I n d . Eng. Chem. Res. 1992,31, 549-561 Industrial Waste-waters and Residues; Information Transfer, Inc.: Rockville, MD, 1977; pp 97-106. Mahalingam, R.; Biyani, R. K.; Shah, J. T. Simulation of Solidification Temperature Profiles in the Polyester Process for Immobilization of Hazardous Wastes. Znd. Eng. Chem. Process Des. Deu. 1981a,20,8590. Mahalingam, R.; Jain, P. K.; Biyani, R. K.; Subramanian, R. V. Mixing Alternatives for the Polyester Process for Immobilization of Hazardous Residuals. J . Hazardous Mater. 1981b,5 , 77-91. Middleman, S. Drop Size Distributions Produced by Turbulent Pipe Flow of Immiscible Fluids through a Static Mixer. Znd. Eng. Chem. Process Des. Dev. 1974,13,78-83. Morris, W. D. An Experimental Investigation of Mass Transfer and Flow Resistance in the Kenics Static Mixer. Znd. Eng. Chem. Process Des. Dev. 1974,13,270-5. Nauman, E. B. Enhancement of Heat Transfer and Thermal Homogeneity with Motionless Mixers. AZChE J. 1979,25,247-58. Nauman, E. B.; Nigam, K. D. P. Residence Time Distribution of
649
Power-Law Fluids in Motionless Mixers. Can. J. Chem. Eng. 1985,63,519-21. Powell, M. R. Process Parameters for the Continuous Solidification/Stabilization Disposal of Hazardous Wastes Through Polymeric Microencapsulation. M.S. Thesis, Department of Chemical Engineering, Washington State University, Pullman, WA, 1990. Sherman, P. Emulsion Science; Academic Press: New York, NY, 1968. Strieff, F. In-Line Dispersion and Mass Transfer Using Static Mixing Equipment. Sulzer Tech. Rev. 1977,3,108-13. Subramanian, R. V.; Mahalingam, R. Immobilization of Hazardous Residuals by Polyester Encapsulation. In Tolcic and Hazardous Waste Disposal; Pojasek, R., Ed.; Ann Arbor Science: Ann Arbor, MI, 1979; Vol. I, Chapter 14.
Received for review February 21, 1991 Revised manuscript received August 7, 1991 Accepted October 1, 1991
SEPARATIONS Pressure Drop Correlations and Scale-up of Size Exclusion Chromatography with Compressible Packings Abdul W. Mohammad, Donna G. Stevenson,and Phillip C. Wankat* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Pressure drop measurements were made for compressible gel packings Sephadex G-25 and G-100. A critical velocity at which pressure drop kept increasing was observed. Correlations were developed for pressure drop. Measurements of the chromatographic separation obtained for separating bovine serum albumin and nickel nitrate were correlated as measurements of H. A scaling procedure which keeps separation, pressure drop, and feed throughput constant was developed for each packing. The procedure and results for G-25 are similar to those developed previously for rigid packings. The use of smaller diameter packings can greatly reduce the amount of packing required. For G-100, which is quite compressible, the scaling procedure was different. Results of the scaling analysis for G-100 showed that there is little advantage to decreasing the particle size with very compressible packings. Size exclusion chromatography (SEC) separates molecules based on differential access to pores in the stationary phase and is used as a commercial method for large-scale separation of biomolecules (Janson and Hedman, 1982). SEC is most commonly used on a large scale for desalting usingdextran, polyacrylamide, and agarose based packings. Unfortunately, due to the highly porous nature of the packings which makes them compressible, the maximum flow rate is limited. Several different approaches for scaling of chromatographic systems have been proposed (e.g. Rudge and Ladisch, 1986; Wankat and Koo, 1988). The approach used by Wankat and Koo (1988) has also been used for adsorption and ion-exchange systems (Wankat, 1987) and adiabatic pressure swing adsorption processes (Rota and Wankat, 1990). This approach assumes that the existing old design is satisfactory. A new design, using a different particle size, will be developed while maintaining the same separation, pressure drop, and throughput. For chromatography with rigid particles it was found that reducing the particle size by half reduces the amount of packing
used and the cycle time by 75% if pore diffusion controls and by 60% if film diffusion controls. In order to develop scaling rules for SEC with compressible packings, it is important to develop correlations between pressure drop and measurable parameters such as flow rate, column diameter and length, and particle diameter. Previous works (Joustra et al., 1967; Davies and Bellhouse, 1989) have shown that the pressure drop correlation is highly nonlinear as opposed to the linear correlation for rigid particles (Bird et al., 1960). The previous experimental work measured pressure drop across the entire chromatographic system including tubing, valves, fittings,frits, and packing. Since the extracolumn pressure drop is significant, measurements of Ap across the column alone are needed. In this paper, the scaling approach will be extended to SEC with compressible Sephadex G25 and GlOO packings. The pressure drop correlations will be determined from the experimental data. Then using the experimental relationship for H and flow rates for the separation of bovine serum albumin (BSA) and nickel nitrate (NN), the scaliig
0888-588519212631-0549$Q3.QQ/Q 0 1992 American Chemical Society
550 Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992
1
d2-l
LJ Eluent
.Net
Glass Column Packed Wirh Scphadcx
Rccordcr
Figure 1. Equipment schematic.
analysis will be illustrated for different value functions.
Experimental Method The system for the experiments is shown in Figure 1. During the pressure drop experiments, the sample loop, the spectrophotometer,and the recorder were disconnected from the system. The pressure transducer was disconnected during the separation experiments. Four columns with diameters of 1.0,1.6,2.6, and 5.0 cm were used in the experiments (corresponding to the columns K10, XK16, XK26, and XK50, respectively, from Pharmacia). Three displacement pumps (two Minipumps, Milton Roy Co., with ranges of 0.0-2.67 and 0.767-7.67 mL/min and Masterflex, Cole-Palmer Co., with ranges of 10.0-100 d / m i n ) were used to pump the eluent through the system. For the 1.0- and 1.6-cm columns, the connections between equipment were made with Pharmacia polyethylene capillary tubing, with 0.1-cm i.d. For the 2.6and 5.0-cm columns, slightly larger tubing (0.35 cm) was used in order to reduce the extracolumn pressure drop. Sephadex G25 coarse with a wet mean particle diameter of 0.021 cm was used. Sephadex GlOO regular with a wet mean particle diameter 0.0157 cm and superfine with a wet mean particle diameter of 0.0088 cm were used. The wet particle size distributions were determined with a dark field microscope and an image analyzer. Detailed results for G25 are given by Stevenson (1989) and for GlOO by Mohammad (1991). The experiments that were done in this research are different from previous experiments (Joustra et al., 1967; Davies and Bellhouse, 1989). Joustra et al. and Davies and Bellhouse treated pressure drop as the independent variable and measured flow rate a t a specific pressure drop. They failed to separately measure the extracolumn pressure drop, and thus the actual column pressure drop was unknown. Pressure Drop Experiments. Pressure drop across the column is measured using a pressure transducer, Viatran differential pressure transducer Model 123, with range of 0-125 OOO dyn/cm2. The pressure transducer was connected to the system through two three-way tubing
connectors which are above and below the column (see Figure 1). The transducer is attached to a regulated power source and to a meter (Keithley Model 197 digital multimeter) from which the voltage can be read. In each experiment, after the column is packed, the packing was allowed to gravity settle until the bed height became constant. For G25, this period of settling takes about 30 min, while for GlOO regular and superfine it takes from 1to 2 h. The column adaptor was then inserted down to the top of the packing. This was done carefully to make sure that no bed compression occurred. When the flow rate was increased,the bed shrank due to compression and a small gap was left between the adaptor and the top of packing. If samples were introduced, a large amount of peak spreading will be caused by this small gap. However, pushing down the adaptor may in itself cause additional compression. To ensure that the compression which occurred was solely from the change in flow rate, the column adaptor was fixed at the initial level for pressure drop measurements. Separate experiments were done for chromatographic runs. Flow rate was increased step by step until close to the critical flow rate at which point the pressure drop did not equilibrate. At each flow rate, the bed was allowed to equilibrate until the pressure drop and bed length were constant. For G25, it took from 15 to 30 min before the bed became equilibrated. For G100, it took from 30 to 60 min. The bed length and the reading from the pressure transducer were then recorded. The measured pressure drop includes the extracolumn pressure drop which comes from the tubing, plunger, and net ring. The extracolumn pressure drop as a function of velocity was obtained for each column by measuring the pressure drop across an empty column (without packing). The pressure drop across the empty column is negligible compared to the pressure drop of the tubing, expansion and contraction, and the support net. The extracolumn pressure drop accounts for 30-70% of the total pressure drop depending on the type of column packing, packing length, and velocity. Separation Experiments. During a separation experiment, the sample was injected into the system by two single-channel, four-port valves (SRV-3Pharmacia), as shown in Figure 1. For Sephadex G-25 the 1.0-cm-diameter column was used, and for Sephadex G-100 the 1.6cm-diameter column was used. Concentrations were measured by absorbance at 280 nm. Eluent flow rate was measured by timing the flow of 10 mL of eluent into a graduated cylinder. The objective of these experiments was to measure the separation efficiency of the packings used. The results obtained in these experiments will be used later to develop scaling rules. Samples containing bovine serum albumin (BSA) and nickel nitrate (NN) were used. To prevent excessive zone spreading, the adaptor had to be moved down to the level of the packing every time the flow rate was increased.
Experimental Results Sephadex G25. Experiments were done for column diameters of 1.0,2.6, and 5.0 cm at approximately equal initial column lengths of 11.6, 12.3, and 11.7 cm, respectively. Table I shows the results at the critical velocity. Figure 2 shows the pressure drop vs velocity relationship for the three different column diameters. For most of the velocity range, the pressure drop relationship is linear. This would indicate that the G25 packing is quite rigid with a small degree of compressibility and the pressure drop correlation is linear as predicted by the BlakeKozeny equation. Only at large velocities close to the critical ve-
Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992 551 Table I. Column Dimensions and Critical Velocities init length, crit flowrate, packing and column cm mL/min Seuhadex G25 coarse D = 1.0 cm 11.6 7.8 D = 2.6 cm 12.3 49.9 D = 5.0 cm 179 11.7 Sephadex GlOO regular D = 1.6 cm 12.7 0.98 D = 2.6 cm 0.91 12.9 0.72 12.4 D = 5.0 cm Sephadex GlOO superfine D = 1.6 cm 10.5 0.76 D = 2.6 cm 10.7 2.01 10.6 D = 5.0 cm 6.66
crit velocity, cm/min
length at *o,cri cm
reduction in length, %
Dld,
9.92 9.42 9.15
11.3 12.0 11.3
2.6 2.4 3.4
48 124 240
1.96 4.82 12.7
10.4 10.1 9.7
18.1 21.7 22.6
102 166 318
0.38 0.38 0.34
8.3 8.4 8.3
20.9 21.5 21.7
182 295 568
2owo
h1.6 cm. L42.7 M 18ooo
o b 2 6 om.L-12.8em ~
b S . 0 cm. 4-12.4 an
12ooo (dync/wn’)
m
0
3 V.L
(cm*/mm)
.6
9
I
v. (cm/min)
Figure 2. Pressure drop data at different diameters for Sephadex G25 coarse.
Figure 3. Pressure drop data at different diameters for Sephadex GlOO regular.
locity did the pressure drop increase nonlinearly. After the critical velocity, the pressure drop did not equilibrate. It should also be pointed out that the critical velocity is larger for smaller diameter columns. This may be due to the wall supporting the packing. Since the degree of compressibility is small, this wall effect is significant only at high flow rates at which the packing is deformed. This support will diminish as the column diameter increases, and this will cause the packing to collapse thus attaining higher pressure at the same velocity. Sephadex GlOO Regular: Experiments at Different Diameters, Constant Length. Three columns with diameters of 1.6, 2.6, and 5.0 cm were used in these experiments. The 1.0-cm-diameter column was not used since the adaptor could not be pushed down the column without compressing the packing. Table I shows the dimensions and critical velocities for experiments with GlOO regular. Figure 3 shows the plot of pressure drop vs velocity for all three cases as the velocity increases. Two important observations from this plot are the nonlinear relationship between pressure drop and velocity and the dependence of pressure drop on column diameter. The nonlinearity in the pressure drop curve is probably caused by the compressibility of the packing. When the velocity was decreased, hysteresis was observed in the pressure drop curve (Stevenson, 1989). Hysteresis was also observed by Jowtra et al. (1967) when the pressure drop was decreased. Note that this wall effect is different than the wall effect for rigid particles (Cohen and Metzner, 1981) which is due to changes in porosity near the wall. For rigid particles
the wall effect is significant only for column-to-particle diameter ratios less than 50 to 1. Table I shows that D / d , = 48 for the 1.0-cm4.d. column; thus, this wall effect is essentially negligible. The dependence of the pressure drop on column diameter can be explained by the presence of wall support. As the flow rate increases, the particles are deformed from a spherical shape. This,in turn, creates a force which will be balanced by the support from the wall. As the column diameter increases, the wall support will be diminished. The loss of support will reduce the porosity which causes the pressure drop to be higher at the same flow rate. In addition, the deformed particles will have a higher pressure drop than spheres even at constant porosity. The changes in length are shown in Figure 4. As the critical velocity is approached, a sudden decrease in column length was not observed. Ladisch and Tsao (1978) observed similar effects for ion-exchange resins. They postulated the presence of a small region at the bottom of the column where the particles are highly compressed and porosity is small. In this case there could be a large pressure drop increase with very little change in the column length. This highly compressed region could also explain the hysteresis observed when velocity is decreased. Our data are not sufficiently detailed to confirm or deny the Ladisch and Tsao hypothesis. In contrast with the data for G25, the change in length, (L- Li)/Li, shown in Figure 4 is much more significant for GlOO because it is much more compressible. What is expected is that the more significant the wall support, the
552 Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992
Change in Length (%)
A
D4.6cm
v,
(Cdmin)
Figure 5. Pressure drop data for Sephadex GlOO regular at varying lengths, D = 2.6 cm. 25000
Sephadex GlOO regular column diam, cm 1.6 2.6 5.0
bl 48.7 123.3 112.4
Sephadex GlOO superfine
b2
bl
b2
19.5 45.9 30.9
81.1 41.4 48.1
9.15 2.93 3.23
D=1.6cm.~=105cm A
-2.6
a
D=5.0cm.L,=IO.bcm
cm, L,=10.5 cm
less reduction in length will occur. That seems to be the case. For modeling purposes, the data were fitted with the following equation:
L --- Li -- blv& Li b2 + V&
(1)
The values for constants bl and b2 are shown in Table 11. The solid lines in Figure 4 show the fit of eq 1. Sephadex GlOO Regular: Experiments at Varying Lengths, Constant Diameter. These experiments were done to investigate the effect of column length on the pressure drop. Only the 2.6-cm-diameter column had a long enough plunger to allow for four different bed lengths. For the 5.0-cm-diameter column, only two different bed lengths were investigated. Table I11 shows some of the important results for the experiments with the 2.6-cm column, and Figure 5 shows the pressure drop vs velocity. For rigid packings if the length is doubled, the pressure drop will also double. However, with compressible packing~the pressure drop increases much faster than linearly. This apparently occurs since the longer bed puts more weight on the particles at the bottom of the column. Coupled with the compression force from the flow, the particles at the bottom of the column deform faster and thus the pressure drop increases. Another interesting observation from Table I11 ia that the total percentage reduction of length is approximately the same for all lengths.
5 vo
(cm/min)
Figure 6. Pressure drop data at different diameters for Sephadex GlOO superfine.
Sephadex GlOO Superfine: Experiments at Different Diameters, Constant Length. GlOO superfine packing has an average particle diameter about half of that for GlOO regular. Table I shows the column dimensions and critical velocities, and Figure 6 shows the pressure drop vs velocity. A comparison of Sephadex G-100 superfine to regular in Table I shows that the ratio of the column diameter to particle diameter is now approximately doubled, which makes it harder for the wall to support the particles even for the 1.6-cm4.d. column. Overall, the pressure drop increases faster than for Sephadex GlOO regular, and the critical velocity is about half that of GlOO regular.
Table 111. Results for Critical Velocity with SeDhadex GlOO Regular at Varviniz LenPths. D = 2.6 cm Li, Sephadex GlOO regular Li,Sephadex GlOO superfine 5.0 12.9 20.4 34.5 5.9 10.7 22.5 critical velocity, cm/min 1.55 0.91 0.63 0.40 0.59 0.38 0.19 length at v,,~”,cm 3.9 10.1 15.9 27.0 4.6 8.4 17.9 total reduction of length, % 22 22 21 22 21 21 20 ~
Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992 553
Bovine Scnun Albumin
Change in Length (%)
0 5
0 A
.25 0
10
15
Velum. (ml)
b 1 . 6 cm bl.6cm I
I
I
.5
1
1.5
I
I
I
I
2
2.5
3
35
4
Figure 9. Example of EMG modeling of chromatography peaks for BSA and NN on 1.0-cm column of Sephadex 25 coarse with uo = 2.33 cm/min.
u v.L (cma/min)
Figure 7. Percent change in length for Sephadex GlOO superfine.
by chemical interaction of the sample with the packing. The exponentially modified Gaussian model (EMG) was used to fit the peaks. The EMG model is a Gaussian profile convolved with an exponential decay function to give the following (Grushka, 1972):
EMG(V) = (V-
v, - vy
exp (- V'/
24
T)
d V' (2)
The total variance of the EMG is described by the equation
0.05
.I
.15
.l
25
yo
3
35
.4
A5
5
55
(cm/min)
Figure 8. Pressure drop data for Sephadex GlOO superfine at varying lengths, D = 2.6 cm.
Figure 7 shows the change of length as a function of velocity. The difference in percent length reduction is much smaller than in Figure 4 which is what should be expected since the effect of wall is much smaller. The data were fitted to eq 1. Table 11shows the values for constants bl and b2. The solid lines in Figure 7 are the fit. Sephadex GlOO Superfine: Experiments at Varying Lengths,Constant Diameter. Again these experiments were done to investigate the effect of length on the pressure drop. Table I11 shows the critical velocities and length reductions for the experiment with D = 2.6 cm. Figure 8 shows the pressure drop vs velocity. Again the increase in pressure drop is significantly greater than linear when the length is increased. Also, the critical velocities are lower than for Sephadex G-100 regular. Separation Experiments. The chromatographicpeaks were found to be asymmetrical and skewed. An example is shown in Figure 9 for Sephadex G25. This behavior is most likely caused by the extracolumn effect from the tubing, connectors, frit, and detector (Jonsson, 1987; Sternberg, 1966). It is also possible that the skew is caused
u2
= ug2
+ 72
(3)
The value of the T/U, is a measure of peak asymmetry. Once T and u, are known, the column efficiency can be determined. From eq 2 the following is derived (Hanggi and Carr, 1985): EMG(V,V,,a,,T) = A exp(B)C(V,V,,u,,r)
(4)
A = area/27
(5)
-v---v,
2"-
2,:[
a,
:]
I
(8)
Equations 4-8 are used to fit the experimental data obtained. The IMSL subroutine DUNSLF,which uses a Levenberg-Marquardt algorithm and a finite-difference Jacobian, was programmed to do the fit (Stevenson, 1989). By fitting the equation, one obtains values for T, u , V,, and area. Figure 9 shows examples of data points dtted to the model for Sephadex G25. After fitting the concentration data, other information
654 Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992
0
5
0
vo
10
dV
=
s_:f(V, where f(v) is eq 4. Then
I
I
I
4
.6
.8
vo ( c m / k )
can be determined. The equation is integrated to obtain the mean retention volume, V, (Foley and Dorsey, 1983): l:Vf(V,
I
.2
(CnJmin.1
Figure 10. Has a function of velocity for BSA and NN, Sephadex G25 coarse.
VRi
I
0
=vg+7
(9)
dV
a2 is
Figure 11. H as a function of velocity for BSA, Sephadex G100.
0.07
-
0.06
0.05 -
computed: 0.03
and then
N = V,:/U2
0.01
(11)
From these 0
H = L[a2/VR?]
Sephadex G25. In Figure 10, H i s given for BSA and nickel nitrate as a function of superficialvelocity. Analysis of the data for BSA shows that it could be adequately fit by a constant, HBsA= 0.226, and this solid line is shown. Since the BSA is excluded from the pores, this result is reasonable. For nickel nitrate, H could be adequately fit by a linear model, but a slightly better fit was obtained with the following empirical form (Grushka et al., 1975): HNN
= 0.342~0O.~~'
(13)
The above form of equation can be related to the particle diameter through the following equation (Wankat and Koo, 1988): Hcolum =
AVndpn+l
(14)
The form of this equation corresponds to the usual exponential form of mass-transfer correlations (e.g., Sherwood et al., 1975). For nickel nitrate we found n at one value of d, and then used eq 14 to generate the following expression.
HNN= 184~~0.62'd P 1.628
2
4
(12)
(15)
.6
.a
1
vo ~ C d m i n )
Figure 12. H as a function of velocity for NN, Sephadex G100.
The constant n = 0.6276 is very close to the predicted value for porous packings which is n = 0.6 (Grushka et al., 1975). For BSA, which is excluded, there is no internal mass transfer. Then the van Deemter equation predicts H is linearly dependent on particle diameter (Grushka et al., 1975). Thus HssA
10.8dp
(16)
Sephadex GlOO. Figures 11and 12 show the H values for BSA and NN at different velocities using both the GlOO regular and superfine packings. Note that BSA is not totally excluded from G100. H increases as velocity increases. The values of H for BSA are generally about 10 times higher than that for NN. This is expected since the diffusion constant is much lower for BSA molecules. In general the values for Sephadex GlOO superfine are below the range of those obtained with GlOO regular. For BSA, the increases in H as the velocity increases for GlOO superfine are approximately the same as for GlOO regular. However for NN, H increases much less for GlOO superfine.
Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992 555 in addition to extracolumn effects appears to be causing the skewing of peaks with G100. The more modest skewing of the G25 peaks can probably be explained as an extracolumn effect.
Modeling the Pressure Drop Data In trying to model the pressure drop data, a correlation that relates pressure drop to measurable parameters such as velocity, column length, permeability, column diameter, and particle diameter is the objective. With that in mind, the following equation which follows the approach taken by Joustra et al. (1967) has been found to fit the data well for the curve with increasing velocity.
'8i .6
-
4-
BSA
A 0 "
2-
-
where 1
I
I
0
2
4
A
8
10
l/\(min/cm)
Figure 13. Ratio of ./a, for BSA and NN.
For modeling purposes, the experimental data for H were fitted with linear equations. For Sephadex GlOO regular data, the following equations are used. For BSA H 0.234 + O.321Uo (17) and for NN H = 0.0322 + O.O464Uo (18) For Sephadex GlOO Superfine data, the following equations are used. For BSA H = 0.121 + O.25LUo (19) and for NN H = 0.0339 + O.OO95Uo (20) The use of the exponentially modified Gaussian model to fit the data produced an additional parameter T which is the exponential time constant for the peak in addition to the retention volume, V,, and the Gaussian portion of the variance, uF The ratio of T / U ~is a measure of how badly the peak 18 skewed Figure 13 shows the ratio of T f ug as a function of l / u o for both BSA and NN. If the peak skew is assumed to be due to extracolumn effects, T / U will ~ be the same for Sephadex GlOO regular and GlOO superfine. The ratio appears to increase slightly as l / u o increases, which means that the peak is skewed more at lower velocities. Notice also that the values for BSA are slightly higher than for NN. The data were fitted with the following equations. For BSA
7 = 0.899 + 0.0336(
-$
Qg
and for NN
7 = 0.898 + 0.0175(
k)
ug
(21)
The a term was found to correlate linearly with the ratio of column diameter to initial length, D/Li. Note that Li is the length of the bed at the initial stage when there is no flow. L is the bed length after the compression had taken place at a particular velocity ug. The 4 term takes into account the criticalvelocity at which the pressure drop will increase infinitely. C1is a constant that will make 4 significant only when u, is close to the critical value. For both Sephadex G25 and GlOO data, a value of 200 for C, is adequate. It was found that u,,L can be correlated to L in eq 24. ko is the ratio D/Li,which is why U ? , ~is~used the permeability term at the initial stage when there is no flow. As the flow rate increases, the permeability will change, and this is accounted for in the exponential term. The particle diameter, d is the diameter measured when the particle is a t equil&hm initially with the solvent. However, as the flow rate is increased, the particle will be deformed. This change is also taken into account by the exponential term. Sephadex G26 Packing. Sephadex G25 particles are only slightly compressible. Using eq 23, the a term is approximately zero, which makes the exponential term approximately 1.0. The permeability, k,, and the particle diameter, dp,O,are basically constant since they will not change much. This choice for a means that the entire compressibility effect is included in 4, which implies that compressibility is important only near the critical velocity. From linear regression of the data below the critical region, eq 23 becomes A p = 194u,& (25) Using a mean particle diameter value of 0.021 cm, the external void fraction of 0.385 (Stevenson, 19851, and a water viscosity value of 0.01 g/(cm.s), the permeability equation is (26)
The constant C2 was found to be 77.3, which is somewhat This result that T / U ~is larger for BSA is surprising. different from the reported value of 150 for rigid particles Relative values of u t can be estimated from eq 12 using (Bird et al., 1960). The fitted line was shown in Figure H values from eqa 17-20 and experimentally observed region is taken into account by 9. The retention volumes. Approximately, ~ ~ 2.5 , ug,"'. ~ s 2. ~The critical ~ values for critical velocity u0,, and L at u0,, are shown in When the T / U values ~ are equal, TBSA 1.6~". If T were Table I. due solely to extracolumn effects, we would expect u,, a Sephadex GlOO Packings. For Sephadex GlOO regular 1/D (Jonsson, 1987) and thus T a 1/D. This would give and superfine, the parameters uo, L,and dp,oin eq 23 were TBSA < T", which was not observed. Some phenomenon
- -
556 Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992
87-
654-
3-
4’
0
G1OOng
1
GlMXf 0 ,
r
I
I
I
1
0
.Z
.6
4
.8
1
DLi
Figure 15. Dependence of vo,cril on D/Li for Sephadex G100.
The solid lines in Figure 14 show the fits. One would expect that the dependence on column diameter will diminish as the particle diameter decreases or the column diameter increases. As we can see, the use of the smaller particle size Sephadex GlOO superfine yielded a lower value for the slope for a. Figure 15 shows the value uo,eril as a function of D/Li for Sephadex GlOO regular and Sephadex GlOO superfine. Linear regression of the experimental data gave the following equations. For Sephadex GlOO regular
Vo,c,iL= 11.2 - 9.29(D/Li)
(31)
and for Sephadex GlOO superfine
Vo,criL= 3.42 - 1.24(D/LJ
(32)
The solid lines in Figure 15 show the fits.
Scaling Analysis The scaling procedure that will be shown below is for the separation of bovine serum albumin from nickel nitrate. For Sephadex G25, this type of separation (desalting) is the most common commercial application. Commercial large scale operation using Sephadex G25 packing has been reported as early as in the 1960s (Joustra et al., 1967). Sephadex GlOO packing, meanwhile, due to its highly porous nature, has been used most often to fractionate proteins and other larger molecules. That does not mean, however, that the approach taken here will be irrelevant. I t is hoped that the scale-up procedure for Sephadex GlOO packing that will be presented here will form the basis upon which further scale-up with compressible packings can be made. Obviously different experimental data are needed and it is possible that different conclusions will be reached. Scaling Using Sephadex G25 Packing. It will be assumed that Sephadex G25 particles are rigid particles below the critical velocity, and as such eq 24 will be used for the pressure drop with the exponential constant, a, equal to zero and r#~ = 1.0. The brief scaling example developed below is similar to the example shown for rigid particles by Wankat and Koo (1988). Further details are given in that article. First, the scaling factors are defined:
Ind. Eng. Chem. Res., Vol. 31, No. 2, 1992 667 Table IV. Scaling Results for Sephadex G25 guess U0,nsw
(33) If the scaling factors are substituted into the ratio for pressure drop, the following equation is obtained. (34) The goodness of the separation can be measured by resolution R or by purity. For two componenh with dissimilar H values, R is exprmed by the following equation (Wankat and Koo, 1988):
Taking the ratio of the resolutions
The advantage of using resolution instead of purity is a simple expression results and the scaling procedure is simplified. The use of purity will be shown for G100. There are six unknowns: RR, R,, Rdp,RD, RL, and R If we set any four parameters, the other two can be c 9 culated. Additional assumptions are as follow: (a) The chromatographic system is linear with independent solutes. (b) The chromatographic peaks are Gaussian. (c) Changing the particle diameter, length, or column diameter does not change the equilibrium and the packing characteristics. (d) The chemistry of the system is unchanged as particle diameter is changed. It will be assumed that the old design with Sephadex G25 coarse is adequate. The parameters L, uo, and Q are unknown. For the new design, we intend to use the smaller Sephadex G25 medium particles. Thus the design problem is to find parameters for the new system and then to determine if the new system offers advantages compared to the old design. From the retention volumes, the capacity factors were found to be kkN= 1.42 and kLSA = 0.0. Substituting the ITS (eqs 17 and 18) and capacity factors into eq 36
For the new design the pressure drop and throughput should be the same, which means that R, = 1 and RQ = 1. Sephadex G25 coarse has a mean particle diameter of 0,021 cm while Sephadex G25 medium has a mean particle diameter of 0.0105 cm. That means Rd = 0.5. A case study approach will be used. Tf;e procedure will be to pick uOslewand calculate the scaling factors. If equal or better resolution is obtained for the new system, and the velocity is within the acceptable range (i.e., below the critical velocity), the design calculation is completed. For an example, the old design with Sephadex G25 coarse has the following specifications: d = 0.021 cm, L = 15.8 cm, uo = 5 cm/min, and R = 1.0. ‘fable IV shows the scaling , ~ For the first and second factors when u ~ , is~ picked. guesses we do not reach our target of better or equal resolution. With the third guess, uo = 4.0 cm/min, we obtain
UO,old
O.g~o,~id o*8U0,01d
RL
RD
RR
0.25 0.28 0.31
1.0 1.05 1.12
0.84 0.91 1.00
an equal resolution for the new design. Comparison between the old and the new designs shows that the new design column is shorter, fatter, and operates at lower velocity. The ratio of packing volumes is (volume packing),,, = RLRD2 (volume packing)old
(38)
which is (0.31)(1.12)2= 0.39. The new design uses much less packing than the old design. Since a shorter column is to be used,the peaks will exit earlier. In order to obtain the same resolution, the feed pulse needs to be shortened. This can be done by scaling the feed period in the same way as the retention time of the solutes.
In order to have the same throughput of feed with shorter feed pulses, the cycle time needs to be decreased. The resolution of overlapping peaks from adjacent feed pulses will be constant if the cycle time is scaled in the same way as the retention times.
For the new design above with RQ= 1.0, R f d and Rwcle are equal to 0.39. Is the new design better? The current price of both Sephadex G25 coarse and Sephadex G25 medium is $87/100 g (Pharmacia, 1991). Assuming that the bulk price of G-25 coarse equals that of G25 medium, thismeans the cost of the packing in the new design is 619% less than in the old design. Unless the cost of the column for the new design is too expensive, which should not be the case, the new design should be considered because it offers the same separation at a significantly reduced cost. By manipulating the scaling factors, other effects of changing different parameters on the new design can be seen. If RQ > 1.0, the procedure becomes a scale-up analysis. For G25 the scale-up procedure is very similar to the example shown here since pressure drop is not a function of column diameter. Scaling Using Sephadex GlOO Packings. Scaling a system with more compressible packing such as Sephadex GlOO is much more complicated since the pressure drop is nonlinear with respect to velocity. The equations involved cannot be simplified as in the case with Sephadex G25 packing. Again, a case study approach will be used to do the scaling. One thing to remember is that the scaling rules developed here are restricted in the sense that they are only applicable for Sephadex GlOO regular and GlOO superfine packings for the range of column diameters studied here. For Sephadex GlOO purity will be used instead of resolution. The reason for this is that the peaks for Sephadex GlOO were highly skewed and unless the T / U value ~
