Transport processes Part 5 Ron Zevenhoven bo Akademi University - PDF document

Transport processes (TRP) Transport processes – Part 5 Ron Zevenhoven Åbo Akademi University Thermal and Flow Engineering / Värme- och strömningsteknik tel. 3223 ; ron.zevenhoven@abo.fi VST rz18 2/39 5 Transport processes (TRP) VST rz18

3/39 5 Transport processes (TRP) Steady state: ∂ ../ ∂ t = 0  V     v  0    0   2  T neglecting  2 x VST rz18 4/39 5 Transport processes (TRP) Pr 1/3 = thickness thermal boundary layer / hydraulic boundary layer Sc 1/3 = thickness mass transfer boundary layer / hydraulic boundary layer VST rz18

5/39 5 Transport processes (TRP) T 0 (T-T 0 )/(T 1 -T 0 ) VST rz18 6/39 5 (T-T 0 )/(T 1 -T 0 ) Transport processes (TRP) λ (T 1 -T 0 ) √ … T 0 VST rz18

7/39 5 Transport processes (TRP) (T 1 -T 0 ) VST rz18 8/39 5 T 0 Transport processes (TRP) T 0 T - T 0 = T(x, y=0) - T 0 = q/(T(x,y=0) - T 0 ) = VST rz18

9/39 5 q/(T(x)-T 0 ) = Transport processes (TRP) 0.89 ~ √(½π) 1.77 ~ 2√(½π) VST rz18 10/39 5 Transport processes (TRP) 5.78 ~(2.405) 2 VST rz18

11/39 5 Transport processes (TRP) against direction r : - → +  T  can be used instead of T VST rz18 12/39 5 Transport processes (TRP) VST rz18

13/39 5 Transport processes (TRP) See section 5.1 Ay  Blasius : V x VST rz18 Lévêque problem Transport processes (TRP) & • Lines of constant concentration in a Lévêque problem description   m μ • Note that the substitution of (5.45) into (5.41) requires that  ν   n which with m = 1 gives n = (ν-1)/(μ+2) VST rz18

15/39 5    c n c     x x  2  2  2   c m c m ( m 1 ) c     2 2   2 2   y y y   ( 1 )   m 1 and n  µ 2  m µ 2  the boundary condition requires Transport processes (TRP)   n 1      c ( 1 ) c      x µ 2 x 2 2 2  c   c  2 2 2  y y   VST rz18 16/39 5 Transport processes (TRP)  µ 1 1      x 1 x   µ 2 µ 2 p       t x  1    t x  1 ( x , p ) e t dt ( x ) e t dt 0 0 VST rz18

Transport processes (TRP) Transport processes (TRP) VST rz18 VST rz18 18/39 5 17/39 5

19/39 5 Transport processes (TRP) See end of chapter 4 VST rz18 20/39 5 Transport processes (TRP)   R y   y   y y    2    2   2 2 V  1 ( )  2 V  1 ( 1 )  2 V  2 ( )   R   R   R R  VST rz18

21/39 5 Transport processes (TRP) 4 1/3 = 1.59 1.59·0.539 = 0.855 VST rz18 22/39 5 Transport processes (TRP) Gz corrects for entrance region growing boundary layer thickness VST rz18

23/39 5 Transport processes (TRP) Ideal gas β ~1/T(K) VST rz18 24/39 5 difference with respect to average Transport processes (TRP) VST rz18

Transport processes (TRP) Transport processes (TRP) 1960 p. 297 - 300 Bird Stewart & Lightfoot See also VST rz18 VST rz18 26/39 5 25/39 5

Transport processes (TRP) Transport processes (TRP) Momentum balance Energy balance VST rz18 VST rz18 28/39 5 27/39 5

29/39 5   v p        µ z g ( T T )  2  y z  p    g due to weight  z Transport processes (TRP) T(y) VST rz18 Example Lévêque problem /1question Transport processes (TRP) VST rz18

Example Lévêque problem /2question Transport processes (TRP) VST rz18 Example Lévêque problem /3answer Transport processes (TRP) VST rz18

Example Lévêque problem /4answer Transport processes (TRP) VST rz18 Example Lévêque problem /5answer Transport processes (TRP) VST rz18

Example Lévêque problem /6answer Transport processes (TRP) 110 VST rz18 36/39 A classroom exercise - 6 • Derive equation (5.21) in the course material: Transport processes (TRP) differential equation (5.7) with boundary conditions (5.18, 5.19, 5.20). • Note that for the Laplace transform £ { τ } = p, and £ {T} = T the boundary condition (5.19) becomes . d T q    dy p • And, what would be the result with T = T 0 in boundary conditions (5.18) and (5.20)? VST rz18

Mass transfer, phase equilibrium • Note: For heat transfer problems the driving force is a temperature gradient which is continuous at Transport processes (TRP) material boundaries or phase boundaries, and transport proceeds until thermal equilibrium is reached. For mass transfer the driving force for transfer across phase (or material) boundaries is the deviation from chemical - and phase equilibrium, and transport proceeds until the different concentrations reach this equilibrium. • Example: VST rz18 januari Mass 2018 transfer, phase Transport processes (TRP) equilibrium Driving force = c equilibrium – c, saturation at c = c equilibrium VST rz18

Sources used 39/39 5 (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) • J.R. Welty, C.E. Wicks, R.E. Wilson. “Fundamentals of momentum, heat and mass transfer” Wiley New York (1969) * Earlier versions of Hanjalić et al. book but in Dutch VST rz18