Mixed convection nanofluid flow over microscale forward-facing step

International Communications in Heat and Mass Transfer 78 (2016) 145–154

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Mixed convection nanofluid flow over microscale forward-facing step — Effect of inclination and step heights☆ A. Sh. Kherbeet a,⁎, H.A. Mohammed b, Hamdi E. Ahmed c, B.H. Salman d, Omer A. Alawi e, Mohammad Reza Safaei f, M.T. Khazaal g a

Department of Mechanical Engineering, KBU International College, 47800 Petaling Jaya, Selangor, Malaysia Department of Refrigeration and Air Conditioning Engineering, Technical College of Engineering, Duhok Polytechnic University (DPU), 61 Zakho Road- 1006 Mazi Qr, Duhok-Kurdistan Region- Iraq c Department of Mechanical Engineering, University of Anbar, Ramadi 31001, Iraq d DNV GL, Engineering Department, Las Vegas, NV 89146, USA e Department of Thermofluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, UTM Skudai, 81310 Johor Bahru, Malaysia f Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran g Refrigeration Department, Eng. Division, South Oil Company, Ministry of Oil, Basra, Iraq b

a r t i c l e

i n f o

Available online xxxx Keywords: Mixed convection Microscale forward-facing step Heat transfer enhancement Nanofluids Nanoparticle Base fluid

a b s t r a c t A numerical study of nanofluid flow and heat transfer of laminar mixed convection flow over a three-dimensional, horizontal microscale forward-facing step (MFFS) is reported. The effects of different step heights and the duct inclination angle on the heat transfer and fluid flow are discussed in this study. The straight and downstream walls were heated to a constant temperature and uniform heat flux respectively. The numerical results were carried out for step heights of 350 μm, 450 μm, 550 μm and 650 μm. Different inclination angles were considered to determine their effects on the flow and heat transfer. Ethylene glycol-SiO2 nanofluid is considered with a 25 nm particle diameter and 4% volume fraction. The results reveal that the Nusselt increases as the step heights increase. Additionally, no significant effect of the duct inclination angle is found on the heat transfer rate and the fluid flow. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The separation and reattachment flow region in the microscale forward-facing step (MFFS) plays an important role in the heat transfer performance. The flow separation and the subsequent reattachment are carried out by sudden compression in the geometry of the flow such as the forward-facing step (FFS). The local heat transfer has a large variation within the separation flow region and remarkable augmentation in heat transfer yields in the reattachment region. Thus, it is essential to understand the basic mechanism of heat transfer in thermal engineering applications in such flows, where cooling or heating is required. These applications could be required in energy system equipment, cooling systems for electronic equipment, combustion chambers, cooling passages for turbine blades, high performance heat exchangers and chemical processes. Remarkable mixing of high and low energy fluid takes place in the reattachment region within these devices, and subsequently significant effects occur to their heat transfer performance. In the backward-facing step (BFS), owing to it having the simplest design, a large number of studies have been conducted in ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected] (A.S. Kherbeet), [email protected] (H.A. Mohammed).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.08.016 0735-1933/© 2016 Elsevier Ltd. All rights reserved.

relation to this geometry. Conversely, only a few studies have discussed the flow over the FFS. This may be because the flow separation takes place at the edge of the step in the case of BFS geometry, causing a recirculation region behind the step position. However, when the flow passes over the FFS geometry one or more recirculation regions will be developed. The number of recirculation regions depends on the thickness of the momentum boundary layer at the step and the value of the flow velocity [1]. The existence of more than one recirculation region makes the FFS geometry more complicated compared to the BFS, with a very limited available data about the flow over a FFS. The effect of the step height (SH) on the flow over the BFS geometry in natural, mixed, and forced convection heat transfer has been examined extensively by former investigators such as Abu-Nada [2] and Nie and Armaly [3] and others. However, the effect of SH on the flow over a FFS has been discussed in just one study presented by Abu-Mulaweh [4] to the best knowledge of the authors. In this study, the measurements of heat transfer and fluid flow of turbulent mixed convection boundarylayer air flow over an isothermal two-dimensional, vertical FFS is presented. The experimental investigation was carried out in an existing low turbulence, open circuit tunnel that was oriented vertically. Three values of SHs were utilized to study their effects on the hydrothermal performance, which were 0 mm, 11 mm and 22 mm. The flat plate and both step geometries were supported in the test section of the tunnel and spanned its entire width of 85.1 cm. The upstream and downstream lengths were

146

A.S. Kherbeet et al. / International Communications in Heat and Mass Transfer 78 (2016) 145–154

Nomenclature Cp Dh dp g Gr H h h k Nu P Pr q Re s T T∞ Tw u ui u∞ U v V W X x,y,z Xi Xe Xr Y Z

specific heat, J/kg.K hydraulic diameter, 2 h, m nanoparticles diameter, nm gravitational acceleration, m/s2 Grashof number, gβqws4/(kv2) total channel height, m convective heat transfer coefficient, W/m2·K inlet channel height, m thermal conductivity, W/m.K Nusselt number, h.Dh/k dimensionless pressure, P = (p+ρgx)/ρ u2∞ Prandtl number, vf/αf heat flux, W/m2 Reynolds number, ρu∞ Dh/μ f step height, m fluid temperature, K temperature at the inlet or top wall, K temperature of the heated wall, K velocity component in x-direction, m/s local inlet velocity, m/s average velocity for inlet flow, m/s dimensionless streamwise velocity component, u/u∞ velocity component in y-direction, m/s dimensionless transverse velocity component, v/u∞ channel width, μm dimensionless streamwise coordinate, x/s streamwise, transverse and spanwise coordinates, μm upstream length, μm streamwise coordinate as measured from the step, μm reattachment length, μm dimensionless spanwise coordinate, y/s dimensionless transverse coordinate, z/s

Greek symbols φ nanoparticles concentration thermal diffusion of fluid, N.s/m2 αf β thermal expansion coefficient, 1/K θ dimensionless temperature, density of fluid, kg/m3 ρf ρs density of solid, kg/m3 νf kinematic viscosity of fluid, m2/s μ dynamic viscosity, N.s/m2 Subscripts o outlet eff effective f fluid s solid nf nanofluid w wall ∞ inlet condition

274.3 cm and 81.3 cm respectively. Both the upstream and the downstream walls and the step itself were heated to a constant and uniform temperature. The front edge of the upstream plate was chamfered to ensure a proper development of the boundary layer flow. The results clearly indicated that the introduction of the FFS significantly affects the flow characteristics in the recirculation region. The largest magnitude of maximum mean transverse velocity increases with the SH increase. The magnitude of the negative transverse velocity component in the flow region near the heated downstream wall decreases as the streamwise distance increases downstream from the step. The results showed that the effect

of the FFS somewhat diminishes as the streamwise distance increases downstream from the step. The values of turbulent intensities fluctuations at a streamwise location increase to its maximum value with the increases of distance from the heated wall, then decrease as the distance from the heated wall continues to increase, reaching to its minimum value at the edge of the boundary-layer. The results showed that, the local Nusselt number starts with its minimum value at the step edge and increases to its maximum value at the reattachment region. The magnitude of the local Nusselt number decreases as the distance continues to increase in the streamwise direction. The measured local Nusselt number downstream of the FFS increases with the increasing SH. One technique of heat transfer enhancement is by utilizing nanofluids. These are fluids in which nanometer-size particles are suspended in conventional heat transfer base fluids [5]. Past studies have shown that nanofluids exhibit enhanced thermal properties, such as higher thermal conductivity and convective heat transfer coefficients compared to the base fluid [6–9]. The nanoparticles are either metallic or nonmetallic materials such as Al2O3, SiO2, Cu, CuO, ZnO and TiO2 [10]. Several researchers have investigated the effect of nanofluids on the thermal conductivity enhancement [11–29]. The first investigation of the thermal behavior and nanofluid flow characteristics over the BFS was presented by Abu-Nada [30]. He reported that by increasing the nanoparticles volume fraction the Nusselt number can be enhanced. Mohammed et al. [31,32] studied the effect of nanofluids on mixed convective heat transfer over a vertical and horizontal BFS. Their results showed that the SiO2 nanofluid has the highest primary recirculation region and the diamond nanofluid has the highest Nusselt number in the primary recirculation region. More recently Kherbeet et al. [33] presented a numerical investigation of the nanofluid effect of laminar flow on a mixed convection heat transfer over a twodimensional microscale backward facing step (MBFS). It was revealed that the fluids with SiO2 nanoparticles were shown to have the highest Nusselt number. Moreover, increases in the nanoparticles' volume fraction increased the Nusselt number. As a continuous work, Kherbeet et al. [34] investigated the effect of SH of MBSF on the heat transfer and nanofluid flow characteristics. They outlined that the Nusselt number and skin friction coefficient increased with increasing the SH, while Reynolds number and pressure drop decreased. In addition, Kherbeet et al., [35] examined different types of nanofluids with different volume fractions and particles diameters in the MBFS. They displayed that silica oxide nanofluid provided the highest heat transfer rate, which increased with increasing nanoparticles concentrations and decreasing nanoparticles diameter. Besides, the static pressure and the wall shear stress increased with increasing particle concentration and decreasing particles diameter. In addition, they did not observed any effect for the nanoparticle volume factions, materials and diameters on the skin friction coefficient. From the above literature review, it is clearly shown that the effect of SH on mixed convection heat transfer and nanofluid flow over a threedimensional MFFS has not received any attention yet which motivated the present study. Moreover, there is no existing work discussing the effect of the inclination of MFFS on a mixed convection nanofluid flow. Therefore, the present study focuses on laminar mixed convection nanofluid flow over a 3-D MFFS having several values of SHs with different inclination angles with the horizon. The results of interest, including velocity distribution and skin friction coefficient, wall shear stress, pressure drop and Nusselt number are depicted to illustrate the effect of SH and the inclination angle on these parameters. 2. Channel flow system and implementing equations 2.1. Physical model and assumption The schematic diagram of the adopted geometry and the flow configuration used in this study is shown in Fig. 1. Four values of SH were chosen to study the effect of the step, these being 350 μm, 450 μm,


147

Table 1 Thermophysical properties for pure ethylene glycol and SiO2 nanoparticles. Particle type

ρ (kg/m3)

μ (N.s/m2)

k (W/m.K)

Cp(J/kg.K)

β(1/K)

Pure ethylene glycol SiO2

1114.4 2200

1.57E-02 -

2.52E-01 1.2

2415 703

6.50E-04 5.50E-06

The continuity, momentum and energy governing equations in nondimensional form, in Cartesian coordinates are given as [30]: ∂U ∂V ∂W þ þ ¼0 ∂X ∂Y ∂Z Fig. 1. Schematic diagram of a horizontal 3D microscale forward-facing step (MFFS).

550 μm and 650 μm. To study the effects of the inclination, different angles were considered as shown in Fig. 2. To ensure the fully developed flow in the channel inlet and outlet, the upstream wall length is regarded as 15 × 103 μm and the downstream wall length is considered 50 × 103 μm. The straight wall temperature and the downstream wall heat flux were fixed at 300 K and 12 W respectively. The total channel height and width are maintained at 900 μm and 3 × 103 μm respectively. The flow at the duct entrance was considered to be hydro-dynamically steady and fully developed. Streamwise gradients of all quantities at the duct exit were set to zero. The upstream wall and the step wall were considered to be adiabatic surfaces. The base fluid (i.e., ethylene glycol) and the nanoparticles are assumed to have a thermal equilibrium and no slip condition occurs. The fluid flow is considered to be incompressible and Newtonian. The formulation of the nanofluid thermophysical properties is affected by the buoyancy force, and the Boussinesq approximation is considered. The thermophysical properties of the nanofluid are given in Table 1. 2.2. Governing equations

U

∂U ∂U ∂U 1 ∂P þV þW ¼− ρ ∂X ∂Y ∂Z ð1−φÞ þ φ s ∂X ρf 1 þ Re

2

1 2:5

ð1−φÞ

ρ ð1−φÞ þ φ s ρf

!!

∂ U ∂X 2

2

þ

∂ U ∂Y 2

2

þ

∂ U

!

∂Z 2

þ

Gr x Re2

θ

ð2Þ U

∂V ∂V ∂V 1 ∂P þV þW ¼− ρs ∂Y ∂X ∂Y ∂Z ð1−φÞ þ φ ρf 1 þ Re

2

1 ð1−φÞ2:5


!!

∂ V ∂X 2

2

þ

∂ V ∂Y 2

2

þ

∂ V

!

∂Z 2

þ

Gr y Re2

θ

ð3Þ U

∂W ∂W ∂W 1 ∂P þV þW ¼− ρ ∂X ∂Y ∂Z ð1−φÞ þ φ s ∂Z ρf þ

1 Re

Introducing the following dimensionless quantities: u v w x y z ðp þ ρgxÞ ;V ¼ ;W ¼ ;X ¼ ;Y ¼ ;Z ¼ ;P ¼ ; u∞ u∞ u∞ Dh Dh Dh ρu2 ∞ T−T ∞ θ¼ ðqw s=kÞ

ð1Þ

2

1 ð1−φÞ

2:5


!!

∂ W ∂X 2

2

þ

∂ W ∂Y 2

2

þ

∂ W ∂Z 2

! þ

Gr z Re2

θ

ð4Þ

U¼

0 ∂θ ∂θ ∂θ 1 B B þV þW ¼ U B ∂X ∂Y ∂Z RePr @

knf kf

1

C ∂2 θ ∂2 θ ∂2 θ C þ þ C ðρCpÞs A ∂X 2 ∂Y 2 ∂Z 2 ð1−φÞ þ φ ðρCpÞ f

!

ð5Þ where Re ¼

ρ f um Dh vf Pr ¼ μf αf

Gr y ¼

gβqw s4 kv2

2.3. Boundary conditions The boundary conditions for the above set governing equations are: (i) Upstream conditions at X ¼ −

xi s H u , ≤Y ≤ : Ui ¼ i ; V ¼ 0; Dh Dh u∞ Dh

w w ≤W ≤ ; W ¼ 0; θ ¼ 0 Dh Dh The flow at the duct inlet is considered to be fully developed with an average velocity. Thus, the inlet velocity distribution is parabolic. and −

(ii) Downstream exit conditions at X = Xe: 0 ≤Y ≤ 2

2

2

w∂ U ∂ V ∂ W ∂ θ ¼ 0; 2 ¼ 0; ; ¼0 Dh ∂X 2 ∂X ∂X 2 ∂X 2 x xe H w w (iii) Straight wall at X ¼ − i ≤X ≤ and Y ¼ and − ≤W ≤ Dh Dh Dh Dh Dh U= 0 , V = 0 ,W =0 ,θ = 0 W≤

Fig. 2. Distributions of the inclination angles for the MFFS.

2

H w and − ≤ Dh Dh

148


The effective heat capacity is given as [38]:

(iv) Stepped wall conditions: x s Upstream of the step at − i ≤X ≤0; and Y ¼ : U ¼ 0; Dh Dh ∂θ V ¼ 0; W¼ 0; ¼0 ∂Y s ∂θ ¼0 At the step X = 0 and 0≤Y ≤ : U ¼ 0; V ¼ 0; W ¼ 0; Dh ∂X xe : U ¼ 0; Downstream of the step at Y = 0, and 0≤X ≤ Dh ∂θ V ¼ 0; W ¼ 0; ¼ −1 ∂Y

The effective thermal conductivity equation of nanofluid is presented by Vajjha et al. [36] as: keff ¼ kstatic þ kbrownian where kstatic is given by Ghasemi et al. [37] as:

kstattic

ð6Þ

where ks and kf are the thermal conductivities of the solid particles and the base fluid respectively. While the thermal conductivity due to the Brownian motion is given by Vajjha [36] as:

kbrownian

sffiffiffiffiffiffiffiffiffi KT f ðT; φÞ ¼ 5 10 βφρ f Cp f ρs ds 4

ð7Þ

where T f ðT; φÞ ¼ 2:8217 10−2 φ þ 3:917 10−3 T0 þ −3:0669 10−2 φ−3:91123 10−3 where K is the Boltzmann constant, T is the fluid temperature, T0 is the reference temperature. In the above Eqs. (1)–(4), the viscosity of the nanofluid is approximately considered as viscosity of a base fluid if containing a dilute suspension of fine spherical particles and is given by Corcione [38]: μ eff 1 ¼ −0:3 μf 1−34:87 dp =d f φ1:03 " #1=3 6M df ¼ Nπρfo

ð8Þ

Where μ eff and μ f are the viscosity of nanofluid and base fluid, respectively, dp is the nanoparticle diameter, df is the base fluid equivalent diameter and φ is the nanoparticle volume fraction. M is the molecular weight of the base fluid and N is the Avogadro number, and ρfo is the mass density of the base fluid calculated at temperature T = 293 K. The effective density of nanofluid is given as [39]: ρeff ¼ ð1−φÞρ f þ φρs

ð1−φÞðρCpÞ f þ φðρCpÞs

ð10Þ

ð1−φÞρ f þ φρs

where Cps is the heat capacity of the solid particles, and Cpf is the heat capacity of the base fluid. The effective thermal expansion is expressed as [39]: βeff ¼

ð1−φÞðρβÞ f þ φðρβÞs

ð11Þ

ð1−φÞρ f þ φρs

where βs and βf are the thermal expansion of the solid particles and base fluid respectively. The β equation for SiO2 particle material is expressed in Table 2, as given by Vijjha [39].

2.4. Nanofluid equations

" # ks þ 2k f −2φ k f −ks ¼ kf ks þ 2k f þ φ k f −ks

ðCpÞeff ¼

ð9Þ

where ρeff and ρf are the nanofluid and base fluid densities respectively, and ρs is the density of the nanoparticle.

3. Numerical procedure and grid test The finite volume approach was used to solve Eqs. (1)–(4), with the corresponding boundary conditions. The flow field was solved using the SIMPLE algorithm. The second-order upwind differencing scheme is considered for the convective terms. The diffusion term in the energy and momentum equations is approximated by second-order central difference, which gives a stable solution [40]. In the x-direction, a fine grid is used in the regions near the step and the point of reattachment to resolve the steep velocity gradients, while a coarser grid is used downstream of that point. However, the fine grid in the y-direction is used near the top, the bottom walls and directly at the step to ensure the accuracy of the numerical simulations and to save both the grid size and computational time. A non-uniform and quadrilateral element grid system is considered in the simulations. At the end of the iteration, the residual sum for each of the conserved variables is computed and stored. The convergence criterion required the maximum relative mass residual, based on the inlet mass, to be smaller than 1 × 10−3. A very fine mesh (180 × 34 × 60), considered to represent the base case of finding a better grid density and minimum time, was required for the iteration. The grid independent study was carried out for the case of flow over a FFS using ethylene glycol-SiO2 nanofluid as the working fluid and Reynolds number of 35. Solutions were performed with different grid numbers and densities, which are explained in Table 3. The grid size was 170 × 28 × 50 (170 grid points in x-direction, 28 grid points in y-direction and 50 grid points in z-direction) with 1.025, 1.3 and 1.2, as a successive ratio in x-, y- and z-directions respectively does not present significant variation in u-velocity, when compared to the u-velocity that is calculated with 180 × 34 × 60. Hence, this value of grid density is considered to give accurate results with a minimum iteration time. A detailed description of the grid generation can be found in Saldana and Anand [41]. 4. Code validation To demonstrate the validity and accuracy of the present model output and because of the absence of much related data to the current topic, comparisons with the literature are performed here. From the open literature above, the results of Saldana and Anand [41], which discussed the laminar flow over a 3D FFS geometry, is undertaken in the present research for validation. The results, represented by the

Table 2 β value for SiO2 particles and its boundary conditions. Type of particles

β

Concentration (%)

Temperature (K)

SiO2

1.9526(100ϕ)−1.4594 [39]

1% ≤ φ ≤ 10%

298 K ≤ T ≤ 363 K


149

Table 3 Grid independence study. Grid size x×y×z

Expansion factors ex × ey × ez

u-velocity at x = 103 μm

180 × 34 × 60 170 × 28 × 50 160 × 22 × 40 150 × 16 × 30 140 × 10 × 20

1.02 × 1.3 × 1.08 1.02 × 1.3 × 1.08 1.02 × 1.3–1.08 1.02 × 1.3 × 1.08 1.02 × 1.3 × 1.08

0.002446 0.002411 0.002352 0.002234 0.001945

velocity components in three dimensions, validated for two values of Reynolds number (Re = 400 and 800) as shown in Fig. 3. The figure shows that the present numerical data and the results of literature are in a good agreement.

ðu−velocityÞbase −ðu−velocityÞ %difference 100 ðu−velocityÞ base

Base 1.430907604 3.843008994 8.667211774 20.48242028

5. Results and discussions The numerical study presented in this work considered a nanofluid compound of Ethylene-glycol as a base fluid with SiO2 nanoparticles of

0.007 Re = 400, Saldana and Anand Re = 800, Saldana and Anand Re = 400, Present work Re = 800, Present work

2.8

Re = 400, Saldana and Anand Re = 800, Saldana and Anand Re = 400, Present work Re = 800, Present work

0.006

2.4

0.005 2

v (m/s)

0.003

1.2

0.002 0.8

0.001 0.4

0 0

0

0.25

0.5

0.75

1

0

0.25

0.5

z/W

z/W

(a)

(b)

0.075 Re = 400, Saldana and Anand Re = 800, Saldana and Anand Re = 400, Present work , Present work Re = 400,

0.05

Re = 800,

0.025

w (m/s)

u (m/s)

0.004 1.6

0

-0.025

-0.05

0

0.25

0.5

0.75

z/W

(c) Fig. 3. Comparison distributions with the results of Saldana and Anand [41].

1

0.75

1

150


4% as volume fraction and a diameter of 25 nm immersed in the base fluid. The Reynolds number was 35, while the downstream wall was maintained to a constant heat flux of 12 W. The straight wall temperature was fixed at 300 K. The upstream wall, step wall and the side walls are considered to be thermally isolated. Four values of the SH were considered, to study the step effect, which are 350 μm, 450 μm and 550 μm respectively, on the flow and thermal fields. Twelve angles were considered to study the effect of the inclination on the fluid flow and heat transfer. These were 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300° and 330°. The effects of SH and the inclination on the (velocity in three dimensions, pressure drop, skin friction coefficient, wall shear stress and Nusselt number) are presented and interpreted in this study.

These results were taken at the center width of the downstream section of the duct. As can be seen from this figure, the flow at the step is symmetric, relative to the center of the duct width and these distributions clearly show the development of a maximum (peak) in that velocity component at approximately z/(W/2) = 0. The results showed that the velocity increases with the SH increase. Thus, the duct of step height of 650 μm has the highest velocity profile, while the duct of SH of 350 μm has the lowest velocity. The transverse velocity distributions of different SHs along the spanwise direction for different x-sections and y-plane of 100 μm are shown in Fig. 5. It is found that at x = 65 μm the flow over the step of 350 μm has higher positive velocity, while at S = 450 μm, 550 μm and 650 μm the v-velocity is also positive but it is less than the corresponding velocity for S = 350 μm, as shown in Fig. 5-a. However, by increasing the x-section value to x = 325 μm, x = 650 μm and x = 1300 μm there two peaks of negative velocity are developed adjacent to the duct's side walls, as illustrated in Fig. 5b-c. The results revealed that the maximum peak value of the negative velocity is found to be at x-section of 325 μm

5.1. Effect of different step heights

8

8

The spanwise distributions of the streamwise velocity component (u) are presented in Fig. 4, for different x-locations and y = 100 μm.

7

u (m/s)

3

4

5

6 5 4

u (m/s)

3

Step = 350 Step = 450 Step = 550 Step = 650

6

7


x = 65 Micron, y = 100 Micron

0

0.5

0

-0.5

-1

1

(a)

(b)

1


7

3

5 4 3

4

u (m/s)

5

6

6

7

0.5

8


x = 1300 Micron, y=100 Micron

1

2

x = 650 Micron, y =100 Micron

2

u (m/s)

0

z/(W/2)

1 0

-1

-0.5

z/(W/2)

8

-1

-0.5

0

0.5

1

0

0

1

1

2

2

x = 325 Micron, y = 100 Micron

-1

-0.5

0

z/(W/2)

z/(W/2)

(c)

(d)

Fig. 4. Spanwise distributions of the u-velocity components on different x-sections (Re = 35).

0.5

1

1. 5


0. 2

Step = 350 Step = 450 Step = 550 Step = 650 x= 325 Micron, y= 100 Micron

v (m/s) -0 .1

0. 75

v (m/s)

0

1

0. 1

1. 25


151

0

0.5

1

-1

-0.5

0

z/(W/2)

z/(W/2)

(a)

(b)

0.5

01

-0.5

05

-1

0

0

0.


1


0.

0

-0 .3

0. 25

-0 .2

0. 5

x= 65 Micron, y=100 Micron

1 .0 -0

v (m/s)

5 .0

-0

v (m/s)


-0

.1

-0

.0

2


0.5

1

-1

3

0

.0

-0.5

-0.5

0

0.5

1

-0

-1

z/(W/2)

z/(W/2)

(c)

(d)

Fig. 5. Spanwise distributions of the v-velocity components on different x-sections (Re = 35).

and SH of 650 μm, as presented in Fig. 5-b. This is due to the increases of SH in the recirculation region after the step increase has caused an increase in the negative velocity. However, increasing the x-sections to 650 μm and 1300 μm the peak of the negative velocity value decreases and the SH of 650 μm has the lowest value, while the SH of 350 μm has the highest peak negative value. The spanwise velocity distributions of Eg-SiO2 nanofluid at different SHs along the spanwise direction are presented in this section. The results of the spanwise velocity trend for x = 65 μm showed that at the SH of 350 μm the spanwise velocity reached the higher value. However, for the same x-section and SH of 650 μm, the spanwise velocity had the lowest velocity profile, shown in Fig. 6-a. With the increase of x-section values it is found that the difference in the w-velocity, utilizing different SHs has decreased as shown in Fig. 6b-c. From all the figures it is found that the maximum peak value of the velocity is adjacent to the side wall. In addition, there is another peak opposite to that, also adjacent to the sidewall, indicating that the flow travels towards the sidewalls. At the central plane in the spanwise direction the w-component is equal to

zero along the streamwise direction. For different x-sections the higher spanwise velocity is found at x = 1300 μm, as shown in Fig. 6-c. According to Fig. 6-a, the w-component adjacent to the backstep presents negative values in the positive z-coordinate and positive values in the negative z-coordinate. Therefore, in this zone the flow is directed towards the side walls. This could be the reason why the minimum in u-velocity is located near the side walls and not at the central plane in the z-direction [42]. The effect of different SHs on the static pressure behavior presented over the stepped wall is shown in Fig. 7. It is found that the sudden compression in the duct significantly affects the static pressure behavior. The results show that the maximum static pressure value starts from the upper corner of the step, then continues to decrease in the streamwise direction on the downstream wall until it reaches its minimum value at the channel exit. It is revealed that the pressure drop increases with the SH increases and higher SH results in higher pressure drop. This might be due to the total static pressure of the dynamic fluids being a function of the static pressure and fluid


0. 3

152

0. 2

0.

2

0. 3


x= 65 Micron, y= 100 Micron

0. 1

0

0.5

-0 .2

1

-1

-0.5

0

z/(W/2)

z/(W/2)

(a)

(b)

0.5

1

0.

0.

2

1

-0.5

-0 .3

.2


-0

-1

0

w (m/s)

-0

-0

.1

.1

0

w (m/s)

0.

1



0.

0.

1

05


0

w (m/s)

0

w (m/s)


-0

-0

.0

.1

5

x= 1300 Micron, y= 100 Micron

-1

-0.5

0

0.5

1

-1

-0.5

0

z/(W/2)

z/(W/2)

(c)

(d)

0.5

1

Fig. 6. Spanwise distributions of the w-velocity components on different x-sections (Re = 35).

velocity. However, the Reynolds number decreases with the increase of the SH. The effect of different SHs on the skin friction coefficient and the x-wall shear stress along the downstream wall are presented in Figs. 8–9 respectively. Fig. 8 shows that the skin friction coefficient starts to decrease with increases in the distance downstream from the step edge, due to the separation conditions in this region reaching its minimum value at the reattachment point. Then the trend increases again to reach its maximum value at the fully developed region. In general, the results show that the skin friction coefficient is strongly dependent on the SH and increases with the increase of the SH value. This is because the SH increases led to a decrease in the Reynolds number, thereby having an inverse proportion with the skin friction coefficient [43–44]. The wall shear stress in the streamwise direction for different SHs is plotted in Fig. 9. The results show that the wall shear stresses in the streamwise direction have the same behavior as the skin friction coefficient, due to the wall

shear stress being a function of the skin friction coefficient and that they are positively proportioned. The effect of forward-facing SHs on the local Nusselt number downstream of the forward facing step is illustrated in Fig. 10. For the given forward-facing SHs (S = 350 μm, 450 μm, 550 μm and 650 μm), the results show that the local Nusselt number increases when increasing the distance downstream from the step. Then, in a very short distance it reaches its maximum value in the vicinity of the reattachment region. This is due to the impact of the reattached fluid, which could enhance the heat transfer in this area and cause this rapid increase in the Nusselt number [4]. However, the magnitude of the local Nusselt number decreases with continuous increases in the distance in the streamwise direction. It can be seen from Fig. 10 that, in general, the local Nusselt number (i.e., the heat transfer rate) downstream of the FFS (i.e., the heated wall) increases with the increasing SH value. This is because the increases of the introduction of a FFS increase the region of eddies and the length of the reattachment region.

153

00 35 00 30 00 25 00 20 00

00

0

0.01

0.02

0.03

0.04

50

0

20

00

10

00

15

00 00

60 00

40


00

00

1E 00

80

Static pressure (Pascal)

+0

6

1.

2E


X - W all shear stress (P ascal)

+0

6


0.0

0

x (m)

0.001

0.002

0.003

0.004

0.005

x (m)

Fig. 7. Pressure drop distributions for different step heights at the center of the duct. Fig. 9. X-wall shear stress distribution for different step heights at the center of the duct.

5.2. Effect of inclination

6. Conclusions Detailed numerical results of the flow and thermal fields in laminar mixed convection nanofluid flow adjacent to a horizontal, threedimensional MFFS have been reported. The effect of forward-facing SHs and the inclination angle on the fluid flow and heat transfer behavior has been presented. The present results reveal that the introduction of a FFS enhances the heat transfer and as the SH increases the Nusselt number increases. The results also reveal that the skin friction coefficient and the wall shear stress in the streamwise direction increase

10

0

In this study different inclination angles are considered to study the effect of duct inclination on the fluid flow and heat transfer characteristics. No previous studies have discussed the effect of the duct inclination in relation to the FFS. However, previous studies of the BFS, which considered dimensions in centimeters, showed that there is an effect of the inclination angle on the Nusselt number value and the length of the reattachment (see for example Iwai et al. [45], Hong et al. [46] and Lin et al. [47]). In this study, which considers the microscale dimensions, the results showed that there are no noticeable effects of the duct inclination on the fluid flow and heat transfer values, as shown in Fig. 11. This is due to the Grashof number being a function of the hydraulic diameter power four (i.e. SH), which will give a tiny value of the Grashof number. However, decreases in the Grashof number will lead to decreases in the Richardson number and as the Richardson number

decreases the effect of the buoyancy force is decreased [48]. Thus, the inclination effect in the MFFS is not obvious due to its small hydraulic diameter made by the buoyancy forces having a negligible effect.

4.5

80

4


90

Step = 650 Micron Step = 550 Micron Step = 450 Micron Step = 350 Micron

70 60

3

50

Nu

2.5

40

2

30

1.5 1

20

Skin friction coefficient

3.5

10

0.5 0 0

0.001

0.002

0.003

0.004

0.005

x (m) Fig. 8. Skin friction coefficient distribution for different step heights (Re = 35, z = 0).

0

0.001

0.002

0.003

0.004

0.005

x (m) Fig. 10. Nusselt number distribution for different step heights at the duct center (z = 0).

154


100 Angle = 0 Angle = 30 Angle = 60 Angle = 90 Angle = 120 Angle = 150 Angle = 180

90 80 70

Nu

60 50 40 30 20 10 0

0.001

0.002

0.003

0.004

0.005

x (m) Fig. 11. Nusselt number distribution for different inclination angles.

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