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Ind. Eng. Chem. Res. 1999, 38, 3628-3634
Speeding the Design of Bioseparations: A Heuristic Approach to Engineering Design E. N. Lightfoot Department of Chemical Engineering, University of Wisconsin, Madison, Wisconsin 53706
A variety of economic and social pressures require lowering the costs of equipment and process development, and particular emphasis must be placed on speeding what is basically a heuristic activity. The early stages of conceptualization and screening of alternatives are the most critical, and it is precisely these which have largely been ignored in the public literature and academic curricula. Such early decisions must be made without detailed understanding, and they are essentially evolutionary activities (Lightfoot, E. N. Chem. Eng. Prog. 1998, Jan, 67-74). Evolution dynamics are notoriously unpredictable, and yet observation of biological evolution shows that they can be both surprisingly fast and efficient. Here methods are suggested for achieving high efficiency for equipment and process development. A critical feature of evolutionary dynamics is the need to identify system anomalies, and these can only be discussed effectively through use of specific examples. Here examples will be taken from the general area of separations processes, with particular emphasis on biological separations. Introduction Problem Definition. For our purposes the overall technical activities of engineers can be organized as in Figure 1, and those of particular interest here are represented in the upper left corner: invention. We include as invention the proverbial “flash of genius”, which is the impetus for a new development, and also the early selection processes, which determine the direction of the developmental process. These activities are essentially qualitative and are followed by the quantitative calculations and experiments which lead to a final design. These distinctions will become clearer as our discussion unfolds. As pointed out earlier,1 there are some useful parallels between these inventive processes and biological evolution, and we consider two here: the generally obscure origins and the subsequent combinatorial problems to be faced. Despite a very voluminous literature, no one has yet provided a convincing explanation for the first defining steps in the origin of terrestrial life. However, our understanding of the subsequent steps, though very far from complete, is improving rapidly (see, for example, ref 2), and the utility of this knowledge is already being realized by chemical engineers.1 It was recognized very early that enormous combinatorial problems make predicting the results of evolutionary processes all but impossible in advance. This is easily demonstrated by an update of Jonathan Swift’s old problem: how many ape hours would be required to produce a Shakespeare play by random typing? For a text of 120 000 characters at 8 bits/character, typical of a modern computer, there are 10301 008 combinations. If the apes could match the speed of a good typist, producing these combinations would require about 10301 000 years. Comparing this number with the 1010 years widely accepted for the age of our universe, let alone the much smaller time for existence of higher mammals, one can see that truly random evolutionary processes are exceedingly inefficient. Nevertheless, it is clear that biological evolution has proceeded quite rapidly, considering the nature of the task. Much of
Figure 1. Organization of engineering activity.
evolutionary research has concentrated on finding the techniques employed in biological evolution to minimize the combinatorial problem. Moreover, and this point is critical to our discussion, independently repeated evolutionary processes tend to produce remarkably similar results, a phenomenon known as convergent evolution. There are many striking examples, and one is shown in Figure 2: a comparison of the Australian sugar glider with the American southern flying squirrel. The first is a marsupial and the second a mammal, and their development appears to be entirely independent. Yet both have the same structure. It appears then that the range of effective solutions to an evolutionary problem may not be extremely large and that one can devise a strategy for finding these solutions in a reasonably short time. We suggest below that this is indeed possible for processes of engineering interest. Problem Importance. Rapid growth of worldwide competition and public expectations along with increasing sociopolitical constraints has put U.S. industry in a difficult position (J. A. Miller, du Pont, quoted in ref 1). Economic constraints have, in fact, often led to a decrease in research and developmental effort even though rapid improvements in technology are clearly required. The only way to resolve this paradox is to increase the speed and efficiency of research, and it is
10.1021/ie9900566 CCC: $18.00 © 1999 American Chemical Society Published on Web 07/02/1999
Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3629 Table 1. A Descriptive Hierarchy conceptual orders of magnitude discrete pseudocontinuum continuum molecular
Figure 2. Convergent evolution: (a) American flying squirrel and (b) Australian sugar glider. Drawing by Enrique Ruede-S.
the basic thesis of this paper that the bottleneck is invention, as defined above. Problem-Solving Strategy. Our major concern here will be to deal with the development of a feasible prototype design following an initial inventive “flash of genius”. The unpredictability of evolutionary processes requires that one “begin at the end”: defining the project goal as precisely as possible without eliminating potentially attractive possibilities and proposing qualitative means of reaching the goal. One must start with relatively crude global conceptual models and proceed through increasingly detailed and reliable approximations toward a final design. Only after a promising conceptual system has been devised can one begin the systematic series of calculation and experiment normally needed to produce a final design. Note that this approach eliminates the possibility of true optimization: before any optimization techniques can be employed, many possibly better approaches have already been eliminated. This is unavoidable. The approach used must be tailored to the problem at hand, and we shall provide examples below. However, in general one proceeds through successive levels of description which can often be summarized as in Table 1. The examples here have been chosen for use in the design of separations processes and equipment, which we shall use as a basis for discussion below.
qualitative morphology semiquantitative estimates macroscopic balances, compartmental modeling transfer and dispersion coefficients the equations of motion, energy and diffusion kinetic theory and molecular dynamics
These levels represent a hierarchical set, in which predictions at each successive level can only be justified by analysis at the level immediately below it in the table. In other words, we are always working with a significant element of uncertainty which can only be tested after the fact. This is the essential paradox behind all creative processes, and success of the inventor depends on skill in making good judgments in the face of uncertainty, i.e., on intuition. This has long been recognized by creative people, and it is exemplified by a quote from the French mathematician Henri Poincare´: “It is by logic we prove, it is by intuition we invent” (ref 3, p 351). Intuition, however, is largely some weighted average of past experience, and a major goal of this paper is to illustrate how experience may be put to effective use. The initial “flash of genius” is much more difficult to deal with, and there seems to be no way to systematize this. Most truly original ideas arise only after long study and are triggered by apparently irrelevant observations.3,4 The case of chromatography, which we shall return to below, is typical in this respect. Michael Tswett, a Polish biochemist working with crystallization of plant pigments, noticed the formation of colored rings on drying filter paper, and over a period of 5 years he gradually became convinced that he was seeing a selective migration of individual solutes.5-7 This observation led to the development of a crude exploratory apparatus and ultimately through laboratory columns to modern production techniques. However, most new ideas arise from combinations of familiar ones.8 Applications To illustrate the proposed strategy, we consider the relatively compact field of separations processes. We begin with rather general considerations and then proceed to more detailed examples. Empirical Observations. One of the most insightful observations in the development of separations processes has become known as the Sherwood plot, named after T. K. Sherwood who first suggested it: a plot of selling prices of various chemicals against their concentrations at the start of the separations processes needed to produce marketable material from the naturally occurring source. He found that available data could be correlated by the simple equation
C/P ) K(I/P)
(1)
where C is the selling price, P is the mass of the product, I is the initial mass of impurities, and K is a “universal” constant, essentially independent of product purity. This equation, which suggests that the primary determinant of price is the cost of handling inert material of no value, has since been confirmed many times. An example for processing of biologicals is shown in Figure 3.9 The trend line in this figure is just a line of slope -1, in accord with eq 1. Note that only the two most concentrated feeds, water itself and ethanol, depart significantly from the trend but also that the scale is greatly compressed:
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Figure 3. Economics of separation processes, from Dwyer, J. L. Bio/Technology 1984, 2, 957, as reported in ref 9.
Figure 4. Separation process trajectories.
1 order of magnitude, a factor of 10, is hardly notable here, and a number of individual species depart by more than this from the trend. Nevertheless, the correlation, which covers 10 orders of magnitude, is impressives and surprisingly useful.10 It shows, for example, that such common processes as distillation and solvent extraction are inherently unsuited to recovery from dilute solution. The Sherwood plot also suggests diagramming separations processes on a graph of thermodynamic activity vs concentration (Figure 4), which can be considered independent variables. A specific example is the concentration of proteins where the path FIP is commonly used by laboratory chemists: salting to raise thermodynamic activity and causing a spontaneous precipitation. This makes a great deal of sense to chemists whose time is their greatest expense. To the production engineer, on the other hand, the consumption of large amounts of salt, typically ammonium sulfate, and the subsequent waste disposal problem make such a process quite unattractive. The spontaneous adsorption on an ion-exchange adsorbent, followed by thermodynamic release using a much smaller amount of salt, is far preferable. Concave trajectories on such a diagram tend to be the most attractive for large-scale processing. The Sherwood plot has also led to classification of downstream processing of biologicals into three categories: capture or concentration, fractionation, and polishing. The first is primarily concerned with volume reduction, and it is the most expensive. Polishing is the
inverse of concentration, and relatively small process volumes permit the use of expensive techniques such as size-exclusion chromatography. Fractionation is concerned with separation of the desired product from closely related impurities. Transport Analogies. Equation 1 is just a specialized recognition that materials handling costs are important in large-scale separations processes. One can go an additional step and set up guidelines for separator design, recognizing that most separators require transfer of a desired solute between two adjacent streams moving parallel to each other. This is done in Table 2. The balance between mass and momentum transport is the primary concern. The need to maximize lateral mass transport, i.e., perpendicular to the primary flow direction, is self-evident, and the simultaneous desirability of minimizing momentum transport follows from very widespread experience. Momentum transport manifests itself as pressure drop or forces between the flowing streams and packing or containing walls. Its minimization is important in gravity-flow separators, probably the vast majority, because the momentum source provided by gravitational pull is relatively weak and is the usual limiting factor in separator productivity. It is also important in systems such as chromatographic columns, where fluids are moved by pumps as the high-pressure drops required in high-resolution equipment are potentially damaging to packings and require both expensive pumps and very sturdy construction. Axial dispersion tends to reduce the degree of separation and occurs by a combination of diffusion and convective dispersion, the result of local variations in axial velocity.11-15 Although typically ignored in the modeling of common equipment such as packed column absorbers, it is normally a significant effect. Macroscopic flow nonuniformity has similar, but often even more pronounced, effects. It is being increasingly recognized as an important design consideration. We shall find that both depend on macroscopic equipment morphology and are important but secondary considerations. We shall return to these factors later in our discussion. First, however, we shall examine how efforts to decrease momentum relative to mass transfer have led to what might be called parallel evolutionary lines, now competing actively in a highly competitive market. We shall use chromatographic columns, widely used on a large scale in industrial biotechnology, as a compact set of examples. After many years of incremental development, it was realized, probably first by Steve Matson and his associates,16,17 that there were only a few ways to improve the mass/momentum ratio, and each of these has produced a viable family of commercial columns. These are summarized in Table 3 and discussed, in turn, below. (1) Accepting High Momentum Transfer Loads: Prochrom. These devices are packed and operated at very high pressures (typically 70 and 50 bar), which produce and maintain dense uniform packings and avoid channeling of the percolating fluid. If properly maintained, for example, keeping the lower distributor screen clean, they are capable of very high resolution, exhibiting as many as tens of thousands of equivalent theoretical plates under process conditions.5 They use packings of small mechanically strong and rigid particles, and they are certainly among the highest quality, and most expensive, columns. They require pretreat-
Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 3631 Table 2. Guidelines for Separator Design maximize lateral mass transport minimize momentum transport (pressure drop and crushing stress) minimize axial dispersion maintain flow uniformity
small diffusion distances and large surface areas large distances and small surface areas small lateral dimensions, frequent radial redistribution careful packing, effective distributors, large L/D
Table 3. Dealing with the Mass/Momentum Transport Ratio accept high momentum transport rates modify packing thicken momentum boundary layers decrease momentum load eliminate form drag
small rigid particles, high pressures, and sturdy equipment high capacity and diffusivity, internal convection, structured or cast packings expanded beds small L/D, adsorptive membranes multicapillary columns
ment of the feed stream to remove essentially all particulates. (2) Redesigning Column Packings: Several Manufacturers. Several packing modifications have been used to improve mass transfer relative to momentum transfer, beginning with the use of hydrogel packings, long known to have unusually high solute capacity, by Pharmacia. Later, workers at BioSepra found that imbedding a hydrogel within a branching silica skeleton greatly improved mechanical strength and preserved much of the gel’s high capacity. By pure serendipity, they also found that hydrogels exhibit unusually high diffusivity. Another promising approach, first suggested by Cal Giddings,18 was to incorporate a few large channels within packing particles to permit internal convection or perfusion. This approach was implemented commercially by Perseptive Biosystems, but it is not clear how to determine when effective perfusion has been achieved. More recently, BioRad has produced commercial columns in which packing particles have been replaced by cast porous plugs,19,20 and these have shown excellent mass to momentum transfer ratios. (3) Selectively Increasing Momentum-Transfer Boundary-Layer Thickness: Pharmacia and Upfront Chromatography. Pressure drops can also be reduced relative to mass transfer by the expansion of packed beds in upflow. Here advantage is taken of the differences in location of momentum- and mass-transfer resistance: all momentum transfer occurs in the percolating fluid, and momentum-transfer boundary layers can be greatly thickened by bed expansion. Masstransfer resistance, on the other hand, is concentrated almost entirely inside the packing particles. Success of this technique depends on the expanded bed technology of Howard Chase,21 which, in turn, depends on the use of dense particles with a significant range of diameters, a good flow distributor, and careful vertical alignment. An initial upflow of solvent segregates the particles by size, with the biggest at the bottom, and during subsequent upflow of protein feed there is very little relative movement of individual particles even though the bed volume expands severalfold. Elution is carried out in downflow to recompress the bed, and the breakthrough curves are reasonably sharp, though not truly comparable to those for conventional fixed beds. These systems can handle a heavy load of particulates, including, for example, E. coli cells, and they are well suited to the capture stage. (4) Decreasing the Momentum-Transfer Load: Many Manufacturers. The simplest way to reduce pressure drop is to lower the column length to diameter ratio: improper interpretation of the effluent curve shape has tended to push percolation velocities too
high22,23 so that most mass-transfer resistance is due to intraparticle diffusion. This is not generally desirable because it means that an unnecessarily high pressure drop will be observed, and one can rescale the column by lowering the length and keeping the volume constant. This will not affect operation until boundary-layer masstransfer resistance becomes significantsor until nonuniform flow distribution is observed. The shortest columns are known as adsorptive membranes, and these have such small effective pores that diffusional resistance to adsorption sites is negligible. One can then use very high percolation velocities and obtain high column productivity. Flow distribution tends to be the primary transport problem, but existing systems tend to have rather low adsorptive capacities. So far they are limited to laboratory-scale operation, but they have a promising future. (5) Eliminating Form Drag: Alltech. A very promising recent development is the multicapillary column, which eliminates form drag and produces a very high ratio of mass transfer to pressure drop. Small capillary diameters and extremely small variations in channel dimensions are needed to provide high capillary resolution and uniform flow distribution, and this has required development of a new technology. To date these columns are used only for gas chromatography, but they have a promising future for other applications. (6) Simulated Counterflow: UOP and Other Manufacturers. Finally, one may change the operating mode, and it has been shown that true counterflow is more economical of both adsorbent and solvent than the chromatographic mode (see, for example, ref 24), but it is not possible to obtain high resolution while moving adsorbent particles. Recently, the simulated moving bed (SMB) technology pioneered by UOP has been successfully adapted to the small scales of process chromatography. However, application is currently limited to relatively large volume clean separations such as optical resolution: SMB’s are so complex that they are only justified for continuous processing. Quantitative Calculations. One cannot build even a prototype column on the basis of the above qualitative considerations: one needs some way to determine equipment dimensions, stream rates, and process conditions. We now turn to this problem and begin as suggested above with the crudest but simplest approximations in a hierarchical set of calculations. (a) Preliminary Estimates. The first step is rough sizing for screening purposes, and this is normally done at the order of magnitude level. The utility of this approach resides in the relative insensitivity of transport dynamics to details of geometry and boundary conditions.25 Here we consider just a few illustrative examples.
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(1) Mass vs Momentum Transfer in Packed Beds. Here we back up our previous discussion by comparing these two processes, and we start with mass and momentum transfer in packed beds of spherical adsorbents obeying Fick’s second law. Our purpose is to obtain comparable estimates of mass- and momentumtransfer rates, and we begin by examining mass transfer to a sphere with constant surface composition c0. The volume average or bulk composition for this situation is ∞
∑1 (1/n2) exp[-n2π2t^/R2]
cb/c0 ) 1 - (6/π)2
(2)
and
(1/c0) dcb/dt ≈ (6^/R2) exp[-π2t^/R2]
(3)
Table 4. Prediction of Batch Adsorption Efficiency length, L (cm) diameter, D (cm) velocity, v (cm/h) % capacity 25 100 400 1600
1 0.5 0.25 0.125
250 1000 4000 16000
95.9 96.3 96.4 96.4
for any given chemical system (solute, buffer, and packing chemistry): the diffusional response time just defined, solute or solvent residence time ht, and a time constant characteristic of any slow reaction involved in adsorption Trxn. Under these circumstances performance depends only upon two time-constant ratios: Tdiff/th and Trxn/th. For fast adsorption kinetics, generally observed, for example, in ion-exchange and hydrophobic interaction packings, one need only keep the first of these ratios constant on scale-up
Here R is the particle radius, and ^ is the effective solute diffusivity. It follows that mass transport dynamics scale with a time constant
ht ) V/Q ) πD2L/πD2ν
Tdif ≡ R2/6^
where V is the column volume, Q is the volumetric flow rate, D is the column diameter, is the column void fraction, dp is the packing particle diameter, ^im is the effective solute diffusivity, and v is the superficial fluid velocity. The column can then be scaled by keeping the ratio L^im/vdp2 constant. This is true for nonlinear solute distribution between phases, for volume or mass overloading and even for gradient elution as long as gradients are kept constant relative to the feed volume. This prediction is tested in Table 4 for Hyper-D packing, which exhibits a highly nonlinear distribution using the calculator package supplied by the manufacturer, BioSepra Corp. The fractional adsorptive capacity does not change significantly over a 64-fold increase in the column length. For very short columns one must also consider axial diffusion, with a time constant on the order of Tdif,z ) L2/2^z,eff. Here L is the column length, and ^z,eff is the effective diffusivity in the axial direction. In general, one need consider only time constants within a factor of 10 relative to that of primary interest. (3) Discrete Descriptions and Compartmental Modeling. The most common use of the macroscopic balances is of a book-keeping nature, and applications of this type do not fit naturally into our discussion. However, approximating a complex mass transport system by a network of continuous stirred tank reactors, or CSTR’s, is often surprisingly useful. Thus, the most commonly used measure of chromatographic performance, the number of equivalent theoretical plates, is based on such a discrete model for the column (ref 5, p 171). One can also obtain a surprisingly good estimate of axial dispersion in a packed column by assuming the percolating fluid to mix completely in an axial distance of one particle diameter. See the discussion of dispersion below. Similar approximations are used throughout biotechnology, for example, in pharmacokinetic predictions of drug distribution kinetics. They owe their success to time constant separation, and they prove to be useful when two conditions are met:25 (1) The characteristic time for change of solute concentration in the feed to each compartment is large compared to compartmental response time. (2) The time scale of interest to the observer is larger than the response time.
(4)
Very similar results are found for other boundary conditions, and the important point is inverse dependence of diffusion rates on the square of the particle radius. We now look at momentum transfer resulting from fluid velocity and note that
-dp/dz ) F/V ) d(momentum/V)/dt ≡ dP k/dt
(5)
where P k momentum per unit volume V. Note that
P k [)] Fν
(6)
where v is the fluid velocity, F is force, m is mass, F is the fluid density, and [)] means “has the dimensions of”. Now, for flow in a packed bed of spheres with a void fraction at the low velocities characteristic of chromatography, one can write (ref 26, p 199)
3 4R2 ν ) (-dp/dz) 150µ (1 - )2
(7)
where v is now the superficial velocity and µ is the fluid viscosity. It follows that the fractional rate of change of momentum per unit volume is
d(P/Fν)/dt ) (150/4)[(1 - )2/3](µ/F)/R2
(8)
which is closely analogous to Eq 3 and shows that the rate of transfer of momentum per unit volume from the fluid to the solid surfaces is governed by a momentum time constant
Tmom ≡ (4/150)[3/(1 - )2]R2/(µ/F)
(9)
The kinematic viscosity (µ/F) is analogous to the diffusivity. Both processes show the same dependence on particle radius. However, the void fraction appears only in the expression for momentum transfer, and this is one of the important differences in these two processes. (2) Resolution of a Chromatographic Column. In many modern chromatographic columns the major source of mass-transfer resistance is intraparticle diffusion, and there are only three time constants that need to be considered in correlating performance data
Tdif ) dp2/6^im
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Moreover, when a series of compartments is used to approximate system behavior, the number of such compartments should be large, preferably at least 10. Pseudocontinuum Descriptions and Asymptotic Approximations. The next level of precision is the use of transfer and dispersion coefficients, which, in turn, are obtained as contractions of the equations of change (ref 26, Chapters 6, 13, and 21). These are normally approximations for idealized conditions, but they are surprisingly accurate. Particularly useful are asymptotic solutions for mass transfer in either short or long systems at high Schmidt numbers.25 Schmidt numbers in all liquids of biological interest are large enough to ensure meeting this last requirement, and a surprising number of systems fall in the range of either the short or long asymptotes. Moreover, these equations are surprisingly useful even for conditions not meeting the restrictions for which they were developed.28 Axial dispersion does not lend itself quite so cleanly to such simple asymptotes, but Ananthakrishnan et al. do at least provide an organizational chart,27 and Brenner15 provides numerical results for dispersion in short tubes. Athalye et al.11 review the situation for packed beds, which can usually be considered long, and Edwards et al.12 provide a rather general approach to describing two-dimensional systems. We shall consider here only two examples, both of which permit comparison of packed and multicapillary columns. We shall assume the packing of the first to be spheres with a void fraction of 0.38 and a layer of adsorbent material on the inner surface of the capillaries with a thickness of 0.1 times the bare radius for the second. Now we shall compare mass- and momentum-transfer capacity in terms of the Nusselt number for mass transfer and the analogous quantity Re × f/2 for momentum transfer. We will use here widely accepted approximations as follows:
Packed column:
Re × f/2 )
(
)
(1 - )2 3
Nu ) 10
^eff(cap)/^ ≈ 1 +
Nu/[Re × f/2] ) 0.038
The Nusselt number is estimated on the basis of experience, backed up by limiting calculations.29 The friction factor is based on the Blake-Kozeny equation (ref 26, p 199).
Capillaries: Re × f/2 ) 8 Nu/[Re × f/2] ) 2.37 The Nusselt number is a lower limit obtained from film theory while the friction factor is obtained from the Hagen-Poiseulle equation. It is now seen that the capillary outperforms the packed column by a factor of more than 60 in terms of mass- to momentum-transfer ratio. Moreover, the momentum is now transferred directly to the walls rather than as a crushing pressure building up toward a maximum at the bed outlet. One can then use a highcapacity, high-diffusivity packing such as a hydrogel. However, it remains to consider axial dispersion and macroscopically variable flow.
1 Pe2 192
Pe ≡ 〈v〉D/^
(10) (11)
where 〈v〉 is the flow average velocity and D is channel diameter. For flow in packed beds there is no simple reliable expression, but for many purposes one may use the approximation11
^eff(part)/^ ≈ 0.4Pe
(12)
where v is now the flow average interstitial velocity. Note that the coefficient 0.4 in eq 12 is close to the 1/2 given above as predicted by a simple compartmental analysis. It is immediately seen that capillaries are much more sensitive to the Peclet number. Because Peclet numbers can be over a thousand for particles as small as 50 µm in protein chromatography, axial dispersion can be quite important in capillary systems. From a practical standpoint, capillary diameters must be quite small, preferably under 50 µm. The Alltech Corp. has developed a technique for producing devices in this range, and the results are quite impressive. Continuum Equations and Process Models. The most reliable and accurate descriptions for most systems are obtained by integration of the continuum equations of change, although even here some approximations are required to obtain a useable result.29-31 A one-dimensional pseudocontinuum model of the form
(
(75/2) ≈ 263
Nu ) 2/ln(10/9) ) 18.98
Relative dispersion: It is customary to model chromatographic devices as if the flow were one-dimensional and uniform over the equipment cross section, but of course this is not the case. Therefore, it is also customary to use an effective axial diffusivity which accounts for the nonuniformity if the column is sufficiently long. For flow in capillaries the effective diffusivity is given by (see, for example, ref 25):
)
∂cp ∂cf ∂2cf ∂cf +ν - ^eff,z 2 ) -(1 - ) ∂t ∂z ∂t ∂z
(13)
has proven adequate for the continuous phase under all conditions so far encountered,11 and a linearized masstransfer coefficient has usually been used for the particulate phase:
Vp
∂cp ≈ Akp(cp* - cp) ∂t
(14)
in the absence of chemical reaction. Here the subscripts f and p refer to the fluid and particulate phases, respectively, V is the particle volume, and A is its surface area. The asterisk indicates equilibrium with the fluid phase. Equations 13 and 14 have been found11 to predict the behavior of commercial columns, using only parameters determined separately from the chromatographic process. An example is shown in Figure 5, using the commonly accepted lumped-parameter approximations for the mass-transfer coefficients: for the fluid boundary layer a pseudosteady short-length asymptote is used, and for the particle interior the long-time approximation
kc,p ≈ 10^/D
(15)
The boundary layer occurs in something close to a spatially periodic flow, but there is little interaction
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Figure 5. Comparison of predicted and observed behavior of a commercial chromatographic column. Dotted line A represents the contribution of external boundary-layer resistance, B and C represent the expected range of axial dispersion contributions, and D represents the contribution of internal solute diffusion. The solid lines represent the expected range of the combined contributions, and the points are actual data. From ref 11, where a more complete description is provided.
between any particle and those upstream. It exhibits very short response times, and the boundary-layer approximation should be acceptable. The situation is different for the particle interior, however, where the response time is on the order of Tdif ≈ R2/6^, which may or not be small relative to the time scale over which the ambient concentration changes. For the relatively low percolation velocities and broad solute peaks of the experiments represented in Figure 5, the lumped-parameter approximation was adequate. However, for many fast commercial separations, this is not the case. One must then return to order-of-magnitude arguments. For a Gaussian solute distribution then, one must be sure that the ratio
Tdiff/σ , 1
(16)
where σ is the standard deviation of the solute peak in units of time. Closing Remarks The above discussion is too short to do more than sketch out a strategy for process or equipment development. The approximations suggested here should be continually tested, and it should be remembered that much remains to done even for the specific examples discussed. The useful ranges of the heuristics presented must be continually investigated, and it should be remembered that the final stage of design needs much more detailed attention than is given here. As just one example, the use of short wide columns presents severe challenges in distributor design and packing procedures, both presently being pursued by the author and his associates. Literature Cited (1) Lightfoot, E. N. What Chemical Engineers Can Learn from Mother Nature. Chem. Eng. Prog. 1998, Jan, 67-74. (2) Smith, J. M.; Szathma´ry, E. The Major Transitions in Evolution; Freeman: Oxford, U.K., 1995. Smith, J. M.; Szathma´ry, E. The Major Transitions in Evolution; Oxford University Press: Oxford, U.K., 1997. (3) Miller, A. I. Insights of Genius; Copernicus: New York, 1996.
(4) Coveney, P.; Highfield, R. Frontiers of Complexity: Fawcett Columbine: New York, 1995. (5) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Nonlinear Chromatography; Academic Press: New York, 1994. (6) Berezkin, V. G., Compiler. Chromatographic Adsorption Analysis: Selected Works of M. S. Tswett; Ellis Horwood: London, 1990. (7) Schwab, G. M. Angew. Chem. 1937, 50, 546-553. (8) Holland, J. R. Hidden Order; Addison-Wesley: Reading, MA, 1995. (9) Atkinson, B.; Mavituna, F. Biochemical Engineering and Biotechnology Handbook, 2nd ed.; Stockton Press: New York, 1991; pp 873-904, 906-991. (10) Lightfoot, E. N.; Cockrem, M. C. M. What Are Dilute Solutions? Sep. Sci. Technol. 1987, 22, 165-189. (11) Athalye, A. M.; Gibbs, S. J.; Lightfoot, E. N. Predictability of Chromatographic Separations: Study of Size-exclusion Media with Narrow Particle Size Distributions. J. Chromatogr. 1992, 589, 71-85. (12) Edwards, D. A.; Shapiro, M.; Brenner, H.; Shapira, M. Transp. Porous Media 1991, 6, 337-358. (13) Burnell, J. G.; Graham, J. W.; Young, R. Self-similar Radial Two-phase Flows. Transp. Porous Media 1991, 6, 359-390. (14) Souto, H. P.; Moyne, C. Dispersion in two-dimensional porous media. Part II. Dispersion tensor. Phys. Fluids 1997, 9 (8), 2253-2263. (15) Brenner, H. The diffusion model for longitudinal mixing in beds of finite length. Numerical values. Chem. Eng. Sci. 1961, 17, 29-243. (16) Matson, S. L. Membrane Reactors. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1979. (17) Matson, S. L.; Quinn, J. A. Membrane Reactors in Bioprocessing. Ann. NY Acad. Sci. 1986, 469, 152-165. (18) Giddings, J. C. Dynamics of Chromatography. Marcel Dekker: New York, 1965. (19) Zeng, C.-M.; Hjerten, S.; et al. Hydrophobic-interaction chromatography on continuous beds derivatized with isopropyl groups. J. Chromatogr. A 1996, 753, 227-234. (20) Petro, M.; Svec, F.; Frechet, J. M. J. Molded continuous poly(styrene-co-divinylbenzene) rod as a separation medium for the very fast separation of polymers, etc. J. Chromatogr. A 1997, 752, 59-66. (21) Chase, H. A. Development and Operating Conditions for Protein Purification Using Expanded Bed Techniques: the Effect of the Degree of Bed Expansion on Adsorption Performance. Biotechnol. Bioeng. 1995, 49, 512-526. (22) Jeansonne, M. S.; Foley, J. P. J. Chromatogr. A 1992, 594, 1. (23) Yuan, Q. S.; Rosenfeld, A.; Root, T. W.; Klingenberg, D. J.; Lightfoot, E. N. Flow Distribution in Chromatographic Columns. J. Chromatogr., in press. (24) Roper, D. K.; Lightfoot, E. N. Comparing Steady Counterflow and Differential Chromatography. J. Chromatogr. A 1993, 654, 1-16. (25) Lightfoot, E. N.; Lightfoot, E. J. Mass Transfer. In KirkOthmer Encyclopedia of Separation Technology; 4th ed.; Ruthven, D. M., Ed.; 1997. (26) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (27) Ananthakrishnan, V.; Gill, W. N.; Barduhn, A. J. Laminar Dispersion in Capillaries. AIChE J. 1965, 11, 1063-1072. (28) Baier, G.; Grateful, T. M.; Graham, M. D.; Lightfoot, E. N. Predictions of Mass Transfer Rates in Spatially Periodic Flows. Chem. Eng. Sci. 1999, 54, 343-355. (29) Reis, J. F. G.; Noble, P. T.; Chiang, A. S.; Lightfoot, E. N. Chromatography in Beds of Spheres. Sep. Sci. Technol. 1979, 14 (5), 367-394. (30) Lightfoot, E. N.; Coffman, J. L.; Lode, F.; Perkins, T. W.; Root, T. W. Refining the Description of Protein Chromatography. J. Chromatogr. A 1997, 760, 130. (31) Lode, F.; Rosenfeld, A.; Yuan, Q. S.; Root, T. W.; Lightfoot, E. N. Refining the Scale-up of Chromatographic Separations. J. Chromatogr. A 1998, 796, 3-14.
Received for review January 20, 1999 Revised manuscript received April 23, 1999 Accepted May 5, 1999 IE9900566
