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Unformatted text preview: T4 Chaplcr3 Laplaee'l'ransforms Because this system is linear. multiplying the pulse magnitude. {h} by a factor of four
would yield a maximum concentration of reactant in the second stage of shoot 2.0 {the dif
ference between initial and maximum concentration will be four times as large}. 0n the
other hand. the solutions obtained above strictly apply only for r.. = 0.25 min. Hence. the
effect of a fourfold increase in t... can be predicted only by resolving the model response for
t.» = l min. Qualitatiuely, we know that the maximum value of e: will increase as r..
increases. Because the impulse response model is a reasonably good approximation with
rs = 13.25 min. we expect that .i'mtIll changes in the pulse width will yield an appreedririatel}.r
proportional effect on the maximum concentration change. This argument is based on a
proportional increase in the approximately equivalent impulse input. A quantitative verifi
calion using numerical simulation is left as an exercise. I SUMMARY In this chapter we have considered the application of Laplace transform techniques to solve linear dif~
ferential equations. Although this material ma}r be a review for some readers. an attempt has been
made to concentrate on the important properties of the Laplace transform and its inverse. and to point
out the techniques that make manipulation of transforms easier and less prone to error. The use of Laplace transforms can be extended to obtain solutions for models consisting of simulta—
neous differential equations. However, before addressing such extensions. we introduce the concept of input—output models described by transfer functions. The conversion of differential equation models
into transfer fonction models. covered in the next chapter. represents an important simplification in
the methodology. one that can be exploited extensively in process modeling and control system design. REFERENCES Churchill. R. ‘vﬂ. ﬁgurationrt Merlintwice. 3rd ed . McﬁrawHill. Hanna. 0. "Land 0. C. Sandal], Computational Method: in Chem: N ew 'r'rirlt1 llirll. DigIre, P.R.G.: Ari introduction to Laplace Trertrfnmu and Fourier
Series. Springer Ve1 lag. New Yo: it. 1999. EXERCISES 5.1 Use Eq. 3a] to sheenI that the Laplace transform of
bf . . _ tIJ
ta) e em to: Is s + it —hr '
lb] .9 cos LiJf Is {5 d__ b]: + w: 3.2 A student has lepiace transformed an ordinary differ
ential equation {ODE} and obtained the following
transform: , _ 4
“ti—am The following facts are known:
{i} The original ODE had all zero initial
conditions.
{ill lts only input was sin tar where the radian fre
quency to = {a} What can you say about the original ODE? In
other words, determine what it was to the maxi.
mum extent possible. ml Engineering. PrenticeHall. Englewood Cliffs. NI, 1995. Seliit'l. l. L.. the Laplace Transform Diem} errd Appli'mrton. Springer. New York. 1995' {bl Is your result unique? Ctr are there other possible
forms of the ODE that lead to the same Yls}? {e} Without ﬁnding ylt). what functions of time will he
in the squtionT 3.3 Figure E33 shows a pulse function.
(a) From details in the drawing. emulate the pulse
width,t.,.. h. Slope = — a:
tilt] 0 ft in: Figure 13.3.3 Triangular pulse
function. 3.4 (In; [Tonstruct this function as the sum of simpler time
eiEmema, some perhaps translated in time. whose
transforms can be found directly from Table 3. I. [c] Find Mfr]. [d] What Is the area under the pulse’.’ Derive Laplace transforms of the input signaLﬁ shown
in Figs. tilde: and Elite, by summing component
functions found in Table 3.1. 
0 __L._.__
o 2 +5 timin}
Figure Ell1n fit} at] I, Figure EME
The startup procedure for a batch reactor includes a heating step where the reactor temperature is gradu
ail}r heated to the nominal operating temperature of iii 3.? 3.3 3.9 ExerciSes T5
ET. The desired temperature proﬁle it” is shown in
Fig. Eli. What is Tfs}? Using partial fraction expansion when: required. tind
.tfr} for '1‘” Xi" = r? i airtight];
“1} Km = is} 231:} Fit—(31371].
[c] xii} = (7:: {21
Mi Xe} — —.; :71 J‘ + I ll!“ ‘3’ Xi” = Expand each of the following .rdotnain functions into
partial fractions: ta} no = lﬁ
th} Yrs} _ 1 I men
' :{fﬁ'i'yfi"1]_1[§+'2} {a} For the integrodiffcrcnliai equation I'
e'TdT
a i+3i+2x=2 ﬁnd Jrft}. Note that i: = dxidt. etc.
{b} TWhat is the value ofxllzt} as t H m? For each of the following functions .115]. Iwhat can you
sag].r about If!) [I] E t E #1} without solving for Lit)? In
other words, what are x{l]_] and Jim}? Is xiii} converg
ing or diverging? Is xii} smooth or oscillatory? _ 6 +2]
(3} X“) " 4: + 95‘:— Zt‘ﬂir + 4} tits2 — 3
{b} Xti} = W to Xe} = 13% 3.10 (a) For each of the following cases, detenniae what wartime ol time1 e.g., eortstaat+ e4“, witl appear in yft'].
{Note that you do not have to ﬁnd y[t]!i Which yfi'}
are oscillatorer Which exhibit a constant value of
yft) for large values of e . 2 ...
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