ME 160 Introduction to Finite Element Method. Chapter 5 Finite Element Analysis in Heat Conduction Analysis of Solid Structures

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1 San Jose Sae Unvers Deparmen o Mehanal Engneerng ME 6 Inroduon o Fne Elemen Mehod Chaper 5 Fne Elemen Analss n Hea Conduon Analss o Sold Sruures Insruor a-ran Hsu Proessor Prnpal reerenes: ) he Fne Elemen Mehod n hermomehans.r.hsu Allen & Unwn Boson 986. ISB --6- SJSU Lrar: A8.58 H a 8 h Floor wh one reserved op a s Floor) Chaper. ) Chaper Fne Elemen Euaons or Hea ranser G.P. shov Sprngerhp://sprnger.om/ ) Appled Fne Elemen Analss L. J. Segerlnd John Wle & Sons 976

2 Par Inroduon o Fundamenals o Hea Conduon n Solds

3 Fourer law o hea onduon A naural phenomenon s ha hea lows n a sold s possle onl wh emperaure gradens wh hea rom he loaons a hgher emperaure o he loaons wh lower emperaure. Conseuenl hea wll low rom he le sde o he rgh sde o he sla we manan he suaon o a > wh a and eng he emperaure a le and rgh aes o he sla respevel as llusraed elow: Amoun o hea low Q a Area A Q d = A a ) A a ) Q Q d d where K = hermal onduv o maeral wh uns o: Bu/n-s- o F n he radonal ssem or w/m- o C n he SI ssem.

4 Fourer law o hea onduon-on d Insead o oal hea low a more ommonl used ermnolog n engneerng analss s hea lu dened as hea low n sold per un area and me. Mahemaall s epressed as: Q A d a ) or hea low n a sold sla a veor uan For onnuous varaon o emperaure eween he wo aes and le he oordnae along he lengh o he sla e -as we wll have he aove epresson n he orm o: ) ) Hea low: a > d + Δ) ) ) ) ) a ) wh onguous varaon o emperaure he ollowng epresson prevals: + Δ Δ X d ) ) 5.) d Euaon 5.) s he mahemaal epresson o he Fourer Law o Hea Conduon

5 Fourer law o hea onduon n -D spae r) ) Poson veor: r: ) wh omponens: r r 5.) ) ) ) 5.a) 5.) 5.) where and are he hermal onduv o he maeral along he respeve - - and -dreons. For sorop maerals we wll have = = = he resulan oal hea lu n he sold n Euaon 5.) s he veor sum o he omponens n Euaon 5.) o e: )

6 Hea Conduon Euaon n Solds Qr) r) Gven a sold suaed n a spae dened a oordnae ssem r) or ) Hea lues n and ou o he sold.. and hea generaed n he sold he amoun Q) per un volume and un me. here wll e ndued emperaure dsruon or emperaure eld) n he sold r) or ) n he sold. he hea onduon euaon was derved usng he Fourer law o hea onduon and on he ass o law o onservaon o energes Q ow we susung he hea lues shown n Euaons 5.) and 5.) no he aove epresson o eld: Q 5.5) Euaon 5.) s he hea onduon euaon or solds n whh ρ s he mass dens and s he spe hea o he maeral For sead-sae hea onduon n he sold: Q 5.6) he erm Q) n oh Euaons 5.) and 5.5) s he hea GEERAED he sold suh as Ohm s heang or nulear sson 5.)

7 Hea Conduon Euaon n Solds wh spe ondons he Hea onduon euaon: Q 5.5) he oundar ondons: ) Speed emperaure on he oundar surae S : s = ) on S 5.7a) ) Speed hea low on he oundar surae S : n + n + n =- s on S ) Speed onveve oundar ondon on he oundar surae S : n + n + n =h s - ) on S he nal ondons: ) = o ) n =osne o ouward normal lne n -dreon) In he aove oundar ondons s n Euaon 5.7) s he hea lu aross he oundar rom eernal soures and h s he hea ranser oeen o he surroundng lud a ul lud emperaure or onveve oundar ondon over surae S. 5.7) 5.7) 5.7d)

8 Par Fne Elemen Formulaons

9 Fne elemen ormulaon o hea onduon n sold sruures he prmar unnown uan n ne elemen analss o hea onduon n sold sruures s he EMPERAURE n he elemens and ODES. As usual he ver rs sep n FE analss s o dsree he onnuum sruure no dsreed FE model suh as llusraed elow: r) Qr) Connuum sold r) Qr) Dsreed FE model emperaure: m r) Elemen emperaure odal pal elemen

10 Fne elemen ormulaon o hea onduon n sold sruures on d he Inerpolaon Funon []: he same denon o nerpolaon unon or sress analss s used or he hea onduon analss.e.: Elemen emperaure = Inerpolaon Funon [)] odal emperaure {} where he nerpolaon unon: [)] = { m } he nodal emperaure: {} = { m } 5.8) 5.9) 5.) he emperaure gradens n he elemen ma e oaned n erms o nodal emperaure derenae he relaonshp n Euaon 5.8) as: B m m m 5.) where he mar [B] has he orm: B m m m 5.)

11 Fne elemen ormulaon o hea onduon n sold sruures on d he unonal or dervng elemen euaons: Beause he onduon o hea n solds an e ompleel desred smple derenal euaons suh as Q 5.5) or ransen sae and Q 5.6) or sead-sae and he oundar and nal ondons epressed n Euaons 5.7) Galern mehod suh as desred n Chaper wll e used o derve he elemen euaon. We wll rs revew he Galern mehod n he ne slde.

12 Sep Chaper Galern mehod In onras o he Ralegh-R mehod hs mehod s used o derve he elemen euaons or he ases n whh spe derenal euaons wh approprae mahemaal epressons or he oundar ondons avalale or he analal prolems suh as hea onduon and lud dnam analses Elemen Φr) odal {Φ} Φr) = r)φ Real Suaon on solds r : Appromae suaon: Dsreed Suaon wh elemens Derenal Euaon: DΦ) or he volume V 5.) Derenal Euaon: Dr)Φ) or he elemen volume V Boundar ondon: BΦ) or he real suaon on oundar S 5.5) Boundar ondon: Br)Φ) or he real suaon on elemen oundar Mahemaal model: W D dv W B ds Mahemaal model: W D r where W and W are Galern mehod les v arral W s weghng and W unons r) he same elemen euaon: W and W dv W B ds r ) = R v s are dsreed weghng unons and R s he Re sdual [r)] n Euaon 5.9) and le R o e mnmum or R or good dsreaon resulng n: [K e ] {} = {Q}

13 Fne elemen ormulaon o hea onduon n sold sruures on d Dervaon o Elemen Euaon usng Galern Mehod Usng he Galern mehod we an rewre he as hea onduon euaon n he ollowng orm: dv Q v Euaon 5.) Euaon 5.9) B norporae he oundar ondons n Euaons 5.7) n he aove euaon wll resul n he elemen euaon wh he alaned o hea lus aross he oundar and he ndued emperaure n he elemen n he ollowng euaon: ds h ds ds n Q dv dv dv s s s s v v v ) wh n n n n norrmal o ouward ne he dreon and lu aross oundares hea os 5.)

14 Fne elemen ormulaon o hea onduon n sold sruures on d Dervaon o Elemen Euaon usng Galern Mehod on d he hea alane n Euaon 5.) ma e lumped o he ollowng elemen euaon: C K K R R R where n he oeen mares: he hea apne mar : C h he ondv mar : he onveve mar : and he nodal hermal ore mares: he hea lu aross he oundar he hea generaon mar : he hea lu aross he oundar h K B B dv K h ds h S v v dv R n S : S R Q Q dv v S : R s he onveve hea lu ross he oundar S S : ds ds R h ds h S 5.) 5.5a) 5.5) 5.5) 5.6a) 5.6) 5.6) 5.6d)

15 Par Hea Conduon n Planar Sruures Usng Fne Elemen Mehod

16 Fne elemen ormulaon o hea onduon n sold sruures n planes Hea onduon n a apered plae: H L s ) h s - ) L r s s FE Mesh ο ο ο ο s s h s - ) s X FE ormulaon n a rangular plae elemen: ) Elemen emperaure: ) odal emperaure: ); ); ) ) ) )

17 Fne elemen ormulaon o hea onduon n sold sruures n planes FE ormulaon n a rangular plae elemen-he nerpolaon unon: We assume he elemen emperaure ) s represened a smple lnear polnomal unon ha: R R where α α and α are onsans 5.7) wh 5.8) Beause he oordnaes ) ) and ) o he nodes n a FE mode are ed. We ma susue hese oordnaes no Euaon 5.7) and oan he ollowng epressons or he orrespondng uanes a he hree nodes: or n a mar orm or nodal emperaures: or ode or ode or ode A and he unnown oeens A h 5.9) 5.)

18 he mar [A] n Euaons 5.9) and 5.) onans he oordnaes o he hree nodes as: A Fne elemen ormulaon o hea onduon n sold sruures n planes on d FE ormulaon n a rangular plae elemen he nerpolaon unon - on d: he nverson o mar [A] - = [h] an e perormed o gve: A h where A Is he deermnan o he elemen o mar [A] A wh A= he area o rangle made ) 5.)

19 B susung 5.) no 5.) and hen 5.9) he elemen uan represened ) an e made o eual he orrespondng nodal uanes {}: o e: h R Fne elemen ormulaon o hea onduon n sold sruures n planes on d FE ormulaon n a rangular plae elemen he nerpolaon unon - on d: We wll hus have he nerpolaon unon: ) = {R} [h] wh {R} = { } n Euaon 5.8) and [h} gven n Euaon 5.) We hus have he relaonshp eween he elemen uan o he nodal uanes he ollowng epresson: ) = {)} {} or epress he aove euaon n he orm aordng o Euaon 5.8) as: 5.) 5.) wh a A a A a A A 5.) and

20 a ) ) rangle made o A he elemen he area o A Fne elemen ormulaon o hea onduon n sold sruures n planes on d FE ormulaon n a rangular plae elemen he nerpolaon unon - on d: wh a A a A a A a a 5.) 5.5a) 5.5) 5.5)

21 Fne elemen ormulaon o hea onduon n sold sruures n planes on d FE ormulaon n a rangular plae elemen he elemen oeen mar: he onduv mar [K]: B ollowng Euaon 5.5) we have he onduv mar or a rangular plae elemen o e: A dd B B K 5.6) he emperaure graden mar [B] an e oaned he ollowng ormulaon: A B 5.7) We ma oan he onduv mar susung Euaon 5.7) no Euaon 5.6) leadng o: A K 5.8)

22 Fne elemen ormulaon o hea onduon n sold sruures n planes on d FE ormulaon n a rangular plae elemen he elemen euaons: As n he ase o sress analss n haper he elemen euaons or hea onduon solds o plenar geomer ma e shown o ae he orm: K e 5.9) where [K e } = oeen mar n Euaon.8) {} = nodal emperaure and {} = hermal ores a he nodes he hermal ores a nodes are: {} = { Q } + { } = h } n whh { Q } = hea generaon n he sold wh Q Qdv Q { } = hea lu aross oundar wh ds L or sde - where = hness o he plane v v dv { h } = onveve hea lu aross oundar wh h h ds S Qv ds S S m L m or sde -m L m or sde m- 5.) 5.a) 5.) 5.)

23 Eample 5. Cool HO m Use ne elemen mehod o deermne he emperaure varaon aross he hness o longudnal ns o a uular hea ehanger as shown n he gure on he le. he hea ehanger s desgned o hea up he old lud ousde he ue he ho lud rulang nsde he ue. he ross-seon o a sngle n s llusraed n he gure shown n lower-le o hs slde. he n s made o alumnum wh he properes: Mass dens ρ =.7 g/m Spe hea =.9 J/g- o C and hermal onduv =.6 W/m- o C he dsreed FE model o he n ross-seon s shown elow: Conveve BC h Cold HO s ue wh one longudnal n m Conveve BC h ) Conveve BC h ) ) ) Hea lu s Conveve BC h X Boundar ondons: s = W/m h = W/m - o C = o C

24 Eample 5.-on d Inerpolaons unons or Elemens: We wll use Euaons 5.5a) o deermne he onsan oeens a and I = ) or eah elemen. hese oeens wll hen e used o epress he nerpolaon unon o oh Elemen and as n Euaon 5.). We reale he ollowng nodal oordnaes n he FE model o he n: For elemen ode and ): = = ; = = ; = = he area A o he ross-seon area o Elemen s ompued usng he epresson: 8 A hs leads o A = m We wll urher ompue he onsan oeens he ollowng epressons: a a a

25 Eample 5.-on d Inerpolaons unons or Elemens: For Elemen ode and ): = = ; = = ; = = he area A o he ross-seon area o Elemen s he same as o Elemen = m. he onsan oeens are deermned he same wa as or hose n Elemen. a a a We wll hus have he nerpolaon unons or oh elemen and susung he onsan oeens no Euaon 5.): For Elemen : Leads o: e a) For Elemen : Leads o: e )

26 Eample 5.-on d Elemen oeen mares [K e ]: We wll use Euaon 5.8) o derve hese mares. For Elemen : K e For Elemen : K e.6.6 )) ) ) )) )) )) )) ) ) )) K A 5.8) )) ) ) ) )) )) )) )) ) ) )) )) )).6 )) )).6 ) ode: ) ) )).75 ) )) ) ) ode: C) d)

27 Eample 5.-on d Asseml o elemen oeen mares or Overall oeen onduane) mar Elemens n K e ode We need o assemle he elemen oeen mares o onsru he overall sruure oeen mar summng up he wo elemen oeen mares. We need o add he elemens or he nodes ha are shared varous elemens. In he presen ase we have ode and shared oh hese wo elemens. We esalsh he ollowng map or assemlng he overall oeen mar *K+: Elemens n K e + = where =elemen n he mar n Euaon ) and = elemens n Euaon d) We hus have he overall oeen or onduane) mar n he orm: K ode or he [K ] mar e)

28 Eample 5.-on d Se hermal ores a he nodes We have he ollowng hea aross he oundares o he n: ) Hea lu enerng he n rossng he lne - wh s = W/m ) Hea leavng he n rossng oundar lne - onveon wh h = W/m - o C = - W/m - o C ) Hea leavng he n rossng oundar lne - onveon wh h = W/m - o C = - W/m - o C he sruure has a lengh.e. he hness = m We wll ormulae he euvalen nodal hermal ores or he aove speed oundar hermal ores aordng o he ormulas o: L s or hea lu ross lne - lne - and L h ds h h h S h or hea removal onveon ) Hea lu enerng he n rossng he lne - wh s = W/m : W L s ) ) ) Hea leavng he n rossng oundar lne - and lne - onveon wh h = W/m - o C = - W/m - o C W L h L h L h h h ) ) ). W L h L h L h h h ) ) ) and

29 Eample 5.-on d Se hermal ores a he nodes-on d We hus have he hermal ore mar or he nodes as: h h h h he overall sruure hea onduon euaon: [K]{} = {} )

30 Eample 5.-on d Solve Se hermal or nodal ores emperaures he nodes-on d g) We hus solve or he nodal emperaures o e: = 69.7 o C = 7.8 o C = 57.8 o C and = o C

31 Eample 5. he same Eample.6 o he eoo on A Frs ourse n he Fne Elemen Mehod 5 h edon Darl Logan pulshed Cenage Learnng = o F h= =5 o F Prolem: For he -D od shown n Fgure - deermne he emperaure dsruon. he emperaure a he le sde o he od s mananed a o F. he edges on he op and oom o he od are nsulaed. here s hea onveon rom he rgh sde wh onveve oeen h = Bu/h- - o F. he ree sream emperaure s =5 o F. he oeens o hermal onduv are K =K =5 Bu/h-- o F. he dmensons are shown n he gure. Assume he hness o e. Fgure - -D od sueed o emperaure varaon and onveon 5 Fgure - Dsreed -D od o Fgure - Soluon: he dsreed FE model o he od s shown n Fgure - wh elemens and 5 nodes. odal oordnaes are: = = or ode = = or ode = = or ode = = or ode and 5 = 5 = or ode 5 We wll ormulae he elemen oeen mares or all he elemens n Fgure - usng he euaons 5.5a) and 5.8)

32 Eample 5. Con d For Elemen : wh odes 5 he area A s: A leads o: A =. o nd he onsan oeens n Euaon 5.5a): From Euaon 5.5a): a From Euaon 5.5): a From Euaon 5.5): a

33 Eample 5. Con d We wll use Euaon 5.8) o ormulae he elemen oeen mar: A K 5.8) ) ) )) )) )) )) )) )) ) ) ) ) )) )) )) ) ) )) ) 5 A K e ode 5

34 Eample 5. Con d Elemen oeen mares or Elemen and ollowng he smlar approah as shown elow: ode 5 For Elemen : K e For Elemen : ode 5 K e A noe rom he Insruor: For Elemen : ode 5 K e a) A noe rom he Insruor: he auhor o hs eample Darl Logan) arues he onveve hea leavng he od n Elemen as par o hea onduon av n hs elemen. So he derves anoher onduane oeen mar [K h ] or hs Elemen as wll e shown n he ne slde.

35 Eample 5. Con d Addonal hea onduane mar or onveve hea ranser n Elemen m K h h ds h S S m m m m m m h For he ase wh onveve hea ranser rom Edge - he ollowng epresson s used: For he urren suaon he sde ha has onveve hea ranser s Sde - we wll hus have: )) K h 6 B addng hs mar o he onduane o Elemen n Euaon a) we oan he Conduane mar o Elemen o e: h L ) ) K h 6 5.9) ode 5 K e

36 Eample 5. Con d Assemle he elemen oeen mares or he Overall oeen mar aounng he a ha ode s shred all elemens o K Bu / h F he hermal ores a nodes: We alread now ha emperaure a ode and are speed o e o F ) he hermal ores aross oundar - o elemen s: h L 5 5))) Bu / h

37 Eample 5. Con d he overall hea onduon euaon eomes: ** ** = -5) o F)+-5) o F)=-5 o F on he le sde o he h euaon n he le-hand-sde o he euaon We ma sole he aove euaons and oan: = 69. o F = 59. o F and 5 = 8.6 o F wh speed = = o F

38 Summar on Hea Conduon Analss o Plane Sruures FE Mehod ) An overvew o hea onduon n -D solds was presened n hs Chaper wh hea onduon euaon or he ndued emperaure dsruons n he solds he soures o: a) hea generaon he sold ) he presred surae emperaure ) speed hea lu aross he oundar suraes and d) he onveve hea aross he oundar suraes. ) Fne elemen ormulaon o hea onduon n solds s derved usng he Galern mehod due o he a ha hea onduon n solds an e desred he hea onduon euaons wh presred oundar ondons mahemaal epressons. ) Fne elemen ormulaons egn wh he dervaon o nerpolaon unons [} = { m } or rangular plane elemens wh odes and m. hese unons relae he elemen emperaures and he nodal emperaures. ) he nerpolaon unons or he FE analss were derve on he ass o lnear polnomal unon or he emperaure varaons n he elemen. 5) Speal FE ormulaons o he aoremenoned oundar ondons were presened. 6) hs haper onl presens he FE ormulaon or sead-sae hea onduon n solds o plane geomer.