Laminar Convective Heat Transfer Across a Circular Pipe by using MWCNT + Fe3O4 Hybrid Nanofluid: A Numerical Study

MWCNT + Fe3O4 - water hybrid nanofluid’s heat transfer due to convection through a pipe was studied numerically and presented in this paper. Using the ANSYS FLUENT 22 software, the single phase energy equation of second order, momentum equation, and mass equation are all solved. The diameter of the pipe was 4mm and lthe lengthwas 800mm. Water which was considered the base fluid was used with different volume fractions (0% to t%) of MWCNT + Fe3O4 nanoparticles for a wide variety of Reynolds number (300 to 1000) at a steady heat flux of 2000 W/m2 at the wall of the tubes. The result demonstrates that for both MWCNT + Fe3O4 -H2O nanofluid and water, increasing Reynolds number increases the Surface Nusselt number, skin friction coefficient and also the pressure loss in the evolved region.


Introduction
Our lives depend on heat as a source of energy, but since the advent of technology, heat has posed a problem in modern devices, and engineers and researchers have been trying to find new ways to deal with it.Everything emits heat, from human bodies to automobile engines, and any buildup of heat can cause a device to behave unexpectedly.This is one of the biggest issues faced in a wide range of industries, encompassing vital sectors such as energy production, transportation networks, manufacturing processes, climate control systems, thermal management in various applications, and efficient lubrication mechanisms.Effective addition or removal of heat has been a key priority for researchers.In a lot of applications, heat is carried away using fluid which motivates researchers to find ways to make the fluid thermally more conductive.Since engine oil, ethylene glycol, and water are frequently employed as heat transfer fluids but have relatively low thermal conductivities compared to solids, most researchers are currently focusing more on increasing the thermal conductivity of working fluids by mixing solid particles.And combining a base fluid (water, motor oil, or ethylene glycol) with a negligibly small number of nanoscale-sized (1nm to 100nm) solid particles (Al2O3, CuO, TiO2, SiC, SiO2, Fe2O3, MgO, etc.) the thermal conductivity can be increased significantly.In 2010, Maryam Abareshi presented a fabrication method for Fe3O4 and evaluated thermal conductivity.Their findings demonstrate that as volume concentration and temperature rise, the nanofluids' thermal conductivity ratio also rises.At a temperature of 40 °C, the nanofluid containing 3% volume of nanoparticles exhibited a remarkable enhancement in thermal conductivity, reaching an impressive 11.5%.The results of the experiment were meticulously compared and contrasted with theoretical models, providing a robust and comprehensive analysis of the findings.[1] Wei Yu conducted a thorough investigation on Fe3O4 nanofluids formulated with kerosene as the base fluid.The study encompassed a wide temperature range spanning from 20°C to 60°C, while exploring various volume concentrations ranging from 0.0% to 2.0%.The results revealed a compelling correlation between the increase in particle volume concentration and the simultaneous augmentation of both viscosity and thermal conductivity within the nanofluid.[2] Xuan and Li conducted a comprehensive study in 2003, focusing on convective heat exchange and flow characteristics of Cu-water nanofluids in a tube with a 10-mm inward distance.Their investigation specifically targeted the turbulent region.Remarkably, their findings indicated that the nanofluid's friction factor, particularly those with 1-2% nanoparticle fractions, closely resembled those of water flow.This intriguing result prompted further exploration and experimentation in this area.[3] In 2013, L. Syam Sundar conducted an experimental investigation on Fe3O4/water nanofluid.The study spanned temperatures in the range of 20-60 °C and volume concentrations from 0 -2%.The results demonstrated a direct correlation between increasing particle volume concentration and enhanced thermal conductivity and viscosity in the nanofluid.[4] In 2017, Amin Shahsavar and Mehdi Bahiraei conducted measurements on the thermal conductivity and viscosity of a water-based hybrid nanofluid.This nanofluid consisted of carbon nanotubes (CNTs) and Fe3O4 magnetic particles.By utilizing the collected data, the researchers developed artificial neural networks capable of predicting viscosity and thermal conductivity based on input parameters.[5] In 2019, Mohsen Tahmasebi Sulgani and Arash Karimipour investigated the thermal performance of 10w40 engine oil added with Al2O3-Fe2O3 Nanopowder between temperatures of 25-65°C.They found that the mass fraction of 4% carried out the most improvement.[6]

Physical model and boundary conditions
The rate of heat transfer and accompanying friction factor is evaluated using a two-dimensional tube with a constant heat flux applied to its surface.Numerical research has been conducted using the ANSYS Fluent, a well-known commercial computational fluid dynamics software to examine the performance of nano-fluid through a tube.It is shown how a laminar flow moves through an 800 mm long, two-dimensional, circular pipe with a 4 mm diameter.With the assumption that the tube wall exhibited no-slip condition, a consistent and even heat flux of 2000 W/m2 is imposed at the tube's boundary.The fluid is permitted to flow at an appropriate velocity while maintaining a homogeneous temperature of 303 K at the inlet of the tube.Following the fluid stream's thermal and hydrodynamic improvement, all of the heat exchange and fluid dynamic characteristics were extracted.Temperatures are measured at a line that is 790mm from the intake.

Numerical methods and methodology
In this specific numerical analysis, we used ANSYS(Fluent), which is a commercialized software for computational fluid dynamics.A control volume method was used to resolve all of the governing equations for energy, laminar quantities, mass, and momentum.Boundary pressure at the outlet and laminar velocity at the inlet have been considered in this study.For a circular tube, the relaxation factors for the pressure equation are 0.4, the momentum equation is 0.785, the energy equation is 1, and the density equation is 0.8.A wide variety of Reynolds numbers are used to examine MWCNT + Fe3O4-water nanofluids with various particle volumetric fractions, and the findings are then compared to water as the base fluid.

Governing Equations:
In the presence of laminar flow and under steady-state conditions, the essential equations governing momentum energy and continuity, for forced convection can be succinctly expressed as follows.
The equation for momentum: The momentum equation may be written as follows for laminar flow: The equation for continuity: Since the mass in the control volume under steady flow is constant, the mass conversation may be represented as: The equation for energy: Energy can be transferred through three mechanisms: mass transfer, work transfer, and heat transfer.Consequently, for a control volume experiencing a steady flow, the energy balance equation can be expressed in the following manner:

Nanofluid's thermal and fluid dynamic properties:
According to the data presented in Table 1, it can be observed that the thermophysical properties of the operating fluid exhibit temperature dependency.[7] Table 1: Properties related to the thermal behavior of various working fluids [7] Volume For the nanofluid flow, the Reynolds number is written as: Nanofluid is assumed to completely dissipate the heat transfer rate, Q nf , to the tube wall as it flows along a tube of circular shape, increasing its temperature from the entrance fluid bulk temperature   to the exit fluid bulk temperature   .So, Where m nf = rate of mass flow of the nanofluid And C P nf = nanofluid's specific heat at a constant pressure The formula for the typical heat transfer coefficient, h c , is Where A w = circular tube's surface area The difference in temperature between the walls is calculated as: Therefore, the expression for the average Nusselt number is as follows: The expression for the pressure difference is shown below:

Grid Independency Test:
To build grids that work best, it is necessary to take into account the grids' shape, quality, and quantity.The amount of grids in particular affects the overall processing cost and the precision of the findings of simulation analysis.Due to the large spatial discretization inaccuracy that coarse grids produce, analysis results are less accurate.On the other hand, excessively fine grids could significantly decrease the precision of analysis results by substantially increasing the round-off errors even beyond truncation errors [8].Therefore, choosing the right number of grids is essential [9].We carried out the grid independence test using ANSYS and found out that for 220000 elements, the % of relative error was the lowest, as depicted in Table 2. Therefore, for our study, we used 220000 number of elements.

CODE VALIDATION TEST
In this experiment, water flowed through a tube at a constant heat flux and uniform velocity, resulting in laminar flow.A range of Reynolds numbers (300-1000) has been considered to calculate the Nusselt number and the friction factor.In the fully developed region of the flow, the obtained Nusselt numbers were compared to the values obtained from Lévêque's solution for laminar tube flow as depicted in Fig 2 .This comparison showed that the calculated Nusselt numbers had an error of less than 2% for all the Reynolds numbers tested.Lévêque's solution [10] for laminar tube flow is as follows:  From the figure, it is evident that as the nanofluid's surface Nusselt number increases, the volume fraction and Reynolds number also increase.This happened as a consequence of an increase in the rate of energy exchange and an increase in productive thermal conductivity due to the chaotic and erratic movement of the ultrafine particles in nanofluids, as a result of an increase in volume fraction and convection, which occurred due to an increase in Reynolds number [11].Increased velocity of fluid and temperature difference causes Reynolds number to increase, which eventually causes an increment in the surface Nusselt number as well [12].

Conclusion:
In this particular investigation, the focus was on examining the potential improvements in heat transfer and friction factor when utilizing a Fe3O4-water nanofluid in a conventional circular tube.The outcomes of the study convincingly demonstrate a noteworthy increase in the pressure drop, skin friction coefficient and Nusselt number, as the volume percentage of the nanofluid is augmented in comparison with pure water.Additionally, the adoption of the Fe3O4-water nanofluid provides distinct advantages, including a higher coefficient of heat transfer and an enhanced volumetric flow rate, which make it a highly compelling choice over the traditional use of pure water.

Figure 1 :
Figure 1: Physical model of the simulation and the correlating mesh

Figure 2 :
Figure 2: Comparison of Nusselt number from Lévêque's solution and present work for different Reynolds numbers

Figure 3 :
Figure 3: Surface Nusselt number and Reynolds number comparison for various volume fractions of MWCNT + Fe 3 O 4 -water

Figure 4 :
Figure 4: Pressure loss and Reynolds number comparison for various volume fractions of MWCNT + Fe 3 O 4 -water

Figure 5 :
Figure 5: Skin Friction coefficient and Reynolds number comparison for various volume fractions of MWCNT + Fe 3 O 4 -water

Figure 6 :
Figure 6: Comparison of Nu enhancement percentages between 0.1% and 0.3% with increasing Reynolds number.

Figure 7 :
Figure 7: Comparison of skin friction coefficient between 0.1% and 0.3% with increasing Reynolds number.

Table 2 :
Comparison of number of elements and surface Nusselt number along with % of relative error