Model Mathematics

i_Ks = \frac{1 e -9 F}{C_m} v_Ks \frac{d}{d t} (q_K_i) = - v_Ks \frac{d}{d t} (q_K_o) = v_Ks \frac{d}{d t} (q_mem) = I_mem_Ks Vmem = \frac{q_mem}{C_m}

Am_Ks = zK F V_mem u_K_i = R T \ln (K_K_i q_K_i) u_K_o = R T \ln (K_K_o q_K_o) Af_Ks = mu_S_Ks2 + u_K_i + Am_Ks Ar_Ks = mu_S_Ks2 + u_K_o v_Ks = \{\begin{matrix} 1 kappa_Ks (ⅇ^{\frac{Af_Ks}{R T}} - (ⅇ^{\frac{Ar_Ks}{R T}})) if Am_Ks = 0 \\ \frac{\frac{1 kappa_Ks Am_Ks}{R T}}{ⅇ^{\frac{Am_Ks}{R T}} - 1} (ⅇ^{\frac{Af_Ks}{R T}} - (ⅇ^{\frac{Ar_Ks}{R T}})) otherwise \end{matrix} V_mem = \frac{q_mem}{C_m} I_mem_Ks = F ((- zK) v_Ks + (v_xs1 + v_xs2) (z_xs_r - z_xs_f))

Af_xs1 = mu_S_Ks0 + z_xs_f F V_mem Ar_xs1 = mu_S_Ks1 + z_xs_r F V_mem Af_xs2 = mu_S_Ks1 + z_xs_f F V_mem Ar_xs2 = mu_S_Ks2 + z_xs_r F V_mem v_xs1 = kappa_xs1 (ⅇ^{\frac{Af_xs1}{R T}} - (ⅇ^{\frac{Ar_xs1}{R T}})) v_xs2 = kappa_xs2 (ⅇ^{\frac{Af_xs2}{R T}} - (ⅇ^{\frac{Ar_xs2}{R T}})) v_S_Ks0 = - v_xs1 v_S_Ks1 = v_xs1 - v_xs2 v_S_Ks2 = v_xs2 mu_S_Ks0 = R T \ln (K_S_Ks0 q_S_Ks0) mu_S_Ks1 = R T \ln (K_S_Ks1 q_S_Ks1) mu_S_Ks2 = R T \ln (K_S_Ks2 q_S_Ks2) \frac{d}{d t} (q_S_Ks0) = v_S_Ks0 \frac{d}{d t} (q_S_Ks1) = v_S_Ks1 \frac{d}{d t} (q_S_Ks2) = v_S_Ks2