Model Mathematics

R = \frac{2 π 1}{5} P_L = P_m - (\frac{R omega V_T}{2} \sin (\frac{omega time}{1})) - (E (2.5 - (\frac{V_T 1}{2} \cos (\frac{omega time}{1})))) dP_Ldt = \frac{(- R) {omega}^{2} V_T}{2 1} \cos (\frac{omega time}{1}) - (E (2.5 - (\frac{V_T}{2} \sin (\frac{omega time}{1}))))

q = \frac{P_m - P_A}{R} \frac{d}{d time} (P_A) = \frac{P_m E Q_A}{P_A 1} + dP_Ldt \frac{d}{d time} (V_A) = \frac{P_m - P_L - (\frac{V_A E}{1})}{R}

p_ao = f_o (P_A - P_w) p_ac = f_c (P_A - P_w) Q_A = q + 1 D_c (p_c - p_ac) + 1 D_o (p_o - p_ao) f_oi = \{\begin{matrix} \frac{f_o V_D + f_om (V_T - V_D)}{V_T} if V_T \geq V_D \\ f_o otherwise \end{matrix} f_ci = \{\begin{matrix} \frac{f_c V_D + f_cm (V_T - V_D)}{V_T} if V_T \geq V_D \\ f_c otherwise \end{matrix} \frac{d}{d time} (f_o) = \frac{1}{V_A} (1 D_o (p_o - p_ao) + (f_oi - f_o) q - (f_o (1 D_c (p_c - p_ac) + 1 D_o (p_o - p_ao)))) \frac{d}{d time} (f_c) = \frac{1}{V_A} (1 D_c (p_c - p_ac) + (f_ci - f_c) q - (f_c (1 D_o (p_o - p_ao) + 1 D_c (p_c - p_ac))))

delta = 10^{1.9} h = 1 10^{- 7.4} df_satdp = \frac{(L {(1 + K_T sigma p_o)}^{4} + {(1 + K_R sigma p_o)}^{4}) (3 L {K_T}^{2} {sigma}^{2} p_o 1 {(1 + K_T sigma p_o)}^{2} + L K_T sigma 1 {(1 + K_T sigma p_o)}^{3} + 3 {K_R}^{2} {sigma}^{2} p_o 1 {(1 + K_R sigma p_o)}^{2} + K_R sigma 1 {(1 + K_R sigma p_o)}^{3}) - ((L K_T sigma p_o {(1 + K_T sigma p_o)}^{3} + K_R sigma p_o {(1 + K_R sigma p_o)}^{3}) (4 L K_T sigma 1 {(1 + K_T sigma p_o)}^{3} + 4 K_R sigma 1 {(1 + K_R sigma p_o)}^{3}))}{{(L {(1 + K_T sigma p_o)}^{4} + {(1 + K_R sigma p_o)}^{4})}^{2}} \frac{d}{d time} (p_o) = \frac{D_o}{sigma V_c} {(1 + \frac{4 T_h}{sigma} df_satdp)}^{- 1} (f_o (P_A - P_w) - p_o) \frac{d}{d time} (p_c) = \frac{D_c}{sigma_c V_c} (p_ac - p_c) + \frac{1 delta l_2}{sigma_c} h z - (delta r_2 p_c) \frac{d}{d time} (z) = \frac{delta r_2 sigma_c p_c}{1} - (delta l_2 h z)