A composition analyzer is used to measure the concentration of a pollutant in a wastewater stream. The relationship between the measured composition Cm and the actual compositionCis given by the following transfer function (in deviation variable form): C′m (s)/C′(s) = e^−θs/τs + 1 where θ = 2 min

Question

A composition analyzer is used to measure the concentration of a pollutant in a wastewater stream. The relationship between the measured composition [latex]C_{m}[/latex] and the actual composition [latex]C[/latex] is given by the following transfer function (in deviation variable form):

[latex]\frac{C_{m}^{\prime}(s)}{C^{\prime}(s)}=\frac{e^{- heta s}}{ au s+1}[/latex]

where [latex] heta=2[/latex] min and [latex] au=4 min[/latex]. The nominal value of the pollutant is [latex]\bar{C}=5 ppm[/latex]. A warning light on the analyzer turns on whenever the measured concentration exceeds [latex]25 ppm[/latex].

Suppose that at time [latex]t=0[/latex], the actual concentration begins to drift higher, [latex]C(t)=5+2 t[/latex], where [latex]C[/latex] has units of ppm and [latex]t[/latex] has units of minutes. At what time will the warning light turn on?

Accepted Answer

First, consider then the undelayed response (with θ=0); then apply the Real Translation Theorem to find the desired delayed response.

Denote the undelayed response (for [latex] heta=0[/latex] ) by [latex] ilde{ c }_{m}^{\prime}(t)[/latex]. Then,

[latex]c _{m}^{\prime}(t)= ilde{ c }_{m}^{\prime}(t- heta)[/latex] (1)

Taking the Laplace transforms give,

[latex]C _{m}^{\prime}(s)= e ^{- heta s} ilde{ C }_{m}^{\prime}(s)[/latex] (2)

The transfer functions for the delayed and undelayed systems are:

[latex]\frac{ C _{m}^{\prime}(s)}{C^{\prime}(s)}=\frac{ e ^{- heta s}}{ au s+1}[/latex] (3)

[latex]\frac{ ilde{ C }_{m}^{\prime}(s)}{C^{\prime}(s)}=\frac{1}{ au s+1}[/latex] (4)

For the ramp input, [latex]c ^{\prime}(t)=2 t[/latex]; from Table 3.1 :

[latex]C^{\prime}(s)=\frac{2}{s^{2}}[/latex] (5)

Substituting (5) into (4) and rearranging gives:

[latex] ilde{ C }_{m}^{\prime}(s)=\left(\frac{1}{ au s+1} ight)\left(\frac{2}{s^{2}} ight)[/latex] (6)

The corresponding response to the ramp input is given by Eq. [latex]5-19[/latex] with [latex]K=1, a=2[/latex], and [latex] au=10[/latex] :

[latex]Y(s)=\frac{Ka au ^2}{ au s+1}-\frac{Ka au }{s} + \frac{Ka}{s^2}[/latex] (5-19)

[latex] ilde{ c }_{m}^{\prime}(t)=20\left( e ^{-t / 10}-1 ight)+2 t[/latex] (7)

Let [latex] ilde{t}_{a}[/latex] denote the time that the alarm goes on for the undelayed system; thus, the alarm lights up when [latex] ilde{c}_{m}\left( ilde{t}_{a} ight)=25 min[/latex]; i.e., when [latex] ilde{ c }_{m}^{\prime}\left( ilde{t}_{a} ight)=25-5=20[/latex] min. Substituting into (7) and solving for [latex] ilde{t}_{a}[/latex] by trial and error gives

[latex] ilde{t}_{a}=9.24 min[/latex]

Let [latex]t_{a}[/latex] denote the time that the alarm goes off for the system with time delay. It follows from the definition of a time delay that,

[latex]t_{a}= ilde{t}_{a}+ heta=9.24+2.00=11.24 min[/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}{ au 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}{( au 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}{( au _{1}s+1)( au _{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{ au _{3}s+1}{( au _{1}s+1)( au _{2}s+1)} [/latex]
13. [latex] 1 - e^{-t/τ} [/latex]	[latex] \frac{1}{s( au 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 \ 18. e^{-bt}\sin ωt \end{matrix} ight \}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/ au }sin(\sqrt{1-\xi ^{2}}t/ au ) [/latex] [latex] (0≤\|ζ\|≤1) [/latex]	[latex] \frac{1}{ au ^{2}s^{2}+2\zeta au s+1} [/latex]
20. [latex] 1+\frac{1}{ au _{2}- au _{1}} ( au _{1}e^{-t/ au _{1}}- au _{2}e^{-t/ au _{2}}) [/latex] [latex]( au_{1} ≠ au_{2})[/latex]	[latex] \frac{1}{s( au _{1}s+1)( au _{2}s+1)} [/latex]
21. [latex] 1-\frac{1}{\sqrt{1-\zeta ^{2}} }e^{-\zeta t/ au }sin[\sqrt{1-\zeta ^{2}}t/ au +\psi ] [/latex] [latex]ψ=tan^{-1}\frac{\sqrt{1-\zeta ^{2}} }{\zeta }, (0≤\|ζ\| < 1) [/latex]	[latex]\frac{1}{s( au ^{2}s^{2}+2\zeta au s+1)} [/latex]
22. [latex] 1-e^{-ζt∕τ}[\cos (\sqrt{1-\zeta ^{2}}t/ au )+\frac{\zeta }{\sqrt{1-\zeta ^{2}}} \sin (\sqrt{1-\zeta ^{2}}t/ au )] [/latex] [latex] (0≤\|ζ\|< 1) [/latex]	[latex] \frac{1}{s( au ^{2}s^{2}+2\zeta au s+1)} [/latex]
23. [latex] 1+\frac{ au _{3}- au _{1}}{ au _{1}- au _{2}}e^{-t/ au _{1}}+\frac{ au _{3}- au _{2}}{ au _{2}- au _{1}}e^{-t/ au _{2}} [/latex] [latex]( au_{1} ≠ au_{2})[/latex]	[latex] \frac{ au _{3}s+1}{s( au _{1}s+1)( au _{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^{- heta s}F(s) [/latex]
27. [latex] \int_{0}^{t}{f( au )d au } [/latex]	[latex] \frac{1}{s}F(s) [/latex]
∗Note that f(t) and F(s) are defined for t ≥ 0 only.

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