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Heat transfer enhancement of Fe3O4 ferrofluids in the presence of magnetic field
Author’s Accepted Manuscript Heat Transfer Enhancement of Fe3O4 Ferrofluids in the Presence of Magnetic Field Farzad Fadaei, Mohammad Shahrokhi, Asghar Molaei Dehkordi, Zeinab Abbasi www.elsevier.com/locate/jmmm
PII: DOI: Reference:
S0304-8853(16)32470-2 http://dx.doi.org/10.1016/j.jmmm.2017.01.046 MAGMA62394
To appear in: Journal of Magnetism and Magnetic Materials Received date: 4 October 2016 Revised date: 21 December 2016 Accepted date: 13 January 2017 Cite this article as: Farzad Fadaei, Mohammad Shahrokhi, Asghar Molaei Dehkordi and Zeinab Abbasi, Heat Transfer Enhancement of Fe 3O4 Ferrofluids in the Presence of Magnetic Field, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2017.01.046 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Heat Transfer Enhancement of Fe3O4 Ferrofluids in the Presence of Magnetic Field Farzad Fadaei, Mohammad Shahrokhi, Asghar Molaei Dehkordi*, and Zeinab Abbasi Department of Chemical and Petroleum Engineering, Sharif University of Technology, P. O. Box 11155-9465, Tehran, Iran
Abstract In this article, three-dimensional (3D) forced-convection heat transfer of magnetic nanofluids in a pipe subject to constant wall heat flux in the presence of single or double permanent magnet(s) or current-carrying has been investigated and compared. In this regard, laminar fluid flow and equilibrium magnetization for the ferrofluid were considered. In addition, variations of magnetic field in different media were taken into account and the assumption of having a linear relationship of magnetization with applied magnetic field intensity was also released. Effects of magnetic field intensity, nanoparticle volume fraction, Reynolds number value, and the type of magnetic field source (i.e., a permanent magnet or current-carrying wire) on the forced-convection heat transfer of magnetic nanofluids were carefully investigated. It was found that by applying the magnetic field, the fluid mixing could be intensified that leads to an improvement in the Nusselt number value along the pipe length. Moreover, the obtained simulation results indicate that applying the magnetic field induced by two permanent magnets with a magnetization of 3×105 (A/m) (for each one), the fully developed Nusselt number value can be increased by 196%.
Keywords: Permanent magnet; nanofluid; ferrofluid; convective heat transfer; current carrying wire, heat-transfer enhancement.
*
Corresponding author. Tel.: +98 21 6616 5412; Fax: +98-21-66022853; E-mail address: [email protected] (Asghar Molaei Dehkordi).
1
1. Introduction Magnetic nanofluids (MNFs) are a new class of heat transfer fluids that can be prepared by dispersing superparamagnetic nanoparticles with a typical diameter of less than 20 nm in base fluids such as water, oil, ethylene glycol, etc. MNFs exhibit both the fluid and magnetic properties [1-3]. Nanoparticles used in these nanofluids include metallic materials such as iron, nickel, cobalt, and their oxides such as magnetite (Fe3O4). To prevent the aggregation of nanoparticles due to the interaction between them, the suspended nanoparticles should be covered with a monolayer of insoluble surfactant [4]. The heat-transfer rate of MNFs can be enhanced and controlled by external magnetic field (MF), thus, ferrofluids have found several applications in heat-transfer processes and attracted much attention in recent years [5-7]. For example, applications such as thermosyphons controlled by an MF [8-10] and improved cooling systems of high power electric transformers [11, 12] were successfully reported. Using MNFs in the presence of an external MF can be considered as an appropriate solution for enhancing forced-convection heat transfer and fluid mixing [13, 14]. In an experimental work, Goharkhah et al. [15] studied effects of constant and alternating MFs on the laminar forced-convection heat transfer of water-based ferrofluids in a uniformly heated tube using four electromagnets as the MF source. According to their results, as the magnetic field is turned off, using the base fluid with a φ = 2 vol % of nanoparticles, improves the average heattransfer rate by 13.5% compared to the base fluid. This enhancement could be increased to 18.9% and 31.4% by applying constant and alternating magnetic fields with an MF intensity of 500 G, respectively. Moreover, they reported that the heat-transfer rate could be enhanced with an increase in the intensity of the MF, Reynolds number (Re) value, and nanoparticle concentration. They also reported that constant MF applied causes the migration of nanoparticles to the tube wall, leading to the enhancement of local thermal conductivity and heat-transfer rate in the zones close to the electromagnets. In another work, Goharkhah et al. [16] conducted an experimental study on the
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laminar forced-convection heat transfer of ferrofluids in a uniformly heated parallel plate under an external MF induced by four electromagnets. They reported that the convective heat transfer has a direct relationship with Re value and nanoparticle concentration and at a constant Re value; the MF intensity increases the heat-transfer rate. They also claimed that there is an optimum frequency that can be directly varied with Re value. They reported that a maximum heat-transfer enhancement of 16.4% could be obtained in the absence of an MF. However, using constant and alternating MFs, this enhancement could be increased by 24.9% and 37.3%, respectively. Free and forced-convection heat transfer of ferrofluids in various geometries such as rectangular, cylindrical, etc. have been studied by several research groups [11, 12, 17]. Ganguly et al. [18] studied the effect of MF induced by line source dipole on the forced-convection heat transfer of an MNF in a two-dimensional (2D) duct numerically. They reported that the existence of an MF can intensify the mixing rate, consequently, increases the heat-transfer rate. They also examined the effects of position and number of magnetic dipoles on the heat-transfer rate. Goharkhah and Ashjaee [19] numerically studied forced-convection heat transfer of water-based Fe3O4 nanofluids in the presence of “on and off” non-uniform MFs. The applied MF was induced by eight line dipoles. The effects of alternating MF intensity and frequency on the convective heat transfer were investigated for several Re values in the laminar fluid flow regime. Their results indicate that the heat-transfer rate can be improved with an increase in Re value and a maximum enhancement of 13.9% at Re = 2000 and f = 20 Hz could be obtained. Moreover, they found that at a constant Re value, heattransfer rate has a direct relationship with the MF intensity while there is an optimum value for the field frequency. In addition, the optimum frequency increases with an increase in Re value. Laminar forced-convection heat transfer of Fe3O4 MNFs in an isothermal minichannel in the presence of constant and alternating MFs was numerically studied by Ghasemian et al. [20] using the two-phase mixture approach. Their obtained results indicate that the heat-transfer enhancement becomes more significant when the MF source located in the fully developed region. A maximum
3
value of 16.48% for the heat-transfer enhancement was reported using a constant MF. This value could be increased to 27.72% using an alternating MF with the same intensity at a field frequency of f = 4 Hz. Their results also indicate that the heat-transfer enhancement could be more significant for small Reynolds numbers values. Aminfar et al. [21] used a two-phase approach to study forced-convection heat transfer of ferrofluids passing through a vertical tube applying a static MF with positive and negative gradients. They concluded that in regions with negative magnetic gradient, a magnetic Kelvin body force is exerted that improves the heat-transfer rate, while in positive gradient regions, the heat-transfer rate decreases. They also reported that for MFs with positive gradient, the applied field can act as a pump, whereas for those with negative gradient, the larger power input is required. In another study, Aminfar et al. [22], used principles of magnetohydrodynamics to study heat transfer in an electricconductive ferrofluid. Fluid flow and heat transfer of water-based MNFs passing through an annulus were studied by Bahiraei et al. [23]. The applied MF was non-uniform and a two-phase Euler-Lagrange approach was used to model the system. They reported that not only the concentration distribution of nanoparticles on the surface of the annulus is non-uniform, but also their concentration in the vicinity of the wall becomes small. Hence, the velocity profile under the applied MF becomes flattered. To justify this result, they stated that a magnetic force is exerted on the particles in the opposite direction of the fluid flow and due to the non-uniformity of the concentration distribution of MNPs, a larger force can be exerted on the fluid elements in the central zones. Therefore, the fluid velocity is decreased and increased at the center of the annulus and near the wall, respectively, resulting in a nearly flat velocity profile. They also investigated the effects of size and concentration of nanoparticles and the gradient of the applied MF on the heat-transfer rate in order to optimize the heat-transfer rate and minimize the pressure loss. Aminfar et al. [24] developed a three-dimensional (3D) model based on the two-phase approach
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to analyze heat transfer in a vertical channel using finite volume method. In their study, the fluid flow regime was considered to be laminar and the nanofluid consisted of water containing 4 vol % Fe3O4 nanoparticles. A non-uniform MF induced by an electric wire was applied to the vertical wall. Their obtained results indicate that the Nusselt number value (Nu) increased by 22%. In addition, they observed the creation of some vortices enforcing the fluid motion towards the wall that this phenomenon enhances the heat-transfer rate. A study on the heat transfer of ferrofluids in microchannels was conducted by Xuan et al. [25]. They took into account the fluid-solid interactions and applied the Lattice-Boltzmann technique to solve the developed governing equations. They examined the effects of magnitude and direction of MF on the heat-transfer rate. They finally concluded that heat-transfer rate could increases if the directions of MF gradient and fluid flow are the same. Jafari et al. [26] used a two-phase approach to model heat transfer in a kerosene based ferrofluid system. Their obtained results indicate that magnitude and direction of the MF play an important role and affect the heat-transfer rate. They also reported that increasing the size of nanoparticles results in the formation of agglomerates in the system, hence, has a negative effect on the heat-transfer rate. Pourmehran et al.[27] investigated the magnetohydrodynamics of nanofluid convection flow over a vertical stretching sheet considering the buoyancy effect and thermal radiation. Different base fluids and nanoparticles were examined in their work. Influences of various parameters such as nanoparticle size, nanoparticle concentration, magnetic parameter, buoyancy parameter, and the radiation on velocity and temperature profiles, and Nu value were examined. According to their results, TiO2–ethylene glycol (50%) leads to maximum reduced Nu and minimum skin friction coefficient. They reported that the reduced Nu had an inverse relationship with an increase in the nanoparticle concentration. Rahimi-Gorji et al. [28] investigated fluid flow and thermal analysis of an unsteady squeezing nanofluid in the presence of variable MF by Galerkin Method. Their results indicate that the values
5
of Nu value and skin friction coefficient for CuO nanofluid are larger than those for Al2O3 nanofluid. They also reported that Nu value increases as the nanofluid volume fraction increases and skin friction coefficient decreases at the same time. Increasing the Hartman number resulted in an increase in the velocity and temperature profiles and an increase in squeeze number can be associated with a decrease in the velocity. In addition, Pourmehran et al.[29] conducted an analytical study on this system. Pourmehran et al. [30] studied thermal and fluid flow analysis of a fin-shaped microchannel heat sink cooled by different nanofluids The Forchheimer–Brinkman-extended Darcy model was used to describe the fluid flow and the two-phase model (for solid and liquid phases) with thermal dispersion was applied for heat transfer. Their results show that Cu-water nanofluid is more effective than Al2O3-water nanofluid. Furthermore, they found that inertial effect has no effect on Nu value, friction factor, and total thermal resistance, while the mentioned parameters significantly vary by nanoparticle size and volume fraction, channel width, and volumetric flow rate. In addition, RahimiGorji et al.[31] presented an analytical solution to this problem. Malvandi et al. [32] investigated thermal performance of water-based alumina nanofluid inside a microannular tube numerically. Influences of thermophoresis and Brownian motion were considered as a nanoparticle migration mechanism in their work. They reported that increasing MF intensity and the slip velocity intensify the thermal performance while increasing the nanoparticle volume fraction, ratio of inner wall to the outer wall radius, and the heat flux ratio decreases it. Malvandi et al.[33] conducted a theoretical study on the influence of nanoparticle migration on the heat-transfer enhancement of nanofluid condensate film over a vertical cylinder. The Brownian motion and thermophoretic diffusivity were considered in their model. It was found that nanoparticle migration has a significant influence on the heat-transfer rate. Furthermore, they found that heattransfer enhancement in film condensation strongly depends on the thermophysical properties of nanoparticles.
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Numerous studies recently conducted on the heat transfer of MNFs in the presence of an MF were focused on the magnetohydrodynamic, e.g. [34-36], however, less attention was paid to the ferrohydrodynamic case. In most studies conducted on the later case, free convection was just considered and there are few studies concerning forced-convection heat transfer [14]. In the present work, ferrohydrodynamic forced-convection heat transfer has been investigated. In the previous studies conducted to analyze the forced-convection heat transfer in the presence of MNFs, usually simple MFs were considered in order to reduce the computational burden. Moreover, MFs considered in some reported studies are impractical (like a constant gradient MF) [21, 22, 37] while, the MFs applied in this study are practically applicable. Moreover, in numerous studies conducted so far, it was assumed that the magnetization varies linearly with the applied MF intensity that can be valid just for low MF intensities [38] while, this assumption was released in the present work. In addition, variations of MF in different media (i.e., surrounding air and inside the system) were neglected in most of the previously published works that were taken into account in the present study. Moreover, less attention has been paid to MFs induced by permanent magnets or current carrying wires that are 3D fields [39]. Because of promising potential of these types of MFs in the enhancement of heat-transfer rate, the present work was aimed to investigate forced-convection heat transfer of ferrofluids in a pipe in the presence of permanent magnets or a current carrying wire. The main contributions of the present work can be summarized as follows:
Effects of various types of MF such as single or double permanent magnet(s) and current carrying wire on the forced-convection heat transfer in a pipe have been investigated and compared.
The MFs applied in this study are practically applicable and quite complicated relative to those considered in the previously published works.
Variations of MF in different media have been taken into account.
The assumption of the linear relationship of magnetization with the applied MF intensity has been released.
Effect of MF on the heat-transfer rate has been investigated in a 3D problem.
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2. Geometry Description and Problem Statement In the present work, forced-convection heat transfer of an MNF consisted of water and Fe3O4 nanoparticles in a horizontal pipe subject to a uniform wall heat flux as the boundary condition has been considered. The cold fluid introduces into the pipe and after the heat exchange leaves the system. The fluid flow was supposed to be laminar and at steady state conditions. In addition, the fluid was assumed to be non-electrically conductive and the effect of buoyancy force was neglected. The MF applied can be induced by single or two permanent magnet(s) or a current-carrying wire located parallel to the pipe. In the absence of MF, the problem could be reduced to a 2D one, however, in the presence of an MF, it should be modeled as a 3D problem. A schematic diagram of the system studied is shown in Fig. 1. Note that the symmetric geometry of the system enables us to reduce the computational load by solving the governing equations just for the upper half of the pipe. Diameter and length of the pipe were D and L = 150D, respectively. Moreover, the dimensions of the permanent magnets were 150D×0.5D×0.5D. Furthermore, the wire diameter was 0.2D and the distance between the centers of the wire and the pipe was 0.75D.
q
y
z
x
8
Fig. 1. Schematic diagram of the system (single permanent magnet and current carrying wire). The values of system parameters considered in the present work are summarized in Table 1.
Table 1: Various parameters used in the present study Parameter
Description
Value
Tin (K)
Inlet temperature
298.15
q (W m–2)
Constant wall heat flux
1000
D (m)
Pipe diameter
0.01
L (m)
Length of pipe
1.5
φ (-)
Volume fraction of MNP
0.03
kB (J K–1)
Boltzmann constant
1.38×10–23
Md (kA m–1)
Domain magnetization
446
Dwire (m)
Wire diameter
0.002
d (nm)
Particle diameter
10
3. Governing Equations The governing equations including continuity, momentum, thermal energy balance, and the Maxwell electromagnetic equations can be expressed as follows: Continuity equation is the same as one used for non-magnetic fluids [28]
. v 0 , t
(1)
where ρ is the fluid density and v is the linear velocity. The conservation equation of linear momentum [40]
9
m (
v v.v) p m 2 v 0 M H m g , t
(2)
where p and 0 are the hydrodynamic pressure and the magnetic permeability of free space, respectively. Kelvin body force term in Eq. (2) ( 0 M .H ) stands for the force applied by a nonuniform MF and M is the magnetization supposed to be equal to the equilibrium magnetization. Because the suspended nanoparticles show superparamagnetism, Langevin’s equation can be used to describe the equilibrium magnetization [41, 42] 1 H , M M d coth H
(3)
where is the volume fraction of nanoparticles in the base fluid and α is Langevin’s coefficient can be evaluated by
0 M d H Vc k BT
.
(4)
where Md, kB, and Vc are the bulk magnetization (446 kA m–1), Boltzmann constant (1.38×10–23 J K–1), and the volume of the magnetic core, respectively. To evaluate the MF distribution, electromagnetic equations of Maxwell can be used. The MF sources can be a current-carrying wire and single or two permanent magnets located parallel to the pipe. Maxwell equations were solved in three zones, including pipe inner region, the magnetic source, and the surrounding air. The corresponding governing equations are as follows [1]:
.B 0 ,
(5)
H J f ,
(6)
10
B 0 ( H M ) ,
(7)
where B and Jf are the magnetic flux and current densities, respectively. The thermal energy equation derived and reported by Rosensweig is quite complex [43], hence, for the sake of simplicity, the following assumptions were made: 1) the fluid is electrically nonconductive, 2) the fluid flow is incompressible with constant properties, and 3) the fluid is under equilibrium magnetization. Applying these assumptions, the thermal energy equation can be expressed as [44] 2 DT M DH ) 0T ( ) km2T m v vt , Dt 2 T H Dt
mC p , m (
(8)
where the first and second terms on the right-hand side of Eq. (8) are the conduction heat transfer and viscous dissipation, respectively. Moreover, it was supposed that conduction heat transfer in the axial (fluid flow) direction is negligible. The local Nu value can be evaluated as follows [24]: Nu
q D , k (Tw, avg Tavg )
(9a)
where Tw,avg is the arithmetic average of the wall temperature, Tavg is the local mixing cup fluid temperature, q is the wall heat flux, D is the pipe diameter, and k is the fluid thermal conductivity. Moreover, Tavg could be determined by R
Tavg
rv Tdr . rv dr z
0
(9b)
R
0
z
The density and specific heat capacity of nanofluids can be evaluated by [24]
m (1 ) f p ,
(10)
11
C p ,m
pC p , p (1 ) f C p , f , p (1 ) f
(11)
where ρp, ρf, Cp,p and Cp,f, and φ are, respectively, the solid particle density, base fluid density, the specific heat capacities of particles and the base fluid, and the volume fraction of the nanoparticles. The viscosity and thermal conductivity of the nanofluid were supposed to be a function of the nanoparticle volume fraction as follows [28]:
5
25
m 1 2 f . 4 2
(12)
k p 2 K f 2(k f k p ) km kf . k p 2 K f (k f k p )
(13)
Physical properties of nanoparticles and the base fluid used in the present work are summarized in Table 2. Moreover, governing equations used to simulate the system behavior are summarized in Table 3. 3.1.
Boundary conditions The following boundary conditions were applied:
At the pipe inlet: vr v 0, vz Vinlet , T T0 ,
(21)
At the fluid–wall interface: vr v vz 0 and q k
T . r
(22)
At the pipe outlet, the diffusional heat flux in the fluid flow direction was assumed to be zero and the outlet gauge pressure was also set zero. In the domains far from the applied MF, it was assumed that the MF diminishes and the normal component of B vector and the tangential component of H vector across the boundaries were supposed to be continuous and can be expressed as [41] n.B input n.B output ,
(23)
12
n H n H input output .
(24)
Table 2. Physical properties of nanoparticles and the base fluid used in the present work Properties
Value
Properties
Value
f (kg/m3 )
998
p (kg/m3 )
5200
C p, f (J / kg K)
4182
C p , p (J / kg K)
670
f (kg/ms)
9×10–4
p (kg/ms)
---
k f (W/m K)
0.6
k p (W/m K)
6
Table 3. Simplified governing equations Linear Momentum, r direction Linear Momentum, ϴ direction Linear Momentum, z direction
m vr
vr v vr v v 2 H r p 1 H r H vz r 0 M r M r r z r r r r r
1 1 2v 2 v m rvr 2 2r 2 r r r r r
m vr
v v v v v v H H 1 p 1 H H r vz r 0 M r M M z z r r z r r r r r
1 1 2v 2 v m rv 2 2 2 r r r r r r
m vr
vz v vz v vz z r r z
1 vz m r r r r
Maxwell
H z H z p 1 H z z 0 M r r M r M z z
1 vz 2 2 r
v 1 1 v rvr z 0 r r r z
(14)
(15)
(16)
2
Continuity
Maxwell
H r M z z
.B 0 0. M H 0
1 1 r (M r H r ) M H M z H z 0 r r r z
1 H r 1 H z H ˆ H r H z 1 H J f rˆˆ (rH ) zˆ Jf z r r z r r r
13
(17)
(18)
(19)
Thermal energy balance
M 0T T
mC p ,m vr
H r T v T T M r H r v H r vz vz 0T vr r r z r r z T H H H z H v H M z H z v H z vr r r vz z 0T T vr r r vz z H H
1 T km r r r r
(20)
2 1 T m 2 2 r
3.2. Numerical solution To solve the set of governing equations given in Table 3, the finite element method was used. The calculation domain was divided into tetrahedral elements. Integral equations corresponding to Eqs. (14-20) could be obtained for each element applying the conservation laws. To obtain an algebraic approximation to the control volume conservation equations, the velocity and temperature fields were interpolated quadratically in each element and other variables were linearly interpolated in each one. In other words, the second-order variables were stored at each node and the first-order dependent variables were stored at the vertices of these elements. The discretized equations were algebraic approximation to the control volume conservation equations. An iterative method was also used to solve the nonlinear coupled governing algebraic equations. Starting with guessed temperature and MF, the coefficients in the momentum equations were calculated. The momentum equations were solved, then, the pressure field was obtained. After that, MF and thermal energy equations were solved, respectively. The entire procedure was repeated until convergence could be achieved. The mesh independence was also examined for the system under the following conditions: the values of the Reynolds number, nanoparticle volume fraction, and the permanent magnet magnetization were set to 200, 0.03, and 3×105 (A/m), respectively. The system geometry was spatially discretized using unstructured grids of tetrahedral volume elements. For the base case, 930,000 tetrahedral elements were considered to check the mesh independence. Other cases with different numbers of elements were considered as summarized in Table 4. The average Nu values
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were evaluated for all the cases and, then, the percentage error of the average Nu value relative to that obtained for the base case was determined as presented in Table 4. As can be observed from this table, increasing the number of elements from 930,000 to 1,438,000 resulted in a 0.17% reduction in the average Nu value. Hence, to reduce the computational load, 930,000 elements were used throughout the present work. Table 4. Variations of the average Nu value versus the number of elements Number of elements Nuavg Percentage error (%) 167000
9.683
3.92
506000
9.468
1.62
930000
9.317
0
1438000
9.301
-0.17
Solving the coupled governing equations takes approximately 17 h CPU time, running on an 8 cores Intel (R) Core (TM) i7-4770 CPU processor (3.4 GHz) with 16 GB of RAM these explaneations were added to manuscriot
4. Results and Discussion 4.1. Model validation To validate the developed model, the obtained simulation results were compared with those previously published including experimental and numerical results for a non-magnetic nanofluid. Fig. 2 demonstrates the obtained simulation results concerning the heat-transfer coefficient along with the experimental data reported by Kim et al. [45] in which Re = 1460 and the nanofluid contains 3 Vol % Al2O3 nanoparticles. As can be observed from this figure, there is good agreement between the
simulation and experimental results. Fig. 3 shows the present simulation results of heat-transfer coefficient for different volume fractions and diameters of nanoparticles along with those reported by Ebrahiminia et al. [46]. As can be observed from this figure, both the simulation results are almost
15
the same. In what follows, the influences of various operating conditions and design parameters on the hydrodynamic behavior and heat-transfer characteristics are presented and discussed. In the following subsections, Re = 200, = 0.03, and M = 3×105 (A/m) were considered. 3000 Present study Kim et al. [45] measured data
2000
2
h (W/m K)
2500
1500
1000
500
0
100
200
300
400
x/D
Fig. 2. Comparison of simulation results obtained in the present study with the experimental heattransfer coefficients reported in Ref. [45]. 2400 Ebrahimnia et al. [46] = 0.06, d = 20 nm Present study with = 0.06, d = 20 nm
2000
Ebrahimnia et al. [46] = 0.02, d = 100 nm
h (W/m2K)
Present study with = 0.02, d = 100 nm
1600
1200
800
400
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
z (m) Fig. 3. Comparison of the obtained simulation results with those reported in Ref. [46] as a function
of volume fraction of nanoparticles and their diameter. 16
4.2. Magnetic field distribution Fig. 4 shows the MF distribution (H) for three cases of single permanent magnet, double permanent magnets, and the current carrying wire that were parallel to the pipe with ferrofluid flow. It should be noted that variations of MF in different media, i.e., surrounding air, inside the pipe, and the magnetic source were taken into account in the present study, while this distribution has been ignored in most previously published works [18-20]. As can be observed from the onset shown in Fig. 4 (a) that sketches H magnitude on a centerline cut line shown by the red thick line, one can obviously observe a jump in the MF distribution at the pipe/air interface in the case of single permanent magnet. Note that in numerous
studies conducted in this field, an MF with a constant gradient has been applied that is impractical and the specified MF is not in accordance with Maxwell’s equations [21, 22, 37]. In the case of using double magnets with opposite poles next to each other (SNSN), MF lines leave one pole and enter the other one. Effects of the MF distribution on the heat-transfer rate are explored in the following subsections.
17
200000
H (A/m)
160000
120000
80000
40000
0
0
(a)
0.25
0.5
0.75
1
1.25
1.5
2r/D
(c)
(b)
Fig. 4. MF distribution in a) single permanent magnet; b) two permanent magnets, and c) currentcarrying wire. 4.3. Effect of magnetic field intensity of single permanent magnet Applying MFs with a gradient exert a body force in the radial and azimuthal directions, resulting in the creation of velocity components in these directions. In other words, at any given cross section of the pipe, the MNF undergoes some kind of lateral rotation as shown in Fig. 5. The magnitude of such velocities depends on the magnitude of MF and its gradient. The lateral rotation would change the temperature profile, consequently, affects the heat-transfer rate as shown in Fig. 6. Because the magnetization induced by the MF is proportional to the reciprocal of temperature, the body force exerted on the cold fluid is larger than that on the warmer fluid. This leads to the movement of cold fluid towards the MF source and intensifies the fluid mixing.
18
1 0.8 0.6 0.4
Mesh xMesh yz
2r/D
0.2
x y z
100100200
0
100100200 100100200 100100200 100100200 100100200 100100200 100100200
-0.2 -0.4 -0.6 -0.8 -1
-1
-0.8 -0.6 -0.4 -0.22r/D0
0.2
0.4
0.6
0.8
1
2r/D Fig. 5. Lateral mixing at the cross section of the pipe.
19
1.75 1.65 1.55
1.35
1.55
1.25
1.75
5 45
0
1.15
1.
-0.5
1.4 1.55
1.0 5
1.25
1.55
1.05
1.35
1.55
1.15
1.05
1.15
1.65
1.45
1.85
5 1.4 5 1.5
1.15
1.35
-1
1.25
5 1.0
1.35
1.2 5
1.15
5
5
5 1.0
1.05
1. 5
1.3
1.1 5
1.05
1.25
1.15
1. 4 5
1.4 5
1.25
5 1.3
1.25
1. 45
5 1.3
1.5 5
(b)
1.55
1.35
(a)
0.5
1
-1
-0.5
Dimensionless Radius
1.25
0
0.5
1
Dimensionless Radius
1.01.0 5 5
1.2
1.8
1.5
1.65
1.05
1.35
1.2
1 .5
(c)
1.5
1.2
1.35
1.65
1.05
1.8
1 .5
1.2
-1
-0.5
0
0.5
1
Dimensionless Radius
Fig. 6. Effects of MF of the single permanent magnet on the dimensionless temperature distribution at the cross section of the pipe for a) M = 0 A/m; b) M = 1×105A/m; and c) M = 3×105 A/m at z/L = 150. Fig. 7 shows the effects of the single permanent magnet with different magnetizations on the dimensionless axial velocity profile in the symmetric plane in the fully developed region. As can be observed from this figure, the velocity profile shifts towards the MF source. Because of the aforementioned fluid rotation the mixing efficiency increases and the temperature difference between the bulk mean temperature of the fluid and the wall temperature decreases, resulting in an increase in the Nu value. Moreover, applying an external MF would alter the thermal boundary layer and reduce the resistance
20
against heat transfer resulting in the enhancement of heat-transfer rate. Effects of MF induced by a single permanent magnet on the local Nu value are shown in Fig. 8. As can be observed from this figure, the Nu value increases with an increase in the magnetization of the single permanent magnet. 2
vz/ Vinlet
1.5
1 M (A/m) 5
510 3105 1105 4 310 without magnetic field
0.5
0 -1
-0.5
0
0.5
1
2r/D
Fig. 7. Axial velocity profile as a function of the magnetizations for the single permanent magnet. 14
M(A/m) 5
510 5 310 5 110 4 310 without magnetic field
12
Nu
10 8 6 4 2
0
50
100
150
z/D
Fig. 8. Nusselt number value versus the dimensionless pipe length as a function of the magnetization for the single permanent magnet. 21
4.4. Effect of magnetic field intensity of current carrying wire Effects of MF induced by the current carrying wire on the local Nu value for different electric currents are shown in Fig. 9. Under these conditions, the MF intensity can be precisely controlled by manipulating the electric current passing through the current-carrying wire. As can be seen, increasing electric current increases the Nu values the same as previous cases. 14
I (A) 1000 100 50 5 without magnetic field
12
Nu
10 8 6 4 2
0
50
100
150
z/D
Fig. 9. Nusselt number value versus the dimensionless pipe length for various values of current in the current-carrying wire. 4.5. Effect of volume fraction of magnetic nanoparticles Fig. 10 demonstrates how the volume fraction of magnetic nanoparticles (MNPs) influences the Nu value in the presence of a single permanent magnet. As can be seen, the Nu value increases with an increase in the volume fraction of MNPs. This can be justified as follows: according to Eq. (3), the magnetization of ferrofluid is directly proportional to the volume concentration of MNPs. Therefore, larger Kelvin body force is exerted on the fluid elements leading to the heat -transfer enhancement. It should be also added that the physical properties of ferrofluid can be affected by
22
changing the volume fraction of MNPs. Similar effects of the volume fraction of MNPs on the Nu value can be observed for the current carrying wire or the double permanent magnets that are not reported for the sake of brevity.
16
0.06 0.03 0.015 0.003 0
14 12
Nu
10 8 6 4 2
0
25
50
75
100
125
150
z/D
Fig. 10. Nusselt number value versus the dimensionless pipe length as a function of nanoparticle volume fraction for the single permanent magnet. 4.6. Effect of Reynolds number value The effect of Re value on the heat-transfer rate is shown in Fig. 11 for the single permanent MF. This figure clearly shows that larger Nu values can be obtained for larger Re values. For laminar flow in the absence of an MF, the Nu value reaches a constant value of 4.36 in the fully developed region while in the presence of MF, this fully developed value increases with an increase in the Re value. To examine the effect of MNPs and the type of magnetic source on the Nu value, its value was evaluated for water, nanofluid, and nanofluid in the presence of MF induced by one/two permanent magnet(s) or currentcarrying wire at Re=100 and =0.03 and the obtained results are shown in Fig. 12. As can be seen from this figure, the lowest Nu value belongs to pure water as the heat-transfer fluid. The presence of magnetite 23
nanoparticles (3 Vol %) improves slightly the Nu value in the pipe entrance zone due to the change of physical properties of ferrofluid such as thermal conductivity, however, the corresponding fully developed Nu value is the same as that for pure water as could be expected (i.e., Nu = 4.36 for the constant wall heat flux as the boundary condition). In the case of ferrofluid and in the presence of an electric current carrying wire an improvement in the local Nu value is observed. The fully developed Nu value is also increased that is not the case for heat transfer in the absence of MF. By increasing the electric current from 50 A to 1000 A, a significant enhancement in the Nu value is observed. Almost the same level of improvement can be obtained using a single permanent magnet with M = 3×105 A/m. If double permanent magnets are used, considerable improvement in the Nu value can be observed. When two permanent magnets are used on both sides of the pipe, the effect of MF is intensified. This effect enhances the local fluid mixing, consequently, leading to a redistribution of the fluid temperature profile and an increase in the local Nu value. These results clearly indicate that application of permanent magnets can be more efficient and economical than the current carrying wire. 16
Re 2000 1000 500 200 100
15 14
Nu
13 12 11 10 9
0
50
100
150
z/D Fig. 11. Nusselt number versus the dimensionless pipe length as a function of Reynolds number
value for the single permanent magnet.
24
16 14 12
Nu
10 8 6 5
4
Double permanent magnet (M = 310 A/m) Current carrying wire (1000 A) 5 Single permanent magnet ( M = 310 A/m) Current carrying wire (50 A) Water + Nanoparticles (3 vol.%) Pure water
2 0
0
25
50
75
100
125
150
z/D
Fig. 12. Nusselt number value versus the dimensionless pipe length
5. Conclusions The major aim of the present work was to investigate 3D forced-convection heat transfer of a ferrofluid flowing in a pipe under the influence of external non-uniform MFs induced by permanent magnets or a current carrying wire. The effects of the external magnetic field on the hydrothermal behavior of the ferrofluid flow were also explored. Applying a magnetic field to the pipe carrying an MNF exerts a body force perpendicular to the flow direction and as a result, the lateral velocity components are generated and the axial velocity distribution can be modified. This leads to an increase in the velocity and temperature gradients near the pipe wall, consequently, enhances the local Nu value. Heat transfer rate and Nu value can be increased with an increase in the intensity of permanent magnet magnetization and the electric current; however, using permanent magnets is more efficient. It was found that by using a single or double permanent magnet(s), enhancements of 113% (M = 3×105 A/m) and 196% (M = 3×105 A/m for each one) for the fully developed values of Nu can be obtained, respectively, while the Nu value increases up to 26% in the case of wire carrying a current of 50 A. In the presence of an MF with an increase in the Re value, the average and fully developed Nusselt values
25
also increase that is not the case for heat transfer in the absence of MF. The heat-transfer coefficient becomes larger with an increase in the volume fraction of nanoparticles. This can be attributed to the fact that the volume fraction of MNPs directly affects the magnetization of ferrofluid and its physical properties. In this work, a static MF was applied that could be replaced by oscillating or rotating MFs. This subject is the objective of our future studies. In addition, in the case of using a large volume fraction of MNP, the simulation can be performed applying a two-phase modeling approach. Moreover, the effects of multiple permanent magnets can be also investigated and, thus, the appropriate configuration to reach the maximum Nu value can be determined in the future works.
Acknowledgement The present authors acknowledge the financial support provided by the National Iranian Oil Refining & Distribution Company (NIORDC).
Nomenclature B Cp D d g H I km kB L M Md Nu p q r Re T t v
Magnetic flux density (T) Specific heat (J/kg K) Pipe diameter (m) Particle diameter (m) Gravitational acceleration (m/s2) Magnetic field (A/m) Electric current (A) Thermal conductivity (W/m K) Boltzmann constant, (1.38×10–23 J/K) Pipe length (m) Magnetization (A/m) Domain magnetization (A/m) Nusselt number Pressure (Pa) Constant wall heat flux (W/m2)
r coordinate (m) Reynolds number Absolute temperature (K) Time (s) Linear velocity (m/s) 26
Vc z
Volume of magnetic core (m3) z coordinate (m)
Greek Symbols α μ0 ρ φ
Langevin equation parameter Shear viscosity (Pa s) Vacuum permeability 4π×10-7 (N/A2, Tm/A) Density (kg/m3) MNPs’ volume
Subscripts avg Average f Fluid m Mixture p Particle w Wall Abbreviations MF Magnetic field MNF Magnetic nanofluid MNP Magnetic nanoparticle
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Research Highlights
3D forced-convection heat transfer of magnetic nanofluids is investigated. Effects of single or double permanent magnet on the heat transfer are studied. Influences of magnetic field induced by a current-carrying wire are studied. Effects of magnetic field intensity and Reynolds number value are studied.
Variations of magnetic field in different media are taken into account.
Graphical
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PII: DOI: Reference:
S0304-8853(16)32470-2 http://dx.doi.org/10.1016/j.jmmm.2017.01.046 MAGMA62394
To appear in: Journal of Magnetism and Magnetic Materials Received date: 4 October 2016 Revised date: 21 December 2016 Accepted date: 13 January 2017 Cite this article as: Farzad Fadaei, Mohammad Shahrokhi, Asghar Molaei Dehkordi and Zeinab Abbasi, Heat Transfer Enhancement of Fe 3O4 Ferrofluids in the Presence of Magnetic Field, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2017.01.046 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Heat Transfer Enhancement of Fe3O4 Ferrofluids in the Presence of Magnetic Field Farzad Fadaei, Mohammad Shahrokhi, Asghar Molaei Dehkordi*, and Zeinab Abbasi Department of Chemical and Petroleum Engineering, Sharif University of Technology, P. O. Box 11155-9465, Tehran, Iran
Abstract In this article, three-dimensional (3D) forced-convection heat transfer of magnetic nanofluids in a pipe subject to constant wall heat flux in the presence of single or double permanent magnet(s) or current-carrying has been investigated and compared. In this regard, laminar fluid flow and equilibrium magnetization for the ferrofluid were considered. In addition, variations of magnetic field in different media were taken into account and the assumption of having a linear relationship of magnetization with applied magnetic field intensity was also released. Effects of magnetic field intensity, nanoparticle volume fraction, Reynolds number value, and the type of magnetic field source (i.e., a permanent magnet or current-carrying wire) on the forced-convection heat transfer of magnetic nanofluids were carefully investigated. It was found that by applying the magnetic field, the fluid mixing could be intensified that leads to an improvement in the Nusselt number value along the pipe length. Moreover, the obtained simulation results indicate that applying the magnetic field induced by two permanent magnets with a magnetization of 3×105 (A/m) (for each one), the fully developed Nusselt number value can be increased by 196%.
Keywords: Permanent magnet; nanofluid; ferrofluid; convective heat transfer; current carrying wire, heat-transfer enhancement.
*
Corresponding author. Tel.: +98 21 6616 5412; Fax: +98-21-66022853; E-mail address: [email protected] (Asghar Molaei Dehkordi).
1
1. Introduction Magnetic nanofluids (MNFs) are a new class of heat transfer fluids that can be prepared by dispersing superparamagnetic nanoparticles with a typical diameter of less than 20 nm in base fluids such as water, oil, ethylene glycol, etc. MNFs exhibit both the fluid and magnetic properties [1-3]. Nanoparticles used in these nanofluids include metallic materials such as iron, nickel, cobalt, and their oxides such as magnetite (Fe3O4). To prevent the aggregation of nanoparticles due to the interaction between them, the suspended nanoparticles should be covered with a monolayer of insoluble surfactant [4]. The heat-transfer rate of MNFs can be enhanced and controlled by external magnetic field (MF), thus, ferrofluids have found several applications in heat-transfer processes and attracted much attention in recent years [5-7]. For example, applications such as thermosyphons controlled by an MF [8-10] and improved cooling systems of high power electric transformers [11, 12] were successfully reported. Using MNFs in the presence of an external MF can be considered as an appropriate solution for enhancing forced-convection heat transfer and fluid mixing [13, 14]. In an experimental work, Goharkhah et al. [15] studied effects of constant and alternating MFs on the laminar forced-convection heat transfer of water-based ferrofluids in a uniformly heated tube using four electromagnets as the MF source. According to their results, as the magnetic field is turned off, using the base fluid with a φ = 2 vol % of nanoparticles, improves the average heattransfer rate by 13.5% compared to the base fluid. This enhancement could be increased to 18.9% and 31.4% by applying constant and alternating magnetic fields with an MF intensity of 500 G, respectively. Moreover, they reported that the heat-transfer rate could be enhanced with an increase in the intensity of the MF, Reynolds number (Re) value, and nanoparticle concentration. They also reported that constant MF applied causes the migration of nanoparticles to the tube wall, leading to the enhancement of local thermal conductivity and heat-transfer rate in the zones close to the electromagnets. In another work, Goharkhah et al. [16] conducted an experimental study on the
2
laminar forced-convection heat transfer of ferrofluids in a uniformly heated parallel plate under an external MF induced by four electromagnets. They reported that the convective heat transfer has a direct relationship with Re value and nanoparticle concentration and at a constant Re value; the MF intensity increases the heat-transfer rate. They also claimed that there is an optimum frequency that can be directly varied with Re value. They reported that a maximum heat-transfer enhancement of 16.4% could be obtained in the absence of an MF. However, using constant and alternating MFs, this enhancement could be increased by 24.9% and 37.3%, respectively. Free and forced-convection heat transfer of ferrofluids in various geometries such as rectangular, cylindrical, etc. have been studied by several research groups [11, 12, 17]. Ganguly et al. [18] studied the effect of MF induced by line source dipole on the forced-convection heat transfer of an MNF in a two-dimensional (2D) duct numerically. They reported that the existence of an MF can intensify the mixing rate, consequently, increases the heat-transfer rate. They also examined the effects of position and number of magnetic dipoles on the heat-transfer rate. Goharkhah and Ashjaee [19] numerically studied forced-convection heat transfer of water-based Fe3O4 nanofluids in the presence of “on and off” non-uniform MFs. The applied MF was induced by eight line dipoles. The effects of alternating MF intensity and frequency on the convective heat transfer were investigated for several Re values in the laminar fluid flow regime. Their results indicate that the heat-transfer rate can be improved with an increase in Re value and a maximum enhancement of 13.9% at Re = 2000 and f = 20 Hz could be obtained. Moreover, they found that at a constant Re value, heattransfer rate has a direct relationship with the MF intensity while there is an optimum value for the field frequency. In addition, the optimum frequency increases with an increase in Re value. Laminar forced-convection heat transfer of Fe3O4 MNFs in an isothermal minichannel in the presence of constant and alternating MFs was numerically studied by Ghasemian et al. [20] using the two-phase mixture approach. Their obtained results indicate that the heat-transfer enhancement becomes more significant when the MF source located in the fully developed region. A maximum
3
value of 16.48% for the heat-transfer enhancement was reported using a constant MF. This value could be increased to 27.72% using an alternating MF with the same intensity at a field frequency of f = 4 Hz. Their results also indicate that the heat-transfer enhancement could be more significant for small Reynolds numbers values. Aminfar et al. [21] used a two-phase approach to study forced-convection heat transfer of ferrofluids passing through a vertical tube applying a static MF with positive and negative gradients. They concluded that in regions with negative magnetic gradient, a magnetic Kelvin body force is exerted that improves the heat-transfer rate, while in positive gradient regions, the heat-transfer rate decreases. They also reported that for MFs with positive gradient, the applied field can act as a pump, whereas for those with negative gradient, the larger power input is required. In another study, Aminfar et al. [22], used principles of magnetohydrodynamics to study heat transfer in an electricconductive ferrofluid. Fluid flow and heat transfer of water-based MNFs passing through an annulus were studied by Bahiraei et al. [23]. The applied MF was non-uniform and a two-phase Euler-Lagrange approach was used to model the system. They reported that not only the concentration distribution of nanoparticles on the surface of the annulus is non-uniform, but also their concentration in the vicinity of the wall becomes small. Hence, the velocity profile under the applied MF becomes flattered. To justify this result, they stated that a magnetic force is exerted on the particles in the opposite direction of the fluid flow and due to the non-uniformity of the concentration distribution of MNPs, a larger force can be exerted on the fluid elements in the central zones. Therefore, the fluid velocity is decreased and increased at the center of the annulus and near the wall, respectively, resulting in a nearly flat velocity profile. They also investigated the effects of size and concentration of nanoparticles and the gradient of the applied MF on the heat-transfer rate in order to optimize the heat-transfer rate and minimize the pressure loss. Aminfar et al. [24] developed a three-dimensional (3D) model based on the two-phase approach
4
to analyze heat transfer in a vertical channel using finite volume method. In their study, the fluid flow regime was considered to be laminar and the nanofluid consisted of water containing 4 vol % Fe3O4 nanoparticles. A non-uniform MF induced by an electric wire was applied to the vertical wall. Their obtained results indicate that the Nusselt number value (Nu) increased by 22%. In addition, they observed the creation of some vortices enforcing the fluid motion towards the wall that this phenomenon enhances the heat-transfer rate. A study on the heat transfer of ferrofluids in microchannels was conducted by Xuan et al. [25]. They took into account the fluid-solid interactions and applied the Lattice-Boltzmann technique to solve the developed governing equations. They examined the effects of magnitude and direction of MF on the heat-transfer rate. They finally concluded that heat-transfer rate could increases if the directions of MF gradient and fluid flow are the same. Jafari et al. [26] used a two-phase approach to model heat transfer in a kerosene based ferrofluid system. Their obtained results indicate that magnitude and direction of the MF play an important role and affect the heat-transfer rate. They also reported that increasing the size of nanoparticles results in the formation of agglomerates in the system, hence, has a negative effect on the heat-transfer rate. Pourmehran et al.[27] investigated the magnetohydrodynamics of nanofluid convection flow over a vertical stretching sheet considering the buoyancy effect and thermal radiation. Different base fluids and nanoparticles were examined in their work. Influences of various parameters such as nanoparticle size, nanoparticle concentration, magnetic parameter, buoyancy parameter, and the radiation on velocity and temperature profiles, and Nu value were examined. According to their results, TiO2–ethylene glycol (50%) leads to maximum reduced Nu and minimum skin friction coefficient. They reported that the reduced Nu had an inverse relationship with an increase in the nanoparticle concentration. Rahimi-Gorji et al. [28] investigated fluid flow and thermal analysis of an unsteady squeezing nanofluid in the presence of variable MF by Galerkin Method. Their results indicate that the values
5
of Nu value and skin friction coefficient for CuO nanofluid are larger than those for Al2O3 nanofluid. They also reported that Nu value increases as the nanofluid volume fraction increases and skin friction coefficient decreases at the same time. Increasing the Hartman number resulted in an increase in the velocity and temperature profiles and an increase in squeeze number can be associated with a decrease in the velocity. In addition, Pourmehran et al.[29] conducted an analytical study on this system. Pourmehran et al. [30] studied thermal and fluid flow analysis of a fin-shaped microchannel heat sink cooled by different nanofluids The Forchheimer–Brinkman-extended Darcy model was used to describe the fluid flow and the two-phase model (for solid and liquid phases) with thermal dispersion was applied for heat transfer. Their results show that Cu-water nanofluid is more effective than Al2O3-water nanofluid. Furthermore, they found that inertial effect has no effect on Nu value, friction factor, and total thermal resistance, while the mentioned parameters significantly vary by nanoparticle size and volume fraction, channel width, and volumetric flow rate. In addition, RahimiGorji et al.[31] presented an analytical solution to this problem. Malvandi et al. [32] investigated thermal performance of water-based alumina nanofluid inside a microannular tube numerically. Influences of thermophoresis and Brownian motion were considered as a nanoparticle migration mechanism in their work. They reported that increasing MF intensity and the slip velocity intensify the thermal performance while increasing the nanoparticle volume fraction, ratio of inner wall to the outer wall radius, and the heat flux ratio decreases it. Malvandi et al.[33] conducted a theoretical study on the influence of nanoparticle migration on the heat-transfer enhancement of nanofluid condensate film over a vertical cylinder. The Brownian motion and thermophoretic diffusivity were considered in their model. It was found that nanoparticle migration has a significant influence on the heat-transfer rate. Furthermore, they found that heattransfer enhancement in film condensation strongly depends on the thermophysical properties of nanoparticles.
6
Numerous studies recently conducted on the heat transfer of MNFs in the presence of an MF were focused on the magnetohydrodynamic, e.g. [34-36], however, less attention was paid to the ferrohydrodynamic case. In most studies conducted on the later case, free convection was just considered and there are few studies concerning forced-convection heat transfer [14]. In the present work, ferrohydrodynamic forced-convection heat transfer has been investigated. In the previous studies conducted to analyze the forced-convection heat transfer in the presence of MNFs, usually simple MFs were considered in order to reduce the computational burden. Moreover, MFs considered in some reported studies are impractical (like a constant gradient MF) [21, 22, 37] while, the MFs applied in this study are practically applicable. Moreover, in numerous studies conducted so far, it was assumed that the magnetization varies linearly with the applied MF intensity that can be valid just for low MF intensities [38] while, this assumption was released in the present work. In addition, variations of MF in different media (i.e., surrounding air and inside the system) were neglected in most of the previously published works that were taken into account in the present study. Moreover, less attention has been paid to MFs induced by permanent magnets or current carrying wires that are 3D fields [39]. Because of promising potential of these types of MFs in the enhancement of heat-transfer rate, the present work was aimed to investigate forced-convection heat transfer of ferrofluids in a pipe in the presence of permanent magnets or a current carrying wire. The main contributions of the present work can be summarized as follows:
Effects of various types of MF such as single or double permanent magnet(s) and current carrying wire on the forced-convection heat transfer in a pipe have been investigated and compared.
The MFs applied in this study are practically applicable and quite complicated relative to those considered in the previously published works.
Variations of MF in different media have been taken into account.
The assumption of the linear relationship of magnetization with the applied MF intensity has been released.
Effect of MF on the heat-transfer rate has been investigated in a 3D problem.
7
2. Geometry Description and Problem Statement In the present work, forced-convection heat transfer of an MNF consisted of water and Fe3O4 nanoparticles in a horizontal pipe subject to a uniform wall heat flux as the boundary condition has been considered. The cold fluid introduces into the pipe and after the heat exchange leaves the system. The fluid flow was supposed to be laminar and at steady state conditions. In addition, the fluid was assumed to be non-electrically conductive and the effect of buoyancy force was neglected. The MF applied can be induced by single or two permanent magnet(s) or a current-carrying wire located parallel to the pipe. In the absence of MF, the problem could be reduced to a 2D one, however, in the presence of an MF, it should be modeled as a 3D problem. A schematic diagram of the system studied is shown in Fig. 1. Note that the symmetric geometry of the system enables us to reduce the computational load by solving the governing equations just for the upper half of the pipe. Diameter and length of the pipe were D and L = 150D, respectively. Moreover, the dimensions of the permanent magnets were 150D×0.5D×0.5D. Furthermore, the wire diameter was 0.2D and the distance between the centers of the wire and the pipe was 0.75D.
q
y
z
x
8
Fig. 1. Schematic diagram of the system (single permanent magnet and current carrying wire). The values of system parameters considered in the present work are summarized in Table 1.
Table 1: Various parameters used in the present study Parameter
Description
Value
Tin (K)
Inlet temperature
298.15
q (W m–2)
Constant wall heat flux
1000
D (m)
Pipe diameter
0.01
L (m)
Length of pipe
1.5
φ (-)
Volume fraction of MNP
0.03
kB (J K–1)
Boltzmann constant
1.38×10–23
Md (kA m–1)
Domain magnetization
446
Dwire (m)
Wire diameter
0.002
d (nm)
Particle diameter
10
3. Governing Equations The governing equations including continuity, momentum, thermal energy balance, and the Maxwell electromagnetic equations can be expressed as follows: Continuity equation is the same as one used for non-magnetic fluids [28]
. v 0 , t
(1)
where ρ is the fluid density and v is the linear velocity. The conservation equation of linear momentum [40]
9
m (
v v.v) p m 2 v 0 M H m g , t
(2)
where p and 0 are the hydrodynamic pressure and the magnetic permeability of free space, respectively. Kelvin body force term in Eq. (2) ( 0 M .H ) stands for the force applied by a nonuniform MF and M is the magnetization supposed to be equal to the equilibrium magnetization. Because the suspended nanoparticles show superparamagnetism, Langevin’s equation can be used to describe the equilibrium magnetization [41, 42] 1 H , M M d coth H
(3)
where is the volume fraction of nanoparticles in the base fluid and α is Langevin’s coefficient can be evaluated by
0 M d H Vc k BT
.
(4)
where Md, kB, and Vc are the bulk magnetization (446 kA m–1), Boltzmann constant (1.38×10–23 J K–1), and the volume of the magnetic core, respectively. To evaluate the MF distribution, electromagnetic equations of Maxwell can be used. The MF sources can be a current-carrying wire and single or two permanent magnets located parallel to the pipe. Maxwell equations were solved in three zones, including pipe inner region, the magnetic source, and the surrounding air. The corresponding governing equations are as follows [1]:
.B 0 ,
(5)
H J f ,
(6)
10
B 0 ( H M ) ,
(7)
where B and Jf are the magnetic flux and current densities, respectively. The thermal energy equation derived and reported by Rosensweig is quite complex [43], hence, for the sake of simplicity, the following assumptions were made: 1) the fluid is electrically nonconductive, 2) the fluid flow is incompressible with constant properties, and 3) the fluid is under equilibrium magnetization. Applying these assumptions, the thermal energy equation can be expressed as [44] 2 DT M DH ) 0T ( ) km2T m v vt , Dt 2 T H Dt
mC p , m (
(8)
where the first and second terms on the right-hand side of Eq. (8) are the conduction heat transfer and viscous dissipation, respectively. Moreover, it was supposed that conduction heat transfer in the axial (fluid flow) direction is negligible. The local Nu value can be evaluated as follows [24]: Nu
q D , k (Tw, avg Tavg )
(9a)
where Tw,avg is the arithmetic average of the wall temperature, Tavg is the local mixing cup fluid temperature, q is the wall heat flux, D is the pipe diameter, and k is the fluid thermal conductivity. Moreover, Tavg could be determined by R
Tavg
rv Tdr . rv dr z
0
(9b)
R
0
z
The density and specific heat capacity of nanofluids can be evaluated by [24]
m (1 ) f p ,
(10)
11
C p ,m
pC p , p (1 ) f C p , f , p (1 ) f
(11)
where ρp, ρf, Cp,p and Cp,f, and φ are, respectively, the solid particle density, base fluid density, the specific heat capacities of particles and the base fluid, and the volume fraction of the nanoparticles. The viscosity and thermal conductivity of the nanofluid were supposed to be a function of the nanoparticle volume fraction as follows [28]:
5
25
m 1 2 f . 4 2
(12)
k p 2 K f 2(k f k p ) km kf . k p 2 K f (k f k p )
(13)
Physical properties of nanoparticles and the base fluid used in the present work are summarized in Table 2. Moreover, governing equations used to simulate the system behavior are summarized in Table 3. 3.1.
Boundary conditions The following boundary conditions were applied:
At the pipe inlet: vr v 0, vz Vinlet , T T0 ,
(21)
At the fluid–wall interface: vr v vz 0 and q k
T . r
(22)
At the pipe outlet, the diffusional heat flux in the fluid flow direction was assumed to be zero and the outlet gauge pressure was also set zero. In the domains far from the applied MF, it was assumed that the MF diminishes and the normal component of B vector and the tangential component of H vector across the boundaries were supposed to be continuous and can be expressed as [41] n.B input n.B output ,
(23)
12
n H n H input output .
(24)
Table 2. Physical properties of nanoparticles and the base fluid used in the present work Properties
Value
Properties
Value
f (kg/m3 )
998
p (kg/m3 )
5200
C p, f (J / kg K)
4182
C p , p (J / kg K)
670
f (kg/ms)
9×10–4
p (kg/ms)
---
k f (W/m K)
0.6
k p (W/m K)
6
Table 3. Simplified governing equations Linear Momentum, r direction Linear Momentum, ϴ direction Linear Momentum, z direction
m vr
vr v vr v v 2 H r p 1 H r H vz r 0 M r M r r z r r r r r
1 1 2v 2 v m rvr 2 2r 2 r r r r r
m vr
v v v v v v H H 1 p 1 H H r vz r 0 M r M M z z r r z r r r r r
1 1 2v 2 v m rv 2 2 2 r r r r r r
m vr
vz v vz v vz z r r z
1 vz m r r r r
Maxwell
H z H z p 1 H z z 0 M r r M r M z z
1 vz 2 2 r
v 1 1 v rvr z 0 r r r z
(14)
(15)
(16)
2
Continuity
Maxwell
H r M z z
.B 0 0. M H 0
1 1 r (M r H r ) M H M z H z 0 r r r z
1 H r 1 H z H ˆ H r H z 1 H J f rˆˆ (rH ) zˆ Jf z r r z r r r
13
(17)
(18)
(19)
Thermal energy balance
M 0T T
mC p ,m vr
H r T v T T M r H r v H r vz vz 0T vr r r z r r z T H H H z H v H M z H z v H z vr r r vz z 0T T vr r r vz z H H
1 T km r r r r
(20)
2 1 T m 2 2 r
3.2. Numerical solution To solve the set of governing equations given in Table 3, the finite element method was used. The calculation domain was divided into tetrahedral elements. Integral equations corresponding to Eqs. (14-20) could be obtained for each element applying the conservation laws. To obtain an algebraic approximation to the control volume conservation equations, the velocity and temperature fields were interpolated quadratically in each element and other variables were linearly interpolated in each one. In other words, the second-order variables were stored at each node and the first-order dependent variables were stored at the vertices of these elements. The discretized equations were algebraic approximation to the control volume conservation equations. An iterative method was also used to solve the nonlinear coupled governing algebraic equations. Starting with guessed temperature and MF, the coefficients in the momentum equations were calculated. The momentum equations were solved, then, the pressure field was obtained. After that, MF and thermal energy equations were solved, respectively. The entire procedure was repeated until convergence could be achieved. The mesh independence was also examined for the system under the following conditions: the values of the Reynolds number, nanoparticle volume fraction, and the permanent magnet magnetization were set to 200, 0.03, and 3×105 (A/m), respectively. The system geometry was spatially discretized using unstructured grids of tetrahedral volume elements. For the base case, 930,000 tetrahedral elements were considered to check the mesh independence. Other cases with different numbers of elements were considered as summarized in Table 4. The average Nu values
14
were evaluated for all the cases and, then, the percentage error of the average Nu value relative to that obtained for the base case was determined as presented in Table 4. As can be observed from this table, increasing the number of elements from 930,000 to 1,438,000 resulted in a 0.17% reduction in the average Nu value. Hence, to reduce the computational load, 930,000 elements were used throughout the present work. Table 4. Variations of the average Nu value versus the number of elements Number of elements Nuavg Percentage error (%) 167000
9.683
3.92
506000
9.468
1.62
930000
9.317
0
1438000
9.301
-0.17
Solving the coupled governing equations takes approximately 17 h CPU time, running on an 8 cores Intel (R) Core (TM) i7-4770 CPU processor (3.4 GHz) with 16 GB of RAM these explaneations were added to manuscriot
4. Results and Discussion 4.1. Model validation To validate the developed model, the obtained simulation results were compared with those previously published including experimental and numerical results for a non-magnetic nanofluid. Fig. 2 demonstrates the obtained simulation results concerning the heat-transfer coefficient along with the experimental data reported by Kim et al. [45] in which Re = 1460 and the nanofluid contains 3 Vol % Al2O3 nanoparticles. As can be observed from this figure, there is good agreement between the
simulation and experimental results. Fig. 3 shows the present simulation results of heat-transfer coefficient for different volume fractions and diameters of nanoparticles along with those reported by Ebrahiminia et al. [46]. As can be observed from this figure, both the simulation results are almost
15
the same. In what follows, the influences of various operating conditions and design parameters on the hydrodynamic behavior and heat-transfer characteristics are presented and discussed. In the following subsections, Re = 200, = 0.03, and M = 3×105 (A/m) were considered. 3000 Present study Kim et al. [45] measured data
2000
2
h (W/m K)
2500
1500
1000
500
0
100
200
300
400
x/D
Fig. 2. Comparison of simulation results obtained in the present study with the experimental heattransfer coefficients reported in Ref. [45]. 2400 Ebrahimnia et al. [46] = 0.06, d = 20 nm Present study with = 0.06, d = 20 nm
2000
Ebrahimnia et al. [46] = 0.02, d = 100 nm
h (W/m2K)
Present study with = 0.02, d = 100 nm
1600
1200
800
400
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
z (m) Fig. 3. Comparison of the obtained simulation results with those reported in Ref. [46] as a function
of volume fraction of nanoparticles and their diameter. 16
4.2. Magnetic field distribution Fig. 4 shows the MF distribution (H) for three cases of single permanent magnet, double permanent magnets, and the current carrying wire that were parallel to the pipe with ferrofluid flow. It should be noted that variations of MF in different media, i.e., surrounding air, inside the pipe, and the magnetic source were taken into account in the present study, while this distribution has been ignored in most previously published works [18-20]. As can be observed from the onset shown in Fig. 4 (a) that sketches H magnitude on a centerline cut line shown by the red thick line, one can obviously observe a jump in the MF distribution at the pipe/air interface in the case of single permanent magnet. Note that in numerous
studies conducted in this field, an MF with a constant gradient has been applied that is impractical and the specified MF is not in accordance with Maxwell’s equations [21, 22, 37]. In the case of using double magnets with opposite poles next to each other (SNSN), MF lines leave one pole and enter the other one. Effects of the MF distribution on the heat-transfer rate are explored in the following subsections.
17
200000
H (A/m)
160000
120000
80000
40000
0
0
(a)
0.25
0.5
0.75
1
1.25
1.5
2r/D
(c)
(b)
Fig. 4. MF distribution in a) single permanent magnet; b) two permanent magnets, and c) currentcarrying wire. 4.3. Effect of magnetic field intensity of single permanent magnet Applying MFs with a gradient exert a body force in the radial and azimuthal directions, resulting in the creation of velocity components in these directions. In other words, at any given cross section of the pipe, the MNF undergoes some kind of lateral rotation as shown in Fig. 5. The magnitude of such velocities depends on the magnitude of MF and its gradient. The lateral rotation would change the temperature profile, consequently, affects the heat-transfer rate as shown in Fig. 6. Because the magnetization induced by the MF is proportional to the reciprocal of temperature, the body force exerted on the cold fluid is larger than that on the warmer fluid. This leads to the movement of cold fluid towards the MF source and intensifies the fluid mixing.
18
1 0.8 0.6 0.4
Mesh xMesh yz
2r/D
0.2
x y z
100100200
0
100100200 100100200 100100200 100100200 100100200 100100200 100100200
-0.2 -0.4 -0.6 -0.8 -1
-1
-0.8 -0.6 -0.4 -0.22r/D0
0.2
0.4
0.6
0.8
1
2r/D Fig. 5. Lateral mixing at the cross section of the pipe.
19
1.75 1.65 1.55
1.35
1.55
1.25
1.75
5 45
0
1.15
1.
-0.5
1.4 1.55
1.0 5
1.25
1.55
1.05
1.35
1.55
1.15
1.05
1.15
1.65
1.45
1.85
5 1.4 5 1.5
1.15
1.35
-1
1.25
5 1.0
1.35
1.2 5
1.15
5
5
5 1.0
1.05
1. 5
1.3
1.1 5
1.05
1.25
1.15
1. 4 5
1.4 5
1.25
5 1.3
1.25
1. 45
5 1.3
1.5 5
(b)
1.55
1.35
(a)
0.5
1
-1
-0.5
Dimensionless Radius
1.25
0
0.5
1
Dimensionless Radius
1.01.0 5 5
1.2
1.8
1.5
1.65
1.05
1.35
1.2
1 .5
(c)
1.5
1.2
1.35
1.65
1.05
1.8
1 .5
1.2
-1
-0.5
0
0.5
1
Dimensionless Radius
Fig. 6. Effects of MF of the single permanent magnet on the dimensionless temperature distribution at the cross section of the pipe for a) M = 0 A/m; b) M = 1×105A/m; and c) M = 3×105 A/m at z/L = 150. Fig. 7 shows the effects of the single permanent magnet with different magnetizations on the dimensionless axial velocity profile in the symmetric plane in the fully developed region. As can be observed from this figure, the velocity profile shifts towards the MF source. Because of the aforementioned fluid rotation the mixing efficiency increases and the temperature difference between the bulk mean temperature of the fluid and the wall temperature decreases, resulting in an increase in the Nu value. Moreover, applying an external MF would alter the thermal boundary layer and reduce the resistance
20
against heat transfer resulting in the enhancement of heat-transfer rate. Effects of MF induced by a single permanent magnet on the local Nu value are shown in Fig. 8. As can be observed from this figure, the Nu value increases with an increase in the magnetization of the single permanent magnet. 2
vz/ Vinlet
1.5
1 M (A/m) 5
510 3105 1105 4 310 without magnetic field
0.5
0 -1
-0.5
0
0.5
1
2r/D
Fig. 7. Axial velocity profile as a function of the magnetizations for the single permanent magnet. 14
M(A/m) 5
510 5 310 5 110 4 310 without magnetic field
12
Nu
10 8 6 4 2
0
50
100
150
z/D
Fig. 8. Nusselt number value versus the dimensionless pipe length as a function of the magnetization for the single permanent magnet. 21
4.4. Effect of magnetic field intensity of current carrying wire Effects of MF induced by the current carrying wire on the local Nu value for different electric currents are shown in Fig. 9. Under these conditions, the MF intensity can be precisely controlled by manipulating the electric current passing through the current-carrying wire. As can be seen, increasing electric current increases the Nu values the same as previous cases. 14
I (A) 1000 100 50 5 without magnetic field
12
Nu
10 8 6 4 2
0
50
100
150
z/D
Fig. 9. Nusselt number value versus the dimensionless pipe length for various values of current in the current-carrying wire. 4.5. Effect of volume fraction of magnetic nanoparticles Fig. 10 demonstrates how the volume fraction of magnetic nanoparticles (MNPs) influences the Nu value in the presence of a single permanent magnet. As can be seen, the Nu value increases with an increase in the volume fraction of MNPs. This can be justified as follows: according to Eq. (3), the magnetization of ferrofluid is directly proportional to the volume concentration of MNPs. Therefore, larger Kelvin body force is exerted on the fluid elements leading to the heat -transfer enhancement. It should be also added that the physical properties of ferrofluid can be affected by
22
changing the volume fraction of MNPs. Similar effects of the volume fraction of MNPs on the Nu value can be observed for the current carrying wire or the double permanent magnets that are not reported for the sake of brevity.
16
0.06 0.03 0.015 0.003 0
14 12
Nu
10 8 6 4 2
0
25
50
75
100
125
150
z/D
Fig. 10. Nusselt number value versus the dimensionless pipe length as a function of nanoparticle volume fraction for the single permanent magnet. 4.6. Effect of Reynolds number value The effect of Re value on the heat-transfer rate is shown in Fig. 11 for the single permanent MF. This figure clearly shows that larger Nu values can be obtained for larger Re values. For laminar flow in the absence of an MF, the Nu value reaches a constant value of 4.36 in the fully developed region while in the presence of MF, this fully developed value increases with an increase in the Re value. To examine the effect of MNPs and the type of magnetic source on the Nu value, its value was evaluated for water, nanofluid, and nanofluid in the presence of MF induced by one/two permanent magnet(s) or currentcarrying wire at Re=100 and =0.03 and the obtained results are shown in Fig. 12. As can be seen from this figure, the lowest Nu value belongs to pure water as the heat-transfer fluid. The presence of magnetite 23
nanoparticles (3 Vol %) improves slightly the Nu value in the pipe entrance zone due to the change of physical properties of ferrofluid such as thermal conductivity, however, the corresponding fully developed Nu value is the same as that for pure water as could be expected (i.e., Nu = 4.36 for the constant wall heat flux as the boundary condition). In the case of ferrofluid and in the presence of an electric current carrying wire an improvement in the local Nu value is observed. The fully developed Nu value is also increased that is not the case for heat transfer in the absence of MF. By increasing the electric current from 50 A to 1000 A, a significant enhancement in the Nu value is observed. Almost the same level of improvement can be obtained using a single permanent magnet with M = 3×105 A/m. If double permanent magnets are used, considerable improvement in the Nu value can be observed. When two permanent magnets are used on both sides of the pipe, the effect of MF is intensified. This effect enhances the local fluid mixing, consequently, leading to a redistribution of the fluid temperature profile and an increase in the local Nu value. These results clearly indicate that application of permanent magnets can be more efficient and economical than the current carrying wire. 16
Re 2000 1000 500 200 100
15 14
Nu
13 12 11 10 9
0
50
100
150
z/D Fig. 11. Nusselt number versus the dimensionless pipe length as a function of Reynolds number
value for the single permanent magnet.
24
16 14 12
Nu
10 8 6 5
4
Double permanent magnet (M = 310 A/m) Current carrying wire (1000 A) 5 Single permanent magnet ( M = 310 A/m) Current carrying wire (50 A) Water + Nanoparticles (3 vol.%) Pure water
2 0
0
25
50
75
100
125
150
z/D
Fig. 12. Nusselt number value versus the dimensionless pipe length
5. Conclusions The major aim of the present work was to investigate 3D forced-convection heat transfer of a ferrofluid flowing in a pipe under the influence of external non-uniform MFs induced by permanent magnets or a current carrying wire. The effects of the external magnetic field on the hydrothermal behavior of the ferrofluid flow were also explored. Applying a magnetic field to the pipe carrying an MNF exerts a body force perpendicular to the flow direction and as a result, the lateral velocity components are generated and the axial velocity distribution can be modified. This leads to an increase in the velocity and temperature gradients near the pipe wall, consequently, enhances the local Nu value. Heat transfer rate and Nu value can be increased with an increase in the intensity of permanent magnet magnetization and the electric current; however, using permanent magnets is more efficient. It was found that by using a single or double permanent magnet(s), enhancements of 113% (M = 3×105 A/m) and 196% (M = 3×105 A/m for each one) for the fully developed values of Nu can be obtained, respectively, while the Nu value increases up to 26% in the case of wire carrying a current of 50 A. In the presence of an MF with an increase in the Re value, the average and fully developed Nusselt values
25
also increase that is not the case for heat transfer in the absence of MF. The heat-transfer coefficient becomes larger with an increase in the volume fraction of nanoparticles. This can be attributed to the fact that the volume fraction of MNPs directly affects the magnetization of ferrofluid and its physical properties. In this work, a static MF was applied that could be replaced by oscillating or rotating MFs. This subject is the objective of our future studies. In addition, in the case of using a large volume fraction of MNP, the simulation can be performed applying a two-phase modeling approach. Moreover, the effects of multiple permanent magnets can be also investigated and, thus, the appropriate configuration to reach the maximum Nu value can be determined in the future works.
Acknowledgement The present authors acknowledge the financial support provided by the National Iranian Oil Refining & Distribution Company (NIORDC).
Nomenclature B Cp D d g H I km kB L M Md Nu p q r Re T t v
Magnetic flux density (T) Specific heat (J/kg K) Pipe diameter (m) Particle diameter (m) Gravitational acceleration (m/s2) Magnetic field (A/m) Electric current (A) Thermal conductivity (W/m K) Boltzmann constant, (1.38×10–23 J/K) Pipe length (m) Magnetization (A/m) Domain magnetization (A/m) Nusselt number Pressure (Pa) Constant wall heat flux (W/m2)
r coordinate (m) Reynolds number Absolute temperature (K) Time (s) Linear velocity (m/s) 26
Vc z
Volume of magnetic core (m3) z coordinate (m)
Greek Symbols α μ0 ρ φ
Langevin equation parameter Shear viscosity (Pa s) Vacuum permeability 4π×10-7 (N/A2, Tm/A) Density (kg/m3) MNPs’ volume
Subscripts avg Average f Fluid m Mixture p Particle w Wall Abbreviations MF Magnetic field MNF Magnetic nanofluid MNP Magnetic nanoparticle
27
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Research Highlights
3D forced-convection heat transfer of magnetic nanofluids is investigated. Effects of single or double permanent magnet on the heat transfer are studied. Influences of magnetic field induced by a current-carrying wire are studied. Effects of magnetic field intensity and Reynolds number value are studied.
Variations of magnetic field in different media are taken into account.
Graphical
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