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Effect of an alternating nonuniform magnetic field on ferrofluid flow and heat transfer in a channel
Journal of Magnetism and Magnetic Materials 362 (2014) 80–89
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Effect of an alternating nonuniform magnetic field on ferrofluid flow and heat transfer in a channel Mohammad Goharkhah n, Mehdi Ashjaee Department of Mechanical Engineering, University of Tehran, North Kargar street, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 7 October 2013 Received in revised form 2 March 2014 Available online 15 March 2014
Forced convective heat transfer of water based Fe3 O4 nanofluid (ferrofluid) in the presence of an alternating non-uniform magnetic field is investigated numerically. The geometry is a two-dimensional channel which is subjected to a uniform heat flux at the top and bottom surfaces. Nonuniform magnetic field produced by eight line source dipoles is imposed on several parts of the channel. Also, a rectangular wave function is applied to the dipoles in order to turn them on and off alternatingly. The effects of the alternating magnetic field strength and frequency on the convective heat transfer are investigated for four different Reynolds numbers (Re¼100, 600, 1200 and 2000) in the laminar flow regime. Comparing the results with zero magnetic field case, show that the heat transfer enhancement increases with the Reynolds number and reaches a maximum of 13.9% at Re¼ 2000 and f¼20 Hz. Moreover, at a constant Reynolds number, it increases with the magnetic field intensity while an optimum value exists for the frequency. Also, the optimum frequency increases with the Reynolds number. On the other hand, the heat transfer enhancement due to the magnetic field is always accompanied by a pressure drop penalty. A maximum pressure drop increase of 6% is observed at Re¼ 2000 and f¼5 Hz which shows that the pressure drop increase is not as significant as the heat transfer enhancement. & 2014 Elsevier B.V. All rights reserved.
Keywords: Heat transfer Forced convection Ferrofluid Channel Magnetic field Alternating Non-uniform
1. Introduction The demand for more efficient cooling methods is increasing steadily due to the importance of energy saving and also due to development of modern industrial and scientific applications. Among several methods which have been proposed for cooling and heat transfer enhancement, using of nanofluids has achieved growing interest by many researchers recently. Adding nanoparticles to the carrier fluid improves the transport properties of the fluid such as thermal conductivty and leads to heat transfer enhancement. An innovative enhancement method is the use of magnetic nanofluid called ferrofluid which is a synthesized colloidal mixture of nonmagnetic carrier liquid, typically water or oil, containing single domain permanently magnetized nano particles, typically magnetite [1]. Ferrofluid has the merits of nanoparticles in improving the thermal properties of the fluid. Moreover, the flow field and temperature distribution can be altered by applying an external
n
Corresponding author. Tel.: þ 98 21 61114048. E-mail addresses: [email protected] (M. Goharkhah), [email protected] (M. Ashjaee). http://dx.doi.org/10.1016/j.jmmm.2014.03.025 0304-8853/& 2014 Elsevier B.V. All rights reserved.
magnetic field. Thus, there is additional potential for heat transfer enhancement by flow mixing and disturbing thermal boundary layer specifically in the laminar regime. Ferrofluid can be utilized in mechanical and thermal engineering, low Reynolds flows and micro scale heat exchangers in MEMS devices, aerospace, spacecraft cooling system in microgravity conditions and bioengineering [2–7]. A comprehensive literature review reveals that the majority of numerical works on heat transfer of ferrofluid is devoted to thermomagnetic convection in which ferrofluid is confined in an enclosure and is subjected to a magnetic field [8–21]. Geometries such as cubic enclosure [8,9], rectangular cell with different aspect ratios [10–18], cylindrical [19], annular [20] and triangular [21] enclosures have been considered in numerical and experimental studies with a variety of thermal boundary conditions. By contrast, only a few works have concentrated on the forced convection heat transfer problem [22–32]. Neuringer [22] studied the effect of magnetic field on two cases numerically, the two-dimensional stagnation point flow of a heated ferrofluid against a cold wall and the two-dimensional parallel flow of a heated ferrofluid along a wall with linearly decreasing surface temperature. Wall shear stress, heat transfer, velocity and temperature boundary layer profiles were obtained for both cases and compared to the case with no magnetic field.
M. Goharkhah, M. Ashjaee / Journal of Magnetism and Magnetic Materials 362 (2014) 80–89
Nomenclature B Cp ! FK H i, j m M Nu P Pr r T t Vm u, v X; Y x, y
magnetic flux density (T) specific heat (J/kgK) magnetic body force (N) magnetic field(A/m) unit vectors (horizontal, vertical) strength of the magnetic field, (Am) magnetization (A/m) Nusselt number Pressure (Pa) Prantdl number (Pr ¼ ν=α) radial direction (m) Temperature(K) time(s) scalar magnetic potential fluid velocities (horizontal, vertical, m/s) locations of a single line dipole, m Cartesian coordinates (horizontal, vertical, m)
Ganguly et al. [3] numerically studied the two-dimensional forced convection heat transfer of a hot ferrofluid flowing through a channel with a cold wall under the influence of a line-source dipole. It was shown that the local vortex resulted from the magnetic field alters the advection energy transport and enhances the heat transfer. Strek and Jopek [23] studied the two-dimensional and time dependent heat transfer of a ferrofluid in a channel with isothermal walls under the influence of the magnetic dipole located below the channel. They showed that an imposed thermal gradient produces a spatial variation in the magnetization through the temperature dependent magnetic susceptibility for ferrofluids and therefore renders the Kelvin body force nonuniform spatially. Furthermore, Strek [24] considered the ferrofluid flow in a channel with porous walls and studied the effect of Darcy number on the heat transfer and fluid flow. Xuan et al. [25] calculated the flow and temperature distribution of a ferrofluid in a micro channel with adiabatic and isothermal walls using the lattice-Boltzmann method. In their work the magnetic force was considered a constant and it was shown that heat transfer augmentation depends on the magnitude and direction of the the magnetic force. Aminfar et al. [26] numerically studied the mixed convection of a ferrofluid in a vertical tube in the presence of nonuniform magnetic field using a two phase mixture model and control volume technique. They considered the positive and negative magnetic field gradients and showed that that the magnetic field with negative gradient acts similar to the buoyancy force and enhances the Nusselt number, while the magnetic field with positive gradient decreases it. More recently, Aminfar et al. [27] studied the forced convection of ferrofluid in a duct under the influence of a transverse nonuniform magnetic field which was generated by an electric current going through a wire parallel with the duct. They reported a considerable enhancement of the average heat transfer coefficient for their studied case. Additionally, several numerical works have been carried out on biomagnetic fluid dynamics (BFD) in which blood as a magnetic fluid is under the influence of an external magnetic field [28–32]. The governing equations for these problems are similar to those derived for ferrofluids. In these works magnetization of the fluid is assumed to vary linearly with temperature and magnetic field
81
Greek symbols
α β Θ μ μ0 ρ ϕ χm
thermal diffusivityðm2 =sÞ volumetric thermal expansion coefficient ð1= KÞ viscous dissipation dynamic viscosityðkg=m sÞ magnetic permeability in vacuum ð4π 10 7 N=A2 Þ densityðkg=m3 Þ polar angle (rad) magnetic susceptibility
Subscripts ref 0 f m nf p
reference condition at reference temperature (300 K) fluid mean value nanofluid particle
intensity, and the line dipole is used as the source of magnetic field. The temperature, as well as the skin friction and the rate of heat transfer were shown to be increasing near the magnetic source. The present research addresses the laminar forced convective heat transfer of water based Fe3 O4 ferrofluid in a two-dimensional channel under an alternating nonuniform magnetic field. A constant heat flux is applied at the top and bottom surfaces of the channel. The main objective is to characterize the heat transfer augmentation by periodical attraction of colder fluid to the surface by the magnetic field, transfer of surface heat to the fluid and release of the fluid into the bulk flow. The magnetic field is produced by placing the line dipoles on the top and bottom of the channel. A rectangular wave function is also applied to the dipoles in order to generate an alternating magnetic field. The effects of the magnetic field strength and frequency on the convective heat transfer is examined and the optimum frequency is obtained for different Reynolds numbers. Moreover, temporal variations of the velocity and the temperature profiles in the channel are calculated through a time dependent numerical simulation and then used to explain the heat transfer characteristics of the ferrofluids.
2. Problem description The cold ferrofluid flows through a two-dimensional channel with the size of 0:004 mðHÞ 0:5 mðLÞ which is heated by a uniform heat flux from top and bottom surfaces. The flow is considered homogenous and laminar. The thermal properties of the fluid are assumed to be constant and the buoyancy effects are negligible compared with the forced convection and hydromagnetic effects. The schematic of the problem is shown in Fig. 1. A two-dimensional magnetic field used in the simulations is created by eight identical magnetic dipoles along the top and bottom surfaces. Four line dipoles are placed above the top surface and four line dipoles beneath the bottom surface. Magnetic field of a dipole can be an approximation of the field produced by an edge-dipole permanent magnet or an electromagnet composed of a rectangular current-carrying loop of very high aspect ratio. Other realistic magnetic fields can also be simulated by appropriate arrangment of a
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Fig. 1. Schematic of the studied problem.
number of these line dipoles [3,14]. The distance between dipoles and their distance from the channel surfaces are constant, 0.09 m and 0.001 m, respectively. The first dipole is located at x¼0.1 m from the channel entrance beneath the bottom surface. Also, top dipoles are shifted 0.045 m to the right with respect to the bottom dipoles. The alternating magnetic field is produced by applying rectangular pulse waves with equal connection and disconnection times to the magnetic dipoles. For all simulated cases, a phase shift of pi is imposed to the top dipoles with respect to the bottom dipoles. Hence, the top and bottom dipoles turn on and off sequentially.
3. Governing equations
Gauss's law for the magnetic flux density is expressed as ! ! ! B ¼ μ0 ðM þ H Þ; ð1Þ ! and Ampere's law for the magnetic field, H with zero current density is written as !! ∇ : B ¼ 0: ð2Þ It is assumed that the fluid is non-conducting. Hence, the electromagnetic current resulted from ferrofluid flow in the magnetic field can be neglected. Then, -
∇ H ¼ 0:
Thus, the magnetic field can be obtained from [2] ! H ¼ ∇V m ;
ð3Þ
ð4Þ
where V m is the scalar magnetic potential. For a line dipole it can be expressed as [23] m xX V m ðx; yÞ ¼ ; 2π ðx X Þ2 þ ðy Y Þ2
ð5Þ
where m represents the strength of the magnetic field and X and Y are the locations of a single line dipole. Then, H is obtained in polar coordinates as ! ! ! m 1 U ð cos ð2ϕÞ i þ sin ð2ϕÞ j Þ; H ðr; ϕÞ ¼ 2π r2 where r and ϕ are defined in the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx XÞ2 þ ðy YÞ2 ;
χ m ¼ χ m ðTÞ ¼
χ
0 ; 1 þ βðT T 0 Þ
ð10Þ
Where, χ 0 is the susceptibility at a reference temperature. The corresponding Kelvin body force per volume for a ferrofluid is defined as [2,33] ! ! F K ¼ M :∇B : ð11Þ 3.2. Fluid flow and heat transfer equations The governing equations for conservation of mass, momentum in x, y directions (Navier–Stockes equations) and energy are given by equations:
3.1. Magnetic body force
-
The variation of the total magnetic susceptibility, χ m as function of temperature is of the form [34]
ð6Þ
∂u ∂v þ ¼0; ∂x ∂y
ð12Þ
∂u ∂u ∂u 1 ∂P μ ∂2 u ∂2 u þu þv ¼ U þ þ F K ðxÞ; þ ∂t ∂x ∂y ρ ∂x ρ ∂x2 ∂y2
ð13Þ
∂v ∂v ∂v 1 ∂P μ ∂2 v ∂2 v þu þv ¼ U þ þ F K ðyÞ; þ ∂t ∂x ∂y ρ ∂y ρ ∂x2 ∂y2
ð14Þ
2 ∂T ∂T ∂T ∂ T ∂2 T þ u þv ¼ α þ 2 þ μΘ; 2 ∂t ∂x ∂y ∂x ∂y
ð15Þ
In Eq. 15, Θ is the viscous dissipation and is of the form: 2 2 ∂u ∂v ∂v ∂u 2 þ Θ¼2 þ2 þ ∂x ∂y ∂x ∂y
4. Thermophysical properties of nanofluids The effective thermophysical properties of the ferrofluid used in the governing momentum and energy equations can be evaluated by using different relations from the literature. The following equations have been used for calculation of nanofluid bulk density and specific heat [35]:
ρnf ¼ φρp þ ð1 φÞρf cp;nf ¼
ð7Þ
yY ; ð8Þ ϕ ¼ tan 1 xX ! ! where i and j correspond to the unit vectors in x and y directions, respectively. Also, the relation between the magnetic field and magnetization vector is as follows: ! ! M ¼ χm H : ð9Þ
ð16Þ
φρp cp;p þ ð1 φÞρf cp;f ρnf
ð17Þ ð18Þ
where, ρp is the particle density, ρf is the base fluid density and cp;p and cp;f are the particle and the base fluid specific heats, respectively. The viscosity of the ferrofluid at different concentrations has been estimated by the Einstein equation as
μnf ¼ μf ð1 þ 2:5φÞ
ð19Þ
It is shown in Ref. [36] that Eq.19 is reliable with the experimental viscosity measurements. Furthermore, the thermal conductivity of the ferrofluid is calculated from the following equation, as
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Table1 Thermophysical properties of the magnetite nanoparticle, DI-water and the ferrofluid. Density (kg=m3 Þ
Magnetite nanoparticle (Fe3 O4 ) DI-Water Ferrofluid (2 vol%)
Specific heat (J/kgK)
4950
640
997 1076
4180 3854
recommended by Ref. [20] 2kf þ kp þ 2φðkp kf Þ knf ¼ kf 2kf þ kp þ 2φðkp kf Þ
Thermal conductivity (W/mK)
Viscosity (Ns/m2 )
7
0.61 0.64
0.00065 0.00069
ð20Þ
where, kf and kp are conductivities of the base fluid and particle, respectively. A magnetite water based ferrofluid with a volume fraction of 2% is used in the numerical simulations. The thermophysical properties of magnetite nanoparticles, DI-water and ferrofluid are presented in Table 1.
5. Numerical solution The set of differential Eqs. (12–16) have been discretized with the control volume technique. For the convective and diffusive terms a second order upwind method has been used while the SIMPLEC procedure has been introduced for the velocity–pressure coupling. A structured non-uniform grid has been used for the computational domain. It is finer near the channel entrance and near the top and bottom surfaces where the velocity and temperature gradients are large. In order to ensure that the calculations are accurate and grid independent, several combination of node numbers in longitudinal and lateral directions have been examined. Temperature and axial velocity profiles at the channel exit have been calculated for all the cases. Results are plotted in Fig. 2. As shown, these grid distributions give almost identical results. The 50–1000 grid (50 nodes in vertical and 1000 nodes in horizontal direction) has been used in the calculations. In order to demonstrate the validity and also precision of the numerical procedure, a comparison has been made with the previously published work in the field of ferrohydrodynamics. Fig. 3 indicates the variation of Nusselt number in the channel and its comparison with the results of Ganguly et al. [3] for two different values of magnetic field strength, m1 ¼ 0:19 and m2 ¼ 0:58 Am. As shown, the code has predicted the magnetic field effect appropriately.
Fig. 2. Grid independency test, (a) temperature and (b) axial velocity at the channel exit.
6. Results and discussion A water based ferrofluid containing 2 vol% Fe3 O4 particles flows into the channel with temperatre of 340 K and is heated from the top and bottom surfaces at a uniform heat flux. Passing through the channel, the ferrofluid is influenced by an alternating nonuniform magnetic field produced by eight identical line source dipoles. Fig. 4(a) shows the magnetic flux density distribution calculated from Eqs.1–10 for one of the dipoles located at x ¼0.1 m, beneath the bottom surface. An alternating magnetic field is obtained by applying a rectangular wave function to the magnetic dipoles, as shown in Fig. 4(b). The period of the alternating magnetic field function, T,
Fig. 3. Comparison between calculated Nusselt number and results of the previous work [3].
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Fig. 4. (a) Magnetic flux density distribution for the dipole located at x¼ 0.1, (b) rectangular pulse wave applied to the magnetic dipoles.
is defined as the sum of connection and disconnection times. Also, frequency, f, is the reciprocal of the period of the wave function. Simulations are carried out for four different Reynolds numbers of Re¼100, 600, 1200 and 2000, four magnetic field intensities of m ¼ 0:1; 0:15; 0:2; 0:25 Am and the frequencies of f ¼ 1; 2:5; 5; 6:67; 10; 20 Hz. Moreover, the top dipoles have a phase shift of pi with respect to the bottom dipoles. Temporal variations of the streamlines and the temperature distribution for the magnetic field function period of T ¼1 s (equal to f ¼ 1 Hz), is illustrated at three instants namely,t 1 ¼ 12 s, t 2 ¼ 12:25 s and t 3 ¼ 12:5 s in Fig. 5. The sections A and B are in the vicinity of two adjacent magnetic dipoles located at x ¼0.37 m below and x ¼0.415 top of the channel. Since the alternating magnetic field functions of the top and bottom dipoles have a phase difference of pi, completely different temperature distribution and flow patterns are observed in parts A and B, at each instant. Mechanism of heat transfer enhancement can be inferred from Fig. 5. During each period of magnetic field functioning, first, the bottom dipoles are turned on and the colder fluid is attracted to the heated bottom surface from center of the channel. Thus, the heat can be transferred to the fluid. Next, the fluid is released to the bulk flow at the disconnection time of botton dipole. Finally, it is attracted by the top dipole as it turns on. The fluid attraction and releasing occur along the channel repeatedly. Therefore, the fluid particles are expected to travel in a sinusoidal path, resulting in better mixing, the thermal boundary layer disturbance and enhancement of the forced convection heat transfer. Another point worth mentioning is the formation of two recirculation zones upstream and downstream of the dipoles. This is due to the temperature dependence of the magnetic susceptibility and resultant non-symmetrical magnetic body forces [3]. The vortices have great influence on the thermal boundary layer thickness. Thickening of the thermal boundary layer is observed in locations where direction of the flow in the vortex is along the
Fig. 5. Variation of streamlines and temperature distribution around two adjacent magnetic dipoles with time, t1 ¼ 12 s, t2 ¼ 12.25 s and t3 ¼12.5 s.
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Fig. 6. Variation of local heat transfer coefficient on the (a)bottom surface and (b) top surface at the time sequence of t1 ¼12 s, t2 ¼12.25 s and t3 ¼12.5 s.
direction of the mean flow. By contrast, thermal boundary layer shrinks in the locations where the mean flow is in opposite direction of flow in the vortex region. These two phenomena result in decrease and increase of heat transfer rate, respectively. However, the second effect is dominant and heat transfer is expected to increase due to the magnetic field. In order to quantify the above discussion, variation of local heat transfer coefficient on the bottom and top surfaces of the channel are depicted in Figs. 6 and 7(a) and (b), respectively. As shown in Fig. 6(a) and (b), four dipoles near the top and bottom surfaces observed in Fig. 5 have led to four large peaks. Four small peaks are also seen on the curves which are due to the manipulation of fluid flow by the line dipoles near the opposite surface. Since the magnetic field of the bottom surface turns on at t 1 , the local heat transfer coefficient increases with the elapse of time while the trend is opposite for the top dipole, due to the phase shift of pi, as expected. The insets in Fig. 6 which magnify the regions around the dipoles clarify the significant role of downstream vortices in heat transfer enhancement. Growth of the downstream vortex at t 3 , as shown in part A of Fig. 5 has decreased the thermal boundary layer thickness and increased local heat transfer coefficient as a result. Moreover, slight decreases in the local heat transfer coefficient are observed when the magnetic field is turned on. This can be due to thickening of the thermal boundary layer as mentioned before. However, the local heat transfer coefficient increases sharply when the boundary layer thickness decreases.
Fig. 7. Axial variation of velocity profile at (a) t1 ¼12 s, (b) t2 ¼ 12.25 s and (c) t3 ¼ 12.5 s.
It is observed that when the ferrofluid passes over the section of magnetic field it accelerates noticeably along the surface due to the magnetic force. This is another reason for sharp increase of the local heat transfer coefficient at magnetic field locations. Moving a
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Fig. 9. Effect of magnetic field intensity on the forced convective heat transfer, m1 ¼ 0.1, m2 ¼0.15, m3 ¼ 0.2 and m4 ¼ 0.25 Am.
Fig. 8. Axial variation of temperature profile at (a) t1 ¼12 s, (b) t2 ¼12.25 s and (c) t3 ¼12.5 s.
little away from the dipoles, axial velocity reaches negative values near the surface, which implies the change of flow direction in the larger recirculation zone. Additionally, the magnetic field gradients
are weak and far from the dipoles and the velocity profiles almost merge. Axial variation of the temperature profile at the same time sequence is plotted in Fig. 8(a)–(c). Cooling effect of the ferrofluid is obvious near the dipoles. The effect of the magnetic field strength on forced convection heat transfer is demonstrated in Fig. 9. It shows the variation of the Nusselt number ratio, Nu=Nuref , on the bottom surface with time for different values of magnetic field strength, m1 ¼ 0:1, m2 ¼ 0:15, m3 ¼ 0:2 and m4 ¼ 0:25 Am, at Re ¼600 and f¼ 10. Nu and Nuref are the average Nusselt number on the surface in the presence and the absence of the magnetic field, respectively. It can be seen that forced convection heat transfer has a direct relation with magnetic field strength. Since the Kelvin magnetic force increases with the magnetic field strength, more colder fluid is attracted to the heated surface. Therefore, heat is transferred more effectively from the heated surfaces to the fluid. Fig. 10 illustrates the dependence of the convective heat trasnfer on the frequency of the alternating magnetic field. Variations of the Nusselt number ratio and the average surface temperature of the bottom surface with time at Re ¼600 and m ¼0.2 Am have been depicted for different frequencies in Fig. 10 (a) and (b), respectively. As shown, the steady state has been reached after elapse of 12 sec. The average surface temperature and Nusselt number ratio are both periodic functions of time with the same frequency of the external magnetic field. Although the alternating magnetic field has led to increase of convective heat transfer for all the values of frequencies, magnitude of enhancement depends strongly on the frequency of the alternating magnetic field. For this case (Re¼600 and m ¼ 0.2), the maximum heat transfer of 8.5% has been obtained at f 4 ¼ 5 Hz. The Nusselt number ratios on the botom and top surfaces as a function of magnetic field intensity and frequency are shown in Fig. 11(a) and (b) for Re¼600, respectively. It is apparent from the figure that the optimum frequency is f¼ 5 Hz at Re¼600 and is independent of the magnetic field intensity. It is also shown that the convective heat transfer is slightly greater for the bottom surface. For the constant heat flux boundary condition, the main goal of the cooling system is to lower the temperature of the heated surface as much as possible. Therefore, the mean temperature of
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Fig. 10. Variation of (a) the Nusselt number ratio and (b) average surface temperature of the bottom surface with time at Re¼ 600, m¼ 0.2 Am for f1 ¼20 Hz, f2 ¼ 10 Hz, f3 ¼ 66 Hz, f4 ¼5 Hz, f5 ¼ 2.5 Hz, and f6 ¼1 Hz.
Fig. 11. Variation of heat transfer enhancement with the frequency for different values of magnetic field strength for (a) bottom surface and (b) top surface.
the heated surface can be a measure of efficiency of the system. Values of the mean temperature on the heated surface have been calculated for all the cases. Result for the bottom surface at Re¼600 is shown in Fig. 12. As shown, frequency of the magnetic field is a key parameter by which surface temperature can be controlled Variation of the Nusselt number ratio on the bottom surface with time for different frequencies at Re¼ 2000 and m ¼ 0.2, is shown in Fig. 13. In contrast to the case of Re¼600, it can be observed that the forced convective heat transfer increases with the frequency. Furthermore, a maximum heat transfer of 13.9% is obtained at f 1 ¼ 20 Hz. In order to show the dependency of the optimum frequency on the Reynolds number, the heat transfer enhancement is plotted as a function of frequency in Fig. 14 for different Reynolds numbers and the magnetic field intensity of m ¼ 0.2. Clearly, the heat transfer enhancement increases with the Reynolds number and reaches the maximum value of 13.9% at
Re¼2000 and f 1 ¼ 20 Hz. It is also observed that an optimum frequency exists which increases with the Reynolds number. With Reynolds number increase, the ferrofluid passes over the magnetic dipoles with a higher velocity and the time that ferrofluid particles are influenced by the magnetic field is decreased. Thus, faster attracting and releasing of the particles is required to disrupt the thermal boundary layer effectively which means to increase the frequency of the magnetic field function. Another important parameter in the application of nanofluids in a heat exchanging equipment is the pressure drop. In such systems, there is a trade-off between pressure drop and heat transfer enhancement. However, for the ferrofluid flow in the presence of a magnetic field, the increase of pressure drop is not as significant as heat transfer enhancement [37]. Pressure drop of the ferrofluid across the channel has been calculated for all studied cases. Pressure drop ratio (ΔP=ΔP ref ) as a function of magnetic field frequency and strength for Re ¼2000 has been depicted in Fig. 15. Where ΔP ref is the pressure drop in the absence of the magnetic field.
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Fig. 12. Variation of average bottom surface temperature with frequency for different magnetic field strengths.
Fig. 14. Variation of the heat transfer enhancement with frequency for different Reynolds numbers at m¼0.2.
Fig. 13. Variation of the Nusselt number ratio with time at Re ¼ 2000 and m¼ 0.2 Am for f1 ¼ 20 Hz, f2 ¼10 Hz, f3 ¼67 Hz, f4 ¼5 Hz, f5 ¼ 2.5 Hz, f6 ¼ 1 Hz.
Fig. 15. Variation of pressure drop increase with the frequency at Re¼ 100 and 2000 for m¼ 0.2.
As shown, the pressure drop increases with magnetic field strength. However, the maximum pressure drop ratio is 6% for m¼0.25. Hence, the heat transfer enhancement due to the magnetic field dominates pressure drop increase.
to the case with no magnetic field. A maximum of 13.9% enhanement has been obtained at Re ¼ 2000 and f ¼20 Hz. Periodic attraction of the cold fluid to the heated surface and release of it to the bulk flow disturbs the thermal boundary layer, improves the flow mixing and increases heat transfer as a result. Magnetic force causes the ferrofluid to accelerate noticeably along the surface when it passes over the section of magnetic field. This is another reason for a sharp increase of the local heat transfer coefficient at magnetic field locations. Heat transfer enhancement increases with the Reynolds number and the intensity of the magnetic field. Frequency of the alternating magnetic field has an important role on the heat transfer. Results show that an optimum value exists for the frequency. It is independent of the magnetic field intensity but increases with the Reynolds number. The alternating magnetic field causes the increase of the pressure drop up to a maximum of 6% at Re¼2000 and f¼5 Hz. However, it is not as significant as the heat transfer enhancement.
7. Concluding Remarks Laminar forced convection heat transfer of a ferrofluid in a channel subjected to a uniform heat flux from top and bottom surfaces has been studied numerically. The possibility of heat transfer enhancement utilizing an alternating magnetic field by eight line source dipoles has been investigated at different Reynolds numbers, magnetic field intensities and frequencies. The following results have been obtained.
Forced convection heat transfer of ferrofluid can be enhanced noticably by applying an alternating magnetic field compared
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Effect of an alternating nonuniform magnetic field on ferrofluid flow and heat transfer in a channel Mohammad Goharkhah n, Mehdi Ashjaee Department of Mechanical Engineering, University of Tehran, North Kargar street, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 7 October 2013 Received in revised form 2 March 2014 Available online 15 March 2014
Forced convective heat transfer of water based Fe3 O4 nanofluid (ferrofluid) in the presence of an alternating non-uniform magnetic field is investigated numerically. The geometry is a two-dimensional channel which is subjected to a uniform heat flux at the top and bottom surfaces. Nonuniform magnetic field produced by eight line source dipoles is imposed on several parts of the channel. Also, a rectangular wave function is applied to the dipoles in order to turn them on and off alternatingly. The effects of the alternating magnetic field strength and frequency on the convective heat transfer are investigated for four different Reynolds numbers (Re¼100, 600, 1200 and 2000) in the laminar flow regime. Comparing the results with zero magnetic field case, show that the heat transfer enhancement increases with the Reynolds number and reaches a maximum of 13.9% at Re¼ 2000 and f¼20 Hz. Moreover, at a constant Reynolds number, it increases with the magnetic field intensity while an optimum value exists for the frequency. Also, the optimum frequency increases with the Reynolds number. On the other hand, the heat transfer enhancement due to the magnetic field is always accompanied by a pressure drop penalty. A maximum pressure drop increase of 6% is observed at Re¼ 2000 and f¼5 Hz which shows that the pressure drop increase is not as significant as the heat transfer enhancement. & 2014 Elsevier B.V. All rights reserved.
Keywords: Heat transfer Forced convection Ferrofluid Channel Magnetic field Alternating Non-uniform
1. Introduction The demand for more efficient cooling methods is increasing steadily due to the importance of energy saving and also due to development of modern industrial and scientific applications. Among several methods which have been proposed for cooling and heat transfer enhancement, using of nanofluids has achieved growing interest by many researchers recently. Adding nanoparticles to the carrier fluid improves the transport properties of the fluid such as thermal conductivty and leads to heat transfer enhancement. An innovative enhancement method is the use of magnetic nanofluid called ferrofluid which is a synthesized colloidal mixture of nonmagnetic carrier liquid, typically water or oil, containing single domain permanently magnetized nano particles, typically magnetite [1]. Ferrofluid has the merits of nanoparticles in improving the thermal properties of the fluid. Moreover, the flow field and temperature distribution can be altered by applying an external
n
Corresponding author. Tel.: þ 98 21 61114048. E-mail addresses: [email protected] (M. Goharkhah), [email protected] (M. Ashjaee). http://dx.doi.org/10.1016/j.jmmm.2014.03.025 0304-8853/& 2014 Elsevier B.V. All rights reserved.
magnetic field. Thus, there is additional potential for heat transfer enhancement by flow mixing and disturbing thermal boundary layer specifically in the laminar regime. Ferrofluid can be utilized in mechanical and thermal engineering, low Reynolds flows and micro scale heat exchangers in MEMS devices, aerospace, spacecraft cooling system in microgravity conditions and bioengineering [2–7]. A comprehensive literature review reveals that the majority of numerical works on heat transfer of ferrofluid is devoted to thermomagnetic convection in which ferrofluid is confined in an enclosure and is subjected to a magnetic field [8–21]. Geometries such as cubic enclosure [8,9], rectangular cell with different aspect ratios [10–18], cylindrical [19], annular [20] and triangular [21] enclosures have been considered in numerical and experimental studies with a variety of thermal boundary conditions. By contrast, only a few works have concentrated on the forced convection heat transfer problem [22–32]. Neuringer [22] studied the effect of magnetic field on two cases numerically, the two-dimensional stagnation point flow of a heated ferrofluid against a cold wall and the two-dimensional parallel flow of a heated ferrofluid along a wall with linearly decreasing surface temperature. Wall shear stress, heat transfer, velocity and temperature boundary layer profiles were obtained for both cases and compared to the case with no magnetic field.
M. Goharkhah, M. Ashjaee / Journal of Magnetism and Magnetic Materials 362 (2014) 80–89
Nomenclature B Cp ! FK H i, j m M Nu P Pr r T t Vm u, v X; Y x, y
magnetic flux density (T) specific heat (J/kgK) magnetic body force (N) magnetic field(A/m) unit vectors (horizontal, vertical) strength of the magnetic field, (Am) magnetization (A/m) Nusselt number Pressure (Pa) Prantdl number (Pr ¼ ν=α) radial direction (m) Temperature(K) time(s) scalar magnetic potential fluid velocities (horizontal, vertical, m/s) locations of a single line dipole, m Cartesian coordinates (horizontal, vertical, m)
Ganguly et al. [3] numerically studied the two-dimensional forced convection heat transfer of a hot ferrofluid flowing through a channel with a cold wall under the influence of a line-source dipole. It was shown that the local vortex resulted from the magnetic field alters the advection energy transport and enhances the heat transfer. Strek and Jopek [23] studied the two-dimensional and time dependent heat transfer of a ferrofluid in a channel with isothermal walls under the influence of the magnetic dipole located below the channel. They showed that an imposed thermal gradient produces a spatial variation in the magnetization through the temperature dependent magnetic susceptibility for ferrofluids and therefore renders the Kelvin body force nonuniform spatially. Furthermore, Strek [24] considered the ferrofluid flow in a channel with porous walls and studied the effect of Darcy number on the heat transfer and fluid flow. Xuan et al. [25] calculated the flow and temperature distribution of a ferrofluid in a micro channel with adiabatic and isothermal walls using the lattice-Boltzmann method. In their work the magnetic force was considered a constant and it was shown that heat transfer augmentation depends on the magnitude and direction of the the magnetic force. Aminfar et al. [26] numerically studied the mixed convection of a ferrofluid in a vertical tube in the presence of nonuniform magnetic field using a two phase mixture model and control volume technique. They considered the positive and negative magnetic field gradients and showed that that the magnetic field with negative gradient acts similar to the buoyancy force and enhances the Nusselt number, while the magnetic field with positive gradient decreases it. More recently, Aminfar et al. [27] studied the forced convection of ferrofluid in a duct under the influence of a transverse nonuniform magnetic field which was generated by an electric current going through a wire parallel with the duct. They reported a considerable enhancement of the average heat transfer coefficient for their studied case. Additionally, several numerical works have been carried out on biomagnetic fluid dynamics (BFD) in which blood as a magnetic fluid is under the influence of an external magnetic field [28–32]. The governing equations for these problems are similar to those derived for ferrofluids. In these works magnetization of the fluid is assumed to vary linearly with temperature and magnetic field
81
Greek symbols
α β Θ μ μ0 ρ ϕ χm
thermal diffusivityðm2 =sÞ volumetric thermal expansion coefficient ð1= KÞ viscous dissipation dynamic viscosityðkg=m sÞ magnetic permeability in vacuum ð4π 10 7 N=A2 Þ densityðkg=m3 Þ polar angle (rad) magnetic susceptibility
Subscripts ref 0 f m nf p
reference condition at reference temperature (300 K) fluid mean value nanofluid particle
intensity, and the line dipole is used as the source of magnetic field. The temperature, as well as the skin friction and the rate of heat transfer were shown to be increasing near the magnetic source. The present research addresses the laminar forced convective heat transfer of water based Fe3 O4 ferrofluid in a two-dimensional channel under an alternating nonuniform magnetic field. A constant heat flux is applied at the top and bottom surfaces of the channel. The main objective is to characterize the heat transfer augmentation by periodical attraction of colder fluid to the surface by the magnetic field, transfer of surface heat to the fluid and release of the fluid into the bulk flow. The magnetic field is produced by placing the line dipoles on the top and bottom of the channel. A rectangular wave function is also applied to the dipoles in order to generate an alternating magnetic field. The effects of the magnetic field strength and frequency on the convective heat transfer is examined and the optimum frequency is obtained for different Reynolds numbers. Moreover, temporal variations of the velocity and the temperature profiles in the channel are calculated through a time dependent numerical simulation and then used to explain the heat transfer characteristics of the ferrofluids.
2. Problem description The cold ferrofluid flows through a two-dimensional channel with the size of 0:004 mðHÞ 0:5 mðLÞ which is heated by a uniform heat flux from top and bottom surfaces. The flow is considered homogenous and laminar. The thermal properties of the fluid are assumed to be constant and the buoyancy effects are negligible compared with the forced convection and hydromagnetic effects. The schematic of the problem is shown in Fig. 1. A two-dimensional magnetic field used in the simulations is created by eight identical magnetic dipoles along the top and bottom surfaces. Four line dipoles are placed above the top surface and four line dipoles beneath the bottom surface. Magnetic field of a dipole can be an approximation of the field produced by an edge-dipole permanent magnet or an electromagnet composed of a rectangular current-carrying loop of very high aspect ratio. Other realistic magnetic fields can also be simulated by appropriate arrangment of a
82
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Fig. 1. Schematic of the studied problem.
number of these line dipoles [3,14]. The distance between dipoles and their distance from the channel surfaces are constant, 0.09 m and 0.001 m, respectively. The first dipole is located at x¼0.1 m from the channel entrance beneath the bottom surface. Also, top dipoles are shifted 0.045 m to the right with respect to the bottom dipoles. The alternating magnetic field is produced by applying rectangular pulse waves with equal connection and disconnection times to the magnetic dipoles. For all simulated cases, a phase shift of pi is imposed to the top dipoles with respect to the bottom dipoles. Hence, the top and bottom dipoles turn on and off sequentially.
3. Governing equations
Gauss's law for the magnetic flux density is expressed as ! ! ! B ¼ μ0 ðM þ H Þ; ð1Þ ! and Ampere's law for the magnetic field, H with zero current density is written as !! ∇ : B ¼ 0: ð2Þ It is assumed that the fluid is non-conducting. Hence, the electromagnetic current resulted from ferrofluid flow in the magnetic field can be neglected. Then, -
∇ H ¼ 0:
Thus, the magnetic field can be obtained from [2] ! H ¼ ∇V m ;
ð3Þ
ð4Þ
where V m is the scalar magnetic potential. For a line dipole it can be expressed as [23] m xX V m ðx; yÞ ¼ ; 2π ðx X Þ2 þ ðy Y Þ2
ð5Þ
where m represents the strength of the magnetic field and X and Y are the locations of a single line dipole. Then, H is obtained in polar coordinates as ! ! ! m 1 U ð cos ð2ϕÞ i þ sin ð2ϕÞ j Þ; H ðr; ϕÞ ¼ 2π r2 where r and ϕ are defined in the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ ðx XÞ2 þ ðy YÞ2 ;
χ m ¼ χ m ðTÞ ¼
χ
0 ; 1 þ βðT T 0 Þ
ð10Þ
Where, χ 0 is the susceptibility at a reference temperature. The corresponding Kelvin body force per volume for a ferrofluid is defined as [2,33] ! ! F K ¼ M :∇B : ð11Þ 3.2. Fluid flow and heat transfer equations The governing equations for conservation of mass, momentum in x, y directions (Navier–Stockes equations) and energy are given by equations:
3.1. Magnetic body force
-
The variation of the total magnetic susceptibility, χ m as function of temperature is of the form [34]
ð6Þ
∂u ∂v þ ¼0; ∂x ∂y
ð12Þ
∂u ∂u ∂u 1 ∂P μ ∂2 u ∂2 u þu þv ¼ U þ þ F K ðxÞ; þ ∂t ∂x ∂y ρ ∂x ρ ∂x2 ∂y2
ð13Þ
∂v ∂v ∂v 1 ∂P μ ∂2 v ∂2 v þu þv ¼ U þ þ F K ðyÞ; þ ∂t ∂x ∂y ρ ∂y ρ ∂x2 ∂y2
ð14Þ
2 ∂T ∂T ∂T ∂ T ∂2 T þ u þv ¼ α þ 2 þ μΘ; 2 ∂t ∂x ∂y ∂x ∂y
ð15Þ
In Eq. 15, Θ is the viscous dissipation and is of the form: 2 2 ∂u ∂v ∂v ∂u 2 þ Θ¼2 þ2 þ ∂x ∂y ∂x ∂y
4. Thermophysical properties of nanofluids The effective thermophysical properties of the ferrofluid used in the governing momentum and energy equations can be evaluated by using different relations from the literature. The following equations have been used for calculation of nanofluid bulk density and specific heat [35]:
ρnf ¼ φρp þ ð1 φÞρf cp;nf ¼
ð7Þ
yY ; ð8Þ ϕ ¼ tan 1 xX ! ! where i and j correspond to the unit vectors in x and y directions, respectively. Also, the relation between the magnetic field and magnetization vector is as follows: ! ! M ¼ χm H : ð9Þ
ð16Þ
φρp cp;p þ ð1 φÞρf cp;f ρnf
ð17Þ ð18Þ
where, ρp is the particle density, ρf is the base fluid density and cp;p and cp;f are the particle and the base fluid specific heats, respectively. The viscosity of the ferrofluid at different concentrations has been estimated by the Einstein equation as
μnf ¼ μf ð1 þ 2:5φÞ
ð19Þ
It is shown in Ref. [36] that Eq.19 is reliable with the experimental viscosity measurements. Furthermore, the thermal conductivity of the ferrofluid is calculated from the following equation, as
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83
Table1 Thermophysical properties of the magnetite nanoparticle, DI-water and the ferrofluid. Density (kg=m3 Þ
Magnetite nanoparticle (Fe3 O4 ) DI-Water Ferrofluid (2 vol%)
Specific heat (J/kgK)
4950
640
997 1076
4180 3854
recommended by Ref. [20] 2kf þ kp þ 2φðkp kf Þ knf ¼ kf 2kf þ kp þ 2φðkp kf Þ
Thermal conductivity (W/mK)
Viscosity (Ns/m2 )
7
0.61 0.64
0.00065 0.00069
ð20Þ
where, kf and kp are conductivities of the base fluid and particle, respectively. A magnetite water based ferrofluid with a volume fraction of 2% is used in the numerical simulations. The thermophysical properties of magnetite nanoparticles, DI-water and ferrofluid are presented in Table 1.
5. Numerical solution The set of differential Eqs. (12–16) have been discretized with the control volume technique. For the convective and diffusive terms a second order upwind method has been used while the SIMPLEC procedure has been introduced for the velocity–pressure coupling. A structured non-uniform grid has been used for the computational domain. It is finer near the channel entrance and near the top and bottom surfaces where the velocity and temperature gradients are large. In order to ensure that the calculations are accurate and grid independent, several combination of node numbers in longitudinal and lateral directions have been examined. Temperature and axial velocity profiles at the channel exit have been calculated for all the cases. Results are plotted in Fig. 2. As shown, these grid distributions give almost identical results. The 50–1000 grid (50 nodes in vertical and 1000 nodes in horizontal direction) has been used in the calculations. In order to demonstrate the validity and also precision of the numerical procedure, a comparison has been made with the previously published work in the field of ferrohydrodynamics. Fig. 3 indicates the variation of Nusselt number in the channel and its comparison with the results of Ganguly et al. [3] for two different values of magnetic field strength, m1 ¼ 0:19 and m2 ¼ 0:58 Am. As shown, the code has predicted the magnetic field effect appropriately.
Fig. 2. Grid independency test, (a) temperature and (b) axial velocity at the channel exit.
6. Results and discussion A water based ferrofluid containing 2 vol% Fe3 O4 particles flows into the channel with temperatre of 340 K and is heated from the top and bottom surfaces at a uniform heat flux. Passing through the channel, the ferrofluid is influenced by an alternating nonuniform magnetic field produced by eight identical line source dipoles. Fig. 4(a) shows the magnetic flux density distribution calculated from Eqs.1–10 for one of the dipoles located at x ¼0.1 m, beneath the bottom surface. An alternating magnetic field is obtained by applying a rectangular wave function to the magnetic dipoles, as shown in Fig. 4(b). The period of the alternating magnetic field function, T,
Fig. 3. Comparison between calculated Nusselt number and results of the previous work [3].
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Fig. 4. (a) Magnetic flux density distribution for the dipole located at x¼ 0.1, (b) rectangular pulse wave applied to the magnetic dipoles.
is defined as the sum of connection and disconnection times. Also, frequency, f, is the reciprocal of the period of the wave function. Simulations are carried out for four different Reynolds numbers of Re¼100, 600, 1200 and 2000, four magnetic field intensities of m ¼ 0:1; 0:15; 0:2; 0:25 Am and the frequencies of f ¼ 1; 2:5; 5; 6:67; 10; 20 Hz. Moreover, the top dipoles have a phase shift of pi with respect to the bottom dipoles. Temporal variations of the streamlines and the temperature distribution for the magnetic field function period of T ¼1 s (equal to f ¼ 1 Hz), is illustrated at three instants namely,t 1 ¼ 12 s, t 2 ¼ 12:25 s and t 3 ¼ 12:5 s in Fig. 5. The sections A and B are in the vicinity of two adjacent magnetic dipoles located at x ¼0.37 m below and x ¼0.415 top of the channel. Since the alternating magnetic field functions of the top and bottom dipoles have a phase difference of pi, completely different temperature distribution and flow patterns are observed in parts A and B, at each instant. Mechanism of heat transfer enhancement can be inferred from Fig. 5. During each period of magnetic field functioning, first, the bottom dipoles are turned on and the colder fluid is attracted to the heated bottom surface from center of the channel. Thus, the heat can be transferred to the fluid. Next, the fluid is released to the bulk flow at the disconnection time of botton dipole. Finally, it is attracted by the top dipole as it turns on. The fluid attraction and releasing occur along the channel repeatedly. Therefore, the fluid particles are expected to travel in a sinusoidal path, resulting in better mixing, the thermal boundary layer disturbance and enhancement of the forced convection heat transfer. Another point worth mentioning is the formation of two recirculation zones upstream and downstream of the dipoles. This is due to the temperature dependence of the magnetic susceptibility and resultant non-symmetrical magnetic body forces [3]. The vortices have great influence on the thermal boundary layer thickness. Thickening of the thermal boundary layer is observed in locations where direction of the flow in the vortex is along the
Fig. 5. Variation of streamlines and temperature distribution around two adjacent magnetic dipoles with time, t1 ¼ 12 s, t2 ¼ 12.25 s and t3 ¼12.5 s.
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Fig. 6. Variation of local heat transfer coefficient on the (a)bottom surface and (b) top surface at the time sequence of t1 ¼12 s, t2 ¼12.25 s and t3 ¼12.5 s.
direction of the mean flow. By contrast, thermal boundary layer shrinks in the locations where the mean flow is in opposite direction of flow in the vortex region. These two phenomena result in decrease and increase of heat transfer rate, respectively. However, the second effect is dominant and heat transfer is expected to increase due to the magnetic field. In order to quantify the above discussion, variation of local heat transfer coefficient on the bottom and top surfaces of the channel are depicted in Figs. 6 and 7(a) and (b), respectively. As shown in Fig. 6(a) and (b), four dipoles near the top and bottom surfaces observed in Fig. 5 have led to four large peaks. Four small peaks are also seen on the curves which are due to the manipulation of fluid flow by the line dipoles near the opposite surface. Since the magnetic field of the bottom surface turns on at t 1 , the local heat transfer coefficient increases with the elapse of time while the trend is opposite for the top dipole, due to the phase shift of pi, as expected. The insets in Fig. 6 which magnify the regions around the dipoles clarify the significant role of downstream vortices in heat transfer enhancement. Growth of the downstream vortex at t 3 , as shown in part A of Fig. 5 has decreased the thermal boundary layer thickness and increased local heat transfer coefficient as a result. Moreover, slight decreases in the local heat transfer coefficient are observed when the magnetic field is turned on. This can be due to thickening of the thermal boundary layer as mentioned before. However, the local heat transfer coefficient increases sharply when the boundary layer thickness decreases.
Fig. 7. Axial variation of velocity profile at (a) t1 ¼12 s, (b) t2 ¼ 12.25 s and (c) t3 ¼ 12.5 s.
It is observed that when the ferrofluid passes over the section of magnetic field it accelerates noticeably along the surface due to the magnetic force. This is another reason for sharp increase of the local heat transfer coefficient at magnetic field locations. Moving a
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Fig. 9. Effect of magnetic field intensity on the forced convective heat transfer, m1 ¼ 0.1, m2 ¼0.15, m3 ¼ 0.2 and m4 ¼ 0.25 Am.
Fig. 8. Axial variation of temperature profile at (a) t1 ¼12 s, (b) t2 ¼12.25 s and (c) t3 ¼12.5 s.
little away from the dipoles, axial velocity reaches negative values near the surface, which implies the change of flow direction in the larger recirculation zone. Additionally, the magnetic field gradients
are weak and far from the dipoles and the velocity profiles almost merge. Axial variation of the temperature profile at the same time sequence is plotted in Fig. 8(a)–(c). Cooling effect of the ferrofluid is obvious near the dipoles. The effect of the magnetic field strength on forced convection heat transfer is demonstrated in Fig. 9. It shows the variation of the Nusselt number ratio, Nu=Nuref , on the bottom surface with time for different values of magnetic field strength, m1 ¼ 0:1, m2 ¼ 0:15, m3 ¼ 0:2 and m4 ¼ 0:25 Am, at Re ¼600 and f¼ 10. Nu and Nuref are the average Nusselt number on the surface in the presence and the absence of the magnetic field, respectively. It can be seen that forced convection heat transfer has a direct relation with magnetic field strength. Since the Kelvin magnetic force increases with the magnetic field strength, more colder fluid is attracted to the heated surface. Therefore, heat is transferred more effectively from the heated surfaces to the fluid. Fig. 10 illustrates the dependence of the convective heat trasnfer on the frequency of the alternating magnetic field. Variations of the Nusselt number ratio and the average surface temperature of the bottom surface with time at Re ¼600 and m ¼0.2 Am have been depicted for different frequencies in Fig. 10 (a) and (b), respectively. As shown, the steady state has been reached after elapse of 12 sec. The average surface temperature and Nusselt number ratio are both periodic functions of time with the same frequency of the external magnetic field. Although the alternating magnetic field has led to increase of convective heat transfer for all the values of frequencies, magnitude of enhancement depends strongly on the frequency of the alternating magnetic field. For this case (Re¼600 and m ¼ 0.2), the maximum heat transfer of 8.5% has been obtained at f 4 ¼ 5 Hz. The Nusselt number ratios on the botom and top surfaces as a function of magnetic field intensity and frequency are shown in Fig. 11(a) and (b) for Re¼600, respectively. It is apparent from the figure that the optimum frequency is f¼ 5 Hz at Re¼600 and is independent of the magnetic field intensity. It is also shown that the convective heat transfer is slightly greater for the bottom surface. For the constant heat flux boundary condition, the main goal of the cooling system is to lower the temperature of the heated surface as much as possible. Therefore, the mean temperature of
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Fig. 10. Variation of (a) the Nusselt number ratio and (b) average surface temperature of the bottom surface with time at Re¼ 600, m¼ 0.2 Am for f1 ¼20 Hz, f2 ¼ 10 Hz, f3 ¼ 66 Hz, f4 ¼5 Hz, f5 ¼ 2.5 Hz, and f6 ¼1 Hz.
Fig. 11. Variation of heat transfer enhancement with the frequency for different values of magnetic field strength for (a) bottom surface and (b) top surface.
the heated surface can be a measure of efficiency of the system. Values of the mean temperature on the heated surface have been calculated for all the cases. Result for the bottom surface at Re¼600 is shown in Fig. 12. As shown, frequency of the magnetic field is a key parameter by which surface temperature can be controlled Variation of the Nusselt number ratio on the bottom surface with time for different frequencies at Re¼ 2000 and m ¼ 0.2, is shown in Fig. 13. In contrast to the case of Re¼600, it can be observed that the forced convective heat transfer increases with the frequency. Furthermore, a maximum heat transfer of 13.9% is obtained at f 1 ¼ 20 Hz. In order to show the dependency of the optimum frequency on the Reynolds number, the heat transfer enhancement is plotted as a function of frequency in Fig. 14 for different Reynolds numbers and the magnetic field intensity of m ¼ 0.2. Clearly, the heat transfer enhancement increases with the Reynolds number and reaches the maximum value of 13.9% at
Re¼2000 and f 1 ¼ 20 Hz. It is also observed that an optimum frequency exists which increases with the Reynolds number. With Reynolds number increase, the ferrofluid passes over the magnetic dipoles with a higher velocity and the time that ferrofluid particles are influenced by the magnetic field is decreased. Thus, faster attracting and releasing of the particles is required to disrupt the thermal boundary layer effectively which means to increase the frequency of the magnetic field function. Another important parameter in the application of nanofluids in a heat exchanging equipment is the pressure drop. In such systems, there is a trade-off between pressure drop and heat transfer enhancement. However, for the ferrofluid flow in the presence of a magnetic field, the increase of pressure drop is not as significant as heat transfer enhancement [37]. Pressure drop of the ferrofluid across the channel has been calculated for all studied cases. Pressure drop ratio (ΔP=ΔP ref ) as a function of magnetic field frequency and strength for Re ¼2000 has been depicted in Fig. 15. Where ΔP ref is the pressure drop in the absence of the magnetic field.
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Fig. 12. Variation of average bottom surface temperature with frequency for different magnetic field strengths.
Fig. 14. Variation of the heat transfer enhancement with frequency for different Reynolds numbers at m¼0.2.
Fig. 13. Variation of the Nusselt number ratio with time at Re ¼ 2000 and m¼ 0.2 Am for f1 ¼ 20 Hz, f2 ¼10 Hz, f3 ¼67 Hz, f4 ¼5 Hz, f5 ¼ 2.5 Hz, f6 ¼ 1 Hz.
Fig. 15. Variation of pressure drop increase with the frequency at Re¼ 100 and 2000 for m¼ 0.2.
As shown, the pressure drop increases with magnetic field strength. However, the maximum pressure drop ratio is 6% for m¼0.25. Hence, the heat transfer enhancement due to the magnetic field dominates pressure drop increase.
to the case with no magnetic field. A maximum of 13.9% enhanement has been obtained at Re ¼ 2000 and f ¼20 Hz. Periodic attraction of the cold fluid to the heated surface and release of it to the bulk flow disturbs the thermal boundary layer, improves the flow mixing and increases heat transfer as a result. Magnetic force causes the ferrofluid to accelerate noticeably along the surface when it passes over the section of magnetic field. This is another reason for a sharp increase of the local heat transfer coefficient at magnetic field locations. Heat transfer enhancement increases with the Reynolds number and the intensity of the magnetic field. Frequency of the alternating magnetic field has an important role on the heat transfer. Results show that an optimum value exists for the frequency. It is independent of the magnetic field intensity but increases with the Reynolds number. The alternating magnetic field causes the increase of the pressure drop up to a maximum of 6% at Re¼2000 and f¼5 Hz. However, it is not as significant as the heat transfer enhancement.
7. Concluding Remarks Laminar forced convection heat transfer of a ferrofluid in a channel subjected to a uniform heat flux from top and bottom surfaces has been studied numerically. The possibility of heat transfer enhancement utilizing an alternating magnetic field by eight line source dipoles has been investigated at different Reynolds numbers, magnetic field intensities and frequencies. The following results have been obtained.
Forced convection heat transfer of ferrofluid can be enhanced noticably by applying an alternating magnetic field compared
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