Table 1.

List of equations and solutions for Logistic‐Monod growth in batch culture

Equations and solutions for Logistic‐Monod growth in batch culture
$$\frac{1}{X}\frac{dX}{dt}=\mu =\frac{{\mu }_{m}S}{K_S+S}$$ (1)
$$\frac{1}{X}\frac{dX}{dt}=\mu ={\mu }_m\left(1-\frac{X}{X_m}\right)$$ (2)
$$\frac{1}{X}\frac{dX}{dt}=\mu ={\mu }_m\left(\left(,\frac{S}{K_S+S},\right)\right)\left(\left(,1-\frac{X}{X_m},\right)\right)$$ (3)
$$\frac{dS}{dt}=-\frac{1}{Y_{\text{x/s}}}\frac{dX}{dt}=-\frac{\mu X}{Y_{\text{x/s}}}$$ (4)
$$\ln \left(\left(,\frac{X}{X_0},\right)\right)={\mu }_{m}t+\frac{K_S{Y}_{\text{x/s}}}{X_0+S_0{Y}_{\text{x/s}}}\ln \left(\left(,\frac{X_0\left(X_0-X+S_0{Y}_{\text{x/s}}\right)}{S_{0}X{Y}_{\text{x/s}}},\right)\right)$$ (5)
$$S=\left\{\left\{,S_0+\frac{X_0-z}{Y_{\text{x/s}}}∣\mid ,z\in \text{solve}\left(\left(,\ln \left(\left(,\frac{X}{X_0},\right)\right)={\mu }_{m}t+\frac{K_S{Y}_{\text{x/s}}}{X_0+S_0{Y}_{\text{x/s}}}\ln \left(\left(,\frac{X_0\left(\left(,X_0-X+S_0{Y}_{\text{x/s}},\right)\right)}{S_{0}X{Y}_{\text{x/s}}},\right)\right),X,\right)\right),\right\}\right\}$$ (6)
$$X=\frac{X_0{X}_m{e}^{{\mu }_{m}t}}{X_m-X_0+X_0{e}^{{\mu }_{m}t}}$$ (7)
$$S=\frac{X_0{X}_m-{X_0}^2+{X_0}^2{e}^{{\mu }_{m}t}-S_0{X}_0{Y}_{\text{x/s}}+S_0{X}_m{Y}_{\text{x/s}}-X_0{X}_m{e}^{{\mu }_{m}t}+S_0{X}_0{Y}_{\text{x/s}}{e}^{{\mu }_{m}t}}{Y_{\text{x/s}}\left(\left(,X_m-X_0+X_0{e}^{{\mu }_{m}t},\right)\right)}$$ (8)
$$\begin{array}{l}\ln \left(\frac{X}{X_0}\right)\left(\frac{1}{X_m}+\frac{K_S{Y}_{\text{x/s}}}{X_0{X}_m+S_0{X}_m{Y}_{\text{x/s}}}\right)+\ln \left(\frac{X_0-X_m}{X-X_m}\right)\left(\frac{1}{X_m}+\frac{K_S{Y}_{\text{x/s}}}{X_0{X}_m-{X_m}^2+S_0{X}_m{Y}_{\text{x/s}}}\right)\\ +\frac{K_S{Y}_{\text{x/s}}\ln \left(\frac{X\hspace{0.17em}-\hspace{0.17em}{X}_0\hspace{0.17em}-\hspace{0.17em}{S}_0{Y}_{\text{x/s}}}{-S_0{Y}_{\text{x/s}}}\right)}{{S_0}^2{Y_{\text{x/s}}}^2+2S_0{X}_0{Y}_{\text{x/s}}-X_m{S}_0{Y}_{\text{x/s}}+{X_0}^2-X_m{X}_0}=\frac{{\mu }_{m}t}{X_m}\end{array}$$ (9)
$$\begin{array}{l}\frac{K_S\ln \left(\left(,\frac{S}{S_0},\right)\right)}{{S_0}^2{Y_{\text{x/s}}}^2+2S_0{X}_0{Y}_{\text{x/s}}-X_m{S}_0{Y}_{\text{x/s}}+{X_0}^2-X_m{X}_0}+\frac{\left(\left(,X_0-X_m+Y_{\text{x/s}}\left(K_S+S_0\right),\right)\right)}{-{X_m}^2{Y}_{\text{x/s}}+S_0{X}_m{Y_{\text{x/s}}}^2+X_0{X}_m{Y}_{\text{x/s}}}\ln \left(\left(,\frac{X_0-X_m}{X_0-X_m-S{Y}_{\text{x/s}}+S_0{Y}_{\text{x/s}}},\right)\right)\\ +\frac{\left(\left(,X_0+Y_{\text{x/s}}\left(K_S+S_0\right),\right)\right)}{S_0{X}_m{Y_{\text{x/s}}}^2+X_0{X}_m{Y}_{\text{x/s}}}\ln \left(\left(,\frac{X_0-S{Y}_{\text{x/s}}+S_0{Y}_{\text{x/s}}}{X_0},\right)\right)=\frac{{\mu }_{m}t}{X_m{Y}_{\text{x/s}}}\end{array}$$ (10)