Free-Convection-1 - Free Free Convection Physical...

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Unformatted text preview: Free Free Convection Physical Mechanisms External & Internal Flows The Film Conductance Governing Parameters M. Van Dyke, An Album of Fluid Motion Van Dyke An Album of Fluid Motion Free Free convection... • Buoyant forces produce the fluid motion. motion. • These forces are proportional to the fluid temperature. temperature. • Thus, the velocity and temperature profiles are tightly coupled. • The boundary layer concept remains valid, but T and are not easily interpreted. interpreted. Typical values of h.... Convective Process Free convection Gases Liquids Forced convection Gases Liquids Convective phase change h (W/m2-K) 2 - 25 50-1000 25-250 50 - 20 ,000 2,500 - 200,000 External External free convection Tw or q”w x y Tambient < Tw Free convection from a vertical flat surface. Free convection from a heated cylinder. External free convection x y Free convection from two surfaces in close proximity. Free convection from an array of heated, parallel tubes. Internal Internal free convection L T1 H T1 < T0 H y x T2 < T1 T0 L Free convection in a flat, horizontal layer of aspect ratio H:L << 1. Free convection in a vertical slot of aspect ratio H:L > 1. Physical mechanism - Buoyant forces Buoyant Force = F = g(ρ −ρ)Δϑ Z B 0 Mass of fluid, m, at densityρ Gravity force (weight) = - gm =gρΔϑ Weight of displaced fluid in the surroundings at 0(T0) is g 0 Equation Equation of state Net force on the fluid mass: Δ F = g(ρ0 − ρ )Δ ϑ (33.1) The buoyant force is connected to temperature and thus heat transfer through the equation of state . at constant total pressure: ρ = ρ 0 [1 − β ( T − T0 )] β =− 1 ⎛ ∂ρ ⎞ ρ ⎝ ∂T ⎠ P (33.2a) (33.2b) Isobaric coefficient of thermal expansion ρ = ρ 0 [1 − β (T − T0 )] (33.2a) 1 ⎛ ∂ρ ⎞ β =− ρ ⎝ ∂T ⎠ P (33.2b) β is the isobaric coefficient of therml expansion of the fluid. For ideal gases, β =T-1 ~ 3 x 10-3 K-1. The implication is that the free convection is associated with small temperature differences, and thus characteristic velocities are small due to buoyancy. External External flow and heat transfer in free convection - the vertical plate Temperature is coupled to buoyant forces that drive the forces that drive the flow in the boundary layer. Thus the flow and heat transfer problems are tightly coupled. In gases, the thermal and velocity boundary layer are almost identical. Turbulent boundary layer flow and heat transfer heat transfer Vertical heat plate plate at either constant temperature or constant heat flux. Transition region. Laminar boundary layer flow and heat transfer Ambient at Ta and constant total pressure. The vertical plate T ( x, y) Turbulent boundary layer flow and heat transfer u( x , y) Laminar boundary layer flow and heat transfer Vertical heat plate at either constant temperature or constant heat flux. X y Note: the thermal boundary layer, δT(x), and velocity boundary layer, δ(x), are nearly identical for air. Film Film conductance & the heat transfer coefficient Newton’s Law of Cooling is used as in forced convection. All of the quantities that determine the heat transfer coefficient in forced convection heat transfer coefficient in forced convection are also present in free convection. V m Fluid particles move with velocity V that is due to the buoyancy created by the heated wall. y The buoyant force is due to the difference in density between fluid particles near the wall and far from the wall. This density difference is produced by the volumetric expansion of the fluid, which depends on the local fluid temperature through the equation of state. X The general equation for heat transfer Consider the fluid particle at an average temperature Tavg in the the boundary layer. V Tavg = T0 + ΔT 2 T0 y m T0 + ΔT X The change in density of the particle is given by given by ρ = ρ 0 [1 + β ( Tavg − T0 )] ρ − ρ0 = ρ0β Δ T 2 (33.3) The The general equation for heat transfer ΔT Thus Δρ is given by Δ ρ = ρ0 β 2 The buoyant force is given by (33.4) FB = gΔρ = ρ0 gβ ΔT 2 (33.5) The work done by the buoyant force over a distance work done by the buoyant force over distance L in the direction of FB is FB L = gΔ ρL = ρ 0 gβL ΔT 2 (33.6) The general equation for heat transfer δWB = ρ0 gβL ΔT 2 L V Assume that the work done by buoyancy produces a change in kinetic energy of the fluid particle, or X δWB ≈ δKE = ρ0V 2 y 2 (33.7) The The general equation for heat transfer V δWB ≈ δKE L ρ0 V 2 2 ≈ ρ 0 gβL ΔT 2 2 (33.8) X y V ≈ gβL ΔT (33.9) The general equation for heat transfer From the analysis of forced convection and and Buckingham Pi Theorem, we have a b hL = C ⎛ LVρ ⎞ ⎛ cμ ⎞ ⎜ ⎜ μ ⎟⎜k⎟ ⎟ k ⎝ ⎠⎝ ⎠ 2 2 2 (33.10) where a and b are constants, and L is the vertical distance. Rewrite Eq. (33.10) in the following way hL = C ⎛ L V ρ ⎜ ⎜ μ2 k ⎝ ⎞ ⎟ ⎟ ⎠ a/2 ⎛ cμ ⎞ ⎜⎟ ⎝k⎠ b (33.11) The The general equation for heat transfer Using the approximate value of velocity from Eq. (33.9), we get hL = C ⎛ gβL ρ ΔT ⎞ ⎜ ⎟ 2 ⎜ ⎟ k μ ⎝ ⎠ 3 2 Nusselt Number Grashof Number a/2 ⎛ cμ ⎞ (33.12) ⎜⎟ ⎝k⎠ b Prandtl Number Note that the distance L here becomes a characteristic dimension of the heat transfer surface, e.g., length or diamter. Grashof and Rayleigh numbers Consider a plate at constant wall temperature. The local Grashoff number is the grouping: Grx = gβ (Tw − T∞ ) x 3 ν 2 (33.13) The Rayleigh number is Ra = GrPr. or Ra x = Grx Pr = gβ (Tw − T∞ ) x 3 να (33.14) Grashof Grashof and Rayleigh numbers For an overall characteristic dimension L, the overall Grashof number is, Grx = gβ (Tw − T∞ ) L3 ν2 gβ (Tw − T∞ ) L3 (33.15) The overall Rayleigh number is, Ra x = Grx Pr = να (33.16) The general equation for heat transfer From Eq. (33.12), the general result is, Nu L = CGr L a/2 Pr c (33.17) where L is the distance or length scale of choice. Eq. (33.17) is successfully applied to heated flat plates of length L, long horizontal cylinders of diameter D, and spheres of diameter, D. For long vertical cylinders, the appropriate length is the height of the cylinder. Experimental Experimental heat transfer correlations The vertical plate at Tw = constant: Nu L ,avg = CRa L n (33.18) Laminar flow and heat transfer: C = 0.59 n = 0.25 RaL < 109 Turbulent flow and heat transfer: flow and heat transfer C = 0.10 n = 1/3 RaL > 109 Properties at mean film temperature, (Tw+Ta)/2. Experimental heat transfer correlations Horizontal cylinders at constant temperature: n (33.18) L , avg L Nu = CRa 104 < RaL < 109 C = 0.53 n = 1/4 109 < RaL < 1012 C = 0.13 n = 1/3 Properties at mean film temperature, (Tw+Ta)/2. Experimental Experimental heat transfer correlations Horizontal cylinders at constant temperature: Design guideline: A vertical cylinder may be treated as a plate if D 35 ≥ 1 L GrL / 4 (33.19) The Churchill -Chu correlations for flat plates Nu avg = 0 .68 + 0 .670 Ra 1 / 4 ⎡ ⎛ 0 .492 ⎞ 9 / 16 ⎤ ⎟ ⎢1 + ⎜ ⎥ ⎝ Pr ⎠ ⎥ ⎢ ⎣ ⎦ 0 .387 Ra 1 / 6 ⎡ ⎛ 0 .492 ⎞ 9 / 16 ⎤ ⎟ ⎢1 + ⎜ ⎥ ⎢ ⎝ Pr ⎠ ⎥ ⎣ ⎦ 8 / 27 4/9 , Ra < 10 9 (33.20) Nu 1/ 2 avg = 0 .825 + , 0 .10 < Ra < 10 7 (33.21) Eq. (33.20) can be used for constant heat flux or constant temperature. Eq. (33.21) can be used for constant temperature. Properties are evaluated at the mean film temperature for both. Nusselt Nusselt numbers for vertical plates at g β q ′′x Gr = Gr Nu = constant heat flux kν * x x x 2 4 Laminar heat transfer for air or water Nu x = 0 .60 ( Gr x* Pr) 1 / 5 10 5 < Gr x* < 10 11 (35.4) Turbulent heat transfer for air or water Nu x = 0.17 (Grx* Pr) 1 / 4 2 x10 13 < Grx* Pr < 10 16 For both correlations: (35.5) (1) Constant wall heat flux (2) Properties at the film temperature Cylinders... Vertical Vertical cylinders Design guideline: A vertical cylinder may be treated as a vertical plate if as vertical plate if D 35 ≥ 1/ 4 L GrL (35.6) Nusselt numbers for horizontal cylinders Horizontal cylinders at Tw = Constant. n L , avg L Nu = CRa 104 < RaL < 109 C = 0.53 n = 1/4 109 < RaL < 1012 Ra 10 C = 0.13 n = 1/3 (35.7) Properties at mean film temperature, (Tw+Ta)/2. Nusselt Nusselt numbers for horizontal cylinders Constant wall temperature: 1/ 2 Nu avg ⎧ ⎫ ⎪ ⎪ Gr Pr = 0.60 + 0.387⎨ 16 / 9 ⎬ ⎡ ⎛ 0.559⎞ 9 / 16 ⎤ ⎪ 1+ ⎪ ⎥ ⎢ ⎝ Pr ⎠ ⎦ ⎩⎣ ⎭ 1/ 6 , 10 −5 < Gr Pr < 10 12 (35.8) −6 9 Nud,avg = 0.36 + 0.518(Grd Pr) ⎡ ⎛ 0.559⎞ 1+ ⎢ ⎝ Pr ⎠ ⎣ 1/ 4 9 /16 4 / 9 ⎤ ⎥ ⎦ , 10 < Grd Pr < 10 G (35.9) Properties evaluated at the film temperature. Horizontal surfaces… Horizontal Horizontal surfaces Constant wall temperature: (1) Use a characteristic length dimension of L = A/P A/P (2) Properties evaluated at the film temperature Constant wall heat flux: Nu L ,avg = 0.13(GrL Pr) 1 / 3 , GrL Pr < 2 x10 8 Nu L ,avg = 0.16(GrL Pr) 1 / 3 , 2 x10 8 < GrL Pr < 1011 (35.8a,b) Horizontal surfaces For Eqs. (35.8a,b), the following apply: (1) All properties are at an effective film temperature, Te, except β. Te = Tw , avg − 0.25 (Tw , avg − T∞ ) where the average wall temperature(35.9) is obtained from the Newton cooling law with wall heat flux specified. Spheres... Spheres... Nusselt numbers for spheres Constant temperature surface: Nu d , avg = 2 + 0 .589 Ra 1 / 4 d ⎡ ⎛ 0 .469 ⎞ ⎟ ⎢1 + ⎜ ⎢ ⎝ Pr ⎠ ⎣ 9 / 16 ⎤ ⎥4 / 9 ⎥ ⎦ , Ra d < 10 11 , Pr > 0 .5 (35.10) Properties at the film temperature. ...
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