Characterization the effects of nanofluids and heating on flow in a baffled vertical channel

Abstract

The laminar 2-D blended convection of the nanofluids at different volume fractions has gained interest in the last decade due to an enormous application in technology. The laminar-flow stream system can be further modified by changing the geometry of the channel, adding an external heating source, and changing the initial conditions at which the stream is being influenced. The investigation of this system includes the variation of the geometrical parameters of the channel, Reynolds number, Nusselt number, and type of the nanoparticles used in preparing the nanofluid with water as the base fluid. These parameters constitute a very successful leading to utilize the numerical solutions by using a finite volume method. Regarding heat flow, one side of the channel was supplied by the heat while the temperature of the other side was kept steadily. The upstream walls of the regressive confronting step were considered as adiabatic surfaces. The nanofluids were made by adding aluminum oxide (Al₂O₃), copper oxide (CuO), silicon dioxide (SiO₂), or zinc oxide (ZnO) nanoparticles to various volume fractions in the scope of 1 to 4% and diverse nanoparticle diameters of 25 to 80 nm. The calculations were performed with heat flux, Reynolds numbers (Re), and step height (S) at a range of 100 <  < 600 W/m², 100 <  Re  < 500, and 3 ≤ S ≤ 5.8, respectively. The numerical study has shown that the nanofluid with SiO₂ has the highest value of the Nusselt number (Nu). The distribution area and the Nu increase as Reynolds number increases and diminish as the volume fraction diminishes with the increase of the nanoparticle diameter. The outcome of this paper has shown that assisting flow has shown superiority over the opposing flow when Nu increases.

Introduction

The regressive confronting step stream is a basic idea that provides a straightforward geometrical model of complex solutions for stream detachment and reattachment (D’Adamo, Sosa, & Artana, 2014). Some application examples include cooling turbine blades, chemical processes, cooling of nuclear reactors, wide angle diffusers, high-performance heat exchangers, energy system equipment, and flow in valves. Heat transfer in separated flows is frequently encountered in various engineering applications centrifugal compressor blades, gas turbine blades, and microelectronic circuit boards (Hashim et al., 2018; Ma, Mohebbi, Rashidi, & Yang, 2018a, 2019; Mohebbi & Heidari, 2017; Mohebbi, Izadi, & Chamkha, 2017; Mohebbi, Izadi, Delouei, & Sajjadi, 2019; Mohebbi, Rashidi, Izadi, Sidik, & Xian, 2018; Ranjbar, Mohebbi, & Heidari, 2018). A thorough comprehensive detail of these streams constitutes a prime significance outlining constructing devices such as combustors, airfoils, turbines, and others (Bomela, 2014). Previous work has utilized the geometry of a backward-facing step to consider the partition and the possibility of reattaching the flowing streams. In an alternative model of more perplexing geometries such as airfoils or other three-dimensional shapes, the geometric smoothness of a regressive confronting step stream guarantees the exact purpose of detachment. A large portion of the most punctual investigations of the regressive confronting step streams is tentatively seen in view of stream representation procedures. As a rule, the essential distribution zone was no longer effective, and the outcomes acquired for various geometries are strongly related to the basic stream (Denham & Patrick, 1974; Kumar & Dhiman, 2012). One of the techniques to deal with enhancing separating regions in the confined areas is to use the nanofluids (NFs) (Abchouyeh, Mohebbi, & Fard, 2018; Delouei, Sajjadi, Izadi, & Mohebbi, 2019; Hasan, Sopian, Jaaz, & Al-Shamani, 2017; Izadi, Hoghoughi, Mohebbi, & Sheremet, 2018; Izadi, Mohebbi, Chamkha, & Pop, 2018; Izadi, Mohebbi, Karimi, & Sheremet, 2018; Ma, Mohebbi, Rashidi, Yang, & Sheremet, 2019; Mohebbi & Heidari, 2017; Mohebbi & Rashidi, 2017). NFs are those fluids with suspended nanoparticles commonly metal oxides (Saidur, Leong, & Mohammad, 2011) which can be suspended in the base fluid which is commonly water (Yu & Xie, 2012). Along these lines, there is no drop in the stream field. It is found that more testing exhibits that NFs enhance the properties of the heat flux by increasing the flux conductivity and the convective heat transfer coefficients (Al-Aswadi, Mohammed, Shuaib, & Campo, 2010; Al-Shamani et al., 2015; Jaaz et al., 2017; Jaaz, Sopian, & Gaaz, 2018; Ma, Mohebbi, Rashidi, & Yang, 2019; Mohammed, Al-Aswadi, Shuaib, & Saidur, 2011; Mohammed, Al-Aswadi, Yusoff, & Saidur, 2012; Mohebbi, Lakzayi, Sidik, & Japar, 2018; Mohebbi, Nazari, & Kayhani, 2016; Murshed, De Castro, Lourenço, Lopes, & Santos, 2011; Nazari, Mohebbi, & Kayhani, 2014; Sheikholeslami & Ganji, 2016).

The estimation of the flow over a regressive confronting step have been broadly considered in the most recent decades for both constrained and blended convection, see, for instance, Armaly, Durst, Pereira, and Schönung (1983), Gartling (1990), Tylli, Kaiktsis, and Ineichen (2002), and subsequent related references therein. Likewise, a few different investigations (Ma, Mohebbi, Rashidi, Manca, & Yang, 2018; Nie & Armaly, 2002; Williams & Baker, 1997) tried the 3-D laminar convection stream in reverse geometrical opposing step. Abu-Mulaweh (2003) led a broad survey of research on laminar blended convection over a backward-facing step (BFS). The reverse confronting step flow while exchanging heat was investigated at uniform wall heat flux and uniform wall temperature variation (Nikkhah et al., 2015; Sheikholeslami & Chamkha, 2016). Blackwell and Armaly (1993) explored various approaches to numerically tackle the issues for consistent 2-D-one-stage blended convection flow in a vertical channel (Selimefendigil & Öztop, 2015).

In another investigation, an exploratory numerical study of a blended convection laminar flow over a regressive confronting step was performed by Abu-Mulaweh, Armaly, and Chen (1993). Other studies found that the inclination angle influences both the flow and its relevant heat stream fields (Alawi, Azwadi, Kazi, & Abdolbaqi, 2016; Kherbeet et al., 2015; Safaei, Togun, Vafai, Kazi, & Badarudin, 2014). It was concluded that the expansion in the inclination edge from the vertical reduces the adjacent Nu estimation. This expansion is a direct result of a lessening the streamwise caused by the extension of the inclination edge. Hong, Armaly, and Chen (1993) and Iwai, Nakabe, Suzuki, and Matsubara (2000) have anticipated numerically that the laminar blended convection flow over a regressive confronting advance in two-dimensional and three-dimensional configurations, respectively. They observed that the reattachment length increases as the inclination angle increases gradually from 0° to 180° and, hence, a reduction in the wall friction coefficient and Nu was observed (Nazari, Kayhani, & Mohebbi, 2013). The opposite was looked for both reattachment length and Nu as the inclination edge increases from 180° to 360°. Various examinations on turbulent flow over a backward-facing step have been investigated mostly in common and constrained convection; see, for instance (Abe, Kondoh, & Nagano, 1995; Chen, Nie, Armaly, & Hsieh, 2006; Park & Sung, 1995; Rhee & Sung, 1996). A few researchers considered the convective heat flux exchange throughout utilizing NFs, see, for instance, Trisaksri and Wongwises (Daungthongsuk & Wongwises, 2007; Trisaksri & Wongwises, 2007; Wang & Mujumdar, 2007), and the subsequent references therein. It ought to be noted, in any case, that the most investigations of a regressive confronting step employ regular non-NFs (Hashemi, Abouali, & Ahmadi, 2017; Ma, Mohebbi, Rashidi, & Yang, 2018b; Mohammed, Alawi, & Sidik, 2016; Mohammed, Alawi, & Wahid, 2015). Adversely, other investigations with NFs in reverse confronting step geometry have been developed. The primary numerical examination to investigate the stream and heat flux exchange over a regressive confronting step utilizing NFs was performed by Abu-Nada (2008). In this study, the volume fraction nanoparticles (∅) and Re were taken at 0 ≤ ∅ ≤ 0.2 and 200 ≤ Re ≤ 600, respectively using the nanoparticles of Al₂O₃, CuO, SiO₂, or ZnO. It was suggested that the high Nu number inside the distribution zone generally depends on the thermophysical properties of the nanoparticles (Ma, Mohebbi, Rashidi, Yang, & Sheremet, 2019) while it shows no Re relation (Ma, Mohebbi, Rashidi, & Yang, 2018c).

Numerical examination of forced and mixed convection in even and vertical pipe of various NFs was studied (Al-Aswadi et al., 2010; Mohammed et al., 2011, 2012). The effects of Re number (75 ≤ Re ≤ 225), temperature difference between 0 and 30 °C using Au, Ag, Al₂O₃, Cu, or CuO as nanoparticles with water as the base fluid were studied, while SiO₂ and TiO₂ were inspected on the fluid stream and heat flux step characteristics. It is found that a dispersion region develops a straight backward-facing step between the edge of the wall and within 1 mm or so before the corner where the downstream divider is located (Alawi et al., 2016). In that couple of millimeters zone between the appropriation region and the downstream divider, the stream shows a reversing U-turn opposing the circulation stream where circulated and uncirculated streams come together in the channel. In these systems, the highest peak of Nu is obtained using Au as the nanoparticle, while the most astonishing slightest peak in the circulation district was obtained using diamond nanoparticles.

The literature review has shown that the steady 2-D-NF laminar mixed convection flowing over a backward-facing step in a vertical configuration under uniform heat flux has not been thoroughly investigated and this case will be a very convincing argument for this study. The present examination, for instance, oversees particular NFs such as CuO, ZnO, and SiO₂ at various volume fractions and various nanoparticle ranges. The impacts of heat flux and Re on the velocity distribution, skin friction coefficient, and Nu are analyzed and offered an explanation to the peak under the effect of NFs on these parameters for facilitating the restricting streams.

The aim is to study the effects of different types of nanoparticles such as Al₂O₃, CuO, SiO₂, and ZnO and base water and to analyze different volume fractions (∅) of nanoparticles in the range of 1–4%. The calculations were performed with heat flux, Reynolds numbers (Re), and step height (S) at a range of 100 <  < 600 W/m², 100 <  Re  < 500, and 3 ≤ S ≤ 5.8, respectively.

Methodology

Numerical model

Physical model

Figure 1 shows the geometrical arrangement for the direct step backward facing where the two parameters of the stepping height and expansion ratio were set at 4.8 mm and 2, respectively. Additionally, the upstream and downstream walls were assembled at 24 mm and 144 mm, respectively. At the wall, a uniform heat flux (q_w) was applied to the step downstream wall (X_e) while the opposite wall of the channel was kept isothermally consistent with the fluid temperature of T₀. The wall upstream is characterized by two steps: X_i and S; both were kept under adiabatic surface condition. The reattachment length (X_r) is shown on the right side of the bottom of the channel. At the channel entrance, the nanofluid stream is hydrodynamically persistent and the flow achieves the fully developed conditions at the edge of the movement which results in zero speed at the internal surface of the pipe. The nanofluids (nanoparticles and water) were adjusted for specific heat flux to attain a no-slip condition. The flow was assumed to be incompressible and follows Newtonian fluid in regard to the viscosity. As an approximation, the exchange radiation heat flux, its counterpart distribution term, and the inward heat flux were dismissed. The thermophysical properties of the nanofluids are constant while the buoyancy was constrained as an opposing motion to the gravity. The Boussinesq equation is employed for this system.

Fig. 1

Schematic diagram for 2D BFS in a vertical channel

FLUENT solver is based on the finite volume method. In this case, domain of the model to be examined is discretized into a finite set of the control volumes, which are termed meshes or cells. Then, the general conservation equations for mass, momentum, and energy solved on this set control volumes. Other transport equations such as the equations for species may also be applied. Subsequently, partial differential equations are discretized into a system of algebraic equations. All algebraic equations are solved numerically to render the solution field. Computational fluid dynamic (CFD) theories, the CFD modeling process, physical model, and assumptions such as model, governing equations and boundary conditions, thermophysical properties of nanofluids, finite volume method (FVM), and fluid computation by the SIMPLE algorithm are presented in the following sections. CFD techniques are used to study and solve complex fluid flow and heat transfer problems in the backward and forward double steps. CFD is a computer-based tool for simulating the behavior of systems involving fluid flow, heat transfer, and other physical-related processes. It works by solving the equations of fluid flow (in a special form) over a region of interest, with specified (known) conditions on the boundary of that region. CFD has facilities in the industry as it can solve several system configurations in the same time and at a cheaper cost compared with other experimental studies. In the current study, a CFD technique is used to investigate the effect of using nanofluids as work fluid instead of the conventional fluids over backward and forward double steps. The interest region where flow simulation is to be done is the computational domain of the problems.

Cost analysis of using nanofluids

To investigate the cost-effectiveness of using nanofluids as coolants, computational fluid dynamics method is employed to directly simulate the flow and heat transfer of the nanofluids in a 2-dimensional micro channel in the present paper. There are two different numerical methods for doing these. One is based on molecular dynamics which directly focuses on the molecular behaviors of the nanoparticles. This method needs more CPU time and computer memory. The other is based on Navier-Stokes questions introducing the thermal and dynamic parameters of the nanofluids obtained from the mixture fluid theory and experimental measurements. Nanofluids are prepared by either one-step or two-step methods. Both methods require advanced and sophisticated equipment. This leads to higher production cost of nanofluids. Therefore, high cost of nanofluids is a drawback of nanofluid applications (Xu, Pan, & Yao, 2007).

Governing equations

To prepare the CFD investigation of the reverse confronting step, it is imperative to set up the basic three equations which describe the continuity (Eq. 1), momentum (Eq. 2), and the energy (Eq. 3). It is also important to set the non-dimensional variables. In addition, the Boussinesq equation was used to deal with the viscous dispersal and compressibility impact. The conditions for two-dimensional laminar incompressible streams can be composed as suggested by (Abu-Nada, 2008):

$$ \mathrm{The}\ \mathrm{continuity}\ \mathrm{equation}:\kern0.5em \frac{\mathit{\partial U}}{\mathit{\partial X}}+\frac{\mathit{\partial V}}{\mathit{\partial Y}}=0 $$

(1)

$$ \mathrm{The}\ \mathrm{momentum}\ \mathrm{equations}{\displaystyle \begin{array}{ll}X\hbox{-} \mathrm{motion}:& U\frac{\partial U}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{\mu_{nf}}{\rho_n{\upsilon}_{nf}}\frac{1}{\mathit{\operatorname{Re}}}\left(\frac{\partial^2U}{\partial {X}^2}+\frac{\partial^2U}{\partial {Y}^2}\right)\\ {}Y\hbox{-} \mathrm{motion}:& U\frac{\partial U}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{\mu_{nf}}{\rho_n{\upsilon}_{nf}}\frac{1}{\mathit{\operatorname{Re}}}\left(\frac{\partial^2U}{\partial {X}^2}+\frac{\partial^2U}{\partial {Y}^2}\right)+\frac{{\left(\rho \beta \right)}_{nf}}{\rho_{nf}{\beta}_{nf}}\frac{Gr}{{\mathit{\operatorname{Re}}}^2}\theta \end{array}} $$

(2)

$$ \mathrm{The}\ \mathrm{energy}\ \mathrm{equation}:\kern0.5em U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{\alpha_{nf}}{\alpha_f RePr}\left(\frac{\partial^2\theta }{\partial {X}^2}+\frac{\partial^2\theta }{\partial {Y}^2}\right) $$

(3)

The dimensionless variables are defined as follows,

$$ X=\frac{x}{S}\kern0.5em Y=\frac{y}{S}\kern0.5em \theta =\frac{T-{T}_0}{T_w-{T}_0}\kern0.5em U=\frac{u}{U_0}\kern0.5em V=\frac{v}{V_0}\kern0.5em \mathit{\operatorname{Re}}=\frac{\rho {u}_0{D}_h}{\mu } $$

Boundary conditions

The boundary conditions as shown below reflect the limiting situations at the walls in regard to the flow, no-slip condition, and temperature. The conditions at the channel segment include the temperature and the speed. The overall arrangement is governed by the following conditions (Ghasemi & Aminossadati, 2010; Öztop, 2006):

Upstream inlet:
$X=-\frac{\ {X}_i}{\ {D}_h}$	$\frac{S}{\ {D}_h}\le Y\le \frac{H}{\ {D}_h}$	${U}_1=\frac{u_i}{\ {u}_o}$	V = 0	θ = 0
Downstream exit:
$X=\frac{\ {X}_e}{\ {D}_h}$	$0\le Y\le \frac{H}{\ {D}_h}$	$\frac{\partial^2U}{\partial {X}^2}=0$	$\frac{\partial^2V}{\partial {X}^2}=0$	$\frac{\partial^2\theta }{\partial {X}^2}=0$
Top wall:
$-\frac{1}{\ {D}_h}\le X\le \frac{\ {X}_e}{\ {D}_h}$	$Y=\frac{H}{\ {D}_h}$	U = 0	V = 0	θ = 0
Sidewalls:
$-\frac{\ {X}_i}{\ {D}_h}\le X\le \frac{\ {X}_e}{\ {D}_h}$	$0\le Y\le \frac{H}{\ {D}_h}$	U = 0	V = 0	θ = 0
Stepped wall condition: (a) Upstream:
$-\frac{X_i}{\ {D}_h}\le X\le 0$	$Y=\frac{S}{\ {D}_h}$	U = 0	V = 0	$\frac{\partial \theta }{\partial Y}=0$
Stepped wall condition: (b) At the step:
X = 0	$0\le Y\le \frac{S}{\ {D}_h}$	U = 0	V = 0	$\frac{\partial \theta }{\partial Y}=0$
Stepped wall condition: (c) Downstream:
$0<X\le \frac{\ {X}_i}{\ {D}_h}$	Y = 0	U = 0	V = 0	$\frac{\partial \theta }{\partial Y}=-1$

Code validations and grid testing

• i. Preliminary evaluation of the grid

A preliminary evaluation was adopted to coordinate the grid geometrical conditions and the ability of the grid to show the impact of heat transfer of the fluids with various NFs. The cross investigation was driven with the working fluids, NFs, at Re = 50 and 175, while keeping the downstream wall and the channel stream at temperature difference of ∆T = 15 °C. Three grid size of 140 × 50, 128 × 50, and 140 × 60 were used for testing a self-governance system while a fourth grid size of 128 × 50 was maintained for self-governing independence trials. Several local trials have shown that using other grid sizes reduces the performance of the grid up to 4% in evaluating speed and temperature. The partition between the center points was extended to reach 14 in the x-direction while it reaches 3.5 in the y-direction. The current examination has adopted a non-uniform structure and quadrilateral sides. The system was specifically tested near the steps at the edge and the walls to guarantee the precision of the numerical deviation which helps remarkably to save both the computational time and cross assessment.

The second cross tests have improved the situation of air flow at Re = 100 while a uniform heat flux progress of q_w = 200 W/m² was applied to the downstream wall. The points of the cross-sectional center are set at different sizes of the grid while the framework numbers in the x-direction were 100, 120, and 140 and 50, 60, and 70 in the y-heading to ensure the system flexibility test. A grid size of 100 × 50 was used to examine and then to validate the cross self-sufficient course of action. It shows under 3% reduction in Nu and other grid sizes that are mentioned in Table 1, a fine structure is used as a part of the x-coordinate which is close to the divider motion section, while it increases with expansion of 0.1 in x-bearing along the way from the wall position. Other fine grids of the improvement extents at the ratio of 0.285 in the y-heading are utilized very close to the base divider and the edge of the wall to ensure the accuracy of the framework free course of action.

• ii. Validation of the code

Table 1 Grid tests for the Nusselt number at Re = 100

The numerical approval strategy that comprises the numerical code under particular conditions was tested and then used as a benchmark for afterward contrasting. The main experiment was performed by creating the constrained convective nanofluids stream over an even BFS in a pipe (Al-Aswadi et al., 2010). For this stream, Re numbers for the laminar flow were taken at 50 and 175. Figures 2, 3, and 4 demonstrate the consequences of the correlation with the findings of the various NFs as presented by Al-Aswadi et al. (2010). The velocity distribution for different stepped wall increments (X/S) in the distribution along the pipe is shown in Figs. 2 and 3 for Al₂O₃ and CuO under same conditions, respectively. The distribution location shows up and the measure of the local distribution diminishes as the separation between the step X/S achieves the reattachment condition at zero-speed flow as shown in Figs. 2a–d and 3a–d. Figures 2a–c and 3a–c demonstrate that a distribution location grew continuously downstream. Additionally, the downstream of flow begins to end up completely by creating fluid stream toward the pipe exit as shown in Figs. 2 and 3d.

Fig. 2

Comparison of velocity distribution with the results of Al-Aswadi et al. (2010) in the recirculation region for S = 4.8 mm, and ER = 2 at Re=175. For Al₂O₃ at different X/S: a 1.04, b 1.92, c 2.6, d 32.8

Fig. 3

Comparison of velocity distribution with the results of Al-Aswadi et al. (2010) in the recirculation region for S = 4.8 mm, and ER = 2 at Re=175. For CuO at different X/S: a 1.04, b 1.92, c 2.6, d 32.8

Fig. 4

Comparison of skin friction coefficient of SiO₂ nanofluids for different Reynolds numbers a Re = 50 and b Re = 175 at the bottom wall downstream the step

The skin friction coefficients shown in Fig. 4 were demonstrated at the base of the wall step showing that they achieve the separation downstream from the step showing the highest extreme base peak until achieving a consistent friction coefficient along the rest of the base wall. The condition of the flow to reach fully developed flow depends on the asymptotic variation of the skin coefficient. The second experiment was prepared to reproduce blended convective wind stream over BFS in the channel according to the work performed by Hong et al. (1993). The conditions of this experiment include setting Re at 100 and applying a uniform wall heat flux at 200 W/m². The velocity distribution and Nu at the point where θ^° = 0 are explained in Fig. 5a, b.

Fig. 5

Comparison of the present results with the results of Hong et al. (1993) (S = 4.8 mm, and ER = 2) for Re=100, and q_w = 200 W/m² (X = 3) at 0° angle. a Velocity distribution. b Nusselt number

Procedure of numerical calculation

The procedure of the numerical calculations was executed through governing equations and the conditions alongside the limit as depicted in Eqs. (1) to (3). The conditions for the strong and fluid stage were considered as a single domain. The discretization of the conditions in the liquid and strong regions has come to an end by utilizing FVM. The diffusion term under energy and momentum conditions is approximated continuously by arranging the focal distinction until achieving a steady state condition. However, a second-arranged upstream procedure was prepared to show the relation to the convective terms. The calculation of the stream field was completely utilized by the SIMPLE calculation (John & Anderson, 1995). This was an iterative arrangement methodology where the calculation is introduced through speculating the weight field. At this point, the momentum equations were utilized to determine the speed parts followed by refreshing the weight factor to reach the continuity state. Despite the fact that the steady state does not contain any weight factor, it can be changed effortlessly into a weight adjustment condition (Patanker, 1980).

Thermophysical characteristics of nanofluids

The zinc oxide (ZnO) nanoparticles were synthesized using a simple precipitation method with zinc sulfate and sodium hydroxide as starting materials. The synthesized sample was calcined at different temperatures for 2 h (Kumar, Venkateswarlu, Rao, & Rao, 2013); copper oxide (CuO) is the inorganic compound with the formula CuO. As a black solid, it is one of the two stable oxides of copper, the other being Cu₂O or cuprous oxide. As a mineral, it is known as tenorite. It is a product of copper mining and the precursor to many other copper-containing products and chemical compounds (Richardson, 2002). Silicon dioxide (SiO₂), also known as silica, is an oxide of silicon with the chemical formula SiO₂, most commonly found in nature as quartz and in various living organisms (Fernández, Lara, & Mitchell, 2015; Iler, 1979). In many parts of the world, silica is the major constituent of sand. Silica is one of the most complex and most abundant families of materials, existing as a compound of several minerals and as a synthetic product. Notable examples include fused quartz, fumed silica, silica gel, and aerogels. It is used in structural materials and microelectronics (as an electrical insulator) and as components in the food and pharmaceutical industries. Aluminium oxide (Al₂O₃) (IUPAC name) or aluminum oxide (American English) is a chemical compound of aluminum and oxygen with the chemical formula (Al₂O₃). It is the most commonly occurring of several aluminum oxides and specifically identified as aluminum (III) oxide. It is commonly called alumina and may also be called aloxide, aloxite, or alundum depending on particular forms or applications. It occurs naturally in its crystalline polymorphic phase α-Al₂O₃ as the mineral corundum, varieties of which form the precious gemstones ruby and sapphire. Al₂O₃ is significant in its use to produce aluminum metal, as an abrasive owing to its hardness and as a refractory material owing to its high melting point (Davis, 1993).

With a specific end goal to complete simulation for NFs, the compelling thermophysical characteristics of NFs must be first determined. For this situation, the nanoparticles are Al₂O₃, CaO, ZnO, and SiO₂. The required properties for the recreations are to set the effective heat conductivity (k_eff), to calculate the effective viscosity (μ_eff) and effective density (ρ_eff), and to assume that the effective coefficient of heat expansion (β_eff) and effective particular heat capacity (c_peff) are following the incompressible fluid and their estimation depends on the volume fraction of their nanofluid (∅). By utilizing Brownian movement of nanoparticles in reverse confronting step, the effective heat flux conductivity can be acquired as following mean experimental requirements (Ghasemi & Aminossadati, 2010; Heidari, Mohebbi, & Safarzade, 2016):

$$ {k}_{\mathrm{eff}}={k}_{\mathrm{static}}+{k}_{\mathrm{Brownian}} $$

(4)

where k_static is defined as,

$$ {k}_{\mathrm{static}}={k}_{bf}\left[\frac{k_{np}+2{k}_{bf}-2\left({k}_{bf}-{k}_{np}\right)\varnothing }{k_{np}+2{k}_{bf}+2\left({k}_{bf}-{k}_{np}\right)\varnothing}\right] $$

(4.1)

and the Brownian thermal conductivity,

$$ {k}_{\mathrm{Brownian}}=5\times {10}^4\beta \varnothing {c}_{p, bf}\sqrt{\frac{kT}{2{\rho}_{np}{R}_{np}}}f\left(T,\varnothing \right) $$

(4.2)

where Boltzmann constant:

$$ {\displaystyle \begin{array}{c}k=1.3807\times {10}^{-23}\ \mathrm{J}/\mathrm{K}\\ {}{\beta}_{\mathrm{Al}2\mathrm{O}3}=8.4407{\left(100\varnothing \right)}^{-1.07304}\\ {}{\beta}_{\mathrm{CuO}}=9.881{\left(100\varnothing \right)}^{-0.9446}\end{array}} $$

Modeling function, β (Vajjha & Das, 2009):

$$ {\displaystyle \begin{array}{c}{\beta}_{\mathrm{SiO}2}=1.9526{\left(100\varnothing \right)}^{-1.4594}\\ {}{\beta}_{\mathrm{ZnO}}=8.4407{\left(100\varnothing \right)}^{-1.07304}\end{array}} $$

Modeling function

$$ f\left(T,\varnothing \right)=\left(2.8217\times {10}^{-2}\varnothing \right)+3.917\times {10}^{-3}\Big)\left(\frac{T}{\ {T}_0}\right)+\left(-3.0699\times {10}^{-2}\varnothing -3.91123\times {10}^{-3}\right) $$

and μ_eff can be empirically estimated (Corcione, 2010):

$$ {\mu}_{\mathrm{eff}}=\frac{\mu_f}{1-34.87{\left(\frac{d_p}{d_f}\right)}^{-0.3}{\varnothing}^{1.03}} $$

(5)

where,

$$ \mathrm{The}\ \mathrm{equivalent}\ \mathrm{diameter}\ \mathrm{is}\ {d}_{\mathrm{eff}}={\left(\frac{6M}{N\pi {\rho}_{bt}}\right)}^{1/3} $$

(6)

$$ \mathrm{The}\ \mathrm{density}\ \mathrm{of}\ \mathrm{NF}\ \left[68\right]\ \mathrm{is}\ {\rho}_{nf}=\left(1-\varnothing \right){\rho}_f+\varnothing {\rho}_{nf} $$

(7)

The effective heat capacity at constant pressure of NF (Ghasemi & Aminossadati, 2010) is:

$$ {\left({c}_p\right)}_{nf}=\left(1-\varnothing \right){\left({c}_p\right)}_f+\varnothing {\left({c}_p\right)}_{np} $$

(8)

The effective coefficient of thermal expansion of nanofluid (Ghasemi & Aminossadati, 2010):

$$ {\left(\beta \right)}_{nf}=\left(1-\varnothing \right){\left(\beta \right)}_f+\varnothing {\left(\beta \right)}_{np} $$

(9)

The experimental values of the thermophysical properties for pure water with different NFs at 300 K are listed in Table 2.

Table 2 Thermophysical properties for pure water and different nanofluids at T = 300 K (Corcione, 2010)

Results and discussion

Reattachment zone

The zone was allocated to display the numerical results to test the 2-D vertical mixed convective stream by standing up the step. The results of speed distribution and skin friction coefficient were taken at different values of Nu at various nanofluids at hand, various nanoparticle volume fractions, Reynolds numbers, and varying heat flux and the height of the wall, and finally with varying nanoparticle diameters. The Reynolds number was taken between 100 and 500 while the heat flux was considered between 100 and 600 W/m². The stream reaches out from a totally developed upstream to outline a basic circulation region. Starting forward, the speed profile is reattached and redeveloped pushing toward a totally developed stream as fluid streams toward channels exit.

Nanofluid parameters’ impacts

The nanoparticles with water as a base fluid of Al₂O₃, CaO, ZnO, and SiO₂ were used. The impact of various nanofluids on the heat transfer enhancement was tested under the following parameters of ∅ = 0.04, d_p = 25 nm, Re = 100, and q_w = 500 W/m² along the channel. Figure 6a shows that Nu acquires the highest value near the wall due to an increasing heating of the wall which decreases steadily as the detachment continues. This is a result of the temperature refinement when the exchanged stream is joined to the essential separated stream which has a lower temperature. It is found that SiO₂ nanofluid has the highest Nu peak followed by Al₂O₃, CuO, and finally ZnO. The results mentioned can be explained in terms of how much is the reduction in the fluid viscosity and the effect of the heat flux since the viscosity is very critical to these two factors.

Fig. 6

The effect of nanofluid parameters along the downstream wall at Re = 100, q_w = 500 W/m², ∅ = 4%, and d_p = 25 nm for a different nanoparticle type, b different volume fractions, and c different nanoparticle diameters

This study also shows the effect of ∅ between 1 and 4% on the characteristics of the heat transfer as illustrated in Fig. 6b under the conditions of nanoparticle d_p = 25 nm, Re = 100, and q_w = 500 W/m². As ∅ increases, the Nu of SiO₂ nanofluids ends up higher than that of unadulterated fluids. This is because of the density increasing which prompts the heat transfer enhancement as showed in Fig. 6b. A possible explanation to this could be referred to the effect of the decreasing temperature refinement as ∅ increases which results in increasing the heat transfer improvement. The impact of the diameter of nanoparticles on the heat transfer reveals that the heat transfer varies significantly with the mean nanoparticle diameter between 25 and 80 nm, at Re = 100, ∅ = 4%, and q_w = 500 W/m².

Figure 6c shows the eventual outcome of Nu when SiO₂ nanofluids were used. It has been shown that the mean nanofluid diameter increases due to the increasing Nu. This is because the nanofluid properties such as density and heat capacity increase with a decreasing nanoparticle volume ratio which affects the surface area. Thus, the small diameter of the nanoparticles results in increasing the heat transfer improvement. Consequently, Nu increases as the diameter of the nanoparticles increases.

Different Reynolds numbers’ impacts

Distribution of the velocity

Figure 7 shows the speed circulations of SiO₂ nanofluid at Re numbers in the scope of 100 ≤ Re ≤ 500, q_w = 500 W/m², d_p = 25 nm, and ∅ = 4% at various X/S sections along the stepped wall. In Fig. 7a, as Re increases, the speed profile for SiO₂ nanofluid becomes more profound and illustrative. The speed increases as Re increases until reaching the distribution at which the flow reached the reattachment. It has been noted that as the separation between the step and the stepped wall increase, the flow achieves the reattachment faster and the span of the distribution area diminishes as Re expands further as shown in Fig. 7b, c. Beyond this point, the downstream shows that the flow begins to redevelop until reaching completely developed flow at the exit as shown in Fig. 7d.

Fig. 7

Velocity distributions of SiO₂ nanofluid with ∅ = 4%, and d_p = 25 nm at different Re for q_w = 500 W/m² at a X/S = 0, b X/S = 1.8, c X/S = 12.82, and d exit

Coefficient of skin friction

Figure 8a explains the behavior of the skin friction coefficient of the SiO₂ nanofluid at Re in the scope of 100 ≤  Re  ≤ 500, ∅ = 4%, q_w = 500 W/m², and d_p = 25 nm along the downstream wall. It is shown that the skin friction coefficient decreases rapidly as Re increases. Additionally, it is seen that the crests at and close to the essential local distribution are displaced along the way from the wall divider in the streamwise bearing as Re increases. This increase comes with the increasing of the length differential of the essential distribution region. The behavior of the coefficient of friction is inversely proportional.

Fig. 8

The effect of Re of SiO₂ nanofluid with ∅ = 4%, and d_p = 25, q_w = 500 W/m² along the downstream wall. a Skin friction coefficient. b Nusselt number

Nusselt number

The Nu of SiO₂ nanofluid at Re between 100 and 500, ∅ = 4%, q_w = 500 W/m², and d_p = 25 nm along the downstream wall is plotted in Fig. 8b. It is shown that Nu increases as Re increases exhibiting stronger convective current which is caused by the reduction in the isotherms. The behavior of Nu shows an exponential reduction from its optimal level until the vortex is started forming. At the highest Re, the stream decreases to the minimum apex of the Nu number at the region where the flows are mixed. Along this zone, the Nu SiO₂ nanofluid was found to possess higher values as Re number increases.

Various heat fluxes’ impacts

Distribution of the velocity

Figure 9 shows the speed distribution of SiO₂ nanofluid at q_w varies between 100 and 600 W/m², Re = 100, d_p = 25 nm, and ∅ = 4% along the X/S sections. It seems that there is an insignificant effect at the edge region, as the stream profiles show no pattern difference as shown in Fig. 9a.

Fig. 9

Velocity distributions of SiO₂ nanofluid with φ = 4%, and d_p = 25 nm at different heat fluxes for Re = 100 at a X/S = 0, b X/S = 1.8, c X/S = 12.82, and d exit

However, the degree of the circulation area expands as the heat transfer increases between the heated stepped wall and the stream of the nanofluid as shown in Fig. 9b, c. It was also found that the buoyancy force slightly pushes up the fast stream and cutting down heat flux and increasing the reattachment length. As the divisions, X/S along the wall increases the circulation zone decreases as shown in Fig. 9b, c. At this location, the flow starts to redevelop approaching full developing stream flow at the exit according to Fig. 9d.

Skin friction coefficient

The testing of the skin friction of SiO₂ nanofluid was carried out with ∅ = 4%, and d_p = 25 nm, heat flux in the scope of 100 ≪ q_w ≪ 600 W/m², and Re = 100 along the downstream divider are shown in Fig. 10a. It is observed that the increase in the heat flow enhances the skin friction coefficient. However, at a particular heat flux of q_w = 600 W/m², the skin friction coefficient shows higher values due to the diminishing shear stress at the vortex.

Fig. 10

The effect of heat flux of SiO₂ nanofluid with ∅ = 4%, and d_p = 25, Re = 100 along the downstream wall. a Skin friction coefficient. b Nusselt number

Nusselt number

Figure 10b explains the behavior of Nu-SiO₂ with ∅ = 4%, d_p = 25 nm, heat flux of 100 ≪ q_w ≪ 600 W/m², and Re = 100 along the downstream divider. As q_w increases, it was noticed that the highest value of Nu decreases due to the increasing effect of the expanding buoyancy force.

It is important to note that the reduction of optimized Nu continues until roughly the exiting point. The results show that the stream with high heat flow possesses the most astounding Nu least peak in the region where the flows are mixed. It was also found that alongside this territory, Nu with SiO₂ nanofluid possesses the highest value when the heat flow is at the highest value.

Conclusions

Numerical simulation for laminar blended convection stream in a vertical 2-D backward-facing step channel configuration was investigated under a variety of parameters and initial conditions. This topic has recently gained special importance especially due to consolidating Re and nanofluids such as aluminum, copper(II) oxide, zinc oxide, and silicon dioxide with water. Reynolds number was chosen within 100 ≪  Re  ≤ 500, diameter of nanoparticles d_p: [25, 80] nm, and ∅: [1%, 4%]. The backward-facing step downstream flow has reached its uniform condition at the uniform heat flux between 100 and 600 W/m² while the channel particular propel statures were taken in the extent of 3.0 ≪ S ≪ 5.8 mm. The governing equations have shown very good agreement with the solution using FVM with specific suppositions coupled with proper limit conditions. The current investigation examined the influence of the stream and its relevant opposing flow on the properties of the heat flux. The following conclusions are drawn from this investigation:

• 1. SiO₂ nanoparticles show the most astounding Nu followed by Al₂O₃, CaO, and ZnO.

• 2. Nu increased continuously by expanding Re and by increasing the step height as well. It was also observed that Nu increases as the heat flux increases. However, Nu is slowly diminishing as the nanofraction ∅ diminishes.

• 3. The present work creates a region of a straight downstream distribution for blended convection stream at various Re and heat flux. It was found that this distribution is highly dependent on increasing Re. The best location of this distributing region was allocated at the interface of the corner where the stepping height joins the downstream wall.

• 4. The heat flux was found very dependent on the conditions that correlate the preferred flux exchange rate over restricting stream condition.

Nomenclature

A Area (m²)

Al₂O₃ Aluminum oxide

A_r Aspect ratio

BFS Backward-facing step

c_p Specific heat (kJ/kg K)

CFD Computational fluid dynamic

CuO Copper oxide

d_p Diameter of nanofluid particles (nm)

FVM Finite volume method

D_h Hydraulic diameter at inlet (m)

ER Expansion ratio, H/S

e Expansion successive ratio

FFS Forward-facing step

f Elliptic relaxation function

Gr Grashof number, gβ∆TS³/v²

g Gravitational acceleration (m/s²)

H Total channel height (m)

h Inlet channel height (m)

h Convective heat transfer coefficient (W/m² K)

H₂O Water

k Thermal conductivity (W/m K)

k Turbulent kinetic energy (m²/s²)

S Step height (m)

SiO₂ Silicon dioxide

L Total length channel (m)

Nu Nusselt number, hs/k

Nu_{s, max} Maximum Nusselt number, hs/k

P Nodal point number of processors

p Pressure (p_a)

p_a Pressure at the outlet (p_a)

Pr Prandtl number, μ C_p/k

q_w Wall heat flux (W/m²)

Re Reynolds number, ρu_o D_h/μ

Ri Richardson number, Gr/Re²

T Temperature (K)

T_w Heat wall temperature (K)

T_o Temperature at the inlet, outlet, or top wall (K)

UHF Uniform heat flux

U Dimensionless streamwise velocity component, u/u_o

u Velocity component x-direction (m/s)

u_o Average velocity for inlet flow (m/s)

u_∞ Free-stream velocity (m/s)

V Dimensionless transverse velocity component, v/u_o

v Velocity component y-direction (m/s)

X Dimensionless length at x-coordinate, x/S

x x-Coordinate direction (m)

X_e Downstream wall length (m)

X_i Upstream wall length (m)

X_n Nusselt number peak length (m)

X_o Secondary recirculation length (m)

X_r Reattachment length (m)

Y Dimensionless length at y-coordinate, y/S

y y-Coordinate direction (m)

ZnO Zinc oxide

Greek symbols

ρ Density (kg/m³)

β Thermal expansion coefficient, (1/K)

υ Kinematic viscosity (m²/s)

φ Volume fraction (%)

μ Dynamic viscosity (N m/s)

α Thermal diffusivity of the fluid (m²/s)

∆ Amount of difference

휽 Dimensionless temperature, (T − T_o)/(T_w − T_o)

Subscripts

bf Base fluid

i Inlet

o Outlet

f Fluid

nf Nanofluid

s Solid (nanoparticles)

W Wall