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Free Convection: Chapter 9

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Free Convection: Chapter 9 Free Convection: Chapter 9 Free Convection 1 General Considerations • Free convection refers to fluid motion induced by buoyancy forces. • Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density. • In heat transfer, density gradients are due to temperature gradients and the body force is gravitational. • Stable and Unstable Temperature Gradients Free Convection 2 General Considerations (cont.) Free Boundary Flows Ø Occur in an extensive (in principle, infinite), quiescent (motionless at locations far from the source of buoyancy) fluid. Ø Plumes and Buoyant Jets: • Free Convection Boundary Layers Ø Boundary layer flow on a hot or cold(TTs ≠ surface∞ ) induced by buoyancy forces. General Considerations (cont.) Combustion in Microgravity • Effective combustion occurs due to effective transfer of fresh oxidizer (air) by combined effect of buoyancy-driven convection molecular diffusion. • In space (microgravity), only molecular diffusion provides the reaction zone with fresh oxidizer (air) http://www.spaceflight.esa.int/ impress/text/education/ Microgravity/Why %20Do_Microgravity_Research. html On Earth In space Free Convection • In the previous discussions, a free stream velocity set up the conditions for convective heat transfer. • Due to friction with the surface, the flow must be maintained by a fan or pump– thus it is called forced convection. • An alternate situation occurs when a flow moves naturally due to buoyancy forces • This so called “free ” or “natural” convection and it is illustrated in the figure. Free Convection 5 Free Convection [2] • Buoyancy is the result of difference in density between materials. • In this case, the difference in density is due to the difference in temperature. • In the figure, the air next to the plate is heated, its density decreases, and the resulting buoyancy forces the air to rise. • However, note that only the flow inside the thermal boundary layer moves– the velocity is zero both at the wall and far away from it. Free Convection 6 Non-dimensional Coefficients Ø Grashof Number: g T T L3 β ( s − ∞ ) Buoyancy Force GrL = 2 ∼ ν Viscous Force L → characteristic length of surface β → thermal expansion coefficient (a thermodynamic property of the fluid) Perfect Gas: ρ=p/RT 1 ⎛⎞∂ρ β = ⎜⎟ ρ ⎝⎠∂T p Liquids: β → Tables A.5, A.6 Perfect Gas: β =1/T ( K) • The Grashof plays the same role in free convection that the Reynolds number plays in forced convection. • Practically, the Grashof number is the ratio of forcing (buoyancy) forces to restraining (viscous) forces. Free Convection 7 Non-dimensional Coefficients Ø Rayleigh number: g T T x3 β ( s − ∞ ) Rax = Grx Pr = να Free Convection 8 Vertical Plates Vertical Plates • Free Convection Boundary Layer Development on a Hot Plate: x-component velocity temperature Ø Ascending flow with the maximum velocity occurring in the boundary layer and zero velocity at both the surface and outer edge. Ø How do conditions differ from those associated with forced convection? TT? Ø How do conditions differ for a (colds < ∞plate) Vertical Plates Vertical Plates: Laminar Boundary Layer • Boundary layer approximation: x momentum dp 2 ∂u ∂u 1 ∞ µ ∂ u u + v = − − g+ 2 (9.1) ∂x ∂ y ρ dx ρ ∂ y • The pressure far from the wall is hydrostatic: dp ∞ = −ρ g (9.2) dx ∞ • Substitute Eq. (9.2) into (9.1): ∂u ∂u µ ∂2u u v g / (9.3) Δρ = ρ − ρ + = (Δρ ρ)+ 2 ∞ ∂x ∂ y ρ ∂ y • The volumetric expansion coefficient: 1 ⎛ ∂ρ ⎞ 1 ρ − ρ β = − (9.4) β = − ∞ ρ ⎝⎜ ∂T ⎠⎟ ρ T −T p ∞ (TTs < ∞ )? Vertical Plates Vertical Plates: Laminar Boundary Layer • Boussinesq Approximation 1 ⎛ ∂ρ ⎞ 1 ρ − ρ β = − ≈ − ∞ ρ − ρ = ρβ T −T ρ ⎝⎜ ∂T ⎠⎟ ρ T −T ∞ ( ∞ ) p ∞ • The x-momentum equation becomes: ∂u ∂u µ ∂2u u + v = ρβ T −T + (9.5) ( ∞ ) 2 ∂x ∂ y ρ ∂ y • All together: The laminar free convection boundary layer equations ∂u ∂v + = 0 (9.6) ∂x ∂ y ∂u ∂u µ ∂2u u + v = ρβ T −T + (9.7) ∂x ∂ y ( ∞ ) ρ ∂ y2 ∂T ∂T ∂2T u + v = α (9.8) ∂x ∂ y ∂ y2 (TTs < ∞ )? Vertical Plates (cont.) • Form of the x-Momentum Equation for Laminar Flow ∂u ∂u µ ∂2u u + v = ρβ T −T + (9.5) ( ∞ ) 2 ∂x ∂ y ρ ∂ y Net Momentum Fluxes Buoyancy Force Viscous Force ( Inertia Forces) Ø Temperature dependence requires that solution for u (x,y) be obtained concurrently with solution of the boundary layer energy equation for T (x,y). ∂T ∂T ∂2T u + v = α 2 (9.8) ∂x ∂ y ∂ y – The solutions are said to be coupled. Vertical Plates (cont.) Similarity Solution Ø Based on existence of a similarity variable, η , through which the x-momentum equation may be transformedη, from a partial differential equation with two-independent variables ( x and y) to an ordinary differential equationη expressed exclusively in terms of η . 1/4 1/4 y ⎛ Gr ⎞ ⎡ ⎛ Gr ⎞ ⎤ x ⎢ x ⎥ η ≡ ⎜ ⎟ (9.13) and ψ (x, y)≡ f (η) 4ν ⎜ ⎟ (9.14) x ⎝ 4 ⎠ ⎢ ⎝ 4 ⎠ ⎥ ⎣ ⎦ Ø Transformed momentum and energy equations: 2 ''' ' ' * f +3ff −2( f ) +T = 0 (9.17) η = 0: f = f ' = 0; T * = 1. *'' *' η → ∞: f ' → 0; T * → 0. T +3Pr fT =df0 x −∗1/2 (9.18) TT− ∞ fGruT′(η) ≡=x ≡ dTTην2 ( ) − ∂ψ 2ν T −T s ∞ u = = Gr1/2 f '(η); T * ≡ ∞ ∂ y x x T −T s ∞ Ø Solve these coupled ODEs numerically Vertical Plates (cont.) Similarity Solution: Numerical Integration fT′(η) and ∗ : dimensionless x-component velocity dimensionless temperature Ø Velocity boundary layer thickness −1/4 ⎛ Gr ⎞ x Pr 0.6: 5x x 7.07 x1/4 > δ = ⎜ ⎟ = 1/4 ∼ ⎝ 4 ⎠ Gr ( x ) Vertical Plates (cont.) Nusselt number • The local Nusselt number 1/4 1/4 hx ⎛ Gr ⎞ dT * ⎛ Gr ⎞ Nu x x g(Pr) (9.19) x = = −⎜ ⎟ = ⎜ ⎟ k ⎝ 4 ⎠ dη ⎝ 4 ⎠ η=0 0.75Pr1/2 g(Pr)= 1/4 (9.20) 0.609+1.221Pr1/2 +1.238Pr ( ) • The average Nusselt number 1/4 hL 4 ⎛ Gr ⎞ Nu L g(Pr) (9.21) L = = ⎜ ⎟ k 3⎝ 4 ⎠ 4 NuL = NuL 3 Transition to Turbulence Transition in a free convection boundary layer depends on the relative magnitude of the buoyancy and viscous forces in the fluid. It is customary to correlate its occurence in terms of the Rayleigh number. Ø Rayleigh Number: 3 gTTxβ ( s − ) 9 Ra== Gr Pr∞ ≈ 10 xc,, xc να Free Convection 16 Empirical Correlations: The Vertical Plate Ø All Conditions (Churchill-Chu) 2 ⎧ 1/6 ⎫ ⎪⎪0.387 RaL Nu L =+0.825 ⎨⎬9/16 4/9 ⎪⎪⎡⎤1+ 0.492/ Pr ⎩ ⎣⎦( ) ⎭ 9 Ø Laminar Flow ( Ra L < 10 ) : (More accurate for laminar flow) 1/4 0.670 RaL Nu L =+0.68 9/16 4/9 ⎡⎤1+ 0.492/ Pr ⎣⎦( ) Free Convection 17 Mixed Convection Ø A condition for which forced and free convection effects are comparable. Ø Exists if 2 (GrLL/Re) ≈ 1 - Forced convection → Gr / Re2 ≪1 ( L L ) - Free convection → Gr / Re2 ≫ 1 ( L L ) Ø Heat Transfer Correlations for Mixed Convection: nn n Nu≈ NuFC± Nu NC +→ assisting and transverse flows - → opposing flows n ≈ 3 Free Convection 18 .
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