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