Model Mathematics

mtime = \frac{time - (167 ⌊ \frac{time}{167} ⌋)}{1} Ca_i = \{\begin{matrix} 1 1.8535847759 e -5 {mtime}^{3} + (- 0.001590341066858) {mtime}^{2} + 0.043645898640826 {mtime}^{1} + 0.167079436392938 if mtime \geq 1.17 \land mtime < 30.84 \\ (1 (- 5.7458491 e -8) {mtime}^{3} + 3.1122215647 e -5 {mtime}^{2} - (0.006618494920814 {mtime}^{1})) + 0.720441888689583 if mtime \geq 30.84 \\ 0.216 otherwise \end{matrix}

permtot = \sqrt{{(\frac{TRPN}{perm50})}^{nperm}} \frac{d}{d time} (xb) = k_xb (permtot (1 - xb) - (\frac{1}{permtot} xb))

Ca_50 = Ca_50ref (1 + beta_1 (lambda_m - 1)) \frac{d}{d time} (TRPN) = k_off ({(\frac{Ca_i}{Ca_50})}^{n_TRPN} (1 - TRPN) - TRPN)

lambda = 1 dlambdadt = 0

lambda_m = \{\begin{matrix} 1.2 if lambda > 1.2 \\ lambda otherwise \end{matrix} lambda_s = \{\begin{matrix} 0.87 if lambda_m < 0.87 \\ lambda_m otherwise \end{matrix} overlap = 1 + beta_0 (lambda_m + lambda_s - 1.87)

T_0 = T_ref xb overlap

Q = Q_1 + Q_2 Tension = \{\begin{matrix} \frac{T_0 (a Q + 1)}{1 - Q} if Q < 0 \\ \frac{T_0 (1 + (a + 2) Q)}{1 + Q} otherwise \end{matrix} \frac{d}{d time} (Q_1) = A_1 dlambdadt - (alpha_1 Q_1) \frac{d}{d time} (Q_2) = A_2 dlambdadt - (alpha_2 Q_2)