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Ind. Eng. Chem. Res. 2004, 43, 1734-1739
SEPARATIONS Centrifugal Separation for Cleaning Well Gas Streams Michael Golombok* Shell International Exploration and Production, Volmerlaan 8, 2288 GD Rijswijk, The Netherlands
Les Chewter Shell International Chemicals, Badhuisweg 3, 1003 BM Amsterdam, The Netherlands
An examination of centrifugal gas separation applied to natural gas wells shows that it has some technical advantages for bulk removal of souring components in highly contaminated streams. Different criteria from those used in traditional UF6 processing apply, and we develop and describe these. In particular, the distribution of components means that enrichment occurs much closer to the center, which would imply easier separation. Symmetry considerations during analysis of a binary system show that the previous implicit assumption of steady-state pressure gradient requiring zero component convection during the setting up of the concentration gradient is incorrect, although the correction for our case is relatively small. Analysis of times for batch spin-up as well as radial pressure distributions indicate that high throughput operation at elevated pressures may be an option. 1. Introduction Natural gas often cannot be distributed from the reservoir directly to customerssit needs to be cleaned up. The contaminants are usually carbon dioxide (CO2) and hydrogen sulfide (H2S). The former does not need to be completely removed, but the latter is highly poisonous and needs to be removed to 1 ppm levels. The standard technology for doing this is solution absorption usually with an amine-based solvent.1 This is a fairly energy-intensive process because the solvent needs to be heated for regeneration and the rates for processing in a gas well can be high, for example, 1000 MMscf/d (equivalent roughly to 800 t/h). In addition, the acid gas stream is generated at atmospheric pressure, which means that extra compression is required if the waste is to be reinjectedsand this in itself is not currently practicable because of corrosion problems. With increasing gas demand (as witnessed recently in U.S. gas price fluctuations), attention is turning to fields where there are significant levels of contamination with H2S and CO2. Examples of the former are to be found in Kazakhstan where H2S can constitute as much as 33% of the reserve.2 Examples of the latter can be found off Northwest Australia where CO2 concentrations in excess of 50% are found, but this is compensated by the extremely large size of the reserve.3 These new reserves cannot be treated economically by amine solvent processes. This can be seen from the consideration that even to clean up a gas stream containing 10% contaminant will have an energy cost on the order of 10% of the gas value produced.1 Clearly higher levels of contaminant cannot be economically handled. Attention therefore turns to alternative sepa* To whom correspondence should be addressed. Tel: +31 70 447 2327. Fax: +31 70 447 3366. E-mail: michael.
[email protected].
ration processes that would remove the bulk of the contaminant to the level where it can be treated with lower-scale conventional processes. In addition, with current environmental and sustainability considerations, it is highly desirable to have a separation process that automatically produces the CO2 or H2S at a pressure where it can be used for reinjection either for sequestration or for enhanced hydrocarbon recovery underground. One such process that would address these requirements is centrifugal gas separation.4 This is the process whereby gas is rotated at high speeds (typically >30000 rpm) in order to generate a strong gravitational field within which heavier molecular weight components are pushed out to the wall at high pressure, leaving lighter components at lower pressure near the center of the rotor. If we consider a cylinder filled at a gas feed pressure of two components pf1 and pf2 (Figure 1a) and then spun up to speed rotating about its axial axis (Figure 1b), then the forces on an annular shell of gas will be such that the net pressure equals the centrifugal force on the element. We then have
∂p ) Fω2r ∂r
(1)
By substituting in the ideal gas law and integrating, we obtain the radial partial pressure profile for a component given by5,6
pi ) p0ieAir
2
(2)
where the coefficient for component i is given by
Miω2 Ai ) 2RgT
10.1021/ie030691i CCC: $27.50 © 2004 American Chemical Society Published on Web 02/28/2004
(3)
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Figure 1. Schematic of batch centrifuge showing (a) stationary cylinder with feed pressures of components 1 and 2sthese have been set as equal so that the effect of the different molecular weights on the roating profiles is more clearsand (b) spinning cylinder with different partial pressure distributions of two components 1 and 2 in the radial direction.
(r is the radial distance, ω is the angular velocity, M is molecular weight, and Rg is the gas constant). Figure 1b shows that the centerline pressure (p0i) is less than the feed pressure (pfi) (we derive the equations below) and that the wall pressure will be higher than the feed pressure. The actual amounts by which centerline and wall pressure deviate from the feed pressure depends on the molecular weight. (To make this effect qualitatively more clear, we have schematically shown equal feed pressures of the two components 1 and 2 in Figure 1.) This technology is well-known and established in its application to the separation of isotopes of the gas UF6.7,8 However for that application the throughputs are very low because of the following: (1) The molecular weights are quite high with the result that there is a high radial pressure gradient, which leads to desublimation of the UF6 gas. This means that feeds pressures need to be kept low for the UF6 case. (2) The molecular weight differences between the UF6 components to be separated are quite small, so the separation is very small requiring many stages and long separator residence times. On the other hand, the components found in natural gas wells (CH4, CO2, H2S, N2) result in much lower pressure gradients, and there is no risk of solid formation. Also the molecular weight differences in natural gas are on the order of 10 times those for UF6 separation. Finally eq 2 shows that the waste streams of concentrated contaminant are generated near the wall of the centrifuge and at elevated pressure, making them ideal for re-injection as identified above. The only specific chemical application of gas centrifuges of which we are aware is the work by Auvil and Wilkinson, who examined SO2/H2 mixtures,9 and by Williams, who considered steady-state profiles for syngas separation.10 That work showed that an important limit is the time taken for the gas components to differentially diffuse down the concentration gradients represented by eq 1. We have used a similar starting point for calculating equilibrium spin-up times for well gases; however we use different criteria for physically separating light and heavy gas streams. Section 2 develops this theory to identify the radial crossing point where the steady-state rotating concentration profile crosses that of the nonrotating feed. We show that
efficient recovery from a rotating profile requires us to collect all gas radially beyond this point. The technology of designing such a system is not addressed for this batch study because in practice one would use a countercurrent flow system, which leads to typically a factor of 2-5 improvement. However the basic physics of the separation is determined by the spatial concentration profiles, which can be more easily modeled in a batch process. In section 3, we examine the kinetics of the evolution of these profiles and the assumptions for obtaining the spin-up times (i.e., the assumption of changes in concentration gradient without changes in pressure profile). Similarly to the previous section, we do not consider the more complex counter-current scenario since we wish here to illustrate the spatial dimensions of the concentration profilesshence, the use of the simpler scenario. In section 4, we apply the new criteria we have developed as being more relevant from an industrial chemical viewpoint to obtain the effective separation times for different mixtures. The results are reviewed and discussed. 2. Equilibrium Profiles 2.1. Pressure. We consider a cylinder filled with gas to a feed pressure (pf) where the mole fraction of contaminating component is xf (as in Figure 1). When the gas spins up, we obtain the pressure gradient for each component as specified in eq 1. The question is how do we relate this pressure gradient to the initial pressure at the start of the experiment before the rotor is spun up. (Recall that we are considering a batch process here; however, equivalently we note that in a continuous system this effectively determines the pressure at a feed inlet. It is the spin-up time that determines the rotor length in order to determine equipment size for a stable rotating medium to be reached with the concomitant separations.) If we consider the stationary position, then we have effectively a feed pressure pf. The total number of molecules is given by pV/RgT summed for all the volumes inside the cylinder of radius R and length l. For a feed pressure pf, then this simply gives
n)
πR2lpf RgT
(4)
In the case of the spinning rotor, we must sum the pV contributions for each component in each elemental volume. By integrating eq 2 between r ) 0 and r ) R and equating the total number of moles in eq 4, we obtain the relationship between the centerline pressure and the feed pressure for each component:
p0i )
AiR2 2
pfi
eAiR - 1
(5)
This constitutive relationship relates the separation process feed pressure to the operating centerline pressuresit has not, to our knowledge, been previously published. It determines the centerline pressure of the reactor and, hence, the wall to center pressure, which we identified as an important factor above. Figure 2 shows a plot of the centerline pressure (p0i) normalized against feed pressure (pfi) as a function of molecular weight for a 5 cm radius centrifuge at 70000 rpm. It can be seen that for low molecular weights the center-
1736 Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004
Figure 2. Centerline pressure (p0i) normalized against feed pressure (pfi) as a function of molecular weight for a centrifuge operating at 70000 rpm with a radius of 5 cm.
line pressure varies very quickly but that at higher molecular weights there is little variation. Physically this means that, for low molecular weight materials, the pressure does not decrease that much at the center (with a corresponding increase in the outer portion). Only for molecular weights above ca. 150 g do we start to get order of magnitude pressure changes and greater. This is a similar effect as observed above for wall to centerline pressures and explains why it is so difficult to separate the heavy isotopessa combination of a small molecular weight differencesand from a reactor point of view a negligible difference in centerline pressures. In Figure 2, we have shown the ratio of centerline pressure to feed pressure only for one equipment operating condition (i.e., speed and radius). We can more easily conceptually separate intrinsic feed from equipment operational effects by recognizing that the argument of the exponential (A) in eq 1 actually represents a characteristic length given by rA ) 1/xA. Indeed we can generalize the representation of centrifuge performance by combining the variables for centrifuge radius and speed in one variable γi ) AiR2 so that eq 5 becomes
p0i )
γi
pfi
e -1 γi
(6)
γ can increase as a result either of (1) increasing molecular weight of feed or (2) increasing rotation rate or rotor radius (i.e., the peripheral speed of the rotor, the maximum value of which is set by the tensile strength of the rotor material). The net effects are the samesthe radial pressure gradients increase dramatically for any of these parameters. It is also useful to use the combined dimensionless γ parameter to reconsider more schematically the radial pressure distribution (normalized against centerline pressure) for three typical values of γ. This is shown in Figure 3 and allows us to optimize performance for a particular set of combined parameters. Typically for isotope separation, γ is O(10) whereas for well gases it is O(0.1). Thus for natural gas components, the wall pressures are typically less than three times the centerline pressure determined by eq 5 from the feed pressure. 2.2. Concentration. Whereas in the isotope separating centrifuges almost all the mass is concentrated near the wall, resulting in the need for a spatially highly resolving gas scoop (i.e., stream splitting) system, there is a much more uniform mass distribution for well gases like methane, hydrogen sulfide, nitrogen, and carbon dioxide. We also have the advantage of molecular weight differences larger than 3 mass units. The result is that
Figure 3. Nondimensionalized pressure profile as a function of radius (nondimensionalized) for different values of the combined gas property-centrifuge parameter γ.
Figure 4. Cumulative fraction of gas contained within a fractional distance of axis of centrifuge at 70000 rpm.
spatial resolution of different streams can be geometrically configured more efficiently. This is shown in Figure 4 where we have compared the cumulative fraction of feed component at a particular fraction radius for methane and UF6. The lighter component is much more dispersed along the radius as compared to the heavier component that concentrates near the wall. We need all the same to make a clear conceptual difference between simply an augmented partial pressure and a really enriched stream (i.e., concentration). For a twocomponent system, the mole fraction at radius r is given by9
x1 )
x10e(A1-A2)r
2
x10(e(A1-A2)r - 1) + 1 2
(7)
Close to the wall, the concentration of heavy component is high; however, the total recovery is small. Further in toward the center, the recovery is higher; however, the average enrichment is lower. The optimum position for recovery can be determined by control of the parameters in turn dependent on the original feed gas composition, which we have previously identified. This means identifying the radial feed crossing point (i.e., that point rx along the rotor radius at which the desired (heavy) component exceeds the feed concentration x1 > x1f). This is defined by
p1(rx) p1(rx) + p2(rx)
) x1f
(8)
Using this definition and the relation with centerline
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by Fick’s law then determines the time for the equilibrium concentration profile to be reachedsso during evolution to the equilibrium profile, the behavior is governed by the balance between the driving force and the centrifugal fluxsthe latter being given by5 Figure 5. Schematic profiles for concentration during centrifuge operation as compared to feed concentration for (a) typical well gas mixture and (b) isotope separation.
pressures and after some manipulation, one obtains
p20 -(A1-A2)rx2 p2f e ) p10 p1f
(9)
The expressions for the centerline pressures of the different components p0i have been derived above in eq 5. Inserting this into eq 9, we obtain the solution for the mole fraction crossing point:
Fiui ) 2DiFiAir
Consideration of the flux in and out of the radial surfaces of an annular section leads to an equation in terms of the density of each component:
(
2
M2(eA1R - 1) 2
which defines the point where x1 ) x1f. The concentration cross point is dependent on both componentssthis can be seen from a comparison between a typical natural gas profile (our interest) and the isotope profilesboth shown schematically in Figure 4. For efficient recovery, we must (1) minimize the concentration crossing point where the desired component starts to concentrate (i.e., minimizing rx) and (2) maximize the area of Figure 5 above rx (i.e., maximizing the area for r > rx). The first condition means solving eq 10 for the radial crossing point rx. This is
rx )
x[
2
]
M2(eA1R - 1) 1 ln A1 - A2 M (eA2R2 - 1) 1
(11)
Note the similarity of this expression to the abovedefined characteristic length scale (rA ) 1/xA), but in this case it is modified by the log factor dependent on molecular mass weighted separation factors. Previous studies of centrifugal gas separation always stress that the process is dependent purely on molecular weight difference (unlike others such as differential diffusion).11,12 This is true for extreme cases such as the isotopic case presented in Figure 5b. However from a stream recovery viewpoint the situation is more complex. The expression in eq 11 for the radial crossing point is a function of the molecular weight difference, molecular weight ratios, rotation rate, and rotor radius. One can thus in theory optimize the stream splitting to a design for a minimum value of rx. 3. Evolution of Concentration Profile So far, we have been looking at an established (spunup) equilibrium concentration profile operating under laminar conditions in order to suppress turbulent mixing.13 The crucial question is how long it takes to generate this profile, as this ultimately determines the size of the unit required to process the gas. This time is a function of the diffusion of the gas components. The initial rotor rotation quickly establishes a pressure profile given by eq 1, which automatically leads to a different density profile for each component. Diffusion
(13)
(Note that we are dealing with a batch reactor so there is no imposed axial or radial convective flow; i.e., u ) w ) 0.) For each component from the ideal gas equation we have
Fi )
(10)
M1(eA2R - 1)
)
∂F1 D ∂ ∂F1 ) r - 2A1F1r ∂t r ∂r ∂r
2
e(A1-A2)rx )
(12)
xiMip RT
(14)
and using the spatial differential form of eq 1, this leads to the equation in a two-component system 1 and 2:
(
)
∂x1 ∂(x1p) D ∂ rp ) + 2(A2 - A1)r2px1(1 - x1) ∂t r ∂r ∂r
(15)
The classical solution of this equation has involved the assumption that Dp is constant;5 however, for our purposes we do not need to do this. (At any rate, we are particularly interested in higher pressures where Dp is not necessarily constant.) From symmetry considerations, eq 15 must also apply to the other (2) component so that
(
)
∂x2 ∂(x2p) D ∂ rp ) + 2(A1 - A2)r2px2(1 - x2) ∂t r ∂r ∂r
(16)
Using the fact that x1 + x2 ) 1 and the resulting differential relationship between x1 and x2, we can add eqs 15 and 16 to obtain ∂p/∂t ) 0. We know that this is not truesand we shall demonstrate it explicitly in the next sectionsbecause diffusion of components along the density gradient established by the centrifugal motion must lead to changes in the partial pressures (i.e., the partial pressures and concentrations, which vary, are coupled to one another). In fact, we shall see in the next section that this assumption does not make much of a differencesit would only be significant in much weaker fields, for example, a gravitational one. In our case, a substantial portion of the gas is confined to an area near the periphery so that the spatial variations are in effect negligible on the result. The correction is relatively small. We can show that this corresponds to neglecting convection (so-called “drift” terms) and that this leads to the conclusion that the flux is equal in both directions, which is untrue since the pressure gradients of each component are not equal. To show this, we consider an alternative derivation of the flux equations above. This starts from the equations relating concentration and molecular flux of two species 1 and 2 at a particular position:14
∂c1 1 ∂ + (rN1r) ) 0 ∂t r ∂r
(17)
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Now considering only the radial direction, the total molecular flux N1 is given by
N1 ) x1(N1r + N2r) - cD12
(
[
)]
∂x1 M1x1 vm1 1 ∂p + ∂r RT M1 F ∂r (18)
where vm1 is the molar volume of component 1 and c is the total concentration of species. The first term on the right-hand side is the convection termsoften referred to in the literature as the “drift”.5,14 The second term includes cross-coupling terms arising from the variation in the total number of species per unit volume. If we assume that the convective term is zero, then it is a relatively simple matter to show that this leads to eq 15 with the above-demonstrated implicit assumption of static pressure. Inserting eq 18 into eq 17 yields
( [
(
)] )
∂c1 1 ∂ ∂x1 M1x1 vm1 1 ∂p + rcD12 ) 0 (19) ∂t r ∂r ∂r RT M1 F ∂r Using the ideal gas law to replace the molar volume, the differential form of eq 1, and the constitutive relationship eq 14, we can restate the second term in brackets on the right-hand side to obtain
( [
∂x1 ω2r ∂c1 1 ∂ rcD12 + x (x - 1)(M1 - M2) ∂t r ∂r ∂r RT 1 1
])
)0 (20)
which on replacing concentrations with pressure yields eq 15. One fundamental assumption here is that the product of diffusivity and pressure Dp is now a constant. The other is the omission of the convective term from eq 18. If this were included, it would lead to a convective correction to eq 15 of the form (N1 + N2)dx1/dr. However when we calculate this, it turns out to be a relatively small correction termsnonetheless, strictly speaking the pressure distribution does change somewhat during the diffusion of components to the centrifugal equilibrium. 4. Evolution Times The reason that we have indulged in a fairly detailed discussion of the evolution of the concentration profile and related parameters lies in the fact that this is what ultimately will determine the vessel size. The standard technology for removal of CO2 and H2S is the amine treatersan unwieldy and chemically intensive process that moreover only yields the waste streams at around atmospheric pressure.1 Subsequent compression of CO2 waste streams for reinjection in gas wells adds substantially to the total cost of the process. In our case however, the CO2 would emerge at already enhanced pressures. The crucial question therefore is the size of the units and whether unlike the benchmark amine treater standard, the gas centrifuge, can yield a waste gas stream at elevated pressures suitable for reservoir reinjection. This depends substantially on the fluid dynamical aspects, which would seek to suppress turbulent mixing. We have indicated that this problem can be solved, and we do not handle it further here. Rather we now proceed to consider the effective separator size. Rather than proceeding to a full countercurrent calculation, we follow the procedure of Auvil and Wilkinson9 to get a measure of the size requirement and behavior by doing a relatively simple batch calculation and
Figure 6. (a) τ90 as a function of feed pressure xCO2 ) 0.1 for two rotational speeds and rotor radii. (b) Effect of rotation rate on τ90 for two different rotor radii, xCO2 ) 0.1, pfi ) 1 bar.
looking at the time for the concentration profile to evolvesor rather for our particular case, we introduce the idea of the time to get reasonably close to the equilibrium concentration profile. Starting off as the flat line corresponding to the feed concentration in Figure 5, we may dynamically model the evolution of the equilibrium profile. This was carried out using Aspen Custom Modeler using an automatic differential equation solver. In their original work, Auvil and Wilkinsons who were concerned with other materials and much lower contaminant concentrations than in the current casestook a parameter related to separative power based on the development of the axial concentration difference with the feed as a fraction of the equilibrium difference.9 However, we have shown in section 2 that for efficient recovery we need to consider a cumulative yield with respect to the radial crossing point (rx) as defined in eq 11. Accordingly we define the time τ90 as the time for the integrated area to the right of the radial crossing point to reach 90% of its equilibrium value. Physically this corresponds to recovering 90% of all CO2 that could ultimately end up in the enriched region. Figure 6 shows the effect of the feed pressure on the evolution time of a 10% CO2 in methane mixture at 40000 rpm. This is clearly a linear effectsarising from the formalism for calculating diffusion constants at different pressures with the assumption that the product of diffusivity and density DF is a constant. Given that the time for any diffusive effect will be determined by the inverse of the diffusivity, it is not surprising that we calculate a linear dependence of τ90 on pressure. For the smaller radius then, an increase in speed from 40000 to 100000 rpm has hardly any effect. At higher radius, τ90 is reduced as we increase the speed over that range. (These speed range and radii choices are determined by currently available commercial ultra-centrifuge technology and rotors.) Similarly τ90 is independent of the feed gas composition. As the amount of CO2 in
Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004 1739
the feed increases, the diffusive gradient set up is higher; however, there is also a greater amount of material that needs to be transferred into the region r > rx. The net effect is that the time required is independent of composition in this respect. Such results would seem to indicate that very large units will be required for realistic throughputs. The amount of separation is greater at higher radii for a particular speed, and so the concentration distribution has further to evolve from the fully mixed input feed conditionsmore separation will take longer. However, an interesting effect is observed in Figure 6b when the rotor radius is changedsthe evolution time shows a stronger decrease at higher radius (i.e., at larger peripheral velocities). This is really a reflection of the effect observed in Figure 6a. The separation time decreases more quickly with rotation rate at larger radius. (Note that the limits are on the peripheral velocity as this determines the centrifugal force, which must not exceed the tensile strength of the rotor material.) This decrease in τ90 is one of the effects we are pusuing for optimizationsnot in the context of the batch centrifuge described here, but rather applied to countercurrent operation where the comparable separation times are considerably lower in magnitude. 5. Conclusion (1) Aside from the issue of turbulent mixing (soluble by channeling), there are a number of concentration distribution issues that suggest that, contrary to previous assumptions, centrifugal gas separation can work in a high-pressure high-throughput mode when low molecular weight gases are used. In particular, the components of natural gas have much lower pressure gradients than are found in isotope systems, and also the differences in molecular weights are higher. (2) For chemical processing, rather than concentrating on separative power and cascade structures, the best performance is obtained if one collects all gas beyond the radial crossing point where the concentration increases beyond that of the feed. (3) Changes in the pressure profile do occur during the separation, although neglect of convective terms is the reason that these relatively small changes have been ignored. (4) Calculated batch times remain rather long for our applications but are expected to be considerably less in countercurrent mode. (5) The major assumption in the standard treatment of the gas centrifuge is the constant behavior of the product term Dp (i.e., diffusivity and pressure). Although this holds good for the lower pressures required (for completely different reasons) in isotope processing, the higher pressures we are dealing with here imply that the behavior of this transport term needs to be examined at higher pressures. (6) The work in this paper has looked at behavior in a simple batch (or equivalently) co-current centrifuge. The results developed here are currently being used to develop an experimental program for cleaning natural gas.
Nomenclature A ) molecular argument coefficient (eq 2) c ) concentration D ) diffusion constant l ) centrifuge length M ) molecular weight n ) number of moles N ) molecular flux p ) pressure r ) radius variable rx ) radial point where composition matches that of feed R ) centrifuge radius Rg ) gas constant T ) temperature u ) radial flux Vm ) molar volume x ) mole fraction Greek Letters F ) density γ ) centrifuge performance variable µ ) viscosity ω ) angular velocity Subscripts 0 ) centerline f ) feed i ) generic component r ) radial vector component 1,2 ) particular components
Literature Cited (1) Kohl, A. L.; Nielsen, R. B. Gas Purification; Gulf: Houston, 1997. (2) Samoilov, B. V. Developing Kazakhstan’s Tengiz field will be a tough task. World Oil 1993, 214 (7), 16. (3) Hanif, A.; Suharito, T.; Green, M. H. Possible utilisation of CO2 on Natuna’s gas field using reforming of methane to syngas; SPE Paper 77926; SPE: Richardson, TX, 2002. (4) Olander, D. R. The Gas Centrifuge. Sci. Am. 1978, 239 (2), 27. (5) Cohen, K. The theory of isotope separation; McGraw-Hill: New York, 1951. (6) Olander, D. R. The theory of uranium enrichment by the gas centrifuge. Prog. Nucl. Energy 1981, 8 (1), 1. (7) Soubbarameyer, J. Centrifugation. In Uranium Enrichment; Villani, S., Ed.; Springer-Verlag: Berlin, 1979. (8) Los, J. De scheiding van zware isotopen in een centrifugaal veld. Ph.D. Dissertation, Leiden University, The Netherlands, 1963. (9) Auvil, S. R.; Wilkinson, B. W. The steady and unsteadystate analysis of a simple gas centrifuge, AIChE J. 1976, 22 (3), 564. (10) Williams, L. O. Application of centrifugal separation to the production of hydrogen from coal. Appl. Energy 1980, 6, 63. (11) Auvil, S. R. A general analysis of gas centrifugation with emphasis on the countercurrent production centrifuge. Ph.D. Dissertation, Michigan State University, East Lansin, MI, 1974. (12) Pratt, H. R. C. Countercurrent separation processes; Elsevier: Amsterdam, 1967. (13) Brouwers, J. J. H. Phase separation in centrifugal fields with emphasis on the rotational particle separator. Exp. Therm. Fluid Sci. 2002, 26 (2-4), 325. (14) Bird, R. B.; Stuart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960.
Received for review September 2, 2003 Revised manuscript received January 6, 2004 Accepted January 27, 2004 IE030691I
