HEAT TRANSFER CHAPTER 6 Introduction to convection 们au #1 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 1 HEAT TRANSFER CHAPTER 6 Introduction to convection
Boundary Layer Similarity Parameters The boundary layer equations(velocity, mass energy continuity ) represent low speed, forced convection flow. Advection terms on the left side and diffusion terms on the right side of each equation such as Advection u OT aT aT Diffu usion Non-dimensionalize the equations by setting x≡ ane L L where l is characteri stic length of the surface where v is the freestream velocity =U) and T*= T-T and P=p/pv Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 2 Boundary Layer Similarity Parameters • The boundary layer equations (velocity, mass, energy continuity) represent low speed, forced convection flow. • Advectionterms on the left side and diffusion terms on the right side of each equation, such as: Advection Diffusion • Non-dimensionalize the equations by setting: * s * 2 and / T T -T and where V is the freestream velocity ( ) where L is characteristic length of the surface P p V T T U V v and v V u u L y and y L x x s = − = 2 2 y T y T v x T u = +
Boundary Layer Similarity Parameters(Contd The boundary layer equations can be rewritten in terms of the non-dimensional variables Continuity au av 0 x-momentum* au au aP ax at"* aT a a2T ene With boundary conditions Wall u (x 0)=0;v(x,y=0)=0 T( U(x) Freestream: u(x,y [≡lifv=Ul T(x Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 3 Boundary Layer Similarity Parameters (Cont’d) • The boundary layer equations can be rewritten in terms of the non-dimensional variables Continuity x-momentum energy • With boundary conditions 0 * * * * = + y v x u 2* 2 * * * * * * * * * y u x VL P y u v x u u + = − + 2* 2 * * * * * * * y T y VL T v x T u = + ( , ) 1 [ 1if ] ( ) Freestream: ( , ) ( , 0) 0 Wall: ( , 0) 0 ; ( , 0) 0 ; * * * * * * * * * * * * * * * * = = = = = = = = = = = T x y V U V U x u x y T x y u x y v x y
Boundary Layer Similarity Parameters(Contd From the non-dimensionalized boundary layer equations, dimensionless groups can be seen Reynolds#ReL≡ Prandtl# PP≡ Substituting gives the boundary layer equations Continuity k au au aP X-momentum ax ReL ay 2 OX Energy aT *aT 1 a-T R eL Pr #4 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 4 Boundary Layer Similarity Parameters (Cont’d) • From the non-dimensionalized boundary layer equations, dimensionless groups can be seen Reynolds # Prandtl # Substituting gives the boundary layer equations: VL Re L 0 * * * * = + y v x u 2* 2 * * * * * * * * * Re 1 y u x P y u v x u u L + = − + Re Pr 1 2* 2 * * * * * * * y T y T v x T u L = + Continuity: x-momentum: Energy: Pr
Back to the convection heat transfer problem Solutions to the boundary layer equations are of the form dP dP x y Re L where O for flat plate dx dx dP Re. p L Rewrite the convective heat transfer coefficient aT k T-T h T-T L v=0 k「ar L Define the Nusselt number as aT x Re. D dP u Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 5 Back to the convection heat transfer problem… • Solutions to the boundary layer equations are of the form: • Rewrite the convective heat transfer coefficient • Define the Nusselt number as: = = = * * * * * * * * * * * * , ,Re ,Pr, , ,Re , where: 0 for flat plate d x d P T f x y d x d P d x d P u f x y L L L T T L y T T T T T T k T T y T k T T q h s y s s s f s y f s x x − − − − − = − − = − = = = 0 0 0 * * * = = y f x y T L k h = = = * * * 0 * * ,Re ,Pr, * dx dP f x y T L y Nu
Nusselt number for a prescribed geometry hL f(x,Rex, Pr) Lo oca khk f f(re, Pr) Average dP (For a prescribed geometry is known) =o Many convection problems are solved using Nusselt number correlations incorporating Reynolds and prandtl numbers The Nusselt number is to the thermal boundary layer what the friction coefficient is to the velocity boundary layer ou dP x Re Re L dx Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Nusselt number for a prescribed geometry (For a prescribed geometry, is known) Many convection problems are solved using Nusselt number correlations incorporating Reynolds and Prandtl numbers • The Nusselt number is to the thermal boundary layer what the friction coefficientis to the velocity boundary layer. ( ) (Re ,Pr) Average k hL ,Re ,Pr Local k hL f * f L L Nu f Nu f x = = = = * * dx dP = = = = * * * 0 * * 2 ,Re , Re 2 2 dx dP f x y u V C L L y s f
Heat transfer coefficient, simple example Given 100%C. Experimental measurements ofar Air at 20C flowing over heated flat plate temperatures at various distances from the surface are as shown Experimental measurements EXperinental incasurcnicnts 4r,8o=6.2mm 2 mm ayly-o Ay L,2 mm Air I rnrn Su plate 2U 50 100°C Find: convective heat transfer coefficient. h Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 7 Heat transfer coefficient, simple example • Given: Air at 20ºC flowing over heated flat plate at 100ºC. Experimental measurements of temperatures at various distances from the surface are as shown • Find: convective heat transfer coefficient, h Experimental measurements
Heat transfer coefficient, simple example Solution aT Recall that h is computed by From Table a-4 in appendix, at a mean fluid temperature T +t (average of free-stream and surface temperatures) 20+100)/2=60°C the air conductivity k is =0.028 W/m-K Temperature gradient at the plate surface from experimental data is -66.7K/mm =-66, 700 K So. convective heat transfer coefficient is -0.028×(-66700) 80 W =23.345 m K Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 8 Heat transfer coefficient, simple example • Solution: Recall that h is computed by • From Table A-4 in Appendix, at a mean fluid temperature (average of free-stream and surface temperatures) the air conductivity, k is 0.028 W/m-K • Temperature gradient at the plate surface from experimental data is -66.7 K/mm = -66,700 K/m • So, convective heat transfer coefficient is: 0 = − − = T T y T k h s y f x m K W 23.345 80 - 0.028 ( 66700) 2 = − h = 2 m T Ts T + = T C m = (20 +100) 2 = 60
Example: Experimental results for heat transfer over a flat plate with an extremely rough surface were found to be correlated by an expression of the orm Nu=0.04 Re. Pr 1/3 where Nu, is the local value of the nusselt number at a position x measured from the leading edge of the plate Obtain an expression for the ratio of the average heat transfer coefficient to the local coefficient KNOWN: Local Nusselt number correlation for flow over a roughened surface FIND: Ratio of average heat transfer coefficient to local coefficient SCHEMATIC: Nuy =0.04Re .Pr Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 9 Example: Experimental results for heat transfer over a flat plate with an extremely rough surface were found to be correlated by an expression of the form where is the local value of the Nusselt number at a position x measured from the leading edge of the plate. Obtain an expression for the ratio of the average heat transfer coefficient to the local coefficient. 0.9 1/3 0.04Re Pr Nux = x Nux
ANALYSIS: The local convection coefficient is obtained from the prescribed correlation, l=Nux-=0.04-Rex pr hx=0.04k 13 ≌分+ To determine the average heat transfer coefficient for the length zero to x, X dx=-C1 x X 0 0 11x 0.1 x09 Hence, the ratio of the average to local coefficient is 1. 11C1X COMMENTS: Note that Nu. /Nuy is also equal to 1. ll. Note, however, that Nux dx Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 10