#### Abstract

As research shows, in new and renewable energy systems, including solar energy, the study of turbulent flow is of great importance due to its high efficiency in heat transfer. It is also used in petrochemical and oil industries and cooling systems. Therefore, this paper focuses on the turbulent heat transfer of nanofluid between two parallel plates and the effect of the volume fraction of nanoparticles on turbulent heat transfer is investigated. The nanofluid applied in the study was alumina-water. The beginning and the end of the walls were insulated, and the middle part was considered as the heat source. The two-equation - model was used to model viscosity of turbulent flow. The governing equations were solved simultaneously using the control volume method based on SIMPLER algorithm. In this study, the effects of the Reynolds’ number in the range of 10^{4} to , volume fraction of 0.01 to 0.04, and nanoparticle diameter of 20 nm to 100 nm on field flow and rate of heat transfer were examined. The influence of Brownian movement on heat performance was considered. Evaluation showed that increasing the Reynolds’ number decreased the thickness of the laminar sublayer in turbulent flow and increased temperature and velocity differences. These greater temperature and velocity differences resulted in increased heat transfer and decreased skin friction. The findings imply that heat performance improves when nanoparticles are added to basic fluid. With increasing volume fraction of nanoparticles, shear stress of the channel wall increases, and consequently, skin friction increases too. In addition, the effect of nanoparticle diameter on thermal and hydraulic performance was studied. It was found that heat transfer and skin fraction decreased in the presence of the larger nanoparticles.

#### 1. Introduction

Investigations on forced convection of flow in channels are significant because of their multiple applications in industry, renewable energy systems, and solar energy. Researchers have conducted numerous studies on heat transfer in channels, including Yang et al. [1], Hibachi and Acharya [2] and Tehran and Abed [3] for laminar flow, Vijayan and Bali [4] and Fedora and Visconti [5] for turbulent flow, and Rasool et al. [6–8] for heat transfer of Darcy–Forchheimer nanofluid flow over plates. Krishna and Chamkha [9] studied the heat transfer of micropolar fluid flow past an infinite vertical porous surface. Naseem et al. [10] numerically studied heat transfer in a Newtonian fluid over a porous stretched surface in the presence of constant magnetic and thermal radiation, with considering the role of Dufour and Soret numbers. Ahmed et al. [11] numerically investigated the mixed convective of Williamson fluid flow over a curved surface. Zonta et al. [12] examined the impact on turbulent heat transfer of micro-sized particles dispersed in the flow between two parallel plates. They found the heat flux in walls increases by almost 2% when the flow composed of smaller particles, while the heat transfer is reduced when larger particles are added.

Cooling strategies using fluids with low thermal conductivity limit convection. Nanofluids provide a new environment which causes heat transfer to increase. Thus, nanofluids have many engineering applications, such as cooling systems of electronic components, building heating, solar collector energy storage, nuclear reactor systems, and microbial fuel cells [13–18]. The effect of using nanofluids on convection heat transfer has been studied by various researchers [19–23]. Nonlinear radiation effects on magnetic/nonmagnetic nanoparticles with different base fluids over a flat plate was studied by Saranya et al. [24]. Heat transfer of different nanofluid flows over a Riga plate was studied by several articles [25–29]. Much research has been carried out related to forced convective heat transfer of nanofluids under laminar flow, for example, the studies by Xu and Pan [30], Tsai and Chein [31], Hedayati and Domairry [32], Malvandi and Ganji [33], and Mirzaei et al. [34]. Santra et al. [35] studied the effect of water-copper nanofluid on heat transfer between two parallel plates under laminar flow. It was shown that when volume fraction of nanoparticles increases, heat transfer increases. Raisi et al. [36] carried out a numerical investigation of thermal performance of microchannels cooled by nanofluid laminar flow of copper and water. In their study, in addition to the Reynolds’ number and volume fraction of nanoparticles, the effect of slip factor on the heat transfer rate was studied. Manay and Sahin [37] conducted an experimental study on heat transfer of forced convection in nanofluid laminar flow. The volume fraction of nanofluid and height of the microchannel were found to be important factors on pressure drop and rate of heat transfer in their study. Their results showed that, with increasing microchannel height, heat transfer rate decreased but pressure drop increased.

Numerical and experimental studies have also been carried out on forced convective heat transfer in channels under turbulent flow in nanofluids. Turbulent flow and heat transfer of 3 nanofluids (CuO, SiO_{2}, and Al_{2}O_{3}) in water and ethylene glycol with constant heat flux in a cylindrical pipe were studied by Namburu et al. [38]. They concluded that when volume fraction increases and the diameter of nanoparticles decreases, heat transfer improves. Bianco et al. [39, 40] studied heat transfer of turbulent convection of alumina-water nanofluids in tubes with different boundary conditions. Results showed that the heat transfer coefficient for the nanofluid was greater than that of the fluid and increasing the volume concentration and Reynolds’ number the of nanoparticles improved heat performance. Bayat and Nikseresht [41] numerically investigated turbulent nanofluid heat transfer in pipes with constant heat flux boundary conditions. They concluded that when nanoparticle volume fraction increases, heat transfer and pressure drop also increase. Saha and Paul [42] studied turbulent forced convection in horizontal pipes in which the basic fluid was water and the nanoparticles were Al_{2}O_{3} and TiO_{2}. Increasing the volume fraction of nanoparticles and Reynolds’ number and reducing the nanoparticle diameter were found to cause a reduction in the heat transfer rate. Numerical analysis of heat transfer forced convection for turbulent flow of copper-water nanofluid was conducted by Behroyan et al. [43] in a horizontal pipe. This study compared five types of computational fluid dynamics with experimental results of previous studies. Other similar numerical investigations were conducted by Yang et al. [44], Maiga et al. [45], Aghaei et al. [46], and Leong et al. [47].

Williams et al. [48] and Fotukian and Esfahany [49] studied heat transfer of turbulent forced convection of nanofluid in pipes experimentally. Heyhat et al. [50] evaluated heat transfer of horizontal pipes under alumina-water turbulent flow, experimentally. Their experimental results showed that the heat transfer coefficient for a nanofluid is greater than that for a basic fluid and increases by volume concentration of particles. They also found that the effect of Reynolds’ number on heat transfer rate is not significant.

Some studies have examined forced convection using non-Newtonian fluids. Usman et al. [51, 52] numerically studied heat transfer of the power-law fluid over and between stretchable rotatory disks. Usman et al. [53] numerically investigated the entropy generation in a power-law nanofluid flow over a stretchable rotatory porous disk. Exploration of temperature dependent thermophysical characteristics of yield exhibiting non-Newtonian fluid flow under gyrotactic microorganisms was studied by Sohail et al. [54]. Heat and mass transmission on mixed convection boundary layer flow of Casson liquid over a linear elongating surface in porous medium was studied by Sohail et al. [55]. They reported that augmenting values of magnetic parameter reduces the fluid velocity and upsurges the temperature and concentration profiles. Sohail and Naz [56] numerically investigated heat and mass transmissions of boundary layer flow of non-Newtonian Sutterby nanofluid by a stretched cylinder by incorporating the revised models by engaging Cattaneo–Christov theory. Mathematical model and rheological aspects of chemically reacting Casson-type nanofluid flow considering ethylene glycol-based nanoparticles with thermophoretic diffusion and Brownian motion was studied by Osman et al. [57]. Shamshuddin and Ghaffari [58] investigated the heat and mass transfer in a steady flow of Sutterby nanofluid over the surface of a stretching wedge.

Apart from these studies on circular channels or tubes, other studies have been carried out on the cross. Hussein et al. [59] evaluated friction coefficient and heat transfer in 3 different segmented surfaces, while Mohammed et al. [60] studied hydraulic and thermal parameters of turbulent nanofluid flow in rib-groove channels numerically. In these studies, the rib-grooves were of different forms and the effect of different parameters, including nanofluid type, volume fraction, nanoparticle dimension, and Reynolds’ number, was evaluated. They concluded that the highest rate of heat transfer is associated with rectangular grooves. Ziaei-Rad [61] investigated heat transfer of forced convection for two horizontal and parallel plates under turbulent nanofluid flow of alumina-water. The - model was used to calculate turbulent viscosity. He found that effect of adding nanoparticles to basic fluid is greater for hydraulic and thermal parameters.

According to recent research, there is a lack of investigation regarding heat transfer of forced convection under turbulent flow of a nanofluid between 2 parallel plates. Despite the heat transfer of pure fluid in forced convection under turbulent flow through two parallel plates and pipes, the heat transfer of nanofluid forced convection has been observed to be lower in these channels. Therefore, we decided to study heat transfer of forced convection of nanofluid between two parallel plates. The thermal conditions imposed in this case have not been observed in previous studies. The effect of Brownian motion of nanoparticles on thermal performance is included. The main subject of this study is to evaluate the effect of various parameters of alumina-water nanofluid on thermal and hydraulic performance under turbulent flow. A single-phase model is applied in this study.

#### 2. Problem Description

The schematic of the present problem is shown in Figure 1. A horizontal channel of length and height is assumed. Nanofluid flow with constant temperature and horizontal velocity enters channel from left and exits from right. The channel is divided into 3 parts: input and output of a channel with length , which is insulated and the middle of the channel is a thermal source with temperature and length of . All mentioned parameters are dimensionless, with channel height (). Height of channel , length , length of entry and output , and length of middle part . The present study examines Reynolds’ number, volume fraction, and diameter of nanoparticles on flow and heat transfer rate.

#### 3. Formulation of the Problem

In this study, forced convective heat transfer in a two-dimensional Cartesian system is investigated. The nanofluid is Newtonian and incompressible, and the flow is turbulent and steady state. The condition is considered nonslip at the walls. There is a thermal equilibrium between the fluid and the nanoparticles in the channel. The geometry and diameter of nanoparticles are the same. The effect of viscous dissipation is ignored, and there is no generation and storage of energy. The governing equations on turbulent flow, including continuity, momentum, and energy are as follows [62].

Connection equation:

Momentum equation:

Energy equation:

To stimulate turbulent flow, the - two-equation model is used. Equations are presented below for turbulent kinetic energy () and dissipation rate of turbulent kinetic energy (), as proposed by Jones and Launder [63].

The constant values of the - model presented by Launder and Sharma [64] are as follows.

The terms Reynolds’ stresses are presented by the Boussinesq assumption as follows [65].

In addition, the turbulent heat flux can be written as the turbulent Prandtl number. The turbulent Prandtl number mainly considered here is

Density, heat capacity, volume expansion coefficient, and thermal diffusivity of the nanofluid can be defined as suggested in [66, 67]

The Corcoran model was used for thermal conductivity and viscosity of nanofluid. The results of this model were closer to the experimental results. In this model, thermal conductivity and dynamic viscosity depend on temperature, volume fraction of nanofluids, the diameter of the nanoparticles, and thermal conductivity of pure fluids and nanoparticles. The relationships used are according to those presented in [68].

The Reynolds’ number of nanoparticles and diameter of a water molecule is defined as follows:

In which, is a fixed value of Boltzmann’s constant and is the Avogadro number in the formulas presented here. is freezing point and is the molecular weight of the basic fluid. In this model, the effect of Brownian motion of the nanoparticles has been considered. The thermophysical properties of the fluid and Al_{2}O_{3} nanoparticles are presented in Table 1.

With dimensionless equations, a wide range of materials and various streaming modes can be examined in the form of a general problem. We used the following dimensionless variables to obtain the main dimensionless equations.

The equations of energy, momentum, and continuity were rendered dimensionless by substituting dimensionless parameters, as follows.

Connection equation:

-momentum equation:

-momentum equation:

Energy equation:

- two-equation model:

In these equations, represents the generation of turbulence kinetic energy due to the mean velocity gradients:

The hydrodynamic boundary conditions of the equations include nonslip condition on the walls and uniform input velocity. The upper and lower walls are insulated, except the middle part of the walls where their temperatures are constant. Output velocity and temperature are developed. The boundary condition for at the channel walls follows from the assumption of equivalence between production and dissipation of kinetic energy of the turbulent fluid in the near wall region (see Launder and Spalding for details) [70].

The input of turbulent kinetic energy () and dissipation rate of turbulent kinetic energy () is provided as follows [71].

In which the turbulent intensity and , which is the length scale of turbulence.

The boundary conditions for the dimensionless equations are showed in Table 2.

is the nearest point to the wall along the axis. The Nusselt number is a parameter that indicates the amount of heat transfer. The local Nusselt number on the hot source is defined as follows [72].

By integrating local Nusselt numbers during the flow along the warm part of the wall, the average Nusselt number can obtained as follows:

The local and average skin friction is presented as follows:

#### 4. Numerical Method and Validation

The governing equations and boundary conditions are discretized by the finite difference method based on finite volume. The power-law scheme is used for convection-diffusion terms. The SIMPLER algorithm is used to solve equations simultaneously. First, the velocity equations are solved by initial conjecture and the corrected pressures are obtained. Subsequently, the temperature equations and the turbulent flow equations - are solved; the details of which are given in Reference [73]. A computer program was written using FORTRAN programming language to run the algorithm, and the convergence criteria were as follows:

and are nodes of the grid in directions and . represents the dimensionless dependent parameters that are solved, and is the number of repetitions. To ensure the accuracy of the written code, it was compared with results presented in previous studies. First, skin friction was compared with the relationship found by Halleen et al. [74]. In this paper, the turbulent flow is investigated experimentally in the channel and the skin friction coefficient is obtained in this way (. Figure 2 shows the graph at different Reynolds’ numbers for a pure fluid. As it turns out, there is little difference between the results.

In addition, for the accuracy of thermal performance, the consistency of the present work with the findings of Hatton and Quarmby [75] was investigated. In their study, the Nusselt number for an area between two parallel plates where one side is insulated and the other is at a constant temperature for various Reynolds’ and Prandtl numbers was calculated. According to Table 3, it can be seen that there is little difference between the results and the accuracy of our computer program, which is thus acceptable.

To perform the computerized calculations, first, a suitable grid should be chosen. Velocity in the turbulent boundary layer is divided in two parts of the viscous sublayer and fully turbulent sublayer. Coles and Hirst [76] have modeled velocity in these two parts as follows.

Figure 3 shows values of based on , showing little difference between the results of the present research and the relationships found in Coles and Hirst [76].

depends on the one hand on the number of points in the -direction and on the other hand is dependent on the Reynolds’ number. So, for different Reynolds’ numbers, is considered. A suitable grid for different Reynolds’ numbers is shown in Table 4.

#### 5. Results and Discussion

After confirming the reliability of the program, we extracted the results. The effect of different parameters, including Reynolds’ number (), nanoparticle volume fraction (), and nanoparticle diameter on hydraulic and thermal performance were investigated. Hot wall temperature and cold wall temperature and Prandtl number () were assumed to be constant in the calculations.

##### 5.1. Reynolds’ Number

It this section, the volume fraction of nanoparticles and diameters of nanoparticles were fixed and the effect of the Reynolds’ number was studied. Velocity profiles for different Reynolds’ numbers in Figure 4 show that a higher Reynolds’ number causes a steeper slope near the wall. Figure 5 shows that by increasing the Reynolds’ number the temperature profile slope rises and thus the temperature gradient increases in the boundary layer.

A larger Reynolds’ number shows higher entry velocity. Physically, when velocity increases, the convection terms become more significant in the energy equation. Therefore, the heat transfer rate increases. We observe in the temperature graph that when the Reynolds’ number increases, temperature gradient increases near the wall. Temperature change has a direct relationship with the average Nusselt number. When temperature gradient increases near the wall, the average Nusselt number also increases. Figure 6 shows average Nusselt number variations for different Reynolds’ numbers. In this figure, it is observed that with increasing Reynolds number, heat transfer or Nusselt number has increased. By increasing the Reynolds number, in fact, the speed of the inlet flow to the channel increases, which increases the heat transfer from the surface. In fact, in the energy equation, the convection terms increase, which increases the heat transfer.

The local skin friction coefficient for different Reynolds’ numbers is shown in Figure 7. At the entrance of the channel, has maximum value and this reduces until it reaches a fixed amount (hydraulic developed area). As can be seen, with increasing Reynolds’ number, has declined. The velocity profiles in Figure 4 show that, with increasing velocity, velocity gradient and therefore the shear stress increase. Moreover, the friction coefficient has an inverse relationship with the input speed or Reynolds’ number. On the other hand, the effect of Reynolds’ number on the friction coefficient is greater than on the shear stress. Therefore, by increasing the Reynolds’ number, the skin friction coefficient is reduced.

##### 5.2. Volume Fraction of Nanoparticles

Due to the impact of volume fraction of nanoparticles on thermophysical properties of nanofluids, it was decided to evaluate this parameter. Here, too, the diameter of the nanoparticles is constant . Changes in the average skin friction coefficient based on Reynolds’ number are shown for different volume fractions in Figure 8. Physically, with increasing the volume fraction of nanoparticles, the wear of nanoparticles with the wall increases, and as a result, the shear stress on the wall increases and skin friction increases.

Values of average Nusselt number and its increase were compared with those for pure fluid for different Reynolds’ numbers and different volume fractions, as shown in Table 5.

According to the results presented in this table, it can be seen that for a fixed volume fraction of nanoparticles with increasing Reynolds’ numbers, convection flows are amplified and heat transfer rate increases. By increasing , the conductivity of nanofluids increases, which increases the heat transfer rate and therefore the average Nusselt number. In addition, Brownian motion effects, applied in the thermal conductivity of a nanofluid, increase the heat transfer rate. Increase in and Reynolds’ number are factors affecting the rate of heat transfer. Adding nanoparticles at high Reynolds’ numbers causes heat transfer to increase much more. The reason for this is the faster heat transmission speed in high Reynolds’ numbers. In other words, fluid flow captures heat from the heat source more quickly and transfers it to the outlet then cold flows are quickly replaced. This increases the temperature gradient, thereby increasing the average Nusselt number.

##### 5.3. Nanoparticle Diameter

The diameter of the nanoparticles affects the viscosity and thermal conductivity of nanofluids. Figure 9 shows the average Nusselt number, depending on the diameter of the nanoparticles, for different Reynolds’ numbers. Physically, the smaller the diameter of the nanoparticles, the more the heat transfer rate increases, because the surface area between the nanoparticles and the base fluid increases. In addition, according to the relationship of the thermal conductivity of nanofluids, reducing the diameter of the nanoparticles increases the thermal conductivity and thus the heat transfer rate increases. The larger the Reynolds’ number, the greater the effect of particle size on the rate of heat transfer: when Reynolds’ number increases, variations in size of the nanoparticles also have a greater effect.

A graph of average skin friction coefficient is shown in Figure 10, based on variation in the diameter of the nanoparticles. When nanoparticle diameter increases, the viscosity of the nanofluid decreases, thereby reducing shear stress. As can be seen in Figure 10, with a constant Reynolds’ number, is reduced with increasing nanoparticle diameter. The laminar sublayer is thicker in turbulent flow when the diameter of the nanoparticles is larger, so the velocity gradient is reduced.

#### 6. Conclusion

The present study evaluated turbulent forced convection heat transfer of an alumina-water nanofluid in a horizontal channel with two fixed constant temperature heaters on each side.

Governing equations were solved using the SIMPLER algorithm. Variations in Reynolds’ number, volume fraction, and diameter of nanoparticles were studied. In general, the study of turbulent flow inside the channels gives a better physical view of the flow conditions and close adaptation to the actual state of the flow inside the channel. The results can be summarised as follows: (a)Higher Reynolds’ numbers increase flow rate and reduce laminar sublayer thickness in turbulent flow. Consequently, temperature and velocity gradients increase in this area. Higher Reynolds’ numbers increase heat transfer and reduce skin friction coefficient(b)Increasing the volume fraction of nanoparticles causes the average Nusselt number to increase, and thus, a higher heat transfer rate occurs. For higher Reynolds’ numbers, the volume fraction of nanoparticles has a greater effect on heat transfer(c)When nanoparticles are added to a base fluid, wall shear stress and consequently the skin friction coefficient increase(d)With smaller diameters of nanoparticles, the contact area of the nanoparticles with the base fluid increases, and the heat transfer ratio becomes greater. Moreover, the skin friction coefficient has a higher value with small diameters

The following topics are suggested to the readers of this article to complete the study: use of other proposed models for the thermophysical properties of nanofluids, the effects of magnetic field on turbulent flow, channel rotation at different angles and the effect of free convection on channel flow, and the application of channels with convergent and divergent geometries.

#### Nomenclature

: | Skin friction |

: | Specific heat (J/kgK) |

: | Dimensionless dissipation rate of turbulent kinetic energy |

: | Channel height (m) |

: | Dimensionless channel height () |

: | Dimensionless turbulent kinetic energy |

: | Channel length (m) |

: | Dimensionless channel length () |

: | Avogadro number |

: | Nusselt number |

: | Fluid pressure (Pa) |

: | Dimensionless pressure |

: | Prandtl number |

: | Reynolds number |

: | Temperature (K) |

: | Inlet flow temperature (K) |

: | Thermal source temperature (K) |

: | Velocity components (m/s) |

: | Dimensionless velocity components |

: | Inlet velocity (m/s). |

*Greek Symbols*

: | Thermal diffusivity |

: | Dissipation rate of turbulent kinetic energy |

: | Dimensionless temperature |

: | Turbulent kinetic energy |

: | Thermal conductivity |

: | Dynamic viscosity |

: | Turbulent molecular viscosity |

: | Kinematic viscosity |

: | Density |

: | Solid volume fraction. |

*Subscripts*

eff: | Effective |

f: | Pure fluid |

m: | Mean |

nf: | Nanofluid |

s: | Nanoparticle. |

#### Data Availability

All required data is available in the text of the paper.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.