studied numerically by using lattice Boltzmann method. Temperature of microchannel walls is higher than that of the inletfluid. Effects of change in nanoparticles volume fraction and slip coefficient are investigated. Slip velocity, temperature jump and velocity and temperature profiles are presented at different cross sections. Moreover, a correlation is developed to predict nanofluid Nusselt number through the microchannel. As a result, higher value of slip coefficient corresponds to less Nusselt number and more slip velocity and temperature jump.

©2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

Investigation offluidflow and heat transfer in micro scales has become one of the most attractive topics in recent years. Micro- pumps, microvalves, microchannels, etc. are a few examples of micro-devices which many researches have reported on [1e3].

Higher efficiency along with their small sizes increases the appli- cation of these micro devices more than before. Cooling the hot walls of a microchannel is a fundamental topic in the area of microflows due to its widespread applications. Surface effect is more important in the micro scale level, so that the well-known no- slip boundary condition fails and slip flow regime must be considered[4e7]. As a result, slip velocity and temperature jump can be observed along thefluid-solid boundaries. In addition to the slipflow, the transient and free molecular regimes might occur in the micro gas flows. It should be mentioned that Navier Stokes equations work well only for slip flow; however particle-based methods must be used for other flow regimes. Molecular dy- namic (MD), lattice Boltzmann method (LBM) and direct simulation of Monte Carlo (DSMC) are some of these approaches[8e10]. LBM requires lower computation cost and uses easier equations

compared with MD or DSMC; it also deals with simple parallel al- gorithms and well describes the complex boundaries.

One time step of LBM is achieved by Collision and propagation between the fictive fluid particles located on a specified lattice points [11e17]. Collision must satisfy the conservation laws for which several models have been presented so far. Among them, BGK model shows suitable stability and accuracy[18e21]. He et al.

[22]presented the thermal lattice Boltzmann method (TLBM) to simulatefluidflow and heat transfer. They introduced TLBM based on the internal energy distribution function. TLBM is able to take into account the pressure work and viscous heat dissipation terms [23]. Karimipour et al.[24]showed that TLBM worked well for the mixed convection of air in a microchannel. Investigations on LBM are still going on in order to attain more progress for better simu- lation offluidflow and heat transfer. Thus, several researchers have been trying tofind out if it is applicable for different conditions and geometries[25,26].

An innovative approach to increase heat transfer rate is using

“nanofluids”which is a homogeneous mixture of liquid and solid nanoparticles. Nanofluid conduction is higher than that of the base fluid due to the large thermal conductivity of nanoparticles. Suit- able performance of nanofluid at macro scales levels have been shown in several previous works[27e29]. Santra et al.[30]studied the heat transfer of laminar copperewater nanofluidflow through two isothermally heated parallel plates and also Esfe et al. [31]

E-mail addresses:arashkarimipour@gmail.com,arash.karimipour@pmc.iaun.ac.

ir,arash.karimipour@uniroma1.it,arash.karimipour@yahoo.com.

Contents lists available atScienceDirect

International Journal of Thermal Sciences

j o u r n a l h o m e p a g e : w w w . e l s e v ie r . c o m / l o c a t e / i j t s

http://dx.doi.org/10.1016/j.ijthermalsci.2015.01.015 1290-0729/©2015 Elsevier Masson SAS. All rights reserved.

investigated the natural convection around an obstacle placed in an enclosurefilled with different types of nanofluids.

The slipflow regime might happen for the liquidflows at micro scales. Aminossadati et al.[32]studied the effects of magneticfield on nanofluid forced convection in a partially heated microchannel;

however without taking into account the slip velocity and tem- perature jump boundary conditions. The investigation of nanofluid flow in a microchannel in the absence of slipflow regimes along the walls can also be referred in other works [33,34]. However, in several of them the slip velocity (not the temperature jump) was presented for different states of nanofluid microflow [35,36].

Simulation of nanofluidflow using LBM has become an attractive topic for researchers at last few years[37,38]. Among these works, some are concerned with the investigation of nanofluid flow through a microchannel by LBM; however they all ignored the simulation of slip velocity and temperature jump by lattice Boltz- mann method for the nanofluid[39,40,50,51].

Therefore, the force convection of nanofluid through a micro- channel is studied here for thefirst time by using LBM considering slip velocity and temperature jump boundary conditions. To the best of authors' knowledge, this approach has not been imple- mented in any previous works.

2. Problem statement

Fig. 1shows the physical geometry of microchannel. The inlet temperature of the Newtonian nanofluid is less than that of the walls (Ti¼0.5Tw); while Reynolds number is keptfixed as Re¼rnf

uiDH/mnf¼0.01. At micro scales levels, the effects of gravity can be ignored due to very small value of characteristic length; hence the gravity effects and buoyancy motions are not involved. To exert the fully developed condition for both hydrodynamic and thermal domains, microchannel length is chosen long enough in compari- son with its height. The incompressibleflow regime is laminar and it is assumed that the spherical nanoparticle diameter is

dp¼10 nm. The mixture is a homogeneous substance of water and nanoparticles. Nanofluid forced convection is studied numerically by applying LBM-BGK and using hydrodynamic (f) and thermal (g) distribution functions.

The effects of different kinds of nanofluid such as water-Ag, water-Cu and water-Al₂O₃ are investigated for three values of nanoparticles volume fraction (4¼0,4¼0.02 and4¼0.04). The slip coefficient value is taken into account as 0.005<B<0.05 in order to illustrate the slip velocity and temperature jump. Simu- lation of nanofluid temperature jump by LBM is presented for the first time. Moreover, it will be tried to develop a correlation for prediction of nanofluid Nusselt number through the microchannel.

The present work nanofluid is supposed as a homogeneous single phase mixture of water and nanoparticles. As a result, some characteristics of nanofluid such as local variations of volume fraction and actions and re-actions of nanoparticles are ignored.

However the nanoparticles' Brownian motion effects on the ther- mal conductivity are considered here by using Eq.(5).

3. Mathematical formulation 3.1. Nanofluid

A homogeneous mixture of liquid as the base fluid and sus- pended solid nanoparticles inside, is called nanofluid. Using mixture model, the effective density and heat capacity can be achieved as follows:

r_nf¼4r_sþ ð14Þr_f (1)

rCp

nf¼ ð14Þ rCp

fþ4 rCp

s (2)

where subscripts f and s show thefluid and solid nanoparticles, respectively. The subscript nf is also referring to nanofluid while4 Nomenclature

B¼b/h non-dimensional slip coefficient c¼(cx,cy) microscopic velocity

Cp heat capacity, J kg¹K¹ DH¼2 h hydraulic diameter, m dp diameter of nanoparticles, nm

f density-momentum distribution function g internal energy distribution function h, l height and length of microchannel, m H¼h/h, L¼l/h non-dimensional height and length k thermal conductivity, W m¹K¹

Ls length of slip, m Nu Nusselt number Pr¼ynf/anf Prandtl number Re¼r^nfuiDH/mnf Reynolds number

t time, s

T temperature, K Ti inlet temperature, K Tw wall temperature, K u¼(u,v) macroscopic velocity, m s¹ (U,V)¼(u/ui,v/ui) non-dimensional velocity ui inlet velocity, m s¹

Us slip velocity x, y coordinates, m

(X,Y)¼(x/h,y/h) non-dimensional coordinates Z heat dissipation

Greek symbols

a thermal diffusivity, m²s¹ b ^{slip coef}ficient

4 volume fraction of nanoparticles m dynamic viscosity, Pas

q¼T/Ti non-dimensional temperature qs temperature jump

r density, kg m³

tf,tg momentum and internal energy relaxation times y kinematics viscosity, m²s¹

z temperature jump distance

U ^collision

Super- and Sub-scripts e equilibrium f fluid (water)

i inlet, lattice directions nf nanofluid

out outlet

s solid nanoparticles

w wall

a ^xey direction components

is the volume fraction of nanoparticles. Now, thermal diffusivity can be obtained by applying nanofluid density, heat capacity and thermal conductivity as follows[41]:

anf¼k_nf. rCp

nf (3)

Using Brinkman's model[42]and Chon^'s model[43], the effec- tive viscosity and thermal conductivity can be handled as:

mnf¼mf

.ð14Þ^2:5 (4)

k_nf

k_f ¼1þ64:74^0:7460 d_f

d_p

0:3690 k_s k_f

!_0:7476 m r_faf

!_0:9955

r_fBcT

3pm²l_BF

_1:2321 (5)

where

m¼A10^TCB;C¼140ðKÞ;B¼247ðKÞ;A¼2:41410⁵ðPasÞ (6) A large number of models have been presented to determine the nanofluid thermal conductivity. However the one in Eq.(5), has shown more accuracy as a result of considering nanoparticle diameter and Brownian motion. The mean free path of basefluid is shown as lBF and Boltzmann constant equals to B_c¼1.380710²³J/K.

3.2. Lattice Boltzmann method

Boltzmann equation is written in two forms showing the hy- drodynamic and thermal domains, respectively:

vf_i vtþc_iavf_i

vx_a¼UðfÞ ¼ 1 t_f

f_if^e_i

(7) vg_i

vtþc_iavg_i

vx_a¼UðgiÞ f_iZ_i¼0:5jcuj²UðfiÞ f_iZ_i

¼ g_ig^e_i

tg f_iZ_i (8)

Density-momentum and internal energy distribution functions are shown as f and g while their equilibrium terms are f^eand g^e. The microscopic and macroscopic velocity vectors are indicated by u¼(u,v) andc. Moreover, the symbols ofU^,tfandtgrepresent the collision operator, hydrodynamic and thermal relaxation times, respectively. It is noteworthy that arepresents x or y geometry direction components and i expresses direction of lattice velocity for D2Q9 model (Fig. 2).

c_i¼

cosi1

2 p;sini1 2 p

;i¼1;2;3;4

c_i¼ ffiffiffi p2

cos ði5Þ

2 pþp 4

;sin ði5Þ

2 pþp 4

;i¼5;6;7;8 c₀¼ ð0;0Þ

(9) The following equation corresponds to heat dissipation:

Z_i¼ ðc_iau_aÞ du_a

dt þc_iavu_a vx_a

(10) Hydrodynamic and thermal equilibrium distribution functions can be obtained by:

f^e_i ¼uir

"

1þ3ðci$uÞ þ9ðci$uÞ² 2 3u²

2

#

; i¼0;1;…;8 u0¼4=9; u1;2;3;4¼1=9;u5;6;7;8¼1=36

g^e₀¼ 2 3reu²

(11)

g^e_1;2;3;4¼1 9reh

1:5þ1:5

c_1;2;3;4$u þ4:5

c_1;2;3;4$u2

1:5u²i g^e_5;6;7;8¼ 1

36reh 3þ6

c_5;6;7;8$u þ4:5

c_5;6;7;8$u₂

1:5u²i (12) Discretize of Eqs.(7) and (8)lead to:

f_iðxþc_iDt;tþDtÞ f_iðx;tÞ ¼ Dt

2t_f f_iðxþc_iDt;tþDtÞ f^e_iðxþc_iDt;tþDtÞ Dt

2t_f f_iðx;tÞ f^e_iðx;tÞ

(13)

g_iðxþc_iDt;tþDtÞ g_iðx;tÞ ¼ Dt

2tg g_iðxþc_iDt;tþDtÞ g^e_iðxþc_iDt;tþDtÞ Dt

2f_iðxþc_iDt;tþDtÞ Z_iðxþc_iDt;tþDtÞ Dt

2tg g_iðx;tÞ g^e_iðx;tÞ Dt

2f_iðx;tÞZ_iðx;tÞ

(14)

Fig. 1.The physical model of the microchannel.

Fig. 2.D2Q9 lattice model.

The new functions of~f_iandg~_i are applied to change the last equations into explicit forms:

~f_i¼f_iþ Dt 2t_f

f_if^e_i

(15)

~g_i¼g_iþ Dt 2tg

g_ig^e_i þDt

2f_iZ_i (16)

As known, LBM consists of two parts of collision and streaming at each time step. Thus, by integrating Eq.(13)into Eq.(16)these parts can be written as one equation for hydrodynamic or thermal domains separately as follows:

~f_iðxþc_iDt;tþDtÞ~f_iðx;tÞ ¼ Dt tfþ0:5Dt

h~f_iðx;tÞ f^e_iðx;tÞi (17)

~g_iðxþciDt;tþDtÞ~g_iðx;tÞ ¼ Dt

tgþ0:5Dt ~g_iðx;tÞg^e_iðx;tÞ tgDt

tgþ0:5Dtf_iZ_i (18) Now the value of macroscopic variables can be estimated as [44]:

r¼P

i

~f_i

ru¼P

i

c_i~f_i

re¼rRT¼P

i

~ g_iDt

2 X

i

f_iZ_i

(19)

3.3. Boundary conditi006Fns of inlet and outlet of the microchannel

Having more accuracy and stability, non-equilibrium bounce back model is used at the inlet and outlet for hydrodynamic domain [45]:

~f₁¼~f₃þ2 3r_iu_i

~f₅¼~f₇þ1 2

~f₄~f₂ þ1

6r_iu_i

~f₈¼~f₆1 2

~f₄~f₂ þ1

6riu_i

(20)

~f₃¼~f₁2 3routuout

~f₇¼~f₅1 2

~f₄~f₂ 1

6r_outu_out1 2r_outv_out

~f₆¼~f₈þ1 2

~f₄~f₂ 1

6routuoutþ1 2routvout

(21)

The same method is applied to determine the unknown thermal distribution functions at the inlet and outlet:

~ g₅¼

6reþ3dtP

i

f_iZ_i6ð~g₀þ~g₂þg~₃þg~₄þg~₆þg~₇Þ 2þ3u_iþ3u²_i

h

3:0þ6u_iþ3:0u²_ii 1 36~g₁

¼

6reþ3dtP

i

f_iZ_i6ðg~₀þg~₂þ~g₃þ~g₄þ~g₆þ~g₇Þ 2þ3u_iþ3u²_i

h

1:5þ1:5uiþ3:0u²_ii1 9~g₈

¼

6reþ3dtP

i

f_iZ_i­6ð~g₀þ~g₂þ~g₃þ~g₄þ~g₆þ~g₇Þ 2þ3u_iþ3u²_i

h

3:0þ6u_iþ3:0u²_ii 1 36

(22)

Eqs. (20) and (22) are applied for inlet hydrodynamic and thermal distribution functions using“u_i”which is known from the specified value of Re¼0.01. Eqs.(21) and (23)are also used for the outlet flow in which uout, vout show the horizontal and vertical macroscopic velocities in x and y directions at the last horizontal grid of domain. Unlike ui, the values of uout, voutare not constant and should be determined at each time step during the solution process.

3.4. Boundary conditions along the walls of the microchannel

Using Lsas the slip length, uwallas the wall velocity and uliquidas the velocity of the liquid on the wall; slip velocity along the microchannel walls can be simulated as follow[46]:

Du_wall¼u_fluidðy/wallÞ u_wall¼L_svu_fluidðyÞ vy

wall

(24)

where recalls the linear Navier boundary condition. However more shear rates correspond to fail in Navier conditions. Therefore in this study, it is assumed that shear rates own low values of which shows the appropriate consistency between the slip length and linear Navier model. It plays a rule like Knudsen number (Kn) in a gas microflow which concerns the linear slip boundary condition as D^Uwall¼KnvU/vY.

The following equation is used to simulate the nanofluid slip velocity on the stationary walls[47]:

~g₆¼

6ð~g₁þ~g₅þ~g₈Þ 3dtX

i

c_ixZ_if_i6reuout

23uoutþ3u²_out h

3:06:0uoutþ6:0voutþ3:0u²_outþ3:0v²_out9:0uoutvout

i 1 36

~ g₃¼

6ð~g₁þ~g₅þ~g₈Þ 3dtX

i

c_ixZ_if_i6reuout

23uoutþ3u²_out h

1:51:5uoutþ3:0u²_out1:50v²_outi1 9

~g₇¼

6ð~g₁þ~g₅þ~g₈Þ 3dtX

i

c_ixZ_if_i6reuout

23uoutþ3u²_out h

3:06:0uout6:0voutþ3:0u²_outþ3:0v²_outþ9:0uoutvout

i 1 36

(23)

us¼±bvu vy

y¼0;h (25)

wherebshows the coefficient of slip length. Using this equation in dimensionless form, leads to:

U_s¼±BvU vY

Y¼0;1

(26) However to be used in LBM, the recent equation should be written based on the density-momentum distribution function.

Hence, a combination model of bounce back and specular boundary condition is used which is called specular reflective bounce back model[12]. This model shows a desired accuracy to estimate the slip velocity in LBM. For the bottom wall:

~f₂¼~f₄

~f_5;6¼r~f_7;8þ ð1rÞ~f_8;7 (27) The same equations can be written for the top wall.

Using the same procedure done for Eq. (24), the values of temperature jump along the mirochannel walls can be determined by Ref.[48]:

DT_wall¼T_fluidðy/wallÞ T_wall¼zvT_fluidðyÞ vy

wall

(28) In the recent equation,zis the temperature jump distance. In addition, it can be written in non-dimensional form as:

qqwall¼ B Pr

vq vY

Y¼0;1

(29) Eq.(29)is written based on internal energy distribution function for LBM by applying diffuse scattering boundary condition (DSBC) to simulate the temperature jump[49]. For example for the bottom wall:

~

g_2;5;6¼ 3

rweg^e_2;5;6ðr_w;u_w;e_wÞð~g₄þ~g₇þ~g₈Þ (30) Because of using lattice Boltzmann method, it is not possible to apply Eqs.(26) and (29)directly for the current study. These two equations were used in some of the previous works as well[47,48]

in order to derive the first order of liquid slip flow along the boundaries. Hence, the aim of presenting them here is just to emphasize on considering the slipflow regime and providing the appropriate formulation based on the hydrodynamic and thermal distribution functions which can be used for LBM formulation.

Hence Eqs.(27) and (30)are applied to simulate the slip velocity and temperature jump of nanofluid. More details regarding the treatment of temperature jump can be found in Ref.[49].

Present work intends to take into account the temperature jump and its effects which have been mostly excluded in previous re- ported papers. This implies that there will be a certain amount of temperature gradient (even a minor value) between the wall and the nanofluid layer over. Consequently, the bulk temperature should be used to estimate the Nusselt number as follows:

Nu¼ k_nf.

k_fD_HðvT=vyÞ_w

T_wT_balk (31)

Eq.(31)implies that Nu changes asymptotically to a constant value along the walls. This value is named outlet Nusselt number (Nu) which is analyzed during the following parts.

4. Grid study and validation

Force convection of nanofluid through a microchannel is simu- lated by LBM. To do this, a FORTRAN computer code is developed.

Table 1a and b show different lattice nodes for grid independency

Table 1a

Grid study at4¼0 andВ¼0.015.

90045 80040 70035

CfRe 21.13 21.10 21.04

Nu 7.25 7.23 7.18

Table 1b

Grid study in the point of x¼0.04L, y¼0.5 at4¼0.04 (for Al2O3) andВ¼0.03.

90045 80040 70035

U 1.263 1.261 1.257

q 1.570 1.567 1.562

Fig. 3.Validation of velocity and temperature profiles for the compressible air slip flow in a microchannel with those of Kavehpour et al.[5].

study which among them the grid of 80040 is selected for the following computations.Fig. 3shows the validation of velocity and temperature profiles for the compressible air slipflow in a micro- channel presented by Kavehpour et al.[5]at two different cross sections which shows an acceptable concordance between the results.

Moreover, Comparison of the current results by LBM versus analytical solution of Kandlikar et al.[1]is shown inFig. 4a. Their study focused on a planeflow between parallel plates with the width of 2 h considering first order of slip velocity which was

solved analytically as u^þ¼1-y^*2þ8Kn where u^þ¼u/((-h²/2m^)(dp/

dx)) and y^*¼y/h[1]. Good agreement is observed between the LBM numerical results of the present work with those by Kandlikar et al.

[1]. The last selected case (Fig. 4b) for validation is the nanofluid flow in a channel (not in a microchannel and for higher values of Re) which was reported by Santra et al.[30]. Their chosen nano- fluid, is a mixture of water-Cu and the channel walls are in higher temperature than the inletfluid.

5. Results and discussions

The effects of different slip coefficients (B) and different kinds of nanofluid at various4are investigated. Thermophysical properties of Ag, Cu and Al2O3nanoparticles as well as pure water ones are presented inTable 2. It is reminded that Reynolds number, Re¼rnf

u_iD_H/mnfand Prandtl number, Pr¼ynf/anfare also demonstrated for the nanofluid mixture.

Liquid slipflow regime would occur at 0.1<B<0.001. However, at low values of B (like B¼0.001), slip velocity and temperature jump amounts are not significant along the microchannel walls.

Therefore, to show the slip effects on the walls more clearly, the lower limit is selected as B¼0.005. On the other hand, for higher values of B than the upper limit (B>0.1), thefirst order slip velocity and temperature jump equations (Eqs. (26) and (29)) would not work accurately. Moreover at B¼0.1, the results from Eqs.(26) and Fig. 4.a Comparison of fully developed velocity profiles from the LBM numerical

approach versus analytical solution of Kandlikar et al.[1]where u^þ¼u/((-h²/2m)(dp/

dx)) and y^*¼y/h. b Numfrom present work with those of Santra et al.[30]for nanofluid channelflow.

Table 2

Thermophysical properties of Cu, Ag, Al2O3and water.

cp(J/Kg K) r(Kg/m³) K (W/mK) m(Pas)

Water 4179 997 0.6 8.9110⁴

Cu 383 8954 400 e

Ag 235 10,500 429 e

Al2O3 765 3970 40 e

Fig. 5.U profiles of water-Al2O3nanofluid along the microchannel atВ¼0.03 and B¼0.05 for4¼0.04.

(29) show very little detour in comparison with LBM results.

Consequently, to avoid inaccuracy in the results, the higher limit is selected as B¼0.05.

5.1. Effects of different slip coefficients

Fig. 5indicates U¼u/uiprofiles for water-Al2O3nanofluid along the microchannel atВ¼0.03 and B¼0.05 for4¼0.04. The profiles of X¼0.08L and X¼0.16L belong to the fully developed condition which means a short entrance length for X¼0.02L and X¼0.04L.

Moreover, more slip coefficient (B) corresponds to higher signifi- cant changes in the profiles of the entrance length.Fig. 5also shows that Umaxof the fully developed region is less than 1.5 in contrary to macroflows. This can occurred due to the existence of slip velocity on the walls. Fig. 6 shows the q ¼T/T_i profiles of water-Al₂O₃ nanofluid atВ¼0.03 and B¼0.05 for4¼0.04. Heat transfer from hot walls to the cold nanofluid leads to the increase the fluid temperature. Larger temperature jump is also seen at X¼0.02L and X¼0.04L although its value decreases at higher Xs. Dimensionless temperature of the microchannel wall equals toqw¼Tw/Ti¼2 which means in the lack of temperature jump, thefluid's temper- ature should be equals to 2 along the whole length (0<X<L) for Y¼0 and Y¼1. However, as it is seen inFig. 6, thefluid temper- atures adjacent to the walls (at Y¼0 and Y¼1) are significantly

less than 2 at X¼0.02L and X¼0.04L. These imply of temperature jump existence at these areas which has decreasing trend along the walls with X. An arrow is drawn inFig. 6(for X¼0.02L) in order to have more visibility.

U and q ^profiles of water-Al₂O₃ at different values of B for x¼0.08L (fully developed condition) and4¼0.04 are shown in Fig. 7. Larger amounts of Umaxare seen at horizontal centerline as well as less slip velocity at B¼0.005 than those of B¼0.03 and B ¼0.05. Larger temperature jump at more B can be also seen obviously in thisfigure. Water-Al2O3 nanofluid streamlines and isotherms at B¼0.005 and B¼0.05 for4¼0.04 are presented in Fig. 8. Inlet cold nanofluid from the left side of microchannel cools the hot walls and then leaves it from the right outlet. Streamlines are completely horizontal and symmetric through the microchannel.

Fig. 9andFig. 10 delineate the variations of slip velocity, Us, temperature jump,qs, and local Nusselt number, NuX, of water- Al₂O₃nanofluid along the microchannel walls at different Bs for 4¼0.04. Us,qsand NuXstart off from their maximum values at the inlet and then decrease asymptotically with X, reaching to the constant values at the end. Rise in the values of B, leads to increase in bothqsand Us, however it causes reduction in NuX. According to Eq.(31), Nusselt number can be estimated with (vT/vy) at y¼0 and y¼H. The largest temperature gradient ofvT/vy is achieved at inlet (at X¼0 and nearby) where nanofluid temperature is equal to

Fig. 6.qprofiles of water-Al2O3nanofluid along the microchannel atВ¼0.03 and B¼0.05 for4¼0.04.

Fig. 7.U andqprofiles of water-Al2O3nanofluid at different B for x¼0.08L and 4¼0.04.

T_nanofluid ¼ T_i¼T_cold. Along the microchannel walls and by increasing X, nanofluid temperature would increase so that Tw¼Thot>Tnanofluid>Ti¼Tcoldwhich leads to decrease invT/vy. As a result, Nusselt number would have its maximum value at the inlet

and then reaches to a constant value along the walls. Moreover, this phenomenon can occur more substantially for nanoparticles with higher thermal conductivity at higher volume fractions (4) which is to be investigated in the next section.

Fig. 8.Streamlines (top) and isotherms (bottom) of water-Al2O3nanofluid at B¼0.005 (line) and B¼0.05 (dash) for4¼0.04.

Fig. 9.Slip velocity, Us, and temperature jump,qs, of water-Al2O3nanofluid at different B for4¼0.04.

Fig. 10.NuXof water-Al2O3nanofluid along the microchannel wall at different B for 4¼0.04.

Fig. 11.Usof pure water (4¼0) and water-Ag nanofluid (4¼0.04) at different B.

5.2. Effect of types of nanofluid and volume fraction of particles

Fig. 11andFig. 12show Usandqsof pure water (4¼0) and water-Ag nanofluid (at 4 ¼0.04) at different values of B. The change in4does not have significant effects on U_s, whereasqsis more sensitive to it.Fig. 12illustrates the entrance region which has the highest temperature gradient occurring close to the walls. This fact implies the importance of temperature jump and its effects, especially at entrance length, in spite of findings by previous studies[32e34]. NuXof water-Ag nanofluid decreases with B and increases with 4 based on the results brought in Fig. 13. In thisfigure, local Nusselt number profiles are shown at different 4 for В ¼ 0.03 and B ¼ 0.05. Finally, outlet Nusselt number for nanofluid, defined in Eq.(31), is presented inFig. 14 with values of 4 and B for different types of nanoparticles. It shows the substantial effect of using nanofluid to increase the heat transfer rate where addition of 2% nanoparticles of Ag or Cu leads to an increase of almost 30% in Nusselt number.

However, this phenomenon occurs with lower intensity at more B. Moreover, the following correlation is developed to deter- mine the nanofluid Nusselt number in a microchannel as a function of 04 0.04, 0.005B0.05 and 2.4Pr¼ynf/ anf6.2:

Nu¼ ð26:45Þð0:64⁴Þ

169:93^B 0:85^Pr

(32)

The averaged and maximum deviations of this correlation are 2.4% and 7.6%, respectively; with only three points with deviations exceeding 5%. According to this correlation and its statistical quantities of4, B and Pr,Fig. 15is drawn for the Nusselt number.

Moreover, Nusselt number values from Eq.(32)versus numerical data are presented inFig. 16which implies appropriate accuracy of presented correlation.

6. Conclusion

Nanofluid forced convection in a microchannel was studied numerically by thermal lattice Boltzmann method (TLBM). Effects of different types of nanofluid (water-Ag, water-Cu and water- Al2O3)were investigated for three values of nanoparticles volume fraction (4¼0,4¼0.02 and4¼0.04). Slip coefficient was selected as 0.005< B < 0.05 in order to consider the slip velocity and temperature jump effects.

It was observed that fully developed condition was achieved after a small entrance length. However, Umax at fully developed region was less than 1.5 unlike the macroflows. Us,qsand NuX

started off from their maximum values at the inlet and then decreased asymptotically with X, reaching to the constant values at the end. Rise in the values of B, led to increase in bothqsand Us, however it caused reduction in Nu_X.

The change in4did not have significant effects on U_s, whereas qs was more sensitive to it. This elucidated the importance of

Fig. 12.qsof pure water (4¼0) and water-Ag nanofluid (4¼0.04) at different B.

Fig. 13.NuXof water-Ag nanofluid along the microchannel wall at different4for В¼0.03 and B¼0.05.

temperature jump and its effects at entrance length. Using 2%

nanoparticles of Ag or Cu led to an increase of almost 30% in Nusselt number. However, this phenomenon occurred mildly at more B. Moreover, to predict the nanofluid Nusselt number in a

microchannel, a correlation was developed as a function of 4, B and Pr.

It is recommended to apply nanofluid with lower B and higher4 and Pr to increase Nu; however, the effects of4and Pr are more significant.

Fig. 14. Nanofluid outlet Nu with values of4and B for different types of nanoparticles.

Fig. 15.Outlet Nu from Eq.(32)at4¼0.0, 0.02, 0.04 and B¼0.01, 0.02, 0.03, 0.04, 0.05 for different types of nanoparticles.

Fig. 16.Outlet predicted Nusselt number from Eq.(32)versus numerical data.

Acknowledgment

The author gratefully acknowledges the Research&Technology affairs of Najafabad Branch, Islamic Azad University, Isfahan, Iran, for the support in conducting this research work.

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