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Abstract

The gaseous low-pressure nanofluid flow of a steady-state two-dimensional laminar forced convection heat transfer in the entrance region of pipes is numerically investigated. Such flows are of interest for many engineering applications like the nuclear reactor and electronic equipment cooling, heat exchangers, and many others. Physical parameters considered in this study are Reynolds number (Re), Prandtl number (Pr), nanosolid particles volume fraction $(ϕ)$ , Knudsen number (Kn), and the aspect ratio (AR). These parameters ranges are as follows: $500 \leq Re \leq 2000$ , $0.76 \leq \Pr \leq 0.95$ , $0 \leq ϕ \leq 0.5$ , $0 \leq Kn \leq 0.1$ , and $1 \leq AR \leq 10$ . The outcome of this study shows that by increasing Kn, velocity slip and temperature jump at the solid boundaries increase. In addition, heat transfer is enhanced by dispersing Al₂O₃ nanoparticles in the base low-pressure gaseous flow. Results show that there is no effect of the nanoparticles volume fraction with values below 0.03 on the average Nusselt number. The average Nusselt number increases $(\bar{Nu})$ as the value of the nanoparticles volume fraction exceeds 0.03. For instance, at Re = 1000, results show that when dispersing Al₂O₃ nanosolid particles with volume fractions of 0.3 and 0.5; there is an enhancement in the average Nusselt number of 30.35% and 136.74%, respectively, when compared to the case of dispersing Al₂O₃ nanosolid particles of 0.03 volume fraction.. Moreover, it is concluded that the average Nusselt number $(\bar{Nu})$ depends directly on Reynolds (Re), Prandtl (Pr) numbers, and the nanoparticles volume fraction $(ϕ)$ and inversely on Knudsen number (Kn) and the aspect ratio (AR) for the investigated range of parameters considered in this study. Finally, a correlation of Nusselt number among all the investigated parameters in this study is proposed as $\bar{Nu} = 0.976 A R^{- 0.0536} P r^{2.39} R e^{0.22} K n^{- 0.378} ϕ^{0.07}$ .

Keywords

Forced convection low-pressure pipe flow entrance region nanofluid

Introduction

Dispersing nanosolid particles in the base fluid has attracted great attention and motivated researchers to investigate such flows in the past few decades. This is mainly due to their role in heat transfer enhancement that can be found in many engineering applications such as automotive and electronic equipment’s cooling, heat exchangers, reduction of the flue gases temperature in boilers, and many others.

The goal of this study is to investigate the effect of adding Al₂O₃ nanoparticles on the flow and heat characteristics on the low-pressure laminar slip gaseous air flow in the entrance region of a circular pipe. Moreover, the effect of Knudsen number, aspect ratio, Reynolds number, Prandtl number, and volume fraction of the nanoparticles on these characteristics will be addressed and discussed thoroughly.

A low-pressure flow is classified based on Knudsen number (Kn). According to Schaaf and Chambre;¹ Cercignani and Lampis,² four different flow regimes are identified—continuum regime, slip regime, transitional regime, and free molecular regime. For slip flows, Navier–Stokes equations can be used along with both slip velocity and temperature jump boundary conditions applied at the surfaces.

Akbari et al.³ studied forced turbulent convection of Al₂O₃–water and Cu–water inside horizontal tubes. They showed that dispersing the nanosolid particles in the flow improves thermophysical properties and hence enhances heat transfer. However, some penalties are paid due to the increase in pressure drop. Balandin et al.⁴ reported that by adding graphene to the base fluid, the resulting thermal conductivity will enhance heat transfer.

Many experimental and numerical works investigated and discussed different types of nanoparticles such as metals and oxide metals that have been used in order to enhance the heat transfer. In this research, we will consider dispersing Al₂O₃ to the base fluid (low-pressure air) in circular tubes with constant surface temperature.

Wang and Mujumdar,⁵ Yu et al.,⁶ Sarkar et al.,⁷ Saidur et al.,⁸ Suresh et al.,⁹ and Hussien et al.¹⁰ presented an excellent review of the heat transfer characteristics of nanofluid in forced and free convection flows. Al-Kouz et al.¹¹ investigated numerically the effect of dispersing nanoparticles of Al₂O₃ to the low-pressure air flow inside a square cavity in which the hot wall is attached to two fins, and they concluded that dispersing such particles will enhance heat transfer. Moreover, a correlation that describes the relationship of Nusselt number among all the investigated parameters is proposed.

Characteristics of the nanoparticles such as the type and concentration play a very important role in determining the thermal behavior of the resulting nanofluid. Many researches have tackled this issue such as Akbari et al.,³ Balandin et al.,⁴ Kalteh et al.,¹² and Hussien et al.¹⁰

In the work carried out by Labib et al.,¹³ the effect of base fluid on convective heat transfer utilizing Al₂O₃ nanoparticles is numerically investigated. They concluded that ethylene glycol base fluid would give better heat transfer enhancement than that of water.

Heris et al.¹⁴ investigated experimentally convective heat transfer of Al₂O₃ water nanofluid in circular tube. They studied the effect of Reynolds number, Peclet number, and the nanoparticle concentration on the heat transfer characteristics. Their experimental results show that by adding nanoparticles to the base fluid, a better heat transfer is achieved.

Moghadassi et al.¹⁵ used a computational fluid dynamics model to study the effect of nanofluids on laminar forced convective heat transfer in horizontal circular tube. They used water-based Al₂O₃ and Al₂O₃–Cu hybrid nanofluid. Their results show that the hybrid nanofluid is superior as far as that heat transfer enhancement is concerned.

In Salman et al.,¹⁶ a numerical solution of the laminar convective heat transfer in a two-dimensional microtube with constant heat flux is presented. They investigated the effect of the type, size, and volume fraction of nanoparticles on the heat transfer characteristics. They concluded that Nusselt number is directly proportional to the volume fraction and Reynolds number and inversely proportional with the nanoparticle size for the investigated range they considered. Experimental work to investigate the turbulent convective heat transfer and pressure loss of aluminum water nanofluid in horizontal tube with different particle concentrations was carried out by Williams et al.¹⁷ A comparison of their results with the available traditional single-phase correlations for fully developed flow is presented. It was found that there is no abnormal heat transfer enhancement is observed in the study.

It is true the fact that there are lots of research that deals with the heat transfer characteristics utilizing nanofluid. However, there is lack of studies that consider heat transfer characteristics of low-pressure gaseous nanofluid. In this article, further insight on adding nanoparticles to the low-pressure gaseous flow and heat transfer characteristics in the entrance region of laminar forced convective heat transfer for isothermal circular pipes is provided. Computational fluid dynamics (CFD) is used to obtain the solution for such flows. Effects of Knudsen number (Kn), Reynolds number (Re), Prandtl number (Pr), aspect ratio (AR), and nanoparticles volume fraction on the flow and heat transfer characteristics are investigated.

Mathematical formulation

Governing equations

In this study, two-dimensional laminar and steady-state forced convection gas flow in the entrance region of isothermal pipes with T = 70°C is investigated. Figure 1 represents flow in the geometry under investigation, with diameter D and length L, continuum and slip flow regimes are considered.

Figure 1.

Schematic diagram of the problem under investigation.

Following Hussien et al.,¹⁰ the properties of the resulting nanofluid can be calculated based on the following equations assuming that the suspension of the nanoparticles is homogeneous with well-dispersed nanoparticles, and the nanosolids are considered to be stable. The nanoparticles are introduced in the base fluid by certain techniques such as loading the particles using wall jet:

Viscosity

μ_{nf} = \frac{μ_{f}}{{(1 - \emptyset)}^{2.5}}

(1)

Density

ρ_{nf} = (1 - \emptyset) ρ_{f} + \emptyset ρ_{s}

(2)

Heat capacitance

C_{P_{nf}} = (1 - \emptyset) {(C_{P})}_{f} + \emptyset {(C_{P})}_{s}

(3)

Thermal expansion coefficient

β_{nf} = β_{f} [\frac{1}{1 + \frac{(1 - \emptyset) ρ_{f}}{\emptyset ρ_{s}}} \frac{β_{s}}{β_{f}} + \frac{1}{1 + \frac{\emptyset}{1 - \emptyset} \frac{ρ_{s}}{ρ_{f}}}]

(4)

Thermal conductivity

k_{nf} = k_{f} \frac{k_{s} + 2 k_{f} - 2 \emptyset (k_{f} - k_{s})}{k_{s} + 2 k_{f} + \emptyset (k_{f} - k_{s})}

(5)

Table 1 summarizes the thermophysical properties used to obtain the resulting properties of the Al₂O₃–air nanofluid.

Table 1.

Thermophysical properties used in the study.

Physical properties	Air	Al₂O₃
C_p (J/kg K)	1006.43	765
ρ (kg/m³)	1	3970
k (W/m² K)	0.025	40
β (1/K)	0.00333	0.0000085
α (m²/s)	0.000019	0.00001317

It should be noted that the resulting physical properties of air–Al₂O₃ nanofluid were calculated by introducing user-defined functions (UDF) in Fluent 18 commercial software.

By referring to Incropera et al.,¹⁸ the governing equations that describe the problem under investigation are summarized below:

Continuity

\frac{1}{r} \frac{\partial (vr)}{\partial r} + \frac{\partial u}{\partial z} = 0

(6)

z-momentum

{ρ_{n}}_{f} (u \frac{\partial u}{\partial z} + v \frac{\partial u}{\partial r}) = - \frac{\partial P}{\partial z} + μ_{nf} [\frac{\partial^{2} u}{\partial z^{2}} + \frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r}]

(7)

r-momentum

ρ_{nf} (u \frac{\partial v}{\partial z} + v \frac{\partial v}{\partial r}) = - \frac{\partial P}{\partial r} + μ_{nf} [\frac{\partial^{2} v}{\partial z^{2}} + \frac{\partial^{2} v}{\partial r^{2}} + \frac{1}{r} \frac{\partial v}{\partial r} - \frac{v}{r^{2}}]

(8)

Energy

{ρ_{n}}_{f} C_{p} f (u \frac{\partial T}{\partial z} + v \frac{\partial T}{\partial r}) = k_{nf} [\frac{\partial^{2} T}{\partial z^{2}} + \frac{\partial^{2} T}{\partial r^{2}} + \frac{1}{r} \frac{\partial T}{\partial r}]

(9)

Referring to Karniadakis et al.,¹⁹ Lockerby et al.,²⁰ and Kandlikar et al.,²¹ the boundary conditions applied for the slip flow regime are slip velocity and the temperature jump at the surfaces are summarized as follows

u_{w} - u_{g} = (\frac{2 - σ_{v}}{σ_{v}}) λ \frac{\partial u}{\partial n} \approx (\frac{2 - σ_{v}}{σ_{v}}) Kn (u_{g} - u_{c})

(10)

v_{g} = 0

(11)

\begin{array}{l} T_{w} - T_{g} = (\frac{2 - σ_{T}}{σ_{T}}) \frac{2 γ}{γ + 1} \frac{k}{μ c_{v}} λ \frac{\partial T}{\partial n} \\ \approx (\frac{2 - σ_{T}}{σ_{T}}) \frac{2 γ}{γ + 1} \frac{k}{μ c_{v}} K_{n} (T_{g} - T_{c}) \end{array}

(12)

where $σ_{v}$ and $σ_{T}$ represent the momentum and thermal accommodation coefficients, and Kn is defined as follows. It should be noted that the values of the momentum and thermal accommodation coefficients are set to 1 in all simulations carried out in this study. Authors had studied the effect of these parameters on the results and it was found that their effect is negligible

Kn = \frac{λ}{D}

(13)

where $λ$ is the mean free path and D is the diameter of the pipe. Thermal boundary condition is imposed at surfaces such that

At r = D / 2, T = 70 ° C

(14)

The local heat transfer coefficient and Nusselt number are computed using equations (15) and (16), respectively

h (x) = \frac{q ″}{(T_{s} - T_{m} (x))}

(15)

Nu (x) = \frac{h (x) D}{k_{nf}}

(16)

where T_s(x) and T_m(x) are the nanofluid temperature at the surface and the mean fluid temperature in the axial location, respectively. The mean fluid temperature can be calculated as follows

T_{m} (x) = \frac{\int_{0}^{R} uTdr}{\int_{0}^{R} udr}

(17)

Then, the average heat transfer coefficient along the wall is calculated by combining equations (15)–(17)

\bar{h} = \int_{0}^{L} h (x) dx

(18)

From equation (18), the average Nusselt number can be calculated as follows

\bar{Nu} = \int_{0}^{L} Nu (x) dx = \frac{\bar{h} D}{k_{nf}}

(19)

The relative enhancement of heat transfer coefficient computed as

\bar{N u}, e n h a n c e m e n t % = \frac{{\bar{N u}}_{n f} - {\bar{N u}}_{a i r}}{{\bar{N u}}_{a i r}} \times 100 %

(20)

Solution methodology

A finite volume technique is used to solve the problem under investigation in which the semi-implicit method for pressure-linked equations (SIMPLE) algorithm presented by Versteeg and Malalasekera²² and Patankar and Spalding²³ is utilized for pressure–velocity decoupling purposes. Moreover, to calculate the pressure field, PRESTO algorithm is used. In addition, a hybrid second-order accuracy scheme of central and upwind difference is used to differentiate the convective terms. A starting mesh of 100 × 100 elements is considered. In this study, all momentum and thermal accommodation coefficients are assumed unity for all simulations. Effects of momentum and thermal accommodation coefficients were investigated for different values, and it was found that their effects on Nusselt number is negligible compared to other parameters considered in this study. The convergence criteria are set so that the maximum of the normalized absolute residual across all nodes is less than 10⁻⁶.

Sample of the grid that was used in the simulations is illustrated in Figure 2. It consists of a two-dimensional mesh. Initially and in order to capture the gradients near the solid-fluid interface, the grid step sizes are increasing in the x and r directions with expansion factors of 1.06 and 1.15, respectively, after that the mesh was adapted such that velocity gradients near the solid surfaces are calculated. If required, additional cells were added to reduce the gradients below a certain value as illustrated in the case of the right outlet in the figure. Grid independency test is performed and shown in Figure 3 by monitoring the average Nusselt number at the wall in which different numbers of grid nodes were considered. It was found that adding more cells will not change the value of the average Nusselt number at the surface of the pipe with a maximum error of 1% compared to the minimum number of elements chosen for the initial mesh. It is worth mentioning here that the mesh that was utilized for all simulations is of 650,000 nodes.

Figure 2.

Adaptive grid system technique used in the simulations.

Figure 3.

Grid independency test.

Figure 3 demonstrates that the solution is mesh-independent for a grid of 100 × 100 nodes. This grid size is used for all simulations conducted in this research.To verify the numerical code, results of the current code are compared with the results obtained by De Vahl Davis and Jones,²⁴ De Vahl Davis,²⁵ and Nag et al.²⁶ for the case of gaseous Boussinesq laminar steady-state air flow inside a square cavity with no fins attached to the hot wall and Pr = 0.71. Boundary conditions considered for this case are as follows: adiabatic upper and lower walls of the cavity. The hot left wall dimensionless temperature is set to one while the cold right wall dimensionless temperature is set to zero.

Table 2 shows a comparison between the mean Nusselt number at the cold surface obtained by the current code and that obtained by De Vahl Davis and Jones,²⁴ De Vahl Davis,²⁵ and Nag et al.²⁶ at Ra = 10⁴, 10⁵, and 10⁶; where Ra = (gβ(T₁ – T₂)L³/αν). Comparisons show excellent agreement with an error of less than 1%.

Table 2.

Comparison between the mean Nusselt number at the cold surface obtained by the current code and that obtained by De Vahl Davis and Jones,²⁴ De Vahl Davis,²⁵ and Nag et al.²⁶ at Ra = 10⁴, 10⁵, and 10⁶.

Ra	10⁴	10⁵	10⁶
De Vahl Davis and Jones²⁴ and De Vahl Davis²⁵	2.243	4.519	8.800
Nag et al.²⁶	2.24	4.51	8.82
Present study	2.241	4.52	8.83

Also, Figure 4 shows a comparison of the current code results with the results obtained by Taamneh and Bataineh²⁷ for the case of a lid driven cavity in which nanofluid particles of Al₂O₃ is dispersed inside the cavity, ϕ = 0.01 and different Richardson numbers were considered. Where Ri = gβ(T₁ – T₂)L/V². The hot wall dimensionless temperature which corresponds to T₁ of the cavity is set to 1, and the cold wall dimensionless temperature that corresponds to T₂ of the lid driven cavity is set to 0; while the other two walls are adiabatic. The figure illustrates that there is a maximum error of 3% in the average Nusselt number between the results obtained from the current code to those obtained by Taamneh and Bataineh.²⁷ Moreover, the code was validated with correlation given by Sieder and Tate²⁸ for the case of developing thermal and hydrodynamic boundary layers

\bar{N u_{D}} = 1.86 {(\frac{R e_{D} \Pr}{L / D})}^{1 / 3} (\frac{μ}{μ_{s}})

(21)

In this validation, air is used as the base fluid, Re = 1000, Pr = 0.7, and L/D = 10; the correlation yields a value of 7.655 for the average Nusselt number while the current code yields a value of 7.99 with a maximum error of 4% compared to the correlation result.

Figure 4.

Comparison between results obtained from the current code to results obtained by Taamneh and Bataineh.²⁷

In addition, a comparison between results obtained from the current code for the local Nusselt number and the experimental results obtained by Akhavan-Zanjani et al.²⁹ is presented in Figure 5. The comparison is for the case of laminar convective heat transfer of water and graphene nanofluid fully developed flow in circular pipe at different Reynolds numbers and constant surface heat flux. The nanofluid flowed in a straight copper tube with inner diameter, outer diameter, and length of 4.20, 6.00, and 2740.2 mm, respectively. The volume fraction of the nanosolid particles of graphene is set to 0.01%. The figure shows that there is a maximum error of less than 3% between the obtained results from the current code and these obtained by Akhavan-Zanjani et al.²⁹

Figure 5.

Comparison between results obtained from the current code to results obtained by Akhavan-Zanjani et al.²⁹

Finally, a comparison between the average Nusselt number against Reynolds number obtained from the current code and the experimental results obtained by Trivedi and Johansen³⁰ is presented in Figure 6. The comparison is carried out for the forced convective heat transfer in Al₂O₃–air nanoaerosol for three different cases, namely, air flow, nanofluid with mass fraction of 0.02%, and nanofluid with mass fraction of 0.1%. The figure shows an excellent agreement between the experimental results and the results obtained by the current code with a maximum error of less than 4%.

Figure 6.

Comparison between results obtained from the current code to results obtained by Trivedi and Johansen.³⁰

Authors believe that the error in all the comparisons is because of the discretization method used as well as the mesh. In order to reduce the error, higher order discretization techniques can be used.

Results and discussion

Figure 7 shows the velocity magnitude along the radial direction for the case where AR = 10 at different axial locations x of 0.1, 0.5, and 1. Reynolds number (Re) is fixed to 1000, Knudsen numbers (Kn) of 0, 0.05, and 0.1 to cover the continuum, and slip flow regimes were considered. Nanoparticles volume fractions $(ϕ)$ of 0, 0.3, and 0.5 were considered as well. By inspecting the plots in the figure, it is obvious from plots that for the case of Knudsen number (Kn) is equal 0, the no slip boundary condition holds for all values of nanoparticles volume fraction $(ϕ)$ and all locations. Moreover, as Knudsen number (Kn) increases, then the slip velocity at the boundaries increases for all values of Knudsen number and locations. The graph also illustrates the developing velocity profile for a fixed volume fraction and a fixed Knudsen number. It is clear that at x = 0.1, the velocity profile is flatter than x = 0.5 and x = 1. This flatness of the velocity profile decreases as x increases. Moreover, the graph shows that the velocity magnitude is higher for a fixed Knudsen number when φ = 0 at x = 0.1, same trend occurs for x = 0.5 and x = 1.

Figure 7.

Variation of the velocity magnitude (V) at different x locations for different Knudsen numbers (Kn) and different values of nanoparticles volume fraction $(ϕ)$ , Re = 1000 and AR = 10.

Figure 8 illustrates temperature profiles along the radial direction for the case where AR = 10 at different axial locations x of 0.1, 0.5, and 1. Reynolds number (Re) is fixed to 1000, Knudsen numbers (Kn) of 0, 0.05, and 0.1 to cover the continuum, and slip flow regimes were considered. Nanoparticles volume fractions $(ϕ)$ of 0, 0.3, and 0.5 were considered as well. The graph shows for the case of fixed volume fraction and location, the higher Knudsen number is, the higher temperature jump occurs at the boundaries. In addition, the graph shows that for the case of fixed volume fraction and fixed Knudsen number, the flatness in the temperature profile decreases as the position increasing; this is due to the developing thermal boundary layer evolution with the position.

Figure 8.

Variation of the temperature (T) at different x locations for different Knudsen numbers (Kn) and different values of nanoparticles volume fraction $(ϕ)$ , Re = 1000, and AR = 10.

Figure 9 illustrates the variation in the average Nusselt number $(\bar{Nu})$ along with Prandtl number (Pr) and the nanoparticles volume fraction $(ϕ)$ for the case where AR = 10 and Re = 1000. The graph shows that there is no effect of the volume fraction of the nanoparticles $(ϕ)$ on the average Nusselt number $(\bar{Nu})$ for values less that 0.03, and as this value increases, the average Nusselt number increases. Moreover, the graph shows that as Prandtl number (Pr) increases then the average Nusselt number increases as well.

Figure 9.

Variation in the average Nusselt number $(\bar{Nu})$ along with Prandtl number (Pr) and the nanoparticles volume fraction $(ϕ)$ for the case where AR = 10 and Re = 1000.

In Figure 10, the average Nusselt number $(\bar{Nu})$ is plotted against Reynolds number (Re) for different values of nanoparticles volume fraction $(ϕ)$ for the case where Kn =0.05 and AR = 10; the graph illustrates that as Reynolds number increases, the average Nusselt number $(\bar{Nu})$ increases for all values of the nanoparticles volume fraction. Moreover, the graph shows that there is no effect of the nanoparticles volume fraction with values below 0.03 on the average Nusselt number. The average Nusselt number increases $(\bar{Nu})$ as the value of the nanoparticles volume fraction exceeds 0.03. For instance, the graph shows that the value of the average Nusselt number $(\bar{Nu})$ at Reynolds number (Re) equals to 1000 and volumetric fraction of 0.03 is 4.381. This value at the same Reynolds number and nanoparticles volume fraction $(ϕ)$ of 0.3 is 5.711 with an enhancement of 30.35% in the average Nusselt number $(\bar{Nu})$ . In addition, the graph shows that for Reynolds number equals 1000 and nanoparticles volume fraction $(ϕ)$ of 0.5, the average Nusselt number is 10.372 with an enhancement in of 136.74% in the average Nusselt number.

Figure 10.

Variation of the average Nusselt number $(\bar{Nu})$ with Reynolds number (Re) for different values of nanoparticles volume fraction $(ϕ)$ for the case where Kn = 0.05 and AR = 10.

Figure 11 demonstrates the average Nusselt number $(\bar{Nu})$ variation along the aspect ratio (AR) for the case where Kn = 0.05 and Reynolds number = 1000. The graph shows that there is a declining effect of the aspect ratio on the average Nusselt number.

Figure 11.

Variation of the average Nusselt number $(\bar{Nu})$ with the aspect ratio (AR) for the case where Kn = 0.05 and Re = 1000.

Finally, a correlation for the average Nusselt number $(\bar{Nu})$ among all the parameters taken into consideration in this study in the slip flow regime is proposed in the following equation with R² = 0.87

\bar{Nu} = 0.976 A R^{- 0.0536} P r^{2.39} R e^{0.22} K n^{- 0.378} ϕ^{0.07}

(22)

Conclusion

A steady two-dimensional analysis of low-pressure gaseous laminar nanofluid flow in the entrance region of pipes in which a constant surface temperature of 70°C is carried out. This type of flow serves in many engineering applications such as those found in nuclear reactor and electronic equipment cooling and many others. Rarefaction, Reynolds number (Re), aspect ratio (AR), and the volume fraction of the nanoparticles $(ϕ)$ effects on both flow and heat characteristics of such flows are investigated. Ranges of these parameters considered in this study are as follows: $500 \leq Re \leq 2000$ , $0.76 \leq \Pr \leq 0.95$ , $0 \leq ϕ \leq 0.5$ , $0 \leq Kn \leq 0.1$ , and $1 \leq AR \leq 10$ . Results show that as Knudsen number (Kn) increases, the slip velocity and the temperature jump at the boundaries will increase and the average Nusselt number $(\bar{Nu})$ decreases. Moreover, it is found that as Reynolds number (Re) increases, the average Nusselt number $(\bar{Nu})$ increases. In addition, it is found and for the investigated range of the nanoparticles volume fraction $(ϕ)$ that there is no effect on the heat characteristics for volume fractions less than 0.03. As this fraction increases beyond 0.03, an increase in the average Nusselt number is observed. Also, it was concluded that as the aspect ratio (AR) increases, the average Nusselt number decreases. Moreover, it was concluded that as Prandtl number (Pr) increases, the average Nusselt number increases. Finally, a correlation for the average Nusselt number and the parameters investigated in this study is proposed.

Footnotes

Appendix 1

Handling Editor: Bo Yu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Wael Al-Kouz

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