Laminar forced convection flow over a backward facing step using nanofluids

International Communications in Heat and Mass Transfer 37 (2010) 950–957

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

Laminar forced convection flow over a backward facing step using nanofluids☆ A.A. Al-aswadi a, H.A. Mohammed a,⁎, N.H. Shuaib a, Antonio Campo b a b

Department of Mechanical Engineering, College of Engineering, Universiti Tenaga Nasional, Km 7, Jalan Kajang-Puchong, 43009 Kajang, Selangor, Malaysia Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX 78249, USA

a r t i c l e

i n f o

Available online 6 July 2010 Keywords: Forced convection Backward facing step Heat transfer enhancement Nanofluids Recirculation flow

a b s t r a c t Laminar forced convection flow of nanofluids over a 2D horizontal backward facing step placed in a duct is numerically investigated using a finite volume method. A 5% volume fraction of nanoparticles is dispersed in a base fluid besides using various types of nanoparticles such as Au, Ag, Al2O3, Cu, CuO, diamond, SiO2, and TiO2. The duct has a step height of 4.8 mm, and an expansion ratio of 2. The Reynolds number was in the range of 50 ≤ Re ≤ 175. A primary recirculation region has been developed after the sudden expansion and it starts to change to become fully developed flow downstream of the reattachment point. The reattachment point is found to move downstream far from the step as Reynolds number increases. Nanofluid of SiO2 nanoparticles is observed to have the highest velocity among other nanofluids types, while nanofluid of Au nanoparticles has the lowest velocity. The static pressure and wall shear stress increase with Reynolds number and vice versa for skin friction coefficient. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The phenomena of flow separation and subsequent reattachment which occur due to a sudden expansion in flow passages such as backward facing steps have been recognized as important industrial situations. This complex flow structure present in heating or cooling applications such as cooling electronic equipments, cooling turbines blades, combustion chambers, chemical processes, cooling of nuclear reactors, wide angle diffusers, high performance heat exchangers, energy systems equipment, and flow in valves. In many instances separation of flow is undesirable and leads to unwanted pressure drops and energy losses which require additional fan or pumping power. However, in other circumstances flow separation may be encouraged, such as in burner flame stabilization use for turbulence promotion leading to enhanced mixing or heat and mass transfer rates. It has been reported from the literature that the first efforts for studying the separation and reattachment flow over a backwardfacing step were done in the late 1950's. The advent of new instrumentation and the improvement of the numerical codes increase the number of new research in such problem and facilitate the complex study of three-dimensional flow in the separation and reattachment zone. The horizontal, inclined, vertical cases were investigated for different geometrical, boundary conditions and fluids properties. Abu-Mulaweh [1] made an extensive review on the fluid flow and heat transfer of single-phase laminar mixed convection flow over different orientations backward and forward facing steps. ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (H.A. Mohammed). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.06.007

Simulations and measurements of flow adjacent to the 2D backward-facing step geometry have appeared extensively in the literature. Experimental and numerical studies for 2D forced convection flow where there is no heat transfer effect has been investigated by Denham and Patrick [2], and Armaly et al [3,4]. They found that the reattachment point moves downstream far from the step as Reynolds number increases. Moreover, they demonstrated that when Re ≲ 400 the flow is indeed 2D over large parts of the width of the test section while Re N 400 is indeed 3D due to the onset of transitional fluctuation. The 3D forced convection flow has been extensively investigated by many researchers [5–13]. They found that the reattachment zone is larger near the side walls compared to the centerplane of the duct. Some studies focused on the heat transfer phenomena over the backward-facing step which exhibit an influence on the fluid flow behaviors due to the buoyancy force. The horizontal flow has an insignificant effect in buoyancy effect compared to inclined and vertical flow due to the height limitation. Aung [14], Sparrow and Chuck [15], Khanafer et al [16], and Chen et al [17], and Kanna and Das [18] established and reported experimental and numerical results on heat transfer for 2D airflow passing a horizontal backward facing step channel heating at either uniform wall temperature or uniform heat flux from below. They concluded that the Nusselt number was independent of Reynolds number. The maximum local Nusselt number was noticed to be near the reattachment point. While the minimum was noticed at the bottom between the stepped wall and the step. The Nusselt number and the recirculation region were decreased as the buoyancy force increases. The effects of the flow and heat transfer in a 3D backward-facing step ducts have been extensively studied using different parameters [19-25]. They

A.A. Al-aswadi et al. / International Communications in Heat and Mass Transfer 37 (2010) 950–957

Nomenclature AR Cp ER Gr g H h h k L Nu P q Re s T U u um u v Xr Xi, Xe x, y W

aspect ratio, W / s specific heat, kJ / Kg. K expansion ratio, H / h Grashof number, gβΔTs3 / v2 gravitational acceleration, m/s2 total channel height, m convective heat transfer coefficient, W / m2. k inlet channel height, m thermal conductivity, W / m. k total length of the channel, m local Nusselt number, hL / k pressure, Pa heat flux, W / m2 Reynolds number ρumh / μ step height, m temperature, K dimensionless velocity inlet velocity, m/s average velocity for inlet flow, m/s velocity component x direction, m / s velocity component y direction, m / s reattachment length, m upstream, downstream lengths, m coordinate directions width, m

Greek symbols β thermal expansion coefficient, 1/ K ρ density, kg / m3 μ dynamic viscosity, N. m / s φ nanoparticles volume fraction Δ amount of difference ∅ inclination angle v kinematic viscosity, m2 / s

Subscripts 0 outlet f base fluid nf Nanofluid s solid w wall ∞ inlet condition

fraction used were in the range of 200 ≤ Re ≤ 600 and 0 ≤ φ ≤ 0.2, respectively, for five types of nanoparticles which are Cu, Ag, Al2O3, CuO, and TiO2. It was found that the high Nusselt number inside the recirculation zone is mainly depended on the thermophysical properties of the nanoparticles and it is independent of Reynolds number. Both the Reynolds number and thermophysical properties of the nanoparticles have an influence on the value of Nusselt number outside the recirculation region. There was two recirculation regions found for Re N 300, primary recirculation at the stepped wall and a secondary recirculation at the top wall of the channel. It is clear from the literature review that only the above study is available. Thus, this study attempts to fulfill the existing gap in this area. It is obvious from the foregoing review that most of the studies are performed considering the conventional fluids. Very little research is performed considering various types of nanofluids for heat transfer enhancement over a backward facing step. In the present study, 2D numerical simulations for forced convective flows over a backward facing step in a duct are carried out using different types of nanofluids with 5% nanoparticles volume fraction of Au, Ag, Al2O3, Cu, CuO, diamond, SiO2, and TiO2. The aim of this study is to obtain an understanding of the velocity distribution, pressure drop, wall shear stress, and skin friction coefficient encountered along both top and bottom walls downstream from the step. Besides that, to study the performance of the backward facing step at various nanofluids types and Reynolds number.

2. Model description and governing equations The geometry considered and the flow configuration used in this study is shown in Fig. 1. The step height, and expansion ratio were fixed at 4.8 mm and 2, respectively. The upstream wall (Xi) and downstream wall (Xe) were 50 mm and 1000 mm, respectively. The flow at the duct entrance was considered to be hydrodynamically steady and fully developed; streamwise gradients of all quantities at the duct exit are set to be zero. All the walls were fixed to be adiabatic. The nanoparticles and the base fluid (i.e. water) are assumed to have a thermal equilibrium and no slip condition occurs between them. The formulation of density and dynamic viscosity properties of the nanofluids are assumed to be constant and given in Table 1. Based on these simplifying assumptions the steady 2D mass conservation and momentum equations governing the fluid flow problem are reduced to the following forms [31]: Continuity equation: ∂u ∂v + =0 ∂x ∂y

observed that the maximum local Nusselt number appears at the vicinity of the reattachment point between the sidewall and the centerplane. The effects of backward facing step channel orientation were investigated by Lin et al [26,27], Hong et al [28], Abu-Mulaweh et al [29], and Iwai et al [30]. They noticed that as the inclination angle increases from 0º to 180º, the reattachment length increases, but the wall friction coefficient and the Nusselt number at the heated wall decrease. The opposite behavior was observed for both reattachment length and Nusselt number as the inclination angle increases from 180º to 360º. The first numerical effort to investigate the flow and heat transfer influence over a backward facing step using nanofluids is conducted by Abu-Nada [31]. The Reynolds number and nanoparticles volume

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Fig. 1. Schematic diagram for backward facing step.

ð1Þ

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Table 1 Properties of various types of nanofluids. Al2O3 Au

Property Pure water

Ag

ρ μ

10500 3970 – –

997.1 1×10− 3

Where; Re¼ Cu

CuO

Diamond SiO2

19300 8933 6500 3510 – – – –

ρf u m h μf

ð4Þ

TiO2

2200 4250 – –

X momentum equation:   ∂v ∂u 1 ∂p =− u +v ð2Þ ρs ð1−φÞ + φ ρf ∂x ∂y ∂x  z z  1 1 ∂u ∂u    + + Re ð1−φÞz;5 ð1−φÞ + φ ρs ∂xz ∂yz ρf

In Eqs. (1)–(3), the viscosity of the nanofluid is approximated as viscosity of a base fluid μf containing dilute suspension of fine spherical particles as given by Brinkman [32]: μnf =

μf ð1−φÞ2:5

The effective density of the fluid is given as: ρnf = ð1−φÞρf + φρs

Y momentum equation:   ∂v ∂u 1 ∂p =− u +v ð3Þ ρs ð1−φÞ + φ ρf ∂x ∂y ∂y  z z  1 1 ∂v ∂v    + + Re ð1−φÞz;5 ð1−φÞ + φ ρs ∂xz ∂yz ρf

ð5Þ

ð6Þ

3. Numerical procedure and code validation A finite volume technique was implemented for discretizing Eqs. (1)–(3) inside the computational domain. The SIMPLE algorithm

Fig. 2. Velocity distributions (Re = 50, ΔT = 15°C), at different X / s, (a) 1.04, (b) 1.92, (c) 2.6, (d) 32.8.

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was used to link the pressure and velocity fields. For the stability of the solution, the diffusion term in the momentum equations is approximated by second order central difference. Moreover, a second order upwind differencing scheme is adopted for the convective terms. Quadrilateral elements and non-uniform grid system are employed in the simulations. The grid is highly concentrated close to the step and near the step corners, in order to insure the accuracy of the numerical simulations and for saving both the grid size and computational time. At the end of each iteration, the residual sum for each of the conserved variables is computed and stored, thus recording the convergence history. The convergence criterion required that the maximum relative mass residual based on the inlet mass be smaller than 1 × 10− 3. A grid independence tests were performed using several grid densities and distributions for Re = 50 with ΔT = 15°C between the downstream wall and the bulk fluid, and they were used as a criteria. The working fluid for the grid testing is air, where a grid size of 128 × 50 confirms a grid independent solution. Grid densities of 140 × 50, 128 × 60, and 140 × 60, where selected to perform a grid independence test and they show less than 3% difference in velocity compared to the chosen grid (128 × 50). The geometry and its

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boundary conditions are based on a study done by Lin et al [26] for validation purposes. The distance between the nodes has a last/first ratio of 14 in the x-direction and 3.5 in the y-direction. The present results show a very good agreement with Lin et al [26] as shown in Fig. 2a–d. This geometry and grid size are used to carry out the results for the nanofluid case. Fig. 2a–d show the velocity distribution of airflow at different X/s along the bottom heated wall. The recirculation region is clearly appeared in Fig. 2a and b, where the recirculation region size decreases as the flow transfer far downstream from the step. As the flow reaches the reattachment point the flow initiate to become fully developed and it become more pronounced as the flow transfer farther along the wall downstream from the reattachment point as shown in Fig. 2c and d. 4. Results and discussion Simulations are performed for different Reynolds numbers of 50, 100, and 175, and various types of nanoparticles with 5% volume fraction of nanofluid compared to its pure base fluid (water). According to the earlier studies for forced convective flow [1–13], it is known that the flow over a backward facing step channel is extremely sensitive to the sudden geometrical expansion at the step.

Fig. 3. Velocity distributions (Re = 175) for different nanofluids at different X = s, (a) 1.04, (b) 1.92, (c) 2.6, (d) 32.8.

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(b)

(a) 0.01

y (m)

y (m)

0.01

Re = 50 Re = 100 Re = 175

Re = 50 Re = 100 Re = 175

0.00

0.00 -0.02

0.00

0.02

0.04

-0.02

0.06

0.00

0.02

u (m/s)

0.04

0.06

u (m/s)

(c)

(d)

0.01

y (m)

y (m)

0.01

Re = 50 Re = 100 Re = 175

Re = 50 Re = 100 Re = 175

0.00

0.00 -0.02

0.00

0.02

0.04

0.06

u (m/s)

0.00

0.01

0.02

0.03

u (m/s)

Fig. 4. Velocity distributions of SiO2 nanofluid for different Reynolds numbers at different X = s, (a) 1.04, (b) 1.92, (c) 2.6, (d) 32.8.

The flow expands from the fully developed parabolic upstream from the step and separates to form a primary recirculation region. After that, the velocity profile reattached and redeveloped approaching a fully developed flow as fluid flows towards the channel exit. Fig. 3a–d shows the distribution of the velocity at four positions downstream the step (x/s = 1.04, 1.92, 2.6, and 32.8), respectively. The velocity distribution of different types of nanofluid with 5% nanoparticles are compared with pure base fluid (water) at Re = 175. Fig. 3a-c reveals that there is a recirculation region developed downstream the step. The size of the recirculation region decreases as the distance between the step and the stepped wall increases until the flow reaches the reattachment point where the flow exhibits zero velocity. Downstream of this point shows that the flow starts to redevelop and then approach fully developed flow as the fluid flows towards the exit as shown in Fig. 3d where the effect of the sudden expansion is detached. It is noticed that SiO2 has the highest absolute velocity while Au has the lowest absolute velocity. This results occur due to the difference in nanofluids density based on Eq. (4), where all the parameters are fixed except the density which is indeed responsible for changing the velocity to backup the constant Reynolds number (Re = 175). It is observed that the pure water and nanofluid with diamond nanoparticles have almost the same trend and close velocity peak values. This is because of their close ρ / μ ratio (kinematic

viscosity). Noting that the results shown in Fig. 3a–d are not presented in dimensionless form due to the insignificant difference between different nanofluids where the velocity ratios of the nanofluids are close. The effect of SiO2 nanofluid flow for different Reynolds numbers at different positions downstream the step (x/s = 1.04, 1.92, 2.6, and 32.8) is shown in Fig. 4a–d. It is found that the velocity increases as Reynolds number increases, which enhances to increase the recirculation size and the flow reattached farther from the step. Furthermore, the peaks become sharper as Reynolds number increases due to the free hydrodynamic flow far from the walls friction and the friction between the opposite fluid flow directions. The sudden expansion affects the static pressure which produces due to the hydrodynamic fluid flow from the inlet and vent out of the channel. It is noticed in Fig. 5 that the static pressure monotonically increases start from the lower corner between the stepped wall and the step. After that, it continues to increase in the streamwise direction on the stepped wall until it reaches its maximum peak value where the reattachment length ends and the velocity distribution reaches to zero, then it decreases linearly until it reaches zero Pascal at the exit. It is indicated that the SiO2 nanofluid has the highest static pressure compared to other nanofluids, while Au has the lowest static

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Fig. 5. Static pressure for different nanofluids at the bottom wall downstream the step at Re = 175.

Fig. 7. Wall shear stress for different nanofluids at the bottom wall downstream the step at Re = 175.

pressure. Moreover, pure water is slightly higher than CuO and less than TiO2 nanofluids. This occurs due to the effect of density and dynamic viscosity on the specific weight gravity. In Fig. 6, it is observed that the static pressure increases and its peak size in the recirculation region becomes wider as Reynolds number increases. This phenomenon occurs due to the increase of the recirculation size and the hydrodynamic force of the fluid flow. In the recirculation region, the wall shear stress decreases monotonically on the bottom wall downstream the step until it reaches the minimum point as shown in Fig. 7. Then, it increases until it reaches a zero wall shear stress where the recirculation region ends and the flow is reattached. Moreover, the wall shear stress continues to increase until it reaches its maximum point where it remains constant along the streamwise direction, which indicates that the flow approaches the fully developed channel flow. SiO2 nanofluid is found to have the highest wall shear stress. While Au has the lowest wall

shear stress compared to other nanofluids. Pure water shows to be slightly higher than CuO but lower than TiO2. Furthermore, it is found in Fig. 8 that wall shear stress increases with Reynolds number; this is due to the increase of the friction resistance between the fluid flow and the bottom wall. Fig. 9 illustrates the effect of skin friction coefficient at the bottom wall downstream the step. It is found that the skin friction coefficient increases monotonically and reaches its maximum peak as the distance downstream from the step increases. Then, it decreases monotonically until it reaches the minimum peak. This peak occurs due to the recirculation flow where there is change in the velocity distributions, and the minimum peak occurs due to the reattachment point where the velocity is almost equal to zero. After that, it increases until it reaches a point where the skin friction coefficient remains constant along the rest of the bottom wall to the exit. This shows that the skin friction coefficient asymptotically approaches

Fig. 6. Static pressure of SiO2 nanofluid for different Reynolds numbers at the bottom wall downstream the step.

Fig. 8. Wall shear stress of SiO2 nanofluids for different Reynolds numbers at the bottom wall downstream the step.

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5. Conclusions Numerical simulations are reported for 2D laminar forced convection over a horizontal backward facing step in a duct using various types of nanofluids. The effect of Reynolds number and nanofluids type is investigated on the fluid flow behavior. It is inferred that the recirculation size and the reattachment length increase as Reynolds number increases. Nanofluids with low dense nanoparticles such as SiO2 have higher velocity than those with high dense nanoparticles like Au. The static pressure increases as the Reynolds number increases and SiO2 has the highest static pressure among other nanofluids types. Moreover, at the bottom wall downstream the step the wall shear stress increases with Reynolds number and SiO2 has the highest wall shear stress compared to other nanofluids. The skin friction coefficient decreases as Reynolds number increases. At the top wall, both the wall shear stress and skin friction coefficient have the same trend with different values. References Fig. 9. Skin friction coefficient of SiO2 nanofluids for different Reynolds numbers at the bottom wall downstream the step.

the fully developed channel flow. Furthermore, it is found that the skin friction coefficient decreases as Reynolds number increases. This is because the skin friction coefficient is inversely proportional to the velocity. The effect of skin friction coefficient for different nanofluids is insignificant and the difference cannot be seen easily because they are close to each other with the same trend as shown in Fig. 10. It is observed that both the wall shear stress and skin friction coefficient have the same trend but with different values at the top wall. They are not presented in this paper due to the space limitation. They are found to decrease monotonically until the minimum point and then increase in streamwise direction until they reach the reattachment point at zero value. After that, they increase until they reach a point where they are remain constant along the rest of the top wall.

Fig. 10. Skin friction coefficient for different nanofluids at the bottom wall downstream the step at Re = 50.

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