Chapter 7(I) External Flow: The Flat Plate in Parallel Flow Chapter 7 Section 7.1 through 7.3
External Flow: The Flat Plate in Parallel Flow Chapter 7 Section 7.1 through 7.3 Chapter 7(I)
The Problem of Convection Heat transfer Mass transfer Nu,=f(x',Re,,Pr)Sh,=f(x',Re,.Sc Nux=f(Rex:Pr) Shx =f(Res:Sc) Experimental (Empirical)approach----perform measurements Theoretical approach----solve equations
The Problem of Convection ( ) *,Re ,Pr Nu f x x x = Nu f x = (Re ,Pr x ) Heat transfer Mass transfer ( ) *,Re , x x Sh f x Sc = Sh f Sc x = (Re , x ) Experimental (Empirical) approach ----perform measurements Theoretical approach ----solve equations
The Empirical Method o T I-E=q=hA (Ts-T) Experiment for measuring -Ts As NuL =CRe"Pr" Insulation 7=☑+Z Film T for properties E 2 NuL C RetmPr NuL C Re Log NuL Log ( Log ReL Log ReL (a) (b)
The Empirical Method Experiment for measuring h L Re Pr m n L Nu C= L 2 s f T T T + ∞ = Film T for properties
Physical Features Physical Features Laminar Turbulent 8(x) External flow,the boundary layers develop freely without constraint. Boundary layer conditions may be entirely laminar,laminar and turbulent,or entirely turbulent
Physical Features Physical Features • External flow, the boundary layers develop freely without constraint. • Boundary layer conditions may be entirely laminar, laminar and turbulent, or entirely turbulent
Flow conditions To determine the conditions,compute Re,=PuLut U V and compare with the critical Reynolds number for transition to turbulence, Relaminar flow throughout Re>Re>transition to turbulent flow ·Value of Rex.e depends on free stream turbulence and surface roughness. Nominally, Rexe≈5xl03
Flow conditions • To determine the conditions, compute and compare with the critical Reynolds number for transition to turbulence, ReL ρuL uL μ ν ∞ ∞ = = , Re Re laminar flow throughout L xc → transition to turbulent flow • Value of Rex,c depends on free stream turbulence and surface roughness. Nominally, 5 , Re . 5 10 x c ≈ ×
o Surface thermal conditions are commonly idealized as being of uniform temperature T or uniform heat flux Thermal boundary layer development may be delayed by an unheated starting length. Ts=To Ts>T x=L X Equivalent surface and free stream temperatures forx
• Surface thermal conditions are commonly idealized as being of uniform temperature or uniform heat flux . Ts s q′′ • Thermal boundary layer development may be delayed by an unheated starting length. Equivalent surface and free stream temperatures for and uniform (or ) for x ξ
Similarity Solution Similarity Solution for Laminar. Constant-Property Flow over an Isothermal plate Based on premise that the dimensionless x-velocity component,u/u, and temperature,T=(T-T)/(T-T),can be represented exclusively in terms of a dimensionless similarity parameter n=y(u /vx)v2 Similarity permits transformation of the partial differential equations associated with the transfer of x-momentum and thermal energy to ordinary differential equations of the form df *f an 2 an d'f=0 where (ulu)=df/dn,and dn2 2
Similarity Solution Similarity Solution for Laminar, Constant-Property Flow over an Isothermal Plate • Based on premise that the dimensionless x-velocity component, , and temperature, , can be represented exclusively in terms of a dimensionless similarity parameter u u/ ∞ ( ) ( ) * / T TT T T s s ∞ ≡− − ⎡ ⎤ ⎣ ⎦ ( )1/2 η ≡ y u x ∞ /ν • Similarity permits transformation of the partial differential equations associated with the transfer of x-momentum and thermal energy to ordinary differential equations of the form 3 2 3 2 2 0 df df f d d η η + = where / / , ( ) u u df d ∞ ≡ η and 2 * * 2 Pr 0 2 d T dT f d d η η + =
Similarity Solution(cont.) Subject to prescribed boundary conditions,numerical solutions to the momentum and energy equations yield the following results for important local boundary layer parameters: with u/u =0.99 at n=5.0, 5.0 5x 6= 1/2 (Re,)v2 with ,yl Bu =uuu。/x dn 7=0 and df1dnl。=0.332, C.=2=064Re pu12 -with h.=gg1(.亿-T)=kaT'/a,。=k(u./w)dr'/d切，o and dTd.332 P for Pr.6 =0.332Re2prv3 and Nus=k =Prl/3
Similarity Solution (cont.) • Subject to prescribed boundary conditions, numerical solutions to the momentum and energy equations yield the following results for important local boundary layer parameters: ( ) ( ) 1/2 1/2 - with / 0.99 at 5. 5.0 5 / 0, Rex x u vx u u η δ ∞ ∞ = = = = 2 2 0 0 - with / s y u d f u u vx y d η τμ μ η ∞ ∞ = = ∂ = = ∂ 2 2 0 and / 0.332, df d η η = = , 1/2 , 2 0.664Re / 2 x s x Cf x uτρ − ∞ ≡ = ( ) ( ) * * 1/2 0 0 - with / / / / xss y h q T T k T y k u vx dT d η ∞ ∞ η = = = − =∂ ∂ = ′′ * 1/3 0 and / 0.332 Pr for Pr 0.6, dT d η η = = > 1/2 1/3 0.332 Re Pr x x x h x Nu k = = 1/3 r and P t δ δ =
Similarity Solution(cont.) Average Boundary Layer Parameters: C.=1.328 Re2 瓦=h,dx Nux =0.664 Rel2 Pr/3 The effect of variable properties may be considered by evaluating all properties at the film temperature. T= s+To 2
Similarity Solution (cont.) • Average Boundary Layer Parameters: , 0 1 x s x sdx x τ ≡ ∫ τ 1/2 , 1.328 Re x f x C − = 0 1 x x x h x = ∫ h dx 1/2 1/3 x 0.664 Re Pr Nu = x • The effect of variable properties may be considered by evaluating all properties at the film temperature. 2 s f T T T + ∞ =
Turbulent Flow Turbulent Flow ·Local Parameters: Empirical Cf.x=0.0592Re,15 Correlations Nu,=0.0296Re5pr/ ·Average Parameters: 瓦.=（hnk+hnsr) Substituting expressions for the local coefficients and assuming Res.=5x105, 0.0741742 For Res=0 or L>(Re>Rex.), ReL C.L=0.074 Rez5 NuL=(0.037Re5-871）Pr3 NuL =0.037 Re4/5 Pr/3
Turbulent Flow Turbulent Flow • Local Parameters: 1/5 , 4/5 1/3 0.0592 Re 0.0296 Re Pr f x x x x C Nu − = = Empirical Correlations • Average Parameters: ( 0 1 ) 1 c c x L L am t x urb h h dx h dx L = + ∫ ∫ Substituting expressions for the local coefficients and assuming 5 Re 5 10 x,c = × , , 1/5 0.074 1742 Re Re f L L L C = − ( ) 4/5 1/3 L 0.037 Re 871 Pr Nu = − L , , ( ) 1/5 , 4/5 1/3 For Re 0 or Re Re , 0.074 Re 0.037 Re Pr x c c L x c f L L L L L x C Nu − = = =