Ind. Eng. Chem. Res. 1988,27, 823-830
(C1-, Clm, C20, and C22-0acid methyl esters). But the overall solubility is decreased with a decrease of the density of c02. The apparatus is equipped with the chamber for the selective removal of the dissolved materials from the loaded supercritical phase. Clel acid methyl ester is separated to a certain degree from a mixture of ClW, Clel, Cle2, and Cle3acid methyl esters by holding a higher temperature inside the chamber, which is packed with Rashchig rings. The utility of the separation chamber, which is packed with AgN03 supported on silica gel as an enhancement agent for the fractionation, is confirmed on the extraction of a mixture of C18+ C18-1, CIg2, and ClS3 acid methyl esters. It is found that ClW acid methyl ester can be selectively separated from the mixture, and the concentration is maximally above 90%. Further, ethyl acetate is used as an entrainer for the separation of C18-3 acid methyl ester held in the chamber after the treatment described above. A fractionation takes place with the addition, and the concentration reaches ca. 70%. Thus, these results demonstrate that, when one makes the SC-C02extraction by using a proper entrainer and a gas-effusion-type apparatus, it is possible to isolate specific components in a desired purity and a large solubility from a mixture of higher fatty acid methyl esters. We think that this technique can be easily applied to isolate useful components from the natural products which contain valuable acids. Registry No. AgN03, 7761-88-8; COz, 124-38-9; tristearin, 555-43-1; triolein, 122-32-7; ethanol, 64-17-5; ethyl ether, 60-29-7; acetone, 67-64-1; methylene chloride, 75-09-2; ethyl acetate, 141-78-6; methyl hexadecanoate, 112-39-0; methyl stearate, 11261-8; methyl eicosanoate, 1120-28-1; methyl docmanoate, 929-77-1;
823
methyl oleate, 112-62-9; methyl linoleate, 112-63-0; methyl linolenate, 301-00-8.
Literature Cited Brunner, G . Fluid Phase Equilib. 1983, 10, 289. Brunner, G.; Peter, S.; Retzlaff, B.; Riha, R. "High Pressure Science and Technology". Sixth AIRART Conference 1979, Vol. 1, p 565. Eisenbach, W. Ber. Bunsenges. Phys. Chem. 1984, 88, 882. Handbook of Oil Chemistry;The Japan Oil Chemists' Society: Tokyo, 1971; Chapter 1. Hirai, A.; Hamazaki, T.; Terano, T.; Nishikawa, T.; Tamura, A.; Kumagai, A.; Sajiki, J. Lacent 1980, 11, 1132. Ikushima, Y.; Arai, M.; Nishiyama, Y. Appl. Catal. 1984, 11, 305. Ikushima, Y.; Saito, N.; Hatakeda, K.; Ito, S.; Asano, T.; Goto, T. Bull. Chem. SOC.J p n . 1986,59, 3709. Lossonczy, T. 0. Am. J . Clin. Nutr. 1978, 31, 1340. Markley, K. S. Fatty Acids Part I,2nd ed.; Interscience: New York, 1960; Chapter 6. Nagahama, K. Bunrigizyutsu 1981, 11, 23. Paul, P. F.; Wise, W. S. The Principles of Gas Extraction; Mills and Boor: London, 1971. Peter, S.; Brunner, G. Extraction with Supercritical Gases; Verlag Chemie: Weinheim, 1980. Randall, L. G. Sep. Sci. Technol. 1982, 17, 1. Sagara, H. Kemikaru Enginiyaringu 1981, 461. Saito, S. Petrotech 1982, 5, 115. Sako, T.; Yokochi, T.; Sugeta, T.; Nakazawa, N.; Hakuta, T.; Suzuki, 0.;Sato, S.; Yoshitome, H. J. Jpn. Oil Chem. SOC.1986,35, 463. Sanders, T. A. B.; Younger, K. M. Br. J . Nutr. 1981, 45, 613. Subramanian, B.; McHugh, M. A. Ind. Eng. Process Des. Dev. 1986, 25, 1.
Tsekhanskava, Yu.: Jomter, M. B.: Mashkiya, E. V. 2.Fiz. C h i n . 1964, 38,2166. Yamaguchi. K. Eighth Technical Seminar. The Societv of Chemical Eniineering of-Japan, Nagoya, 1985. ' Received for review May 12, 1987 Revised manuscript received December 11, 1987 Accepted December 21, 1987
Mathematical Analysis of Crossflow Magnetically Stabilized Fluidized Bed Chromatography J. Carl Pirkle, Jr.,* and Jeffrey H. Siegell Corporate Research Laboratories, Exxon Research and Engineering Company, Annandale, N e w Jersey 08801
A mathematical model was developed for a crossflow magnetically stabilized fluidized bed (MSB) chromatograph with the pertinent continuity equations derived in partial differential equation form. In the case of linear adsorption isotherms, Fourier transform analysis yielded an expression for the resolution of the elution curves in terms of system parameters and operating conditions. Calculated predictions of the first moment and the variance of the elution curves were in good agreement with experimental results. A systematic parametric study conducted using the model showed that two factors, resistance to mass transfer and the width of the feed zone, have dominant effects on the size of the chromatographic bed. If smaller particles are utilized, which can be accomplished without excessive pressure drop by using an MSB, the mass-transfer resistance is reduced significantly. This allows an increase in the fraction of the bed width devoted to the feed zone and, thus, results in a lower desorbent (carrier fluid) to feed ratio and more concentrated product solutions. Recently, a crossflow magnetically stabilized fluidized bed (MSB) has been proposed for the continuous chromatographic separation of multicomponent feeds (Siegell et al., 1985, 1986). As shown in Figure 1,the magnetically stabilized fluidized solids, which have adsorbent properties, move horizontally, while the fluidizing fluid moves through the bed with a large vertical and small horizontal velocity. The mixture to be separated is introduced into the bed at the bottom, near the solids entrance. In the fluid phase, adsorbable components of the feed move largely in a vertical direction and only slightly in a horizontal direction. 0888-5885/88/2627-0823$01.50/0
They move significant horizontal distances in the solid phase. When the adsorbable components elute from the top surface of the bed, they are located at different positions, depending on their adsorption equilibrium characteristics, the system parameters, and the operating conditions. These same factors determine the shapes and degrees of overlap of the outflow concentration profiles. The MSB is ideally suited for chromatographic operation. Previous studies have shown the absence of significant fluid and solids backmixing (Rosensweig, 1979; Rosensweig et d., 1981; Siegell, 1982; Siegell and Coulaloglou, 0 1988 American Chemical Society
824 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 CROSSFLOW MSB CHROMATOGRAPHY
'
Components Move to Bed Suriace with Carrier Fluid Y
Product Streams
/t
4
Typical Pellet (Not Io Scale)
Direction 01 Carrier Fluid
L I
I
Moving --c Sollds Y=0
X
1
Carrier Fluid
F e e d Mixture Components A,B,C.D
Figure 1. Diagram of crossflow magnetically stabilized bed chromatography.
1984). Because the bed is fluidized, the pressure drop is limited to the weight of the solids per unit bed area, independent of the particle size and fluid throughput. Thus, smaller particle size and higher carrier fluid rates may be used without suffering the high pressure drops of the previous packed bed chromatographic separations devices. In addition, since the beds are freely flowing fluidized solids, they are mechanically simple, with the solids being the only moving part. In chromatographic-type separations, it is desirable to keep the overlaps of the product concentration profiles minimal. For analytical purposes, however, the overlaps need only be small enough to allow the different species to be detectable (i.e., all peaks must be distinguishable). For commercial chromatography, overlap of the product, or elution, curves decreases product purity. It would be useful, therefore, if the overlap of adjacent elution curves could be determined as a function of system parameters and operating conditions. Then the chromatograph could be designed and operated to give the desired separation. This paper presents a mathematical model that is used to relate the separation effectiveness of crossflow MSB chromatography to gas and solid velocities, adsorbent and adsorbate characteristics, bed height, pressure, temperature, feed characteristics, and nature of the fluidizing medium. It is rigorously valid for cases in which the adsorption isotherms are linear and independent, such as in light loading. Even in more highly loaded cases, the adsorption isotherms can be linearized approximately. From the mathematical relationships, the critical experiments that lead to thorough characterization of the process are easily identified. The validity of the model is checked by comparison of its prediction with experimental results. Finally, important terms and parameters are identified, and parametric sensitivity studies are performed for two limiting cases: (1)when separation effectiveness is controlled by mass-transfer resistance, and (2) when separation effectiveness is controlled by throughput. Mathematical Model Mathematical models for crossflow separation chromatography and the closely related rotating annular chromatograph have been presented in recent years (Wankat, 1977; Wankat et al., 1976; Scott et al., 1976; Begovich and Sisson, 1984). The model presented here differs mainly in the inclusion of detailed mass-transfer processes and fluid mechanical dispersion effects (departure from plug flow) as separate causes of overall dispersion of the elution curves. The adsorption isotherms are assumed to be linear, which is valid in light loading and allows utilization of powerful mathematical techniques developed for linear
C
W
+X
Components Move Crossllow wlth Sollds
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 825
,/--..
t-
Gas Flow
Horizontal Dispersion
111) Desorption
/
External Diffusion
-I--/ Pore Diffusion
Figure 3. Diagram of mass-transfer processes in adsorption; no gas is being injected from the left-horizontal gas flow due to solids motion.
the particle equals the rate of local accumulation of i in the selective and nonselective regions of the particle. N, and Q, represent concentrations, based on intraparticle pore volume, in the selective and nonselective regions of the pellet, respectively. Although trivial in appearance, eq 7 suffices due to conditions of symmetry. Diffusion across the external stagnant film is equated to diffusion into the particle interior by eq 8. The entrance of fresh particles (no adsorbable species present in the nonselective regions) is expressed by eq 9. The rate at which species i migrates from the nonselective to the selective regions of the particles as they move horizontally is given by
v,"/ax
= kp,(N,*- N,)
(10)
ficult separations and "e the ones for which the crossflow MSB chromatograph is intended. In the case of the linear isotherm, eq 1-12 and 13 represent a linear system. An extremely powerful mathematical apparatus is available for analyzing this system, as will be shown in the next section. This mathematical method will yield, in concise algebraic form, the answer to the question posed earlier: How does the quality of separation in crossflow MSB chromatography depend on system parameters and operating conditions? This method involves the use of Fourier transforms and is similar to a method using Laplace transforms to analyze a fixed-bed chromatograph (Kubin, 1965). The partial differential equation system describing the MSB crossflow chromatograph makes the Fourier transform method the logical choice in this case.
Method of Moment Analysis via Use of Fourier Transforms It is easy to relate the momenta of the elution of the type curves shown in Figure 1 to the Fourier transform with respect to X of those curves (Seinfeld and Lapidus, 1974). Let c&,Y) represent the Fourier transform of C,(X,Y): c(jo,Y) = s m C ( X , Y ) -m
11.0' = --o)
Nt' = Nt*(Qi,Q2,-,Qd
(12)
(14)
where the subscript for species has been dropped and j represents the imaginary (-l)ll2. The zeroth, first, and second noncentral moments of the elution curve C,(X,L) are given by
with the boundary condition
N,(X,Y,r)= 0 at X = (11) In eq 10, the intrinsic adsorption rate coefficient k,, (Schneider and Smith, 1968) is proportional to the rate of collisions between species i molecules and the selective surface of the particle as well as the probability of a molecule adhering to the surface. Equation 11 expresses the fact that fresh particles enter from the left without any adsorbable component in the selective regions of the particle. The mass-transfer processes are summarized in Figure 3. The horizontal gas flow indicated here is due to the solids motion. No gas is injected from the left. The exact form of the equilibrium adsorption isotherm may depend on the values of intraparticle fluid-phase concentrations of more than one species; i.e.,
dX
Ml'
c(j~,Ullw=o
= -a4 j w , L ) /a( jo)1jw=0
M i =
a2C(jo,L)/a(jo)21,w=0
(15) (16) (17)
The variance u2 is related to the noncentral moments by a2 = M 2 / M d = [Mz'
- (McL1')21/M01
(18)
Let q(ju,Y,r) and n( jw,Y,r) represent the Fourier transforms of Q, and Ni,respectively, with respect to X. Transforming eq 1-13 and setting 5 = j w gives Dy dc/dY - Vfy dc/dY - [DxE2- EVfxlc = 3Dp,4/Rp aq/drlr=Rp (19) c(0,E) = [CO+ D Y / V ~dc/dYI~=01[1 Y - exp(-EW)I/E (20) dC/dYly,L = 0
(21)
where m is the number of adsorbable components. In cases where more than one of the species i = 1,2,...,m are adsorbed by the same sites, the values of N,* will be affected not only by the the value of Q, but also by values of Q,, j # i, as well. In this paper, the model will be restricted to cases where N,* depends only on Q, in a linear fashion; i.e.,
V&(q + n) = Dp,[d2q/dr2+ 2/r dq/dr]
(22)
(13) N, * = K A8,~ where K& is the linear adsorption coefficient. Linear isotherms are usually valid in the case of light loading, Le., when the fraction of occupied sites on the adsorbent is less than half of the total (Brunnauer, 1945). For sufficiently small values of feed concentrations Cb0or feed band width W, the light loading condition prevails. Even in the case of high concentrations, however, the use of the linear isotherm model is a good approximation for those situations involving components with similar adsorption coefficients. Such situations are those corresponding to dif-
Solving eq 25 for n(f,Y,r) in terms of q((,Y,r) and substituting the result into eq 22 yields a partial differential equation for q only. This can be solved, using the boundary conditions given by eq 23 and 24, for q([,Y,r)in terms of c([,Y). Substituting the solution of q([,Y,r) into eq 19 yields an ordinary differential equation for c(f,Y) which can be solved using the boundary conditions given by eq 20 and 21. For sufficiently large values of L , we obtain the following approximate solution for c([,L):
q = finite at
r=0
(23)
3Dp&/Rp dq/drlr=Rp = kfi[c - qlr=Rpl
(24)
EVsn = kp,(KA,4- n)
(25)
826 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988
experiment or from reliable correlations. Interphase mass-transfer coefficients are computed for the external film by
where
Y=
a =
[
[b2 + 4a]'/' - b 2
kfl = (3/2)ShDl/Rp2
b = VfY/DY Vfx( - DXF2+
(36)
where the Sherwood number is obtained from (Wakao and Funazkri, 1978)
3 4 y B RP
Sh = 2.0
+ 1.1(Rep)06(S~)1/3
(37)
In eq 36, D,is the effective binary diffusivity of species i in the carrier gas. In eq 37, Sc denotes the Schmidt number
s c = I*f/(DlPf)
[
B = kfiRp3/ 3Dpi[XR, cosh (XR,)- sinh (Ut,)] + kfi
- sinh (XR,)R,2 4
and = [V,t[1
+ kp,Kh/(kpI+ 5V,)l/Dp11'2
1
and Re, is the Reynolds number based on the particle radius Re, = 2PfVfYRP/Yf
(30)
(31)
Substituting eq 26-31 into eq 15-19 yields expressions for the first moment X , and the variance cr2 of the elution curve. These relations, normalized with respect to bed depth L, are as follows: x1/L = '#'(I + KAG)(va/vfY) + (1/2)w/L +
v f X / v f Y (32) vertical dispersion + (2/15)(V,2/Vfy)@(1+ KA,)2R,2/(D,,L) intraparticle diffusion + 2(1 + K~)2(V,2/Vfy)@2/(knL) external film resistance + 2KA,(V,Z/ Vfy)'#'/(kp,L) resistance to adsorption on selective sites + 2/Pex horizontal dispersion + (l/12)(W/L)2 width of feedband +2(Vfx)/VfY)*/Pey coupling of horizontal convection with vertical dispersion (33)
+ KA,)(V,/Vfy)]z/Pey
u2/L2 = 2[@(1
The first term on the right side of eq 32 is the contribution of adsorption to the horizontal displacement of the eluting species. As expected, this displacement increases with 4, V,, and KA, and decreases with VfF The second term on the right shows the contribution of the feed band width W to the first moment. Finally, the third term gives the effect of horizontal convection on the horizontal displacement. In eq 33, Pe and Pex denote the vertical and horizontal fluid Peclet numbers, respectively, of the moving, crossflow magnetically stabilized fluidized bed. These dimensionless numbers are a measure of the relative effects of convection and dispersion on mass transport in the interstitial space; i.e., Pey = VfyL/Dy (34) Pex = VfyL/Dx
(38)
(35)
Each term in eq 33 is labeled with the physical phenomena contributing to the variance. The first four terms on the right side of eq 33 consist of the equilibrium adsorption coefficient KALcoupled with vertical dispersion and the resistances of intraparticle diffusion, mass transfer across the external film, and the rate of adsorption of the molecules on the selective sites of the particle, respectively.
Determination of Model Parameters As shown in eq 33, several parameters appear in the expression for the variance of the elution curve. Values for the parameters can be obtained from independent
(39)
For values of Re, exceeding 2.0, the magnitudes of the Peclet numbers Pey and Pex are approximately constant with Re, in fixed beds. In magnetically stabilized fluidized beds, experimental measurement has shown that the horizontal (or transverse) fluid Peclet number is about one-half of that in fixed beds (Siegell, 1982). In liquid MSB's, the vertical (or axial) fluid Peclet number has been measured to range from 10% to 100% of the corresponding values in fixed beds (Siegell, 1987), depending on how far Vfyis above the minimum fluidization velocity. Based on these previous results, in the simulations that follow, we choose Pey and Pex to be one-seventh and one-half, respectively, of the corresponding values in fixed beds, or Pey = 0.07L/RP
(40)
Pex = 2.5L/RP
(41)
and
For a particle with sufficiently large pores, the expression recommended for D,, is (Satterfield, 1970) D,, = PDJr
(42)
where T is the tortuosity factor for the pellet and is the intraparticle porosity. A typical tortuosity for particles with large pores is T 2.5. For particles with small pores, the following formula for intraparticle diffusivity is suggested (Satterfield, 1970):
-
D,, = P2DPl
(43)
where =
l/Di
+
l/DK~
(44)
and DK1is the coefficient for Knudsen diffusion (45)
Separation, Resolution, and HETP Definitions. From the expressions for the variances and the first moments of the elution curves, one can determine the parametric values and operating conditions that allow good separation of the adjacent components. The criterion that is generally used to ensure adequate separation between two components, say A and B, is that the resolution R,, defined as
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 827 be greater than two. This criterion is valid for elution curves with nearly Gaussian shapes (Giddings, 1965), which is the rule for most chromatogramsoperating at conditions of light loading and low mass-transfer resistance. Thus, by combining the definition of the resolution, eq 46, with the expressions for the first moments and the variances, eq 32 and 33, the efficiency of chromatographicseparations can be determined as a function of system parameters and operating conditions. Another measure of the efficiency of a given chromatograph is the height equivalent to a theoretical plate (HETP), a quantity given by the following expression (Giddings, 1965): (47)
Table I. Parameters and Operating Conditions in Separations of COz and SF6 in Air at P = 0.1 MPa, T = 298 K. Particles Used Were Stainless Steel/Alumina quantity W, cm e
P V,, cm/s R,, cm L , cm Vfy, cm/s
value 0.159 0.5 0.693 0.58 0.0152 14 60
Comparison of Model Predictions with Experimental Data In a series of experiments, a horizontally flowing, magnetically stabilized bed (approximately 60 cm long and 7.6 cm wide) was used to separate (1) COPand SF, and (2) COz and argon, with air at room temperature and 1-atm pressure as the carrier gas. The experimental apparatus and methods are described, and several typical chromatograms are presented elsewhere (Siegell et al., 1985,1986). The purpose of these experiments was to demonstrate chromatographic separation with solids moving in a horizontal direction, transverse to the ascending fluid flow. Nearly uniform solids flow profiles for crossflow MSB's have been reported previously (Siegell and Coulaloglou, 1984). However, conditions under which a Bignificant solids flow gradient may exist have been studied (Cheremisinoff et al., 1985). With large deviations from the plug flow behavior, the separation efficiency of the unit could be greatly impaired. Since the mathematical model derived assumes uniform horizontal flow of the bed, a comparison of the theoretical predictions with the experimental results should serve as an additional test of this assumption. The MSB solids used in these experiments consisted of smaller particles of stainless steel embedded in larger alumina pellets (30% alumina, 70% stainless steel by weight). A mercury porosimetry measurement indicated that the mean average pore radius in the adsorbent was 81.5 A and the pellet porosity was 0.50. A sieve analysis showed that the mean average values of R, and RP2were 0.0152 cm and 2.79 X cm2, respectively. The operating conditions and required parameters are given in Table I for the separation of SF6and COz from a mixture using air as the carrier fluid. Also given are the experimental values Xes, of the first moments of the two components. From X e s , W, p, e, V,, Vfx, V f , and L, the adsorption coefficients are calculated from eq 32 and are also given in Table I. The values of DPiwere calculated from eq 43-45, and the values of kf, were computed from eq 36-39. Vertical and horizontal dispersion Peclet numbers were calculated from eq 40 and 41. Glancing at eq 33, it is noted that Table I contains nearly all the information needed for computing the variances. The only parameter missing is the adsorption rate coefficient k,,. In the case of hydrocarbon separation on silica gel, the contribution of the adsorption rate term to u z / L 2
k&
value 0.58 1.19 x 10-3 1.7 X 12.76 2300 64.5
value quantity M ,g / m d Di, cmz/s sc
Sh
The greater the value of HETP, the less efficient is the chromatograph as a separator. HETP is only an approximate measure of separation effectiveness, since its definition involves only one species, usually the last species of interest to elute from the chromatograph.
cuantity Vfx, cm/s P h g/cm3 pf,g/(cm.s) Re, PeX peY
kf,,
DK,, cm2/s D,,,cmz/s D,,, cm2/s Xexp, cm KAL
SF6 146.1 0.103 1.39 4.39 2940 0.0113 0.0102 4.90 x 10-3 1.95 17.5
COZ 44.0 0.165 0.866 4.04 4330 0.0205 0.0182 8.76 x 10-3 5.28 53.0
Table 11. Comparison of Experimental and Theoretical Values of Standard Deviations of Elution Curves: Separation of COz and SF6 in Air at P = 0.1 MPa, T = 298 K. Particles Used Were Stainless SteeVAlumina contribution to u2/L2 SFc co, vertical dispersion intraparticle diffusion external film resistance horizonal dispersion width of feedband horizontal convection total of a2/L2 utheor, cm uexpp cm
0.000 476 0.000 597 0.000 045 0.000 870 0.000011 0.000 003 0.002 002 0.63 0.52
0.004 058 0.002 846 0.000 259 0.000 870 0.000011 0.000 003 0.008 047 1.26 1.35
appeared to be negligible. Thus, k,, must be sufficiently large so that the adsorption step is not the bottleneck in this case. In the comparisons here, the adsorption term in eq 33 is ignored. Substituting the values from Table I into eq 33 and ignoring the term involving kpL,the results shown in Table I1 were obtained. The contributions of the vertical dispersion, resistance to intraparticle diffusion, and horizontal dispersion terms to the variance are important for SF,, but only the first two dispersion terms are important for COz. The vertical dispersion and intraparticle diffusion terms are proportional to (1 + KA,)2,whereas the horizontal dispersion term is independent of Kk Calculated standard deviations dthe,,r are 0.63 and 1.26 for SF, and coz,respectively. The experimental values uexpare 0.52 and 1.35 cm for SF6and COz, respectively. Considering the a priori nature of the model, the agreement between experiment and theory is excellent. Similar results hold in the case of COz and argon separation. Table I11 lists the parameters used in eq 33, whereas Table IV gives the calculated results. Vertical dispersion and resistance to intraparticle diffusion are major contributors to the variance of both argon and COP For the same reasons as cited above in the case of SF,, however, the contribution of horizontal dispersion is important only for argon. The calculated standard deviations are 0.56 and 1.33 cm for argon and COP,respectively. These results are in good agreement with the experimental values of 0.78 for argon and 1.52 cm for COz. The good agreement between experiment and theory provides sufficient confidence in the model to use it as an evaluative tool in assessing the importance of physical phenomena and in parametric sensitivity studies.
828 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 Table 111. Parameters and Operating Conditions i n Separations of COz and Argon i n Air at P = 0.1 MPa, T = 298 K. Particles Used Were Stainless SteeVAlumina quantity value quantity value W, cm Vfx, cm/s 0.54 0.159 0.5 Pf? g/cm3 1.19 x 10-3 0.693 j q , g/(cm.s) 1.7 X 12.97 V,, cm/s 0.54 Re, 2300 R, cm 0.0152 PeX L, cm 14 P eY 64.5 V,, cm/s 61 value
;
argon
coz
33.9 0.193 0.740 3.95 4950 0.0216 0.0194 9.32 x 10-3 1.82 17.8
44.0 0.165 0.866 4.06 4350 0.0205 0.0182 8.74 x 10-3 5.59 61.7
quantity
Table IV. Comparison of Experimental and Theoretical Values of Standard Deviations of Elution Curves: Separation of COz and Argon in Air at P = 0.1 MPa, T = 298 K. Particles Used Were Stainless SteeVAlumina contribution to u2/L2 argon coz vertical dispersion 0.000412 0.004 588 0.000 276 0.003 279 intraparticle diffusion 0.000023 0.000 296 external film resistance 0.000 870 O.OO0 870 horizontal dispersion 0.000 011 o.OO0011 width of feedband horizontal convection 0.000 003 0.000 003 0.009 047 0.001 595 total of 0 2 / L 2 1.33 0.56 atheor, cm 1.52 0.78 ‘Jexp, cm Table V. Parametric Values and Operating Conditions Used to Determine the Relative Importance of Terms i n the Expression for t h e Variance of the Elution Curve, Equation 3 3 Separation of Ethane, Propane, and Butane. Helium Carrier, P = 1.0 MPa, T = 323 K, Silica Gel Adsorbent operating operating condition or condition or parameter value parameter value V, cm/s 0.5 PeX 21 400 50 R,, cm 0.035 V , , cm/s 30 Vfx, cm/s 0.5 W, cm 300 Pf, g/cm3 0.001 51 L, cm t 0.5 p,, g/(cm.s) 1.90 x PeY
600
parameter KAi
kf,, s - ~
D,,, cm2/s k,
s-l
B
0.486
ethane
propane
butane
34.9 353 0.001 41 11.6
155 289 0.001 54 4.1
758 276 0.002 93 4.9
Importance of Physical Phenomena Smaller Pore Adsorbent. To determine the relative importance of various physical phenomena to separation efficiency, typical values of the operating conditions for the crossflow MSB chromatograph were substituted in eq 32 and 33. In addition, experimental values of the system parameters for the case of chromatographic separation of light hydrocarbons (ethane, propane, and butane) from helium by adsorption on silica gel (Schneider and Smith, 1968) were substituted into the model. Silica gel is an example of a smaller pore adsorbent. Table V lists the values substituted into eq 33, and Table I1 gives the resulting values for a2/L2 and each of its
Table VI. Relative Contributions of Various Terms in Equation 33 to Total Variance of Elution Curve: Separation of Ethane, Propane, Butane. Helium Carrier, P = 1.0 MPa, T = 323 K, Silica Gel Adsorbent sDecies total u2/L2 ethane 0.002 314 propane 0.025 019 butane 0.325 282 contribution to u2/L2 term ethane propane butane vertical dispersion 0.000 101 0.001 916 0.045 356 intraparticle diffusion 0.001 209 0.020 902 0.260 061 external-film diffusion 0.000029 0.000 663 0.016 433 adsorption rate resistance 0.000 049 0.000 612 0.002 506 horizontal dispersion 0.000 093 0.000093 0.000 093 horizontal convection 0.000 000 0.000000 0.000 000 width of feedband 0.000 833 0.000 833 0.000 833 Table VII. First Moments and Standard Deviations of Elution Curves, Resolutions, HETP’s, and Required Bed Length Corresponding to Conditions i n Table V: Obtained from Eauations 32 and 33 and Eauations 46 and 47 value quantity ethane propane butane first moment, fill, cm 70 245 1125 std dev, u, cm 14 47 171 HETP,cm 19.4 12.7 7.12 quantity value resolution, R, (ethane-propane) 2.0 resolution R, (ethane-butane) 5.7 resolution R, (propane-butane) 4.0 bed length: cm 1638 ~~~
“Bed length obtained from p1’ + 3a.
contributing terms on absolute and percentage bases. As indicated in Table VI, the contributions of vertical dispersion and resistances to diffusion are responsible for 57.9%, 93.8%, and 98.9% of the variances for ethane, propane, and butane, respectively. The width of the feed band (W = 0.1L) makes a 36% contribution to a2f L2 for ethane, the first component to elute. For components that elute farther from the feed point, the width of the feed band has a less important effect on the magnitude of the variance. Likewise, the horizontal dispersion is important only in the case of components that elute earlier. Since Vfx is of the same order of magnitude as V,, horizontal convection makes negligible contribution to the variance. The resistance to intraparticle diffusion appears to be important for all three species. For silica gel, the pores are extremely small, on the order of 10-20 A. This results in a diffusion process that is in the Knudsen region (Satterfield, 1970), which is reflected in the low values of the effective intraparticle diffusivity D, (see Table V). For large pores, on the order of 1000 A, bulk diffusion will dominate, and the value of D, will begin to approach about one-third of that for free diffusion. Table VI1 gives the values of the first moments and standard deviations of the elution curves corresponding to separation of ethane, propane, and butane under the conditions specified in Table V. Also given are the resolutions and HETP. As indicated, the first moment for butane is 1125 cm. To ensure 99.8% recovery of butane, the length (defined as the magnitude of the horizontal dimension) of the crossflow bed should be at least 1125 + 3(171) = 1638 cm, or 3 standard deviations longer than the first moment for butane. Of course, the slight asymmetry (tailing of the elution curve) normally present in chromatographswill cause the butane remaining in the bed to exceed 0.2% by a small amount. For the conditions in
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 829 Table VIII. First Moments, Standard Deviations, Resolutions, Total Bed Height, and HETP's Corresponding to R, = 2.0 for Ethane-Propane and Large-Pore Particles with T = 2.5; Conditions the Same As in Table V Except for DKi(Computed by Equation 45) and L" (Adjusted To Ensure R. = 2.0 for Ethane-Prouane) a uantits first moment, pl', cm std dev, u, cm HETP,cm "Bed length obtained from
ethane 34.0 9.1 16.8 fill
value propane 119 32.7 12.4
Table IX. Base Values of Parameters Used in Sensitivity Studies parameter KA1 KA2
butane 544 119 7.12
+ 30 for butane.
Table V, the resolution of ethane and butane is 2.0, which barely meets the criterion for good separation. From an economic standpoint, just meeting the separation criterion is desirable since the bed volume is minimized under these conditions. Also, the amount of carrier gas required for separation is minimized. Larger Pore Adsorbent. If the adsorbent used in the separation of ethane, propane, and butane from helium has the same properties as silica gel except for pore size, then DPiwill be larger and the contribution to intraparticle diffusion will be less. To show the effect of larger pore size on the separation of ethane, propane, and butane in helium, the values of Dpiare calculated from eq 42 and are substituted into eq 33. All other values required in eq 33 are taken from Table V, except for the bed height L, which is left as a variable. If the resulting expression for the variance is substituted into the equation for the resolution, the bed height required for all values of R, to exceed 2.0 is found by iterative solution. As indicated in Table VIII, the required bed height is only 145 cm and the corresponding bed length is 901 cm. It is quite obvious that the large-pore particle offers benefits with respect to bed volume. Adsorbent capacity, however, decreases as average pore size increases for surface adsorbing material (Froment and Bischoff, 1979). Furthermore, the mechanical strength of larger pore particles is less than that of particles with smaller pores (Froment and Bischoff, 1979).
Parametric Sensitivity In the parametric sensitivity studies, two types of operations were examined: (1)mass-transfer-limited and (2) feedband-width-limited. In the former case, over 90% of the variance calculated by eq 33 is due to the resistance to mass transfer within the particle. The use of large particles allows higher throughput of gas without exceeding the transition velocity for a magnetically stabilized bed (Siege11 and Coulaloglou, 1984). In the latter case, over 90% of the variance calculated by eq 33 is due to the width W of the feed band. This would be the case if either smaller particles or particles with thin active outer layers were used and, in addition, the width of the feed band were made larger in order to utilize a higher fraction of the bed volume. Table IX gives the base values of the parameters used in the mass-transfer-limited case. Here, the adsorbent particle radius is rather large (0.03 cm), and the effective intraparticle diffusion D, is rather small cm2/s). The feedband width, however, is much larger (100 cm). Each of the values in Table IX were changed, one at a time, by 4=20%. The resulting percentage changes in resolution R, were then calculated. Parametric Sensitivity in Mass-Transfer-Limited Case. The first and second columns of calculated results in Table X give the changes in resolution R, resulting from f20% changes in the parameters listed in Table IX. Most of the changes in the parameters were made one at a time.
Dpl,Cm2/s D,z, Cm2/s kn, s-' kn, s-l kpi, 8-l k,,, s-l d V,, cmls V f , cmls VK, cm/s W , cm L , cm Rp, cm
values in mass-transferlimited case 100 200 10-3 10-3 165 165 10 10 0.5 0.5 50 0.5 1 600 0.03
values in feedband-widthlimited case 100
200 0.035 0.035 16 500 16 500 10 10
0.5 0.5 50 0.5 100 300 0.005
Table X. Sensitivity of Resolution to *20% Changes in Parameters in Table IX
Darameter
DP RP kf kP
ve
Vrx W L
d K A
VfY
percentage change in resolution mass-transferfeedband-width limited limited +20% -20% +20% -20% change change change change 8.25 -9.46 0.02 -0.03 -15.50 22.16 -0.36 0.35 -0.02 -0.55 0.01 0.37 -0.32 -0.12 0.21 0.08 18.48 -19.14 0.05 -0.10 0.00 0.00 0.00 0.00 0.00 0.00 -15.89 22.94 9.54 -10.56 19.28 -19.52 8.42 -9.62 18.86 -19.32 0.24 -0.38 18.83 -19.31 -8.27 10.97 -16.08 23.52
However, the parameters Kh, DF kfi, and kpi were changed at the same time for both species 1 and 2. Results show that Dpi,R,, L, 4, and Vf, were the most influential parameters on the value of R,. Twenty percent changes in these parameters caused 8.25-22.16% changes in R,. As indicated, resolution increased with Dpi,4, and L but decreased with R, and Vfp These effects can be explained by the influence of these parameters on resistance to mass transfer (the dominant contributor to the variance) and the residence time of the gas in the chromatograph. Parametric Sensitivity in Feedband-Width-Limited Case. The third and fourth columns of calculated results in Table X give the changes in resolution R, resulting from &20% changes in the parameters listed in Table IX. Parameters R,, V,, V,, W ,L, 4, and Vfywere changed one at a time. Parameters Dpi,kfi,and KAiwere also changed one at a time, but for both species at once. Results show that V,, W ,L, 4, K h , and Vf, have the most influence on resolution. Twenty percent changes in these parameters caused 15.89-23.52% changes in R,, whereas similar changes in D,, R,, kfi, kpi,and V , caused less than 0.36% changes in R,. As indicated, resolution increased with L, 4, V,, and Kh but decreased with V,, W , and VfF These effects can be explained by the influence of these parameters on the feedband width and the residence time of the gas in the chromatograph. Economically, one would expect the cost of the chromatographic process to increase with the size of the adsorbent bed and with the amount of desorbent (carrier fluid) used. Since the amount of feed processed per unit volume of bed and desorbent flow is greater in the feedband-controlled case, the commercial chromatograph
830 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988
should be operated under this condition. Conclusions In the case of linear adsorption isotherms, the methods of moments generated by Fourier transforms is an effective way to analyze the crossflow magnetically stabilized fluidized bed chromatograph. Good agreement between experiment and the model was obtained. This indicates that the model can be used as a predictive tool. Most of the parameters required in the model can be specified a priori or obtained from correlations. A parameter that is difficult to obtain is the adsorption rate constant kpr,but sensitivity studies indicate that this parameter is of minor importance to separation effectiveness. In commercial operation of the crossflow MSB chromatograph, it is advantageous to increase the feedband width (thus increasing the fraction of the bed utilized and the feed to desorbent ratio) until its contribution to the variance of the elution curves begins to cause unacceptably poor product resolution. In this mode of operation, improvements in resolution can be made by increasing solids velocity, bed height, the adsorption coefficients, or the parameter 4 = (1 - t)P/t (through the intraparticle voidage P). Improvements in resolution can also be made by decreasing vertical gas velocity. The resolution also increases with solids holdup 1 - E, but changing t is complicated by its dependence on gas velocity, particle size, density, etc. For economic operation, the resolution should be kept at approximately 2.0 Gust enough to separate the components) to achieve separation with minimum bed length and desorbent-to-feed ratio. Nomenclature a = symbol for expression defined by eq 29 and used in eq 27 b = symbol for expression defined by eq 28 and used in eq 27 B = symbol for expression defined by eq 30 and used in eq 29
C , ( X , Y ) = external fluid-phase concentration of species i, kmol/m3 c ( j w , Y ) = Fourier transform of C, D, = diffusion coefficient for species i in desorbent, m2/s DK,= Knudsen diffusivity for species i, m2/s D,, = effective intraparticle diffusivity for species i, m2/s D x = horizontal fluid dispersion coefficient, mz/s D y= vertical fluid dispersion coefficient, mz/s HETP = height equivalent to a theoretical plate, m (eq 47) j = imaginary KAt = equilibrium adsorption coefficient for species i kf,= interphase mass-transfer coefficient between external fluid and adsorbent pellet, l / s k, = interphase mass-transfer coefficient between nonselective and selective regions of adsorbent pellet, l / s L = bed depth of adsorbent in fluidized state, m M = molecular weight of desorbent, kg/kmol m = total number of species to be separated N,(X,Y,r) = concentration of species i in selective region of adsorbent pellet, kmol/m3 of pore volume N,* = equilibriumconcentration of species i in selective region of adsorbent pellet, kmol/m3 of pore volume (see eq 12) n ( j w , Y , r ) = Fourier transform of N, P = pressure, MPa Pex = horizontal fluid Peclet number (see eq 34) Pey = vertical fluid Peclet number (see eq 35) Q,(X,Y,r) = concentration of species i in nonselective compartments in adsorbent pellet, kmol/m3 of pore volume q ( j w , Y , r ) = Fourier transform of Q ,
R, = universal gas constant, kJ/(kmol.K) R, = radius of adsorbent particle, m R, = resolution of two elution curves (eq 46) r = radial position in adsorbent pellet, m p = average pore radius in adsorbent, m Re, = fluid Reynolds number based on particle radius (eq 39) Sc = Schmidt number (see eq 38) Sh = Sherwood number for mass transfer across external film (eq 37) T = temperature, K U ( X ) = unit step function of Heaviside V f y= interstitial fluid-phasevelocity in vertical direction,m/s Vfx = interstitial fluid-phase velocity in horizontal direction, m/ s V , = solid velocity in horizontal direction, m/s W = feedband width, m X = horizontal position in fluidized bed, m X I j = first moment of elution curve of species 1, m Y = vertical position in fluidized bed, m Greek Symbols 4 = (1 - e)@/€ = intraparticle pore volume per volume of
external fluid intraparticle porosity = symbol for expression defined by eq 27 and used in eq
@= y
26 = interpellet bed voidage X = symbol for expression defined by eq 31 and used in eq 29 and 30 t
K,,’ =
nth noncentral moment
F~ = nth central moment E = j w in Fourier transform expression pf = fluid density, kg/m3 p = adsorbent pellet density, kg/m3
2 = variance of elution curve, m2
w
= frequency variable in Fourier transform, m-l
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