6. Laminar and turbulent boundary layers

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1 6. Laminar and turbulent boundary layers John Richard Thome 8 avril 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

2 6.1 Some introductory ideas Figure 6.1 A boundary layer of thickness δ Boundary layer thickness on a flat surface δ = fn(u, ρ, µ, x) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

3 6.1 Some introductory ideas The dimensional functional equation for the boundary layer thickness on a flat surface. δ x = fn(re x) ν = µ ρ :kinematic viscosity. Re x : Reynolds number. Re x ρu x µ For a flat surface, where u remains constant. δ x = 4.92 (6.2) Rex = u x ν (6.1) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

4 6.1 Some introductory ideas Figure 6.4 Boundary layer on a long, flat surface with a sharp leading edge. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

5 6.1 Some introductory ideas (u av ) crit : Transitional value of the average velocity. Re critical ρd(u av) crit µ Re xcritical = u x crit ν Transition from laminar to turbulent flow. (6.3) (6.4) Re x,c = John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

6 Thermal boundary layer The wall is at temperature T w k f ( T y y=0) = (T w T )h (6.5) Where k f is the conductivity of the fluid. The following condition defined h within the fluid instead of specifying it as known information on the boundary. ( ) T w T T w T ( ) y y L =0 = hl = Nu L (6.5a) k L f The physical significance of Nu is given by Nu L = hx k f = L δ t (6.6) The nusselt number is inversely proportional to the thickness of the thermal boundary layer δ t. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

7 Thermal boundary layer Figure 6.5 The thermal boundary layer during the flow of cool fluid over a warm plate. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

8 6.2 Laminar incompressible boundary layer on a flat surface Figure 6.7 A steady, incompressible, two-dimensional flow field represented by streamlines, or lines of constant ψ. For an incompressible flow, continuity becomes u x + v y = 0 (6.11) u = u x + v y + w z = 0 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

9 Conservation of momentum Figure 6.9 Forces acting in a two-dimensional incompressible boundary layer. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

10 Conservation of momentum The external forces are : ( τ yx + τ yx y dy ) dx τ yx dx + pdy ( p p ) ( x dx τyx dy = y p ) dxdy x The rate at which A loses x-directed momentum to its surroundings is : ) [ (ρu 2 + ρu2 x dx dy ρu 2 dy + u(ρv) + ρuv ] y dy dx ρuvdx ( ρu 2 = x + ρuv ) dxdy y John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

11 Conservation of momentum We obtain one form of the steady, two-dimensional, incompressible boundary layer momentum equation. u 2 x + uv y A second form of the momentum equation. u u x + v u y = 1 ρ = 1 dp ρ dx + v 2 u y 2 (6.12) dp dx + v 2 u y 2 (6.13) If there is no pressure gradient in the flow, if p and u are constant as they would be for flow past a flat plate, so we obtain, u 2 x + (uv) = u u y x + v u y = v 2 u y 2 (6.15) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

12 The skin friction coefficient The shear stress can be obtained by using Newton s law of viscous shear. τ w = µ u y y=0 = µu Rex x The skin local friction coefficient is defined as : C f τ w ρu /2 2 = (6.33) Rex The overall skin local friction coefficient is based on the average of the shear stress : C f = ReL (6.34) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

13 6.3 The energy equation Using Fourier s law q k T h = = T w T T w T y y=0 (6.35) Pressure variations in the flow are not large enough to affect thermodynamic properties. Density changes result only from temperature changes and will also be small (incompressible behaviour). Temperature varaitions in the flow are not large enough to change k significantly. Viscous stresses do not dissipate enough energy to warm the fluid significantly. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

14 6.3 The energy equation We write conservation of energy in the form. d ( ) ρû dr = ρĥ u n ds dt R S }{{}}{{} rate of internal energy increase in R rate of internal energy and work out of R ( k T ) n ds + q dr (6.36) S }{{} net heat conduction rate out of R For a constant pressure flow field. ρc p T t }{{} energy storage + u T }{{} enthalpy convection = R }{{} rate of heat generation in R k 2 T }{{} + q }{{} heat conduction heat generation (6.37) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

15 6.3 The energy equation In a steady two-dimensional flow field without heat sources, equation 6.37 takes the following forme. With this assumption, 2 T / x 2 2 T / y 2, so the boundary layer form is u T x + v T y = T α 2 y 2 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

16 6.4 The Prandtl number and the boundary layer thickness To look more closely at the implications of the similarity between the velocity and the thermal boundary layers. h = fn(k, x, ρ, c p, µ, u ) We can find the following number by dimension analysis. Prandtl number Pr ν α Relative effectiveness of momentum and energy transport by diffusion in the velocity and thermal boundary layers where for laminar flow. Relationship with other dimensionless number. Nu x = fn(re x, Pr) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

17 6.4 The Prandtl number and the boundary layer thickness For simple monatomic gases, Pr = 2 3. For diatomic gases in which vibration is unexcited, Pr = 5 7. As the complexity of gas molecules increase, Pr approaches an upper value of unity. For liquids composed of fairly simple molecules, excluding metals, Pr is of the order of magnitude of 1 of 10. For liquid metals, Pr is of the order of magnitude of 10 2 or less. Boundary layer thickness, δ and δ t, and the Prandtl number When Pr > 1, δ > δ t, and when Pr < 1, δ < δ t. This is because high viscosity leads to a thick velocity boundary layer, and a high thermal diffusivity should give a thick thermal boundary layer. δ ( ν ) = fn δ t α only. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

18 6.5 Heat transfer coefficient for laminar, incompressible flow over a flat surface The following equation expresses the conservation of thermal energy in integrated form. d δt u(t T )dy = q w (6.47) dx 0 ρc p Predicting the temperature distribution in the laminar thermal boundary layer T T = 1 3 y + 1 ( ) 3 y (6.50) T w T 2 δ t 2 δ t John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

19 Predicting the heat flux in the laminar boundary layer Figure 6.14 The exact and approximate Prandtl number influence on the ratio of boundary layer thicknesses. δ t δ = Pr [ 1/3 1 ( δt 2 /14δ 2)] 1 (6.54) 1/ Pr 1/3 So we can write for 0.6 Pr 50. δ t δ = Pr 1/3 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

20 Predicting the heat flux in the laminar boundary layer The following expression gives very accurate results under the assumptions on which it is based : a laminar two-dimensional boundary layer on a flat surface, with T w constant and 0.6 Pr 50 Nu x = 0.332Re 1/2 Pr 1/3 (6.58) High Pr At high Pr, equation 6.58 is still close correct. The exact solution is Nu x 0.339Rex 1/2 Pr 1/3, Pr John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

21 Some other laminar boundary layer heat transfer equations Figure 6.15 A laminar boundary layer in a low-pr liquid. The velocity boundary layer is so thin that u u in the thermal boundary layer. Low Pr In this case, δ t δ, the influence of the viscosity were removed from the problem and for all pratical purposes u = u everywhere. With a dimension analysis, we can find : Nu x = hx k John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

22 Some other laminar boundary layer heat transfer equations We can define a new dimensionless number. Peclet number Pe x Re x Pr = u x (6.61) α Peclet number can be interpreted as the ratio of heat capacity rate of fluid in the b.l. to axial heat conductance of b.l. The exact solution of the boundary layer equations gives, in this case : For Pe x 100, Pr 1 100, Re x Nu x = 0.565Pe 1/2 x (6.62) Nu L = 1.13Pe 1/2 L John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

23 Some other laminar boundary layer heat transfer equations Churchill-Ozoe correlation : For laminar flow over a flat isothermal plate for all Prandtl numbers is the following for Pe x > 100 Nu x = Re 1/2 x Pr 1/3 (1 + (0.0468/Pr) 2/3) 1/4 (6.63) And Nu x = 2Nu x John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

24 Some other laminar boundary layer heat transfer equations Boundary layer with an unheated starting length Figure 6.16 A b.l. with an unheated region at the leading edge. For laminar flow, with x > x 0 Nu x = 0.332Re1/2 x Pr 1/3 (1 (x 0 /x) 3/4) 1/3 (6.64) Uniform wall temp. : h Uniform heat flux : h q T = 1 L L q 1 L L T (x)dx 0 q T = 0 h(x)dx John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

25 The problem of uniform wall heat flux The exact result for Pr 0.6 is Nu x = 0.453Re 1/2 x Pr 1/3 (6.71) Nu L = Re 1/2 L Pr 1/3 Churchill and Ozoe equations for the problem of uniform wall heat flux. For Pe x > Rex 1/2 Pr 1/3 Nu x = (1 + ( /Pr) 2/3) (6.73) 1/4 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

26 6.6 The Reynolds analogy The analogy between heat and momentum transfer can now be generalized. For a flat surface with no pressure gradient : C f is the skin friction coefficient. [ d 1 ( ) u u ( y ) ] δ 1 d = 1 dx u δ 2 C f (x) (6.25) 0 0 u For constant wall temperature case : [ d 1 ( ) ( ) ] u T T y q w δ d = dx u T w T δ t ρc p u (T w T ) (6.74) But the similarity of temperature and flow boundary layers to one another suggests the following approximation, which becomes exact only when Pr = 1. ( T T δ = 1 u ) δ t 1 T w T u 2 C q w f (x) = ρc p u (T w T ) φ 2 (6.75) The result is one instance of the Reynolds-Colburn analogy. h Pr 2/3 = C f ρc p u 2 (6.76) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

27 6.6 The Reynolds analogy For use in Reynold s analogy, C f must be a pure skin friction coefficient. Stanton number : actual heat flux to the fluid St heat flux capacity of the fluid flow St h = Nu x ρc p u Re x Pr (6.77) Stanton mass transfer number : St m Sh ReSc We obtain C f 2 = St = St m This equation is known as the Reynolds analogy. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

28 6.6 The Reynolds analogy For application over a wider range, some corrections are necessary. In particular, the Chilton-Colburn analogies are : For 0.6 < Pr < 60 C f 2 = StPr 2/3 j H For 0.6 < Sc < 3000 C f 2 = St msc 2/3 j m Where j H and j m are the Colburn j factors for heat and mass transfer. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

29 6.7 Turbulent boundary layers Figure 6.17 Fluctuation of u and other quantities in a turbulent pipe flow. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

30 6.7 Turbulent boundary layers We define the actual local velocity : u = u + u. u is the average term and u is instantaneous magnitude of the fluctuation. ū = 1 T T 0 ūdt + 1 T T 0 u dt = ū + ū (6.82) Similary, we have the total shear stress and total fluxes as : ( τ tot = µ ū ) ( y ρu v q tot = k ) T y ρc pv T And the following eddy diffusivities for these processes : ρε M ū y = ρu v τ tot = ρ(ν + ε M ) ū y ε T H y = v T q tot = ρc p (α + ε H ) T y C A ε m y = u C A N A,tot = (D AB + ε m ) C A y John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

31 Turbulence near the wall We define the actual local velocity : u = u + u. u is the average term and u is instantaneous magnitude of the fluctuation. For steady, incompressible, constant property flow with time-averaged variables, the x-momentum, energy and species conservation equations are : ρ ( ū ū ) ū + v x y ( ρc p ū T T + v x y ( ū C A x + v C ) A y = dp dx + ( µ ū ) y y ρu v ) = y = y ( k T y ρc pv T ( C A D AB y v C A ) ) John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

32 6.8 Heat transfer in turbulent boundary layers For turbulent flow with Re x up to about 10 7 and above , the local friction factor is correlated by : C f,x = Re 1/5 x For turbulent flow, boundary layer development is dependent on random fluctuation of the fluid, not molecular diffusion, and thus the thermal and species boundary layers do not depend on Pr and Sc. Thus : δ δ t δ c With the Chilton-Colburn analogy, the local Nusselt number in turbulent flow is : for 0.6 < Pr < 60 Nu x = StRe x Pr = Rex 4/5 Pr 1/3 The increase in mixing of the fluid causes the turbulent boundary layer to grow more rapidly than the laminar boundary layer, and have larger friction and convection coefficients. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

33 Mixed boundary layer conditions For a laminar layer flowed by a turbulent layer, integrating the global convection coefficient over the laminar zone (0 < x x c ) and then over the turbulent zone (x c < x L) : h L = 1 { x c L } h lam dx + h turb dx L 0 Re x,c is the critical Reynolds number for transition. Setting Re x,c = , for 0.6 < Pr < 60 and < Re L 10 8 Nu L = (0.037Re 4/5 L 871)Pr 1/3 C f,l = Re 1/5 Re L L For situations where L x c and Re L Re x,c, the average Nusselt number reduce to : Nu L = 0.037Re 4/5 L Pr 1/3 C f,l = 0.074Re 1/5 L In the foregoing equations, the fluid physical properties are evaluated at the film temperature. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34 x c

34 Guidelines of application of convection methods Identify the flow geometry (flat plate, cylinder, etc.) ; Decide whether the local or surface average heat transfer coefficient is required for the problem at hand ; Choose correct reference temperature and evaluated fluid properties at that temperature ; Calculate the Reynolds number to determine if the flow is laminar or turbulent ; Calculate the Prandle number ; Select the appropriate correlation that respects the restrictions on its use ; Double-check your design with a second correlation if application is critical to operation when possible. John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril / 34

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