A process consists of five perfectly stirred tanks in series. The volume in each tank is 30 L, and the volumetric flow rate through the system is 5 L/min. At some particular time, the inlet concentration of a nonreacting species is changed from 0.60 to 0.45 (mass fraction) and held there. (a) Write

Question

A process consists of five perfectly stirred tanks in series. The volume in each tank is $30 L$ , and the volumetric flow rate through the system is $5 L / min$ . At some particular time, the inlet concentration of a nonreacting species is changed from $0.60$ to $0.45$ (mass fraction) and held there.

(a) Write an expression for $c_{5}$ (the concentration leaving the fifth tank) as a function of time.

(b) Simulate and plot $c_{1}, c_{2}, \ldots, c_{5} .$ Compare $c_{5}$ at $t=30 min$ with the corresponding value for the expression in part (a).

(a) Write an expression for [latex]c_{5}[/latex] (the concentration leaving the fifth tank) as a function of time.

(b) Simulate and plot [latex]c_{1}, c_{2}, \ldots, c_{5} .[/latex] Compare [latex]c_{5}[/latex] at [latex]t=30 min[/latex] with the corresponding value for the expression in part (a).

Accepted Answer

a) The transfer function for each tank is

[latex]\frac{C_{i}^{\prime}(s)}{C_{i-1}^{\prime}(s)}=\frac{1}{\left(\frac{V}{q} ight) s+1} \quad, i=1,2, \ldots, 5[/latex]

where

[latex]i[/latex] represents the [latex]i ^{ ext {th }}[/latex] tank.

[latex]c_{o}[/latex] is the inlet concentration to tank [latex]1 .[/latex]

[latex]V[/latex] is the volume of each tank.

[latex]q[/latex] is the volumetric flow rate.

[latex]\frac{C_{5}^{\prime}(s)}{C_{0}^{\prime}(s)}=\prod\limits_{i=1}^{5}{\left[\frac{C_{i}^{\prime}(s)}{C_{i-1}^{\prime}(s)} ight]}=\left(\frac{1}{6 s+1} ight)^{5}[/latex]

Then, by partial fraction expansion,

[latex]c_{5}(t)=0.60-0.15\left[1-e^{-t / 6}\left\{1+\frac{t}{6}+\frac{1}{2 !}\left(\frac{t}{6} ight)^{2}+\frac{1}{3 !}\left(\frac{t}{6} ight)^{3}+\frac{1}{4 !}\left(\frac{t}{6} ight)^{4} ight\} ight][/latex]

b) Using Simulink,

The value of the expression for [latex]c_{5}(t)[/latex] verifies the simulation results above:

[latex]c_{5}(30)=0.60-0.15\left[1-e^{-5}\left\{1+5+\frac{5^{2}}{2 !}+\frac{5^{3}}{3 !}+\frac{5^{4}}{4 !} ight\} ight]=0.5161[/latex]

Question 6.10: A process consists of five perfectly stirred tanks in series...

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