For this CSTR the appropriate design equation is
\tau=\frac{V_R}{V_0}=\frac{C_{ C 0} \int_0^{f_e} d f_c}{-r_{ CF}}=\frac{C_{ C 0} f_c}{-r_{ C F}} (A)
with
-r_{ C F}=k_1 C_{ C F}-k_2 C_{ B F} (B)
From stoichiometric considerations
C_{ C F}=C_{ C 0}\left(1-f_c\right) (C)
C_{ B F}=C_{ C 0} f_c (D)
Combining equations (A) to (D) gives
\frac{V_R}{ V _0}=\frac{f_c}{k_1\left(1-f_c\right)-k_2 f_c}=\frac{f_c}{k_1-f_c\left(k_1+k_2\right)}
To minimize the required reactor volume, one may set the temperature derivative of V_{R} equal to zero:
\left(\frac{\partial V_R}{\partial T}\right)_{f_c}=\frac{-V_0 f_c \frac{\partial}{\partial T}\left[k_1-f_c\left(k_1+k_2\right)\right]}{\left[k_1-f_c\left(k_1+k_2\right)\right]^2}=0
or
\frac{\partial k_1}{\partial T}-f_c\left(\frac{\partial k_1}{\partial T}+\frac{\partial k_2}{\partial T}\right)=0
However, from the Arrhenius relation,
\frac{d \ln k}{d T}=\frac{E_A}{R T^2} \quad \text { or } \quad \frac{d k}{d T}=\frac{k E_A}{R T^2}
Thus, the optimum temperature will be that at which
\frac{k_1 E_1}{R T^2}\left(1-f_c\right)-f_c \frac{k_2 E_2}{R T^2}=0
or
\frac{k_1}{k_2}=\frac{E_2}{E_1}\left(\frac{f_c}{1-f_c}\right)
Substitution of the Arrhenius form of the rate constants yields
\frac{A_1 e^{-E_1 / R T}}{A_2 e^{-E_2 / R T}}=\frac{E_2}{E_1}\left(\frac{f_c}{1-f_c}\right)
One can then solve for the optimum temperature:
\frac{1}{T}\left(-\frac{E_1}{R}+\frac{E_2}{R}\right)=\ln \left[\left(\frac{E_2 A_2}{E_1 A_1}\right)\left(\frac{f_c}{1-f_c}\right)\right]\equiv \ln \beta
or
T=\frac{E_2-E_1}{R \ln \left\{\left[\left(E_2 A_2\right) /\left(E_1 A_1\right)\right]\left[f_c /\left(1-f_c\right)\right]\right\}}=\frac{E_2-E_1}{R \ln\beta}
At this temperature
k_1=A_1 e^{-\left(E_1 \ln \beta\right) /\left(E_2-E_1\right)}=A_1 \beta^{-E_1 /\left(E_2-E_1\right)}
and
k_2=A_2 \beta^{-E_2 /\left(E_2-E_1\right)}
Thus, the minimum reactor volume is given by
V_R=\frac{ V _0 f_c}{A_1 \beta^{-E_1 /\left(E_2-E_1\right)}\left(1-f_c\right)-A_2 \beta^{-E_2 /\left(E_2-E_1\right)} f_c}