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TableĀ 1 Batch and fed batch kinetic equations

From: Optimal feed profile for the Rhamnolipid kinetic models by using Tabu search: metabolic view point

\(\frac{dx}{dt} = \mu_{m} \left[ {1 - \frac{x}{{x_{m} }}} \right]x\) Equation (1)
\(x = \frac{{x_{m} x_{0} e^{{\mu_{m} t}} }}{{x_{m} - x_{0} + x_{0} e^{{\mu_{m} t}} }}\) Equation (2)
\(p = \alpha \frac{{x_{m} x_{0} e^{{\mu_{m} t}} }}{{x_{m} - x_{0} + x_{0} e^{{\mu_{m} t}} }} + \beta \frac{{x_{0} }}{{x_{m} }}\ln \left( {1 - \left( {\frac{{x_{0} }}{{x_{m} }}} \right)} \right)\left( {1 - e^{{\mu_{m} t}} } \right)\) Equation (3)
\(p = \alpha A(t) + \beta B(t)\) Equation (4)
\(s = \gamma \frac{{x_{m} x_{0} e^{{\mu_{m} t}} }}{{x_{m} - x_{0} + x_{m} x_{0} e^{{\mu_{m} t}} }} - \eta \frac{{x_{0} }}{{\mu_{0} }}\ln \left( {1 - \left( {\frac{{x_{0} }}{{x_{m} }}} \right)} \right)\left( {1 - e^{{\mu_{m} t}} } \right)\) Equation (5)
\(s = \gamma A(t) - \eta B(t)\) Equation (6)
\(A(t) = \frac{{x_{m} x_{0} e^{\mu }_{m} t}}{{x_{m} - x_{0} + x_{m} x_{0} e^{\mu }_{m} t}}\) Equation (7)
\(B(t) = \frac{{x_{0} }}{{\mu_{0} }}\frac{{x_{0} }}{{x_{m} }}\ln \left( {1 - \left( {\frac{{x_{0} }}{{x_{m} }}} \right)} \right)\left( {1 - e^{{\mu_{m} t}} } \right)\) Equation (8)
\(q_{{s_{1} }} = \frac{1}{{Y{x \mathord{\left/ {\vphantom {x s}} \right. \kern-0pt} s}}}U_{s} + M\) Equation (9)
\(q_{{s_{2} }} = \frac{1}{{Y{x \mathord{\left/ {\vphantom {x s}} \right. \kern-0pt} s}}}U_{s}\) Equation (10)
\(q_{{s_{3} }} = \frac{1}{{Y{x \mathord{\left/ {\vphantom {x s}} \right. \kern-0pt} s}}}U_{s}\) Equation (11)
\(q_{{p_{{}} }} = \frac{1}{{Y{p \mathord{\left/ {\vphantom {p s}} \right. \kern-0pt} s}}}U_{s}\) Equation (12)