Calculate the ratio of the volumes of a CSTR and a PFR (VST/PF) required to achieve,for a given feed rate in each reactor, a fractional conversion (fA) of (i) 0.5 and (ii) 0.99 for the reactant A, if the liquid-phase reaction A → products is (a) first-order, and (b) second-order with respect to A.

Question

Calculate the ratio of the volumes of a CSTR and a PFR ([latex] V_{ST} /V_{PF} [/latex] ) required to achieve,for a given feed rate in each reactor, a fractional conversion ([latex] f_A [/latex] ) of (i) 0.5 and (ii) 0.99 for the reactant A, if the liquid-phase reaction A → products is (a) first-order, and (b) second-order with respect to A. What conclusions can be drawn? Assume the PFR operates isothermally at the same T as that in the CSTR .

Accepted Answer

This example can be solved using the E-Z Solve software (file ex17-2.msp) or as follows . The volumes for a CSTR and a PFR are determined from the material-balance equations :

[latex]V_{ST} = F_{Ao}f_A /( -r_A) [/latex] (14.3-5)

[latex]V_{PF} = F_{Ao} \int_{0}^{f_A}{df_A /( -r_A)} [/latex] (15.2-2)

Thus ,

[latex]\frac {V_{ST}}{V_{PF}} = \frac {f_A /( -r_A)}{\int_{0}^{f_A}{df_A /( -r_A)}} [/latex] (17.1-9)

n accordance with the discussion in Section 17.1.1, this is the same result as for [latex] V_{ST} / V_{BR} [/latex] given by equation 17.1-4 with a = 0 (i.e., with no down-time).

[latex] \frac{V_{BR}}{V_{ST} } = \frac{(1 + a)\int_{0}^{f_A}{df_A /(-r_A)}}{f_A/( -r_A)} (constant ho T) [/latex] (17.1-4)

(a) For a first-order reaction, [latex] ( -r_A) = k_Ac_A [/latex] , occurring at the same T in each reactor, equation 17.1-9 becomes

[latex]\frac {V_{ST}}{V_{PF}} - \frac {f_A /k_Ac_{Ao}(1 - f_A)}{\int_{0}^{f_A}{[df_A /k_Ac_{Ao}(1 - f_A)]}} [/latex]

[latex] = \frac {f_A}{(1 - f_A) \ln [1/ (1 - f_A)]} (n = 1) [/latex] (17.1-10)

(b) Similarly, for a second-order reaction, equation 17.1-9 is

[latex]\frac {V_{ST}}{V_{PF}} - \frac {f_A /k_Ac_{Ao}(1 - f_A)^2}{\int_{0}^{f_A}{[df_A /k_Ac_{Ao}^2(1 - f_A)^2]}} [/latex]

Table 17.1 Values of [latex] V_{ST} / V_{PF} [/latex] for Example 17-2
[latex]f_A[/latex]	n
[latex]f_A[/latex]	1	2
0	1	1
0.5	1.4	2
0.99	21.5	100
1.0	∞	∞

which, on integration and simplification, becomes

[latex] \frac {V_{ST}}{V_{PF}} = \frac {1}{1 - f_A} (n = 2) [/latex] (17.1-11)

From these equations, values of [latex] V_{ST} / V_{PF} [/latex] for [latex]f_A = (i) 0.5 and (ii) 0.99 [/latex] are given in Table 17.1, together with the limiting values for [latex]f_A [/latex] = 0 , 1 .
The conclusions illustrated in Table 17.1 are: (1) for a given order, 12, the ratio increases as [latex]f_A [/latex] increases, and (2) for a given [latex]f_A [/latex] , the ratio increases as order increases. In any case, for normal kinetics, [latex]V_{ST} \gt V_{PF} [/latex], since the CSTR operates entirely at the lowest value of [latex]c_A [/latex], the exit value. (Levenspiel, 1972, p. 332, gives a more detailed graphical comparison for five values of n. This can also be obtained from the E-Z Solve software.)

Figure 17.1, for both normal and abnormal kinetics, can also be interpreted as a comparison between a CSTR and a PFR, with the PFR replacing the BR. Again, area A + B is a measure of [latex] au _{ST}/c_{Ao} ( or V_{ST}/ F_{Ao}) [/latex] . Area A, for normal kinetics, is a measure of [latex] au _{PF}/c_{Ao} or t_{PF}/c_{Ao} ( or V_{PF}/ F_{Ao}) [/latex], where [latex] t_{PF} [/latex] is the residence time of fluid in the PFR, corresponding to reaction time t in a BR. Area A + B + C corresponds to [latex] t_{PF}/c_{Ao} [/latex] for abnormal kinetics.

Question 17.2: Calculate the ratio of the volumes of a CSTR and a PFR (VST/...

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