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Ind. Eng. Chem. Res. 2004, 43, 6626-6628
Convective Flow during Gas Centrifugation Les Chewter Shell International Chemicals, Badhuisweg 3, 1003 BM Amsterdam, The Netherlands
Michael Golombok* Shell Exploration and Production, Kessler Park 1, 2288 GS Rijswijk, The Netherlands
In this addendum, supplemental to recently reported work on centrifugal separation of natural gas, we consider the effect of nonzero convective flow, which we had previously neglected as it had been assumed to be nearly zero. Here, we show that it is not necessary to assume that convective flow is zero, which also leads to the conclusion that, contrary to what we stated earlier, the pressure distribution changes during the diffusion of components to centrifugal equilibrium. 1. Introduction
maintained. The pressure gradient is then given by
We are currently reexamining gas centrifugation both experimentally and theoretically, as an option for cleaning heavily contaminated gas fields. These reservoirs can be very largeson the order of 1012 standard cubic feet. However, unlike other producing gas fields, they contain considerable quantities of the contaminating components carbon dioxide (CO2) and hydrogen sulfide (H2S). Current applied technology1 (principally amine treaters) is only economically and environmentally acceptable if the contamination levels are less than 10%. Previous analyses of the gas centrifuge were almost exclusively oriented toward its application for isotope enrichment using the gas UF6.2-4 Very little work was geared toward application for the separation of other more standard gas mixtures.5,6 Recently, we have shown that the assumptions applicable to isotope separation were not valid for centrifugal separation of lighter gases7sindeed, the possibility of higher-pressure operation and higher throughputs could be more readily tackled than for heavier gases.8,9 In this work, we address one of the assumptions inherent in all previous analyses of the time-dynamical behavior of the separation process in a gas centrifuge, namely, that net convective flux can be neglected. In any realistic scenario, this is not the case because, during the process of gas component separation in a centrifuge, the motion of the molecules arises from various forces. On one hand, centrifugation gives rise to a radial pressure gradient in the gas. On the other hand, concentration gradients for each component cause differential diffusive flux to occur. In previous work,8 we considered only the motion due to the pressure gradient and concentration gradient while neglecting any bulk motion of the gassas is done in the standard reference works. In standard literature, this is often referred to as “drift”. Here, we examine the consequences for the governing equations if we no longer neglect the drift term. An analytical solution can be obtained by some basic mathematical manipulations that we report here for the sake of completeness. The starting point is to assume that, at all times during the separation, the balance of centripetal force and pressure on an annular cylindrical element is * To whom correspondence should be addressed. *Tel.: 31 70 447 2327. Fax: 31 70 447 2289. E-mail: michael.golombok@ shell.com.
∂p ) Fω2r ∂r
(1)
where p refers to the total gas pressure and F refers to the total mass of gas molecules per unit volume, which is given by
F)
1 (M p + M2p2) Rg T 1 1
(2)
This implies that this balance is established on a time scale much shorter than the diffusive times scale for the separationsan assumption that was used previously to establish the pressure profile in the separator at time t ) 0. We also consider the net flux, Ni, of a component i into the annular element in unit time. This gives rise to a time dependence of the concentration (ci moles per unit volume) of a component i as follows
∂ci 1 ∂ )(rNi) ∂t r ∂r
(3)
Alternatively, using the ideal gas law to express the concentration in terms of partial pressure, one can write
ci )
x ip Rg T
(4)
where xi is the mole fraction of component i (all symbol definitions are collected at the end of this paper). We thus have
RgT ∂ ∂pi )(rNi) ∂t r ∂r
(3a)
Taking the time derivative of eq 1 and combining the result with eq 3 for the sum of both components 1 and 2, we obtain
∂ 1 ∂ ω2r ∂F (rN) ) ∂r r ∂r RgT ∂t
[
]
(5)
Differentiating eq 2 with respect to time yields an expression for the time dependence of the total gas density
10.1021/ie049628z CCC: $27.50 © 2004 American Chemical Society Published on Web 08/25/2004
Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004 6627
M1 ∂p1 M2 ∂p2 ∂F ) + ∂t RgT ∂t RgT ∂t
(6)
Inserting eq 6 into the right-hand side of eq 5, we obtain
[
]
[
]
∂(rN1) ∂(rN2) ω2 ∂ 1 ∂ (rN) ) + M2 M1 ∂r r ∂r RgT ∂r ∂r
(7)
This completes the first part of the analysis. The lefthand side represents the total molar flux including all terms, and this is related to the centrifugally induced flow on the right-hand side. In the second part of our derivation, we integrate with respect to r to obtain the expression for the total molar flux, i.e., not just that due to diffusive and centrifugal forces as we assumed in our previous work. Because the differential equation (eq 7) is only with respect to the radial coordinate, integrating twice gives
Nr )
3
2
2
Rr r ω (M N + M2N2) + +β 3 Rg T 1 1 2
(8)
where R and β are the constants of integration with respect to r. (Time was eliminated from eq 5 by the use of eqs 3a and 6). Recognizing that, at the center of the centrifuge where r ) 0, the net radial flux, by symmetry, must also be zero, yields β ) 0. Similarly, at the outer wall where r ) R, the radial flux is also zero, so we obtain
R)-
2R ω2 (M N + M2N2) 3 Rg T 1 1
(9)
convection term, and the second term allows for diffusive coupling due to spatial variations in the number concentration of the two components. In earlier work,8 we assumed that the first term on the right was identically zero. Now, however, this equation along with eq 9 can be simultaneously solved to yield the individual fluxes. Doing so yields
N1 ) N2 )
Y(XM2 - 1) (1 - XM2) + x1X(M2 - M1) Y(1 - XM1)
(14)
(1 - XM2) + x1X(M2 - M1)
This now gives us the expressions for the total molar flux. Inserting eq 12 into eq 3 gives
∂c1 1 ∂ 1 ∂ + (rx N) (rY) ) 0 ∂t r ∂r 1 r ∂r
(15)
Expanding the left-hand side of eq 15 and regrouping common terms yields
∂c1 N ∂x1 x1 ∂(rN) 1 ∂(rY) + + )0 ∂t r ∂r r ∂r r ∂r
(16)
We can obtain an expression for the first term on the left-hand side by differentiating eq 4 with respect to time, which yields
(
)
∂x1 ∂c1 ∂p 1 ) + x1 p ∂t RgT ∂t ∂t
(17)
Using this solution in eq 8, we obtain the total molecular flux as
We can couple this expression to the version of eq 3a applicable to all components, i.e.
N ) N1 + N2 ) X(M1N1 + M2N2)
RgT ∂ ∂p )(rN) ∂t r ∂r
(10)
where X is a mechanical system constant given by
X(r) )
ω (r2 - rR) 3RgT
(11)
(18)
Finally, inserting eq 18 into eq 16 and canceling terms yields
∂x1 1 ∂(rY) p ∂x1 +N )0 RgT ∂t ∂r r ∂r
(19)
This final equation can be compared with the equation for the case of zero convective flux, which we have derived elsewhere8
∂c1 1 ∂(rY) )0 ∂t r ∂r
(19a)
(12)
where Y is also a spatial system variable given by
(
∂c1 p ∂x1 x1 ∂ ) (rN) ∂t RgT ∂t r ∂r
which is equivalent to
N1 ) x1(N1 + N2) - Y
[
Inserting eq 3b into eq 17 gives
2
Notice that, in the centrifuge, the total molecular flux is directed inward. This is due to the decrease in gas density toward the center as the heavier component diffuses to the outer region of the separator. This leads to a lowering of the gas density in the center and a concomitant drift of material to maintain the pressure balance. Because the term X(r) has to be negative (i.e., inside the centrifuge r < R), the remainder of the righthand side of eq 10 is positive. Equation 10 thus gives the total molar flux relative to the stationary state. To solve for the flux of each component, N1 and N2, we need another equation linking the two. This can be obtained from the expression for the individual component fluxes
Y(r) ) cD12
(3b)
) ]
∂x1 M1x1 vm1 1 ∂p + ∂r RT M1 F ∂r
(13)
The first term on the right-hand side of eq 12 is the
x1 ∂p 1 ∂(rY) p ∂x1 + )0 RgT ∂t RgT ∂t r ∂r
(20)
The first and third terms in eqs 19 and 20 are the same. The second terms differ as a result of the inclusion of the drift. Equation 3b shows that there is also a change in total pressure during the redistribution of
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Ind. Eng. Chem. Res., Vol. 43, No. 20, 2004
components if the drift, N, is nonzero. In our previous paper, the middle term on the left-hand side of eq 20 was therefore zero, yielding the time dependence of the concentration during centrifugation “spin-up” with a fixed (time-independent) pressure profile. The drift term is the sum of the two individual terms in eq 14
N)
XY(M2 - M1) (1 - XM2) + x1X(M2 - M1)
(21)
and is also a function of the function Y, which is the overall driving force, being a balance between the two “driving forces”: concentration gradient and pressure gradient. Y will tend to zero at long times as these two driving forces balance and the system comes to equilibrium. Note that, to rigorously solve for the time dependence of the separation of two components in a batch centrifuge, it is necessary to integrate two equations (eqs 3b and 19) rather than just one (eq 20 with zero second term). In conclusion, we have shown that it is possible to obtain the governing equations for a batch centrifuge for a two-component mixture without neglecting the drift term. Using eq 21, the magnitude of this term can be calculated, and this allows for a more rigorous evaluation of whether the term can be neglected. For the system investigated in our recent paper, we have indeed found that N is small enough for the second tem in eq 19 to be assumed equal to zero and that the pressure drop profile (eq 3b) is virtually static. For other systems with a lower pressure gradient and/or larger difference in the molecular weights of the two components, these terms might have to be included for a sufficiently accurate calculation of the batch separation time. Appendix: Derivation of Eq 12 Equation 12 follows from the equation of change for a binary system,1,10 ignoring the terms for temperature and gravitational field gradients and assuming an ideal system. In the notation of ref 10, we have
()
jA ) - j B ) -
[
(
) ]
MAxA vA c 1 MAMBDAB ∇xA + - ∇p F RT MA F (A1)
where jA is the mass flux of component A relative to the mass-average velocity. The molar flux relative to the centrifuge is given by
NA ) J /A + xA(NA + NB)
(A2)
where J /A is the molar flux relative to the molaraverage velocity. Using the three equalities
J /A
j /A M F ) , j /A ) j , M) MA MB A c
(A3)
where j /A is the mass flux relative to the molar-average velocity, we obtain
NA ) j A
1 F + xA(NA + NB) c (MAMB)
Substituting for jA yields
[
NA ) xA(NA + NB) - cDAB ∇xA +
(A4)
(
) ]
MAxA V hA 1 - ∇p RT MA F (A4a)
which, with a change in notation (1 ) A and 2 ) B), is equivalent to eqs 12 and 13 for a cylindrical batch centrifuge. 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 R ) centrifuge radius Rg ) gas constant t ) time T ) temperature x ) mole fraction vM ) specific molar volume X ) mechanical system constant (defined by eq 12) Y ) spatial diffusion variable (eq 14) R, β ) integration constants F ) density ω ) angular velocity Subscripts 0 ) center line f ) feed 1, 2 ) particular components
Literature Cited (1) Kohl, A. L.; Nielsen, R. B. Gas Purification; Gulf Publishing Co.: Houston, TX, 1997. (2) Cohen, K. The Theory of Isotope Separation; McGraw-Hill: New York, 1951. (3) Olander, D. R. The theory of uranium enrichment by the gas centrifuge. Prog. Nucl. Energy 1981, 8 (1), 1. (4) Soubbarameyer, J.; Centrifugation. In Uranium Enrichment; Villani, S., Ed.; Springer-Verlag: Berlin, 1979; pp 183-244. (5) Auvil S. R.; Wilkinson, B. W. The steady and unsteady state analysis of a simple gas centrifuge. AIChE J. 1976, 22 (3), 564. (6) Williams, L. O. Application of centrifugal separation to the production of hydrogen from coal. Appl. Energy 1980, 6, 63. (7) Golombok, M.; Morley, C. Thermodynamic factors governing centrifugal separation of natural gas. Chem. Eng. Res. Des. 2004, 82 (A4), 513. (8) Golombok, M.; Chewter, L. Centrifugal separation for cleaning well gas streams. Ind. Eng. Chem. Res. 2004, 43(7), 1734. (9) Cracknell, R.; Golombok, M. Monte Carlo simulations of centrifugal gas separation. Mol. Simul. 2004, 30(8), 501. (10) Bird, R. B.; Stewart W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; p 568, Equations 18.4-15.
Received for review May 5, 2004 Revised manuscript received August 12, 2004 Accepted August 12, 2004 IE049628Z
