Module 2: Introduction to Numerical Analysis

Transcription

1 CY00 Itroducto to Computtol Chemtr Autum 00-0 Module : Itroducto to umercl Al Am of the preet module. Itroducto to c umercl l. Developg mple progrm to mplemet the umercl method opc of teret. Iterpolto: let qure fttg. umercl tegrto- rpezodl d Smpo oe-thrd rule. umercl oluto of ordr dfferetl equto Ruge-Kutt method. Soluto of o-ler equto ug ewto-rpho method 5. Soluto of ler tem LU decompoto d Gu elmto 6. Ege vlue d Ege vector ote he method re merel outled the workheet. For detled decrpto, plee refer to tetook o elemetr umercl l. Severl uch ook re vlle t the Ittute Cetrl Lrr. Referece:. A Itroducto to umercl Al, K.E. Atko ( d edto), J Wle & So(978).. umercl Method ug MALAB, J.H. Mthew d K.D. Fk ( th Edto), Pero (00). [ h good referece for thoe tereted ug MALAB for umercl method] Both the ook re vlle low prce edto.

2 Workheet 5 opc : o ft et of dt to trght le ug the method of Let Squre Fttg Am: o model the epermetl dt d to terpolte d/or etrpolte the vlue of phcl qutt from the meured dt. Let u coder propert tht deped o depedet vrle. I epermetl et up, everl dcrete vlue of c e recorded vrg. he m of the preet leo to how ou mple w of epreg ccurtel fucto of term of mple equto. ote: I the fgure o the rght, two uch tpcl dt et re how for whch the et ft to the dt re requred. Whle the et ft provded trght le the left pel, the me ot true for the ce how the rght hd pel. herefore, t m e ecer to ue hgher order poloml for ccurte repreetto of the dt. Workg Prcple: o drw the et ft le wth mmum vertcl dplcemet of ll the dt pot. Squre devto: Let u ume tht we hve dt et (, ) (, ) would lke to wrte. For th purpoe, let u defe R d we [ ] ( ) o tht the qure devto R fucto of d. For R to e mmum., we mut hve R R 0. he ove codto led to the followg relto: ; hee two equto m e olved to ot

3 ; W5_. Wrte progrm to ft gve et of dt to trght le ug the method of let qure. W5_. he het cpct of grphte meured t hgh temperture d preeted the tle elow. (K) C P,m (J K - mol - ) Aume tht the temperture depedece of het cpct m e modeled. Ue our progrm to fd the het cpct of grphte t 950 K. c c C m P 0, ) ( W5_. I the tle gve elow, the reult of ketc tud o the g-phe decompoto of O 5 t 67 o C re preeted. he recto tke plce. 5 O O O t (m) [ O 5 ] (mol L ) Clculte the frt order rte cott for th recto t 67 o C. W5_. ote the followg three defto o tdrd devto. () the tdrd devto out regreo: S m S r, () the tdrd devto of the lope: r m S (c) the tdrd devto of the tercept: r where d. Ug thee defto, clculte ll the three devto the dt preeted the two emple cte ove. S S

4 opc : umercl tegrto Am Let u coder the followg tegrl I f ( ) d. he m of th eerce to ppromte the tegrl I [tht, the re uder the curve f()] evlutg the tegrd f() t fte umer of mple pot the tervl [,]. Method he de ehd umercl tegrto to terpolte the tegrd f() d the to tegrte the terpoltg poloml p ltcll. rpezodl Rule f ( ) d [ f ( ) f ( )] Smpo Rule f ( ) d f ( ) f f ( ) 6 where the tegrl h ee ppromted ummg the tegrd t d equdtt pot, repectvel. he former method replce the curve f() trght le whle the ltter ue qudrtc ppromto. o decree the error cued uch evere ppromto, everl uch rule re lked together to gve the compote rule whch re follow: Compote rpezodl Rule I th method, the tervl [,] u-dvded to tervl d ech utervl, f() replced trght le. f ( ) d h f ( ) f ( ) f ( ), where h, h,

5 Compote Smpo Oe-thrd Rule h ppromte method vld ol for eve umer of tervl. f ( ) d h f ( ) f ( ) f ( ), where h, h, ote : for for eve odd f ( ) d h [ f ( ) f ( ) f ( ) f ( ) f ( )... f ( ) f ( ) f ( ) ] W5_5. Wrte progrm to evlute the tegrl () the trpezodl rule d () Smpo oe-thrd rule. I 0 d ug W5_6. he lumou effcec (rto of the eerg the vle pectrum to the totl eerg) of lck od rdtor m e epreed percetge the followg epreo 5 7 X0 5. ep 5 X E d where the olute temperture degree Kelv, the wvelegth cm d the rge of tegrto over the vle pectrum. kg 500 K, wrte progrm tht ue Smpo oe-thrd rule to compute E. W5_7. It well kow epermetll tht t hgh temperture, rrepectve of the ture of the old, t het cpct of cott. h kow the Dulog-Pett lw. O the other hd, t low temperture, the het cpct ecome fucto of temperture d C v ~ t ver low temperture. Uder thee codto, ol mll mout of therml eerg vlle to the tem for the tom/molecule to eecute vrtol moto. herefore, the low temperture vrtol ptter of the old domted low-eerg (.e. log wvelegth) vrtol mode. It h ee how Dee tht thee mode c e ppromted the log wvelegth vrtol mode of cotuou eltc od d the followg epreo oted for C v

6 C v k B π 5 D / D 0 d ep( ) [ ep( ) ] Eq. Sce the tegrl Eq. cot e evluted ltcll, therefore, the method of umercl tegrto re ppled to etmte C v t gve temperture. D chrctertc cott of the old kow the Dee temperture. Wrte progrm tht () red the temperture ut of D, Cv () clculte evlutg the ove tegrl umercll ug k B Smpo oe-thrd rule d () gve output tle cotg the dmeole temperture / D the Cv frt colum d the ecod colum. k B