Convective heat transfer in fluids in the presence
of gravitational and electromagnetic fields (*)
M. F. HAQUE(1) and S. ARAJS(2) †(1) Physics Department, Abubakar Tafawa Balewa University - Bauchi, Nigeria
(2) Physics Department (Laboratory), Clarkson University - Potsdam, New York, 13699, USA (ricevuto il 10 Luglio 1995; approvato l’1 Aprile 1996)
Summary. — We report measurements of convective heat transfer in gases (air,
O2, N2O and freon-22), liquids (kerosene, silicon oil and distilled water) and a colloidal suspension (hematite (a-Fe2O3) in distilled water) under the influence of electric, magnetic and electromagnetic fields. The corresponding heat transfer coefficients indicate a complicated development of heat convections which appear to be strongly interacting. A quantitative interpretation of the results is furnished using dimensional analysis.
PACS 44.25 – Convective and constrained heat transfer.
1. – Introduction
Buoyancy-driven flows in enclosures are of importance in various applications such as in nuclear-reactor design, aircraft cabin insulation, cooling of electronic equipment, and thermal storage systems. In particular, natural-convection heat transfer in the annular space between concentric cylinders and spheres has drawn considerable attention. Several numerical and experimental studies of natural convection within a spherical annulus for various values of the Prandtl and Rayleigh numbers are available.
Natural convection of heat in a fluid often shows itself to be a relatively inefficient means of thermal transfer with many industrial processes using mixed convection or other outside body forces (electrical, magnetic or electromagnetic) to make flows more turbulent. Therefore, the idea of imposing such constrainsts on thermal flows has the aim of enhancing mixing and thus heat transfer within the fluid. In particular, using electrical forces to this end is referred to as electro-thermohydrodynamics (ETHD). Work on this subject dates back to the mid-1930s with Senftleben’s experiments using
(*) The authors of this paper have agreed to not receive the proofs for correction.
M.F.HAQUEandS.ARAJS
666
gases [1-3] and then those of Ahsman and Kronig [4] in 1950 on liquids. Senftleben presented an analysis of heat transfer from a heated wire mounted along the axis of a cylinder. An increase in heat transfer coefficient was observed under the influence of a non-uniform electric field. The effect was first discovered by Senftleben and is called the electroconvectional heat transfer and depends on the various transport properties of the fluid as well as the electric field. Senftleben [2] assumed that the enhanced heat transfer in the presence of electric field is due to electrostrictive force which alters the convection currents. The presence of electric field induces a dipole moment in a spherically symmetric molecule. In the case of a molecule possessing a permanent dipole moment, it will tend to align with the field. Since at constant pressure electric susceptibility is temperature dependent, it turns out that a cold fluid in a non-uniform field would experience more forces than a hot fluid in the same region. As a result, a pressure gradient is developed which forces the cold fluid to replace the hot fluid, thereby generating a circulating current, which is the cause of increased heat transfer. Kronig and Schwartz [5] made a quantitative interpretation of the above phenomena using the theory of similarity. Subsequently, interpretations were also given by Senftleben and Bultmann [6], Senftleben and Lange-Hahn [7] and, more recently, by Lykoudis and Yu [8]. The experimental studies of electroconvectional heat
TABLE I. – Magnetic and electromagnetic Nusselt number in oxygen, nitrous oxide, kerosene and hematite particle suspension in distilled water (vertical cylinder). a) Oxygen 1) Num»p 4 50 cm Hg, TQ4 35 7C, E 40; 2) Nuem»p 450 cm Hg, TQ4 35 7C, E 41.48 kV, 60 Hz. b) Nitrous oxide 1) Num»p 471 cm Hg, TQ4 36 7C, E 4 0; 2) Nuem»p 471 cm Hg, TQ4 36 7C, E 41. 50 kV, 60 Hz. c) Kerosene 1) Num»TQ4 29 .31 7C, E 4 0; 2) Nuem»TQ4 29 .31 7C, E 450 V d.c. d) Hematite in distilled water 1) Num»TQ4 26 .23 7C, E 4 0 ; 2) Nuem: TQ4 26 .23 7C, E40.5 V d.c. H (kG) Oxygen p 450 cm Hg Nitrous oxide p 471 cm Hg Kerosene Hematite in distilled water Num Nuem Num Nuem Num Nuem Num Nuem 0 0 22 0 1.92 839 807 514 765 1 0 24 0 2.57 836 797 480 739 2 0 29 0 3.21 825 792 449 713 3 2 31 0 4.17 818 779 428 687 4 5 36 0.64 5.13 809 772 406 670 5 10 42 1.56 8.02 800 765 390 644 6 18 50 1.91 9.62 793 758 373 622 7 28 66 3.84 11.56 786 753 355 605 8 37 70 5.09 13.47 779 747 342 583 9 48 77 6.36 16.04 774 744 324 570 10 56 84 9.54 19.25 770 740 320 553 11 65 90 12.72 22.46 — — — — 12 78 96 16.18 25.02 — — — — 13 84 100 19.94 27.59 — — — — 14 84 102 23.99 29.51 — — — — 15 84 102 28.03 31.44 — — — — 16 84 102 29.19 32.72 — — — — 17 84 102 30.35 33.68 — — — — 18 84 102 31.21 34.00 — — — —
TABLEII. – The efficiency of magneto- and electromagneto-convection in oxygen, nitrous oxide, kerosene and hematite particle suspension in distilled water (vertical cylinder). a) Oxygen, p 4 50 cm Hg, Td4 24 .5 7C. b) Nitrous oxide, p 4 71 cm Hg, Td4 30 .0 7C. c) Kerosene, Td4 16 .5 7C. d) Hematite in distilled water, Td4 10.5 7C.
H (kG) Oxygen p 450 cm Hg Nitrous oxide p 471 cm Hg Kerosene Hematite in distilled water gm( % ) gem( % ) gm( % ) gem( % ) gm( % ) gem( % ) gm( % ) gem( % ) 0 0 27 0 6 0 2 4 0 33 1 0 29 0 8 21 2 6 2 7 30 2 0 33 0 10 22 2 6 215 28 3 4 35 0 13 23 2 8 220 25 4 8 38 2 15 24 2 9 227 23 5 14 42 5 22 25 210 232 20 6 24 47 6 25 26 211 238 17 7 32 53 12 29 27 212 245 15 8 39 55 15 32 28 213 251 12 9 45 57 18 36 29 213 259 10 10 49 59 25 40 29 214 261 7 11 53 61 31 44 — — — — 12 57 62 36 46 — — — — 13 59 63 41 49 — — — — 14 59 64 45 51 — — — — 15 59 64 49 52 — — — — 16 59 64 50 53 — — — — 17 59 64 51 54 — — — — 18 59 64 52 54 — — — —
transfer from horizontal cylinders were dealt with by Ahsman and Kronig [4, 9], De Haan [10], Arajs and Legvold [11] and Schnurmann and Lardge [12]. Likewise, the experimental studies of magnetoconvectional heat transfer were studied by Senftleben and Pletzner [13] and subsequently studied by different investigators, but the effects are still not fully understood. Furthermore, simultaneous application of electric and magnetic field is an unexplored area.
This paper is devoted to a study of convective heat transfer in gases (air, and freon-22), liquids (silicon oil and distilled water) and a colloidal suspension (a–Fe2O3in
distilled water) under the influence of a.c. and d.c. fields. Convective heat transfer coefficient has been evaluated under different geometries such as horizontal or vertical cylinders and detailed measurements are furnished for different Td, the temperature
difference between the wire and the surrounding medium. A quantitative
interpretation of the experimental results is furnished using dimensional analysis. The experimental results have been compared with the empirical relations and analytical
expressions and a good correlation is obtained up to the Rayleigh number 5 Q 108.
Additionally, magnetic and electromagnetic Nusselt number are evaluated for O2, N2O, kerosene and a colloidal suspension (a-Fe2O3 in distilled water) and the
results are presented in table I. Likewise, the efficiency of magnetic and electromagnetic convection has been evaluated using an empirical relationship and the results are presented in table II.
M.F.HAQUEandS.ARAJS
668
2. – Experimental arrangement
The detailed experimental arrangement used in this investigation has been described elsewhere [14]. However, a brief description of the equipment is furnished here
(
fig. 1a))
. A hot-wire cell made from a single platinum wire (diameter40.025 mm), centered in a copper cylinder (diameter453 mm), was used for the heat transfer studies. This cell was placed into one arm of a Wheatstone bridge and was then immersed in a constant-temperature bath. This cell could be positioned at any angleFig. 1. – a) Simplified form of experimental arrangement used for the investigation of electro-convective and electromagnetoelectro-convective heat transfer in fluids; b) transmission electron micrographs of spherical hematite particles (rp4 0 .15 mm ) used in the convection experiment.
between the vertical and horizontal orientations. Electric fields in the cell were created by an applied electrical potential (d.c. or a.c.) between the central wire and the surrounding cylinders (H.V.). Heat transfer coefficient was obtained by calibrating the wire as a platinum resistance thermometer and then measuring the voltage across it and a standard resistor in series with it.
In order to measure magnetoconvective heat transfer coefficient, non-uniform transverse magnetic fields (with respect to the platinum wire) were obtained by an electromagnet whose cylindrical work space (30.48 cm diameter37.62 cm gap) was filled with two wedge-shaped iron pole pieces each having dimensions 25 .40 311.433 3 .18 30.318 cm. The hot-wire cell was placed between the wedges. In order to obtain the heat transfer data, the cell was also placed in one arm of a Wheatstone bridge. The bridge was kept balanced for any experimental situation. If I0 is the current flowing
through the wire in the absence of any field and Inis the current necessary to keep the
bridge balanced in the presence of any field (electric, magnetic or electromagnetic), then the rate of heat transfer is given by
Qn4 ( 2 I0In1 In2) R ,
where R is the resistance of the wire. The convective specific-heat transfer coefficient is then obtained by using the relation
qn4
Qn
A ,
where A is the surface area of the wire. Similarly, the convective heat transfer coefficient is obtained by using the relation
hn4
Qn
ATd
,
where Tdis the temperature difference between the wire and the surrounding medium.
The symbol n stands for when the applied field is either electric (el), magnetic (m) or electromagnetic (em).
The spherical hematite particles, each of radius rp4 0 .15 mm (density rp4
5 .24 g cm23) used in this investigation were prepared by the method described
earlier [15]. The transmission electron micrographs of these particles are presented in fig. 1b). All measurements were carried out at a fixed particle concentration (np4 4 .4 Q
1015particles Om3) and with a pH value 11, unless otherwise stated.
3. – Results and discussion
3.1. Convection in the presence of electric field. – Convection in the presence of electric field is carried out in air, freon-22, silicon oil, distilled water and colloidal hematite particle suspension in distilled water. The results for air and freon-22 are presented in fig. 2a), b) and c), respectively. As seen in the figures, an absence of convection is noted at a lower electric field. As the field is increased gradually, the convective heat transfer coefficient increases approximately linearly with the electric field and then exhibits a saturation effect at higher electric fields. It then decreases again and approaches zero and becomes negative as may be seen especially in fig. 2a) on further increase in electric field. The heat transfer profile is Gaussian in most of the
M.F.HAQUEandS.ARAJS
670
Fig. 2. – Electroconvective specific-heat transfer coefficients in gases as a function of a.c. and d.c. fields in vertical cylinder. a) Air; d.c. field, p 476 cm Hg, 1) Td4 22 .89 7C, 2) Td4 21 .97 7C, 3) Td4 0 .03 7C, 4) Td4 5 .10 7C; b) Air; a.c. field, 1) Td4 10 .0 7C, p 4 29 .0 cm Hg, 2) Td4 9 .80 7C, p 4 49 .0 cm Hg, 3) Td4 10 .0 7C, p 4 76 cm Hg; c) freon-22; a.c. field, Td4 8 .51 7C, p 4 34 .1 cm Hg.
cases, with the exception that a minimum in heat transfer profile is noted in an a.c. field. The results for silicon oil, distilled water and hematite particle suspension in distilled water are shown in fig. 3 and 4, respectively. In the case of silicon oil, an inhibition in heat transfer is noted at lower electric field, but it increases again as the field is increased further and then approaches towards a saturation value (fig. 3). In the case of distilled water (curve 1, fig. 4), an absence of convection is noted at lower electric field. The heat transfer coefficient then increases gradually with an increase in
Fig. 3. – Electroconvective specific-heat transfer coefficients in silicon oil as a function of d.c. fields. Horizontal cylinder, Td4 12 .85 7C.
electric field, reaches a saturation value and then decreases again as the field is further increased. A similar trend is noted for a colloidal suspension (a-Fe2O3 in distilled
water), but the increase in heat transfer coefficient is almost linear as seen by curve 2 in fig. 4. As the pH of the suspension is increased (pH 411), a rapid increase in heat transfer is noted at a weak field strength. In the case of a d.c. field, the increase is almost vertical near the origin (curve 4, fig. 4). The heat transfer coefficient reaches its peak value and then rapidly approaches towards zero as the field is slightly increased.
On a more fundamental level, we may distinguish between three types of electrical forces acting on a fluid when a potential difference is applied across it [16]. Firstly, there is the Coulomb or the electrophoretic force
Fe4 rsEr,
(1)
where rs is the charge density and Eris the electric field intensity. Secondly, we have
the dielectrophoretic force given by
Fd4 2(Er2O2 ) N˜eN ,
(2)
where e is the permittivity field of the fluid, and thirdly, the electrostrictive force, Fes4 N˜[ (Er2O2 )(ˇeOˇr) ] N ,
(3)
where r is the fluid density. It is usual to include the dielectrophoretic term with the liquid. The dominant electrical body force in the gases arises mainly from the electro-phoretic and electrostrictive forces.
In electrohydrodynamic systems the electrodynamic equations comprise Maxwell’s relations and appropriate constitutive relations. For a single known species of charge
M.F.HAQUEandS.ARAJS
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Fig. 4. – Electroconvective heat transfer coefficients in distilled water and colloidal hematite particle suspension in distilled water as a function of a.c. and d.c. fields. Horizontal cylinder, Td4 6 .5 7C. 1) Distilled water, a.c. field, pH45.40, 2) a-Fe2O3 in distilled water, a.c. field, pH45.40, 3) a-Fe2O3in distilled water, a.c. field, pH411.0, 4) a-Fe2O3 in distilled water, d.c. field, pH4 11 .0.
carrier the constitutive relation for current in a d.c. system assumes the following form:
j 4rsbEr1 qu 1 DcN˜rsN ,
(4)
where, except in ultra-pure liquids in which free electrons may occur, charge carriers are thought to exist as positive or negative ions [17, 18]. The first term represents the current which results from the drift of ions relative to surrounding host fluid molecules, b being the ion mobility while the other terms account for the convective transport and diffusion of ions, respectively. Dc is the ion diffusion coefficient and
except in regions very close to the electrodes (u K0) the diffusion term is small compared with the other two terms and can be neglected.
From the foregoing it is clear that as a body force the electrophoretic component, rather than acting on all the constituents of the medium, operates only on individual ions which then transfer momentum to surrounding molecules as they are driven through the host medium.
In the case of gases, the convective specific-heat transfer coefficient is found to be zero for electric fields less than the critical electric field (fig. 2a), b) and c). As the electric field is applied, the molecules in the gas attain an induced dipole moment and tend to align with the direction of the field. When the applied electric field is less than the critical electric field (E EEc), the induced dipole moment is small, resulting in the
weak electrophoretic force and hence no convection is observed. When E DEc, the
dipolar interaction increases, which in turn, increases the electrophoretic force and enhances the convection. In the saturation region, an energy balance is attained due to Joule heating which develops gradually as the field is increased and is responsible for the reduction of heat transfer as observed in each case. The electric-field intensity inside the cylinder is estimated as
Er4
0 .13 E
r ( voltOcm ) , (5)
where r is the radial distance from the centre of the wire and E is the voltage applied to the cylinder. The maximum conduction current flowing through the cylinder of maximum electric-field strength and pressure was recorded as 100 and 150 mA for air and freon-22, respectively. An estimate of the Joule heating near the immediate vicinity of the platinum wire gives a very significant figure, implying the fact that Joule heating is responsible for the reduction of heat transfer as observed in each case.
In fig. 2a), the convective specific-heat transfer coefficient is also plotted for various
values of Td, the temperature difference between the wire and the surrounding
medium. In these measurements, the air pressure was kept constant, while the temperature difference was varied from positive to negative values, including zero. A Gaussian distribution of heat transfer profile is noted in each case and is associated with a critical electric field such that Ec1E Ec2E Ec3E Ec4E Ec5, where Ec1REc5are the
various critical electric fields at Td45.10 7C, 0.03 7C, 21.97 7C, 22.89 7C and 25.14 7C,
respectively. The result at Td4 25 .14 7C is not shown here since the convective heat
transfer profile at this value of Td is very negligible. As the magnitude of the Td
changes from positive to negative values, the corresponding peaks in the heat transfer profile move towards the origin, while the critical electric field moves away from the origin. The increase in critical electric field with a decrease in Tdresulted from the law
of convection.
A negative value of the electroconvective specific-heat transfer coefficient, qel,
implies a suppression of the free convective specific-heat transfer coefficient, qf, since
qf1 qelE qf, whereas a positive value implies an enhancement of heat transfer
coefficient where qf1 qelD qel. This has been observed in our experiment. It is to be
noted that the last data point presented in curves 3 and 4 in fig. 2a) represents the point at which the cylinder begins to conduct and no measurement was carried out beyond that point. However, in the case of curves 1 and 2, it was not possible to carry out the measurement beyond the last data point presented due to experimental difficulties encountered in balancing the bridge. It was not possible to rebalance the bridge even after removing all the heating current. The effect was more significant at Td4 25 .14 7C.
An estimate of the heat lost by radiation, per unit area from the surface of the heated wire has also been calculated using the relation
qrad4 e 8 s (Tw42 TB4) ,
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674
where e8 is the emissivity, s is the Stefan-Boltzmann constant (s45.67Q 1028WOm2 K4),
Twis the temperature of the wire and TBis the temperature of the surrounding medium. It
is found that the radiative heat loss for air and freon-22 is less than 1% of the heat lost by electroconvection. Under this experimental condition radiation effect is not at all significant.
The behaviour of heat transfer coefficient under the influence of a.c. field is noteworthy
(
fig. 2b) and c))
. One trend that is apparent in an a.c. field is that as the electric field increases with constant frequency, the heat transfer reaches a minimum. The minimum may be due to resonance excited by the a.c. field. At resonant condition, the free convection is suppressed by electroconvection. The suppression in free convection is due to the fact that when the heat transfer reaches a maximum, a small recirculating flow induced by the cylinder walls combines with the downstream recirculation, resulting in two counter-rotating recirculations making the heat transfer coefficient minimum. The minimum thus observed is found to depend on the pressure of the gas, the strength of the electric field, as well as various transport properties of the fluid, such as molecular weight, thermal conductivity, the electric-dipole moment, the viscosity and the polarizability of the fluid.The minimum in heat transfer coefficient under the influence of an acoustic field has been reported by Fand and Cheng [19]. They measured heat transfer from a cylinder in cross-flow at sound levels up to 150 dB and frequencies of 1100 and 1500 Hz. They observed that the enhancement in heat transfer has a minimum, depending on the frequency and concluded that the phenomenon was due to a resonance induced by the sound wave.
Convective heat transfers for various liquids (silicon oil and distilled water) are shown in fig. 3 and 4. If a non-uniform electric field is applied to an inhomogeneous dielectric liquid, regions of lower dielectric constant will experience a force directed towards regions of lower field strength. This force is known as dielectrophoretic force and is given in eq. (2). It is proportional to the difference in dielectric constants and to the gradient of the square of the electric-field strength. If there is a decrease in the dielectric constant as a result of heating, and if this occurs in a region of high field strength, the heated fluid will be forced into regions of lower field strength. Watson [20] has demonstrated the effect of this force on the natural convection process from a horizontal wire. In all cases the effect of the field was to increase heat transfer rates from the wire, the results being virtually identical for d.c. fields of negative and positive polarity. Gross [21] has shown that in a dielectric liquid, a net circulatory force will exist provided curl fc0, where
curl f4 1
8pgrad e3grad E
2
r1grad rs3Er,
(7)
f is the electric body force per unit volume.
The inhibition in heat transfer coefficient observed in silicon oil is due to the electrophoretic force, resulting from the action of the electric field on the free charges. Since the contribution to the net circulatory force caused by the inhomogeneous dielectric cannot be neglected, therefore, the enhancement in heat transfer coefficient observed in silicon oil and distilled water is due to the dominance of the dielectrophoretic force over the electrophoretic force. The reduction in heat transfer coefficient at higher electric field is due to Joule heating as explained earlier.
In the case of a colloidal suspension (a-Fe2O3 in distilled water), an increase in heat
(curve 2, fig. 4) and is remarkable for pH values largerly different from the isoelectric point (i.e.p.) (curves 3 and 4 of fig. 4) (i.e.p. of hematite`7.4). The increase in heat transfer coefficient noted in colloidal suspension is due to the increase in electrophoretic force resulting from the action of the electric field on the charged particles
(
eq. (1))
. For higher surface-charged particles, the electrophoretic force increases tremendously and as a result, a rapid increase in heat transfer is noted both in an a.c. (curve 3) and d.c. field (curve 4). In the case of a d.c. field, the increase is almost vertical near the origin. The heat transfer coefficient reaches a peak value at very weak field (E`0.1 V) strength and then drops down rather quickly and approaches towards zero as the field is slightly increased. The rapid increase in heat transfer at very weak field strength is attributed to charge effect. The reduction in heat transfer noted in this case is associated with the number of particles involved in the energy transfer mechanism. The total number of suspended particles within the cylinder is fixed. Moreover, the cylinder is maintained at positive potential, while the particles are highly negative for higher pH values. As a result, the number of particles present in the fluid medium decrease gradually with increase in electric field, causing a reduction in heat transfer coefficient. A zero value of heat transfer coefficient implies a complete removal of the charged particles from the fluid medium. It is to be noted that the voltage scale used for an a.c. convection is slightly larger than the d.c. convection and this is due to relaxation phenomena associated with the a.c. field.3.1.1. C o r r e l a t i o n o f m e a s u r e d d a t a . One of the primary objectives of this work is to find a relationship between the dimensionless quantities of Nusselt number, Rayleigh number and an appropriate geometric parameter. The mean Nusselt and Rayleigh numbers are, respectively, defined by
(Nuel)L4 helL l , (8) RaL4 gbTdL3 n2 Pr , (9)
where n is the kinematic viscosity, b the thermal expansion coefficient, L the length of the cylinder, l the thermal conductivity of the fluid and the subscript el denotes electric field. The fluid properties were evaluated at the film temperature Tf, defined
by
Tf4
Tw1 TB
2 ,
(10)
where Twis the temperature of the wire and TBis the temperature of the surrounding
medium. The typical values of the parameters used in eqs. (8) and (9) were taken from ref. [22, 23].
Many previous analyses [24-26] of the enclosure free convective heat transfer data have shown that correlations of the form
NuL4 C1RaC2,
(11)
where C1and C2are constants, could correlate extremely well with vertical cylindrical
M.F.HAQUEandS.ARAJS
676
Fig. 5. – Correlation of data for electroconvective heat transfer from vertical cylinder in air (d.c. field). 1) (Nuel)L4 0 .70(RaL)0 .49, 2) (Nuel)L4 ( 2 .9 )(GrD)20 .05(RaL)0 .44, 3) (Nuel)L4
0 .68 10.67(RaL)0 .51] 1 1 ( 492 OPr)0 .56(20 .44, 4) (Nuel)L4 0 .637(RaL)0 .51( 1 10.861OPr)20 .25,
5) (Nuel)L4 0 .678(RaL)0 .50( 1 10.952OPr)20 .25.
the test fluid reported by Warrington and Powe [27] are given by the formula NuL4 0 .479 RaL0 .171.
(12)
In the present study, the heat transfer is described in terms of average Nusselt number for some typical Rayleigh number which corresponds to the linear part of the experimental data. The experimental data have been correlated by dimensional analysis. The data have been compared with the empirical correlations and analytical
expressions, the correlations used for electroconvection being the modified
correlations for free convection. In fig. 5, 6, 7 and 8, the heat transfer correlation Nuelis
plotted against Ra for both the experimental results and predicted values of various correlations for air, silicon oil, distilled water and colloidal hematite particle suspension in distilled water. The analytical results are also shown on the same plot. The experimental values for air, silicon oil and distilled water are slightly higher than the predicted correlations for Ra D6.75Q106, 11 Q 106 and 3 .25 Q 108, and slightly lower than the above values. However, the experimental values for colloidal hematite particle suspension in distilled water are fairly in good agreement with the predicted relations for the Rayleigh number considered. Curve 1 in fig. 5, 6, 7 and 8 is the predicted
Fig. 6. – Correlation of data for electroconvective heat transfer from horizontal cylinder in silicon oil. 1) (Nuel)D40 .32(RaD)0 .45, 2) (Nuel)D4( 0 .58 )(GrD)20 .05(RaD)0 .44, 3) (Nuel)D4
]0.6010.387(RaD)0.245[11(0.559OPr)0.56]20.30(2, 4) (Nuel)D40.637(RaD)0.405(110.861OPr)20.25,
5) (Nuel)D4 0 .678(RaD)0 .404( 1 10.952OPr)20 .25.
correlation obtained by using the equation
(Nuel)L , D4 m(RaL)n,
(13)
where L or D implies vertical or horizontal cylinders and m and n are constants. In the case of air, silicon oil, distilled water and colloidal hematite particle suspension in distilled water, eq. (13) was found to have the following forms:
.
`
/
`
´
(Nuel)air4 0 .70(RaL)0 .49,
(Nuel)silicon oil4 0 .32(RaD) 0 .45
, (Nuel)DH2O4 0 .32(RaD)
0 .36
, (Nuel)a-Fe2O3in DH2O4 0 .32(RaD)
0 .405.
(14)
Al-Arabi and Khamis [28] have correlated heat transfer data in the absence of electric field for cylinders of various lengths, diameters, and angles of inclination to the
vertical. Their results are of the form NuL4 m(GrLQ Pr)n, where the bar denotes
M.F.HAQUEandS.ARAJS
678
Fig. 7. – Correlation of data for electroconvective heat transfer from horizontal cylinder in distilled water. 1. (Nuel)D4 0 .32(RaD)0 .36, 2) (Nuel)D4 0 .58(GrD)20 .05(RaD)0 .375, 3) (Nuel)D4
] 0 .6010 .387(RaD)0 .20[ 11(0.559OPr)0 .56]20 .30(2, 4) (Nuel)D40 .637(RaD)0 .33( 110.861OPr)20 .25,
5) (Nuel)D4 0 .678(RaD)0 .325( 1 10.952OPr)20 .25.
inclination to the vertical, u. In the laminar regime, they found the relation NuL4 [ 2 .9 2 2 .32( sin u)0 .8](GrD)21 O12(GrLQ Pr)(1 O41 (1O12)( sin u)
1 .2)
(15)
where the Grashof number based in cylinder diameter is restricted to the range 1 .08 Q 104G GrDG 6 .9 Q 105. In the case of vertical (u 40) and horizontal (u4907) cylinder,
eq. (15) becomes NuL4 ( 2 .9 )(GrD)21 O12(RaL)1 O4 (16a) and NuL4 ( 0 .58 )(GrD)21 O12(RaD)1 O4, (16b)
respectively. The empirical correlation for free convection can be used to fit the experimental data for electroconvection if the 1 O4 power law in Ra is adjusted. In the case of air, silicon oil, distilled water and colloidal hematite particle suspension in
Fig. 8. – Correlation of data for electroconvective heat transfer from horizontal cylinder in colloidal suspension (a–Fe2O3 in distilled water). 1) (Nuel)D4 0 .32(RaD)0 .405, 2) (Nuel)D4
( 0 .58 )(GrD)20 .05(RaD)0 .42, 3) (Nuel)D4 ] 0 .60 1 0 .387(RaD)0 .225[ 1 1 (0.559OPr)0 .56]20 .30(2,
4) (Nuel)DH40.637(RaD)0 .37( 110.861OPr)20 .25, 5) (Nuel)D40 .678(RaD)0 .368( 110.952OPr)20 .25.
distilled water, eqs. (16a) and (16b) take the following forms:
.
`
/
`
´
(Nuel)aira (RaL)0 .44,
(Nuel)silicon oila (RaD) 0 .44
, (Nuel)DH2Oa (RaD)
0 .375,
(Nuel)a-Fe2O3in DH2Oa (RaD)
0 .42,
(17)
where the parameter affected by the electric field is shown. This result is represented in curve 2 in fig. 5, 6, 7 and 8. Curve 3 in fig. 5, 6, 7 and 8 resulted from the predicted correlation proposed by Churchill and Chu [29]. In the case of vertical and horizontal cylinder, we have NuL4 0 .68 1 0 .67(RaL)1 O4
k
1 1g
0 .492 Prh
9 O16l
24 O9 , 0 ERaLE 109 (18a)M.F.HAQUEandS.ARAJS 680 and NuD4
m
0 .60 10.387(RaD)1 O4k
1 1g
0 .559 Prh
9 O16l
24 O9n
2 , (18b)respectively. Equations (18a) and (18b) can be used to fit the experimental data for electroconvection if the 1 O4 power law in Ra is adjusted. In the case of air, silicon oil, distilled water and hematite particle suspension in distilled water, the affected parameter in eqs. (18a) and (18b) becomes
.
`
/
`
´
(Nuel)aira (RaL)0 .51,
(Nuel)silicon oila (RaD)0 .245,
(Nuel)DH2Oa (RaD)
0 .20,
(Nuel)a-Fe2O3in DH2Oa (RaD)
0 .225.
(19)
Finally, the experimental data have been compared with the analytical expression available for free convection [22]. For vertical cylinders, the expression is
(Nu)L4 0 .637(RaL)1 O4
g
1 1 0 .861 Prh
21 O4 , (20a) (Nu)L4 0 .678(RaL)1 O4g
1 1 0 .952 Prh
21 O4 . (20b)The experimental data for electroconvection can now be compared with the analytical expressions (20a) and (20b) if the 1 O4 power law in RaLis also adjusted. In the case of
air, silicon oil, distilled water and hematite particle suspension in distilled water, the affected parameter in eqs. (20a) and (20b) becomes
.
`
`
/
`
`
´
(Nuel)aira (RaL)0 .51R
(
from eq . ( 20 a))
,(Nuel)aira (RaL)0 .50R
(
from eq . ( 20 b))
,(Nuel)silicon oila (RaD)0 .405R
(
from eq . ( 20 a))
,(Nuel)silicon oila (RaD)0 .404R
(
from eq . ( 20 b))
,(Nuel)DH2Oa (RaD)
0 .33R
(
from eq . ( 20 a))
,(Nuel)DH2Oa (RaD)
0 .325R
(
from eq . ( 20 b))
,(Nuel)a-Fe2O3in DH2Oa (RaD)
0 .37R
(
from eqs . ( 20 a) and ( 20 b))
.(21)
Turning now to the overall heat transfer from the heated platinum wire as shown above, the predicted value of the average Nusselt number for Rayleigh number from 5 Q 106 to 5 Q 108 agrees very well (` 80 % ) with the experimental value. The experimental
result predicts a steeper rise in Nusselt number with increasing Rayleigh number, than the predicted ones; the exponent of the Ra value lies between 0.20 and 0.51. The increase in exponent of the Rayleigh number with respect to the free convection indicates that the efficiency of convection increases dramatically when an electric field is applied to a free convective motion.
3.2. Convection in the presence of magnetic and electromagnetic field. – The convection of electroconducting fluid receives the Lorentz force in a magnetic field. The Lorentz-force term F is given as follows [30]:
F 4rsEr1 j 3 B ,
(22)
where rs is the electric charge density of the fluid, Erthe electric field intensity, j the
electric current density and B the magnetic field. The Maxwell equation for a magneto-electric field gives
div D 4rs,
(23)
where D is the electric displacement and is related to Er as follows: D 4eEr,
(24)
where e is a dielectric constant. On the other hand, Ohm’s law holds for this system,
j 4rsu 1ss(Er1 u 3 B) ,
(25)
where ss is the electric conductivity and u the fluid velocity vector. In this
electromagnetic field, the high-frequency wave such as light is not treated and the displacement current is zero,
ˇD
ˇe 4 0 . (26)
The fluid is assumed to be electrically neutral and the convective term rsu is neglected
in Ohm’s equation. Equations (22) and (25) become
F 4j3B ,
(27)
j 4ss(Er1 u 3 B) .
(28)
The results of the magnetoconvective and electromagnetoconvective heat transfer coefficient for O2, N2O, kerosene and colloidal hematite particle suspension in distilled
water are presented in table I. As seen in the table, magnetic and electromagnetic Nusselt number is presented as a function of magnetic field. It is apparent from the table that there exists a critical Nusselt number below which no convection is observed. As the magnetic field is increased gradually, magnetoconvective heat transfer coefficient increases and this in turn increases the magnetic Nusselt number. At higher field strength the magnetic Nusselt number approaches a stationary value as noted both in O2and N2O. A similar trend is also noted when electric and magnetic fields are
applied simultaneously. In the case of kerosene and colloidal suspension (a-Fe2O3 in
distilled water), a decrease in Nusselt number is noted with an increase in magnetic or electromagnetic field.
At lower magnetic field, the magnetic torque is small, the flux is expelled from most of the convecting region and concentrated at the axis and periphery of the cell. As the magnetic field is increased, the Lorentz force generates vorticity with the opposite sense to that produced by the buoyancy force, and so reduces the radial inflow towards the axis. Magnetic flux is expelled from the toroidal eddy and the motion is excluded from the central flux rope. However, the velocity is affected only in a narrow region
M.F.HAQUEandS.ARAJS
682
around the axis and the convective heat transfer is increased. For yet larger field strength, the flux rope expands until it fills the cell and convection is ultimately suppressed.
3.2.1. E f f i c i e n c y o f c o n v e c t i o n . The efficiency of free convection is defined as the transformation of the rate of electrical power supplied to the system into the rate of kinetic-energy generation of convective motion [31]. In the present study, the view was taken that the measured heat transfer coefficient in the presence of magnetic and electromagnetic field represented a useful conversion of additional electrial energy into the rate of generation of the kinetic energy of the convective motion. The efficiency
gn of convection in the presence of magnetic and electromagnetic field has been
calculated using the following empirical relation: gn4
hn
hn1 hf
, (29)
where hf is the free convective heat transfer coefficient and hn is the heat transfer
coefficient due to electric, magnetic or electromagnetic fields (i.e. n f el, m or em). In the case of gaseous oxygen and nitrous oxide, an increase in efficiency is noted with an increase in magnetic field and, at higher field strength, it approaches towards a stationary value (table II). A similar trend is noted when electric and magnetic fields are applied simultaneously. A zero value of gmimplies an absence of convection, while a
positive value of gm implies an enhancement of free convection by magnetoconvection.
The efficiency obtained in an electromagnetic field is found to be higher than the
magnetic field. The non-zero value of gem at zero magnetic field resulted from the
application of electric field alone. It is noted that the value of gemat non-zero magnetic
fields is non-additive. The non-additivity of electric and magnetic field implies a complicated thermal interaction between electric and magnetic field. In the case of
kerosene, a negative value of gm is noted. The negative behaviour of gm implies a
suppression of the free convection by magnetoconvection. The efficiency obtained under the influence of electric and magnetic field is also negative, since the electroconvection at lower electric field is negative. Therefore, NgemNDNgmN implies a
further suppression of magnetoconvection by electroconvection. The efficiency obtained for a colloidal suspension under the influence of magnetic field is also negative. However, the efficiency obtained under the influence of electric and magnetic field is positive, since electroconvection at very weak field is positive. The positive value
of gem implies an enhancement of the free convection by magnetoconvection, but the
magnitude decreases gradually as the magnetic field is increased.
4. – Conclusion
Laminar convective heat transfers in cylindrical enclosure have been carried out under the influence of electric, magnetic and electromagnetic fields. Of particular interest is the existence of a critical electric field above which a convective motion can be developed by an electric field. A fully developed laminar convection is observed up to the Rayleigh number 5 Q 108. The experimental results agree very well (` 80 % ) with the empirical correlations and analytical expressions up to the Rayleigh number considered. It is observed that the empirical correlation and analytical expression for electroconvection follow laws analogous to free convection with the exception that the
one-fourth power law in Rayleigh has been changed and adjusted between 0.20 and 0.51. The higher power law in Rayleigh suggests that the efficiency of convection increases dramatically when an electric field is applied to free convective motion. The maximum efficiency noted in air and freon-22 is 60% and 69% while that in silicon oil, distilled water and colloidal hematite particle suspension in distilled water is 52%, 39% and 71%, respectively. However, in the case of air and freon-22, a minimum in efficiency is also noted in an a.c. field. In the case of O2 and N2O, both magnetic and
electro-magnetic Nusselt number increase with an increase in electro-magnetic or electroelectro-magnetic fields and at higher field strengths, it approaches towards a saturation value. However, in the case of kerosene and a colloidal suspension (a–Fe2O3 in distilled water), both
magnetic and electromagnetic Nusselt number decrease with an increase in magnetic or electromagnetic fields and at higher field strength, it approaches towards a stationary value. An increase in efficiency is noted in O2and N2O under the influence of
magnetic field, and similar trend is also noted when electric and magnetic fields are applied simultaneously. However, in the case of kerosene and a colloidal suspension (a–Fe2O3 in distilled water), the efficiency decreases gradually with an increase in
magnetic or electromagnetic field. The maximum efficiency observed under the influence of magnetic field corresponds to oxygen and hence in the present investiga-tion, oxygen is found to be the most suitable fluid for magnetoconvection.
* * *
The experimental part of this study was carried out at the Physics Department, Clarkson University, Potsdam, New York, U.S.A. Research grant awarded by Abubakar Tafawa Balewa University of Technology, Bauchi, Nigeria, in connection with the preparation of this paper is also gratefully acknowledged.
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