Model Mathematics

\frac{d}{d t} (q_K_o) = v_Kp \frac{d}{d t} (q_K_i) = - v_Kp

\frac{d}{d t} (q_mem) = I_mem_Kp

V_mem = \frac{q_mem}{C_m} I_mem_Kp = F ((- zK) v_Kp + z_r_Kp v_gKp - (z_f_Kp v_gKp)) mu_C_Kp = R T \ln (K_C_Kp q_C_Kp) mu_K_o = R T \ln (K_K_o q_K_o) mu_K_i = R T \ln (K_K_i q_K_i) mu_O_Kp = R T \ln (K_O_Kp q_O_Kp) v_gKp = kappa_gKp (ⅇ^{\frac{Af_gKp}{R T}} - (ⅇ^{\frac{Ar_gKp}{R T}})) Af_Kp = mu_K_i + zK F V_mem + mu_O_Kp Af_gKp = R T \ln (K_C_Kp q_C_Kp) + z_f_Kp F V_mem Am_Kp = zK F V_mem Ar_Kp = mu_K_o + mu_O_Kp Ar_gKp = R T \ln (K_O_Kp q_O_Kp) + z_r_Kp F V_mem v_C_Kp = - v_gKp v_O_Kp = v_gKp - (v_Kp - v_Kp) \frac{d}{d t} (q_C_Kp) = v_C_Kp \frac{d}{d t} (q_O_Kp) = v_O_Kp v_Kp = \{\begin{matrix} kappa_Kp (ⅇ^{\frac{Af_Kp}{R T}} - (ⅇ^{\frac{Ar_Kp}{R T}})) if Am_Kp = 0 \\ \frac{\frac{kappa_Kp Am_Kp}{R T}}{ⅇ^{\frac{Am_Kp}{R T}} - 1} (ⅇ^{\frac{Af_Kp}{R T}} - (ⅇ^{\frac{Ar_Kp}{R T}})) otherwise \end{matrix}