Consider a reactor system made up to two vessels in series: a PFR of volume VPF and a CSTR of volume VST, as shown in Figure 17.5. In Figure 17S(a), the PFR is followed by the CSTR, and in Figure 17.5(b), the sequence is reversed. Derive E(θ) for case (a) and for case (b). Assume constant-density

Question

Consider a reactor system made up to two vessels in series: a PFR of volume [latex] V_{PF} [/latex] and a CSTR of volume [latex] V_{ST} [/latex], as shown in Figure 17.5. In Figure 17.5(a), the PFR is followed by the CSTR, and in Figure 17.5(b), the sequence is reversed. Derive E(θ) for case (a) and for case (b). Assume constant-density isothermal behavior.

Accepted Answer

(a) The material-balance method for obtaining E(t) or E(θ), involving introduction of a “virtual” tracer as a pulse, is illustrated in Section 13.4.1.1 and in Example 17-4. We use it here to obtain E(t), and hence E(θ).
Consider the entry of a small amount of fluid as “tracer” into the PFR at time t = 0. No tracer leaves the PFR until [latex]t = V_{PF}/q_o = \bar{t}_{PF}[/latex] , the mean residence time in the PFR, and hence no tracer leaves the two-vessel system, at the exit from the CSTR, during the period [latex]0 \leq t \lt \bar{t}_{PF}[/latex]. As a result,

[latex]E(t) = 0 ; 0 \leq t \lt t_{PF} [/latex] (17.2-7)

All tracer leaves the PFR at [latex] \bar{t}_{PF}[/latex] , and enters the CSTR at this time as a pulse. Once the tracer leaves the PFR, its concentration in the CSTR immediately increases to [latex]c_{Mo} \equiv c_{MST} [/latex] at [latex] \bar{t}_{PF}[/latex] . We then perform a material balance for tracer around the CSTR, for [latex] t \gt \bar{t}_{PF}[/latex]. The input rate is 0, the output rate is [latex] qc_{MST} [/latex] , and the accumulation rate is [latex] V_{ST}dc_{MST} /dt [/latex] . Thus

[latex]0 - qc_{MST} = V_{ST}dc_{MST}/dt[/latex]

or

[latex]\int_{c_{Mo}}^{c_{MST}}{\frac{dc_{MST}}{c_{MST}}} = - \int_{ \bar{t}_{PF}}^{t}{\frac{q}{V_{ST}}dt}[/latex]

which integrates to

[latex]c_{MST}= c_{Mo}e^{-(q/V_{ST})(t-\bar{t}_{PF})}[/latex]

Forming the fraction of tracer in the outlet stream from vessel 2 as in Example 17-4, and identifying this with E(t)dt, we obtain eventually

[latex]E(t) = \frac{1}{\bar{t}_S}\left(-\exp \frac{\bar{t}_{PF}}{t_{ST}} ight) ; t \geq t_{PF}[/latex] (17.2-8)

where [latex] \bar{t}_{ST} = V_{ST}/q_o [/latex], the mean residence time in the CSTR. This result can be visualized as the same as that in equation 13.4-2 for BMF in a single vessel with a time delay of [latex] \bar{t}_{PF}[/latex].

[latex]E(t) = (1/\bar{t})e^{-t/\bar{t}} (BMF)[/latex] (13.4-2)

The results in equations 17.2-7 and -8 together constitute E(t) for the two-vessel configuration in Figure 17.5(a). To convert E(t) to E(θ), where [latex] heta = t/\bar{t}[/latex], and [latex]\bar{t} = \bar{t}_{PF} + \bar{t}_{ST}[/latex], we use [latex] E( heta) = \bar{t}E(t) [/latex], equation 13.3-7. The results are

[latex]E( heta ) = 0 ; 0 \leq heta \lt \bar{t}_{PF}/\bar{t} = V_{PF}/V[/latex] (17.2-9)

where [latex] V = V_{PF} +V_{ST} [/latex] ; and

[latex]E( heta ) = \frac{\bar{t}}{\bar{t}_{ST}}\exp\left(\frac{\bar{t}_{PF} - \bar{t} heta }{\bar{t}_{ST}} ight) ; heta \geq \bar{t}_{PF}/\bar{t}[/latex]

[latex]= \frac{V}{V_{ST}}\exp\left(\frac{V_{PF} - V heta }{V_{ST}} ight) ; heta \geq V_{PF}/V[/latex] (17.2-10)

(b) The method used in part (a) applied to the configuration in Figure 17.5(b) leads to the same results as in equations 17.2-7 to -10 for E(t) and E(θ).
Example 17-5 shows that differences in earliness or lateness of mixing need not result in differences in RTD. Part (a) involves relatively late mixing and part (b) early mixing, but both arrangements lead to the same RTD. The next example, using the same configuration, explores whether earliness or lateness of mixing affects reactor performance, through the fractional conversion.

Question 17.5: Consider a reactor system made up to two vessels in series: ...

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