Chapter 5: Isothermal Reactor Design: Conversion
Derivation for Laminar Flow:
\(\alpha = \frac{2G(1 - \phi)}{A_c \rho_o g_c D_p \phi^3 P_0} \left[ \frac{150\mu (1 - \phi)}{D_p} + \text{small number} \right]\)
\(\alpha = C \cdot \frac{2(1 - \phi)}{\rho_o g_c \phi P_0} \left[ \frac{G}{A_c D_p} \right]\)
\(\alpha \sim \frac{G}{A_c D_p^2}\)
\(\alpha_1 = C \cdot \frac{G_1}{A_{c1} D_{P1}}\)
\(\alpha_2 = C \cdot \frac{G_2}{A_{c2} D_{P2}}\)
Taking the ratio \(\alpha_1\) to \(\alpha_2\):
\(\alpha_2 = \alpha_1 \left( \frac{G_2}{G_1} \right) \left( \frac{A_{c1}}{A_{c2}} \right) \left( \frac{D_{P1}}{D_{P2}} \right)^2\)
\(G \sim \frac{\dot{m}}{A_c}\)
For constant \(\dot{m}\):
\(\alpha_2 = \alpha_1 \left( \frac{A_{c1}}{A_{c2}} \right)^2 \left( \frac{D_{P1}}{D_{P2}} \right)^2\)