PROJECTS WITH APPLICATIONS OF DIFFERENTIAL EQUATIONS AND MATLAB

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1 PROJECTS WITH APPLICATIONS OF DIFFERENTIAL EQUATIONS AND MATLAB David Szrley Francis Marion Universiy Deparmen of Mahemaics PO Box 1547 Florence, SC 95 I. INTRODUCTION Differenial eqaions (DEs) play a prominen role in oday s indsrial seing. Many physical laws describe he rae of change of a qaniy wih respec o oher qaniies. Since rae of change is simply anoher phrase for derivaive, hese physical laws may be wrien as DEs. For example, Newon s Second Law of Moion saes ha he rae of change of momenm of an obec is eqal o he sm of he imposed forces. Normally his is wrien as F ma, where F represens he oal imposed force. However, if we le x denoe he posiion of he obec, hen his may be rewrien as F mx. Hence, Newon s Second Law of Moion is a second-order ordinary differenial eqaion. There are many applicaions of DEs. Growh of microorganisms and Newon s Law of Cooling are examples of ordinary DEs (ODEs), while conservaion of mass and he flow of air over a wing are examples of parial DEs (PDEs). Frher, predaor-prey models and he Navier-Sokes eqaions governing flid flow are examples of sysems of DEs. Many inrodcory ODE corses are devoed o solion echniqes o deermine he analyic solion of a given, normally linear, ODE. While hese echniqes are imporan, many real-life processes may be modeled wih sysems of DEs. Frher, hese sysems may be nonlinear. Nonlinear sysems of DEs may no have exac solions. However, we sill desire some ype of solion. There are many nmerical echniqes o obain an approximaion o he solion of a DE or sysem of DEs. Many scienific packages conain library commands o nmerically approximae he solion of hese. In pariclar, MATLAB conains he ode45 command which will nmerically approximae he solion o a sysem of ODEs sing a medim order mehod. Ths, he scienific package MATLAB may be sed o explore DEs modeling real-world applicaions ha are more complicaed han he DEs presened in a ypical inrodcory corse. Using MATLAB also has he advanage ha he inernal command gide may be sed o creae graphical ser inerfaces (GUIs) o help faciliae his exploraion. The GUIs ensre ha he sdens are no bogged down wih reading and ndersanding MATLAB code. In his paper, we will discss how o se MATLAB o simlae he solion o DEs. Then proecs he ahor has assigned involving real-world applicaions of DEs will be described. One proec models he indsrial process of film casing while he oher models he condcion of hea wihin a one-dimensional rod. Similar proecs will also be briefly described. Finally, sden feedback from he proecs will be given. 34

2 II. SIMULATING SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN MATLAB MATLAB provides many commands o approximae he solion o DEs: ode45, ode15s, and ode3 are hree examples. Sppose ha he sysem of ODEs is wrien in he form y' f, y, where y represens he vecor of dependen variables and f represens he vecor of righ-handside fncions. All of he commands (e.g., ode45) reqire hree argmens: a filename which rerns he vale of he righ-hand-side vecor f, vecor represening he domain of he independen variable, se of iniial condiions. Consider he DE ha models driven, damped spring moion. x x x x F Figre 1 conains a porion of he MATLAB code sed o nmerically approximae he solion o his DE. Figre 1: MATLAB code. Some noes are in order here. The inernal commands ode45, ode15s, ec. only accep firs-order DEs. Many higher-order DEs may be ransformed ino sysems of firs-order DEs. The order of he formal argmens in SpringMass is imporan. T represens he vales of he independen variable generaed by ode45. Each row in X represens he vale of X corresponding o he associaed ime vale in T. Once he solion is obained, he op may be displayed via he inernal command plo; frills may be added as needed. Figre conains he porion of code sed o display he approximaion. 35

3 Figre : MATLAB code sed o prodce he display of he approximaion. The naral nex sep is o provide sdens wih his code and ask qesions. For example, one cold ask sdens wha is he effec of changing he damping coefficien. In order for he sdens o answer his qesion, hey are reqired o deermine he line(s) in he code where he damping coefficien is defined, appropriaely change hose, and rern he code. This process demonsraes a drawback o sing his approach: i is difficl for he sdens o read and ndersand code. One solion o his problem is he se of GUIs. They provide a means of inrodcing sdens o scienific packages wiho enirely involving he sdens wihin he code. Insead of expecing sdens o modify code, hey may simply edi ex and click a bon. The reqired calclaions are hen accomplished for he sdens wiho being visible. The MATLAB inernal command gide is he GUI developmen inerface. I enables simplified arrangemen and sizing of GUI componens as well as ao-generaion of code for hese componens. Examples of GUI componens wold be psh bons, axes, sliders, pop-p mens, among ohers. Figre 3 displays an example GUI for he spring-mass sysem. Figre 3: Example GUI for he spring-mass sysem. 36

4 The sdens may change he qaniies by simply yping he new vale in he edi ex boxes and rern he code by pressing he psh bon. Wih his GUI, i is easier for sdens o answer qesions relaed o changing sysem parameers. Now he sdens may explore he pariclar model by being provided wih he GUI and some axiliary files. III. PROJECTS Sdens enrolled in an inrodcory ordinary differenial eqaions corse were groped p and given differen proecs. Each proec involved an indsrial process ha may be modeled by DEs. The sdens were asked o ndersand he process, why i is sefl, how he process is modeled, and o presen heir resls a a conference. There were for main hrss of he proecs. Expose sdens o a real-world process Expose sdens o a scienific sofware package (MATLAB) Demonsrae he need for nmerical schemes Gain experience in pblic speaking via presenaion of heir resls IV. FILM CASTING Today s sociey has in grea abndance prodcs ha are made from polymers: clohing made from synheic fibers, plasic bags, food wrap, and disposable diapers are among he mos common examples. I has become imperaive for oday s manfacrers o ndersand he processes sed o make hese prodcs as flly as possible. The processes involve complex flid flow of a molen polymer, governed by sysems of DEs. Film casing is an indsrial process in which flid (molen polymer) is exrded from a recanglar die of hickness e and lengh l, a a emperare T, and a velociy. Please see [1], [3], [6], [9] [1], and [13] for a deailed descripion of he process. The flid is hen sreched in he air de o he consan plling force F of he chill roll, locaed a a disance of L from he die. The film is hen cooled on his chill roll o solidify he flid. The film casing process is shown in Figre 4. Figre 4: Schemaic of he film casing process. 37

5 The five dependen variables of he flow are lengh l, velociy, plling force F, emperare T, and hickness e. Under cerain assmpions, he eqaions governing his process may be dl d F 8 A 36 l l F A F A l df A d Ag dt given as follows. C e p h T T T 4 T 4 de d e dl e l d e This is a sysem of five firs order nonlinear ODEs. Frher, while he iniial condiions for lengh, velociy, emperare, and hickness are simply hose a he die exi, he iniial plling force F is no known a priori. The nmerical echniqe of shooing is sed o deermine he vale of F. As opposed o aemping o solve his sysem analyically, i wold be beer o nmerically approximae he solion sing a nmerical package (e.g., ode45). Code was wrien ha will nmerically simlae he solion o hese eqaions given a se of parameers. Moreover, a GUI was designed so ha sdens were reqired o only edi he parameers. Figre 5 displays he GUI provided o he sdens as well as a ypical solion profile. Sdens were asked o change he se of parameers and o deermine he effec he change had on he solion. Figre 5: GUI and ypical solion profile for he film casing problem. 38

6 V. HEAT CONDUCTION IN A ONE-DIMENSIONAL ROD The manner in which hea is ransferred wihin a one-dimensional rod may be modeled wih a PDE. We assme an iniial emperare disribion and desire o know how hea is condced wihin he rod as ime evolves. The PDE ha models hea condcion may be given by k, x where k is he hermal diffsiviy (see [8]). We assme ha he iniial emperare disribion is prescribed and ha he emperare a he ends of he rod ( x and x L ) are given. Hence he bondary and iniial condiions are (, ), ( L, ) ( x, ), f ( x). Since his is a PDE, he sie of ODE solvers in MATLAB are inappropriae. Hence, we choose o nmerically approximae he solion o his PDE via he finie difference mehod (FDM). See [8] for a rogh descripion of he FDM. The FDM firs akes he coninos domain in he x - plane and replaces i wih a discree mesh, as shown in Figre 6. Figre 6: Example of a FDM mesh. Nex, he parial derivaives in he PDE iself are replaced wih approximaely eqivalen ( m) difference qoiens. If we le ( x, m ), choose o approximae wih a forward difference and wih a cenered difference, we obain he following finie difference eqaion. ( m 1) ( m) k ( m) ( m) ( m ) 1 1 x 39

7 Code was wrien o solve he finie difference eqaion and display he resls. Once he code was working properly, a GUI was designed o allow sdens o nmerically approximae he solion for a given parameer se. Figre 7 shows he GUI as well as a solion profile for a parameer se. Figre 7: GUI and solion profile for hea condcion in a one-dimensional rod. Sdens were asked o change he iniial condiion and he hermal diffsiviy o deermine he effec on he solion profile. VI. FIBER SPINNING PROJECTS Two proecs involved he indsrial process of fiber spinning (see [], [4], [5], [7], and [14]).. In his process, an axisymmeric sream of polymer mel is exrded from a spinnere conaining hosands of capillaries. The spinnere may be hogh of roghly as a shower head. The polymer mel is hen drawn coninosly by he ake-p rolls. Along he lengh of he sream, here is a region where cooling air is applied. This process is illsraed in Figre 8. 4

8 Figre 8: Schemaic of he fiber spinning process. Depending pon he processing parameers, he polymer may ener a semi-crysalline phase where he srcre of he molecles of he polymer canno change. One proec explored fiber spinning exclding he semi-crysalline phase, while he oher proec considered boh phases. Boh processes are governed by sysems of nonlinear ordinary differenial eqaions. VII. WAVE PROPAGATION PROJECT The final proec explored he propagaion of waves on a a sring (see [8]). This process is governed by he following PDE. c x Here c represens he propagaion speed of he wave. The following bondary and iniial condiions are enforced. (, ) ( L, ) ( x, ) f ( x) ( x, ) g ( x) Figre 9 displays he GUI provided o he sdens as well as an example solion profile. \ 41

9 Figre 9: GUI and solion profile for wave propagaion on a a sring. Sdens were asked o change he iniial displacemen of he sring and he propagaion speed o deermine he effec on he solion profile. VIII. STUDENT FEEDBACK Sden response from hese proecs differed beween he obecives of he proecs. Sden responses were generally posiive abo exploring real-world applicaions of DEs and he experience gained in pblic speaking. However, he responses were generally mediocre for he nmerical schemes and he exposre o MATLAB. Many sdens were srprised ha DEs modeling real-world applicaions cold no be solved analyically. Overall, he sdens gave he proecs posiive reviews and considered hem a sccess. REFERENCES [1] Acierno, D., L. Di Maio, C. Ammirai, Film Casing of Polyehylene Terephhalae: Experimens and Model Comparisons, Polymer Engineering and Science, Vol. 4, pp ,. [] Advani, S., C. Tcker III, The Use of Tensors o Describe and Predic Fiber Orienaion In Shor Fiber Composies, Jornal of Rheology, Vol. 31, pp , [3] Baird, D., D. Collais, Polymer Processing, John Wiley and Sons, New York, [4] Denn, M., Process Modeling, Longman Inc., New York, [5] Dofas, A., A, McHgh, C. Miller, Simlaion of Mel Spinning Inclding Flow-Indced Crysallizaion Par I: Model Developmen and Predicions, Jornal of Non-Newonian Flid Mechanics, Vol. 9, pp. 7-66,. [6] Eisele, P., R. Killpack, Propene, Ullman s Encyclopedia of Indsrial Chemisry, 1993 ed. [7] Fisher, R., M. Denn, Mechanics of Nonisohermal Polymer Mel Spinning, American Insie of Chemical Engineers Jornal, Vol. 3, No. 1, pp [8] Haberman, R., Elemenary Applied Parial Differenial Eqaions, Prenice Hall, New Jersey, [9] Lide, D., ed. CRC Handbook of Chemisry and Physics, CRC Press, New York, 1. [1] Mark, J., ed. Physical Properies of Polymers Handbook, AIP Press New York

10 [11] Smih, S., D. Solle, Nonisohermal Two-Dimensional Film Casing of a Viscos Polymer, Polymer Engineering and Science, Vol. 4, pp ,. [1] Vargafik, N., Tables on he Thermophysical Properies of Liqids and Gases, John Wiley and Sons, New York, [13] Ziabicki, A., Fndamenals of Fibre Formaion, John Wiley and Sons, New York, [14] Ziabicki, A., L. Jarecki, A. Waiak, Dynamic Modelling of Mel Spinning, Compaional and Theoreical Polymer Science, Vol. 8, pp ,