国上双人李 “-H Heat Transfer Coefficients in Heat Conduction Problem Heat transfer coefficient was often used as a boundary condition in conduction problems: Introduction to Convection: ·Local Heat Flux and Heat Transfer Coef佰icient T Flow and Thermal Considerations 4=(T-T) ·wverage Heat具and Heat Transfer AT. Coefficient for a Uaiform Surface Temperature: 9=4d,-(亿-J、d4 Chapter Six and AppendixD Newton's Cooling Law Sections 6.1 through 6.8 和教侧 and D.1 through D.3 ◆q=4亿-) 用上4生 工雅快物德辆九所 The Objectives of Heat Convection Study Velocity Boundary Layer:Physical Features We have taken the convection coefficient as a given,with little thought of 一m where the particular numbers come from. btedndace The central objective of comective heat trangfer is to understand the hear s山en mysteries of the heat transfer coefficient: What factors affect the convective heat transfer coefficient? -A region between the surface 6→-0.9 。 (correlations of heat transfer coefficieat) -Why does increase in the flow direction? -Manifestod by a surface shear stress that provides a drag force. -How does 霸上4生 工塑路物液研丸所 ©上生 工童前藏祸克所 Thermal Boundary Layer:Physical Features Boundary Laver Transition T F牌em一a laminar region? 自 ·What are the ch of turbulent region - 自身 ciated Aregion between the suface and 4→=1.09 turbulent flow? - -Why does increase in the flow direction? I0sRes3×10 critical Reynolds namber Nomuily:Re5x10 ansf品coefficient。 -0 xlocation at which transition o turbulence begins -If (T,-T.)is constant,how do g;and 闭上生 1
1 Introduction to Convection: Flow and Thermal Considerations Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.3 Heat Transfer Coefficients in Heat Conduction Problem • Local Heat Flux and Heat Transfer Coefficient: s s q hT T Heat transfer coefficient was often used as a boundary condition in conduction problems: 工程热物理研究所 • Average Heat Flux and Heat Transfer Coefficient for a Uniform Surface Temperature: s s q hA T T s A s q q dA s T T hdA s A s 1 s A s s h hdA A Newton’s Cooling Law • We have taken the convection coefficient as a given, with little thought of where the particular numbers come from. • The central objective of convective heat transfer is to understand the mysteries of the heat transfer coefficient: What factors affect the convective heat transfer coefficient? The Objectives of Heat Convection Study 工程热物理研究所 How do these factors influence the convective heat transfer coefficients? (correlations of heat transfer coefficient) How to enhance heat transfer (increase heat transfer coefficient)? Velocity Boundary Layer : Physical Features – A consequence of viscous effects associated with relative motion between a fluid and a surface. – A region of the flow characterized by shear stresses and velocity gradients. 0 99 u y – A region between the surface and the free stream whose 工程热物理研究所 – Why does increase in the flow direction? – How does vary in the flow direction? Why? s 0.99 u and the free stream whose thickness increases in the flow direction. s y 0 u y – Manifested by a surface shear stress that provides a drag force. s – A consequence of heat transfer between the surface and fluid – A region of the flow characterized by temperature gradients and heat fluxes. – A region between the surface and h f h hi k T Ty Thermal Boundary Layer : Physical Features 工程热物理研究所 – Why does increase in the flow direction? t t the free stream whose thickness increases in the flow direction. 0.99 s t s T Ty T T – Manifested by a surface heat flux and a convection heat transfer coefficient h . s q s f y 0 T q k y 0 / '' f y s s s q kT y h TT TT – If is constant, how do and h vary in the flow direction? T T s s q Boundary Layer Transition • What are the characteristics of laminar region? • What are the characteristics of turbulent region? • What conditions are associated with transition from laminar to turbulent flow? 工程热物理研究所 • Transition criterion for a flat plate in parallel flow: , Re critical Reynolds number c x c u x inertia vicous location at which transition to turbulence begins c x • Why is the Reynolds number an appropriate parameter for quantifying transition from laminar to turbulent flow? 5 6 , 10 Re 3 10 x c 5 Normally: Re 5 10 x c,
The Effects of Boundary Layer Transition Convective Heat Transfer Equations .Why does transition provide a significant increase in the houndary layer thickness? Conseranon of mass (Continury Equation): 池，立-0 Why does it increase significantly Physical tion o t understanding of Why does the convfn Canseranon of energy (First Lar of thermodynamicsl decay in the turbulent region? When L is not long enough only lam 信门 section 6S. ®上本1 用上4生 Magnitude Analysis Magnitude Analysis ingtor very mall by orders)value ngefor very smll by order)vale Main flow ocy:一l Main flow velocity: 1；w-l Pressure:p-1 4培会产 Pressuee:p1 Temperature T-一l 6e芳会÷ 号华 Thickness of thermal bouday layer., a号千Tentnl Thickness of thermal boundary layer:66 亚of avarbie sbine e minde 尝) Magnicude Aaalysis (The method to derive houndary layer eqeations): 会： +g] 霸上4生 工圆品物表研完所 工触前藏利克所 Velocity Boundary Layer Equations Thermal Boundary Layer Equations Velocity Bourdary Layer Appeoximationc 。Thermal Boundary Layer appro线imation 山多T 等会 整亭白 ( 房会 .Assuming no heat source and according to +=史+2+ magnitude analysis: 等0→装尝 受}倍} P-1 -信离] o dx 用上水生 周上生 2
2 • Why does transition provide a significant increase in the boundary layer thickness? • Why does the convection coefficient decay in the laminar region? • Why does it increase significantly ith t iti t t b l d it The Effects of Boundary Layer Transition 工程热物理研究所 with transition to turbulence, despite the increase in the boundary layer thickness? • Why does the convection coefficient decay in the turbulent region? Effect of transition on velocity boundary layer thickness and local convection coefficient When L is not long enough, ReL < Rex,c, only laminar flow exists, and flow transition will not occur 2 2 2 2 ) () u u p uu uv X （ Convective Heat Transfer Equations For steady (laminar flow）, two-dimensional, incompressible, flow with constant fluid properties, the governing equations are as follows: • Conservation of mass (Continuity Equation): • Conservation of momentum (Newton’s Second Law of Motion): 0 u v x y Physical 工程热物理研究所 2 2 2 2 2 2 2 2 p T T TT cu v k q x y xy uv u v yx x y 2 2 2 2 2 2 ) () ) () uv X x y x xy v v p vv uv Y x y y xy （ （ • Conservation of energy (First Law of thermodynamics): These equations are derived in section 6S.1 Physical understanding of the governing equations assuming： 1 for large value ， for very small (by orders) value The magnitude of six basic variables: Main flow velocity: u∞~1 Pressure: p～1 x~ l ~1 ; u~u∞~1 y~ The magnitude of variables inside boundary layer: ~ ~1 v u u Magnitude Analysis 工程热物理研究所 Temperature: T ~1 Characteristic length of the plate: l ~1 Thickness of velocity boundary layer: ~ Thickness of thermal boundary layer: t ~ v~ ~ ~1 y xl 2 2 2 ~ 1 ~ ~ s m 2 2 2 ~ ~~ 1 m s Kinetic viscosity Thermal diffusivity 2 2 2 2 11 1 1 1 /1 / 1 1 t t tt uv a x y xy a The derivative of a variable: substitute the magnitude of the variables 1 1 u x assuming： 1 for large value ， for very small (by orders) value The magnitude of six basic variables: Main flow velocity: u∞~1 Pressure: p～1 x~ l ~1 ; u~u∞~1 y~ The magnitude of variables inside boundary layer: ~ ~1 v u u Magnitude Analysis 工程热物理研究所 Magnitude Analysis (The method to derive boundary layer equations)： Compare the relative magnitude of the variables in the equations, get ride of small ones to simplify the equations. Temperature: T ~1 Characteristic length of the plate: l ~1 Thickness of velocity boundary layer: ~ Thickness of thermal boundary layer: t ~ v~ ~ ~1 y xl 2 2 2 ~ 1 ~ ~ s m 2 2 2 ~ ~~ 1 m s Kinetic viscosity Thermal diffusivity Velocity Boundary Layer Equations • Velocity Boundary Layer Approximation: 2 2 2 2 u u x y , , u v u uvv y xyx x~ l ~1 u~u∞~1 y~ v~ 工程热物理研究所 2 2 2 2 2 2 2 2 ) () ) () u u p uu uv X x y x xy v v p vv uv Y x y y xy （ （ 2 2 uu p u 1 u v x y xy • Assuming negligible body forces and according to magnitude analysis : 0 p y 2 2 u u dp u 1 u v x y dx y p dp x dx p～1 2 2 2 ~ 1 ~ ~ s m Thermal Boundary Layer Equations • Velocity Boundary Layer approximation: 2 2 2 2 T T x y • Thermal Boundary Layer approximation: , , u v u uvv T ~1 x~ 1 y~ 工程热物理研究所 2 2 2 2 2 2 2 2 p T T TT cu v k q x y xy uv u v yx x y , , y xyx 2 2 2 p TT T u u v x y yc y : p k thermal diffusivity c • Assuming no heat source and according to magnitude analysis : viscous dissipation normally could be neglected
Boundary Layer Equations for Convective Heat Transfer Boundary Layer Similarity ·Gm'mingEquations: Fora prescribed geometry,the solutions of boundary layer equations are: 会0 What s the p向scal u-f(zy.LV.p.) v=f八L,',） T-f玉L',A品cpk) hef(x.LV.p.mcpR) x=fL,「，A) For a prescribed geometry,the comresponding independent variables are: Hydrodynamic:Main Stream Velocity( Geometrical:Size Location (.y) (Recall how introduction of the Fuid Properties::: similarity narameters fo pemitted generalization of results for transient,one-dimensonal conditionL As plied to the boundary layers,the principle ofsy is based on determining cing one set oc 2)Get Non-dimensionalized Equations Physical Interpretation of Dimensionless Parameters Reymolds Number:Re .The ratio of nertiaore to viscousses 0 吃 ·The crteria to中e the flow type Related to the velocity boundary thickness 是 L- r.I-1 I.-T The ratio of momentum diffusivity to thermal ditusivty Related to the themmal boundary thickness 尝0 .0 For laminar boundary layer. 卧e1÷6=4 1 Far gas: 哥m For liguid 霸上4生 工圆盐德表研党所 ®上生 工前藏病克所 Local and Average Friction coefficient Local and Average Nu(努塞尔数) .Foraprebed geometry,byntheonmom For a prescribed geometry,by solving the nondimensionlized energy equation: w-fx,y,Re） 1T守 T=f(x,v,Rer,Pr) …乱侣周胤 By Re Pr ay .The dimensionless shear stress,o local friction 0乱. T.-T. coefficiem(范宁摩擦系数)，shen .The dimensonless local comvecton coefficient is then 2 ar G"万“R可w 气G 号乱e周 local Ssscl Tumbe G忌e .What is the functional dependence of the average Nusselt number? -是-0e叫 C=f(Re:) 用上水生 用上生 工漫物研充所 3
3 0 u v x y 2 2 u u dp u 1 u v x y dx y What is the physical significance of each term in the equations? 2 2 TT T u v x y y Boundary Layer Equations for Convective Heat Transfer • For a prescribed geometry, the corresponding independent variables are: • Governing Equations： 工程热物理研究所 For a prescribed geometry, the corresponding independent variables are: Geometrical: Size (L), Location (x,y) Hydrodynamic: Main Stream Velocity (V) Fluid Properties: Hydrodynamic: , ; Thermal: , p c k Pressure gradient: dp dx obtained from a separate consideration in free stream 1) Solve momentum equation and continuity equations to get and 2) Solve energy equation to get 3) Get from the defin and ition according to the temperature and velocity fields s h uxy (, ) vxy (, ) Txy (, ) • Solution procedures: Boundary Layer Similarity • For a prescribed geometry, the solutions of boundary layer equations are : Th t f diff t l th L diff t l it V diff t fl id ti th ,,, ,, ,,, ,, ,,,,,, , p u f xyLV v f xyLV T f x y LV c k ,,,,, , ,,,, p s h f xLV c k f xLV 工程热物理研究所 As applied to the boundary layers, the principle of similarity is based on determining similarity parameters that facilitate application of results obtained for a surface experiencing one set of conditions to geometrically similar surfaces experiencing different conditions. • That means, for different length L, different velocity V, different fluid properties, the solutions are different. How to make the solutions general to a prescribed geometry ? • Boundary Layer Similarity: (Recall how introduction of the similarity parameters Bi and Fo permitted generalization of results for transient, one-dimensional condition). • Key similarity parameters may be inferred by non-dimensionalizing the momentum and energy equations. • Recast the boundary layer equations by introducing dimensionless forms of the independent and dependent variables: * * x y x y L L T T Non-dimensionalized Equations * * u v u v V V * 2 p p V 工程热物理研究所 * s s T T T T T 2 2 u u dp u 1 u v x y dx y 2 2 TT T u v x y y 0 u v x y • dimensional equations: * * * 2* * * * * * *2 * * 2* * * * * *2 1 Re 1 Re Pr L L u u dp u u v x y dx y TT T u v xy y * * * * 0 u v x y • non-dimensionalized equations: 2 p V Physical Interpretation of Dimensionless Parameters: • The ratio of inertia force to viscous stress • The criteria to judge the flow type • Related to the velocity boundary thickness Reynolds Number Re ： L VL VL v Pr Prandtl Number p c v k * * * 2* * * * * * *2 * * 2* * * * * *2 1 Re 1 Re Pr L L u u dp u u v x y dx y TT T u v xy y * * * * 0 u v x y 工程热物理研究所 • The ratio of momentum diffusivity to thermal diffusivity • Related to the thermal boundary thickness For laminar boundary layer: Prn t For gas: For liquid metal: For oil: • For a prescribed geometry, by solving the non-dimensionalized momentum equation: * ** , ,ReL u fxy * * * 0 0 s y y u Vu y Ly • The dimensionless shear stress, or local friction coefficient（范宁摩擦系数）, is then * Local and Average Friction coefficient * * * * 0 u v x y * * * 2* * * * * * *2 1 ReL u u dp u u v x y dx y 工程热物理研究所 * * 2 * 0 2 / 2 Re s f L y u C V y * * * * 0 ,ReL y u f x y 2 * ,Re Re f L L C fx • What is the functional dependence of the average friction coefficient? Re C f f L • For a prescribed geometry, by solving the nondimensionlized energy equation: * ** , ,Re ,Pr T fxy L * * * * 0 * * 0 0 / f yf f s s s y y kT y k k T T T T h T T LT T y L y • The dimensionless local convection coefficient is then Local and Average Nu （努塞尔数） * * 2* * * * * *2 1 Re Pr L TT T u v x y y 工程热物理研究所 * * * * 0 ,Re ,Pr L f y hL T Nu f x k y Nu local Nusselt number • What is the functional dependence of the average Nusselt number? Re ,Pr L hL Nu f k
The Reynolds Analogy(雷诺比拟) The Reynolds Analogy(cont.) With the Stanton number新坦顿数)defined, Scr RePr hNi 尝常多离 dy Re dy With Pr1,the Reynolds analogy,which relatesm prameersofte relocity and thermul bowary layers,is 国m学 子-9 Modified Reymolds (Chilton-Colbumn)Anslogy: -An empirical result eendapplicabilry of the Reynoldsanloy .Hence,boundary the solutionsa of the same form 咖品 子-m 0.6ch<60 乱-. →ca 学- 一piicnb旅o laminar fowf中*h-食. ®上西过 工港研丸所 用上4生 Example I:Average Nu Example I:Average Nu ANALYSIS:For a peescribed goomey m-红时 1500W) 1只 Ke3-()-15m2/s/n m(VL:/v2)-1m A· Hence..u值operties(化与.人Rt=RL2Ao,A-P% nw忘，- (6t2k2-(n ASStMPTIONS:(1)Steady-stste conditions.(2) diation,(匀Blade shapesre geomri山yiiu, : The eat rae fohe e小告安 -T w明 92=2066V 霸上4生 工整梅表研完所 G粉正5K红 工童物藏祸九所 Example 2:Reynolds Analogy (ii)If the Reyuolds uumbers were not equal ()knowledge of the specifie C 厂 限0时RTIES:Preveribed.Air:￥=163：104mh.k=0022W的K.升=Q72 可二亿Rec后克 A万 ne,W油R4-V7水-100的046310mn=1.230 用上水生 周上生 工漫物研充所 4
4 The Reynolds Analogy (雷诺比拟） • Equivalence of dimensionless momentum and energy equations for negligible pressure gradient : Advection terms Diffusion * * 2* * * TT T 1 * * 2* * * * * *2 1 Re uu u u v x y y * * * 2* * * * * * *2 * * 2* * * * * *2 1 Re 1 Re Pr L L u u dp u u v x y dx y TT T u v xy y dp*/dx*~0 Pr~1 工程热物理研究所 * * * * *2 1 Re TT T u v x y y • Hence, for equivalent boundary conditions, the solutions are of the same form: * * * * * * * * y y 0 0 u T u T y y L y y * * 2 * 0 2 / 2 Re s f L y u C V y * * * * 0 ,ReL f y hL T Nu f x k y Re 2 C Nu f • With Pr = 1, the Reynolds analogy, which relates important parameters of the velocity and thermal boundary layers, is 2 Cf St With the Stanton number( ) defined as, 斯坦顿数 p h Nu St Vc Re Pr difi d ld ( hil lb ) l The Reynolds Analogy (cont.) 工程热物理研究所 • Modified Reynolds (Chilton-Colburn) Analogy: – An empirical result that extends applicability of the Reynolds analogy: 2 3 Pr 0.6 Pr 60 2 f H C St j Colburn j factor for heat transfer – Applicable to laminar flow if dp*/dx* ~ 0. – Generally applicable to turbulent flow without restriction on dp*/dx*. Example 1: Average Nu 工程热物理研究所 Example 1: Average Nu 工程热物理研究所 工程热物理研究所 Example 2: Reynolds Analogy 工程热物理研究所
Homework (2-16)67 Chapter6:6.5;6.9;6.14;6.20;6.29 02(7)152 w/m2. 2m q-152w1m2,K5-(←-2]rc-4260w/m2 COMMENTS:If the flow is narbulent over the entire airfoil,the modified Reynolds analogy provides a good measure of the relationship between facfrictoand heat transfer.The relation becomes more approximate with increasing vales of the mitude of the pressu gradien ®上1 用上4生 工雅能的德锅九青 5
5 工程热物理研究所 Homework Chapter 6: 6.5; 6.9; 6.14; 6.20; 6.29 工程热物理研究所
