Estimating the Multiplicity of the Steady States of a Fermentation Process for Lactic Acid Production at a Given Concentration of the Component Producing the Main Substrate

Abstract

A computational scheme has been presented for estimating the technological characteristics of the continuous fermentation of lactic acid at a given concentration of the component that produces the main substrate. The computational scheme includes a preliminary analysis, the principal analysis, and a numerical example of the implementation of calculation. In the preliminary analysis, relationships for estimating the boundary characteristics have been given that form the region of values for these characteristics that, in fact, ensure a given concentration of the component producing the main substrate (M₀, g/L). The coordinates of the point of the maximum productivity and singular points have been determined. The positions of these coordinates have been illustrated using a portrait of the dependence of M₀ on the dilution rate D for a productivity of Q_P = 6 g/(L h). This portrait also shows the boundaries of sections I, II, and III, for which computational relationships have been constructed. The scheme of the construction of computational relationships has been presented separately for each of the sections. The boundaries of sections are determined by relationships (22)–(24). Sets have been constructed at a given value of M₀ for the coordinates of singular points, the extremum point Q_P, and each of the sections: Set1 for section I; Set1* and Set2* for section II; and Set1**, Set2**, and Set3** for section III. Theoretical relationships have been used for a numerical estimation of the characteristics of sets. It has been shown that the range of the assignment of M₀ is the widest for section I, whereas the range of values for the dilution rate D is the greatest for section III. Numerical estimates in a comparative variant for sets in sections II and III have shown that the final concentrations of components under the same initial conditions differ in the values of M₀ and S. This makes it possible to evaluate the further processing of synthesis products (the recovery of lactic acid, the recycling and use of unconverted components, and the recovery of a by-product).

APPENDIX

$$A\left( D \right) = {{\left( {1 - \frac{{{{Q}_{P}}}}{{{{X}_{{\max }}}\left( {\alpha D + \beta } \right)}}} \right)}^{{{{n}_{1}}}}}{{\left( {1 - \frac{{{{Q}_{P}}}}{{{{P}_{{\max }}}D}}} \right)}^{{{{n}_{2}}}}}.$$

(A1)

$$S' = {{S}_{0}} + \frac{{{{k}_{M}}{{M}_{0}}}}{{D + {{k}_{M}}}}.$$

(A2)

$$\begin{gathered} S_{{1,2}}^{'} = \frac{1}{{{{Y}_{{{X \mathord{\left/ {\vphantom {X S}} \right. \kern-0em} S}}}}}}\frac{{{{Q}_{P}}}}{{\left( {\alpha D + \beta } \right)}} + \frac{{{{K}_{{\text{i}}}}}}{2}\left[ {A\left( D \right)\frac{{{{\mu }_{{\max }}}}}{D} - 1} \right] \\ \pm \,\,\sqrt {{{{\left( {\frac{{{{K}_{{\text{i}}}}}}{2}} \right)}}^{2}}{{{\left[ {A\left( D \right)\frac{{{{\mu }_{{\max }}}}}{D} - 1} \right]}}^{2}} - {{K}_{{\text{m}}}}{{K}_{{\text{i}}}}} . \\ \end{gathered} $$

(A3)

$$\left\{ \begin{gathered} P = \frac{{{{Q}_{P}}}}{D};\,\,\,\,X = \frac{P}{{\alpha + {\beta \mathord{\left/ {\vphantom {\beta D}} \right. \kern-0em} D}}}; \hfill \\ B = \left( {{{\alpha }_{B}} + {{{{\beta }_{B}}} \mathord{\left/ {\vphantom {{{{\beta }_{B}}} D}} \right. \kern-0em} D}} \right)\frac{P}{{\alpha + {\beta \mathord{\left/ {\vphantom {\beta D}} \right. \kern-0em} D}}} \hfill \\ S = {{S}_{0}} + \frac{{{{k}_{M}}{{M}_{0}}}}{{D + {{k}_{M}}}} - \frac{1}{{{{Y}_{{{X \mathord{\left/ {\vphantom {X S}} \right. \kern-0em} S}}}}}}\frac{P}{{\alpha + {\beta \mathord{\left/ {\vphantom {\beta D}} \right. \kern-0em} D}}};\,\,\,M = \frac{{D{{M}_{0}}}}{{D + {{k}_{M}}}}. \hfill \\ \end{gathered} \right.$$

(A4)