Flow field and heat transfer of ferromagnetic nanofluid in presence of magnetic field inside a corru

Abstract

Keywords: Heat exchanger; Helical ribs; Nanofluid; Two-phase mixture model; Magnetic field; Corrugation

Introduction

The investigation of heat transfer and boundary layer flows due to utilizing corrugated channels or ribbed tubes and also employing inserting obstacles has become more and more important in many engineering processes with industrial applications such as heat exchangers [1], electronic cooling devices [2], thermal regenerators [3], automobile engine cooling [4], solar heating [5] and air conditioning systems [6]. This is the reason that convective heat transfer and fluid flows through tubes containing corrugations are investigated by numerous researchers [7-13]. Moreover, because of the rising requirement for the developments of heat transfer and fluids flow with better thermal characteristics than common fluids, the nanofluid investigations in thermal systems are more and more significant [14-21]. The word nanofluid refers to an especial class of heat fluids, which includes solid particles in nano range of 1 to 100 nm homogeneously dispersed in a common base fluid such as thermal-oil, ethylene glycol or water. Nanoparticles can be metallic solids (such as Al, Ag, and Cu) or nonmetallic solids (such as Fe2O3, CuO, and Al2O3) [22-26]. The addition of metal or metal oxide nanoparticles to a common base fluid increases the thermal conductivity coefficient of the final fluid. In other words, nanofluid is a well-versed heat transfer or coolant fluid which has gained a considerable acknowledgment in different thermal requests and applications. Additionally, nanofluid offers increased thermal conductivity coefficient, which can improve the efficiency of thermal systems without an important upsurge in pressure drop. This will finally decrease the size and dimensions of thermal systems and, therefore, the costs of production [27]. The electric field can be mentioned as one of the effective active techniques for improving heat transfer [28]. This method can be combined with another passive way, namely nanofluid. The examinations of Magneto-Hydro-Dynamic (MHD) flows are so important in industries and have continued requests in different research areas such as metallurgical processes and petroleum productions [29]. MHD is the science of investigating electrically conducting fluid flows in the presence of an external magnetic field, and it has gained considerable attention of researchers because of its protruding role in different thermal systems and engineering processes such as the fluid flows in microelectronic systems, the flow of liquid metals and crystal growth [30, 31]. Zainala et al. [26] studied numerically and employing MATLAB software for the MHD mixed convection heat transfer in a vertical flat plate heat exchanger filled with single-phase water-based Cu-Al2O3 hybrid nanofluid. According to their obtained results, the nanofluid velocity and temperature upsurges and reduced respectively with an increase of magnetic field parameter. Job and Gunakala [27] studied numerically MHD fluid flow and heat transfer

inside two elastic coaxial pipes pulsatile equipped with porous blocks and filled with water-based CuO nanofluid. He understood that thermal efficiency improves with the increase of solid nanoparticles diameter and volume concentration, Reynolds number, and the elastic modulus of the pipes. Daneshvar Garmroodi et al. [28] studied numerically laminar MHD mixed convection of a cavity equipped with two multiple rotating cylinders and filled with twophase water-based Cu nanofluid. They found that thermal efficiency reduces by an increase of Hartmann number. Eid [29] studied numerically effects of chemical reactions on MHD free convection heat transfer inside an exponentially stretching sheet equipped with heat sources and filled with two-phase nanofluid. He reported interesting results such as, skin friction coefficient, Sherwood number, average Nusselt number, dimensionless velocity, dimensionless temperature, and concentration contours of nanoparticles. Ma et al. [30] investigated numerically laminar MHD convective heat transfer in a two dimensional corrugated channel with active heaters and coolers and filled with water-based Ag-MgO hybrid nanofluid. They realized that the highest Nusselt number occurs at the heater junction and the cooler junction. Sheikholeslami and Rokni [31] studied numerically MHD nanofluid flow in the presence of Lorentz forces. They found that the Nusselt number surges by Hartmann number increase, and the temperature gradients reduce by increasing the Eckert number and melting parameter. Jafarimoghaddam [32] studied numerically MHD porous flow of a three-dimensional bidirectional stretching surface filled with twophase Upper-Convected-Maxwell (UCM) nanofluid. Eid and Mahny [33] studied numerically unsteady MHD heat and mass transfer and fluid flow over a permeable stretching wall with heat generation/absorption and filled with non-Newtonian two-phase nanofluid. It is found that the concentration and thermal boundary-layer thickness values increase by the increase of the magnetic field. In a reviewed paper by Sajid [34], they show the effects of concentration of nanoparticles and nanofluid flow rate on pressure drop, friction factor, and Nusselt number from several investigations. Kumar [35] investigated experimentally PEC of a minichannel filled with Al2O3/MWCNT hybrid nanofluid. Izadi et al. [36] present a numerical investigation on laminar MHD heat transfer and fluid flow in the cooling process inside a porous metal CPU system, which is filled with nanofluid. They understood that usage of a magnetic field with a stronger filed or system with higher porous metal foam can improve the impingement cooling heat transfer. Sheikholeslami [37] studied numerically effects of Brownian motions and nanoparticle shapes on fluid flow and heat transfer in a porous media filled with water-based CuO nanofluid in the presence of a magnetic field. He found that convective heat transfer improves with augmentation of Darcy and Reynolds number, but it reduces with the rise of Hartmann number. Elahi et al. [38] investigated mathematically on the heat transfer characteristics inside a

channels. The results display a good idea based on the various methods in channels in order to optimize the thermal characteristics. Selimefendigil and Oztop [39] investigated numerically MHD pulsating forced convective heat transfer over two parallel plates equipped with a blocks in channel, and filled with nanofluid. They found that the heat transfer of nanofluid at highest particle volume fraction are higher in pulsating flow as compared to base fluid and they are slightly different than the ones obtained in the steady flow. For more results in this filed one can refer to the exist investigation in literature [40-43].The literature review indicates that the effects of magnetic field on the flow field and heat transfer of ferromagnetic nanofluid with consideration of a two-phase model for nanofluid in a heat exchanger equipped with helical ribs is not studied by researchers. In the present study, three methods are employed to enhance the thermal efficiency of heat exchanger (HE): • Usage of corrugations • Utilizing of nanofluid as Heat Transfer Fluid (HTF) • Using of a magnetic field Different magnetic fields intensities and geometrical parameters are studied in this paper. But employing of corrugations and nanofluid can also increase the pressure drop penalty in heat exchanger. Therefore the Performance Evaluation Criteria (PEC) index is employed in the present investigation to analyze the thermal-hydraulic characteristics of heat exchanger. The main aim of the current paper is to achieve an optimum model with the highest PEC value.

For all the investigated models, the initial nanofluid temperature is Tin=300K, and the constant heat flux of q=70,000 W/m2 is adopted. Besides, four different constant magnetic fields (B= 0, 200, 400, and 600 G) are determined to analyze the effects of employing MHD fluid flow in HE under a magnetic field. The HTF is Fe3O4/H2O nanofluid at nanoparticles volume fraction of ϕ = 1.0% and diameter of dnp = 20 nm. Table 1 reports geometrical parameters of studied HE. In the current study, effects of changing three geometrical parameters are analyzed, such as corrugation height, width, and pitch. Three different corrugation height values (b = 6.0, 10.0, and 14.0), three different corrugation width values (r=5.0, 9.0, and 13.0) and three different corrugation pitch values (p = 12.5, 20.0, and 27.5) are studied in this paper. Governing Equations The governing equations, including continuity, momentum, and energy equation for the nanofluid mixture, have been employed instead of employment of the governing equations of each solid- and fluid-phases separately [45]. The continuity: (1) where the mass-averaged velocity ( lated as [46]:

Numerical Model

Physical Model The present paper investigates effects of employing a magnetic field on the flow field and heat transfer of ferromagnetic nanofluid with consideration of a two-phase model for nanofluid in a helix-corrugated heat exchanger. Fig. 1 illustrates the schematic diagram of configuration and geometrical parameters of the studied corrugated tube filled with HTF and under constant heat flux and magnetic field. The length of HE is 200.0 mm, and the tube diameter is 50.0 mm. Also, the inlet length of 830.0 mm and outlet length of 66.0 mm are determined because it is important that the inlet flow must be fully developed, and there not be any flow comeback at the exit section of the channel [1, 44]. The HE is made of stainless steel 304 with 2.0 mm thickness. The velocity inlet boundary condition is applied in the entrance of the inlet section with the certain temperature (Tin) and velocity (uin), and the pressure outlet boundary condition has been adopted in the output of the exit section with zero gage pressure. Also, five different Reynolds numbers in the range of 4,000 to 20,000 are studied. These investigated Reynolds numbers are in the turbulent flow regime.

where is the solid nanoparticles velocity, is the base fluid velocity, and is the mixture density for the whole suspension and is written as [47]: The momentum equation [47]: (3) where is the mixture dynamic viscosity, p is pressure, and are the drift velocity of solid phase and fluid phase [48]: (4)

Figure 1. Schematic diagram of configuration and geometrical parameters of studied corrugated tube filled with HTF under constant heat flux and magnetic field.

where hs and hbf are the enthalpy of nanoparticles and base fluid, respectively. The nanoparticles volume concentration equation for suspension is as follows [50]: (7) The slip velocity is written as: (8)

(11) (12) where, and are the solid phase and base fluid gravity accelerations, respectively. The Reynolds of solid nanoparticles (Res) is calculated as [46]: (13)

where the relations between the relative and drift velocities are written as [47]: (9) The Schiller and Naumann [49] drag relative velocity is written as follow: (10)

where dp is the average diameter of solid nanoparticles and is assumed about 20 nm. The uniform magnetic field ( ) of constant magnitude is applied, where and are unit vectors. The orientation of the magnetic field forms an angle θ with the horizontal axis such that . The electromagnetic force and the electric current are defined by and , respectively. In current investigation, the magnetic field orientation is set to be horizontal (θ = 0)

Table 2. Most imortant non-dimensional number in MHD fluid flow [52-54] Non- dimensional parameter Magnetic Reynolds number Reynolds number Alfven number Hartmann number Batchelor number (magnetic Prandtl number) Stuart number (interaction parameter)

RNG k- �, present study Relizable k- �, present study Standard k- �, present study Sundar et al. (2012), Exp.

where µt,m is the turbulent viscosity and G is the production rate of k [60-62]:

Figure 2. Code validation with experimental data of Sundar et al. [57] due to choosing the appropriate turbulent model. [51]. Table 2 reports the most important non-dimensional parameters in MHD fluid flow. In all simulated models during the current investigation, the HTF flow inside the HE tube is in a turbulent regime since the studied Reynolds numbers are more than 2,300. Due to simulate the resulting turbulent flow inside the absorber tube identically, separately from the continuity, momentum, and energy equations, different k-ε turbulent models equations have also been used in the ANSYS-Fluent commercial software [45]. The choice of the k-ε turbulence model is according to its wide acceptance [55]. At the same time this is successfully used in various relevant numerical studies in HEs and also our validation with experimental study of Sundar et al. [56] which is shown in Fig. 2 leaded to employing the Standard k-ε turbulent model. Fig. 2 illustrates the code validation with experimental results of Sundar et al. [56] due to choosing the appropriate turbulent model. As it is realized in Fig. 2, the error values with Standard k-ε, Realizable k-ε and RNG k-ε models are 8.93%, 14.44%, and 9.72%, respectively. Because of the validated results of the Standard k-ε turbulent model, especially at higher Reynolds numbers, the Standard k-ε turbulent model is adopted in this work for turbulent simulation. As it is presented in the article, selected reference for comparison with the present results, is an experimental investigation. Due to selected reference, the difference between the present results and experimental data is logic. Also, the temperature dependent thermo-physical properties of HTF have been taken into account in simulations [57]. The relations which define the Standard k-ε model are as follow [58]: (14)

The standard constants are employed, Cµ = 0.09, c1 = 1.44, c2 = 1.92, σk = 1.00, σε = 1.30 and σt = 0.85. The coupled steady-state governing equations have been employed, and higher-order spatial discretization arrangements have been determined. The convergence criterion value for all solver variables of the nanofluid flow and heat transfer filed is the RMS residual to be less than 10-6. As it is noted previously, the Reynolds number is defined as [44]:

(18) Where um refer to the fluid or nanofluid average velocity. The average Nusselt number [1] is defined as: (19) The pressure drop between the inlet and outlet of the test section [44]: (20) The friction factor coefficient for fully developed flow is computed as [1]: (21)

The thermal and hydraulic performance evaluation criterion (PEC) is defined in order to compare the overall hydrothermal performance of corrugated and plain HE, which is obtained as follows [1]: (22) where Nuav and Nuav,s are the predicted mean Nusselt number for the corrugated and plain HE. Additionally, f and fs are the predicted mean friction factors for the corrugated and plain HE.

Table 3. The thermophysical properties of the Newtonian base fluid and solid nanoparticles at T = 300 K [63, 64]. ρ (kg/m3)

Nanofluid The considered Newtonian ferro-nanofluid in the current investigation is Fe2O3/H2O. To achieve the most efficient Newtonian nanofluid in the present study, the solid nanoparticles of Fe2O3 at volume concentration of ϕ = 1 % are added to the base fluid with a mean predicted diameter of 20 nm. Table 3 reports the thermophysical properties of the Newtonian base fluid and solid particles. The nanofluid effective density ρnf and specific heat capacity Cp,nf of the thermal Fe2O3/H2O ferro-nanofluid,

are determined using the following equations, where ϕ is the volume concentration [62, 63]. (23) (24) Also, the effective thermal conductivity and dynamic viscosity of the Fe2O3/H2O ferro-nanofluid can be calculated using the following equations [63-66].

(b) Figure 3. (a) Grid mesh layout of the pipe wall and (b) grid independence test in the present study for different Reynolds numbers for a model with b = 6 mm, r = 5 mm and p = 12.5 mm filled with nanofluid at ϕ = 1%.

To simulate the Fe2O3/H2O ferro-nanofluid flow through a HE, the Eulerian-Eulerian Two-Phase Mixture Model (TPM) is employed in the current investigation [45]. The TPM supposes that the phases-coupling is strong, and particles nearly track the interruption flow [46]. The twophases (solid nanoparticles and base fluid) are assumed to be interpenetrating, which funds that every phase has a certain value of velocity field. Also, there is a volume concentration of fluid phase in every control volume and another volume fraction (ϕ) for the nanoparticle phase. This model is illustrated to give powerful estimating, even for low nanoparticle volume fractions [47]. Validation Grid independency test Fig. 3 illustrates the grid mesh layout of the pipe wall and the grid independence test in the present study for different Reynolds numbers for a model with b = 6 mm, r = 5 mm, and p = 12.5 mm filled with nanofluid at ϕ = 1%. As shown in Fig. 3b, a grid independence test was performed for the corrugated pipe using nanofluid to analyze the effects of grid sizes on the results. By comparing the results, it is concluded that mesh configurations contains a grid number of 1,326,834 nodes is assumed to get a satisfactory agreement between the computational time and the accuracy of results with the maximum error of 3%. Code validation Also, the computational fluid dynamics (CFD) code validation was done by comparing of numerical results obtained from the present study and experimental data of Sha et al. [63]. Fig. 4 shows this code validation with experimental data of Sha et al. [64] versus Reynolds numbers and magnetic fields. It can be realized from this figure that a remarkable agreement exists among the empirical data of Sha et al. [63] and numerical results obtained from the present study employing TPM. It is seen that the TPM simulation in the present work leads to an appropriate validation with the experimental data during all studied Reynolds numbers and magnetic fields.

Results And Discussion

Usage of Corrugations and Nanofluid In this section, the thermal-hydraulic analysis of Basic HE (B.HE) filled with Base Fluid (BF), Corrugated HE

B=0G, Sha et al.(2017), Exp. B=0G, present study B=100G, Sha et al.(2017), Exp. B=100G, present study B=200G, Sha et al.(2017), Exp. B=200G, present study

(C.HE) filled with BF, B.HE filled with Nanofluid (NF) and C.HE filled with NF has been done. Fig. 5 illustrates the average predicted Nusselt number versus Reynolds numbers for B.HE and C.HE (b = 6 mm, r = 5 mm and p = 12.5 mm) which are filled with BF or NF (ϕ = 1%) under no magnetic field (B = 0). As it is seen in this figure, usage of C.HE (filled with BF or NF) can increase the average Nusselt number in tube the significantly. The presence of ribs can destroy the laminar sub-layers in HE and generate local vortexes, enhancing the flow mixing in fluid flow. This phenomenon leads to a higher heat transfer coefficient in HE and therefore increases the Nusselt number values. Also, it is found that usage of NF (for both B.HE and C.HE) can increase the

(b) Figure 4. Code validation with the experimental data of Sha et al. [63] versus Reynolds numbers and magnetic fields.

Figure 5. Average predicted Nusselt number versus Reynolds number for B.HE and C.HE (b = 6 mm, r = 5 mm and p =

Conclusion

This paper studied the effects of using a magnetic field on the flow field and heat transfer of ferromagnetic Fe3O4/ H2O nanofluid with consideration of a two-phase model for nanofluid in heat exchanger equipped with helical ribs. The length of the heat exchanger is 200.0 mm, and the tube diameter is 50.0 mm. Also, the inlet length of 830.0 mm and outlet length of 66.0 mm are determined. Besides, four different constant magnetic fields (B = 0, 200, 400, and 600 G) are determined to analyze the effects of employing MHD fluid flow in heat exchanger under the magnetic field. The

heat transfer fluid is Fe3O4/H2O nanofluid at a nanoparticles volume fraction of ϕ = 1.0% and diameter of dnp = 20 nm. Also effects of changing three geometrical parameters are analyzed: Three different corrugation height values (b = 6.0, 10.0 and 14.0), three different corrugation width values (r = 5.0, 9.0 and 13.0) and three different corrugation pitch values (p = 12.5, 20.0 and 27.5). The performance evaluation criteria index is employed to analyze the thermal-hydraulic characteristics of the heat exchanger. Based on the obtained results, • Usage of the corrugated heat exchanger or nanofluid can increase the average Nusselt number and friction factor in the tube sharply. • It is understood that the presence of a magnetic field has a significant effect on heat transfer enhancement in the heat exchanger. • The model with magnetic field 600 G has the highest Nusselt number ratio among all studied models, which is followed with 400 G, 200 G, and 0 magnetic fields, respectively. • Effects of different corrugation heights, widths, and pitches have been studied. • Usage of the novel corrugated heat exchanger with 14 mm corrugation heights, 9 mm rib width, and 12.5 mm blade pitches filled with nanofluid and under a magnetic field of 600 G it suggested in the present study as the most thermal-hydraulic-efficient configuration. • At the Reynolds number of 4,000, the highest performance evaluation criteria values are achieved.

Declarations

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

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

Ethics

There are no ethical issues with the publication of this manuscript.

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