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Advanced Technology Research Center | Silab Division | Samani International Enterprises
FERMENTATION GROWTH STAGE DETERMINATION: Fourier Series Fit via Marquardt-Levenberg Algorithm
VASOS-PETER JOHN PANAGIOTOPOULOS II [President, Samani International Enterprises]
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@begin(researchcredit)
@center(@b(ABSTRACT))
Bacteria are extensively utilised in some newer chemical processes.
Often, one would wish to approximate the stage of their growth, especially
so that one can monitor and control by computer. A
numerical algorithm is proposed for such determination.
@end(researchcredit)
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@PAGEHEADING( left "Fermentation Growth Stage Determination",
CENTER "@value(page)",
RIGHT "Vasos-Peter John Panagiotopoulos II")
@PAGEFOOTING( left "@value(sectiontitle)",CENTER "@value(date)",
RIGHT "Samani Silab ATRC" )
@Chap(Determination of the Characteristic Curve )
Fitting the expected growth curve @foot{T.J.Parry & R.K.Fausey,
@i,p.19,
UK:Hutchison Educational,1973} to an even Fourier cosine series yields
the following coeffiecients:
@equation(a@-<0>=9.1325;a@-<2>=-.1756;a@-<4>=-.1496)
@foot[Hewlett-Packard HP41C @i*+c@-**)|@ovp[)]\@-(max)=2@ovp[)]\|K@-(i)]
with individual nutrient concentration, @equation[c@-(i)=e@+{-@ovp<)>\t}]
@foot[J.E.Bailey & D.F.Ollis,@i,
pp.346,NY:McGrawHill,1977;J.Monod,The Growth of Bacterial Cultures,
@i:371(1949);Dr. Monod received the 1965 Nobel Prize
for this and other work on enzymatic adaptation, which he began during WW2.
@i O' Sullivan, D., G., Quantitative Potentialities in Enzyme
Cytochemistry. Modified Michaelis-Menten Rate Law Applicable when a
Substrate Diffuses Slowly into an Enzyme Site,@i,@b<2>,
pp.119,123,124(1962) ]
Prof. Levenspiel restates this as
@equation{r@-=kC@-C@-/(C@-+C@-),} where c indicates cells
and A, substrate. C@- is, as above, the Monod constant, or the substrate
concentration at the semi-maximal rate. Due to inhibitory product,R, he
proposes the more generalised form,
@equation{k:=k@-=k(1-C@-/C@-@+<*>)@+.} The asterisk is indicative
of the state at which reproduction is fully inhibited. @foot[O.Levenspiel, The
Monod Equation: A Revisit and a Generalization to Product Inhibition
Situations, @i,pp.1671-1687(1980)]
For a CFSTR,
@equation{t@-=(C@-+C@-)/C@-=([C/A](C@-+C@-)-C@-)/([C/A]C@--C@-)
|C@-=0 & kt@- @b(>)1.}
This is the equation that was developed simultaneously by Monod @foot[Monod,
Jacques, @i,@b<79>,390(1950).] and by Novick and Szilard
@foot[Novick, A., & Szilard, L., @i,@b<36>,708(1950).]
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