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Process Intensification in Artificial Gravity Mugurel C. Munteanu, Ion Iliuta, and Faı1c¸ al Larachi*,† Department of Chemical Engineering, Laval University, Quebec G1K 7P4, Canada
Powerful superconducting magnets constitute adequate proxies for generating artificial gravity environments in earthbound experimentations. The application of microgravity or macrogravity conditions could be interesting for the pharmaceutical and medical domains for discovering and identifying new drugs and their actions. In the chemical engineering area, strong inhomogeneous magnetic fields could potentially open attractive applications. For example, in multiphase catalytic systems, several factors must be optimized for improving process efficiency. Preliminary experimentations and model calculations reveal that inhomogeneous and strong magnetic fields, applied to such systems as mini trickle-bed reactors, are capable of affecting reactor hydrodynamics, which can be taken advantage of for improving process performance. Pressure drops, liquid holdups, and wetting efficiency experimental data have been obtained for two-phase downward gas-liquid trickle beds in the presence of inhomogeneous magnetic fields. Magnetic field effects on trickle bed hydrodynamic properties have been explained using the gravitational amplification factor that commutes the Kelvin body force density into an artificial gravitational body force. Introduction The principal target for process intensification is to produce highly efficient reaction and processing systems using configurations that simultaneously significantly reduce reactor sizes and maximize catalytic mass- and heat-transfer efficiencies. It represents a novelty in chemical engineering, which will potentially transform the notion of chemical processing, leading to small, safe, energy-efficient, and environmentally friendly processes.1 Shortening the development time from laboratory to commercial production through the use of new methods that permit the researcher to obtain better conversion or selectivity is one of the highest priorities of process intensification studies. This approach would be particularly advantageous for the fine and pharmaceutical industries, where production amounts are often smallsless than a few metric tons per year.2 The ability to influence the process behavior through the application of external inhomogeneous magnetic fields is of potential practical interest in the operation of chemical reactors from the point of view of process intensification for catalytic reactions. The magnetization force generated by an inhomogeneous magnetic field is a body force and is analogous to the gravitational force. The main characteristics of the magnetization body force density, compared with gravity, is the direction control to realize artificially microgravity or macrogravity conditions:
F MR )
χR dB B µ0 dz
(1)
where χR is the volume magnetic susceptibility, µ0 the absolute magnetic permeability of a vacuum, and B magnetic induction or magnetic flux density. The (mag* Corresponding author. E-mail:
[email protected]. † Current address: ARKEMA/TOTALsCentre Technique de Lyon Chemin de la LoˆnesBP32 69492 Pierre-Be´nite, Ce´dex, France.
netization) Kelvin body force density acts upon nonelectrically or weakly electrically conducting nonmagnetic fluids and must be distinguished from the Lorentz braking force, which is the cornerstone of magnetohydrodynamics and concerns fluids that are electrically conducting and nonmagnetic (e.g., liquid metals, strong electrolytes, and plasmas). Fluids commonly encountered in chemical engineering applications are organic, often nonelectrically conducting, or aqueous, often exhibiting low to moderate ionic strengths. In addition, such fluids are nonmagnetic; that is, their volume magnetic susceptibilities are extremely low, usually on the order of 10-6-10-7. To make eq 1 yield meaningful Kelvin body force densities that are able to affect fluid flows, strong magnetic fields (several Tesla (T)), along with strong magnetic field gradients (several dozens of T/m) are necessary, which can be achieved only through the use of superconducting magnets. In the past decade, there have been some reports indicating that the magnetization force reduces or increases the effect of gravity. Beaugnon and Tournier3 and Ikezoe et al.4 succeeded in levitating water, acetone, ethanol, and organic materials by applying an inhomogeneous magnetic field. Using an inhomogeneous magnetic field, Wakayama et al.5 confirmed that vertical acceleration can be changed continuously from normal gravity to near zero gravity. The flow and diffusion characteristics for oxygen gas under inhomogeneous magnetic fields were reported by Tagawa et al.6 Recently, Wang and Wakayama7 published a detailed numerical study for the methods of controlling natural convection in nonconducting and low-conducting diamagnetic fluids contained in cubical enclosures under inhomogeneous magnetic fields oriented in different directions. The magnetic field effects on biochemical systems were also studied. Lin et al.8 found that the quality of the protein crystals was enhanced or deteriorated when a magnetization force was applied to protein solutions. This effect on proteins may be of major importance for the biological and pharmaceutical industry.
10.1021/ie050195p CCC: $30.25 © 2005 American Chemical Society Published on Web 06/24/2005
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In the present paper, application of strong inhomogeneous magnetic fields on a miniature trickle-bed reactor was studied. Most commercial trickle-bed reactors operate adiabatically at high temperatures and high pressures and generally involve hydrogenation, oxidation, desulfurization, etc. The most important hydrodynamic properties for trickle-bed reactors are liquid holdup, wetting efficiency, and pressure drop. Liquid maldistribution, the formation of hot spots, and a decrease of selectivity are serious problems that emanate during reactor operation. The numerical simulations concerning the behavior of trickle beds under inhomogeneous magnetic fields published recently by Iliuta and Larachi9 indicated that elevated levels of magnetic-field gradients can either improve or deteriorate appreciably liquid holdup and, consequently, wetting efficiency for the catalytic particles. This latter becomes a crucial issue when the reactor is processing liquid-limited or gas-limited catalytic reactions. For liquid-limited reactions, the highest possible wetting efficiency and particle-liquid mass-transfer rates result in the fastest transport of the liquid-phase reactant to the catalyst. Conversely, for gas-limited reactions, it is advantageous to reduce the extra mass-transfer resistance added by the liquid phase, without the danger of gross liquid maldistribution and hot-spot formation. The purpose of this paper is to provide original experimental data, as well as comparisons for two-phase downward gas-liquid trickle flow characteristics through beds of nonporous particles in the presence and in absence of inhomogeneous magnetic fields. An attempt will be made to model and interpret the magnetic field effects on the hydrodynamics of trickle-bed reactors. Theory To quantify the relative importance of gravity versus magnetization for each fluid phase, a gravitational amplification factor was defined9 for each phase to commute magnetization into apparent artificial gravity acceleration. Assuming the direction of nonhomogeneity of the magnetic field is parallel to the gravitational field (radial magnetic field nonhomogeneities neglected), the gravitational amplification factor can be written as follows:
γR )
FRg + FMa FR g
χR dBz )1+ B FRgµ0 dz
(2)
Depending on the signs of magnetic susceptibility (>0 for paramagnetic and 1, indicating macrogravity or hypergravity, (b) 1 > γR > 0, indicating subgravity (microgravity or hypogravity), or (c) γR ) 0, indicating levitation. When the weight is largely counter-balanced by the magnetization force, then γR becomes negative. This case will not be considered in this work. To estimate the liquid holdup and pressure drop variations in the inhomogeneous magnetic field, the artificial gravity concept was used. The magnetic body force was considered in terms of artificial gravity force. Performing a one-dimensional force balance for both phases gives a couple of relationships between pressure gradient, phase holdups, body force densities, and interfacial liquid-solid and gas-liquid drag force densi-
ties. Assuming complementarity between liquid and gas holdups, equality between gas-side and liquid-side pressure gradients (negligible capillary pressure), and neglecting in the formalism partial-wetting-subtended terms (by assuming full wetting), the force balance equations become
gχg dBz dP -g B + gFgg + - Fgl ) 0 dz µ0 z dz -l
(3)
lχl dBz dP + lFlg + + Fgl - Fls ) 0 B dz µ0 z dz
(4)
After substitution of eq 2 into eqs 3 and 4, one obtains the following for each phase:
dP + gFgγgg ) Fgl -g dz
(5)
dP + lFlγlg ) - Fgl + Fls dz
(6)
-l
In the case of trickle flow, it has been shown that, under certain conditions, the slit flow approximation yields a very satisfactory set of constitutive equations for the gas-liquid and liquid-solid drag forces.10,11 As a matter of fact, the slit flow becomes well-representative of the trickle flow regime when the liquid texture is mainly contributed by catalyst-supported liquid films and rivulets, and the gas-liquid separated flow assumption holds. This generally occurs at low liquid flow rates that allow the transport of filmlike liquids.10 We will assume, without proof though, that such hypotheses also hold in the case of artificial gravity operation. The validity of these assumptions and of the several others outlined above will be evaluated later on in terms of model versus experiment comparisons. Choosing the drag force closures of the simplified Holub slit model,10 the system of equations becomes
Ψg ) -
( )( ( )(
dP 1 +1) dz Fgγgg g
3
dP 1 +1) Ψl ) dz Flγlg l
) )
Reg Reg2 E1 + E2 Gag Gag
(7)
Rel Rel2 E1 + E2 Gal Gal
(8)
3
Note that, in eqs 7 and 8, the Galileo numbers use the corresponding artificial gravity values for the liquid and for the gas phase.
GaR )
FR2γRgdp33 ηR2(1 - )3
(9)
Equations 7-9 are an adapted form of the slit model of Holub et al.10 to the context of a trickle bed experiencing artificial gravity conditions. Experimental Section Figure 1 illustrates the experimental setup. The inhomogeneous magnetic field, up to 9 T, was obtained with a superconducting NbTi solenoid magnet system (American Magnetics, Inc.) that has a bore diameter of 2.5 cm. To reach the superconducting state, the solenoid temperature was decreased at 4.2 K using liquid nitrogen and liquid helium. The magnetic field then was
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Figure 1. Experimental setup for applying an inhomogeneous magnetic field. Table 1. Range of Parameters Used in the Magnetically Controlled Artificial Gravity Experimentation of the Trickle Bed Reactor parameter
value(s)/range
water superficial velocity, vl air superficial velocity, vg water density, Fl air density, Fg water kinematic viscosity, υl air kinematic viscosity, υg water magnetic susceptibility (volume), χl gas magnetic susceptibility (volume), χg superficial velocity of aqueous ethanol solution, ve magnetic susceptibility of aqueous ethanol solution (volume), χe diameter of glass beads, dp bed porosity, bed length, L reactor internal diameter temperature pressure
0-9 × 10-3 m/s 0-50 × 10-3 m/s 1000 kg/m3 1.2 kg/m3 1 × 10-6 m2/s 1.86 × 10-5 m2/s -9.02 × 10-6 0.379 × 10-6 0.1 × 10-3-1.2 × 10-3 m/s -7.26 × 10-6 0.6 × 10-3 and 1 × 10-3 m 0.36 and 0.37 40 × 10-3 m 17 × 10-3 m 25° C 1 atm
generated and controlled by a computerized control system. The direction of the magnetic field was vertical. The system produces a maximum product gradient (B dB/dz) of 650 T2/m around point A or A′, which is 12.5 cm away from the solenoid center. A miniature trickle-bed reactor equipped with nonmagnetic inlet and outlet valves was positioned in the maximum product gradient area of the atmospheric magnet bore. A water-air system was used. The magnetic field strength was controlled by changing the current intensity through the magnet. The magnetic field intensity inside and outside of the bore was measured using a Hall effect Gaussmeter (model GM700, Cryogenics, Inc.). The magnetic susceptibilities of the fluids involved in the experiment were determined using an alternating gradient magnetometer (model MicroMag 2900, Princeton Measurements Corporation). The experiments were performed at different flow rates of water and air controlled by flow meters. The ranges of flow rates, fluids properties, reactor dimensions, and packing diameters are summarized in Table 1. A differential pressure transducer (C 9551, Comark, Ltd.) was used to determine the pressure drop.
For the liquid holdup measurements, the miniature reactor was weighted prior to starting the experiments (dry bed). The magnetic field was turned on and the experiments resumed. After the system reached steady state, the two valves were closed instantaneously and the reactor was weighted again (wetted bed). The same procedure was applied for different gas and liquid flow rates. The wetting efficiency experiments were performed in the same system, using a colorimetric method. The glass beads were colored before the experiment with a crystal violet aqueous solution of known concentration. The miniature reactor with the colored beads inside was inserted into the solenoid bore. In the wetting efficiency runs, an aqueous ethanol solution was used instead of pure water. The liquid volume and gas flow rate were held constant, and the liquid flow rate was increased periodically. Depending on the liquid and gas flow rates, the color of the glass beads was washed out by the aqueous ethanol solution. The solution transmittance was measured at each liquid flow rate change, using a spectrometer (model Spectronic 20, Milton Roy Company). The final concentration (no color on glass beads)
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Figure 2. Simplified Holub slit model liquid holdup data (lines) versus experimental liquid holdup for 1-mm-diameter glass beads (symbols), without magnetic field, at different gas velocities: (s, 9) 0 m/s, (- - -, 0) 0.007 m/s, (- - -, [) 0.02 m/s, and (- - -, ]) 0.03 m/s.
Figure 3. Simplified Holub slit model pressure drop data (lines) versus experimental pressure drop for 1-mm-diameter glass beads (symbols), without magnetic field, at different gas velocities: (s, 9) 0.007 m/s, (- - -, 0) 0.02 m/s, (- - -, [) 0.03 m/s, and (- - -, ]) 0.04 m/s.
was considered to be 100% when the glass beads were fully washed and the wetting was total. The wetting efficiency was calculated as being the quotient between the intermediate and final concentrations. The aqueous ethanol solution physical properties are presented in Table 1. The experiments were realized under normal conditions (0 T) and in the presence of a 9-T magnetic field. Results and Discussion The miniature trickle-bed reactor was placed in the area of the solenoid bore marked “A” in Figure 1. The first experiments were performed to measure the liquid holdup, the pressure drop, and the wetting efficiency for both particle diameters, in the absence of a magnetic field. Before starting the experiments, the bed was prewetted. A sufficient time was allowed for the system to reach steady state before measurements were acquired. A large number of experiments were performed to ensure data reproducibility. Experimental data were compared with the predictions of the Holub et al.10 slit model. The single-phase flow Ergun equation coefficients were determined for each bed diameter, and the values observed were E1 ) 155 and E2 ) 1.14, respectively, for the Kozeny-Karman laminar constant and the Burke-Plummer inertial constant. Figures 2 and 3 show the experimental data versus Holub’s simplified model for the trickle flow regime without a magnetic field. As can be seen, the slit model describes the liquid holdup and pressure drop variations for the system investigated in our work very well. The same conclusion can be formulated for the 0.6-mm-diameter glass beads. In the second part of the experiments, the magnetic field was turned on. A 9-T magnetic field was generated with a +650 T2/m product gradient in the vicinity of point A (see Figure 1). The magnetic body force was calculated for water, which is a diamagnetic fluid (χ < 0), and for air, which is a paramagnetic gas (χ > 0). Figure 4 presents the gravitational amplification factor for both phases, as a function of the magnetic field product gradient. For a B dB/dz value of +650 T2/m, the magnetization force acts upwardly for the liquid phase (γ ) 0.52, hypogravity) and downwardly for the gas phase (γ ) 17.66, hypergravity). The values of the magnetization body forces are FMl ) -4679 N/m3 for water and FMg ) 196 N/m3 for air. As it results from
Figure 4. Gravity values for a water-air system at 9-T magnetic field strength: (s) water and (- - -) air.
Figure 5. Liquid holdup in the absence (open symbols) and in the presence (filled symbols) of a 9-T magnetic field at different gas velocities for 1-mm-diameter glass beads: (], [) 0 m/s, (4, 2) 0.02 m/s, and (O, b) 0.04 m/s.
the calculated values, the liquid-phase magnetization force is more important than the gas magnetization force, and it could be assumed that the liquid magnetization force is controlling. Figure 5 shows the liquid holdup experimental data for 1-mm-diameter glass beads for different liquid and gas flow rates. In the presence of a magnetic field, the
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Figure 6. Pressure drop in the absence (open symbols) and in the presence (filled symbols) of a 9-T magnetic field at different gas velocities for 1-mm-diameter glass beads: (0, 9) 0.007 m/s, (O, b) 0.03 m/s, and (4, 2) 0.04 m/s.
Figure 7. Artificial gravity simplified Holub slit model liquid holdup (lines) versus 9-T experimental data for 1-mm-diameter glass beads (symbols) at different gas velocities: (s, 9) 0 m/s, (- - -, 0) 0.007 m/s, (- - -, [) 0.02 m/s, (- - -, ]) 0.03 m/s, and (- - - -, 2) 0.04 m/s.
liquid holdup increases, because the resistance to liquid flow increases. In this case, the liquid magnetization force is more important than the gas magnetization force and it becomes controlling. For the liquid phase, hypogravity conditions prevail. The same results were observed for both glass-bead diameters. The liquid holdup values for 0.6-mm-diameter glass beads diameter are higher than the 1-mm-diameter glass beads values and the magnetic field influence is less visible (not shown in this paper). Figure 6 presents the pressure drop experimental data for 1-mm-diameter glass beads for the water-air system. For two-phase flow, the pressure drop in the presence of magnetic field was determined to increase comparatively with the pressure drop measured in the absence of magnetic field. For positive magnetic field gradients, the liquid magnetization force, being a controlling factor, reduces the effect of gravity but increases the resistance of the liquid flow and enhances the pressure drop. The pressure drop for 0.6-mm-diameter glass beads is larger than the pressure drop for 1-mmdiameter glass beads for normal and artificial gravity conditions (not shown in this paper). Figures 7 and 8 present a comparison between the liquid holdup and pressure drop experimental data
Figure 8. Artificial gravity simplified Holub slit model pressure drop (lines) versus 9-T experimental data for 1-mm-diameter glass beads (symbols) at different gas velocities: (s, 9) 0 m/s; (- - -, 0) 0.007 m/s; (- - -, [) 0.02 m/s; (- - -, ]) 0.03 m/s and (- - - -, 2) 0.04 m/s.
obtained in artificial gravity conditions, along with prediction from the apparent-gravity modified Holub et al.10 slit model for the 1-mm-diameter glass beads. For single-phase flow (no gas), the model predictions are very similar to the experimental values and, in this case, the magnetic field effect on the hydrodynamics properties is described well by the artificial gravity concept. For two-phase flow, the tendencies are similar, but there are some differences between the experimental data and model predictions. The experimental values of liquid holdup and pressure drop are generally underpredicted by the model. Reasons for such mismatch between experimental and predicted data are the assumptions inherent to the model that might no longer be valid when the magnetic field is turned on. For instance, the slit model neglects the velocity and shear slip factors, which characterize the degree of phase interactions at the gas/liquid interface. Another factor that yields differences between model and experimental measurements is the Moses effect, which is defined as a deformation of the liquid surface profile by strong magnetic fields. The Moses effect was studied previously by Sugawara et al.,12 who observed that the interfacial shape between two immiscible nonmagnetic liquids changed under an applied inhomogeneous magnetic field. In the liquid film, a parabolic liquid velocity profile is used. At the interfaces between the catalyst and liquid phase and between the liquid and gas phases, the Moses effect is present and the surface shapes could be deformed, modifying the liquid velocity profile. Another reason for the mismatch between experimental and predicted data could be the assumption of homogeneity of the magnetic field in the radial direction. Also, the peak value for the axial magnetic field product gradient (+650 T2/m) was used in simulations for the entire bed length and was considered constant, although, in reality, the axial magnetic field product gradient value is not constant and is dependent on point position, relative to the solenoid center (see magnetic field longitudinal profile in Figure 1). Figure 9 presents the wetting efficiency experimental data for the 1-mm-diameter glass beads for the aqueous ethanol solution-air system. It is known that the wetting efficiency increases as the liquid mass velocity increases,13 and, at high liquid velocities, the catalyst surface area becomes totally wetted. To unveil the magnetic field effects and to prevent the system from
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Figure 9. Wetting efficiency data in the absence (open symbols) and in the presence (filled symbols) of a 9-T magnetic field at different gas velocities for 1-mm-diameter glass beads: (0, 9) 0.007 m/s and (4, 2) 0.02 m/s.
attaining full wetting very quickly, very low liquid superficial velocities were used in this work, down to ca. 10-5 m/s. The wetting efficiency increases in the presence of an inhomogeneous magnetic field, and this is ascribed to the magnetization force that corresponds to hypogravity conditions for the liquid phase, thereby allowing a longer liquid residence time inside the trickle-bed reactor. The hypergravity status for the gas phase increases the gas interstitial velocity through the reactor, thus forcing the liquid films to spread more efficiently and cover wider surfaces on the glass beads, thereby increasing wetting efficiency. The same results were observed for the bed of 0.6-mm-diameter glass beads. The average wetting efficiency increase rate is 18%. The gravitational amplification factor value for the aqueous ethanol solution was calculated as being 0.51, which is very similar to the water gravitational amplification factor calculated previously (0.52). Improvement of wetting efficiency due to the magnetic field is directly related to simultaneously increasing the liquid holdup, as shown previously. The wetting efficiency of the catalytic particles has a role in the control of mass and heat transfer between the gas, liquid, and catalyst. Larger wetting efficiency values help the bed to exhibit better mesoscale (or particle-wise) liquid distribution, which is beneficial against the risk of hotspot formation or when liquid-limited reactions are being conducted in the reactor. The wetting efficiency experimental data obtained in the 9-T magnetic field were compared with the data predicted by the correlation of Al-Dahhan and Dudukovic.14 The artificial gravity concept was used in the correlation, and normal gravity was converted to an artificial gravity value. Figure 10 shows the wetting efficiency experimental data in the 9-T inhomogeneous magnetic fields versus the theoretical wetting efficiency values predicted by the Al-Dahhan and Dudukovic14 correlation using the artificial gravity concept for a gas velocity of 0.007 m/s. The experimental errors in the wetting efficiency measurements, the magnetic field presence and its side effects, the reduced dimensions of the reactor, the fact that the correlation normal working conditions are high pressure and one-g conditions are factors that could explain the discrepancies noted at the highest and lowest intervals of the liquid superficial velocity.
Figure 10. Experimental wetting efficiency data in 9-T magnetic field (data points) versus Al-Dahhan and Dudukovic correlation13 (solid line) for 1-mm-diameter glass beads at a gas velocity of 0.007 m/s.
Figure 11. Experimental wetting efficiency data in a 9-T magnetic field (symbols) versus the Larachi neural network correlation14 (lines) at different gas velocities for 1-mm-diameter glass beads: (0, s) 0 m/s, (], - - -) 0.007 m/s, and (4, - - -) 0.02 m/s.
Another comparison was made between the experimental data and the theoretical values obtained using the Larachi et al.15 wetting efficiency correlation. The artificial gravity values were used in the neural network correlation formulas, and the results are shown in Figure 11. Using the artificial gravity concept, the wetting efficiency decreases with increasing gas flow rate, which is opposite to experimental expectations. The impact of gas properties on the wetting efficiency is complex and, even in normal gravity conditions, it is not clearly elucidated. The artificial gravity and the magnetic field could affect and change the expected trends by unknown effects. Moreover, the database that was used for establishment of the Larachi et al.15 correlation does not contain information concerning wetting efficiency in artificial gravity conditions, thus explaining its inaptitude to represent artificial gravity wetting efficiency measurements. We compared, for the same liquid and gas flow rates, the liquid holdup and wetting efficiency data with 0-T and 9-T magnetic fields. This allowed us to see a clear effect of magnetic field on the hydrodynamic parameters under otherwise similar conditions. It is true that
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process intensification, as measured, for example, through wetting efficiency, could be realized by modifying the particle diameter and/or the fluid flow rates. However, these are classical means while the purpose of this work was to show a nonclassical means of improving tricklebed hydrodynamics through inhomogeneous magnetic fields. This goes a contrario to a more pragmatic approach, where the burden of involving magnetic fields is perhaps too excessive when improvements of similar amplitude can be achieved by simply changing the flow rates or particle size. Nonetheless, it is hoped that addressing magnetic fields from a fundamental viewpoint could unfold unsuspected applications in the future from this type of study. Conclusion We have experimentally shown that inhomogeneous high magnetic fields affect trickle-bed reactor hydrodynamics. Depending on magnetic susceptibility, inhomogeneous magnetic fields can produce hypogravity or hypergravity conditions. Under a gradient of B dB/dz ) 650 T2/m, liquid holdup was improved by 11%. This can lead to better contacting between the liquid and the catalyst surface. Accordingly, pressure drop was amplified. In addition, an 18% wetting efficiency increase for paramagnetic gas-diamagnetic liquid systems was revealed experimentally in the explored working intervals. The magnetic field effect was described in terms of an artificial gravity body force, and the modified Holub et al. slit model was observed to be an acceptable simple hydrodynamic model for making preliminary estimations of liquid holdup and pressure drop. In regard to wetting efficiency, no definite tool can be identified, and more work is required to gather more experimental data and a better understanding of flow and appropriate subsequent modeling of this parameter. Nomenclature B ) magnetic flux density (T) E1, E2 ) Ergun constants Fgl ) gas-liquid drag force (N/m3) Fls ) liquid-solid drag force (N/m3) FMa ) magnetization force in R-phase (N/m3) g ) gravity acceleration (m/s2) GaR ) Galileo number for the R-phase H ) magnetic field strength (A/m) P ) pressure (Pa) ReR ) Reynolds number for the R-phase; ReR ) vRdp/[υR(1 - )] Greek Letters χR ) volume magnetic susceptibility
R ) bed holdup of the R-phase µ0 ) absolute magnetic permeability of vacuum (H/m) FR ) R-phase density (kg/m3) υR ) R-phase kinematic viscosity (m2/s) γR ) R-phase gravitational amplification factor
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Received for review February 18, 2005 Revised manuscript received May 25, 2005 Accepted May 31, 2005 IE050195P
