Chapter 5 Principles of Convection heat transfer (Text: J. P. Holman, Heat Transfer, 8 th ed., McGraw Hill, NY)

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1 hapter 5 Principles of onvection heat transfer (Tet: J. P. Holman, Heat Transfer, 8 th ed., McGra Hill, NY) onsider a fluid flo over a flat plate ith different temperatures (Fig 5-1) q A ha( T T ) since depends on velocity of the steram h f ( fluid, flo pattern) We term the heat transfer depends on relative motions as convection heat transfer. The problem is ho to evaluate/predict/estimate the value of h for various flo pattern? Flo lassification: (fig. ) Evaluation of convection heat transfer 1. Analytical solution of the fluid temperature distribution. Analogy beteen heat & momentum transfer 3. Dimensional analysis+eperimental data in terms of dimensionless No. Viscous flo (Fig) aminar boundary layers on a flat plate (Fig.) dp Assumptions: 1 incompressible, steady flo,. dy, 3. constant physical properties, and 4. viscous shear force in y-dir are negligible. onservation of Mass (Fig.) v ρudy + ρvd ρ( u + d) dy + ρ( v + dy) d v +... ontinuity eq. for boundary layer onservation of momentum (fig.) F increase in momentum flu in dir

2 Shear force at the button face isτ μ y d u net shear force μ ddy Shear force at the top face is τ μd[ + ( ) dy] y+ dy pressure force over the left face is Pdy P P net pressure force ddy pressure force over the right face is [ P+ d] dy v u v u u P y y y ρuvd + ρ udyd + ρv d + ρ dyd ( μ ) dyd u u u P + μ y y ρ( u v )... Momentum eq. for B.. onservation of Energy Further assumptions: 1. negligible heat conduction in -dir (Fig.) Viscous shear force: u d μ The distance hich it moves per unit time in respect to the control volume is: u the viscous energy is : μ( ) ddy dy If e neglect the nd order differential terms, the energy equation can be derived as T T u v T u μ ρc[ u v T ( )] ddy ddy ( ) ddy T T T μ u + α + u v ( )... energy eq. for B.. ρc Order of magnitude analysis for the to terms on the right hand side u u, y δ T T α α y δ μ μ U ( ) ρc ρc δ If the ratio of these quantities is small, i.e., μ U << 1, ρα c T The viscous dissipation is small in comparison ith conduction term.

3 ν ρcν cμ Pr Prandtl No. α U Pr << 1 ct T T T u + v α... energy eq. u u u u + v ν... momentum eq. Prandtl number controls the relation beteen the velocity and temperature distributions. Approimate integral boundary layer analysis onsider the control volume in the B.. (Fig) Momentum flo across AB : δ ( ) ρ udy δ( ) ( ) d δ Momentum flo across D : ρu dy + [ ρu dy] d d And from the continuity eq. fluid also enters the.v. across face BD at a rate of d δ ( ) [ udy] d d ρ. The flo of -dir momentum into the.v. across upper face is U d [ ρudy] d d δ ( ) Adding up the -dir momentum d δ( ) ( ) ( ) [ d δ ] [ d δ u dy d U udy] d [ u( U u) dy] d d ρ ρ ρ d d The net forces acting on the.v. is dp dp Pδ ( P + d) δ τd δ d τd d d dp 1 P since P U d [ + ρ & ] d d δ ( ) du ρu( U u) dy τ μ... Integral momentum eq. for B.. dy Evaluation of friction coefficient

4 1. assume u(y) the form of polynomials uy ( ) a+ by+ cy + dy 3 The constants are evaluated by applying the boundary conditions y u, 3 U U y δ u U, a, b, c, d δ δ y δ, u u P P y u v, u v, ρ + μ u 3 y 1 y ( ) 3 U δ δ. Substituting the epression into the integral momentum eq. yields d δ 3 y 1 y 3 3 y 1 y 3 du U [ ( ) ][1 ( ) ] dy d ρ + τ μ δ δ δ δ dy d 39δ 3 U ( ρu ) μ d 8 δ δ 14ν + ( δ at ) 13U δ 4.64 Re 1/ 3. To evaluate friction coefficient du 3U 3 μu τ μ μ dy δ 9.8 f τ.647 1/ 1 ρu Re Re 1/ 3 Similarly, the integral energy equation(fig) We negelect the inetic energy term and shear or term enthalpy enter across AB : δt ( ) ρcutdy δ( ) d δt ( ) enthalpy leaves across D : ρcutdy + [ ρcutdy] d d The enthalpy carried into the. V. across the upper face is

5 ct d [ ρudy] d d δt ( ) Heat conducted across the interface beteen the fluid and the solid surface is dt d dy y The net or done ithin the element is onservation of energy gives δ ( ) [ ( ) ] t du μ dy d dy d δt( ) d δt( ) [ ] [ ] d ρ ρ d ct udy d uctdy d d d δt( ) μ δt( ) du + α u( T T ) dy ( ) dy... integral energy eq. for B.. d ρc dy Evaluation of heat transfer coefficient 4. assume T(y) the form of polynomials T( y) e+ fy+ gy + hy 3 The constants are evaluated by applying the boundary conditions y T T, 3 T T ( T Ts) y δt u T, e, f, c, d 3 δ δ y δt, T u y u v, u v + α θ T T 3 y 1 y ( ) 3 θ T T δ δ t t 5. Substituting the epression into the integral energy eq. yields

6 d d δ t d δt ( T T ) udy ( θ θ) udy d d 3 y 1 y 3 y 1 y dt 3αθ θu + d δt 3 3 [1 ( ) ][ ( ) ] dy α δt δt δ δ dy δ t Assume δ > δ, let ς δ / δ d αθ θu [ δ( ς ς ) d 8 δς 4 for δ < δ ς < 1the term ς << ς t 3 d 3αθ θu ( δς ) d δς 1 U 1 1 U 1 t dς dδ α + d d δς ( δς ς ) 3 ( δς ςδ ) 14 ν 8ν But δdδ d & δ 13 U 13U 3 dς 13 α ς + 4ς d 14 ν 3 3/4 13 α ς + 14 ν ithbsδ at orς at t dς dδ α + d d δς δt 1 ς Pr [1 ( ) ] δ 1.6 t 1/3 3/4 1/3 6. To evaluate the convection heat transfer coefficient h ( ) 3 3 δς T T δt 1/3 U 1/ 3/4 1/3 h.33pr ( ) [1 ( ) ] ν h.33pr Re [1 ( ) ] for h.33pr 1/3 1/ 3/4 1/3 Re 1/3 1/ Average heat transfer coefficient

7 hd h h d h.664pr Re 1/3 1/ The foregoing analysis based on the assumption that the fluid properties ere constant throughout the flo. When there is an appreciable variation beteen all and free-stream condition, it is recommended that the properties be evaluated at the so-called film temperature, T f. T f T + T For constant heat flu all h.453pr q T ( T ) Re 1/3 1/ 1 1 q q / T T ( T ) T d d.6795re Pr 3 or q h ( T T) Other relations hurchill & Ozee: Re Pr 1 + Pr isothermal flat plate 1/ 1/3 1 for /3 1/4 > 1 1 Re Pr ,.468 constant heat flu plate.4637,.7 1/ 1/3 The relation beteen fluid friction and heat transfer

8 ρu τ f μ u 3 y 1 y for ( ) U δ δ 3 μu 3 μu U 1/ τ ( ) δ 4.64 ν f 3 μu U 1 ( ).33Re 4.64 ν h.33pr Re Pr ρcu /3 1/ 3 1/ 1/ ρu Re /3 1/ h, anton No. ρcu Re Pr Pr Pr.33Re /3 f... Reynold olburn Analogy Turbulent boundary layer heat transfer Fig. In turbulent flo, u u + u ', u : mean velocity, u ' : fluctuation τ Shear stress: ( ν + εm), εm : eddy viscosity ρ q Heat transfer: ρc( α + εh), εm : eddy diffusivity A Turbulent heat transfer based on fluid-friction analogy /3 f Pr still holds, Eperimental values for f for turbulent boundary layer.59 Re, 5 1 < Re < 1 1/5 5 7 f.37(log Re ), 1 < Re < Average friction coefficient for a flat plate ith a laminar boundary layer up to Re crit and turbulent thereafter can be calculated from

9 f or f.455 A, Re < (log Re ) Re.74 A, Re < 1 Re Re 1/5 7 9 Re crit A Applying the fluid-friction analogy /3 Pr f Pr.96Re, 5 1 < Re < 1 /3 1/5 5 7 /3 7 9 Pr.185(log Re ).584, 1 < Re < 1 Average heat transfer over the entire laminar-turbulent boundary layer is /3 Pr f for Re 5 1, Re < 1 Pr.37 Re 871Re h 1/3.8 Pr (.37 Re 871) for higher Reynold number 5 7 /3 1/5 1 crit h [.8Re (log Re ) 871]Pr, 1 < Re < 1, Re 5 1 The average heat transfer coefficient is defined as.584 1/ crit ( lam turb ) 1 crit h h d+ h d crit An alternative equation by Whitaer 1/ μ.36pr (Re 9), μ < Pr < 38, 1 < Re < 5.5 1,.6 < < 3.5 μ μ Turbulent boundary layer thicness Outside the laminar sublayer, the velocity profile can be described by u y 1/7 ( ) U δ

10 This profile fails to derive τ since y y f ρu 7 1/5 and for Re 1, < f.59re τ ν τ.96( ) U ρu 1/5 substitute into integral mom.eq. d δ y y ν [1 ( ) ]( ) dy.96( ) d δ δ U dδ 7 ν.96( ) d 7 U 1/7 1/7 1/5 1/5 1/5 1. fully turbulent from the leading edge. laminar up to δ δ at,.381re ν R 5 1, δ δ at 5 1 δ 5. (5 1 ) / crit lam crit lam crit U 7 ν 5 δ δlam 7 4 1/5 4/5 4/5.96( ) ( crit ) U δ.381re 156 Re 1/5 1 1/5 Heat transfer in laminar tube flo (Fig) Velocity profile π τπ π μ rdp rd r d r 1 dp du r dr μ d 1 dp + 4μ d u r B.. atr r, u 1 dp ur () ( r r ) 4μ d

11 the velocity at the centerline r dp u 4μ d u r 1 u r Heat transfer: (Fig) dq Assume a constant heat flu at all, d The heat flo conducted into and out of the annular element are qr πrd r qr qr+ dr qr+ dr r The net heat convected out of the element is πrdrρcu d The energy balance gives 1 T 1 T ( r ) ur r r α Assume heat flu qconstant, then the temperature increases linearly ith, i.e. T Boundary conditions: T at r r T q const. at r r r The solution is 4 1 r r T u ( ) + lnr+ α 1 1 T Tc [( ) ( ) ] α 4 r 4 r T : centerline temperature c The bul temperature ocal heat transfer r ur r r 4

12 q" h( T T ), T b T b r r ρπruc Tdr p ρπruc dr p is so-called bul temperature or energy average fluid temperature across the tube. 7 ur Tb Tc + 96 α 3 ur T Tc + 1 α q ha( T Tb) A r r r ( r) r r 4 48 h T Tb 11 r 11 d hd d Turbulent heat transfer in tube /3 f Pr, f : friction factor 8 3/4 1/3.395Re Pr d d d evaluated at bul temperature.8.4.3red Pr ( )

6. Laminar and turbulent boundary layers

6. Laminar and turbulent boundary layers John Richard Thome 8 avril 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 1 / 34 6.1 Some introductory ideas Figure 6.1 A boundary