(a) Individually finned tubes (b) flat (continuous) fins on an array of tubes

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1 (a) Individually finned tues () flat (continuous) fins on an array of tues

2 Eample Fins

3 Fins on Stegosaurus 3

4 Rektangulär fläns, Rectangular fin. Z t d t f 4

5 Rektangulär fläns, Rectangular fin. Z t d t f d ϑ αc ϑ d λ t t ( ϑ f ) (3 3) m αc λ α Z λz α λ Randvillkor, Boundary conditions: : t t ϑ ϑ t tf dt : λ α (t tf ) d tunn och lång fläns, long and thin fin dt d 5

6 Rektangulär fläns, rectangular fin ösning, Solution: ϑ C e m + C e m C 3 cosh m + C 4 sinh m e (cosh m e sinh m m m + e e m m ) 6

7 Rektangulär fläns, rectangular fin ϑ ϑ t t t t f f cosh m( ) cosh m (3 38) För, for ϑ ϑ ϑ ϑ cosh m från flänsen, from the fin? m dϑ λ d λϑ αc λ sinh m ( m) cosh m αcλ ϑ αλ Zϑ tanh m tanh m (3 4) 7

8 Rektangulär fläns, rectangular fin Cu, λ 385 W/m K ϑ ϑ.8.6 Rostfritt stål, λ 7 W/m K Stainless steel.4. Glas, λ.8 W/m K [cm] Glass α 5 W/m K, cm, cm 8

9 Rektangulär fläns, rectangular fin Om vi använder villkoret, If the condition elow is used dϑ λ α ϑ d fås istället, one has cosh m( ) + ϑ ϑ cosh m + och, and α sinh m( ) mλ α sinh m mλ (3 4) ϑ (3 4) ϑ α cosh m + sinh m mλ och, and α + tanh m m mλϑ λ (3 43) α + tanh m mλ 9

10 Rektangulär fläns, rectangular fin funktion function ( ) ( ). t Z lönsamt, preferale if d d > Fig rrangemang av rektangulära flänsar. arrangement of rectangular fins

11 önsamhetskrav, criterion for enefit: d > d mλϑ α + tanh m mλ α + tanh m mλ d d d d + eller, or m α ( + tanh m) cosh m mλ mλϑ N α m α tanh m m + λ cosh m mλ N > α tanh m mλ α m λ m α > λ α tanh m > mλ

12 Rektangulär fläns, rectangular fin m αc λ α λ ntag, assume α α eller, or α λ α λ α > λ λ > α > (3 46) Tumregel, rule of thum: λ α > 5 (3 47)

13 Flänsverkningsgrader, fin effectiveness, fin efficiency : η η från flänsen från asytan utan fläns from the fin from the asearea without the fin : ϕ ϕ från flänsen från en likadan fläns med λ from the fin from a similar fin ut with λ 3

14 Optimal fläns, optimal fin Kriterium, Criterion: Ma. värmeflöde vid given vikt, maimum heat flow at a given weight Z M ρ Z ρ Z, Z, ρ givna, are given. Sök ma, find maimum constant. för, for konstant, (3-4): αcλ ϑ tanh m C Z, Z m αc λ α λ α αλ Zϑ tanh λ 4

15 Optimal rektangulär fläns, optimal rectangular fin Villkor, condition d d ger optimum, gives optimum några mellan räkningar ger; after some algera one otains /.49 λ α (3 55) 5

16 Flänsarrangemang, Fin arrangement mλϑ αλ Zϑ tanh m tanh α λ (3 5) { α u (3 58) λ tanh u 3u cosh (3 54) u ( 3 54) ur villkoret, from the condition d / d } Efter lite räknearete finner man; after some algera one finds: 3 u (3 6a) 3 3 ϑ Z tanh u 4α λ 6

17 7 Flänsens vikt, Weight of the fin M ρ Z ρ Z ρ/λ materialparameter, is the material parameter se taell 3-, see Tale 3-. λ ρ α ϑ u tanh u Z

18 Rak triangulär fläns, Straight triangular fin t f δ d t ϑ t t f Värmealans, Heat alance d ϑ + d dϑ d α β λ ösning, Solution: ϑ I β + BK( K då α ϑ λ ( ) ) β (3 6) Z B ty, as ϑ ändlig, finite för, for ϑ ϑ ( ) ϑ I β 8

19 9 triangulär fläns, triangular fin ) ( I β ϑ 65) (3 ) ( I ) ( I β β ϑ ϑ d dt λ d ) ( di ) ( I Z β β ϑ λ

20 triangulär fläns, triangular fin. ξ β ξ ξ β ξ d d d d d d d d Inför / introduce, ξ ξ β β d ) ( di / d ) ( di ) ( I ) ( I / β λ α β β 66) (3 ) ( I ) ( I Z β β αλ ϑ

21 Triangular fin Optimal triangulär fläns, optimal triangular fin. Ma. värmeflöde vid given vikt, maimum heat flu at given weight. /.39 λ α (3 67)

22 Sammanställning rektangulär och triangulär fläns, summary of formulae for rectangular and triangular fins ϑ ϑ t tf t t f cosh m( ) cosh m (3 38) ϑ ϑ I ( I ( β) β) (3 65) m α λ β α λ η αλ Zϑ tanh m λ α tanh m (3 4) η I( αλ Zϑ I( λ I ( β) α I ( β) β) β) ϕ tanh m m ϕ I( β) / I ( β β) Optimal fläns, optimal fin (ma värmeflöde vid given vikt) λ.49 (3 55) / α Optimal fläns, optimal fin (ma värmeflöde vid given vikt) λ.39 (3 67) / α

23 Formler för flänsverkningsgrader. Formulas for fin efficiencies Några enkla räkningar ger följande resultat, Some simple calculations give: Rektangulär fläns, rectangular fin η λ α tanh m ϕ tanh m m Triangulär fläns, triangular fin η λ α I I ( ( β) β) ϕ I( β) / I( β β) 3

24 Cirkulära flänsar, circular or annular fins r r Värmeledande ytan, heat conducting area πr Konvektiv omkrets, convective perimeter C πr 4πr 4

25 Fin efficiency circular fins 8 ϕ (%) 6 4 r c r + / c + / r c /r 3 4 r r c α /( λ) 5

26 6 nvändning av flänsverkningsgraden ϕ, How to use the fin efficiency in engineering calculations s flänsar area oflänsad fin unfinned area + + ) ( ) ( ) ( ) ( f fins f fläns t t fins f t t flänsar f t t t t + + α λ α λ ϕ α ϕ α { } { } ) 7 (3 ) ( 7) (3 ) ( + + fins f fläns f t t t t ϕ α ϕ α fläns

(b) flat (continuous) fins on an array of tubes

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