Application of Finite Element Method (FEM) Instruction to Graduate Courses in Biological and Agricultural Engineering

Transcription

1 Sesson: 08 Applcaton of Fnte Element Method (FEM) Instructon to Graduate Courses n Bologcal and Agrcultural Engneerng Chang S. Km, Terry H. Walker, Caye M. Drapcho Dept. of Bologcal and Agrcultural Engneerng Lousana State Unversty, Baton Rouge Abstract The applcaton of Fnte Element Methods (FEM) to a graduate level course n Bologcal Engneerng, Advanced Transport Phenomena n Bologcal Engneerng, s presented. Frst, the Galerkn Weak Statement (GWS) was ntroduced to the class to show the fundamental theory of FEM by solvng a D steady state heat transfer problem. Ths technque provdes a more accurate soluton wth the estmaton of error. The concept of error reducton through mesh refnement was also ntroduced. Each student was requred to conduct an ndependent semester project ncorporatng mathematcal modelng and smulaton of a bologcal engneerng problem. One of these projects, fxed bed on exchange modelng, s dscussed n ths paper. The outputs from these class projects llustrate that the students ganed experence n usng FEM to solve dynamc bologcal engneerng problems. Introducton Computer aded modelng of new products has allowed ndustry to quckly optmze desgn whle spendng less tme and money on physcal prototypes. Boprocess and food process engneers often deal wth complex heterogeneous system characterzed by non-newtonan behavor. Solutons to partal dfferental equatons that descrbe these complex systems are dffcult to obtan. Advantages of usng Computer-Aded Engneerng (CAE) prototypng n food and boprocess development (Datta 998; Baker et al. 999) and applcaton to mechancs of materals (Hllsman 994) have been prevously addressed They nclude: ) quck and nexpensve testng of alternatve scenaros that can result n reduced costs and ncreased profts, ) clear understandng of the nteractons between the physcal processes and ther senstvty to varous operatonal parameters, and 3) front-end engneerng before prototypng, makng the prototypes closer to optmum and reducng ther number. Development of computer models to descrbe these complex boprocessng systems s needed. Ths paper addresses the experence n ntroducng Fnte Element Methods (FEM) to a graduate level course n Bologcal Engneerng n LSU, BE 735, Advanced Transport Phenomena n Bologcal Engneerng, durng the fall semester of 00. The course ncluded Page 7..

2 two aspects of FEM development. Frst, the fundamentals of FEM usng the Galerkn Weak Statement (GWS) were ntroduced to the class wth dscusson of error analyss. Ths technque provdes a more accurate soluton wth the estmaton of error. The concept of error reducton through mesh refnement was also ntroduced. An example one-dmensonal steady state heat transfer problem was used to llustrate these concepts. Secondly, the students ganed frst-hand experence wth FEM by applcaton to ndependent semester projects n bologcal engneerng. Each project was requred to nclude computatonal modelng and smulaton of a bologcal engneerng problem, prmarly from the student s research nterest. The fundamental theory of FEM One dmensonal heat transfer was used to present the fundamental theory of FEM. Smple -D heat transfer through a plane wall s shown n Fgure. In ths problem, the equaton s gven n terms of T (temperature) on the doman S, on the boundary a < x < b wth correspondng Neumann (constant flux, q) and Drchlet (constant temperature) boundary condtons. Governng Equaton d æ dt ö - çk - s = 0 over a < x < b dx è dx ø Boundary Condtons dt - k = q at x = a dx T = Tb at x = b Fgure Conducton through a plane wall wth unform thermal conductvty Page 7..

3 The steps to obtan the fnte element soluton presented to the class are descrbed as follows (Baker and Pepper, 99):. State the governng equaton, and ntal and boundary condtons. Defne the approxmate soluton n terms of known spatal functons multpled by unknown expanson coeffcents. Determne that the sutable spatal functons exst and determne the unknown expanson coeffcents. T N N 3 3 N N å Nf N = ( x) = a ( x) + a f ( x) + a f ( x) + + a f ( x) = a ( x) f K () where T N = approxmate soluton, a s the expanson coeffcent, and N N s the tral spatal functon. The relatonshp between the exact soluton and approxmate soluton s T N N ( x) T ( x) + e ( x) = () where e N (x) s the approxmate soluton error.. Defne the Galerkn Weak Statement (GWS) to determne the expanson coeffcents that ensure an absolute mnmum approxmate error. Ths procedure mnmzes the dstance between the T(x) and T N (x) for any specfed number of terms n the approxmaton, N. To form the GWS, the weghted resdual approach s used to meet the Galerkn crteron that the weght functons must be equal to the selected tral functons to mnmze the error by makng the set of functons orthogonal. N ( x) L( T ) GWS = ò f dx º 0 (3) W where f (x) s defned below and L(T N ) s governng dfferental equaton of the system n terms of the approxmaton varable T N. The GWS for the gven -D heat transfer problem wth lnear tral space functons s: x - x - f ( x ) = for x - x x (4) x - x + - x + - x f ( x ) = for x x x + (5) x - x ( x) = 0 for x > x + or x < x - f (6) 3. Select pecewse contnuous spatal functons (lnear, quadratc, or cubc). 4. Defne a dscretzaton of the soluton spatal doman n terms of fnte elements to Page 7..3

4 evaluate the weak statement ntegrals and apply boundary condton to solve for the unknown expanson coeffcents. 5. Determne the approxmaton accuracy by determnng the error to valdate soluton qualty. The approxmate soluton of a -D heat transfer equaton obtaned through these steps can be obtaned and compared to the analytcal soluton, f the analytcal soluton exsts. Asymptotc Error Estmaton The theoretcal estmate of performance of the lnear bass Fnte Element mplementaton of the GWS s descrbed by the error sem-norm e h E, whch cannot be determned unless the exact soluton s known. Thus, t s not possble to estmate error usng a sngle soluton on mesh doman, W h. In practce, a mesh refnement process s followed usng a sequence of nested meshes, W h, W h/, W h/4, etc. The defnton of approxmaton error for two such solutons yelds T + h h h / h / + e = T = T e (7) Smlarly, the energy sem-norm relaton s gven as: T h E h h / h / + e = T = T + e. (8) E E E E The energy norm of the estmated soluton for -D heat transfer equaton can be obtaned usng followng equaton; h h h h T = E T, T º ò kt dx W. (9) M T = å ({ Q} [ DIFF]{ Q} e ) e e= The relatonshp between these two energy sem-norms s derved as h k h / e = e. (0) Thus, the asymptotc error estmate s defned as h / h / DT e = () k - D T = T - T = - e. h / h / h k h / where ( ) Page 7..4

5 The calculated error estmate for the example D heat transfer problem s tabulated n Table, whch explans the relatonshp between mesh refnement and the mprovement of approxmate soluton. Table The calculated error estmate Number of mesh T h D T h/ e h/ 4.50e e e e e e FEM software The FEM program that was selected was FEMLAB (Comsol, Inc), whch s an nteractve MATLAB-based envronment for modelng and solvng scentfc and engneerng problems based on partal dfferental equatons (PDEs). FEMLAB ntegrates computaton, vsualzaton, and programmng n an easy-to-use envronment. FEMLAB frequently uses MATLAB s syntax and data structures. One beneft of ths ntegraton s that you can save and export FEMLAB models as MATLAB programs that run drectly n that envronment, whch allows the freedom to combne FEM-based modelng, smulaton, analyss wth other engneerng algorthms. Applcaton of FEM to Bologcal Engneerng Problems Student Project An on exchange model wth lnear drvng force was developed to descrbe color removal from a bologcal mxture usng on exchange resns. Color removal n on exchange resns can be modeled as an adsorpton process of a dlute speces. Lttle nteracton between dfferent molecules n the flud s assumed, whch allows the applcaton of sngle component adsorpton models. Materal s ntroduced as a bulk flud at the top of a fxed-bed of sphercal on exchange resn beads. Local equlbrum s assumed around the adsorbent bead,.e. the adsorpton reacton s fast, allowng the amount of materal adsorbed onto the bead surface to be determned by an equlbrum sotherm relatonshp. The temperature of the columns s controlled so sothermal condtons are assumed. A concentraton boundary layer (flm) forms around the bead and wll be accounted for by the classcal lnear drvng force (LDF) approxmaton (Rce 98). The flux of materal to the bead s determned by a mass transfer coeffcent multpled by the dfference n concentraton between the bulk and flm. The beads have a bdsperse pore dstrbuton, beng an agglomeraton of many mcro-beads. It s assumed that the bead structure may be reduced nto a lumped parameter, effectve mass transfer coeffcent (Rce 98). The LDF mass transfer coeffcent now becomes an effectve parameter that s assumed to take nto account the pore dffuson effects. Page 7..5

6 Assumng that the bed porosty s constant and that there s no concentraton gradent n the radal drecton, a one-dmensonal, tme dependent axal dsperson model may be used for the fxed bed: C C C q - D + e + - e (3) ( ) 0 v 0 ax = z z t t where n 0 s velocty, C s the bulk flud concentraton, z s the axal drecton, D ax the dsperson coeffcent n the axal drecton, e s the porosty, and q s the concentraton of the adsorbed speces. Ths partal dfferental equaton (PDE) s coupled to the LDF approxmaton of the flm. q * ( - e) = k a( C - C ) c (4) t where k c s the effectve mass transfer coeffcent, a the nterfacal area, C * s the concentraton n the flm. The parameters k c and a wll be lumped together as k c a, for ths dscusson. The concentraton of the adsorbed speces, q, s related to the concentraton n the flm, C *, by an * sotherm, descrbed n general as q = f ( C ). Ths may be nverted to yeld q and substtuted nto (Equaton 4): q - ( - e) = k ca( C - f ( q) ). (5) t Two hyperbolc PDE s descrbng the adsorpton under the LDF approxmaton have been developed. These two equatons were converted to dmensonless form to reduce the number of parameters nvolved: f f f - + h Pe h q y q + ( f - x ( y) ) - ( - e) C 0 x e y q = 0 (6) = St (7) C z v 0L bulk convecton where f =, C 0 = concentraton at tme 0; h =, Pe = =, q = C 0 L D ax axal dsperson el q e - e q t =, L = total bed length, y =, x = C 0 (.e. y = ), and v 0 x - e e C 0 k c al flm mass transfer St = =. v bulk convecton 0 t, t Page 7..6

7 The model posed was solved usng FEMLAB, a plug-n to the mathematcs applcaton of MATLAB. The problem defned here s a -dmensonal, tme-dependent system wth two dependent varables (f,y) and one ndependent varable (h). The senstvty of the breakthrough curve to a parameter can be analyzed by conductng repeated smulatons. For example, the breakthrough curve s very senstve to changes to Peclet (Pe) number and nsenstve to the Stanton (St) number (Fgures 3 and 4). Fgure 3. Senstvty of adsorpton breakthrough curves to the Peclet number Page 7..7

8 Fgure 4. Senstvty of adsorpton breakthrough curves to the Stanton number Student feedback The evaluaton from the students at the end of fall semester of 00 was conducted concernng nstructon n the applcaton of computatonal fnte element methods usng FEMLAB. The evaluaton questons gven to the student are summarzed at Table. The evaluaton results are shown n Fgure 5. The majorty of the students responded that applyng computatonal methods to bologcal engneerng problems was benefcal and expect to use computatonal FEM n the future for ther other project, labs, employment, and etc. Page 7..8

9 No Somewhat Yes 0 Queston Queston Queston 3 Queston 4 Queston 5 Table Evaluaton questons, whch were asked to answer based on the scale: -no, -somewhat, and 3-yes Practcal Tranng Evaluaton USDA Hgher Educaton Challenge Grant. Dd you lke applyng computatonal fnte element methods (FEM) usng FEMLAB (and MATLAB)?. Dd the applcaton of computatonal FEM beneft you n the learnng of transport phenomena? 3. Do you understand the basc concept of numercal methods or the fnte element method? 4. Dd the nstructor (nstructors) explan the use of FEM and FEMLAB adequately? 5. Do you expect to use computatonal FEM n the future (other projects, labs, employment, etc)? Fgure 5. Student evaluatons. Summary The ntroducton of fundamental theory of FEM and the use of FEM software, FEMLAB, has been a postve step n the ntroducton of numercal technques to the Bologcal Engneerng graduate currculum. Page 7..9

10 Thus far, only lnear bass functons n fnte element analyss have been utlzed. The concept of bass functon wll be expanded to nclude hgher polynomal-degree, quadratc, and cubc fnte element bass functons. The accuracy ssue of these bass functons wll be compared usng error estmate wth energy sem-norm. Acknowledgments Ths work was supported by an USDA Hgher Educaton Challenge Grants program FY 000. The authors acknowledge the students who partcpated n the courses utlzng FEMLAB n ther projects, especally Hugh Broadhurst s contrbutons from LSU Audubon Sugar Insttute on the on exchange model. Reference Baker, A. J. and Pepper, D. W. 99. Fnte Elements --3. McGraw-Hll, New York, NY. Baker, A. J., Chambers, Z., and Taylor, M. B.999. Fnte Element Analyss for the Engneerng Scence: a Web-Based, Vdeo-Streamed Educaton Envronment at a Dstance, 999 ASEE Annual Conference Proceedngs Datta, A. K.998. Computer-Aded Engneerng n Food Processng and Product Desgn, Food Technology, 5(0):44-5. Hllsman, V. S.994. Combnng Fnte Element Analyss Software wth Mechancs of Materals, 994 ASEE Annual Conference Proceedngs Rce, RG. 98. Approxmate solutons for batch, packed tube and radal flow adsorbers comparson wth experment. Chem. Eng. Sc. 37:83-97 Bographcal nformaton Chang-Sk Km receved hs BS n Mechancal Engneerng at Kangwon Natonal Unversty, South Korea, and MS and PhD n Engneerng Scence at Lousana State Unversty. He s currently a Post Doctoral Assocate n the Department of Bologcal & Agrcultural Engneerng at Lousana State Unversty. He has research nterests n computer vson, pattern recognton, artfcal neural network, and bonformatcs. Terry H. Walker receved hs BS n Engneerng Scence and Mechancs, and MS and Ph.D. degrees n Bosystems Engneerng at the Unversty of Tennessee. He s currently an Assstant Professor n the Department of Bologcal and Agrcultural Engneerng at Lousana State Unversty. He has research nterests n boprocessng ncludng fungal fermentatons and supercrtcal flud extracton technologes. Page 7..0

11 Caye M. Drapcho receved her BS and MS degrees The Pennsylvana State Unversty. She receved her Ph. D. n Agrcultural and Bologcal Engneerng at Clemson Unversty n 993. She s currently an Assocate Professor n the Bologcal and Agrcultural Engneerng Department n the LSU AgCenter. Her areas of specalzaton are assessment of agrcultural nonpont source pollutants and bologcal reactor desgn for agrcultural waste treatment. Page 7..