(a) For the liquid-phase autocatalytic reaction A + . . . → B + . . . taking place isothermally at steady-state in a recycle PFR, derive an expression for the optimal value of the recycle ratio, $R_{opt}$ , that minimizes the volume or space time of the reactor. The rate law is $( -r_A) = k_Ac_Ac_B$ .

(b) Express the minimum volume or space time of the reactor in terms of $R_{opt}$ .

Question

(a) For the liquid-phase autocatalytic reaction A + . . . → B + . . . taking place isothermally at steady-state in a recycle PFR, derive an expression for the optimal value of the recycle ratio, $R_{opt}$ , that minimizes the volume or space time of the reactor. The rate law is $( -r_A) = k_Ac_Ac_B$ .

(b) Express the minimum volume or space time of the reactor in terms of $R_{opt}$ .

(b) Express the minimum volume or space time of the reactor in terms of [latex] R_{opt} [/latex].

Accepted Answer

(a) With [latex] c_{Ao} \gg c_{Bo} , c_B = c_{Ao} - c_A [/latex] , and the rate law becomes

[latex] ( -r_A) = k_Ac_A(c_{Ao} - c_A) [/latex] (A)

From the material-balance equation 15.3-4, on substitution for [latex] ( -r_A) [/latex] from (A) ,

[latex]V / q_o = au = - (1 + R) \int _{\acute{c_Ao} }^{c_{A1}}{\frac {dc_A}{( -r_A)}} [/latex] (15.3-4)

[latex] \frac{V }{q_o} = au = - (1 + R) \int_{\acute{c_{Ao}} }^{c_{A1}} \frac {dc_A}{k_Ac_A(c_{Ao} - c_A)} [/latex]

[latex] = \frac {1 + R}{k_Ac_{Ao}}\ln \left[\frac {\acute{c_{Ao}} (c _{Ao} - c_{A1} )} {c_{A1}(c_{Ao} - \acute{c_{Ao}} )} ight] [/latex]

[latex] = \frac {1 + R}{k_Ac_{Ao}}\ln \left(\frac {c_{Ao} + Rc_{A1}}{Rc_{A1}} ight) [/latex] (B)

on substitution for [latex] \acute{c_{Ao}} [/latex] from equation 15.3-3. Equation (B) gives [latex] V(R) /q_o [/latex], or τ(R) as a function of R for fixed [latex] c_{Ao}[/latex] and [latex]c_{A1} [/latex] , and the given rate law. To obtain [latex]R_{opt} [/latex]for minimum V/q,, or τ, we set dτ(R)/dR = 0. This leads to a transcendental, implicit equation for [latex]R_{opt} [/latex] :

[latex] \acute{c_{Ao}} = \frac {c_{Ao} + Rc_{A1}}{1 + R} [/latex] (15.3-3)

[latex]\ln \left(\frac {c_{Ao} + R_{opt}c_{A1}}{R_{opt}c_{A1}} ight) = \frac {c_{Ao}( 1 + R_{opt})}{R_{opt} (c_{Ao} + R_{opt}c_{A1})} [/latex] (15.3-6)

(b) To obtain an expression for the minimum volume or space time, we combine equations (B) and 15.3-6 to obtain :

[latex] \frac {V_{min}}{q_o} = au _{min} = \frac {(1 + R_{opt})^2}{k_AR_{opt}(c_{Ao} + R_{opt}c_{A1})} [/latex] (15.3-7)

Question 15.9: (a) For the liquid-phase autocatalytic reaction A + . . . → ...

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