Model Mathematics

A_B = {r_B}^{2} π

\frac{d}{d time} (r_B) = \frac{k_for}{X_b} xb + \frac{1}{X_y} xy - (\frac{k_res}{X_c} xc + k_rB r_B)

\frac{d}{d time} (xy) = k_byp (xb - X_bss) - (k_yd (xy - X_yss))

F_sti = \frac{F_s xy}{1 + ⅇ^{- (k_Fs F_s + k_y xy)}}

F_s = \frac{F_a}{A_B}

F_a = \{\begin{matrix} 10000.0 if time \geq 100.0 \land time < 130.0 \\ 100.0 otherwise \end{matrix}

\frac{d}{d time} (x_no) = k_yno F_sti + X_noe - (k_nod x_no)

\frac{d}{d time} (x_pge) = k_ypge F_sti + k_nopge x_no + X_pgex - (k_pged x_pge)

\frac{d}{d time} (x_opg) = \frac{K_o_p}{1} pi_c xr + Io + k_nopg x_no - (k_opgd x_opg)

\frac{d}{d time} (x_kl) = rl + Il - (k_nokl x_no + rl \frac{1 + \frac{k1}{k2} x_opg + \frac{k3}{k4} K}{K_l_p pi_p xb} x_kl)

pi_L = \frac{\frac{k3}{k4} K_l_p pi_p xb}{1 + \frac{k3 K}{k4} + \frac{k1}{k2 ko} (\frac{\frac{K_o_p}{1}}{pi_p} xr + Io)} (1 + \frac{Il}{rl})

\frac{d}{d time} (xr) = D_R pi_c + k_pger x_pge - (\frac{D_B}{pi_c} xr)

\frac{d}{d time} (xb) = \frac{D_B}{pi_c} xr - (k_B xb)

\frac{d}{d time} (xc) = D_C pi_L - (D_A pi_c xc)

D_B = f0 dB pi_c = \frac{xc + f0 C_s}{xc + C_s} pi_p = \frac{P + P_0}{P + P_s} P = \frac{IP}{kP} P_0 = \frac{SP}{kP} P_s = \frac{k6}{k5}