Seady Hea Conducion Chaper 3 Zan Wu zan.wu@energy.lh.se Room: 53
Ojecives Seady-sae hea conducion Wihou inernal hea generaion - Derive emperaure profile for a plane wall - Derive emperaure profile for a circular layer - Inerpre hermal resisance and apply his concep o calculae hea ransfer rae Wih inernal hea generaion - Derive emperaure profile for a plane wall wih Q
Hea conducion equaion isoropic maerial + + 8 If consan 9 Thermal diffusiviy
Simple plane wall 0 0 0-9 0 consan BC: x = 0, = ; x =, = General soluion: = c x + c
Simple plane wall x 3 3 Hea flow, Fourier s law 3-4 Alernae formulaion poenial = resisance curren
Composie wall Serial circui 4 hermal resisence Q Q A A A 3 3 4 6 3 A A A Q 3 3 4
Convecive hermal resisance Newon s law of cooling A A Convecive hermal resisance:
Composie wall, convecive BC f Ho liquid 3 f Cold liquid 3 Q A f 3 il i f i A A 3 7
Circular ue or layer Shell Q i lnr r o o i 3 0
Composie circular walls Q f i r r3 ln ln r r r r i 3 0 f o
Conac resisance Temperaure drop due o hermal conac resisance
Fouling The accumulaion and formaion of unwaned maerials on he surfaces of processing equipmen One of he major unsolved prolems in hea ransfer
Plane wall wih inernal hea generaion f Q' f x Uniform hea generaion per uni volume
Governing equaion 8 Q z z y y x x c 0 d Q dx ' Q x c x c General soluion
Boundary condiions A he plane of symmery d dx x 0 0 Adiaaic or insulaed BC d x dx wall f
Soluion Q Q x f max f Q Q
Recap: seady-sae hea conducion Sar wih he hea conducion equaion, simply i wih proper assumpions Then ge a general soluion, comining wih BCs o oain a specific soluion for emperaure disriuion Use he Fourier s aw o oain he hea ransfer rae ased on he emperaure disriuion
Hea Transfer from Fins, Exended Surfaces
Ojecives Derive governing equaions and formulae oundary condiions for recangular and riangular fins Calculae fin efficiency and fin effeciveness Undersand opimal fin crieria for recangular and riangular fins Apply fin efficiency in hea ransfer rae calculaions
Example fins a Individually finned ues fla coninuous fins on an array of ues
Example fins
Example microfins Microfin copper ue Caron nanoue microfins on a chip surface
Fins on Segosaurus Asor radiaion from he sun or cool he lood?
Recangular fin
Recangular fin Boundary condiions: long and hin fin, hea ransfer a he fin ip is negligile 3 3 0 A C dx d f Z Z A C m dx d : x f x 0 dx d x f : 0 x x dx Q. f Z
Recangular fin General soluion: mx mx C e C e C cosh mx C sinh mx 3 4 Hyperolic funcions cosh mx mx mx e e sinh mx mx mx e e
Recangular fin f cosh m x 3 38 cosh m f For x = = Q cosh m hea ransfer from he fin? d sinh m Q A A m cosh m dx x 0 m C A anh anh 3 40
Recangular fin Eq. 3-38 = 5 W/m K, = cm, = 0 cm
Fin performance evaluaion : Fin effeciveness Q from he fin Q from he ase area wihou he fin : Fin efficiency from he fin from a similar fin u wih λ
Opimal fin: maximum hea ransfer a a given fin weigh Z M = Z = Z A A =, Z, are given. Find maximum Q for A =, consan. Q CA anh m 3-40 C A C Z, A = Z m 3-35 A Q Z anh 3-5
Opimal recangular fin Condiion dq 0 gives opimum d afer some algera one oains /.49 3 55
Sraigh riangular fin f Q x dx A = f x Z Hea alance Soluion: d dx x d dx x x BK x AI 0 0 0 3 6 Bessel differenial equaion I 0 and K 0 are he modified Bessel funcions of order zero K 0 as x 0 B = 0 ecause is finie for x = 0 x = = AI 0
Triangular fin I A 0 65 3 I x I 0 0 x dx d A Q x 0 0 dx x di I Z Q 66 3 I I Z Q 0 Tale 3. for numerical values of I 0 and I
Opimal riangular fin: maximum hea ransfer a given weigh /.309 3 67
Summary of formulae for recangular and riangular fins opimal fin opimal fin 38 3 m cosh x m cosh f f m 40 3 m anh Z Q m anh m m anh 55 3 49. / 65 3 0 I x 0 I 0 I I Z Q 0 I I 0 I / I 67 3 309. /
Formulas for fin performance Some simple calculaions give: Recangular fin anh m Fin effeciveness anh m m Fin efficiency Triangular fin I I 0 I / I 0
Circular or annular fins Hea conducing area A = r Convecive perimeer C = r = 4r r r 37
Fin efficiency for circular fins
How o use he fin efficiency in engineering calculaions Q Q Q Q oflänsad unfinned area area Q Q flänsar fin s a Q A f Q fins A fins f Q A A 3 7 a f fins