Introduction

Due to the global energy crisis, many different methods were used to increase the thermal performance in heat-exchanging equipment. Modification in flow geometry is one of common technique which is used to enhance heat transfer rate such as expansion or contraction in channel, use rib on wall of channel, insert twist tape through passage flow, etc. In the recent years, nano-fluids have received extensive attention due to its higher thermal conductivity compared with normal cooling fluid. Addad et al. (2003) presented numerical study on fluid flow over forward–backward-facing step using Large Eddy Simulation (LES). They found that the separation length and reattachment offset were about 1.2 and 0.6 from step height, respectively. While Yu et al. (2009) presented numeircal study in 2D and 3D dimension of fluid flow over a backward facing step at low and high Reynolds number with assume two-phase. In their study, LES was used where good agreement noted between 2D and 3D numerical data with experimental data in profile of both velocity and temperature. Laminar, transition, and turbulent fluid flow over backward-facing step were experimentally and numerically studied by Armaly et al. (1983). The results revealed the increase in separation length with increase of Reynolds number for Re < 1200 but decrease at Re between 1200 and 5550. Khanafer et al. (2008) used finite element method for analysis of laminar heat transfer convection of pulsatile flow over a backward-facing step. Enhancement of heat transfer rate was observed with increase of Reynolds number while decrease in thickness of the thermal boundary layer. Laminar fluid flow and heat transfer between parallel plates through baffles were numerically investigated by Kelkar and Patankar (1987). Increase in size recirculation regions, Nusselt number, and friction coefficient was found with increase of Reynolds number. Lima et al. (2008) investigated numerically study of two-dimensional laminar air flows over backward-facing step using two CFD commercial codes, the first one based on finite element method (COMSOL MULTIPHYSICS) and the other based on finite volume method (FLUENT) and the range of Reynolds number. The numerical results agree with previous experimental data and those result also appeared there are non-linear increase in reattachment length.

Different equipments have been used to measure heat transfer near the reattached point of the separated flows, and Mori Yuaks (1986) used the thermal tuft probe, but Kawamura et al. (1987) designed new heat flux probe to find the time and spatial characteristics of heat transfer at the reattachment region of a two-dimensional backward-facing step deign. Oyakawa et al. (1996) also used jet discharge at reattachment region downstream of backward-facing step.

Turbulent heat transfer and fluid flow through annular pipe with sudden expansion were experimentally and numerically studied by Togun et al. (2011) and Oon et al. (2012). The results showed that the highest enhancement of heat transfer was about 18 % at step height 18.5 mm compared to without step.

Recently, effect of use nano-fluid has been studied in many researches due to increase in thermal performance. Abu-Nada (2008) can be considering as a pioneer in numerical study of heat transfer to nano-fluid over backward-facing step. The type of nanoparticles in this study was represented by Cu, Ag, Al2O3, Cuo, and TiO2 with volume friction between 0.05 and 0.2 and range of Reynolds number from 200 to 600. Momentum and energy equations were solved using finite volume method which observed increase of Nusselt number at the top and bottom of the backward-facing step. Also the investigations found high thermal conductivity of nanoparticles as outside of recirculation zones. Later, Togun et al. (2014a) presented numerical study of nano-fluid flow and heat transfer over a backward-facing step. The higher thermal performance was about 26 and 36 % for turbulent and laminar range, respectively, compared with pure water. Turbulent heat transfer of water/functionalized multi-walled carbon nanotube (FMWCNT) nano-fluids over a forward-facing step was numerically investigated by Safaei et al. (2014). The results showed that increase for both Reynolds number and FMWCNT volume fraction leads to increase in local heat transfer coefficient for all cases. Turbulent nano-fluid flow and heat transfer over double forward-facing steps were studied numerically by Togun et al. (2015). They found that the maximum Nusselt number happened at the second step compared to the first step and the biggest thermal improvement occurred at volume fraction (4 %) of Al2O3 compared with others.

The goal of the current research was to study the laminar nano-fluid flow and heat transfer over backward-facing step with and without obstacle. In this research, effort to present new data for using nano-fluids and obstacle with backward-facing step will be more useful to design heat exchanger with higher thermal performance.

Geometry and boundary conditions

Figure 1a, b show the geometry which used in this investigation with same dimensions as established by Togun et al. (2014b). The considered geometry was represented by backward-facing step of channel with and without obstacle. The diameter of the inlet and outlet channel is 4.8 and 9.6 mm, respectively with unheated upstream length is 50 mm and heated downstream length is 1000 mm at 2000 W/m2. Three different heights of obstacle of 1.5, 3, 4.5, and 1.5 mm width were fixed at 200 mm from the step with expansion ratio 2. Volume fraction of CuO nano-fluid varied between 1 and 4 % at Reynolds numbers of 75, 125, 175, and 225.

Fig. 1
figure 1

Flow configuration of the computational domain of backward-facing step (a) without obstacle (b) with obstacle

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Numerical procedure

Governing equations

Two-dimensional continuity, momentum (X, Y), and energy equations with assumption laminar, steady state, and incompressible are used in this model and are given as (1, 2, 3, 4).

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
(1)
$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = - \frac{1}{\rho }\frac{\partial p}{\partial x} + \upsilon \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right)$$
(2)
$$u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = - \frac{1}{\rho }\frac{\partial p}{\partial y} + \upsilon \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right)$$
(3)
$$u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = - \frac{1}{\rho }\frac{\partial p}{\partial y} + \propto \left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right)$$
(4)

The hydrodynamic boundary conditions are At the channel wall surface: u = v = 0 (no slip), Inlet: u = based on Re, v = 0, y = L; P = P out at the outlet. The Reynolds number is calculated based on inlet channel height (H).

$$Re = \frac{u H}{v}$$
(5)

Grid testing and model validation

Two-dimensional viscous laminar model with energy dialog box was used to solve continuity, momentum, and energy equations. The second-order upwind method of Patankar (1980) was selected for the discretization of all terms in equations. The SIMPLE algorithm for pressure–velocity coupling was used by Emad et al. (2015). In this simulation, the residual of solution was smaller than 10−4 for continuity equation, 10−7 for momentum equations, and 10−8 for energy equation. Non-uniform quadrilateral grids were used to increase accuracy of solution which increases density of mesh near regions of backward and obstacle highly than other parts. The criteria of grid independent were represented using three different size of grid at Re = 175, see Fig. 2. Due to the difference in Nusselt number for grid number 3 and 2 was about 2 % then the grid number 2 can be considered as a grid independent. For more validations, compared with the work of Al-aswadi et al. (2010) for computed X-velocity of Cu/water nano-fluid at Re = 175 where the results showed good agreement as shown in Fig. 3.

Fig. 2
figure 2

Grid independent for Reynolds number 175 and pure water

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Fig. 3
figure 3

Comparison velocity profile of present study with Al-aswadi et al. (2010)

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Thermophysical properties of the nano-fluid

Thermophysical properties of the nano-fluid are calculated by specific correlations. The effective density of nano-fluid is written as (Vajjha and Das 2009)

$$\rho_{\text{nf }} = (1 - \phi )\rho_{\text{f}} + \phi \rho_{\text{np}},$$
(6)

where ρ f and ρ np define the density of base fluid and the solid nanoparticles, respectively.

The heat capacity of the nano-fluid is presented by (Vajjha and Das 2009)

$$(\rho C_{\text{p}} )_{\text{nf}} = (1 - \phi )(\rho C_{\text{p}} )_{\text{f}} + \phi (\rho C_{\text{p}} )_{\text{np}},$$
(7)

where \((\rho C_{\text{p}} )_{\text{f}}\) and (ρC p)np define the heat capacities of the base fluid and the nanoparticles, respectively.

As observed by Koo and Kleinstreuer (2005), the effective thermal conductivity of nano-fluid contains static and Brownian effects and calculated with the following empirical correlations:

$$K_{\text{eff}} = K_{\text{static}} + K_{\text{Brownian}}$$
(8)
$$K_{\text{static}} = K_{\text{f}} \left[ {\frac{{\left( {K_{\text{np}} + 2K_{\text{f}} } \right) - 2\phi (K_{\text{f}} - K_{\text{np}} )}}{{\left( {K_{\text{np}} + 2K_{\text{f}} } \right) + \phi (K_{\text{f}} + K_{\text{np}} )}}} \right]$$
(9)
$$K_{\text{Brownian}} = 5 \times 10^{4} \beta \phi \rho_{\text{f}} C_{{{\text{p}},{\text{f}}}} \sqrt {\frac{KT}{{2\rho_{\text{np}} d_{\text{p}} }}} {\text{f}}(T,\phi ),$$
(10)

where k = 1.3809 × 10−23 J/K is the Boltzmann constant, and β is given as

$$\beta = 8.4407\left( {100\phi } \right)^{ - 1.07304}$$
(11)

and f(T, ϕ) is given as

$${\text{f}}\left( {T,\phi } \right) = \left( {2.8217 \times 10^{ - 2} \phi + 3.917 \times 10^{ - 3} } \right)\left( {\frac{T}{{T_{^\circ } }}} \right) + ( - 3.0669 \times 10^{ - 2} \phi - 3.91123 \times 10^{ - 3} )$$
(12)

The effective dynamic viscosity for the nano-fluid could be calculated by the following equations (Corcione 2010):

$$\mu_{\text{eff}} = \mu_{\text{f}} \times \frac{1}{{(1 - 34.87\left( {d_{\text{p}} /d_{\text{f}} } \right)^{ - 0.3} \times \phi^{1.03} )}}$$
(13)
$$d_{\text{f}} = \left( {\frac{6M}{{N\pi \rho_{{{\text{f}}0}} }}} \right)^{1/3},$$
(14)

where d p and d f represented the mean diameter of the nanoparticles and equivalent diameter of a base fluid molecule, respectively; M represented the molecular weight; N represented the Avogadro number = 6.022 × 1023 mol−1; and ρ f0 is the density of the base fluid found at Temperature = 293 K.

Table 1 shows the thermophysical properties of the nano-fluid Corcione (2010) and water Incropera (2007).The Nusselt number is defined as

$$Nu = \frac{hd}{k},$$
(15)

where h is the heat transfer coefficient.

Table 1 Thermophysical properties of nanoparticles (CuO) and water at T = 300 K
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Result and discussion

Effect of Reynolds number

The effects of Reynolds number on local Nusselt number for flow over backward-facing step without obstacle are presented in Fig. 4. It can be seen that the local Nusselt number increases with increase of Reynolds number for all cases where the maximum Nusselt number observed with higher Reynolds number due to increase of recirculation region which created after step. Figure 5 shows the variation of pressure drop with axial distance at different Reynolds number. The results showed the rise of pressure drop with Reynolds number due to the hydrodynamics of the flow.

Fig. 4
figure 4

Effect of Reynolds number on local Nusselt number for backward-facing step without obstacle

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Fig. 5
figure 5

Effect of Reynolds number on pressure drop for backward-facing step without obstacle

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Effect of height obstacle

Four different configurations of backward-facing step (without obstacle, with obstacle of 1.5 mm, with obstacle of 3 mm, with obstacle of 4.5 mm) are conducted in this study. Figures 6, 7 illustrate the effect of height obstacle on local Nusselt number and pressure drop with axial distance at Reynolds number of 225 and heat flux 2000 W/m2, respectively. Generally, local Nusselt number and pressure drop found increases with increased of height obstacle due to recirculation regions as created after and before the obstacle and at inlet region of backward facing step. The highest Nusselt number and pressure drop are observed at backward-facing step with height obstacle of 4.5 mm and Reynolds number of 225 compared with others.

Fig. 6
figure 6

Effect of height obstacle on local Nusselt number

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Fig. 7
figure 7

Effect of height obstacle on local pressure drop

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Effect of volume fraction

Figures 8 and 9 show effect of volume fractions of CuO nanoparticles and pure water on local Nusselt number and pressure drop with axial distance for backward-facing step without obstacle and Reynolds number of 225, respectively. It is clear observed that for all cases increases in Nusselt number with increased of volume fractions of CuO nanoparticles where the highest Nusselt number obtained with 4 % of volume fractions of CuO nanoparticles due to increase of thermal conductivity of base liquid which represented the augmentation in heat transfer. Also increase in pressure drop has been seen with rise of volume fractions of CuO nanoparticles, and maximum pressure drop occurred at 4 % volume fractions of CuO nanoparticles.

Fig. 8
figure 8

Variation of local Nusselt number with different volume fraction of nano-fluids

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Fig. 9
figure 9

Variation of local pressure drop with different volume fraction of nano-fluids

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Average Nusselt number and pressure drop

In this section, the average Nusselt number and pressure drop with different of Reynolds number and four different configurations of backward-facing step for 4 % of volume fractions of CuO nanoparticles are presented in Figs. 10, 11. It is noticed that increase in average Nusselt number with increase both Reynolds number and height obstacle in flow channel. The maximum thermal performance was about 22 % for backward-facing step with obstacle of 4.5 mm, Reynolds number of 225, and 4 % of CuO nanoparticles compared with backward-facing step without obstacle. Increase in pressure drop observed with increase of Reynolds number and height obstacle where the maximum pressure drop found with Reynolds number of 225 and height obstacle of 4.5 m compared with others.

Fig. 10
figure 10

Average Nusselt number with different of Reynolds number and four different configurations

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Fig. 11
figure 11

Average pressure drop with different of Reynolds number and four different configurations

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Streamline of velocity

Figure 12 shows the contour streamline of velocity for backward-facing step (without obstacle, with obstacle of 1.5 mm, with obstacle of 3 mm, with obstacle of 4.5 mm) for Reynolds number of 225. The results show that the main recirculation region is seen for all cases after inlet region of backward-facing step, but two recirculation regions appeared after and before each obstacle. The size of recirculation region observed increase after obstacle with increased height obstacle while decreases before the obstacle due to, as the flow velocity rises, the recirculation regions after obstacle are enlarged while before the obstacle are compressed. The maximum recirculation region found with height obstacle of 4.5 mm compared with others which has significant effect on the thermal performance.

Fig. 12
figure 12

Contour streamline of velocity for (a) without obstacle, (b) with obstacle of 1.5 mm, (c) with obstacle of 3 mm, (d) with obstacle of 4.5 mm

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Conclusion

Numerical simulation of two-dimensional laminar heat transfer in a backward-facing step with and without obstacle using nano-fluids was studied. Finite volume method was used for solving governing equations with suitable boundary conditions. The numerical results were presented for four different configurations of backward-facing step (without obstacle, with obstacle of 1.5 mm, with obstacle of 3 mm, with obstacle of 4.5 mm), Reynolds number range varied from 75 to 225, and volume fractions of nano-fluids varied from 1 to 4 %, at constant heat flux. The obtained results found that Nusselt number increases with increase of Reynolds number and height of obstacle for all cases where the maximum Nusselt number was observed with higher Reynolds number and 4.5 mm height obstacle. Effect of volume fractions of CuO nanoparticles on local Nusselt number was investigated where the maximum Nusselt number found with 4 % of volume fractions of CuO nanoparticles. Also Effects of Reynolds number, height obstacle, and volume fractions of CuO nanoparticles on pressure drop were considered. Recirculation regions at inlet of backward-facing step and before and after each obstacle were observed in counter streamline of velocity. The biggest thermal augmentation was found for Reynolds number of 225, height obstacle of 4.5 mm, and 4 % of volume fractions of CuO nanoparticles.