A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the water flow and adds caustic solution at the same volumetric flow rate q, but with concentration ci.

Question

A continuous, stirred-tank reactor is initially full of water with the inlet and exit volumetric flow rates of water having the same numerical value. At a particular time, an operator shuts off the water flow and adds caustic solution at the same volumetric flow rate q, but with concentration $\bar{c}_{i}$ . If the liquid volume V is constant, the dynamic model for this process is

V \frac{d c}{d t}+q c=q \bar{c}_{i} \quad c(0)=0

where c(t) is the exit concentration. Calculate c(t) and plot it as a function of time.

Data: $V=2 m ^{3} ; \quad q=0.4 m ^{3} / min ; \quad \bar{c}_{i}=50 kg / m ^{3}$

[latex]V \frac{d c}{d t}+q c=q \bar{c}_{i} \quad c(0)=0[/latex]

where c(t) is the exit concentration. Calculate c(t) and plot it as a function of time.

Data: [latex]V=2 m ^{3} ; \quad q=0.4 m ^{3} / min ; \quad \bar{c}_{i}=50 kg / m ^{3}[/latex]

Accepted Answer

[latex]V \frac{d c}{d t}+q c=q c_{i}[/latex]

Since V and q are constant, taking Laplace transforms give

[latex]s V C( s )+q C( s )=q C_{i}( s )[/latex]

Note that c(t = 0) = 0

Also, [latex]\begin{array}{lll}c_{i}(t) & =0 & , & t \leq 0 \c_{i}(t) & =\bar{c}_{i} & , & t>0\end{array}[/latex]

Taking Laplace transform of the input function, a constant, gives

[latex]C_{i}(s)=\frac{\bar{c}_{i}}{s}[/latex]

so that

[latex]s V C(s)+q C(s)=q \frac{\bar{c}_{i}}{s} \quad \text { or } \quad C(s)=\frac{q \bar{c}_{i}}{(s V+q) s}[/latex]

Dividing numerator and denominator by q

[latex]C(s)=\frac{\bar{c}_{i}}{\left(\frac{V}{q} s+1\right) s}[/latex]

Use Transform pair #3 in Table 3.1 to invert [latex](\tau=V / q)[/latex]

Table 3.1 Laplace Transforms for Various Time-Domain Functions∗
f(t)	F(s)
1. δ(t) (unit impulse)	1
2. S(t) (unit step)	[latex] \frac{1}{s} [/latex]
3. t (ramp)	[latex] \frac{1}{s^{2}} [/latex]
4. [latex] t^{n-1} [/latex]	[latex] \frac{ (n−1)! }{s^{n}} [/latex]
5. [latex] e^{−bt} [/latex]	[latex] \frac{ 1 }{s+b} [/latex]
6. [latex] \frac {1}{τ}e^{-t/τ} [/latex]	[latex]\frac{1}{\tau s+1} [/latex]
7. [latex] \frac {t^{n-1}e^{-bt}}{(n-1)!} (n>0) [/latex]	[latex] \frac{1}{(s+b)^{n}} [/latex]
8. [latex] \frac {1}{τ^{n}(n−1)!}t^{n-1}e^{-t/τ}[/latex]	[latex] \frac{1}{(\tau s+1)^{n}} [/latex]
9. [latex] \frac {1}{b_{1}-b_{2}}(e^{-b_{2}t}-e^{-b_{1}t}) [/latex]	[latex] \frac{1}{(s+b_{1})(s+b_{2})} [/latex]
10. [latex] \frac {1}{τ_{1}-τ_{2}}(e^{-t/τ_{1}}-e^{-t/τ_{2}}) [/latex]	[latex]\frac{1}{(\tau _{1}s+1)(\tau _{2}s+1)} [/latex]
11. [latex] \frac{b_{3}-b_{1}}{b_{2}-b_{1}}e^{-b_{1}t} +\frac{b_{3}-b_{2}}{b_{1}-b_{2}}e^{-b_{2}t} [/latex]	[latex]\frac{s+b_{3}}{(s+b_{1})(s+b_{2})} [/latex]
12. [latex] \frac{1}{τ_{1}}\frac{τ_{1}-τ_{3}}{τ_{1}-τ_{2}}e^{-t/τ_{1}} +\frac{1}{τ_{2}}\frac{τ_{2}-τ_{3}}{τ_{2}-τ_{1}}e^{-t/τ_{2}} [/latex]	[latex] \frac{\tau _{3}s+1}{(\tau _{1}s+1)(\tau _{2}s+1)} [/latex]
13. [latex] 1 - e^{-t/τ} [/latex]	[latex] \frac{1}{s(\tau s+1)} [/latex]
14. sin ωt	[latex] \frac{w}{s^{2}+w^{2}} [/latex]
15. cos ωt	[latex] \frac{s}{s^{2}+w^{2}} [/latex]
16. [latex] \sin(ωt + \Phi )[/latex]	[latex] \frac{ ω \cos ϕ+s sin ϕ }{s^{2}+w^{2}} [/latex]
[latex]\left . \begin{matrix} 17. e^{-bt}\sin ωt \[0.5cm] 18. e^{-bt}\sin ωt \end{matrix} \right \}b, ω real[/latex]	[latex]\frac{w}{(s+b)^{2}+w^{2}} [/latex]
	[latex] \frac{s+b}{(s+b)^{2}+w^{2}} [/latex]
19. [latex] \frac{1}{τ\sqrt{1-\xi ^{2}} }e^{−ζt/\tau }\sin(\sqrt{1-\xi ^{2}}t/\tau ) [/latex] [latex] (0≤\|ζ\|≤1) [/latex]	[latex] \frac{1}{\tau ^{2}s^{2}+2\zeta \tau s+1} [/latex]
20. [latex] 1+\frac{1}{\tau _{2}-\tau _{1}} (\tau _{1}e^{-t/\tau _{1}}-\tau _{2}e^{-t/\tau _{2}}) [/latex] [latex](\tau_{1} ≠\tau_{2})[/latex]	[latex] \frac{1}{s(\tau _{1}s+1)(\tau _{2}s+1)} [/latex]
21. [latex] 1-\frac{1}{\sqrt{1-\zeta ^{2}} }e^{-\zeta t/\tau }\sin[\sqrt{1-\zeta ^{2}}t/\tau +\psi ][/latex] [latex] ψ=\tan^{-1}\frac{\sqrt{1-\zeta ^{2}} }{\zeta }, (0≤\|ζ\| < 1) [/latex]	[latex]\frac{1}{s(\tau ^{2}s^{2}+2\zeta \tau s+1)} [/latex]
22. [latex] 1-e^{-ζt∕τ}[\cos (\sqrt{1-\zeta ^{2}}t/\tau )+\frac{\zeta }{\sqrt{1-\zeta ^{2}}} \sin (\sqrt{1-\zeta ^{2}}t/\tau )] [/latex] [latex] (0≤\|ζ\|< 1) [/latex]	[latex] \frac{1}{s(\tau ^{2}s^{2}+2\zeta \tau s+1)} [/latex]
23. [latex] 1+\frac{\tau _{3}-\tau _{1}}{\tau _{1}-\tau _{2}}e^{-t/\tau _{1}}+\frac{\tau _{3}-\tau _{2}}{\tau _{2}-\tau _{1}}e^{-t/\tau _{2}} [/latex] [latex](\tau_{1} ≠\tau_{2})[/latex]	[latex] \frac{\tau _{3}s+1}{s(\tau _{1}s+1)(\tau _{2}s+1)} [/latex]
24. [latex] \frac{df}{dt} [/latex]	sF(s) - f(0)
25. [latex] \frac{d^{n}f}{dt^{n}} [/latex]	[latex] s^{n}F(s)-s^{n-1}f(0)-s^{n-2}f^{(1)}(0)-...-sf^{(n-2)}(0)-f^{(n-1)}(0) [/latex]
26. f(t−θ )S(t−θ )	[latex] e^{-\theta s}F(s) [/latex]
27. [latex] \int_{0}^{t}{f(\tau )d\tau } [/latex]	[latex] \frac{1}{s}F(s) [/latex]
∗Note that f(t) and F(s) are defined for t ≥ 0 only.

[latex]c(t)=\bar{c}_{i}\left(1-e^{-\frac{q}{V} t}\right)[/latex]

Using MATLAB, the concentration response is shown in Fig. S3.18. (Consider [latex]\left.V=2 m ^{3}, c_{i}=50 Kg / m ^{3} \text { and } q=0.4 m ^{3} / min \right)[/latex]

Question 3.18: A continuous, stirred-tank reactor is initially full of wate...

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