Three stirred-tanks in series are used in a reactor train (see Fig. E3.20). The flow rate into the system of some inert species is maintained constant while tracer tests are conducted. Assuming that mixing in each tank is perfect and the volumes are constant: (a) Derive expressions for the

Question

Three stirred-tanks in series are used in a reactor train (see Fig. E3.20). The flow rate into the system of some inert species is maintained constant while tracer tests are conducted. Assuming that mixing in each tank is perfect and the volumes are constant:

(a) Derive expressions for the concentration of tracer leaving each tank; [latex]c_{i}[/latex] is the concentration of tracer entering the first tank.

(b) If [latex]c_{i}[/latex] has been constant and equal to zero for a long time and an operator suddenly injects a large amount of tracer material in the inlet to tank 1, what will be the form of [latex]c_{3}[/latex] (t) (i.e., what types of time functions will be involved) if

[latex] ext { 1. } V_{1}=V_{2}=V_{3}[/latex]

[latex] ext { 2. } V_{1} eq V_{2} eq V_{3} ext {. }[/latex]

(c) If the amount of tracer injected is unknown, is it possible to back-calculate it from experimental data? How would you do it?

Accepted Answer

a) Given constant volumes, overall balances on the three tanks indicate that the flow rate out of each tank is equal to q

Component balance for tracer over each tank,

[latex]V_{1} \frac{d c_{1}}{d t}=q\left(c_{i}-c_{1} ight)[/latex]

[latex]V_{2} \frac{d c_{2}}{d t}=q\left(c_{1}-c_{2} ight)[/latex]

[latex]V_{3} \frac{d c_{3}}{d t}=q\left(c_{2}-c_{3} ight)[/latex]

b) Taking Laplace transform of above equations and eliminating [latex]C_{1}(s)[/latex] and [latex]C_{2}(s)[/latex] gives

[latex]C_{3}(s)=\frac{\left(\frac{q}{V_{1}} ight)\left(\frac{q}{V_{2}} ight)\left(\frac{q}{V_{3}} ight)}{\left(s+q / V_{1} ight)\left(s+q / V_{2} ight)\left(s+q / V_{3} ight)} C_{i}(s)[/latex]

Since [latex]c_{i}(t)=\delta(t), \quad C_{i}(s)=1[/latex]

[latex]V_{1}=V_{2}=V_{3}=V[/latex]

[latex]C_{3}(s)=\frac{(q / V)^{3}}{(s+q / V)^{3}}=\frac{\alpha_{1}}{(s+q / V)}+\frac{\alpha_{2}}{(s+q / V)^{2}}+\frac{\alpha_{3}}{(s+q / V)^{3}}[/latex]

[latex]c_{3}(t)=\alpha_{1} e^{-(q / V) t}+\alpha_{2} t e^{-(q / V) t}+\alpha_{3} t^{2} e^{-(q / V) t}[/latex]

2. [latex]V_{1} eq V_{2} eq V_{3} eq V_{1}[/latex]

[latex]c_{3}(t)=\alpha_{4} e^{-\left(q / V_{1} ight) t}+\alpha_{5} e^{-\left(q / V_{2} ight) t}+\alpha_{6} e^{-\left(q / V_{3} ight) t}[/latex]

(c) Yes, amount of tracer can be calculated by measuring [latex]c_{3}(t)[/latex],

amount of tracer [latex]=\int_{0}^{\infty} q c_{3}(t) d t[/latex], which can be evaluated numerically

Question 3.20: Three stirred-tanks in series are used in a reactor train (s...

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