Investigation of Magneto-Convection Characteristics in A Sudden Expanding Channel with Convex Surface Geometry Under Thermally Developing Flow Conditions

Abstract


Introduction
Global warming is one of the top issues that requires an urgent action.To reduce emission and leave a clean world for future generations 151 countries have taken considerable actions such as the "loss and damage" motto and net-zero emission by 2050 at the COP26 summit, Glasgow Conference in 2021 [1,2].For this purpose, there has been significant increment in the scientific studies carried out on the use of renewable energy sources [3,4] Most researchers have conducted a great deal of study on it.One of these studies is the heat exchanger, which is used in lots of industrial applications.The performance enhancement techniques for heat exchanger have been investigated employing passive [5] and active techniques [6] which are heat transfer improvement techniques.
For instance, rough surface, which includes dimpled fins (DFs), is one of the prominent passive techniques in the geometric improvement, and this phenomenon has been investigated by numerous researchers in the literature [7,8].Malapur et al. [9] conducted a computational analyses on five different dimpled tubes (DT).Researchers have aimed to investigate the effect of DF and compare it with the smooth tube (ST).According to the results, the heat transfer properties of DT override on ST, and the helical pattern of DF with a smaller pitch angle exhibits more heat transfer than other types of fins.
Dagdevir [10] has numerically investigated the convective heat transfer (h) and pressure drop (ΔP) Re=3000.It was reported that the influence of PL is higher than the effect of H and d in terms of h and ΔP.Furthermore, it was found that d=7 mm, PL=30 mm, and H=1.0 mm present the optimum performance regarding hydrothermal behavior.A study aimed to determine the performance of DT has been carried out by Kaood et al. [11].The numerical study has been examined under Re=3000-40000, constant heat flux of q"=40000 W/m 2 and using H2O as HTF.As a result, the DT may enhance the thermal performance of the system up to 121.4% compared to ST.Another study, which focused on the effect of DF on thermohydraulic performance (THP), has been conducted by Zheng et al. [12].
Researchers firstly examined the actual applicability of the elliptical DF to be used as a passive method and created the tube with a reasonable error rate.According to the results, Performance Evaluation Criterion (PEC) has reached a maximum value as 2.4 using the elliptical DF.
Along with the improvement of the nano technology, NFs have been created with the help of nanoparticles, which size is less than 100 nm [13] and dispersed in conventional HTFs such as H2O, ethylene glycol, oil, etc. [14,15].In fact, NFs have been examined together with DF in both experimental and numerical studies in the literature.Gürsoy et al. [16] theoretically analyzed the THP of ferrofluid (FF) in a sudden expansion tube (SET) retrofitted with DF.The analyses have been performed under 100≤Re≤2000, q"=600 W/m 2 , and 1.0%≤φ≤2.0%.The results show that the increment in Re affects Nu positively.It is also found that the flow geometry, which has an expansion ratio (ER) is 2.5 and a pitch ratio (P/d) is 3.0, at the Re=2000 and φ =2.0% case provides the highest PEC.With the addition of NPs to conventional HTFs, a few improvements are observed in the THP.The thermal and hydrodynamic behavior of the existing fluids might change in this way.Thus, a less increase in heat transfer coefficient and viscosity are among the results encountered by researchers [18].To enhance thermal performance of the system, researchers have started to activate the magnetic field (MF) on the system as well as the DFs.Utilization of the MF is placed as one of the active heat transfer enhancement techniques [19] in the literature.Many experimental and numerical studies have been carried out on this subject, especially using Fe3O4 NPs.Recently, Gürdal et al. [20] have presented valuable studies in the literature about MagnetoHydroDynamics (MHD) area.A particular experimental study, in which the scope of MHD, has been carried out by Shahsavar et al. [21].The researchers have utilized two active methods consisting of rotating MF and vibration to enhance the heat transfer rate of Fe3O4/H2O FF flowing in a rifled tube.According to the results, vibration enhances the PEC up to 1.28.
Along with applying the rotating MF, PEC shows an increment of 21.32%.Halawa and Tanious [22] realized a numerical study to determine the optimum MF arrangement, which will provide the highest convective heat transfer rate in the rectangular channel using Fe3O4/H2O FF.Researchers have presented detailed results about the problem.MF has reciprocatively been applied on the channel at four different locations and the magnets have been placed in N-N, S-N, and S-S orientations.While the MF intensity was ranged from 250 to 1000 Gauss, N-N magnet orientation was performed with the optimum results.
The results indicated that the average Nu exhibits an increment range of 16.44%-24.46%applying the MF up to 1000 Gauss.
When the literature review is examined, it is seen that MHD applications are made for different flow geometries and application purposes.However, the effect of DC MF on the THP of FF flowing through SET, which is modernized with DF, has not been investigated so far.This determined purpose reveals the novelty of this study.Based on this situation, the flow geometry of which the best PEC has been obtained in the study (ER=2.5 and P/d=3.0)conducted by Gürsoy et al. [16] has been used for this investigation.It has numerically been handled in this study as the fundamental flow problem, and the effect of MF on the sudden expansion geometry with FF has been examined in detail.In the study, Re=1000, 1500, and 2000 have been used, and FF has been prepared as φ=1.0% and 2.0%.DC MF has been applied at z= 937.5 mm with the intensity of B0=0.03, 0.05, 0.3, and 0.5T.The results are presented in the form of graphs and discussed in detail.

Geometry Description and Boundary Conditions
In this study, MHD convective heat transfer and flow characteristics of FF have been considered for the SET subjected to uniform and constant wall heat flux and DC MF.The flow chart of the problem, definitions, boundary conditions, and process stage of this analysis are given in Fig. 1.Besides, Fig. 2 shows description of the geometry and boundary conditions for investigated computational domain.The HTF enters the inlet tube of SET with D1=8 mm and L1=375 mm under laminar flow regime at 300 K.
HTF is suddenly expanded at z=375 mm and starts flowing in the outlet tube with D2=20 mm and L2=1125 mm.Furthermore, DC MF has been applied at z=937.5 mm with the diversified intensities as B0=0.03,0.05, 0.3, and 0.5T.The affected area length by MF has been considered as LM=100 mm electromagnet diameter used in the experimental study carried out by Tekir et al. [23].The geometric parameters and boundary conditions of the numerical study are summarized in Table 1.

Hartmann number
Ha (-) 0.32-0.52-3.18-5.30* Since Re is constant, the inlet velocity varies according to the THP of the HTF.

Mathematical Modeling and Solution Procedure
In this section, mathematical formulas, thermophysical properties of HTFs, and solution procedure are given for solving numerical studies with the properties defined in Section 2.

Governing equations
The fluid data has been solved via Finite Volume Method.With this approach, the continuity equation is presented in Eq. ( 1) [24].
Any change is not seen in the continuity equation with applying the MF [25].The momentum equation, which analyzed the movement of HTF and including the Lorentz force, is given in Eq. (2-4) as component [26,27].
z-component of Momentum equation: Lorenz Force acting in the counter effect on flow [28] is obtained as follows [29].
While the j (A/m 2 ) symbolizes the current density [29] given in Eq. ( 6) and obtained from Ohm's law.( ) where  (S/m), E (V/m), and V (m/s) describe the electrical conductivity of FF, electric field, and HTF velocity, respectively.The Lorentz force, also known as the magnetic field force, is solved with the Navier-Stokes equations.In this scope, the MHD equation [30,31] defined by Ohm's law and Maxwell's Equations and given in Eq. ( 7) is solved to be able to calculate the MHD effect.
( ) ( ) The energy equation used for single-phase numerical solutions is given in Eq. ( 8) [32].As in the momentum equation, the heat is generated as a result of applying a MF and passing an electric current through conductive materials should also be included in the energy equation.This effect, which is included in the literature as the term Joule Heating [33], is expressed by Eq. ( 9) and added to the right side of Eq. ( 11) in scope of MHD.
where  (W/m 3 ) is the dissipation energy, which indicates the consumed power against viscous forces.

Thermophysical properties of Fe3O4/H2O Ferrofluid and data reduction
In this study, Fe3O4 NPs dispersed with φ=1.0% and 2.0% in H2O, and pure H2O were used as HTF.
The basic thermophysical properties of H2O and Fe3O4 NP at T=300 K are given in Table 2. To calculate the thermophysical properties of FF, well-known correlations in literature have been used.These thermo-physical properties calculation equations and data reduction equations to be used to evaluate the numerical analysis results are given in Table 3.

Reynolds number [-]
Re Nusselt number

Solution method
The iterative solutions have been conducted via ANSYS Fluent 2020 R2 software.Single-phase fluid properties have been accepted for the analyses.Whereas the no-slip condition has been assumed for all tube walls, the heat flux of q"=600 W/m 2 has been just applied to the outlet tube.Analyses have been conducted using the steady-state flow condition.Pressure-based solver has been used for the solutions.Also, absolute velocity formulation has been assumed.The SIMPLE algorithm has been preferred as a pressure-velocity coupling method due to being faster than other solvers.As the Spatial Discretization, Squares Cell Based, Second Order, and Second Order Upwind methods have been used as the discrimination of the governing equations.

Mesh Independency Study and Validation
Since the element number of a mesh directly affects the solution time, the mesh independency study must be performed for the fluid domain to obtain the optimum element number [43].For this purpose, the analysis has been carried out under Re=1000 using H2O as the HTF.The fluid domain has meshed using global and local mesh settings to reach the suitable mesh quality.After that, mesh independency analyses have been conducted using variable mesh values.The Nu and ff results obtained from various mesh element numbers are presented in Fig. 3.It is seen that the results exhibited stability after the 2499684 element numbers.The mesh settings at this point have been approved as the base mesh settings for all fluid domains.The obtained mesh structures are presented in Fig. 4.

Element number (b)
0 1x10 6 2x10 6 3x10 6 4x10 6 5x10 6 6x10 6 7x10 6 8x10 6 9x10 6 10x10 6 To demonstrate the validity of the analyses, the results have been compared with previously published studies in the literature.For this purpose, the fluid domain without DF has been formed and results have been obtained at Re=290 using H2O as HTF.As a result of this, the numerical results have been compared with Ref. [44].The validation has been carried out with the help of the velocity profile given in Fig. 5 at the sudden expansion point (z=375 mm).It is concluded that there is a good agreement between the present study and the data obtained from the literature.Avg.Error 4.56% Fig. 5. Validation of the results with literature using velocity profile at z=375 mm.

Effect of dimple fin and ferrofluid on heat transfer and flow characteristics
The effect of DF and Fe3O4/H2O FF on THP has comprehensively been given in Fig. 6.Generally, the average Nu and average ff exhibit an increment and decrement as the increment of Re, respectively.
Moreover, the average Nu and ff show a rising up in the case of using DT compared to ST.For example, the average Nu presents an enhancement of 22.62% whereas an increase of 3.28% is realized for the average ff at Re=2000.When investigating the effect of Fe3O4/H2O FF on THP, the average Nu shows an augmentation of 5.43% and 6.27% for φ=1.0% and 2.0%, respectively compared to H2O at Re=2000.
Furthermore, a drastic soaring is realized in ff as 3.16% and 3.21% for φ=1.0% and 2.0% at Re=2000, respectively.It is concluded that using DT is more effective than using Fe3O4/H2O FF in terms of heat transfer enhancement, and this result is supported in a study published by Pourfattah et al. [45].In the case of using DT and 2.0% Fe3O4/H2O FF, 30.31% raise is observed for average Nu at Re=2000 while an boost of 6.60% is occurred for average ff.It has already been mentioned that the presence of DF is more efficient than the use of NF for enhancing heat transfer.To demonstrate the heat transfer and flow characteristics mechanism of DT, a visual description is presented in Fig. 7 (a-c).In Fig. 7 (a As a result, the pressure and inertia forces are more dominant than the viscous force, and the thermal boundary layer thickness becomes thinner [47].Thus, the THP of the SET exhibits an enhancement as the decrement of the field synergy angle (α) [48].As the HTF flows over the protrusion section of the DF, an increase in HTF's velocity takes place due to narrower cross-section, and then the HTF spreads over a larger volume.Due to this spread, secondary flow occurs, which prevents the convection of heat flux applied downstream of the DF to the HTF.Furthermore, Fig. 7   According to the contours, the top wall realizes more heat transfer than the bottom wall for judging by the entire tube and this result presents similarity with the data in Ref. [49].Investigation of the PEC presents the applicability and feasibility of the designed thermal system.Within the scope of this concept, the PEC results have been presented in Fig. 8 to exhibit the THP.The trends show that obtained thermal performance using the DF and FF come to the forefront compared to the adverse effect of hydraulics.Besides, it is seen that the using DF dominates the using Fe3O4/H2O FF in terms of THP.When the evaluation has made for DF alone, it was determined that PEC rises by 21.31% at Re=2000.When the evaluation has made for Fe3O4/H2O FF alone, it was determined that PEC soars by 4.85% and 9.79% at Re=2000 for φ=1.0% and 2.0%, respectively.In the case of the use of DF and FF together, PEC exhibits an increment of 26.58% and 27.56% at Re=2000 for φ=1.0% and 2.0%, respectively.The increment in convective heat transfer that occurs with the use of Fe3O4/H2O FF has been attributed to the Brownian motion and Thermophoresis phenomena, which are also referred to as slip mechanisms by many researchers [50,51].The Brownian motion defines as the random motion of NPs and collides of these NPs with each other.As a result of this interaction, heat is transferred from one NP to another.
While the NP flows in the HTF, heat transfer realizes in two different ways.The first phenomenon is called direct contribution, which can explain by the heat transfer among the NPs.In contrast, the second phenomenon is called indirect contribution (micro-convection), which is realized between NP and HTF that surrounds the NP [52].The Brownian motion disrupts the thermal boundary layer, and this phenomenon contributes to enhancing the convective heat transfer performance of the system [53].
Another effect is the Thermophoresis phenomenon (also known as the Soret effect [54]), which has been studied by many researchers and is claimed to be worth investigating [55,56].Thermophoresis can be defined as the time-averaged force formed by the Brownian motion of the NPs in the HTF under the influence of temperature gradient [57].With the effect of this force, the particles in the hot region of the flow area tend to flow toward the cold regions.Because the Thermophoretic force, which is proportional to the temperature gradient, has higher values in the higher temperature gradient [58].

Effect of magnetic field on heat transfer and flow characteristic
The application of the MF in a SET put forward the novelty of this study.Within this scope, different DC MF intensities (B0=0.03T,0.05T, 0.3T, and 0.5T) have been applied on the outlet tube wall at z=937.5 mm.The THP behavior of DT in both presence and absence of MF is given in Fig. 9.The results exhibit a closeness to each other.However, it can be mentioned that as the intensity of MF growths, the average Nu and average ff show an increase.The highest average Nu has been obtained as 5.28 for the case of B0=0.5T and Re=2000 using φ=2.0%Fe3O4 FF.When a comparison is carried out between DC MF intensities and the absence of MF, the average Nu performs an increment of 0.42%, 0.74%, 1.22%, and 1.42% for B0=0.03T,0.05T, 0.3T, and 0.5T, respectively.On the other hand, the average ff tends to decrease with the increment of Re, gradually.Compared to the absence of MF, the average ff shows an increment of 0.30%, 0.50%, 1.30%, and 2.50% for B0=0.03T,0.05T, 0.3T, and 0.5T, respectively.Many factors can explain the variation in the THP of the system with the application of MF.
The one of the most important effects is the Lorentz force, which occurs due to the electrical conductivity of the HTF.Based on the MHD, when an external MF is applied to the magnetizable NPs, creates an effect called the Lorentz force between the fluid and the MF source [59].The Lorentz force acts HTF flow at the opposite direction, which causes occurrence of vortex in the MF applied region [34,60].
Another factor affecting THP is the magnetophoresis transport mechanism [61] formed by Fe3O4 NP acting in MF.Magnetophoresis is a heterogeneous behavior from quasi-homogeneous behavior when magnetizable NPs such as Fe3O4 NP are attracted to the tube´s wall under the influence of an MF.The NPs are attracted to the tube's wall and disperse the boundary layers there leading an increase in heat transfer rate.
The effect of DC MF on PEC is presented in Fig. 10 using Fe3O4/H2O FF with φ=2.0%.
According to the results, the PEC shows an increment up to B0=0.3T.When the Fe3O4/H2O FF is subjected to B0=0.5T DC MF, the PEC exhibits a decrement.A similar result has been acquired by Ghofrani et al. [62] and Yarahmadi et al. [63].When a detailed investigation carries out on the results, it is obtained that the PEC of Fe3O4/H2O FF flowing in DT performs an increment of 27.97%, 28.30%, 28.57%, and 28.31% with exposed to B0=0.03T, 0.05T, 0.3T, and 0.5T DC MF, respectively.To demonstrate the DC MF effect on heat transfer, the surface Nu distribution at MF applied surface is presented in Fig. 12 for B0=0T and 0.3T DC MFs at φ=2.0% and Re=2000.The surface Nu distributions are demonstrated for x and -x direction surfaces.The distributions show that MF has a positive effect on heat transfer not only on the MF applied surface but also after this region.The main reason underlying this result is based on the effect of MF on flow motion.• The average Nu and ff show an increment of 22.62% and 3.28% in DT compared to ST using H2O at Re=2000.• Using Fe3O4/H2O FF shows an enhancement of 5.43% and 6.27% in average Nu for φ=1.0% and 2.0%, respectively, compared to H2O in the same conditions.Besides, Fe3O4/H2O FF increments the average ff by 3.16% and 6.27% compared to H2O, respectively.
• The use of DF provides a 21.31% increment in PEC, while the use of Fe3O4/H2O FF with φ=1.0% and 2.0% results in an increment of 4.85% and 9.79%, respectively.• The results represent that using DF is more effective than Fe3O4/H2O FF in terms of THP.
• The highest average Nu is obtained in B0=0.5TDC MF.However, the highest PEC is acquired in B0=0.3TDC MF.
), the streamlines and local Nu distribution are shown flowing through the tube.While the HTF flows in the DF at Re= 2000, HTF impinges the upstream side of the DF, and flow is interrupted.The velocity profile is approached to the center of the tube due to the effect of this interruption, and flow separation is formed on the upstream side of DF[46].This phenomenon positively affects the flow in terms of THP by thinning the velocity boundary layer.
(b) and (c) show the comparison of local Nu distribution on outlet tubes for ST and DT.As can be seen from the figures, DT provides superiority over ST in terms of convective heat transfer thanks to the enhancement in local Nu values.

Fig. 7 (
Fig. 7 (c) explicitly presents the difference of top and bottom walls in terms of surface Nu distribution.

Fig. 7 .
Fig. 7. Effect of DF on THP at Re=2000; (a) interaction of heat transfer and flow characteristics in DT, (b) comparison to Nulocal in ST and DT as a trend (c) comparison to Nulocal distribution in ST and DT.

"Fig. 9 .
Fig. 9. Variation of average Nu and ff with Re under different DC MF intensities.

Fig. 10 .
Fig. 10.Variation of PEC with Re under different DC MFs intensities.

Fig. 12 .Fig. 13 .local
Fig. 12. Exhibiting the surface Nu distribution.The effect of DC MF on temperature gradient and vorticity formation at Re=2000 and z=927.5 mm is presented in Fig. 13.It can be seen that along with increments of the DC MF intensity, the temperature gradient becomes a more uniform situation shown in Fig 13 (a).However, the same explanation cannot be made for the vorticity formation given in Fig. 13 (b).As the DC MF intensity increases, no visible change in vorticity formation is observed.

Table 1 .
Geometric parameters and boundary conditions of the numerical study.

Table 3 .
Data reduction equations used in the study.
Variation of average Nu and ff with Re under DT and FF effect.
Variation of PEC with Re under DT and FF effect.