Transport processes (TRP) Transport processes – Part 3a Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik tel. 3223 ; ron.zevenhoven@abo.fi VST rz18 2/38 3a Transport processes (TRP) VST rz18
Transport processes (TRP) Transport processes (TRP) more general: T=T 1 VST rz18 VST rz18 4/38 3a 3/38 3a
Transport processes (TRP) Transport processes (TRP) θ = (T –T 1 )/(T 0 –T 1 ) more general: VST rz18 VST rz18 6/38 3a 5/38 3a
7/38 3a Transport processes (TRP) VST rz18 8/38 3a Orthogonality Transport processes (TRP) 1 2 cos ( ) d sin( ) cos( ) k k k k 0 VST rz18
Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 10/38 3a 9/38 3a
11/38 3a Transport processes (TRP) VST rz18 12/38 3a Transport processes (TRP) Note: if the source gives transport in only one direction, then take 2x in that direction and 0x in opposite direction VST rz18
Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 14/38 3a 13/38 3a
Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 16/38 3a 15/38 3a
17/38 3a Transport processes (TRP) VST rz18 18/38 3a Transport processes (TRP) c T p Q c Tdx dx 1 ( x ) dx ( Dirac ) p Q Q Q T ( x ) ( x ) T ( x , t ) ( x , t ) c c p p VST rz18
19/38 3a b is a position ! Transport processes (TRP) But: note the Second Law of Thermodynamics!!! VST rz18 20/38 3a dQ " dT " c w p dt ' dt ' " dT Transport processes (TRP) w t dT dt ' c G ( x , t t ' ) dt ' p dt ' 0 VST rz18
21/38 3a ~ (3.22) Transport processes (TRP) ~ (3.23) 2 r q 4 a ( t t ' ) VST rz18 22/38 3a Transport processes (TRP) 2 2 2 2 x y x y Q " 1 Q " 1 y ( 3 . 39 ) : e 4 at e 4 at dy e 4 at 4 at e 4 at d c 4 at c 4 at 4 at p p 2 2 x x Q " 1 Q " 1 e 4 at 4 at erf ( ) erf ( ) e 4 at 4 at 1 1 c 4 at 2 c 4 at 2 p p VST rz18
23/38 3a Transport processes (TRP) VST rz18 24/38 3a Transport processes (TRP) 2 x dT 2 1 dT 2 1 T T e 4 at @ x 0 : T 1 1 1 dx dx 2 at 2 at at derf ( x ) 2 derfc ( x ) Penetration 2 x e depth dx dx VST rz18
25/38 3a Transport processes (TRP) VST rz18 26/38 3a Transport processes (TRP) " d d dc d dc m D D dt dt dx dx dt VST rz18
Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 ~ (2.87) 28/38 3a 27/38 3a
Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 30/38 3a 29/38 3a
31/38 3a Transport processes (TRP) Example contact temperature VST rz18 Contact temperature – example /1 • A (human) tongue may freeze to a bridge-rail depending on the material. Transport processes (TRP) • Consider the contact temperature for two types of bridge-rail material: steel: λ b = 40 W/(m.K), a b = 10 -5 m 2 /s wood: λ b = 0.2 W/(m.K), a b = 1.2·10 -7 m 2 /s • Tongue data: T a0 = 35°C, λ a = 0.55 W/(m.K), a a = 1.7·10 -7 m 2 /s • Bridge temperature: T b0 < 0°C • Calculate T b0 giving contact temperature 0°C VST rz18
Contact temperature – example /2 • Use (equation 3.59): Transport processes (TRP) 6 . a T . 35 0 55 1 7 10 b b 0 b 0 6 0 . 55 a 1 . 7 10 b b • This gives for steel: T b0 = - 3.7°C for wood: T b0 = - 81°C • This shows that a wooden bridge need not be a problem. VST rz18 34/38 3a Transport processes (TRP) VST rz18
Transport processes (TRP) Transport processes (TRP) x = 0 x = b x = L VST rz18 VST rz18 36/38 3a 35/38 3a
37/38 3a Transport processes (TRP) VST rz18 38/38 3a Sources used (besides course book Hanjalić et al. ) • Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena” Wiley, 2nd edition (1999) Transport processes (TRP) • R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley, New York (1960) • * C.J. Hoogendoorn ”Fysische Transportverschijnselen II”, TU Delft / D.U.M., the Netherlands 2nd. ed. (1985) • * C.J. Hoogendoorn, T.H. van der Meer ”Fysische Transport- verschijnselen II”, TU Delft /VSSD, the Netherlands 3nd. ed. (1991) • D. Kaminski, M. Jensen ”Introduction to Thermal and Fluids Engineering”, Wiley (2005) • S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006) * Earlier versions of Hanjalić et al. book but in Dutch VST rz18
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