Fundamental Concepts of Convection : Flow and Thermal Considerations Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.3
6.1 Boundary Layers: Physical Features Velocity Boundary Layer A consequence of viscous effects associated with relative motion between a fluid and a surface. A region of the flow characterized by shear stresses and velocity gradients. A region between the surface and the free stream whose thickness δ increases in the flow direction. Why does δ increase in the flow direction? ( ) u y δ = u 0.99 Manifested by a surface shear stress that provides a drag force,. τ How does s vary in the flow direction? Why? F D τ s u τ s = µ y= 0 y F = τ da D s s As
Thermal Boundary Layer A consequence of heat transfer between the surface and fluid. A region of the flow characterized by temperature gradients and heat fluxes. A region between the surface and the free stream whose thickness δ t increases in the flow direction. Ts T( y) Why does δ t increase in the δt = 0.99 Ts T flow direction? T Manifested by a surface heat flux q s and q s = kf y a convection heat transfer coefficient h. y ( ) q If Ts T is constant, how do s and h vary in the flow direction? h = 0 k T / y f T s T y= 0
6.2 Local andaverage Heat Transfer Coefficients Local Heat Flux and Coefficient: ( ) q = h T T s s Average Heat Flux and Coefficient for a Uniform Surface Temperature: ( ) q = has Ts T q 1 h = hda A = A s q da ( ) s = s A s s A s s T T hda For a flat plate in parallel flow: s h 1 L o = L hdx
6.3 Boundary Layer Transition What conditions are associated with transition from laminar to turbulent flow? Why is the Reynolds number an appropriate parameter for quantifying transition from laminar to turbulent flow? Transition criterion for a flat plate in parallel flow: ρu xc Rexc, critical Reynolds number µ xc location at which transition to turbulence begins 10 < Re < 3 x 10 5 6 xc, ~ ~
What may be said about transition if Re L < Re x,c? If Re L > Re x,c? Effect of transition on boundary layer thickness and local convection coefficient: Why does transition provide a significant increase in the boundary layer thickness? Why does the convection coefficient decay in the laminar region? Why does the convection coefficient decay in the turbulent region? Why does it increase significantly with transition to turbulence, despite the increase in the boundary layer thickness?
6.4 The Boundary Layer Equations Consider concurrent velocity and thermal boundary layer development for steady, two-dimensional, incompressible flow with constant fluid properties ( µ, c p, k) and negligible body forces. Apply conservation of mass, Newton s 2 nd Law of Motion and conservation of energy to a differential control volume and invoke the boundary layer approximations. Velocity Boundary Layer: Thermal Boundary Layer: 2 2 u u, p dp 2 2 x y x dx T x T y 2 2 2 2
Conservation of Mass: u x v + = y 0 In the context of flow through a differential control volume, what is the physical significance of the foregoing terms, if each is multiplied by the mass density of the fluid? Newton s Second Law of Motion: x-directi on : u x y ρ dx y 2 u u 1 dp u + v = + ν 2 What is the physical significance of each term in the foregoing equation? Why can we express the pressure gradient as dp /dx instead of p / x?
Conservation of Energy: u 2 2 T T T ν u + v = α + 2 x y y c p y What is the physical significance of each term in the foregoing equation? What is the second term on the right-hand side called and under what conditions may it be neglected?
6.5 Boundary Layer Similarity As applied to the boundary layers, the principle of similarity is based on determining similarity parameters that facilitate application of results obtained for a surface experiencing one set of conditions to geometrically similar surfaces experiencing different conditions. Dependent boundary layer variables of interest are: τ s and q or h For a prescribed geometry, the corresponding independent variables are: Geometrical: Size (L), Location (x,y) Hydrodynamic: Velocity (V) Fluid Properties: Hydrodynamic: ρ, µ Thermal : cp, k ( ρ µ ) ( x, L, V,, ) Hence, u = f x, y, L, V,, τ = f ρµ s
( ρ µ p ) ( ρ µ p ) and T = f x, y, L, V,,, c, k ; h= f x, L, V,,, c, k Key similarity parameters may be inferred by non-dimensionalizing the momentum and energy equations. Recast the boundary layer equations by introducing dimensionless forms of the independent and dependent variables. x y u v T Ts x y u v T L L V V T T Neglecting viscous dissipation, the following normalized forms of the x-momentum and energy equations are obtained: u u u u dp 1 u 2 + v = + 2 x y dx ReL y 2 T T 1 T + v = 2 x y ReL Pr y s
ρvl VL Re L = the Reynolds Number µ v cpµ v Pr = the Prandtl Number k α How may the Reynolds and Prandtl numbers be interpreted physically? For a prescribed geometry, u = f ( x, y,re L ) u µ V u τ s = µ = y L y y= 0 y = 0 The dimensionless shear stress, or local friction coefficient, is then C f τ = 2 u s 2 ρ V /2 ReL y y = 0 u = f ( x,re L ) y y = 0 2 Cf = f x Re L (,ReL) What is the functional dependence of the average friction coefficient?
For a prescribed geometry, T = f x y (,,Re,Pr) L h ( T Ts ) T T ( ) kf T / y y= 0 kf kf = = =+ T T L T T y L y s s y = 0 y = 0 The dimensionless local convection coefficient is then hl T Nu = = f x kf y y = 0 Nu local Nusselt number (,Re,Pr) L What is the functional dependence of the average Nusselt number? How does the Nusselt number differ from the Biot number?
6.6 Physical Significance of the Dimensionless Parameters Re L 2 ρvl ρv inertial force = = = µ µ V / L viscous force δ δ t Pr for a gas: for a liquid metal: for an oil: n δ t t δ δ δ δ t δ
6.7 The Reynolds Analogy Equivalence of dimensionless momentum and energy equations for negligible pressure gradient (dp/dx~0) and Pr~1: u u u 1 u x y Re y 2 + v = 2 Advection terms Diffusion u T T 1 T 2 + v = 2 x y Re y Hence, for equivalent boundary conditions, the solutions are of the same form: u u = T = y y = 0 y = 0 C f Re 2 T y = Nu
or, with the Stanton number defined as, h Nu St = ρvc Re Pr p With Pr = 1, the Reynolds analogy, which relates important parameters of the velocity and thermal boundary layers, is C f 2 = St Modified Reynolds (Chilton-Colburn) Analogy: An empirical result that extends applicability of the Reynolds analogy: C f 2 2 3 = St Pr j 0.6 < Pr < 60 Colburn j factor for heat transfer Applicable to laminar flow if dp/dx ~ 0. H Generally applicable to turbulent flow without restriction on dp/dx.
Problem 6.19: Determination of heat transfer rate for prescribed turbine blade operating conditions from wind tunnel data obtained for a geometrically similar but smaller blade. The blade surface area may be assumed to be directly proportional to its characteristic length A L. ( ) s SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Constant properties, (3) Surface area A is directly proportional to characteristic length L, (4) Negligible radiation, (5) Blade shapes are geometrically similar. ANALYSIS: For a prescribed geometry, hl Nu = = f( Re L,Pr ). k
The Reynolds numbers for the blades are Re 2 2 L,1 = ( V1 L 1 / ν1 ) = 15m /s ν1 ReL,2 = ( V2 L 2 / ν2 ) = 15m /s ν2. Hence, with constant properties ( v v ) 1= 2, ReL,1 = Re L,2. Also, Pr 1 = Pr 2 Therefore, Nu = Nu 2 1 ( h 2 L 2 /k 2) = ( h 1 L 1 /k 1) L1 L1 q h 1 2 = h1 = L2 L2 A1 Ts,1 T ( )
The heat rate for the second blade is then ( Ts,2 T ) ( ) L1 A q 2 2 = h2a2 ( Ts,2 T ) = q1 L2 A1 Ts,1 T ( ) ( ) ( ) Ts,2 T 400 35 q2 = q1 = 1500 W Ts,1 T 300 35 q2 = 2066 W. COMMENTS: (i) The variation in ν from Case 1 to Case 2 would cause Re L,2 to differ from Re L,1. However for air and the prescribed temperatures, this non-constant property effect is small. (ii) If the Reynolds numbers were not equal ( L,1 L2 ) ( ReL, Pr) f would be needed to determine h 2. Re Re, knowledge of the specific form of