Supercritical fluid extractions in biotechnology

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TIBTECH-

10 Klein, J. and Kressdorf, B. (1985) in Proceedings of the 3rd European Congress on Biotechnology (Vol. 2) (Neijssel, O. M., van der Meer, R. R. and Luyben, K. C. A. M., eds), pp. 375-379, Elsevier 11 Musgrave, S. C., Kerby, N. W., Codd, G. A. and Stewart, W. D. P. (1982) Biotechnol. Lett. 4,647-652 12 Decleire, M., van Huynh, N., Motte, J. C. and De Cat, W. (1985) Appl. Microbiol. Riotechnol. 22,438-441 13 Durand, M., Lahrnani, P. and Durand, S. (1987) French Patent Application No. 258625 CL:C12-N11/10 1987 14 Baillez, C., Largeau, C. and Casadevall, E. (1985) Appl. Microbiol. Biotechnol. 23, 99-105 15 Brodelius. P. (1983) in Immobilized Cells and Organelles (Vol. 1) (Mattiasson, B., ed.), pp. 27-55, CRC Press []

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16 Draget, K., Myhre, S., Skj&k-Bra~k, G. and Ostgaard, K. (1988) J. Plant Physiol. 132,552-556 17 Jarvis, A. P. and Grdima, T. A. (1983) Rio-Techniques 1, 24-27 18 Martinsen, A., Skj&k-Br~ek, G. and Smidsrod, O. (1989) Biotechno]. Bioeng. 33, 79-89 19 Tanaka, H., Masatose, M. and Veleky, I. A. (1984) Biotechnol. Bioeng. 26, 53-58 20 Klein, J., Stock, J. and Vorlop, K. D. (1983) Eur. J. Appl. Microbiol. Biotechno]. 18, 86-91 21 Smidsrod, O. (1974) Faraday Discuss. Chem. Soc. 57,263-274 22 Smidsrod, O., Haug, A. and Lian, B. (1972) Acta Chem. Scand. 26, 71-78 23 Grasdalen, H. (1983) Carbohydr. Res. 118, 255-260 24 Skjfik-Brmk, G., Larsen, B. and []

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Supercritical fluid extractions in biotechnology Theodore W. Randolph Over the past decade, supercritical fluid (SCF) extraction has been shown to deserve consideration as an alternative to liquid-liquid extraction or distillation. Most current commercial applications of SCF extraction involve biologically produced materials; the technique may be particularly relevant to extraction of biological compounds in cases where there is a requirement for lowtemperature processing, high mass-transfer rates and negligible carry over of solvent into the final product. New advances, in which extraction is combined with reaction or crystallization steps, may further increase the attractiveness of SCFs in the bioprocessing industries. SCFs are materials that exist as fluids at t e m p e r a t u r e s and pressures above their critical t e m p e r a t u r e (Tc) and critical pressure (Pc)*, respectively. Critical pressures and t e m p e r a t u r e s of some fluids are p r e s e n t e d in Table 1. The main attraction of SCF * Tc and Pc: the highest temperature and pressure where a substance can exist in a vapor-liquid equilibrium. T. W. Randolph is at the Department of Chemical Engineering, Yale University, New Haven, CT 06520, USA.

solvents is that their solvent properties are highly sensitive to changes in both pressure and t e m p e r a t u r e (unlike liquid solvents, w h e r e v e r y large pressure changes are r e q u i r e d to affect solvent properties). The ability to change solvent properties in this w a y provides the o p p o r t u n i t y of tailoring the solvent strength to a given application. The use of SCFs as solvents for extraction processes is well d o c u m e n t e d ; reviews i n c l u d e those by Randall 1, Paulaitis et al. 2, Brunner and Peter 3, M c H u g h and Krukonis 4, and Eckert et al. 5. Recent d e v e l o p m e n t s in SCF t e c h n o l o g y are

© 1990, Elsevier Science Publishers Ltd (UK) 0167 - 9430/90/$2.00

25 26

27 28

29

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M A R C H 1 9 9 0 [Vol. 8]

Smidsrod, O. (1986) Int. J. Biol. Macromo]. 8,330-336 Skj&k-Br~ek, G., Grasdalen, H. and Larsen, B. (1986) Carbohydr. Res. 154, 239-250 Andresen, I-L., Skipnes, O., Smidsr~d, O., Ostgaard, K. and Hemmer, P. C. (1977) ACS Syrup. Ser. 48,361-381 Skj~k-Br~ek, G., Grasdalen, H. and Srnidsrod, O. (1989) Carbohydr. Polymers 10, 31-54 Skj&k-Br~ek,G., Draget, K., Grasdalen, H. and Smidsrod, O. in Recent Developments in Industrial Polysaccharides (Crescenzi, V., Dea, I. C. M., Paoletti, S., Stivala, S. S. and Sutherland, I., eds), Gordon and Breach (in press) Scott, C. D. (1987) Enzyme Microb. Technol. 9, 66-73 []

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described in the collection edited by Johnston and P e n n i n g e r 6. Most (though not all) properties of SCFs are i n t e r m e d i a t e b e t w e e n those of gases and those of liquids (Table 2). Densities of SCFs can a p p r o a c h liquid-like densities; and diffusivities, while being about three orders of m a g n i t u d e smaller t h a n diffusivities of low-pressure gases, are still over an order of m a g n i t u d e higher than typical liquid diffusivities. An e x a m p l e of the strong pressured e p e n d e n c y of three properties is s h o w n for carbon d i o x i d e (Fig. 1 ) 7 .

Advantages and disadvantages of SCF solvents for extraction The p r e s s u r e - d e p e n d e n c y of solvent capability and the favorable properties of SCFs c o m b i n e to confer special advantages for extraction processes, especially for the biotechnology industries: • High diffusivity r e d u c e s masstransfer limitations. For extraction from p o r o u s solid matrices (e.g. plant material), w h e r e m a n i p u l a t i o n of external fluid d y n a m i c s has no effect on mass-transfer rates, the high diffusivities exhibited in SCFs m a y yield greatly e n h a n c e d extraction rates. • Low surface tension allows penetration and wetting of pores smaller than those accessible w i t h liquid solvents - of i m p o r t a n c e for extraction of chemicals from cellular material.

TIBTECH- MARCH 1990 [Vol. 8] DTable

79

I

Critical properties of several fluids Compound

Ammonia Butane Carbon dioxide Ethane Ethylene Dinitrogen oxide Pentane Propane Water

Critical temperature (°C)

Critical pressure (bar)

Critical d e n s i t y

132.4 135.0 31.3 32.2 9.2 36.5 196.6 96.6 374.2

112.5 37.5 72.9 48.1 49.7 71.7 37.5 41.9 217.6

0.235 0.228 0.443 0.203 0.218 0.45 0.232 0.217 0.322

• The dramatic sensitivity of solubility to changes in pressure and temperature provides the possibility of manipulating the selectivity of an extraction. Extraction may be coupled with further purification steps by fractionation with changing pressure or temperature. With the addition of a small amount of specific cosolvent, extraction and fractionation may be used to separate compounds according to functionality. • Solid compounds may be crystallized from SCFs and the size of the crystals manipulated by changing process pressures and temperatures 8. The ability to make small crystals is of interest to the pharmaceutical industry, where product morphology can be critical to drug uptake rates, and where mechanical fragmentation procedures may be unacceptable, owing to thermal instability or contamination risks. • SCF solvents allow low-temperature processing, offering a means of separating compounds that cannot be distilled, owing to their thermal instability. • Capacity' for solutes can be very high; enhancement factors (ratio of actual solubility to ideal gas solubility) can be 103-1012 (Ref. 9). • Concern about solvent toxicity in foods and pharmaceuticals is increasing. The low reactivity and toxicity of potential SCF solvents (such as carbon dioxide or ethane), and the gaseous character of many of them at atmospheric conditions (i.e. negligible residual solvent in the processed product) may make SCFs acceptable solvents for bioprocessing. However, SCFs are not suitable for all

(g cm -3)

extraction processes. The main disadvantages are: • Solubilities, while much higher than those predicted by ideal-gas considerations, are still much lower than those achievable in many liquid solvents. • Capital costs are high for highpressure equipment; and, unlike the chemical process industries, biotechnology industries in general lack expertise in high-pressure processing technology. • Insufficient data exist on the physical properties of many biomolecules, making prediction of phase behavior difficult. • Selectivity based on solute functionality can be achieved by use of cosolvents, but this may obviate the advantage of minimal solvent residues in the final product. SCF t e c h n o l o g y - a brief history SCFs, and their abilityto solubilize compounds at low vapor pressures, have been known since the pioneering work of Hannay and Hogarth in 1879 l° . However, there was only sporadic research in the field until the 1950s, when a number of patents appeared for applications of SCF

solvents in extraction (mostly of petroleum-based species). Interest in SCFs in the late 1970s was sparked, at least in part, by concern over dwindling reserves of energy resources. The developments in SCF technology from 1977 to 1987 have been described as belonging to three phases: 'the promises', 'the loss of luster', and 'the monotonic recovery '11. In the late 1970s, SCFs were depicted as miraculous solvents, as evidenced by the title 'The magic of supercritical fluids', which headlined an article in Chemical Engineering 12. Perhaps misled by such overenthusiastic reports, many companies directed research efforts towards SCF technologies that, in retrospect, seemed doomed to failure. The 'loss of luster' phase followed the initial burst of enthusiasm. More realistic phase diagrams, solubility data, and cost estimates for SCF processes produced a sobering effect, especially in applications dealing with high-volume, low-cost chemicals. When feasible, liquid-liquid extraction or distillation usually proved to be cheaper than SCF extraction methods. For most extraction processes for bulk chemicals, SCF technology was found to be uneconomical. However, a number of commercially viable extraction operations were developed for higher value, difficult-toseparate biological compounds - the best-known and most successful process is the HAG (now General Foods) method for extraction of caffeine from coffee using supercritical carbon dioxide. SCF extraction has also been used commercially for the extraction of hop oils from hops (Pfizer). These commercial examples paved the w a y for the current 'monotonic recovery' phase of interest in SCFs.

- - T a b l e 2,

Comparison of physical properties of gases, liquids and SCFs Fluid state Gas Liquid SCF

Density (g cm -3)

Viscosity (g cm -1 S-1 )

(0.6-2)x 10-3 0.6-1.6 0.2-0.9

( 0 . 2 - 3 ) x 10 -2

(1-3)x10 -4 (1-9)x10 -4

Kinematic viscosity

Diffusivity

( c m 2 S-1 )

( c m 2 s -1 )

l x 1 0 -1 1X 10 -2 1x10 -3

0.1-0.4 (0.2-2) x 10-s (2-7)x 10-4

80

TIBTECH - MARCH 1990 [Vol. 8]

--Fig. 1

L1ooo

Density Although proprietary considerations still restrict most of the information regarding new SCF extraction pr ocesses to patent disclosures, SCF research is clearly active in the bioprocess industries. A partial survey of potential applications under development includes extraction of flavors and fragrances from biological materials 13, extraction of cholesterol from food 11, extraction of ~o-3 fatty acids from fish oils 14, extraction of thermally labile pharmaceuticals 15 and extraction of natural pesticides 16. Applications where SCF extraction may be coupled with other SCF processing steps (such as comminution of pharmaceuticals via SCF-crystallization 17, or reaction using enzymatic catalysis 18) may provide commercially viable processes.

,°°l I

lOJ

- 100

~

<:

v

/.'-

/ X

/ ',,

o o > (o

\

/ \

Viscosity ,,z8 x~.

,i

.,.,"

\

zB

\ -10

t-

Kinematic viscosity

Considerations for process design Phase equilibria The single most important consideration for rational design of an SCF extraction process is the phase behavior. This is also crucial if other processes (e.g. crystallization or reaction) are to be coupled with the extraction. Several recent publications have reviewed research into phase behavior in SCFs 5'9'19. Thermodynamic modeling of SCF phase equilibria presents a great challenge. The singular nature of the critical point, the difficulties of obtaining accurate data at high pressures, and the lack of data on physical parameters for biological molecules of interest make advances difficult. Nonetheless, a wide variety of equations of state (EOSs) have been developed and applied to modeling SCF equilibria (Table 3). Among the most commonly applied are the Peng-Robinson EOS and the Redlich-Kwong EOS; additional EOSs include the Hard Sphere-van der Waals, the Augmented van der Waals, and the Perturbed Hard Chain Theory. While more sophisticated EOSs may fit data better and represent the physical reality more closely, parameter estimates are problematic. This is especially true for complex mixtures of biochemicals - for which physical property data (such as critical temperatures, critical pressures, acentric factors, normal boiling points and molar liquid volumes) are unavailable. For practical pur-

0.1 I 0

I 40

I 80

t 120

I 160

1 200

Pressure (bar) Density, viscosity and kinematic viscosity of carbon dioxide at 310 K. Note that the properties are most sensitive to pressure near the critical pressure o f 72.9 bar. ( R e p r o d u c e d , w i t h p e r m i s s i o n , f r o m Ref. 7.)

poses, it is probably best to choose a cubic EOS that uses a minimum of variable parameters. In a few cases these parameters are available in the literature; otherwise they may be estimated from other physical property data; or, most likely, estimated by fitting the parameters to experimentally obtained binary solubility data. Extending the equation to more complic~ed mixtures may be achieved by using limited ternary equilibrium data to estimate interaction parameters for unlike solute pairs; for example, using the PengRobinson EOS for naphthalene solubility in carbon dioxide (Fig. 2). Together with an EOS, a 'mixing rule' must be applied if phase equilibria for mixtures are to be estimated. Unfortunately, results are rather sensitive to the choice of mixing rule, and the best fits of models to data occur w h e n additional parameters are introduced in the mixing rule. Choice of mixing rule is discussed in greater detail by

Prausnitz 23. For the complicated mixtures that are of practical industrial interest, it is recommended that the number of fitted parameters be minimized; one widely used set of mixing rules applied by Redlich 24 is: amixture = ~ ~ y i y j a q ij

aij

=

(aiaj) 1/2

and bmixture = Z yibi i

An additional parameter is often added, giving:

aii= (aiai) 1/2 (1-kij) The binary interaction term kii must be estimated from binary phase equilibrium data. The addition of small amounts of cosolvents, or entrainers, to an SCF

MARCH 1990[Vol. 8]

TIBTECH --Table

81 mBox

3,

I

Symbols

C o m m o n e q u a t i o n s o f state

van der Waals p-

RT V-b

__

a V2

Carnahan-Starling-van der Waals 2° P=RT

~ (1--~+~2--~3)

b

(1-;~)3

a ~V(V+b)

C a r n a h a n - S t a r l i n g - R e d l i c h - K w o n g 2°

P=RT ~ (1--~+~2--~3)

b

(1-~)3

a

T~/2V(V+b)

Peng-Robinson 22 p=

RT V-b

S//

t Re )°356 G~

(ScGt)I/4 - 1.692

Gr

i,j k

a

V2

Redlich-Kwong 21 RT P= V-b

a b D dp g

maximum in the Grashof number 27. They present the following correlation for mass transfer in SCFs:

a(T) V(V+b)÷b(V-b)

may greatly enhance the solubility of a solute. For example, addition of 3.5 mole % of methanol to carbon dioxide increases the solubility of sterols like cholesterol, stigmasterol and ergosterol by about an order of magnitude 25. Such solubility increases may be correlated using the framework of the EOSs mentioned above. Potential cosolvents may be rapidly screened using gas chromatography 26, although the final choice must also consider the cosolvent's overall appropriateness for the process (i.e. its toxicity, reactivity and cost). Mass-transfer consideration s The high diffusion rates found in SCFs cause mass transfer to be more rapid than in liquids. Mass-transfer rates in SCFs may be further enhanced by natural convection 7. Natural convection is expected to be a function of the Grashof number Gr, and the overall mass transfer may then be correlated as Sh = f(Re, Sc, Gr), where S h is the Sherwood number, a dimensionless number useful for correlating the mass transfer coefficient, kc. Because of the low kinematic viscosity and highly concentration-dependent density of SCFs near the critical point, the Grashof number becomes large at these conditions (values 0f3.25 x 107 have been reported) 27. Lim et al. report a maximum in the mass transfer coefficient near the critical point, correlated with a

Further data on mass transfer in SCFs are necessary to confirm these observations, and to delineate the effects of other phenomena that may affect mass transfer in SCF mixtures (e.g. convection induced by critical density fluctuations). However, an optimum operating pressure - due to the counterbalancing effects of enhanced mass transfer at lower pressures and higher solubilities at elevated pressures - may exist. Because the solubility of many compounds is low in SCFs [for example, solubility of vegetable oil from canola seeds is about 1% w/w in carbon dioxide at 40°C and 350 bar (Ref. 28)], and because mass transfer is generally rapid, extraction processes may be equilibrium-limited rather than mass-transfer-limited. Small-scale extraction experiments are necessary to permit rational design of full-scale equipment;

kc

P Re

R Sc Sh

T v V y p ~,

energy parameter in EOS size parameter inEOS molecular diffusivity particle diameter gravitational acceleration Grashof number - g l 3 (Ap/p)v -2 mixtu re component indices mixing parameter mass transfer coefficient pressure Reynolds number = pdpv~ -1 gas constant Schmidtnumber = #p-lD-1 Sherwood number = kcdpD -~ temperature superficial velocity volume mole fraction viscosity density normalized volume = b/V

methodologies for small-scale extraction experiments have been published 29. E q u i p m e n t design High-pressure equipment represents high capital costs. These costs, coupled with the difficulties of feeding solids into an extractor at

--Fig. 2

54 40

i,,,O-'-" f

30 _

...........

.--o-

6

.....

22 o 0 X cO 0

16 12 8 6

0

~ , ~a'l/~ I ~

• O

303.15 K 313.15K}Experimental 323.15 K } Values 303.15 K Calculated 313.15 Curves 323.15

3;

.... >

O 50

I

100

I

I

150

,

I

,

200

I

250

,

I

300

~

I

350

Pressure (bar) Solubility of naphthalene in supercritical carbon dioxide. Data were correlated using the Peng-Robinson EOS. Note the high temperature- and pressuresensitivity, especially near 72.9 bar, the critical pressure of carbon dioxide.

82

TIBTECH - M A R C H 1990 [Vol. 8]

high pressure, force most SCF extractions to be o p e r a t e d in a batch or semi-batch mode. T h e usual operating pressures of 100-300 bar are quite easily h a n d l e d , but the design s h o u l d attempt to m i n i m i z e the a m o u n t of e q u i p m e n t that is pressurized. Econ o m i c s favor the use of longer, smaller-diameter extraction vessels, and parallel networks of small-diameter reactors s h o u l d be considered.

Conclusions SCFs offer m a n y advantages for extraction in the b i o p r o c e s s industries: l o w - t e m p e r a t u r e processing, high mass-transfer rates, and pressure-sensitive solubilities are a m o n g the attractions of this technology. SCF extraction is not a p p r o p r i a t e for every process, however, and limitations i m p o s e d by the phase behavior and e q u i p m e n t costs s h o u l d be carefully w e i g h e d against potential benefits. This n e w t e c h n o l o g y m a y be particularly appropriate for 'difficult' separations of high value products, where considerations such as thermal stability and p u r i t y m a y make c o n v e n t i o n a l extraction or distillation steps impractical. References 1 Randall, L. G. (1982) Separ. Sci. Tech. 17, 1-118

-

~;,

~

'

'

2 Paulaitis, M. E., Krukonis, V. J., Kurnik, R. T. and Reid, R. C. (1982) Rev. Chem. Eng. 1, 179-250 3 Brunner, G. and Peter, S. (1981) Chem. Ing. Tech. 53, 529-542 4 McHugh, M. A. and Krukonis, V. J. (1986) Supercritical Fluid Extraction: Principles and Practice, Butterworths 5 Eckert, C. A., Van Alsten, J. G. and Stioicos, T. (1986) Environ. Sci. Technol. 20, 319-325 6 Johnston, K. P. and Penninger, J. M. L. (eds) (1989) Supercritica] Fluid Science and Technology, American Chemical Society Symposium Series, 406 7 Debenedetti, P. G. and Reid, R. C. (1986) Am. Inst. Chem. Eng. J. 32, 2034-2046 8 Mohamed, R. S., Debenedetti, P. G. and Prud'homme, R. K. (1989) Am. Inst. Chem. Eng. J. 35, 325-328 9 Brennecke, J. F. and Eckert, C. A. (1989) Am. Inst. Chem. Eng. J. 35, 1409-1427 10 Hannay, J. B. and Hogarth, J. (1879) Proc. R. Soc. (London) 29, 324-326 11 Krnkonis, V. J. (1988) Am. Chem. Soc. Symposia Series 97, 26-43 12 Basta, No and McQueen, S. (1985) Chem. Eng. 92, 16 13 Caragay, A. B. (1981) Perfum. Flavor. 6, 43-55 14 Rizvi, S. S. H., Chao, R. R. and Liaw, Y. J. (1988) Am. Chem. Soc. Symposia Series 97, 90-108 15 Larson, K. A. and King, M. L. (1986) Biotech. Prog. 2, 73-82 16 Sch~iffer, Ko and Baumann, W. (1989)

'~ . . . . . ~ ~

18 19 20 21 22 23 24 25 26 27 28 29

~.~i!~ ~ ' ~ .......~ ...... ~ -

Estimating the immeasurable without mechanistic models We w e l c o m e the recent concise r e v i e w by van der Heijden et al. entitled 'State estimators (observers) for the on-line estimation of nonmeasurable process variables '1. The authors justifiably m a d e a critical analysis of the various approaches. However, in attempting to evaluate the merits of A d a p t i v e Inferential Estimation (AIE) 2, two major criticisms, resulting from misinterpretation, were made. van der Heijden stated first that it was a 'serious drawback' of AIE that the offline assays m u s t be regular (although infrequent).

17

Fresenius Z. Anal. Chem. 332, 884-889 Tavana, A. and Randolph, A. D. (1989) Am. Inst. Chem. Eng. J. 35, 1625-1630 Randolph, T. W., Clark, D. S., Blanch, H. W. and Prausnitz, J. M. (1988) Science 239,387-390 Rizvi, S. S. H., Benado, A. L., Zollweg, J. A. and Daniels, J. A. (1986) Food Technol. 40, 55-65 Johnston, K. P. and Eckert, C. A. (1981) Am. Inst. Chem. Eng. J. 27, 773-779 Redlich, O. and Kwong, J. N. S. (1949) Chem. Rev. 44, 233-244 Peng, D. Y. and Robinson, D. B. (1976) Ind. Eng. Chem. Fund. 15, 59-64 Prausnitz, J. M. (1969) Molecular Thermodynamics of Fluid Phase Equilibria, Prentice-Hall Redlich, O., Ackerman, F. J., Gunn, R. D., Jacobson, M. and Lau, S. (1965) ~nd. Eng. Chem. Fund. 4, 369-375 Wong, J. M. and Johnston, K. P. (1983) Biotech. Prog. 2, 29-39 Tavana, A., Chang, J. and Randolph, A. D. (1989) Am. Inst. Chem. Eng. J. 35,645-648 Lim, G-B., Holder, G. D. and Shah, Y. T. (1989) Am. Chem. Soc. Symposia Series 97,379-395 Bulley, N. R., Fattori, M. and Meisen, A. (1984) J. Am. Oil Chem. Soc. 61, 1362-1365 Wright, B. W., Fulton, J. L., Kopriva, A. J. and Smith, R. D. (1988) Am. Chem. Soc. Symposia Series 97, 44-62

van der Heijden et al. will be aware that the d e v e l o p m e n t of AIE has enc o m p a s s e d two parallel approaches: the I n p u t - O u t p u t a p p r o a c h and the State Space estimator. T h e I n p u t O u t p u t a p p r o a c h is s o m e t i m e s governed by the n e e d for regular offline assay data, a l t h o u g h this is not always the case. H o w e v e r , the State Space estimator incorporates a m o d e l w h i c h is not autoregressive on the p r i m a r y variable, biomass (i.e. the current biomass c o n c e n t r a t i o n is not m o d e l l e d as a f u n c t i o n of past concentrations), and c o n s e q u e n t l y does not require regular assays!

Moreover, this algorithm is curr e n t l y u n d e r g o i n g trials in industrial mycelial f e r m e n t a t i o n w h e r e irregular off-line assays are the norm; this suggests that the 'serious drawback' is non-existent. The s e c o n d point of c o n t e n t i o n arose from our use of a general linear m o d e l structure, the p a r a m e t e r s of w h i c h are identified on-line: van der Heijden et al. state that ' A n o t h e r disadvantage of the m e t h o d [AIE] is that o n l y those variables that are truly m e a s u r e d can be estimated, because of the lack of any m e c h a n i s t i c model'. The objectives and targeted areas of application of A d a p t i v e Inferential Estimation have b e e n m i s u n d e r stood. AlE was c o n c e i v e d to address the p r o b l e m of estimating, and u l t i m a t e l y controlling, k e y process variables at a greater f r e q u e n c y than t h e y m a y be m e a s u r e d in the absence of any realistic m e c h a n i s t i c model. If a reliable p h y s i c o c h e m i c a l process