Consider a liquid-phase reaction, A → products, taking place in the reactor systems shown in Figure 17.5(a) and (b), with $V_{ST} / V_{PF} = 4$ .

(a) Calculate $f_A$ , if the reaction is first-order, with $k_A\bar{t}_{PF} = 1$ , for (i) the arrangement in Figure 17.5(a), and (ii) that in Figure 17.5(b) .

(b) Repeat part (a) above for a second-order reaction with $k_Ac_{Ao}\bar {t}_{PF} = 1$ .

Question

Consider a liquid-phase reaction, A → products, taking place in the reactor systems shown in Figure 17.5(a) and (b), with [latex] V_{ST} / V_{PF} = 4 [/latex] .

(a) Calculate [latex]f_A[/latex] , if the reaction is first-order, with [latex] k_A\bar{t}_{PF} = 1[/latex] , for (i) the arrangement in Figure 17.5(a), and (ii) that in Figure 17.5(b) .

(b) Repeat part (a) above for a second-order reaction with [latex] k_Ac_{Ao}\bar {t}_{PF} = 1 [/latex] .

Accepted Answer

(a) (i) If [latex] f_{A1} [/latex] is the fractional conversion at the exit of vessel 1 (PFR), from equation 17.1-14,

[latex] f_A = 1 - \exp(-M_{A1}) [/latex] (17.1-14)

[latex] f_{A1} = 1 - e^{-k_A \bar {t}_{PF}} = 0.632 [/latex]

If [latex] f_{A2} [/latex] is the fractional conversion at the exit of vessel 2 (CSTR), from equation 14.4-1

[latex] V_i = F_{Ao} (f_{Ai} f_{A,i-1}) / ( - r_A)_i [/latex] (14.4-1)

[latex] V_{ST} = F_{Ao} (f_{A2} f_{A1}) /( - r_A)_2 [/latex] (A)

On substitution of the rate law, we obtain

[latex] V_{ST} = \frac {F_{Ao}(f_{A2} - f_{A1})} {k_A(F_{Ao}/q)(1 - f_{A2})} [/latex]

and on substitution of [latex] q = V_{ST} \bar{t}_{ST} [/latex] , this leads to

[latex] f_{A2} = \frac {f_{A1} + k_A\bar{t}_{ST} }{1 + k_A\bar{t}_{ST}} = \frac {0.632 + 4}{1 + 4} = 0.926 [/latex]

Since [latex] k_A\bar{t}_{ST} = k_A(4\bar{t}_{PF}) = 4(1) [/latex].

(ii) If [latex] f_{A1} [/latex] is now the fractional conversion at the exit of the CSTR as vessel 1, from equation 14.3-21 (Table 14.1, nonsegregated flow, for n= l),

[latex]\frac {M_{A1}}{1 + M_{A1}}[/latex] (14.3-21)

[latex] f_{A1} = \frac { k_A\bar{t}_{ST} }{1 + k_A\bar{t}_{ST}} = \frac{4}{1 + 4 } = 0.8 [/latex]

For the PFR, since

[latex]V_{PF} = F_{Ao} \int_{f_{A1}}^{f_{A2}}{\frac{df_A}{ ( -r_A) }} [/latex]

on integration and rearrangement, we have

[latex] f_{A2} = 1 - (1 - f_{A1}) e^{-k_A\bar{t}_{PF}} = 0.926 [/latex]

The performance results for the two configurations are the same for a first-order, or linear, process, even though the earliness and lateness of mixing differ .

(b)(i) Using equation 17.1-15 for the PFR, to calculate [latex] f_{A1} [/latex] we obtain

[latex]f_A = M_{A2} / (1 + M_{A2})[/latex] (17.1-15)

[latex]f_{A1} = k_Ac_{Ao}\bar{t}_{PF} /(1 + k_Ac_{Ao}\bar{t}_{PF}) = 0.500 [/latex]

From equation (A), corresponding to the treatment in (a)(i), but for n= 2 , [latex] f_{A2} = 0.750 [/latex]

(ii) Using the same procedure as in part (a)(ii) for the CSTR (but with n = 2 and equation 14.3-24) and for the PFR in turn, we obtain

[latex] 1 - \frac {(1 + 4M_{A2})^{1/2} - 1 } {2M_{A2}}[/latex] (14.3-24)

[latex] f_{A1} = 0.610 [/latex] and [latex] f_{A2} = 0.719[/latex]

In this case, the results for [latex] f_{A2} [/latex] are different, and this is typical for nonlinear processes. The lower result is for case (ii) in which earlier mixing occurs; that is, when the CSTR precedes the PFR. Thus, although the kinetics, RTD, and degree of segregation (microfluid) are the same, a different mixing pattern in terms of location of mixing may result in a different performance.

Table 14.1 Comparison of nonsegregated and segregated
flow in a [latex]CSTR^a[/latex]

Order

n

[latex]f_A[/latex]

[latex]Nonsegregated^b[/latex]

[latex]Segregated^c[/latex]

0

[latex] M_{A0} ; M_{A0}\lt 1 [/latex]

[latex] 1;M_{A0} \geq 1 [/latex]

(14.3-22)

[latex]M_{A0}(1 - e^{1/M_{A0}} )[/latex]

(all values of [latex]M_{A0}[/latex]

(14.3-23)

1

[latex] \frac {M_{A1}}{1 + M_{A1}} [/latex]

(14.3-21)

[latex] \frac {M_{A1}}{1 + M_{A1}} [/latex]

(14.3-21)

2

[latex] 1 - \frac {(1 + 4M_{A2})^{1/2} - 1 } {2M_{A2}}[/latex]

(14.3-24)

[latex]\frac {e^{1/M_{A2}}}{M_{A2}}E_1\left(\frac {1}{M_{A2}}\right) ^d[/latex]

(14.3-25)

“Constant-density; A + products; [latex](-r_A) = k_Ac_A^n.[/latex]
[latex]^b[/latex]From equation 14.3-19, with [latex]M_{An}[/latex], defined by equation 4.3-4.
‘From equation 13.52,

[latex]1-f_A=\frac{c_A}{c_{A0}}=\int_{0}^{\infty}\left[\frac{c_A(t)}{c_{A0}}\right]_{BR}E(t)dt[/latex] (13.52)

with 3.4-10 or 14.3-20 and 13.4-2.
[latex]^dE_1[/latex] is an exponential integral defined by [latex]E_1(x) =\int_{x}^{\infty}y^{-1}e^{-y}dy,[/latex]
where y is a dummy variable; the integral must be evaluated numerically (e.g., using E-Z Solve); tabulated values also exist

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