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1 (a) Individually finned ues () fla (coninuous) fins on an array of ues

2 Eample Fins

3 Fins on Segosaurus 3

4 Rekangulär fläns, Recangular fin. Z d f 4

5 Rekangulär fläns, Recangular fin. Z d f d αc d λ ( f ) (3 3) m αc λ α Z λz α λ Randvillkor, Boundary condiions: : f d : λ α ( f ) d unn och lång fläns, long and hin fin d d 5

6 Rekangulär fläns, recangular fin ösning, Soluion: m m C e + Ce C3 cosh m + C4 sinh m e (cosh m e sinh m m m m + e e m ) 6

7 Rekangulär fläns, recangular fin f f cosh m( ) cosh m (3 38) För, for cosh m från flänsen, from he fin? m d λ d λ αc λ sinh m ( m) cosh m αcλ αλ Z anh m anh m (3 4) 7

8 Rekangulär fläns, recangular fin Cu, λ 385 W/m K Rosfri sål, λ 7 W/m K.4. Glas, λ.8 W/m K [cm] α 5 W/m K, cm, cm 8

9 Rekangulär fläns, recangular fin Om vi använder villkore, If he condiion elow is used d λ α d fås isälle, one has cosh m( ) + cosh m + och, and α sinh m( ) mλ α sinh m mλ (3 4) (3 4) α cosh m + sinh m mλ och, and α + anh m m mλ λ (3 43) α + anh m mλ 9

10 Rekangulär fläns, recangular fin funkion funcion ( ) ( ). Z lönsam, preferale if d d > Fig rrangemang av rekangulära flänsar. arrangemen of recangular fins

11 önsamheskrav, crierion for enefi: d > d α + anh m m m λ λ α + anh m mλ d d d d + m α ( + anh m) cosh m mλ mλ N α m α anh m m + λ cosh m mλ N > α α anh m m λ m λ α > m λ m λ eller, or α anh m > mλ

12 Rekangulär fläns, recangular fin m α C λ α λ nag, assume α α eller, or α λ α λ α > λ λ α > > (3 46) Tumregel, rule of hum: λ α > 5 (3 47)

13 Flänsverkningsgrader, fin effeciveness, fin efficiency : η η från flänsen från asyan uan fläns from he fin from he asearea wihou he fin : ϕ ϕ från flänsen få från en likadan fläns med λ from he fin from a similar fin u wih λ 3

14 Opimal fläns, opimal fin Krierium, Crierion: Ma. värmeflöde vid given vik, maimum hea flow a a given weigh Z M ρ Z ρ Z, Z, ρ givna, are given. Sök ma, find maimum för, for konsan, consan. (3-4): αcλ anh m C Z, Z m αc λ α λ α αλ Z anh λ 4

15 Opimal rekangulär fläns, opimal recangular fin Villkor, condiion d d ger opimum, gives opimum några M mellan räkningar ger; afer some algera one oains /.49 λ α (3 55) 5

16 Flänsarrangemang, Fin arrangemen m λ anh m αλ Z anh α λ (3 5) { α u (3 58) λ anh u 3u cosh (3 54) u ( 3 54) ur villkore, from he condiion d / d } Efer lie räknearee finner man; afer some algera one finds: 3 u (3 6a) 3 3 Z anh u 4 α λ 6

17 Flänsens vik, Weigh of he fin M ρ Zρ Z λ ρ h u Z ρ/λ maerialparameer, is he maerial parameer λ α 3 4 u anh Z se aell 3-, see Tale 3-. 7

18 Rak kriangulär fläns, Sraigh riangular fin f δ d f Värmealans, Hea alance d + d d d α λ ösning, Soluion: α λ ( ) + BK ( ) I K då (3 6) Z B y, as ändlig, finie för, for ( ) I 8

19 riangulär fläns, riangular fin ) ( I 65) (3 ) ( I ) ( I d d λ d ) ( di ) ( I Z λ 9

20 riangulär fläns, riangular fin ξ Inför inroduce,. ξ ξ ξ ξ d d d d d d d d Inför / inroduce, ξ ξ d ) ( di / d ) ( di ) ( I ) ( I / λ α 66) (3 ) ( I Z αλ 66) (3 ) ( I ) ( Z αλ

21 Triangular fin Opimal riangulär fläns, opimal riangular fin. Ma. värmeflöde vid given vik, maimum hea flu a given weigh. /.39 λ α (3 67)

22 Sammansällning rekangulär och riangulär fläns, summary of formulae for recangular and riangular fins f f cosh m( ) cosh m (3 38) I ( I( ) ) (3 65) m α λ α λ αλ Z anh m η λ anh m α ϕ anh m m (3 4) η ϕ I ( αλ Z I( λ I ( α I( I( ) ) ) / I( ) ) ) / Opimal fläns, opimal fin (ma värmeflöde vid given vik) λ.49 (3 55) α Opimal fläns, opimal fin (ma värmeflöde vid given vik).39 / λ α (3 67)

23 Formler för flänsverkningsgrader. Formulas for fin efficiencies Några enkla räkningar ger följande resula, Some simple calculaions give: Rekangulär fläns, recangular fin η λ α anh m ϕ anh m m Triangulär fläns, riangular fin η λ α I I ( ( ) ) I( ) / I( ϕ ) 3

24 Cirkulära flänsar, circular or annular fins r r Värmeledande yan, hea conducing area πr Konvekiv omkres, convecive perimeer C πr 4πr 4

25 Fin efficiency circular fins 8 ϕ (%) r c /r 6 r c r + / c + / r r c α/( λ) 5

26 nvändning av flänsverkningsgraden ϕ, nvändning av flänsverkningsgraden ϕ, How o use he fin efficiency in engineering calculaions fläns s flänsar area oflänsad fin unfinned area + + ) ( ) ( ) ( f fläns fi f flänsar f + + α λ ϕ α ϕ α 3 { ) ( ) ( f fins fins f + α λ ϕ α { } ) 7 (3 ) ( + fläns f ϕ α 6 { } ) 7 (3 ) ( + fins f ϕ α

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