- Email: [email protected]
thermocapillary convection
International Communications in Heat and Mass Transfer 34 (2007) 523 – 533 www.elsevier.com/locate/ichmt
Effect of magnetic field on two-layered natural/thermocapillary convection☆ D. Ludovisi a , S.S. Cha a,⁎, N. Ramachandran b , W.M. Worek a a
Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Room 2039 ERF (M/C 251), Chicago, Illinois 60607, USA b
Jacobs ESTS Group, ER42, NASA Marshall Space Flight Center, Huntsville, AL 35812, USA Available online 30 March 2007
Abstract Heat transfer in a two-layered fluid system is of great importance in a variety of applications. Control and optimization of convective heat transfer of the immiscible fluids needs complete understanding of all phenomena, especially those induced by surface tension at the fluid interface. The present work is focused on rather complex convective flow and heat transfer phenomena in a cavity, which can be subject to both buoyancy and thermocapillary effects in addition to the influence of magnetic field applied for flow control. With the encapsulant liquid posing magnetic properties, a magnetic force can arise to either enhance or counterbalance the gravity effect when the cavity is placed in a non-uniform magnetic field. In our study, the velocity and temperature distribution of the system can be significantly altered to change the heat transfer by varying intensity and gradient of the applied magnetic field. Preliminary results of numerical computation presented here are for a two-layered liquid cavity MnCl2·4H2O and Fluorinert FC40 under various magnetic fields intensities. © 2007 Published by Elsevier Ltd. Keywords: Marangoni flow; Thermocapillary flow; Interfacial tension; Magnetic field; Two-layered fluid system
1. Introduction The heat transfer study of a double-layered fluid system is of great importance in many engineering applications, which can include metal casting, crystal growth, solar panel manufacturing and heat exchangers with air pockets. In these types of flow, a non-uniform temperature distribution at the fluid interface induces interfacial tension gradients that, in turn, provides a driving force for thermocapillary convection also known as Marangoni convection. The phenomena can cause adverse effects, i.e., imperfections in crystal growth, and thus have attracted interest of various investigators [1–13]. An approach to minimize or enhance the thermocapillary effect is to choose a liquid encapsulant with appropriate thermo-physical and chemical properties [14].
☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (S.S. Cha).
0735-1933/$ - see front matter © 2007 Published by Elsevier Ltd. doi:10.1016/j.icheatmasstransfer.2007.02.003
524
D. Ludovisi et al. / International Communications in Heat and Mass Transfer 34 (2007) 523–533
Nomenclature B
magnetic field [T]
Ca
lU = Capillary number r
cp fmagnetic H k L Ma Y n Nu p Ra Y t T Tcold Thot u U v Y v I II
specific heat [J/kg K] magnetic body force per unit mass [N m3/kg] interface curvature [1/m] thermal conductivity [W/m K] 2 cm = cavity width σ TΔTh / μα = Marangoni number unit vector normal to the interface L AT j Thot −Tcold Ax x¼0 or x¼2 cm = local Nussel number pressure [Pa] βΔTgh3 / να = Rayleigh number unit vector tangent to the interface temperature [K] 298 K = temperature of cold wall 302 K = temperature of hot wall velocity in horizontal direction (x-axis) [m/s] characteristic velocity of the system under investigation [m/s] velocity in vertical direction (y-axis) [m/s] velocity vector [u, v] MnCl2·4H2O Fluorinert FC40
Greek symbols α thermal diffusivity [m2/s] β thermal expansion coefficient [1/K] χ magnetic susceptibility μ0 magnetic permeability of free space [N/A2] μ dynamic viscosity [kg/ms] ν kinetic viscosity [m2/s] ϕ Girifalco/good empirical parameter to estimate interfacial tension [23] ρ density [kg/m3] σ interfacial tension [N/m] σI MnCl2·4H2O surface tension [N/m] σII Fluorinert FC40 surface tension [N/m] σe electric conductivity [1/Ω m] θ interface contact angle with side walls [°] τ¯ stress tensor
For the combined buoyancy and Marangoni convection, the two-layered system poses much more complex and interesting behaviors than the one in a single-layer system due to a strong interaction of two fluid motions. Sparrow et al. [15] performed systematic experiments in a square cavity by using hexadecane overlying water. The numerical investigations by Kimura et al. [16], Myrum et al. [17] and Ramachandran [5] demonstrated that the flow patterns can be altered significantly when the interfacial force becomes comparable to the buoyancy force in a two-layered system.
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525
At now, the application of a magnetic field has been motivated by an interest to further control the flow in a two-layered system [18–21]. As examples, the flow patterns and heat transfer characteristics can be manipulated in devices for material processing and energy utilization. Typically, in crystal growth a magnetic field can be applied to damp the flow in growing single crystals. An option with a fluid of significant electrical conductivity is to exploit Lorentz's force for flow damping. However, the effect of Lorentz's force diminishes with the decrease of convective motion. The option presented here is of a different origin, not being based on the electrical conductivity, to have complete flow control with any flow motion. 2. Physical setup and governing equations Fig. 1 shows a schematic of the physical setup adopted in our initial investigation. The system consists of two immiscible fluids: Fluorinert FC40 and 30% aqueous solution of MnCl2·4H2O by weight. The fluids are contained in a two-dimensional closed cavity with a rectangular cross-section of 2 cm by 1 cm. The volumes of both fluids are equal, with the lighter fluid at the top, thus forming a stable system. A temperature gradient normal to the gravity is imposed on the system by maintaining the vertical side walls at two different isothermal conditions. The top and bottom faces of the cavity are assumed to be adiabatic. The interface between the two fluids is deformable but this initial study, as good approximation, considers only undeformable interface. For the current system of Boussinesq Newtonian fluids in a laminar regime, the governing equations in Cartesian coordinates can be as follows. Continuity: ux þ vy ¼ 0
ð1Þ
Momentum: uux þ vuy ¼ −
px þ mðuxx þ uyy Þ q
ð2Þ
uvx þ vvy ¼ −
pv þ mðvxx þ vyy Þ þ gð1 þ bDT Þ þ fmagnetic q
ð3Þ
Energy: uTx þ vTy ¼ aðTxx þ Tyy Þ
ð4Þ
Here the suffixes denote differentiation with respect to the indicated variables. It should be noted that the governing equations above are for steady state and applicable to each layer of fluids. The boundary conditions are as below. u¼v¼0 u¼v¼0
at at
Ty ¼ 0
y ¼ 0; y ¼ 1 cm
at
x ¼ 0; x ¼ 2 cm 8y y ¼ 0; y ¼ 1 cm 8x
T ¼ Thot ¼ 302 K T ¼ Tcold ¼ 298 K
at at
x¼0 x ¼ 2 cm
ð5Þ ð6Þ ð7Þ ð8Þ
For the interaction of the two fluids at the interface, let the region for MnCl2·4H2O be denoted by I while the one for Fluorinert by II as shown in Fig. 2. The following conditions can then be imposed at the interface. No slip condition: Y vI d Y t¼Y v II d Y t
ð9Þ
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Fig. 1. Schematic of the physical setup.
Kinematic condition: Y n¼Y v II d Y n¼0 vI d Y
ð10Þ
Interface force balance: Y n d ½ ¯s I ¯s II ¼ 2Hr Y n−js r
ð11Þ
Contact angles: h ¼ 90-
ð12Þ
Continuity of temperature: TI ¼ TII
ð13Þ
Continuity of heat flux in the direction normal to the interface: n ÞII ðkjT d Y n ÞI ¼ ðkjT d Y
ð14Þ
Here τ¯ and H represent stress tensor and the mean interface curvature, respectively, ∇s denotes the surface gradient operator that can be written as j− Y nð Y ndjÞ. Since the surface tension strongly depends on the temperature only we can simplify the surface gradient operator in Eq. (11) as follows: js r ¼ rT js T
ð15Þ
With the capillary number Ca of our system fairly small, Eq. (10) together with the tangential component of Eq. (11) is used as boundary conditions for the momentum equations. The normal stress balance, that is, the
Fig. 2. Interfacial unit tangent and normal vectors and liquid phases notation.
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Table 1 Fluid properties: all values are at the average temperature of 300 K (otherwise specified) Property 3
Density (ρ) [kg/m ] Kinetic viscosity (ν) [m2/s] Specific heat (cp) [J/kg K] Thermal expansion coefficient (β) [1/K] Thermal conductivity (k) [W/m K] Dynamic viscosity (μ) [kg/ms] Thermal diffusivity (α) [m2/s] Surface tension at 25 °C (σ) [N/m] Surface tension gradient at 25 °C (σ) [N/m K] Electric conductivity (σe) [1/Ω m] Magnetic susceptibility (χ) [1/Ω m]
MnCl2·4H2O 30% in weight
Fluorinert FC40
1118.25 9.826 · 10− 7 4152.0 1.855 · 10− 4 0.6 1.003 · 10− 3 2.911 · 10− 7 0.0728 − 1.6 · 10− 4 36 5.738 · 10− 6 + 0.05922 / T (K)
1870.0 2.2 · 10− 6 1046.5 1.20 · 10− 3 0.067 4.114 · 10− 3 3.242 · 10− 8 0.016 − 1.22 · 10− 4 0.25 · 10− 13 − 8.273 · 10− 6
normal component of Eq. (11), would be utilized as a distinguished condition for finding the unknown interface location. The magnetic body force per unit mass, experienced by a paramagnetic material, is fmagnetic ¼
vðT Þ jðB2 Þ 2l0 qðT Þ
ð16Þ
where χ(T) and ρ(T) are magnetic susceptibility and density of the fluid, which are functions of temperature, while μ0 is the permeability of free space. If the susceptibility varies with temperature, as in paramagnetic materials, the body force changes with temperature, analogous to the gravitational buoyancy force when the density varies with temperature. To avoid confusion, these two effects will be distinguished as “magnetic” and “gravitational” buoyancy, respectively. By aligning the magnetic field gradient vertically, the magnetic buoyancy can either enhance or counterbalance the gravitational buoyancy. The direction of the net force can even be reversed at some critical value of the magnetic field gradient [22]. The properties of the fluids, which have been employed for our numerical modeling, are assumed to be constant as listed in Table 1. Some properties of the aqueous solution are not readily available in literature, which include viscosity, thermal conductivity, and surface tension. For these, properties of water are assumed. For our two-layered system, the interfacial tension was estimated by using the approach proposed by Girifalco and Good [23]: pffiffiffiffiffiffiffiffiffiffi r ¼ rI þ rII −2/ rI rII ð17Þ This method expressed by Eq. (17) requires the knowledge of an empirical parameter ϕ that depends on the nature of chemical bonds between the two liquids. Due to the composition of MnCl2·4H2O and Fluorinert, ϕ is estimated to
Fig. 3. Stream function in the absence of both the interfacial tension and magnetic field.
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Fig. 4. Stream function with the interfacial tension but without the magnetic field.
be 0.55 [23], which gives an interfacial tension temperature gradient equal to − 1.037 · 10− 4 [N/m K]. However, ϕ greatly affects the results of our numerical investigation and thus it needs to be verified experimentally. In the current study, the effects of Lorentz's force are neglected since its order of magnitude is at least three times lower than either the magnetic body force or the gravitational buoyancy when calculated with the same cavity conditions. The magnetic body force in Eq. (3) is only significant in the solution of MnCl2·4H2O. The variation of the magnetic susceptibility of Fluorinert with respect to temperature is also unavailable in literature: however, it is believed to be negligible for the temperature range of our investigation. 3. Results and discussion The governing equations with the corresponding boundary conditions, Eqs. (1)–(17), are solved with the commercial finite element code FIDAP [24]. Convergence is assumed when the residual of governing equations is less than 10− 4. In an effort to obtain a greater resolution near the walls and interface, where higher velocity and temperature gradients are expected, a non-uniform grid is employed with smaller elements close to the walls and the interface. Systematic evaluation has been made to establish gridconvergence on numerical results. For the present study, a grid with 160 × 80 elements proves to be optimal in terms of accuracy and computation time. As a first step in numerical calculation, a simpler problem of natural convection without the interfacial tension and magnetic field has been calculated for comparison. The steady-state stream function obtained from the numerical modeling is shown in Fig. 3. As expected, each fluid has a large recirculating region. These two cells are driven in the same directions by thermal buoyancy to cause conflicting driving forces in the vicinity of the interface. Fluorinert, which is stronger in buoyancy, creates a small counterclockwise
Fig. 5. Stream function in the presence of both the interfacial tension and magnetic field of ∇(B2) = 50 T2/m.
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Fig. 6. Stream function in the presence of the interfacial tension and a magnetic field of ∇(B2) = − 50 T2/m.
intermediate cell in MnCl2·4H2O. For the current heating conditions, the thermal buoyancy of Fluorinert is about 7 times stronger than that of MnCl2·4H2O due to difference in thermal expansion coefficients. Fig. 4 shows the stream function when the interfacial tension is present between the two fluids. The flow field exhibits the dominating nature of the Marangoni convection that counterbalances the thermal buoyancy effect in MnCl2·4H2O. The interfacial surface tension works in a concerted manner with the lower Fluorinert cell to expand its size. The upper large cell in MnCl2·4H2O is now reduced to two small cells at the upper corners of the cavity. Figs. 5 and 6) show the effect of the magnetic field on the two-fluid system. The magnetic field gradient points in the vertical direction. In the case of Fig. 5, the magnetic force in MnCl2·4H2O is large enough to suppress the thermal buoyancy effect and thus to further promote the Marangoni convection effect. The two cells of the upper corners in Fig. 4 completely disappear. Only two large cells thus circulate in opposite direction. In the case of Fig. 6, the magnetic body force acts in the opposite direction with respect to the Marangoni force at the interface. The counterclockwise cell that has been produced in MnCl2·4H2O by the interfacial tension in Fig. 5 is reduced in size and a large clockwise cell newly forms on top of it. The magnetic force this time is in the direction of buoyancy force and thus the appearance of the large clockwise cell is as expected. The flow pattern becomes very similar to Fig. 3. The overall effect of the magnetic force is transmitted to the Fluorinert layer to slow down the velocity in turn. However, as stated before, the Marangoni convection can play such an important role in a two-fluid layered system that the attempt to completely cancel its effect could be a challenge in fluid flow control. In fact, as observed in Fig. 6,
Fig. 7. Magnitude of the horizontal velocity component along the cavity vertical centerline for various magnetic field intensities.
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Fig. 8. Magnitude of the horizontal velocity component along the interface for various magnetic field intensities.
the intermediate cell in MnCl2·4H2O produced by the interfacial tension is still relatively large and the flow field in Fluorinert appears to be not greatly affected. Figs. 7 and 8 are the plots of the horizontal velocity profiles at the vertical centerline of the cavity and at the interface between the two fluids, respectively, under various magnetic field gradients. As seen, positive magnetic field gradients act in concert with the effect of the Marangoni convection in MnCl2·4H2O. At some typical locations, interfacial velocities are doubled. The opposite occurs when negative magnetic field gradients are applied as stated earlier, to make the fluid flow completely reversed for a strong enough magnetic field. The velocity in Fluorinert is less affected at the centerline, as seen in Fig. 7, except along the interface. This appears to be due to the dominant buoyancy force and ineffectiveness of magnetic force in the Fluorinert region. However, this region is more significantly influenced close to the cold wall, where the velocity peak can be either increased or reduced by 50% for certain values of magnetic field intensity as seen in Fig. 8. It is worth noting that, for the conditions under analysis, our two-layered fluid system works in a manner similar to a gear box, with the magnetic buoyancy cell as a “driving gear”. As seen in Figs. 5–7, when the magnetic effect enhances the
Fig. 9. Temperature distribution along the vertical centerline for various magnetic field intensities.
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Fig. 10. Temperature distribution at the interface between the two fluids for various magnetic field intensities.
velocity of MnCl2·4H2O, the velocity of Fluorinert increases. On the contrary, when MnCl2·4H2O slows down, it has a reverse effect. Figs. 9 and 10 show the distributions of temperature at the vertical centerline and at the interface of the cavity under the influence of magnetic field. Similarly to the behavior of horizontal velocity, the effect of the magnetic field is larger in the MnCl2·4H2O layer and its influence is transmitted to the Fluorinert region again to a lesser degree through the interface where the Marangoni convection dominates. However, the change in temperature near the cold wall is more prominent than the change in velocity. Any variation of the temperature in the vicinity of the two isothermal walls can induce a significant change in heat transfer of the system. That is, even if the temperature might not appreciably differ in net value, its derivative can greatly change to influence the heat transfer. Figs. 11 and 12 show the local Nusselt number along the hot and cold walls. As seen, the local Nusselt number of the hot wall can either decrease or increase by 14% for Fluorinert as compared to the case without a magnetic field. This results in a change of up to 28% in the global value for the range of the tested magnetic field intensity. In MnCl2·4H2O, the change in the local Nusselt number is more pronounced. At the cold wall, the local Nusselt number exhibits a greater change in proximity of the interface, where the value can undergo a variation up to 150% for some magnetic field gradients.
Fig. 11. Local Nusselt number along the hot wall.
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Fig. 12. Local Nusselt number variation along the cold wall.
Finally, Fig. 13 shows the variation of the global heat transfer for the two-layer fluid system. As seen, if a proper magnetic field gradient is applied, the global heat transfer can be either increased or decreased by 100%. This proves the concept that motivated our current study, that is, the application of a magnetic field for fluid flow and heat transfer control.
4. Conclusions The interaction of the Marangoni convection, magnetic body force and thermal buoyancy in a system of two immiscible-fluid layers have been investigated and preliminary results are presented. The flow field and heat transfer for the system has been computed by using a commercial finite element code under various magnetic field gradients. Our investigation reveals that the Marangoni convection can significantly influence the velocity distribution in the cavity. It also shows that introduction of a magnetic field can easily promote or hinder the Marangoni effect, that is, to
Fig. 13. Global heat transfer through the entire cavity under various magnetic field gradient intensities.
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further increase or decrease global convective motion. The effect of the magnetic body force of the upper layer has a difficulty to penetrate through the interface of the two fluids. This appears to be due to the dominant flow field pattern induced by the interfacial tension and the thermal buoyancy of the bottom layer. The difficulty allows the general velocity pattern to be preserved for the bottom layer, except the region close to the cold wall. However, the velocity and temperature fields in the upper paramagnetic fluid can be significantly affected, especially far from the interfacial region. It has been shown that even a small change in the velocity field can produce appreciable changes in temperature distribution, which in turn greatly affect the local Nusselt number and thus the global heat transfer. For the range of magnetic field gradients tested for this study, the heat transfer can be either enhanced or reduced by 100%. References [1] W. Pfann, Zone Melting, second ed.McGraw-Hill, New York, 1966. [2] J. Szekely, M.R. Todd, Natural convection in a rectangular cavity transient behavior and two phase systems in laminar flow, Int. J. Heat Mass Transfer 14 (1971) 467–482. [3] R.W. Knight, M.E. Palmer, Simulation of free convection in multiple fluid layers in an enclosure by finite differences, Numerical Properties and Methodologies in Heat Transfer, Hemisphere, Washington D.C., 1983. [4] T. Doi, J.N. Koster, Thermocapillary convection in a two immiscible liquid layers with free surface, Phys. Fluids, A Fluid Dyn. 5 (1993) 1914–1927. [5] N. Ramachandran, Thermal buoyancy and Marangoni convection in a two fluid layered system, J. Thermophys. Heat Transf. 7 (2) (1993) 352–360. [6] P. Wang, R. Kahawita, Nguyen, Numerical simulation of Buoyancy–Marangoni convection in two superposed immiscible liquid layers with a free surface, Int. J. Heat Mass Transfer 37 (7) (1994) 1111–1122. [7] Q.S. Liu, B. Roux, M.G. Verlarde, Thermocapillary convection in two-layer systems, Int. J. Heat Mass Transfer 41 (11) (1998) 1499–1511. [8] M.Z. Saghir, Convection in two-liquid-layer system with low Prandtl number, Energy Sources 21 (1999) 139–144. [9] M. Hamed, J.M. Floryan, Marangoni convection — Part 1: a cavity with differentially heated sidewalls, J. Fluid Mech. 405 (2000) 79–110. [10] S. Someya, T. Munakata, M. Nishio, K. Okamoto, H. Madarame, Flow observation in two immiscible liquid layers subject to a horizontal temperature gradient, J. Cryst. Growth 235 (2002) 626–632. [11] N.R. Gupta, H. Haj-Hariri, A. Borhan, Thermocapillary flow in double-layer fluid structures: an effective single-layer model, Journal of Colloid and Interface Science 293 (1) (2006) 158–171. [12] Q.S. Liu, G. Chen, B. Roux, Thermogravitational and thermocapillary convection in a cavity containing two superposed immiscible liquid layers, Int. J. Heat Mass Transfer 36 (1) (1993) 101–117. [13] N.R. Gupta, H. Haj-Hariri, A. Borhan, Thermocapillary flow in double-layer fluid structures: an effective single-layer model, J. Colloid Interface Sci. 293 (1) (2006) 158–171. [14] J.J. Bikerman, Physical Surfaces, Academic Press, New York, 1970. [15] E.M. Sparrow, L.F.A. Azevedo, A.T. Prata, Two-fluid and single fluid natural convection heat transfer in an enclosure, ASME J. Heat Transfer 108 (1986) 848–852. [16] T. Kimura, N. Heya, M. Takeuchi, H. Isomi, Natural convection heat transfer phenomena in an enclosure filled with two stratify liquids, JSME Int. J. Ser. B Fluids Therm. Eng. 52 (1986) 617–625 (in Japanese). [17] T.A. Myrum, E.M. Sparrow, A.T. Prata, Numerical solutions for natural convection in a complex enclosed space containing either air–liquid or liquid–liquid layers, Numer. Heat Transf. 10 (1986) 19–43. [18] P. Wang, R. Kahawita, Transient buoyancy–thermocapillary convection in two superposed immiscible liquid layers, Numer. Heat Transf., A Appl. 30 (5) (1996) 477–501. [19] P. Wang, R. Kahawita, Oscillatory behaviour in buoyant thermocapillary convection of fluid layers with a free surface, Int. J. Heat Mass Transfer 41 (2) (1998) 399–409. [20] A.M.L. Bethancourt, M. Hashiguchi, K. Kuwahara, J.M. Hyun, Natural convection of a two-layer fluid in a side-heated cavity, Int. J. Heat Mass Transfer 42 (1999) 2427–2437. [21] S. Madruga, C. Pérez-García, G. Lebon, Instabilities in two-liquid layers subject to a horizontal temperature gradient, Theor. Comput. Fluid Dyn. 18 (2004) 277–284. [22] C.D. Seybert, J.W. Evans, F. Leslie, W.K. Jones Jr., Suppression/reversal of natural convection by exploiting the temperature/composition dependence of magnetic susceptibility, J. Appl. Phys. 88 (7) (2000) 4347–4351. [23] L.A. Girifalco, R.J. Good, A theory for estimation of surface tension and interfacial energies — Part I: derivation and application to interfacial tension, J. Phys. Chem. 61 (1954) 904–909. [24] FIDAP 8.7, Fluid Dynamics Analysis Package, Fluent Inc., 10 Cavendish Court, Lebanon, NH, 03766, USA.
Effect of magnetic field on two-layered natural/thermocapillary convection☆ D. Ludovisi a , S.S. Cha a,⁎, N. Ramachandran b , W.M. Worek a a
Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Room 2039 ERF (M/C 251), Chicago, Illinois 60607, USA b
Jacobs ESTS Group, ER42, NASA Marshall Space Flight Center, Huntsville, AL 35812, USA Available online 30 March 2007
Abstract Heat transfer in a two-layered fluid system is of great importance in a variety of applications. Control and optimization of convective heat transfer of the immiscible fluids needs complete understanding of all phenomena, especially those induced by surface tension at the fluid interface. The present work is focused on rather complex convective flow and heat transfer phenomena in a cavity, which can be subject to both buoyancy and thermocapillary effects in addition to the influence of magnetic field applied for flow control. With the encapsulant liquid posing magnetic properties, a magnetic force can arise to either enhance or counterbalance the gravity effect when the cavity is placed in a non-uniform magnetic field. In our study, the velocity and temperature distribution of the system can be significantly altered to change the heat transfer by varying intensity and gradient of the applied magnetic field. Preliminary results of numerical computation presented here are for a two-layered liquid cavity MnCl2·4H2O and Fluorinert FC40 under various magnetic fields intensities. © 2007 Published by Elsevier Ltd. Keywords: Marangoni flow; Thermocapillary flow; Interfacial tension; Magnetic field; Two-layered fluid system
1. Introduction The heat transfer study of a double-layered fluid system is of great importance in many engineering applications, which can include metal casting, crystal growth, solar panel manufacturing and heat exchangers with air pockets. In these types of flow, a non-uniform temperature distribution at the fluid interface induces interfacial tension gradients that, in turn, provides a driving force for thermocapillary convection also known as Marangoni convection. The phenomena can cause adverse effects, i.e., imperfections in crystal growth, and thus have attracted interest of various investigators [1–13]. An approach to minimize or enhance the thermocapillary effect is to choose a liquid encapsulant with appropriate thermo-physical and chemical properties [14].
☆
Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (S.S. Cha).
0735-1933/$ - see front matter © 2007 Published by Elsevier Ltd. doi:10.1016/j.icheatmasstransfer.2007.02.003
524
D. Ludovisi et al. / International Communications in Heat and Mass Transfer 34 (2007) 523–533
Nomenclature B
magnetic field [T]
Ca
lU = Capillary number r
cp fmagnetic H k L Ma Y n Nu p Ra Y t T Tcold Thot u U v Y v I II
specific heat [J/kg K] magnetic body force per unit mass [N m3/kg] interface curvature [1/m] thermal conductivity [W/m K] 2 cm = cavity width σ TΔTh / μα = Marangoni number unit vector normal to the interface L AT j Thot −Tcold Ax x¼0 or x¼2 cm = local Nussel number pressure [Pa] βΔTgh3 / να = Rayleigh number unit vector tangent to the interface temperature [K] 298 K = temperature of cold wall 302 K = temperature of hot wall velocity in horizontal direction (x-axis) [m/s] characteristic velocity of the system under investigation [m/s] velocity in vertical direction (y-axis) [m/s] velocity vector [u, v] MnCl2·4H2O Fluorinert FC40
Greek symbols α thermal diffusivity [m2/s] β thermal expansion coefficient [1/K] χ magnetic susceptibility μ0 magnetic permeability of free space [N/A2] μ dynamic viscosity [kg/ms] ν kinetic viscosity [m2/s] ϕ Girifalco/good empirical parameter to estimate interfacial tension [23] ρ density [kg/m3] σ interfacial tension [N/m] σI MnCl2·4H2O surface tension [N/m] σII Fluorinert FC40 surface tension [N/m] σe electric conductivity [1/Ω m] θ interface contact angle with side walls [°] τ¯ stress tensor
For the combined buoyancy and Marangoni convection, the two-layered system poses much more complex and interesting behaviors than the one in a single-layer system due to a strong interaction of two fluid motions. Sparrow et al. [15] performed systematic experiments in a square cavity by using hexadecane overlying water. The numerical investigations by Kimura et al. [16], Myrum et al. [17] and Ramachandran [5] demonstrated that the flow patterns can be altered significantly when the interfacial force becomes comparable to the buoyancy force in a two-layered system.
D. Ludovisi et al. / International Communications in Heat and Mass Transfer 34 (2007) 523–533
525
At now, the application of a magnetic field has been motivated by an interest to further control the flow in a two-layered system [18–21]. As examples, the flow patterns and heat transfer characteristics can be manipulated in devices for material processing and energy utilization. Typically, in crystal growth a magnetic field can be applied to damp the flow in growing single crystals. An option with a fluid of significant electrical conductivity is to exploit Lorentz's force for flow damping. However, the effect of Lorentz's force diminishes with the decrease of convective motion. The option presented here is of a different origin, not being based on the electrical conductivity, to have complete flow control with any flow motion. 2. Physical setup and governing equations Fig. 1 shows a schematic of the physical setup adopted in our initial investigation. The system consists of two immiscible fluids: Fluorinert FC40 and 30% aqueous solution of MnCl2·4H2O by weight. The fluids are contained in a two-dimensional closed cavity with a rectangular cross-section of 2 cm by 1 cm. The volumes of both fluids are equal, with the lighter fluid at the top, thus forming a stable system. A temperature gradient normal to the gravity is imposed on the system by maintaining the vertical side walls at two different isothermal conditions. The top and bottom faces of the cavity are assumed to be adiabatic. The interface between the two fluids is deformable but this initial study, as good approximation, considers only undeformable interface. For the current system of Boussinesq Newtonian fluids in a laminar regime, the governing equations in Cartesian coordinates can be as follows. Continuity: ux þ vy ¼ 0
ð1Þ
Momentum: uux þ vuy ¼ −
px þ mðuxx þ uyy Þ q
ð2Þ
uvx þ vvy ¼ −
pv þ mðvxx þ vyy Þ þ gð1 þ bDT Þ þ fmagnetic q
ð3Þ
Energy: uTx þ vTy ¼ aðTxx þ Tyy Þ
ð4Þ
Here the suffixes denote differentiation with respect to the indicated variables. It should be noted that the governing equations above are for steady state and applicable to each layer of fluids. The boundary conditions are as below. u¼v¼0 u¼v¼0
at at
Ty ¼ 0
y ¼ 0; y ¼ 1 cm
at
x ¼ 0; x ¼ 2 cm 8y y ¼ 0; y ¼ 1 cm 8x
T ¼ Thot ¼ 302 K T ¼ Tcold ¼ 298 K
at at
x¼0 x ¼ 2 cm
ð5Þ ð6Þ ð7Þ ð8Þ
For the interaction of the two fluids at the interface, let the region for MnCl2·4H2O be denoted by I while the one for Fluorinert by II as shown in Fig. 2. The following conditions can then be imposed at the interface. No slip condition: Y vI d Y t¼Y v II d Y t
ð9Þ
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Fig. 1. Schematic of the physical setup.
Kinematic condition: Y n¼Y v II d Y n¼0 vI d Y
ð10Þ
Interface force balance: Y n d ½ ¯s I ¯s II ¼ 2Hr Y n−js r
ð11Þ
Contact angles: h ¼ 90-
ð12Þ
Continuity of temperature: TI ¼ TII
ð13Þ
Continuity of heat flux in the direction normal to the interface: n ÞII ðkjT d Y n ÞI ¼ ðkjT d Y
ð14Þ
Here τ¯ and H represent stress tensor and the mean interface curvature, respectively, ∇s denotes the surface gradient operator that can be written as j− Y nð Y ndjÞ. Since the surface tension strongly depends on the temperature only we can simplify the surface gradient operator in Eq. (11) as follows: js r ¼ rT js T
ð15Þ
With the capillary number Ca of our system fairly small, Eq. (10) together with the tangential component of Eq. (11) is used as boundary conditions for the momentum equations. The normal stress balance, that is, the
Fig. 2. Interfacial unit tangent and normal vectors and liquid phases notation.
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Table 1 Fluid properties: all values are at the average temperature of 300 K (otherwise specified) Property 3
Density (ρ) [kg/m ] Kinetic viscosity (ν) [m2/s] Specific heat (cp) [J/kg K] Thermal expansion coefficient (β) [1/K] Thermal conductivity (k) [W/m K] Dynamic viscosity (μ) [kg/ms] Thermal diffusivity (α) [m2/s] Surface tension at 25 °C (σ) [N/m] Surface tension gradient at 25 °C (σ) [N/m K] Electric conductivity (σe) [1/Ω m] Magnetic susceptibility (χ) [1/Ω m]
MnCl2·4H2O 30% in weight
Fluorinert FC40
1118.25 9.826 · 10− 7 4152.0 1.855 · 10− 4 0.6 1.003 · 10− 3 2.911 · 10− 7 0.0728 − 1.6 · 10− 4 36 5.738 · 10− 6 + 0.05922 / T (K)
1870.0 2.2 · 10− 6 1046.5 1.20 · 10− 3 0.067 4.114 · 10− 3 3.242 · 10− 8 0.016 − 1.22 · 10− 4 0.25 · 10− 13 − 8.273 · 10− 6
normal component of Eq. (11), would be utilized as a distinguished condition for finding the unknown interface location. The magnetic body force per unit mass, experienced by a paramagnetic material, is fmagnetic ¼
vðT Þ jðB2 Þ 2l0 qðT Þ
ð16Þ
where χ(T) and ρ(T) are magnetic susceptibility and density of the fluid, which are functions of temperature, while μ0 is the permeability of free space. If the susceptibility varies with temperature, as in paramagnetic materials, the body force changes with temperature, analogous to the gravitational buoyancy force when the density varies with temperature. To avoid confusion, these two effects will be distinguished as “magnetic” and “gravitational” buoyancy, respectively. By aligning the magnetic field gradient vertically, the magnetic buoyancy can either enhance or counterbalance the gravitational buoyancy. The direction of the net force can even be reversed at some critical value of the magnetic field gradient [22]. The properties of the fluids, which have been employed for our numerical modeling, are assumed to be constant as listed in Table 1. Some properties of the aqueous solution are not readily available in literature, which include viscosity, thermal conductivity, and surface tension. For these, properties of water are assumed. For our two-layered system, the interfacial tension was estimated by using the approach proposed by Girifalco and Good [23]: pffiffiffiffiffiffiffiffiffiffi r ¼ rI þ rII −2/ rI rII ð17Þ This method expressed by Eq. (17) requires the knowledge of an empirical parameter ϕ that depends on the nature of chemical bonds between the two liquids. Due to the composition of MnCl2·4H2O and Fluorinert, ϕ is estimated to
Fig. 3. Stream function in the absence of both the interfacial tension and magnetic field.
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Fig. 4. Stream function with the interfacial tension but without the magnetic field.
be 0.55 [23], which gives an interfacial tension temperature gradient equal to − 1.037 · 10− 4 [N/m K]. However, ϕ greatly affects the results of our numerical investigation and thus it needs to be verified experimentally. In the current study, the effects of Lorentz's force are neglected since its order of magnitude is at least three times lower than either the magnetic body force or the gravitational buoyancy when calculated with the same cavity conditions. The magnetic body force in Eq. (3) is only significant in the solution of MnCl2·4H2O. The variation of the magnetic susceptibility of Fluorinert with respect to temperature is also unavailable in literature: however, it is believed to be negligible for the temperature range of our investigation. 3. Results and discussion The governing equations with the corresponding boundary conditions, Eqs. (1)–(17), are solved with the commercial finite element code FIDAP [24]. Convergence is assumed when the residual of governing equations is less than 10− 4. In an effort to obtain a greater resolution near the walls and interface, where higher velocity and temperature gradients are expected, a non-uniform grid is employed with smaller elements close to the walls and the interface. Systematic evaluation has been made to establish gridconvergence on numerical results. For the present study, a grid with 160 × 80 elements proves to be optimal in terms of accuracy and computation time. As a first step in numerical calculation, a simpler problem of natural convection without the interfacial tension and magnetic field has been calculated for comparison. The steady-state stream function obtained from the numerical modeling is shown in Fig. 3. As expected, each fluid has a large recirculating region. These two cells are driven in the same directions by thermal buoyancy to cause conflicting driving forces in the vicinity of the interface. Fluorinert, which is stronger in buoyancy, creates a small counterclockwise
Fig. 5. Stream function in the presence of both the interfacial tension and magnetic field of ∇(B2) = 50 T2/m.
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Fig. 6. Stream function in the presence of the interfacial tension and a magnetic field of ∇(B2) = − 50 T2/m.
intermediate cell in MnCl2·4H2O. For the current heating conditions, the thermal buoyancy of Fluorinert is about 7 times stronger than that of MnCl2·4H2O due to difference in thermal expansion coefficients. Fig. 4 shows the stream function when the interfacial tension is present between the two fluids. The flow field exhibits the dominating nature of the Marangoni convection that counterbalances the thermal buoyancy effect in MnCl2·4H2O. The interfacial surface tension works in a concerted manner with the lower Fluorinert cell to expand its size. The upper large cell in MnCl2·4H2O is now reduced to two small cells at the upper corners of the cavity. Figs. 5 and 6) show the effect of the magnetic field on the two-fluid system. The magnetic field gradient points in the vertical direction. In the case of Fig. 5, the magnetic force in MnCl2·4H2O is large enough to suppress the thermal buoyancy effect and thus to further promote the Marangoni convection effect. The two cells of the upper corners in Fig. 4 completely disappear. Only two large cells thus circulate in opposite direction. In the case of Fig. 6, the magnetic body force acts in the opposite direction with respect to the Marangoni force at the interface. The counterclockwise cell that has been produced in MnCl2·4H2O by the interfacial tension in Fig. 5 is reduced in size and a large clockwise cell newly forms on top of it. The magnetic force this time is in the direction of buoyancy force and thus the appearance of the large clockwise cell is as expected. The flow pattern becomes very similar to Fig. 3. The overall effect of the magnetic force is transmitted to the Fluorinert layer to slow down the velocity in turn. However, as stated before, the Marangoni convection can play such an important role in a two-fluid layered system that the attempt to completely cancel its effect could be a challenge in fluid flow control. In fact, as observed in Fig. 6,
Fig. 7. Magnitude of the horizontal velocity component along the cavity vertical centerline for various magnetic field intensities.
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Fig. 8. Magnitude of the horizontal velocity component along the interface for various magnetic field intensities.
the intermediate cell in MnCl2·4H2O produced by the interfacial tension is still relatively large and the flow field in Fluorinert appears to be not greatly affected. Figs. 7 and 8 are the plots of the horizontal velocity profiles at the vertical centerline of the cavity and at the interface between the two fluids, respectively, under various magnetic field gradients. As seen, positive magnetic field gradients act in concert with the effect of the Marangoni convection in MnCl2·4H2O. At some typical locations, interfacial velocities are doubled. The opposite occurs when negative magnetic field gradients are applied as stated earlier, to make the fluid flow completely reversed for a strong enough magnetic field. The velocity in Fluorinert is less affected at the centerline, as seen in Fig. 7, except along the interface. This appears to be due to the dominant buoyancy force and ineffectiveness of magnetic force in the Fluorinert region. However, this region is more significantly influenced close to the cold wall, where the velocity peak can be either increased or reduced by 50% for certain values of magnetic field intensity as seen in Fig. 8. It is worth noting that, for the conditions under analysis, our two-layered fluid system works in a manner similar to a gear box, with the magnetic buoyancy cell as a “driving gear”. As seen in Figs. 5–7, when the magnetic effect enhances the
Fig. 9. Temperature distribution along the vertical centerline for various magnetic field intensities.
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Fig. 10. Temperature distribution at the interface between the two fluids for various magnetic field intensities.
velocity of MnCl2·4H2O, the velocity of Fluorinert increases. On the contrary, when MnCl2·4H2O slows down, it has a reverse effect. Figs. 9 and 10 show the distributions of temperature at the vertical centerline and at the interface of the cavity under the influence of magnetic field. Similarly to the behavior of horizontal velocity, the effect of the magnetic field is larger in the MnCl2·4H2O layer and its influence is transmitted to the Fluorinert region again to a lesser degree through the interface where the Marangoni convection dominates. However, the change in temperature near the cold wall is more prominent than the change in velocity. Any variation of the temperature in the vicinity of the two isothermal walls can induce a significant change in heat transfer of the system. That is, even if the temperature might not appreciably differ in net value, its derivative can greatly change to influence the heat transfer. Figs. 11 and 12 show the local Nusselt number along the hot and cold walls. As seen, the local Nusselt number of the hot wall can either decrease or increase by 14% for Fluorinert as compared to the case without a magnetic field. This results in a change of up to 28% in the global value for the range of the tested magnetic field intensity. In MnCl2·4H2O, the change in the local Nusselt number is more pronounced. At the cold wall, the local Nusselt number exhibits a greater change in proximity of the interface, where the value can undergo a variation up to 150% for some magnetic field gradients.
Fig. 11. Local Nusselt number along the hot wall.
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Fig. 12. Local Nusselt number variation along the cold wall.
Finally, Fig. 13 shows the variation of the global heat transfer for the two-layer fluid system. As seen, if a proper magnetic field gradient is applied, the global heat transfer can be either increased or decreased by 100%. This proves the concept that motivated our current study, that is, the application of a magnetic field for fluid flow and heat transfer control.
4. Conclusions The interaction of the Marangoni convection, magnetic body force and thermal buoyancy in a system of two immiscible-fluid layers have been investigated and preliminary results are presented. The flow field and heat transfer for the system has been computed by using a commercial finite element code under various magnetic field gradients. Our investigation reveals that the Marangoni convection can significantly influence the velocity distribution in the cavity. It also shows that introduction of a magnetic field can easily promote or hinder the Marangoni effect, that is, to
Fig. 13. Global heat transfer through the entire cavity under various magnetic field gradient intensities.
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further increase or decrease global convective motion. The effect of the magnetic body force of the upper layer has a difficulty to penetrate through the interface of the two fluids. This appears to be due to the dominant flow field pattern induced by the interfacial tension and the thermal buoyancy of the bottom layer. The difficulty allows the general velocity pattern to be preserved for the bottom layer, except the region close to the cold wall. However, the velocity and temperature fields in the upper paramagnetic fluid can be significantly affected, especially far from the interfacial region. It has been shown that even a small change in the velocity field can produce appreciable changes in temperature distribution, which in turn greatly affect the local Nusselt number and thus the global heat transfer. For the range of magnetic field gradients tested for this study, the heat transfer can be either enhanced or reduced by 100%. References [1] W. Pfann, Zone Melting, second ed.McGraw-Hill, New York, 1966. [2] J. Szekely, M.R. Todd, Natural convection in a rectangular cavity transient behavior and two phase systems in laminar flow, Int. J. Heat Mass Transfer 14 (1971) 467–482. [3] R.W. Knight, M.E. Palmer, Simulation of free convection in multiple fluid layers in an enclosure by finite differences, Numerical Properties and Methodologies in Heat Transfer, Hemisphere, Washington D.C., 1983. [4] T. Doi, J.N. Koster, Thermocapillary convection in a two immiscible liquid layers with free surface, Phys. Fluids, A Fluid Dyn. 5 (1993) 1914–1927. [5] N. Ramachandran, Thermal buoyancy and Marangoni convection in a two fluid layered system, J. Thermophys. Heat Transf. 7 (2) (1993) 352–360. [6] P. Wang, R. Kahawita, Nguyen, Numerical simulation of Buoyancy–Marangoni convection in two superposed immiscible liquid layers with a free surface, Int. J. Heat Mass Transfer 37 (7) (1994) 1111–1122. [7] Q.S. Liu, B. Roux, M.G. Verlarde, Thermocapillary convection in two-layer systems, Int. J. Heat Mass Transfer 41 (11) (1998) 1499–1511. [8] M.Z. Saghir, Convection in two-liquid-layer system with low Prandtl number, Energy Sources 21 (1999) 139–144. [9] M. Hamed, J.M. Floryan, Marangoni convection — Part 1: a cavity with differentially heated sidewalls, J. Fluid Mech. 405 (2000) 79–110. [10] S. Someya, T. Munakata, M. Nishio, K. Okamoto, H. Madarame, Flow observation in two immiscible liquid layers subject to a horizontal temperature gradient, J. Cryst. Growth 235 (2002) 626–632. [11] N.R. Gupta, H. Haj-Hariri, A. Borhan, Thermocapillary flow in double-layer fluid structures: an effective single-layer model, Journal of Colloid and Interface Science 293 (1) (2006) 158–171. [12] Q.S. Liu, G. Chen, B. Roux, Thermogravitational and thermocapillary convection in a cavity containing two superposed immiscible liquid layers, Int. J. Heat Mass Transfer 36 (1) (1993) 101–117. [13] N.R. Gupta, H. Haj-Hariri, A. Borhan, Thermocapillary flow in double-layer fluid structures: an effective single-layer model, J. Colloid Interface Sci. 293 (1) (2006) 158–171. [14] J.J. Bikerman, Physical Surfaces, Academic Press, New York, 1970. [15] E.M. Sparrow, L.F.A. Azevedo, A.T. Prata, Two-fluid and single fluid natural convection heat transfer in an enclosure, ASME J. Heat Transfer 108 (1986) 848–852. [16] T. Kimura, N. Heya, M. Takeuchi, H. Isomi, Natural convection heat transfer phenomena in an enclosure filled with two stratify liquids, JSME Int. J. Ser. B Fluids Therm. Eng. 52 (1986) 617–625 (in Japanese). [17] T.A. Myrum, E.M. Sparrow, A.T. Prata, Numerical solutions for natural convection in a complex enclosed space containing either air–liquid or liquid–liquid layers, Numer. Heat Transf. 10 (1986) 19–43. [18] P. Wang, R. Kahawita, Transient buoyancy–thermocapillary convection in two superposed immiscible liquid layers, Numer. Heat Transf., A Appl. 30 (5) (1996) 477–501. [19] P. Wang, R. Kahawita, Oscillatory behaviour in buoyant thermocapillary convection of fluid layers with a free surface, Int. J. Heat Mass Transfer 41 (2) (1998) 399–409. [20] A.M.L. Bethancourt, M. Hashiguchi, K. Kuwahara, J.M. Hyun, Natural convection of a two-layer fluid in a side-heated cavity, Int. J. Heat Mass Transfer 42 (1999) 2427–2437. [21] S. Madruga, C. Pérez-García, G. Lebon, Instabilities in two-liquid layers subject to a horizontal temperature gradient, Theor. Comput. Fluid Dyn. 18 (2004) 277–284. [22] C.D. Seybert, J.W. Evans, F. Leslie, W.K. Jones Jr., Suppression/reversal of natural convection by exploiting the temperature/composition dependence of magnetic susceptibility, J. Appl. Phys. 88 (7) (2000) 4347–4351. [23] L.A. Girifalco, R.J. Good, A theory for estimation of surface tension and interfacial energies — Part I: derivation and application to interfacial tension, J. Phys. Chem. 61 (1954) 904–909. [24] FIDAP 8.7, Fluid Dynamics Analysis Package, Fluent Inc., 10 Cavendish Court, Lebanon, NH, 03766, USA.