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External Convection: Laminar Flat Plate for a Constant Property, Laminar Flow a Similarity Solution Exists for the Flow Field U(Y)

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External Convection: Laminar Flat Plate for a Constant Property, Laminar Flow a Similarity Solution Exists for the Flow Field U(Y) AME 60634 Int. Heat Trans. External Convection: Laminar Flat Plate For a constant property, laminar flow a similarity solution exists for the flow field u(y) Major flow parameters: 5x local boundary layer thickness δx = 1 2 Rex 1 − 2 local skin friction coefficient C f ,x = 0.664Rex € average skin friction coefficient τ 1 x € C = s,x ⇒ τ = τ dx f ,x 2 s,x ∫ s,x 1 ρu x 2 ∞ 0 1 − 2 ⇒ C f ,x =1.328Rex D. B. Go AME 60634 Int. Heat Trans. External Convection: Laminar Flat Plate For a constant property, laminar flow a similarity solution exists for the flow field u(y) Major heat transfer parameters: 1 local thermal boundary layer thickness 3 δx δt,x = Pr uniform surface temperature, Ts 1 1 local Nusselt number Nu = 0.332Re 2 Pr 3 (Pr > 0.6) x x x 1 1 hxx 1 2 3 average Nusselt number x Nux = ⇒ hx = ∫ hx dx ⇒ Nu = 0.664Rex Pr k x 0 € uniform surface heat flux, q”s 1 1 2 3 local Nusselt number Nux = 0.453Rex Pr (Pr > 0.6) L 1 1 q#s#L q## x 2 3 average Nusselt number (Ts −T∞ ) = = ∫ dx ⇒ NuL = 0.680ReL Pr L L kNu € Nu 0 x D. B. Go AME 60634 Int. Heat Trans. External Convection: Turbulent Flat Plate For turbulent flow, only empirical relations exist 1 − 5 local skin friction coefficient C f ,x = 0.0592Rex uniform surface uniform surface local Nusselt number temperature, T heat flux, q” (Pr > 0.6) s s 4 1 4 1 Nu = 0.0296Re 5 Pr 3 5 3 € x x Nux = 0.0308Rex Pr Average parameters x 1 $ c L ' 1 # xc L & τ = & τ dx + τ dx) s,L & ∫ s,x,lam ∫ s,x,turb ) h L = % ∫ hx,lam dx + ∫ hx,turb dx( L 0 € x L € % c ( $ 0 xc ' 5 assuming xc for Rex,c = 5×10 − 1 average skin friction coefficient C = 0.074Re 5 −1742Re−1 € € f ,L L L 4 1 # 5 & 3 uniform surface average Nusselt number NuL = % 0.037ReL − 871( Pr $ ' temperature, Ts € For xc= 0 or L >> xc (Rex,L >> Rex,c) − 1 average skin friction coefficient C = 0.074Re 5 € f ,L L 4 1 uniform surface average Nusselt number Nu 0.037Re 5 Pr 3 L = L temperature, Ts D. B. Go € € AME 60634 Int. Heat Trans. External Convection: Starting Length • The effect of an unheated starting length (USL) can be represented on the local Nusselt number as: Nux 1 ξ = 0 m 3 Nux = where Nux = CRex Pr for Pr > 0.6 a b ξ = 0 * $ξ ' - ,1 −& ) / + % x ( . • Parameters a, b, C, & m depend€ on – thermal boundary condition: uniform surface temperature (UST) or € uniform heat flux (UHF) – flow conditions: laminar or turbulent LAMINAR TURBULENT Ts qs" " Ts qs" " a 3/4 3/4 9/10 9/10 b 1/3 1/3 1/9 1/9 € C €0.332 0.453 € 0.0296 € 0.0308 m 1/2 1/2 4/5 4/5 D. B. Go AME 60634 Int. Heat Trans. External Convection: Starting Length • Uniform Surface Temperature (UST) 2p 2 p 1 L (2p +1) (2p +2) ( ) ( + ) p = 1 (laminar throughout) NuL = NuL 1− (ξ L) ξ = 0 L −ξ [ ] p = 4 (turbulent throughout) 1 $ xc L ' numerical integration for h = & h dx + h dx) L L & ∫ x,lam ∫ x,turb ) laminar/turbulent flow € % ξ xc ( qx" " = hx (Ts − T∞ ) ⇒ q = h L As(Ts − T∞) • Uniform Heat Flux (UHF) € qs# # € Ts(x) = T∞ + ⇒ q = qs# # A s hx • The Nusselt number (and heat transfer coefficient) are functions of the fluid properties (ν,ρ,α,cp,k) € – the effect of variable properties may be considered by evaluating all properties at the film temperature T + T T = s ∞ f 2 – most accurate solutions often require iteration on the film properties D. B. Go € AME 60634 Int. Heat Trans. External Convection: Cylinder in Cross Flow • As with flat plate flow, flow conditions determine heat transfer • Flow conditions depend on special features of boundary layer development, including onset at stagnation point, separation, and onset of turbulence • Stagnation point: location of zero velocity and maximum pressure – boundary layer development under a favorable pressure gradient è acceleration of the free stream flow dp du∞ dx < 0 → dx > 0 • There is a minimum in the pressure distribution p(x) and toward the rear of the cylinder, the pressure increases. – boundary layer development under€ an adverse pressure gradient dp > 0 D. B. Go dx € AME 60634 Int. Heat Trans. External Convection: Cylinder in Cross Flow • Separation occurs when the momentum of the free stream flow is insufficient to overcome the adverse pressure gradient – the velocity gradient reduces to zero – flow reversal occurs accompanied by a downstream wake du = 0 dy y= 0 € • Location of separation depends on boundary layer transition VD Re = D ν note: V ≠ u∞ € D. B. Go € AME 60634 Int. Heat Trans. External Convection: Cylinder in Cross Flow • Force (FD) imposed by the flow on the cylinder is composed of two phenomena – friction è boundary layer shear stress – form drag (pressure drag) è pressure differential due to wake F A is the area projected drag C ≡ D f coefficient D 2 perpendicular to free stream A f (ρV /2) € D. B. Go AME 60634 Int. Heat Trans. External Convection: Cylinder in Cross Flow • Thermal considerations: uniform surface temperature, Ts – heat transfer a function of the angel of separation θ – empirical correlations are used to determine average Nusselt numbers h D NuD = k • Hilpert correlation: Pr ≥ 0.6 Re C m – also suitable for non-circular cylinders D € 0.4-4 0.989 0.330 1 4-40 0.911 0.385 m 3 NuD = CReD Pr 40-4000 0.683 0.466 4000-40,000 0.193 0.618 40,000-400,000 0.027 0.805 € • Churchill and Bernstein: ReDPr > 0.2 4 5 1 2 1 3 ( 5 8+ 0.62ReD Pr " ReD % NuD = 0.3+ *1 + $ ' - 2 3 1 4 [1+ (0.4 Pr) ] )* # 282,000& ,- D. B. Go € AME 60634 Int. Heat Trans. External Convection: Sphere in Cross Flow • Similar flow issues as cylinder in cross flow arise • Thermal considerations: uniform surface temperature, Ts – heat transfer again defined by empirical correlations • Whitaker correlation: – 0.71 < Pr < 380 4 – 3.5 < ReD < 7.6×10 1 4 1 2 2 3 0.4 " µ % NuD = 2 + (0.4ReD + 0.06ReD )Pr $ ' # µs & evaluate fluid properties at T∞ except for µs which is evaluated at Ts € D. B. Go AME 60634 Int. Heat Trans. External Convection: Impinging Jet • Impinging jet consists of a high speed flow impacting a flat surface – generates large convection coefficients • The flow and heat transfer are affected by a number of factors – shape/size of jet, velocity of jet, distance from plate, … • Significant hydrodynamic features: – mixing and velocity profile development in the free jet – stagnation point and zone – velocity profile development in the wall jet D. B. Go AME 60634 Int. Heat Trans. External Convection: Impinging Jet • Local Nusselt number distribution: • Average Nusselt number based on empirical correlations for single nozzles and arrays of nozzles – function of Reynolds number, Pr, distance along wall (r or x), height of jet (H) Nu = f Re ,Pr, r or x D ,H D ( Dh ( ) h h ) € D. B. Go AME 60634 Int. Heat Trans. External Convection: Impinging Jet • Martin correlation: uniform surface temperature, Ts – single round nozzle Nu = G(r D,H D)F1(ReD ) Pr 0.42 valid for 1 2 V D 2000 ≤ Re ≤ 400,000 F (Re ) = 2Re1 2 1+ 0.005Re0.55 ⇒ Re = exit D 1 D D ( D ) D ν 2 ≤ H D ≤12 (D r)(1−1.1(D r)) 2.5 ≤ r D ≤ 7.5 G = 1+ 0.1(H D − 6)(D r) • Martin correlation: uniform surface temperature,€ Ts € – single slot nozzle Nu 3.06 (Rem ) valid for 0.42 = Dh Pr x W + H W +2.78 3000 Re 90,000 ≤ Dh ≤ V D V (2W ) ⇒ Re = exit h = exit 2 ≤ H W ≤10 Dh ν ν −1 4 ≤ x W ≤ 20 +% x ( % H (1.33 . m = 0.695 −-' * + ' * + 3.060 & 2W ) & 2W ) D. B. Go , / € € .
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