A two-way couple of Eulerian-Lagrangian model for particle transport with different sizes in an obstructed channel

    A two-way couple of Eulerian-Lagrangian model for particle transport with different sizes in an obstructed channel Mahla Maskaniyan, Saman Rashidi, Javad Abolfazli Esfahani PII: DOI: Reference:

S0032-5910(17)30161-4 doi:10.1016/j.powtec.2017.02.031 PTEC 12369

To appear in:

Powder Technology

Received date: Revised date: Accepted date:

6 October 2016 24 January 2017 15 February 2017

Please cite this article as: Mahla Maskaniyan, Saman Rashidi, Javad Abolfazli Esfahani, A two-way couple of Eulerian-Lagrangian model for particle transport with different sizes in an obstructed channel, Powder Technology (2017), doi:10.1016/j.powtec.2017.02.031

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ACCEPTED MANUSCRIPT A two-way couple of Eulerian-Lagrangian model for particle transport with

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different sizes in an obstructed channel

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Mahla Maskaniyan, Saman Rashidi*, Javad Abolfazli Esfahani Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran

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*Corresponding author contact details: Mech. Eng. Dep., Ferdowsi University of Mashhad; P.O. Box 91775-1111,

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Mashhad, Iran, Email: [email protected].

Abstract:

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In this paper, a two-way couple of Eulerian-Lagrangian model is used to simulate and account the discrete nature of Al2O3 particles in a particulate channel flow with a built-in heated

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obstruction. The governing equations for fluid flow and particle motions are solved by using the finite volume and trajectory analysis approaches. The simulations are performed for different

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particle sizes and solid volume fractions of particles (φ) in the ranges of 30-500 nm and 0-0.05, respectively and at fixed values of blockage ratio (S=1/8) and Reynolds number (Re=100). The effects of interaction forces acting between the fluid and particles containing the drag, gravity, Brownian, and thermophoresis forces on the particle transport and thermal behaviour of system are investigated. It was found that the nanometer particle does not follow the flow streamline and in fact diffuses across the streamlines. For larger sizes (i.e. 100 and 250 nm), particles concentrate in the vorticity regions around the periphery of the vortices. Moreover, the particle deposition percentage increases with an increase in the particle size. Keywords: Eulerian-Lagrangian; Two-way couple; Particle size; Obstruction; Interaction forces Nomenclature A

surface (m2) 1

ACCEPTED MANUSCRIPT specific heat (J kg-1 K-1)

Cm

constant (=1.14)

Cs

constant (=1.17)

Ct

constant (=2.18)

D

block size (m)

D1-5

sub-domain (-)

dp

nanoparticle diameter (nm)

F, f

force (N)

h

heat transfer coefficient (W m-2 K-1)

H

channel width (m)

k

thermal conductivity (W m-1 K-1)

KB

Boltzmann constant (J K-1)

Kn

Knudsen number (=2λ/d)

Ld

downstream distance of the cylinder (m)

Lu

upstream distance of the cylinder (m)

m

mass (kg)

m

number of grids in the vertical (y) direction (-)

n

number of grids in the horizontal (x) direction (-)

n

normal direction to the channel walls (m)

np

number of particles within a cell volume (-)

Nu

Nusselt number (-)

Nu

surface-averaged Nusselt number (-)

Nu

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Cp

time-averaged Nusselt number (-)

2

ACCEPTED MANUSCRIPT pressure (Pa)

Pr

Prandtl number (-)

Re

flow Reynolds number (-)

S

blockage ratio (-)

St

Strouhal number (-)

Sh, Sv

particles source terms in the base fluid equations

t

time (sec)

T

temperature (K)

x,y

Cartesian coordinate components (m)

u, v

velocity components in x and y directions (m s-1)

U0

velocity at the centerline of the channel (m s-1)

V

velocity vector (m s-1)

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D

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Subscripts/superscripts B

Brownian

c

cold

D

drag

f

base-fluid

h

hot

m

mean

p

particle-pressure

th

thermophoresis

v

viscous

w

wall

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p

3

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non-dimensional variables

Greek symbols particle concentrations

μ

dynamic viscosity (kg m-1 s-1)



kinematic viscosity (m2 s-1)

ρ

density (kg m-3)

α

thermal diffusivity of fluid (m2 s-1)

λ

fluid mean free path (m)

δV

cell volume (m3)

δ

distance between particles (nm)

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1. Introduction

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φ

The study of fluid flow and convective heat transfer in a channel with a built-in heated

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obstruction has received considerable attention due to numerous applications in energy related engineering problems. Such types of applications include compact heat exchangers, air conditioning, electronic cooling, oil and gas flows in reservoirs, chimney stacks, cooling towers, etc. [1]. Moreover, modern heat transfer systems such as the heat sinks for electronic components are based on arrays of short cylindrical pin-fins located inside a channel. In most of these applications, improving the heat transfer rate is an essential challenge for engineers. The heat transfer capacity of the most convective fluids is not sufficient for actual processes due to the low thermal conductivity of these fluids. Adding nanoparticle is a new way to increase the thermal conductivity [2-18]. The particle deposition (particulate fouling) is a destructive phenomenon of nanofluids that could decrease the heat transfer rate. Detail understanding of the transport processes of the nanoparticles leads to prevent this phenomenon.

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ACCEPTED MANUSCRIPT Some researchers simulated nanofluid flow around obstacles. Bovand et al. [19] controlled simultaneously the flow and heat transfer around an equilateral triangular cylinder by using

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nanofluid and changing the orientations of the cylinder. The cylinder was placed against the flow

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by the vertical side, apex, and diagonal side. They named these placements side facing flow, vertex facing flow, and diagonal facing flow, respectively. They found that the maximum

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impacts of nanoparticles on heat transfer improvement was for the case of side facing flow and

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the minimum one was belong to the vertex facing flow. After that, Rashidi et el. [20] optimized this problem by response surface methodology to determine the optimum conditions for the

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maximum heat transfer rate and the minimum drag force. Orientations of the obstacle, values of the solid volume fraction, and Reynolds number were considered as the influencing parameters

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in this study. Their study indicated that the minimum drag force is occurred between the diagonal and vertex facing cases. Rashidi et al. [21] improved the convective heat transfer around a

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triangular cylinder by nanofluids and controlled instabilities of the unsteady flow by magnetohydrodynamics. Their results revealed that the drag force decreases with an increase in Stuart number for small values of Stuart number. Some researchers simulated nanofluid flow in a cavity [22]. Abu-Nada and Oztop [23] simulated nanofluid natural convection in the partially heated rectangular enclosures. They reported that the heat transfer enhances as the volume fraction of nanoparticles increases for all values of Rayleigh number considered in this research. Oztop and Abu-Nada [24] investigated the effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid. They found that inclination angle is a control parameter for nanofluid-filled enclosure. Bahiraei [25] performed a review on different numerical methods for simulating the nanofluid. A more realistic method for modelling the nanofluids is Eulerian-Lagrangian approach. This model

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ACCEPTED MANUSCRIPT treats the individual particles, and considers the forces between the fluid and particles. In this approach, the base-fluid simulates by an Eulerian formulation, while the particles individually

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tracks using a Lagrangian trajectory analysis method. Rashidi et al. [26] used this approach to

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simulate the nanofluid around a triangular obstacle. They considered two different wall boundary conditions as particle boundaries. These boundaries were reflect and trap (absorbing condition).

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They found that for absorbing walls, the concentration drops near the obstacle surface, while for

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reflecting wall the concentration increases slightly near the wall. Some researcher studied the fluid flow and heat transfer in a channel with a built-in square

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cylinder. Valencia [27] investigated numerically the unsteady laminar flow and force convection heat transfer inside a channel with a built-in square cylinder. Their results indicated that the

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square cylinder increases the apparent friction factor in the channel in comparison to the empty

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one. Turki et al. [28] performed a numerical study on unsteady flow field and heat transfer in a

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channel with a built-in heated square cylinder. They showed that the critical value of Reynolds number relative to transition from steady to periodic flow decreases with an increase in the Richardson number.

Some researchers investigated the particle dispersion and distribution inside a closed domain such as cavity [29]. Garoosi et al. [30] modeled the particle deposition with micro size for natural convection flow inside a square cavity by using an Eulerian–Lagrangian approach. Several pairs of heater and cooler were placed inside the cavity. They observed that the chance of particle deposition on the cold wall increases with an increase in the number of the coolers. Finally, Jafari et al. [31] investigated the particle deposition and distribution with micro size inside a channel with a built-in square cylinder by using a lattice Boltzmann method. They

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ACCEPTED MANUSCRIPT showed that the deposition of particles on the front side of the cylinder decreases with a decrease in Stokes number.

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The purpose of the present work is a numerical study on particle transport and convective heat

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transfer in a channel with a built-in heated obstruction by using an Eulerian-Lagrangian approach. The effects of interaction forces acting between the fluid and particles containing the

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drag, gravity, Brownian, and thermophoresis forces on the particle transport in the system are

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examined. Literature review showed that most of researches for nanofluids simulation used oneway couple of Eulerian-Lagrangian approach while, the studies on two-way couple are limited.

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Accordingly, this research focused on this topic.

2. Theoretical formulation

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2.1. Physical model

Computational domain and coordinate system of the problem are presented in Fig. 1. As shown

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in this figure, a 2D channel with a built-in heated obstruction is considered. Fluid flow has a parabolic velocity and uniform temperature, Tc, at the inlet section of the channel. The channel and block walls are kept at constant temperature (Th=310 K) in which is more than the fluid inlet temperature (Tc=300 K). The height of the channel is H and the lengths of the channel at upstream and downstream sides of the block are Ld and Lu, respectively. The side of the block is denoted by D. 2.2. Assumptions Following assumptions are used to simulate this problem: 

The simulations are performed for different values of particle size in the range of 30-500 nm at fixed values of blockage ratio (S=D/H=1/8) and Reynolds number (Re=100).



Unsteady, incompressible, and laminar flow is considered.

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ACCEPTED MANUSCRIPT 

Two-way couple of Eulerian-Lagrangian approach is used to simulate the nanofluid.



The interaction between the continuous phase (water) and the particulate phase (Al2O3)

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are fully considered in the calculations. However, the effect of collision between

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nanoparticles is negligible because the average distances between the particles are less

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than 1000 nm (580 nm for 5% volume fraction) and the chance of particle-particle collision is very low.

The coefficient of restitution for particle collision is considered to be 1.0.



To account the interaction of all discrete particles with the continuous phase, it is not

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

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feasible to track all physically particles. Instead, representative particles, or parcels, are tracked. Each parcel characterizes certain number of actual particles. 5000, 15000 and



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25000 parcels [32] are assumed at φ=0.01, 0.03, and 0.05, respectively for this study. Physical properties of Al2O3 particles are density (ρ=3970 kg m-3), specific heat (Cp=765

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J kg-1 K-1), and thermal conductivity (k=40 W m-1 K-1). 2.3. Governing equations

Two-way couple of Eulerian-Lagrangian approach is used to simulate the nanofluid motion. In this approach, the base fluid is treated as a continuum by using Eulerian model. Moreover, the dispersed particles are tracked by using the Lagrangian model. The particles can exchange mass, momentum, and energy with the continuous phase (base fluid) [33]. The governing equations for the continuous and discrete phases are stated as follows: 2.3.1. Continuous phase 

Continuity equation:

 .(V)  0.



(1)

Momentum equation: 8

ACCEPTED MANUSCRIPT   DV   P    2 V  Sv . t





Energy equation:

C p

DT  k  2T  S h , t

(2)

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

(3)

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 

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where Sv and Sh denote the source terms for the momentum and energy exchanges of particles

np

Sh   np

m p d Vp V dt

mp V

cp

dT p dt

,

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Sv   

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with the fluid, respectively. These terms are calculated by:

,

(4)

(5)

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where subscript ‘‘p’’ denotes the particle. Moreover, mp, np, and δV demonstrate the

respectively.

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nanoparticle mass, the number of particles within a cell volume, and the cell volume,

2.3.2. Discrete phase

As stated earlier, the moving of the particulate phase is tracked in the Lagrangian domain. An integrating method is used to calculate the trajectories of particles. The force balance is integrated on the particulate phase, and it should be equal to the particle inertia with forces acting on the particles. These forces are the drag, Brownian, gravity, and thermophoresis forces. The equations of motion for particles can be determined by [34]:

dX p dt

 Vp ,

(6)

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dV p dt

 FD (V f  V p )  Drag force

g ( p   f ) p



 , f th fB Brownian force Thermophoretic force

(7)

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Gravitational force

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where subscripts "f" and "p" indicate the fluid and particle, respectively.

18 f d p2  p C c

where

,

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FD 

The drag force:

(8)

is the Cunningham correction, calculated by:



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

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The different forces in Eq. 7 are defined by:



2 1.257  0.4e (1.1d / 2 ) , dp

(9)

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Cc  1 

The thermophoresis force [19, 20]:

f th  

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

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where λ denotes the fluid mean free path.

36 2 C s (k f / k p  Ct  Kn)

T ,  f  p d (1  3C m kn)(1  2k f / k p  2Ct Kn) T

where

2 p

,

(10)

and Kn denote the thermal conductivities of fluid and particle and the Knudsen

number, respectively. Moreover, the fixed values of 1.14, 2.18, and 1.17 are considered for Cm, Ct, and Cs, respectively. 

The Brownian force [35-37]:

S 0 , t

fB   where



(11)

indicates the zero-mean, unit-variance independent Gaussian random numbers.

Moreover, S0 is the spectral intensity of the Brownian force, defined by:

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ACCEPTED MANUSCRIPT 216K B T  p  2  f d 5p    f

2

  Cc  

, (12)

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S0 

Boltzmann constant (=1.38×10-23 J K-1), respectively.

   hA p (T f  T p ), 

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 dT p m p C p   dt

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The particle energy equation can be determined by [38]:

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where T, ν, and KB indicate the absolute temperature of the fluid, the kinematic viscosity, and the

(13)

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where Ap, Tf, and Tp denote the particle surface area, the local temperature of the continuous phase, and the particle temperature, respectively. Moreover, the heat transfer coefficient, h, is

kf dp

2  0.6 Re

0.5 p



pr f0.3 ,

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h

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calculated by [37]:

(14)

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where kf and Prf are the thermal conductivity and Prandtl number of the base fluid (water). Finally, Rep denotes the particle Reynolds number based on the relative velocity and the particle diameter (dp).

2.4. Boundary conditions

2.4.1. Boundary conditions for fluid phase 

Inlet boundary conditions (Parabolic velocity with a uniform temperature):

u  U 0 (1  (2 y / D) 2 ), v  0, T  Tc ,

(15)

where U0 is the velocity at the centerline of the channel. 

Channel and block walls (No-slip velocity and constant temperature):

u  0, v  0, T  Th . 

(16)

Outlet boundary condition (Zero gradient boundary): 11

ACCEPTED MANUSCRIPT u v T  0,  0,  0. x x x

(17)

Finally, it is considered that there is no flow inside the channel and the temperature is constant at

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t=0.

(18)

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2.4.2. Boundary conditions for particulate phase

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u  0, v  0, T  Tc .

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The escape type boundary condition is used at the inlet and outlet sections of the channel. The trajectory calculations are terminated when a particle leaves the channel. The particle inlet

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temperature is 300 K and when a particle collides with the wall, the particle temperature increases to 310 K. Finally, the reflect boundary with a restitution coefficient of 1 is used at the

2.5. Parameter definition 

f D , U0

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St 

The Strouhal number:

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walls (Upper and downer walls of the channel and block surfaces).

(19)

where f is the dimensional vortex shedding frequency.  Nu 

The local surface Nusselt number based on the channel width: hH T    kf n

,

(20)

on obstacle

where n and H are the normal direction to the channel walls and the width of the channel, respectively. Note that kf in above equation is the thermal conductivity of fluid. Also, superscript “*” denotes non-dimensional variables. T* and n* are defined by: T* 

T  Tm n , n  , Th  Tm H

(21)

where Th is the temperature on channel walls. Moreover, Tm is mean temperature, defined by: 12

ACCEPTED MANUSCRIPT 

H 2

 uTdy, 

(22)

H 2

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1 Tm  Hum

where um is the mean velocity, calculated by:

1 H

 udy. 

H 2

(24)

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Nu 

T . n

The surface-averaged Nusselt number: 1 NudA, A A

(25)

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

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The heat flux can be evaluated by:  the wall  h(Th  Tm )  k f qon

(23)

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um 

H 2

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

The time-averaged Nusselt number: t

Nu 

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

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where A is the surface area of the channel wall.

1 Nudt , t 0

(26)

where t denotes the time duration.

3. Numerical method of solution The governing equations for fluid and particle phases along the suitable boundary conditions are discretized and then solved by using a finite volume method. A staggered grid arrangement is used to discretize these equations. Note that the pressure and velocity components are stored at cell center and cell faces, respectively for staggered grid arrangement. The third-order accurate QUICK, Green–Gauss, and first order implicit methods are applied to discretize the convective, diffusive, and time derivative terms, respectively [39]. The pressure and velocity terms couple by

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ACCEPTED MANUSCRIPT the SIMPLE algorithm of Patankar [40]. The pressure-correction equation is derived by combining the continuity and momentum equations for fluid to apply the SIMPLE algorithm.

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The convergence criterion of solutions is set to be 10-3 for all parameters with an exception for

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the temperature where this considered value is set at 10−6. 3.1. Grid independence study and validation

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A non-uniform square grid is used inside the computational domain. The grids are finer near the

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channel and block walls. Grid distributions inside the computational domain with a near zone around the block are presented in Fig. 2. As shown in this figure, whole computational domain is

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divided into five sub-domains containing D1, D2, D3, D4, and D5. A grid independence study is performed to insure that the results are not sensitive to grid number. Different grid sizes are

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tested and the results for the average Nusselt number at Re=100, φ=0.01, and t=120 sec are presented in table 1. It should be stated that, n×m in this table denotes to the number of grids in

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the horizontal (x) and vertical (y) directions, respectively. It can be seen that the difference in average Nusselt number between cases 3 and 4 is 0.3%. Therefore, the grid number of case 3 is considered for the rest of simulation. The time history of velocity in y direction at Re=100, φ=0, x/H=1, and y/H=0 is shown in Fig. 3. As shown in this figure, there is the regular fluctuation with approximately constant amplitude after an initial transition. In order to show the validity of the present simulation, the results are compared with experimental data of Heyhat et al. [41]. The experiment was performed for laminar Al2O3–water nanofluid flow in a horizontal circular tube with a constant wall temperature. Note that due to the lack of experimental data for nanofluid flow through a parallel plate channel, the circular tube geometry is selected as a case for validation. Figure 4 displays the results of this comparison for pressure drop ratio at φ=0.01 and dp=40 nm. φ and dp are solid volume fraction and particle size,

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ACCEPTED MANUSCRIPT respectively. Moreover, the ratio of pressure drop is defined as the ratio of this parameter for the nanofluid to that of pure water. The results of both single phase and discrete particle models are

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available in this figure to investigate the accuracies of these models. This figure indicates that the

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discrete phase model predicts a higher pressure drop which is closer to the experimental results because this model accounts for the movements of individual particles, as well as the two-way

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interactions between the particles and the water, and also accounts for the variation of local

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concentration of particles.

4. Results and discussion

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In this section, the results of numerical simulations are presented for different particle sizes and solid volume fractions of particles (φ) in the ranges of 30-500 nm and 0-0.05, respectively and at

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fixed values of blockage ratio (S=1/8) and Reynolds number (Re=100). The effects of different parameters containing particle size and solid volume fraction of particles on the particle

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distribution, particle deposition, particle concentration, velocity, and thermal behaviour of system are investigated.

Figure 5 shows the particle distribution around vortex for different particle sizes at Re=100, φ=0.05, and t=120 sec. As can be seen in this figure, the particle distribution is rather random inside the channel for dp=30 nm. The effect of the Brownian force leads to randomize the particle distribution. This force is created by the random impact of fluid molecules on the suspended particles. Note that the Brownian force is larger in comparison to the inertial forces for smaller particle sizes. It is worth mentioning that the nanometer particle does not follow the flow streamline and in fact diffuses across the streamlines. For larger sizes (i.e. 100 and 250 nm), particles concentrate in the vorticity regions around the periphery of the vortices. Note that the particles are affected by forces generated by the vortices and vortical flow field for these ranges

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ACCEPTED MANUSCRIPT of size (i.e. 100 and 250 nm). However, the inertia forces of particles overcome the centrifugal forces generated by the vortices for the particles with dp=0.5 μm and this leads to exit the

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particles from the vortices path line and tend them toward the channel walls. Figure 6 presents the particle deposition on the bottom wall of the channel for different particle

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sizes at Re=100, φ=0.05, and t=120 sec. As shown in this figure, the particle deposition

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percentage increases with an increase in the particle size. This is due to the effect of gravity force

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on the particle deposition that becomes more significant for higher values of particle size. Note that the particle deposition for 0.5 µm particles is dramatically higher than the other sizes. It is

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worth mentioning that the particle deposition percentages are about 1.1, 2.05, 3.07, and 9.06 for dp= 30 nm, 100 nm, 250 nm, and 0.5 µm, respectively. It should be stated that for this figure, the

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trap boundary condition with a restitution coefficient of 0 is used for walls. Note that for the reflect boundary with a restitution coefficient of 1, the particle deposition is zero. Accordingly,

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we used trap boundary condition to show the effects of particle size on deposition. In other figures, the reflect boundary condition is considered on the walls. Figure 7 shows the particle concentration contours inside the channel at dp=30 nm, Re=100, t=120 sec, and different solid volume fractions. It is observed that the particle concentration is not homogeneous in the domain and there is a slight accumulation near the channel and block walls. This may be due to the rebound assumption on the boundary. It is worth mentioning that the particle concentration increases in the recirculating wake region and in the downstream region. Moreover, the mass diffusion boundary layer is growing along the channel. The growth of concentration boundary thickness increases with an increase in the solid volume fraction. The effects of thermophoresis force on the cross-stream particle velocity (Vp) along the centerline (y =0) at dp=30nm, Re=100, φ=0.01, and t=120 sec are seen in Fig. 8. It should be

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ACCEPTED MANUSCRIPT noted that the cross-stream velocity is plotted on an axial position of at y= 0. The sign of velocity changes from minus to plus in the downstream region due to the effect of block on the flow

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oscillation. For the wall warmer than the fluid, the thermophoresis force is toward the centerline

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of the channel (negative thermophoresis for upper wall and positive thermophoresis for lower wall). Therefore, the effects of thermophoresis force for upper and lower channel walls are

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opposite and the effect of each of them counteracts by otherone. As a result, the differences in

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velocity between two cases, with and without thermophoresis force, are negligible. Figure 9 presents the variations of Strouhal number with particle solid volume fractions for

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dp=30nm, Re=100, and t=120 sec. As shown in this figure, the Strouhal number increases with an increase in the particle solid volume fraction. This augmentation is about 7.44% for

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0<φ<0.05. Increase in the Strouhal number demonstrates a decrease in vortex detachment or flow

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separation time and a decrease in thermal boundary layer thickness that finally leads to an

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increase in the heat transfer rate. It should be noted that the frequency of vortex shedding is determined by Strouhal number and it is influenced by particle solid volume fractions. The isotherm contours for the cases of pure fluid (φ=0) and nanofluid (φ=0.03) at Re=100, dp=30 nm, and t=120 sec are displayed in Fig. 10. The contours for the case of nanofluid are presented for the water and particle phases, separately. Solid and dash lines indicate the pure water and nanofluid, respectively in the same figure. It is worth mentioning that the contours are made non-dimensionally by (T-Tin)/(Tw-Tin). As shown in this figure, the high temperature gradient area is visible near the front face of block, which this region has the thinner thermal boundary layer. Moreover, the isotherm contours extend further in the downstream flow due to develop of the vortex shedding in this region. The fluid temperature decreases for more distance away from the block due to the mixing of the flow in the downstream. However, the thermal

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ACCEPTED MANUSCRIPT boundary layer thickness and subsequently heat transfer increase by adding the particles to the water as the isotherms shifts towards the centre of the channel. The higher thermal conductivity

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of the nanofluid, Brownian diffusion of particles, and their drag are the main reasons for such behaviour. Finally, it is observed that there is a very small difference between the temperature

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contours for the fluid and particle phases in the case of nanofluid.

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The variations of local Nusselt number on the top wall of the channel for the cases of with and

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without block at Re=100, dp=30 nm, φ=0, and t=120 sec is plotted in Fig. 11. As shown in this figure, the Nusselt number increases significantly by placing the block inside the channel. A

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thermally developing flow region with an identical behaviour for both cases is observed in the inlet region of the channel. After location of block, the local Nusselt number for this case shows

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some oscillations due to the vortex shedding after the block with unsteady behaviour. The time-averaged Nusselt number over the top wall of the channel is plotted against the solid

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volume fractions of particles in Fig. 12 for the cases of with and without block at Re=100, dp=30 nm, and t=120 sec. It can be seen that the average Nusselt number increases with an increase in the solid volume fraction of particles for both cases. These enhancements of heat transfer are about 30.77% and 26.66% for the cases of with and without block, respectively at 0<φ<0.05. As a result, the effect of particle on the heat transfer enhancement increases by placing block inside the channel. Moreover, the Nusselt number increases by placing block inside the channel because mixing of the cold fluid in core of the channel with hot fluid near the heated wall improves due to the oscillations created by vortex shedding. This enhancement is about 11.344% at φ=0.05. Figure 13 shows the variations of local Nusselt number along the upper half surfaces of the obstacle for two values of solid volume fraction of particles at dp=30 nm, Re=100, and t=120

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Moreover, the averaged values of Nusselt number over the rear surface of the obstacle are lower

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in comparison to the front and upper surfaces because the temperature contours are less crowded behind the obstacle. Finally, the local Nusselt number increases as the solid volume fraction of

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particles enhances.

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5. Conclusion

In the present research, particle transport and convective heat transfer in a channel with a built-in

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heated obstruction were investigated numerically by using a two-way couple of EulerianLagrangian model. Effects of particle size and particle volume fraction on the particle

highlighted as follows:

The nanometer particle does not follow the flow streamline and in fact diffuses across the

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

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distribution and heat transfer behaviour were examined. The main results of this research are

streamlines. For larger sizes (i.e. 100 and 250 nm), particles concentrate in the vorticity regions around the periphery of the vortices. 

The mass diffusion boundary layer grows along the channel. The growth of concentration boundary thickness increases with an increase in the solid volume fraction.



The particle deposition percentage increases with an increase in the particle size. Moreover, the particle deposition for 0.5 µm particles is dramatically higher than the other sizes.



The effect of thermophoresis force on the cross-stream particle velocity is negligible.



The Strouhal number increases with an increase in the solid volume fraction. This augmentation is about 7.44% for 0<φ<0.05.

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The isotherm contours extend further in the downstream flow due to develop of the vortex shedding in this region. There is a very small difference between the temperature contours for fluid and the

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After location of block, the local Nusselt number for this case shows some oscillations

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particle phases for the nanofluid.

due to the vortex shedding after the block with unsteady behaviour. The effect of particle on the heat transfer enhancement increases by placing the block

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inside the channel.

The Nusselt number increases by placing the block inside the channel. This enhancement

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is about 11.344% at φ=0.05.

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483–489.

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Parabolic inlet velocity T=Tc Y Ld

H

u0

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Lu

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T=T

T=T Cylinder with diameter "D"

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Fig. 1. Computational domain and coordinate system

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(a) (b) Fig. 2. Grid distribution inside the computational domain with a near zone around the obstacle

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0.20

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Fig. 3. Time history of velocity at centerline (y=0) for Re=100, φ=0, and x/H=1

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Fig. 4. Variations of the pressure drop ratio for Al2O3-water nanofluid with Reynolds numbers at φ=0.01 and dp=40 nm

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dp=250 nm

dp=0.5 μm

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.. .. .. .. .. . ... ... . . . . ... . . . . ... .... .. ....... .. . . ... .. ... .... .. . . . .. . .. . . .. . .. .. . . . . . . . .... .. . . .. . . . .... . .. . . .. . .. .. .. ............ ... .. .. .. .. .... . .... . . . . . . .... .. . . .. . . . . . ... .. ... . . . . . . . . . .... . . . . . . . . . . .. . . . .. . ... . . . . .. . .. ... ... ... ..... ...... ................ ... ....... .. . .. ... .. ...... ......... .. . ....... .. . ... ...... .. .. ...... .. .......... ... .. ... .. .. .. .. .. . ... ...... .. .. .. .. .. ..... . ... .... ..... .. ..... . ..... .. .................... .. ... ......... .. ..... .. . .. ... .. ... . . .. .. .. ... . .. .. .. .. .... ... .. ........ .... . ... . .. .. .. ... . .. ... .. ... .... ..... .. ....... .. ............. .......... .................. .................... ............ ................... .. .. . ...... ..... .......... ... ...... .. .. .. .. ........ .. .. ..... . ...... .. ... . .... . .. .. .. .. ........ .. . . . .. .... .. . .. ..... ... .. ... .. ...... ...... ... .. .. .. . ..... . ............ ... . .. . ...... . ..... ... .. .. . ... .... ........ ...... . .. .. ...... . ... .. ... . . . . . . . .. . . . .. . .. . . .............. . . . .. .. .. .. .. .. .. .. .. .. ....... ... ... . .. . .. . . .. .. .. .. .. ..... .... .. .. ........... ............... . .... .. .. .... ..... .. .. . .. . .. . . ... . . ....... . .. . . . .. . .. ........ .... .. .. .... .. ... .... ....... .. . .... ... .. . . .. .. ... .. .. . . .. .. . . .. . . . .. . .. .. . . .. ... . . .. . .. . . .. . . . .. .. .. . . . . . . . . .. . .. . . . ..... .... ... . . . . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... .... . .. . .... .. .. ... .. . . .. . . . .. ... . .. .... .. . .. . . . . . . . .. . . ... . . . .. .. . . . . . . . .. ......... .......... . .. . .. ..... .. .. ............. . . .. . . . .. . . .. . .. . . . . . .... .... . ... .. .. ..... . ...... . .. .. .. . . ... ... . .. .... . . ..... .... . .... ..... . ... . ..... . . . . . .. . . ... .. ... . . . . . .. . . . .. . .. .. .. . . . . . .... . . . .. . . .. . . . . . . . .. .... .. . .. . .. . . . . .. . . . . . . . . . . . .. . . . .... .... . ... . .. . .. . .. . . . . . . . .. . .. . . .. ... . . . . .. . . .. .. .. . .. . . .. . . . . . . . . .... . . . . . .. . . .. ... . ... . .. . .. . . .. . . . . .. . .. .... . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . .. . . . . . . .. . . .. . ..... ................ .. .. .. ... ... . . . .. ....... . ... .. ........ .. ... ... . . . .. .. .. .. .. ..... ... .... ... ............... .. .. .... .. .. . ...... ... .... ... .. .. .... ... ... ...... . .... . .. ......... .... . .... .. . ......... . .. . . . .. .. .. .... .. ............ . . . . . . . . . . . . . .. . .. . ... .. . . . .. .. .. ... ... .... ...... ......... . .. . ............. ... .. ... .................. .. .... .. .. ............. .. ....... .. .... .. ..... .. ......... ... .. ... .......... . ... .... .. . . . . ... . . . . ... .... .. ....... .. . . ... .. ... .... .. . . . .. . .. . . .. . .. .. . . . .. . . . .... .. . .. ... . . ... ..... .. ........... ............. ...................... .... ... ... ... .... . ..... . . . . . . . . . .... . . . . . . . . . . .. . . . .. . ... . . . . .. ... . . . . . . .. . .. . . . . . . . ... .. . . . .. . . . . . .. . ... .. ... . .. . .. ... ... ... ..... ...... ............... ... ....... .. . .. ... .. ...... ......... .. . ........ .. ..... ...... .. .. ...... .............. .... ... ...... ..... ...................... ............................................................ ........................ ....................... ...... ..................... . . . .. .

30

ACCEPTED MANUSCRIPT Fig. 5. Particle distribution around vortex for different particle sizes at Re=100, φ=0.05, and

SC

RI

PT

t=120 sec

4

2

0

MA

D

TE

6

30 nm 100 nm 250 nm 0.5 m

AC CE P

Deposition %

8

NU

10

Fig. 6. Particle deposition for different particle sizes at Re=100, φ=0.05, and t=120 sec

31

φ=0.03

AC CE P

TE

D

MA

NU

φ=0.01

SC

RI

PT

ACCEPTED MANUSCRIPT

φ=0.05

Fig. 7. Nanoparticle concentrations for different solid volume fractions at Re=100, dp=30 nm, and t=120 sec

32

ACCEPTED MANUSCRIPT

Without thermophoresis With thermophoresis

PT

0.6

RI

0.4

SC NU

0.0

MA

Vp

0.2

-0.2

TE

-0.6

D

-0.4

AC CE P

-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

X/D

Fig. 8. Cross-stream particle velocity (Vp) along the centerline (y =0) for the cases of with and without thermophoresis force at dp=30 nm, Re=100, φ=0.01, and t=120 sec

33

ACCEPTED MANUSCRIPT

PT

0.148

RI

0.146

SC

0.142

NU

St

0.144

0.140

MA

0.138

D

0.136

0.01

AC CE P

0.00

TE

0.134

0.02

0.03

0.04

0.05



Fig. 9. Variation of Strouhal number with nanoparticle solid volume fractions for dp=30 nm, Re=100, and t=120 sec

34

PT

ACCEPTED MANUSCRIPT

NU

SC

RI

Pure water and Base liquid phase

MA

Particle phase Fig. 10. Isotherm contours for pure water and nanofluid (base liquid and nanoparticle phases) at

AC CE P

TE

D

Re=100, φ=0.03, and t=120 sec (Solid line: pure water; Dash line: nanofluid)

35

ACCEPTED MANUSCRIPT

20

PT

Without block With block

RI SC NU

10

MA

Nu

15

0 5

10

AC CE P

0

TE

D

5

15

20

25

30

35

x/H

Fig. 11. Variations of local Nusselt number along the channel for the cases of with and without

block installation at φ=0, Re=100, and t=120 sec

36

ACCEPTED MANUSCRIPT

PT

5 Without block With block

SC

RI

4

NU



3

MA

2

0 1

AC CE P

0

TE

D

1

2



3

4

5

Fig. 12. Variations of average Nusselt number on the channel wall with nanoparticle solid volume fractions in the presence of block and without block at dp=30 nm, Re=100, and t=120 sec

37

ACCEPTED MANUSCRIPT

0.5 x

1.5

0

2

SC NU

10

MA



15

RI

 

PT

20

0.0

0.5

1.0

AC CE P

0

TE

D

5

1.5

2.0

Distance along the obstacle waslls (x/D)

Fig. 13. Variations of local Nusselt number along the upper half surfaces of the obstacle for two values of solid volume fraction of particles at dp=30 nm, Re=100, and t=120 sec

38

ACCEPTED MANUSCRIPT Table 1. The effect of grid number on average Nusselt number at Re=100, φ=0.01, and t=120 sec

2( n×m)

3( n×m)

4( n×m)

5( n×m)

40×15 80×30 160×60 320×120

40×5 80×10 160×20 320×40

5×15 10×30 20×60 40×120

75×15 150×30 300×60 600×120

75×5 150×10 300×20 600×40

SC

NU MA D TE 39

Average Nusselt number

Percentage difference

2.7433 2.7845 2.8085 2.8170

1.5% 0.86% 0.3% ----

RI

1( n×m)

AC CE P

1 2 3 4

grid number

PT

No.

ACCEPTED MANUSCRIPT

AC CE P

TE

D

MA

NU

SC

RI

PT

Graphical abstract

40

ACCEPTED MANUSCRIPT Highlights The particle deposition percentage increases with increase in particle size.



Effect of thermophoresis force on cross-stream particle velocity is negligible.



The mass diffusion boundary layer grows along the channel.



The nanometer particle does not follow the flow streamline.

AC CE P

TE

D

MA

NU

SC

RI

PT



41