HEAT TRANSFER CHAPTER 7 External flow 们au #1 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 1 HEAT TRANSFER CHAPTER 7 External flow
External Flow: Flat Plate Topic of the day Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 2 External Flow: Flat Plate Topic of the Day
External Flow: Flat Plate Where we' ve been∴ General overview of the convection transfer equations Developed the key non-dimensional parameters used to characterize the boundary layer flow and convective heat and mass transfer hl N k Where were going Applications to external flow Flat plate Other shapes → Next time Then onto internal flow Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 3 External Flow: Flat Plate Where we’ve been …… • General overview of the convection transfer equations. • Developed the key non-dimensional parameters used to characterize the boundary layer flow and convective heat and mass transfer. Where we’re going: • Applications to external flow – Flat plate Today – Other shapes Next time Then onto internal flow …… f k h L Nu =
Differences between external and internal flow External flow Boundary layer develops freely without constraints Free strea olx Velocity boundar Internal flow: boundary layer is constrained and eventually merges #4 Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 4 Differences between external and internal flow • External flow: Boundary layer develops freely, without constraints • Internal flow: Boundary layer is constrained and eventually merges
How this impacts convective heat transfer Recall the boundary layer convection equations Free stream 6(x) Thermal boundary T≠T oT temperature fluid thermal gradient conductivity As you go further from the leading edge, the boundary layer continues to grow. Assuming the sur face and freestream t do not change with increasing distance x' boundary layer thickness. 8.1 aT ano gs Also Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 5 How this impacts convective heat transfer • Recall the boundary layer convection equations: • As you go further from the leading edge, the boundary layer continues to grow. Assuming the surface and freestream T do not change: with increasing distance ‘x’: – Boundary layer thickness, , – so – and fluid thermal conductivity wall temperature gradient Ts T Also =0 = − y s f y T q k =0 y y T s q
Methods to evaluate convection heat transfer Empirical(experimentalanalysis Use experimental measurements in a controlled lab setting to correlate heat and/or mass transfer in terms of the appropriate non-dimensional parameters Theoretical or Analytical approach Solving of the boundary layer equations for a particular geometry Example · Solve for t* Use evaluate the local nusselt number Nu Compute local convection coefficient, hx Use these(integrate)to determine the average convection coefficient over the entire surface Exact solutions possible for simple cases Approximate solutions also possible using an integralmethod Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Methods to evaluate convection heat transfer • Empirical(experimental) analysis – Use experimental measurements in a controlled lab setting to correlate heat and/or mass transfer in terms of the appropriate non-dimensional parameters • Theoretical or Analytical approach – Solving of the boundary layer equations for a particular geometry. – Example: • Solve for T* • Use evaluate the local Nusselt number, Nux • Compute local convection coefficient, hx • Use these (integrate) to determine the average convection coefficient over the entire surface – Exact solutions possible for simple cases. – Approximate solutions also possible using an integral method
Empirical method to obtain heat transfer coefficient How to set up an experimental test? Let' s say you want to know the heat transfer rate of an airplane wing(with fuel inside) flying at steady conditions wing surface What are the parameters involved? Ⅴ Telocity, wing length. L Prandtl number, Pr -viscosity, u Nusselt number. Nu Which of these can we control easily? Looking for the relation Experience has shown the following relation works well Nu=cRe pr L r Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 7 Empirical method to obtain heat transfer coefficient • How to set up an experimental test? • Let’s say you want to know the heat transfer rate of an airplane wing (with fuel inside) flying at steady conditions…………. • What are the parameters involved? – Velocity, –wing length, – Prandtl number, –viscosity, – Nusselt number, • Which of these can we control easily? • Looking for the relation: Experience has shown the following relation works well: T ,U Twing surface U L Pr Nu m n Nu = C Re L Pr
Empirical method to obtain heat transfer coefficient Experimental test setup Power input L Insulation Measure current(hence heat transfer )with various fluids and test conditions for t. u fluid properties are typically evaluated at the mean film temperature T+t Nur=C Re "mPr/ P CRet APr Log Re Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 8 T ,U = Power input insulation L Empirical method to obtain heat transfer coefficient • Experimental test setup • Measure current (hence heat transfer) with various fluids and test conditions for • Fluid properties are typically evaluated at the mean film temperature T ,U 2 + T T T s f
Analytical Solution- Laminar Flow Assume Steady, incompressible, laminar flow Constant fluid properties For flat plate TU Boundary layer equations 0 Continui ax a au au au Momentum l-+1 Energy aT aT 02T Blasius developed a similarity solution to the hydrodynamic equations in 1908 based on the stream function, y(x,y) Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 9 Analytical Solution – Laminar Flow • Assume: – Steady, incompressible, laminar flow – Constant fluid properties – For flat plate, • Boundary layer equations • Blasius developed a similarity solution to the hydrodynamic equations in 1908 based on the stream function, (x,y) = 0 + y v x u 2 2 y u y u v x u u = + 2 2 y T y T v x T u = + Continuity Momentum Energy T ,U Ts y
Analytical Solution - Laminar Flow(Contd dv≡ ax Define new dependent and independent variables f()≡ ll、/ n≡yVl2/ The momentum equation can be rewritten as And the boundary conditions are df f(o)=0 and Heat Transfer Su Yongkang School of Mechanical Engineering
Heat Transfer Su Yongkang School of Mechanical Engineering # 10 Analytical Solution – Laminar Flow (Cont’d) • Define new dependent and independent variables, • The momentum equation can be rewritten as • And the boundary conditions are y u x v − and u x u f / ( ) y u /x 2 0 2 2 3 3 + = d d f f d d f (0) 0 0 = = = f d df =1 = d df and
