Chapter 8: Multiple Reactions
Example ODE Solver Algorithm
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\[ \boxed{ \begin{aligned} \text{HCHO} + \frac{1}{2} \text{O}_2 \xrightarrow{k_1} \text{HCOOH} \xrightarrow{k_3} \text{CO} + \text{H}_2\text{O} \\ 2\text{HCHO} \xrightarrow{k_2} \text{HCOOCH}_3 \\ \text{HCOOCH}_3 + \text{H}_2\text{O} \xrightarrow{k_4} \text{CH}_3\text{OH} + \text{HCOOH} \end{aligned} } \]
Let \( A = \text{HCHO} \), \( B = \text{O}_2 \), \( C = \text{HCOOH} \), \( D = \text{HCOOCH}_3 \), \( E = \text{CO} \), \( W = \text{H}_2\text{O} \), \( G = \text{CH}_3\text{OH} \).
\[ \frac{dF_A}{dV} = -k_1 C_{T0}^{3/2} \left( \frac{F_A}{F_T} \right) \left( \frac{F_B}{F_T} \right)^{1/2} - k_2 C_{T0}^{2} \left( \frac{F_A}{F_T} \right)^2 \]
\[ \frac{dF_B}{dV} = -\frac{k_1}{2} C_{T0}^{3/2} \left( \frac{F_A}{F_T} \right) \left( \frac{F_B}{F_T} \right)^{1/2} \]
\[ \frac{dF_C}{dV} = k_1 C_{T0}^{3/2} \left( \frac{F_A}{F_T} \right) \left( \frac{F_B}{F_T} \right)^{1/2} - k_3 C_{T0} \left( \frac{F_C}{F_T} \right) + k_4 C_{T0}^{2} \left( \frac{F_W}{F_T} \right) \left( \frac{F_D}{F_T} \right) \]
\[ \frac{dF_D}{dV} = \frac{k_2}{2} C_{T0}^{2} \left( \frac{F_A}{F_T} \right)^2 - k_4 C_{T0}^{2} \left( \frac{F_D}{F_T} \right) \left( \frac{F_W}{F_T} \right) \]
\[ \frac{dF_E}{dV} = k_3 C_{T0} \left( \frac{F_C}{F_T} \right) \]
\[ \frac{dF_W}{dV} = k_3 C_{T0} \left( \frac{F_C}{F_T} \right) - k_4 C_{T0}^{2} \left( \frac{F_W}{F_T} \right) \left( \frac{F_D}{F_T} \right) \]
\[ \frac{dF_G}{dV} = k_4 C_{T0}^{2} \left( \frac{F_W}{F_T} \right) \left( \frac{F_D}{F_T} \right) \]
\[ F_T = F_A + F_B + F_C + F_D + F_E + F_W + F_G \]
\[ F_{A0} = 10, \quad F_{B0} = 5, \quad V_F = 1000, \quad k_1 C_{T0}^{3/2} = 0.04, \quad k_2 C_{T0}^{2} = 0.007, \quad k_3 C_{T0} = 0.014, \quad k_4 C_{T0}^{2} = 0.45 \]