Costraed Cubc Sple Iterpolato or Chemcal Egeerg Applcatos b CJC Kruger Summar Cubc sple terpolato s a useul techque to terpolate betwee kow data pots due to ts stable ad smooth characterstcs. Uortuatel t does ot prevet overshoot at termedate pots, whch s essetal or ma chemcal egeerg applcatos. Ths artcle presets a ew terpolato method that combes the smooth curve characterstcs o sple terpolato, wth the o-overshootg behavour o lear terpolato. Itroducto Iterpolato s used to estmate the value o a ucto betwee kow data pots wthout kowg the actual ucto. Iterpolato methods ca be dvded to two ma categores [,]: Global terpolato. These methods rel o a costructg sgle equato that ts all the data pots. Ths equato s usuall a hgh degree polomal equato. Although these methods result smooth curves, the are usuall ot well suted or egeerg applcatos, as the are proe to severe oscllato ad overshoot at termedate pots. Pecewse terpolato. These methods rel o costructg a polomal o low degree betwee each par o kow data pots. I a rst degree polomal s used, t s called lear terpolato. For secod ad thrd degree polomals, t s called quadratc ad cubc sples respectvel. The hgher the degree o the sple, the smoother the curve. Sples o degree m, wll have cotuous dervatves up to degree m- at the data pots. Lear terpolato result straght le betwee each par o pots ad all dervatves are dscotuous at the data pots. As t ever overshoots or oscllates, t s requetl used chemcal egeerg despte the act that the curves are ot smooth. To obta a smoother curve, cubc sples are requetl recommeded. The are geerall well behaved ad cotuous up to the secod order dervatve at the data pots. Eve though cubc sples are less proe to oscllato or overshoot tha global polomal equatos, the do ot prevet t. Thus, the use o cubc sples chemcal egeerg s lmted to applcatos where oscllato ad overshoot are acceptable or desrable. Tradtoal Cubc Sples Cosder a collecto o kow pots,,,,... -, -,,,,,...,. To terpolate betwee these data pots usg tradtoal cubc sples, a thrd degree polomal s costructed betwee each pot. The equato to the let o pot, s dcated as wth a value o at pot. Smlarl, the equato to the rght o pot, s dcated as wth a value o at pot. Tradtoall the cubc sple ucto,, s costructed based o the ollowg crtera: Curves are thrd order polomals, 3 a b c d - Curves pass through all the kow pots, -
The slope, or rst order dervatve, s the same or both uctos o ether sde o a pot, -3 The secod order dervatve s the same or both uctos o ether sde o a pot, -4 Ths results a matr o - equatos ad ukows. The two remag equatos are based o the border codtos or the startg pot,, ad ed pot,. Hstorcall oe o the ollowg border codtos have bee used [,,3]: Natural sples. The secod order dervatve o the sples at the ed pots are zero. -5a Parabolc ruout sples. The secod order dervatve o the sples at the ed pots are the same as at the adjacet pots. The result s that the curve becomes a parabolc curve at the ed pots. -5b Cubc ruout sples. The curve degrades to a sgle cubc curve over the last two tervals b settg the secod order dervatve o the sples at the ed pots to: -5c Clamped sple. The rst order dervatve o the sples at the ed pots are set to kow values. -5d I tradtoal cubc sples equatos to 5 are combed ad the b trdagoal matr s solved to eld the cubc sple equatos or each segmet [,3]. As both the rst ad secod order dervatve or coectg uctos are the same at ever pot, the result s a ver smooth curve. Eve though tradtoal cubc sples are well behaved or ma applcatos, t does ot prevet overshoot at termedate pots. Ths s llustrated Fgures ad, where a atural cubc sple s tted to hpothetcal ad somewhat uusual dstllato ad pump curves. Clearl ths behavour s uacceptable or chemcal egeerg applcatos, ad the egeer has lttle choce but to revert back to lear terpolato. Proposed Costraed Cubc Sples The prcple behd the proposed costraed cubc sple s to prevet overshootg b sacrcg smoothess. Ths s acheved b elmatg the requremet or equal secod order dervatves at ever pot equato 4 ad replacg t wth speced rst order dervatves. Thus, smlar to tradtoal cubc sples, the proposed costraed cubc sples are costructed accordg to equatos, 3 ad 5a. Equato 4 s replaced b, A speced rst order dervatve, or slope, at ever pot, -6
Fgure Dstllato curve 5 5 5 3 35 3 4 5 6 7 8 9 Volume % Temperature deg C Data Natural Sple Costraed Sple Fgure Pump Curve 5 5 5 3 35 3 4 5 6 7 8 9 Flow m3/h Head m Data Natural Sple Costraed Sple The ke step becomes the calculato o the slope at each pot. Itutvel we kow the slope wll be betwee the slopes o the adjacet straght les, ad should approach zero the slope o ether le approaches zero. A relatvel smple equato that works well ad satses these requremets, s: slope chages sg at pot - 7a Equato 7a s ol vald or termedate pots. The slope at the ed pots s based o rewrtg equato 5a to eld, 3-7b 3-7c As the slope at each pot s kow, t s o loger ecessar to solve a sstem o equatos. Each sple ucto, as gve b equato, ca be calculated based o the two adjacet pots o each sde. Ths s summarzed equatos 8 to 3 below. [ ] 6-8 [ ] 6-9 6 d - c - 3 3 d c b -
3 b c d a - 3 The behavour o the proposed costraed cubc sple s show Fgures ad. I geeral t ts chemcal egeerg eeds well cases where oscllato or overshoot caot be tolerated. A Ecel Vsual Basc or Applcatos VBA eample o ths techque ca be obtaed rom www.kor.co.uk. Ths terpolato method wll also be used the et release o Kor Hdraulcs, whch s a leble ad user-redl ppg etwork solver avalable rom the same webste. Coclusos A moded cubc sple terpolato method has bee developed or chemcal egeerg applcato. The ma beets o the proposed costraed cubc sple are: It s a relatvel smooth curve; It ever overshoots termedate values; Iterpolated values ca be calculated drectl wthout solvg a sstem o equatos; The actual parameters a, b, c ad d or each o the cubc sple equatos ca stll be calculated. Ths permts aaltcal tegrato o the data. Eample The hpothetcal dstllato curve Fgure s represeted b the ollowg data pots: Pot 3 4 5 6,,3,3 3,5 5,5 7,7 9,,3 Calculate the cubc equatos or the rst two segmets. Frst segmet,, or [ ' / / / -7a /3 /5 3 /3 3.88 ' 3/* / - ' / -7b 3/*3 3/.88/ 4.99 " -*' * ' / 6* / - 8 -*.88 *4.99/ 6*3 3/ " **' ' / - 6* / - 9 **.88 4.99/ 6*3 3/ -.4545 d /6 * " - " / - /6 * -.4545 / -.49 c / * *" *" / - / * * *.88/ b c * d * 3 3 / - 3 3 *.49* 3 3 / 4.9 a b * c * d * 3-3 3 3 4.9 -.49 3 or Secod segmet,, or [ ' / 3 / 3 / /5 3/5 5 3 /5 3-7a
Reereces ' / / / -7a.88 " -*' * ' / 6* / - 8 -* *.88/3 6*5 3/3 -.63636 " **' ' / - 6* / - 9 **.88/3 6*5 3/3 -.88 d /6 * " - " / - /6 * -.88.63636/3 -.4545 c / * *" *" / - / * -3*.63636 *.88/3 -.88 b c * d * 3 3 / - 5 3.88*3.4545*3 3 3 /3.388 a b * c * d * 3-3 3.388*.88*.4545* 3 9.9 9.9.388 -.88 -.4545 3 or [. Press, W. H., Teukolsk, S. A., Vetterlg, W. T. ad Flaer, B. P., Numercal Recpes Fortra, The Art o Scetc Computg, Secod Edto, Cambrdge Uverst Press, Cambrdge, Reprted 995.. Herc, P., Essetal o Numercal Aalss, Joh Wle & Sos, New York, 98. 3. McKle, S. ad Leve, M., Cubc Sple Iterpolato, http://ole.redwoods.cc.ca.us/struct/darold/laproj/fall98/skmeg/proj.pdf.