Homework 03b - Textbook Scan Ch 3

# Homework 03b - Textbook Scan Ch 3 - T4 Chaplcr3...

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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'm-tIl-l 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. ﬁguration-rt Merlin-twice. 3rd ed . Mcﬁraw-Hill. Hanna. 0. "Land 0. C. Sandal], Computational Method: in Chem:- N ew 'r'rirlt1 llirll. Dig-Ire, 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. Prentice-Hall. 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 Ell-1n fit} at] I, Figure EME- The start-up 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 .r-dotnain functions into partial fractions: ta} no = lﬁ th} Yrs} _ 1 I men ' :{fﬁ'i'yfi"1]_1[§+'2} {a} For the integro-diffcrcnliai 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} Xt-i} = 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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Homework 03b - Textbook Scan Ch 3 - T4 Chaplcr3...

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