Convection in Boundary Layers

1. Convection in Boundary Layers P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi A tiny layer but very significant………..

2. Momentum Vs Thermal Effects n Potential for diffusion of momentum change (Deficit or excess) created by a solid boundary. a Potential for Diffusion of thermal changes created by a solid boundary. Prandtl Number: The ratio of momentum diffusion to heat diffusion. Other scales of reference: Length of plate: L & Free stream velocity : uoo

4. This dimensionless temperature gradient at the wall is named as Nusselt Number: Local Nusselt Number

5. Average Nusselt Number

6. Computation of Dimensionless Temperature Profile First Law of Thermodynamics for A CV Energy Equation for a CV How to select A CV for External Flows ? Relative sizes of Momentum & Thermal Boundary Layers …

7. u*(y*) q(y*) Liquid Metals: Pr <<< 1 y* 1.0

8. q(y*) u*(y*) y 1.0 Gases: Pr ~ 1.0

9. q(y*) u*(y*) y 1.0 Water :2.0 < Pr < 7.0

10. q(y*) u*(y*) y 1.0 Oils:Pr >> 1

11. The Boundary Layer : A Control Volume For pr < 1

12. Reynolds Transport Theorem • Total rate of change of any extensive property B of a system(C.M.) occupying a control volume C.V. at time t is equal to the sum of • a) the temporal rate of change of B within the C.V. • b) the net flux of B through the control surface C.S. that surrounds the C.V. The relation between A CM and CV for conservation of any extensive property B.

13. Conservation of Mass • Let b=1, the B = mass of the system, m. The rate of change of mass in a control mass should be zero. Above integral is true for any shape and size of the control volume, which implies that the integrand is zero.

14. Conservation of Momentum • Let b=V, the B = momentum of the system, mV. The rate of change of momentum for a control mass should be equal to resultant external force. Momentum equation of per unit volume:

15. For a boundary layer : For an incompressible flow

16. Conservation of Energy • Let b=e, the B = Energy of the system, me. The rate of change of energy of a control mass should be equal to difference of work and heat transfers. Energy equation per unit volume:

17. Using the law of conduction heat transfer: The net Rate of work done on the element is: From Momentum equation: N S Equations

18. Then

19. For an Incompressible fluid: Substitute the work done by shear stress: This is called the first law of thermodynamics for fluid motion.

20. Invoking conservation of mass: First law for a fluid motion:

21. F is called as viscous dissipation.

22. Boundary Layer Equations Consider the flow over a parallel flat plate. Assume two-dimensional, incompressible, steady flow with constant properties. Neglect body forces and viscous dissipation. The flow is nonreacting and there is no energy generation.

23. The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:

24. Boundary Conditions

25. Scale Analysis Define characteristic parameters: L : length u∞: free stream velocity T ∞: free stream temperature

26. General parameters: x, y : positions (independent variables) u, v : velocities (dependent variables) T : temperature (dependent variable) also, recall that momentum requires a pressure gradient for the movement of a fluid: p : pressure (dependent variable)

27. Define dimensionless variables:

28. Similarity parameters can be derived that relate one set of flow conditions to geometrically similar surfaces for a different set of flow conditions:

30. Boundary Layer Parameters • Three main parameters (described below) that are used to characterize the size and shape of a boundary layer are: • The boundary layer thickness, • The displacement thickness, and • The momentum thickness. • Ratios of these thicknesses describe the shape of the boundary layer.

31. Boundary Layer Thickness • The boundary layer thickness, signified by , is simply the thickness of the viscous boundary layer region. • Because the main effect of viscosity is to slow the fluid near a wall, the edge of the viscous region is found at the point where the fluid velocity is essentially equal to the free-stream velocity. • In a boundary layer, the fluid asymptotically approaches the free-stream velocity as one moves away from the wall, so it never actually equals the free-stream velocity. • Conventionally (and arbitrarily), we define the edge of the boundary layer to be the point at which the fluid velocity equals 99% of the free-stream velocity:

32. Because the boundary layer thickness is defined in terms of the velocity distribution, it is sometimes called the velocity thickness or the velocity boundary layer thickness. • Figure  illustrates the boundary layer thickness. There are no general equations for boundary layer thickness. • Specific equations exist for certain types of boundary layer. • For a general boundary layer satisfying minimum boundary conditions: The velocity profile that satisfies above conditions:

33. Further analysis shows that: Where:

34. All Engineering Applications Variation of Reynolds numbers

35. Laminar Velocity Boundary Layer • The velocity boundary layer thickness for laminar flow over a flat plate: • as u∞ increases, δ decreases (thinner boundary layer) • The local friction coefficient: • and the average friction coefficient over some distance x:

36. Laminar Thermal Boundary Layer Boundary conditions:

37. This differential equation can be solved by numerical integration. One important consequence of this solution is that, for pr >0.6: Local convection heat transfer coefficient:

38. Local Nusselt number:

39. Average heat transfer coefficient:

40. A single correlation, which applies for all Prandtl numbers, Has been developed by Churchill and Ozoe..

41. Turbulent Flow • For a flat place boundary layer becomes turbulent at Rex ~ 5 X 105. • The local friction coefficient is well correlated by an expression of the form Local Nusselt number: Local Sherwood number: