Fte Dfferece Method MEL 87 Computatoa Heat rasfer --4) Dr. Praba audar Assstat Professor Departmet of Mechaca Egeerg II Deh
Dscretzato Methods Requred to covert the geera trasport equato to set of agebrac equatos Fte dfferece method Fte voume method Fte eemet method
Itroducto to Fte Dfferece aor seres epaso f ) f ) f f ) f )! gradet curvature
Dscretzato 6 ) ),
Represetato of a Dervatve ), O 6 ) ), Fte dfferece represetato 6 ) ), Forward dfferece
Bacward Dfferece 6 ) ) ), 6 ) ), ), O Bacward dfferece
Cetra dfferece 6 ) ), 6 ) ),,, ) O Subtractg, Cetra dfferece
Forward, bacward ad cetra dfferece stec ) O, ), O,, ) O
Stec drecto ) O ) O,, ) O
d Order ad Med Dervatve ) ) ),, ) ) ) ] ), ) [ 4,,,, O
Boudar Cosderato What d of dfferecg scheme s possbe at the boudares? ) O Cetra dfferece appromato s ot possbe as pot s beeath the boudar How we ca get a secod order accurate scheme? Possbt: Pooma approach
Pooma Approach Assume a b c At grd pot, a At grd pot where, a b c ) At grd pot where, a b c)
Pooma Approach cot d) Sovg these equatos, b 4 Dfferetato of gves At pot boudar), What s the order of appromato?? a b b c b c 4
Order of Appromato aor seres gves, Comparg wth the pooma epresso, we ca sa that our pooma s of O),, ca a be epressed terms of the pooma Represets oe-sded dfferece of d order accurac 6 ) ) ) 4 O O c b a
FDM for D dffuso Uses trucated aor seres epaso to appromate the dervatve of the DE Cosder -D dffuso equato d Γ S Epad aor seres about d pot Subtractg these equatos eds
FDM cot d) Addg the equatos d ) d ) ) Droppg the trucated terms d Γ d Γ ) Γ ) he fa dscretzed equato Γ Secod order trucato error S ) Γ ) S S )
FDM cot d) Γ ) Γ S ) Γ ) We ca wrte oe such equato for each grd pot Boudar codtos gves us boudar vaue Secod order accurate Need to fd a wa to sove the coupe agebrac equato set
D Stead State Coducto Cosder the stead state heat coducto a sab of thcess L, whch eerg s geerated at a costat rate of S W/m. he boudar surface at s mataed at a costat temperature o, whe the boudar surface at dsspates heat b covecto wth a heat trasfer coeffcet h to a ambet at temperature. Compute the temperature sde the sab for h W/m. C), 8 W/m. C), L. m, C, o 5 C, ad S 7. 7. o 5 C C 4 5. m
Souto.m L at h h d d) at ) S d d o B.C. 6 S L / 5) S ) S ) ) ) S ) ) )
reatmet of Boudar Codto o 5 C / cod cov 4 N5. m 6 Appcabe to ode -4 Appg t to ode 5 N N 4 N hn ) S h ) SN N.44 6 4.44 5 h
reatmet of Boudar Codto Aother Wa) o 5 C 4 N5. m App to ode 5 B.C. gves h h 4 5 6.44 6 4.44 4 5 6 5 6 4 6 6 4 h5 ) h 6 5 ) 4
Agebrac Equatos 4 4 5 5 6 6 6 6 6 5 4.44 6 4.44 Ca be soved b homas agorthm Matr verso as show et sde
Matr Form.44 6 6 6 66.44 5 4 [ ][ ] [ ] [ ] [ ] [ ] B A B A
Prescrbed Heat Fu q o X / N- N XL q N Eerg baace gves, q q N N N S SN
FDM represetato N q S ) q S ) N N N N for for N S ) S ) N N N for for For suated or smmetr boudar, Fu boudar,
Ustead Heat Coducto wth FDM Ustead heat coducto - D wth costat therma coductvt Epad the dvdua terms wth aor seres, t α t t t t ) ) 4 4
Ustead Heat Coducto cot d) t α ) )
Epct Soutos t α αt ) ) ) ) Fd -D ustead temperature dstrbuto t stead state w C w C 5 cm 5 cm 5 α 7 Ita temp C, t sec cm / s αt ) w ta ater
Resuts 8 C) 6 4 sec 9sec 6sec 6 4 5 6 7 8 Nodes
D Stead State Heat Coducto ) S,,, ) D stead state wth heat geerato ) S 4,, where
Fu Boudar Codto,, q o W/m /,,- Nodes ) o a prescrbed heat fu boudar q, S After rearragemet, 4 S q o
Fu Boudar Codto aother wa), Appg the fte dfferece equato at the boudar ode ), q o W/m /, S,, 4,- B.C. qo qo,,, S qo,, 4
Covectve Boudar Codto Eerg baace gves,, h ) 4 S After rearragemet, h h,,, 6 S, -,, /, Covecto h,,-
Isuated Boudar 6b cos ) π b b 5 4 6 86.66 5 Isuated Mataed at zero temperature Isuated Node : -4 Node : 4-4 Node : 4 5-4 Node 4: 6 86.66-4 4 Node 5: 6-4 5 Node 6: 4 5 5-4 6 5 86.66 4 4 4 4 4 4 6 5 4 Matr Form