Steady States of a Continuous Fermentation Process for Lactic Acid Production: The Multiplicity for a Given Dilution Rate

Abstract

The results of an analysis of a generalized mathematical model for a continuous fermentation process for lactic acid production have been presented. The mathematical model includes a system of equations for the material balance of the main substrate (S), the substrate produced from raw materials during fermentation (M), biomass (X), a product (P), and a by-product (B). The kinetics of the formation of biomass (the equation for the specific rate μ) takes into account inhibition by the substrate, biomass, and product. An analysis has been performed from the standpoint of the possibility of estimating technological characteristics at a given dilution rate D (D = v/V, where v is the volumetric flow rate through a fermenter, m³/h, and V is the volume of the fermenter, m³). The limiting values of $S{\kern 1pt} '$ and D that ensure the practical implementation of the process have been estimated ($S{\kern 1pt} '$ includes the initial concentrations of the main substrate S₀ and the component that produces the substrate during synthesis M₀). These characteristics have been designated as coordinates of “singular points.” The coordinates of a point that corresponds to the maximum productivity with respect to lactic acid Q_P (${{Q}_{P}} = DP,$ where P is the concentration of lactic acid) have been determined simultaneously. Relationships have been derived for calculating sets of technological characteristics (the initial characteristics D, S₀, and M₀ and the current characteristics X, S, P, B, and M) based on a given value of D within permissible limits. It has been shown that, for the values of D that correspond to singular points (the first, second, and optimal points), there is one set and, for the other values of D, there are two sets. Numerical estimates of technological characteristics for the values of constants that correspond to the basic variant have been given. It has been shown that, with an increase in the productivity Q_P while approaching the value of max Q_P, the region of estimates of technological characteristics narrows. It has been recommended that the results of this study be used to predict the economic estimates of the implementation of particular conditions of a technological process.

Appendices

APPENDIX

$$\frac{D}{{{{\mu }_{{{\text{max}}}}}}} = A\left( D \right)\frac{{{{K}_{{\text{i}}}}S}}{{{{K}_{{\text{m}}}}{{K}_{{\text{i}}}} + {{K}_{{\text{i}}}}S + {{S}^{2}}}}.$$

(A.1)

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

(A.2)

$$S = S{\kern 1pt} ' - \frac{1}{{{{Y}_{{{X \mathord{\left/ {\vphantom {X S}} \right. \kern-0em} S}}}}}}\frac{{{{Q}_{P}}}}{{\left( {\alpha D + \beta } \right)}}.$$

(A.3)

$$S{\kern 1pt} ' = {{S}_{0}} + \frac{{{{k}_{M}}{{M}_{0}}}}{{D + {{k}_{M}}}}.$$

(A.4)

$$\begin{gathered} S = \frac{{{{K}_{i}}}}{2}\left[ {A\left( D \right)\frac{{{{\mu }_{{{\text{max}}}}}}}{D} - 1} \right] \\ \pm \,\,\sqrt {{{{\left( {\frac{{{{K}_{i}}}}{2}} \right)}}^{2}}{{{\left[ {A\left( D \right)\frac{{{{\mu }_{{{\text{max}}}}}}}{D} - 1} \right]}}^{2}} - {{K}_{{\text{m}}}}{{K}_{{\text{i}}}}} . \\ \end{gathered} $$

(A.5)

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

(A.6)

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

(A.7)

The maximum limiting value of the dilution rate is as follows:

$$D_{{{\text{lim}}}}^{{{\text{max}}}} = \frac{{{{\mu }_{{{\text{max}}}}}}}{{2{{{\left( {\frac{{{{K}_{{\text{m}}}}}}{{{{K}_{{\text{i}}}}}}} \right)}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} + 1}}.$$

The equations for calculating max Q_P (the maximum value of Q_P) and the corresponding value of the dilution rate have the following form:

$${{\left( {\frac{{{{K}_{i}}}}{2}} \right)}^{2}}{{\left[ {A\left( D \right)\frac{{{{\mu }_{{{\text{max}}}}}}}{D} - 1} \right]}^{2}} - {{K}_{{\text{m}}}}{{K}_{{\text{i}}}} = 0,$$

(A.8)

$$S_{{{\text{opt}}}}^{'} = \frac{1}{{{{Y}_{{{X \mathord{\left/ {\vphantom {X S}} \right. \kern-0em} S}}}}}}\frac{{{\text{max}}{{Q}_{P}}}}{{\left( {\alpha {{D}^{{{\text{opt}}}}} + \beta } \right)}} + {{\left( {{{K}_{{\text{m}}}}{{K}_{{\text{i}}}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}},$$

(A.9)

$$S_{{{\text{opt}}}}^{'} = S_{0}^{{{\text{opt}}}} + \frac{{{{k}_{M}}M_{{\text{0}}}^{{{\text{opt}}}}}}{{{{D}^{{{\text{opt}}}}} + {{k}_{M}}}},$$

(A.10)

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

(A.11)

NOTATION

B	concentration of the total number of by-products, g/L
D	dilution rate, h–1
K i	inhibition constant, g/L
K m	substrate saturation constant, g/L
k M	constant that determines the amount of produced substrate, h–1
M	concentration of raw materials that additionally produce the substrate, g/L
P	concentration of the product, g/L
Q P	productivity, g/(L h)
S	concentration of the substrate, g/L
X	concentration of biomass, g/L
Y X/S	stoichiometric coefficient, g/g
α, αB, β, βB	constants
μ	specific rate of microorganism growth, h–1

SUBSCRIPTS AND SUPERSCRIPTS

0	initial value
lim	limiting value
max	maximum value
opt	optimum value