1. CHAPTER 5 Principles of Convection
5-1 INTRODUCTION
5-2 VISCOUS FLOW
Shear stress:
boundary layer : the region of flow that develops from the leading edge of a plate in which the effects of the viscosity are observed
The outside boundary of a boundary layer is usually chosen as the point where the velocity of flow is 99 percent of the free-stream value.

2. Three regimes of boundary-layer flow
1. Laminar flow
2. Transitional flow
3. Turbulent flow
The transition occurs when
Renolds number
For most analytical purposes, the critical number for the transition is usually taken as

3. The critical Re for transition is strongly dependent on the surface roughness condition and the “turbulent level”of the free-stream.
The normal range for the beginning of transition is between :
For very large disturbances present in the flow, transition may begin with Renolds number as low as
Foe flows that are very free from fluctuation, the transition may not start until
The transition is completed at Re twice the value at the transition begin.

4. The relative shape for the velocity profiles in laminar and turbulent flow
The laminar profile is approximately parabolic
Structure of turbulent profile :
Laminar sublayer that is nearly linear.
Turbulent portion which is relatively flat in comparison with the laminar profile.

5. The physical mechanism of viscosity in fluids
In laminar flow, the viscosity is attributed to the exchange of momentum between different laminas by the movement of molecules.
For a gas
In turbulent flow, the momentum exchange between different layers is caused by the macroscopic movement of fluid chunks. We can expect a larger viscous-shear in turbulent flow than in laminar flow, due to which the velocity profile is flat in a turbulent boundary layer.

6. Flow in a tube
The critical Re
The range of Re for transition is
Continuity relation in a tube is
Re based on mass velocity is defined as

7. 5-3 INVISCOUS FLOW
The Bernoulli equation for flow along a stream results:
In differential form,
The energy equation for compressible fluid
i is the enthalpy defined by

8. Equation of state of fluid
Relations applicable to reversible adiabatic flow:

9. 5-4 LAMINAR BOUNDARY LAYER ON A FLAT PLATE
Assumptions:
Incompressible and steady flow
No pressure variation in the direction perpendicular to the plate.
Constant viscosity
Viscous-shear in y direction is negligible.
Two methods to study motion of fluid.
1. Newton’s law of motion
2. Force balance
Which applies to a elemental control volume fixed in space.
Which applies to a system of constant mass.

10. Mass continuity equation
Mass in the left face is
Mass out of the left face is
Mass in the bottom face is
Mass out of the top face is
Mass balance on the element is
Mass continuity equation

11. Derivation of momentum equation
Mass in the left face is
Momentum flux in the left face is
Momentum flux out of the left face is
Mass in the bottom face is
Momentum flux in the x direction entering the bottom face
Mass out of the top face is
Momentum flux in the x direction leaving the top face

12. Pressure forces on the left and right faces are and
Net pressure force in the direction of motion is
Viscous-shear force on the bottom face is
Viscous-shear force on the top face is
Balancing force and momentum in x direction gives
Final result
——Momentum equation

13. Integral momentum equation of the boundary layer.
Mass flow through plane 1
[a]
Momentum flow through plane 1
[b]
Momentum flow through plane 2
[c]
Mass flow through plane 2
[d]
Carried momentum in x direction by the flow through plane A-A
The net momentum flow out of the control volume is

14. By the use of or
The pressure force on plane 1 is
The pressure force on plane 2 is
The shear force at wall is
Setting the force on the element equal to the net increase in momentum gives
——Integral momentum equation of the boundary layer.

15. For constant pressure，from Bernoulli equation
One obtains
Integral momentum equation of the boundary layer becomes

16. Evaluation of boundary layer thickness
Boundary conditions are
Inserting the expression into equation [5-17]
For constant-pressure condition
Carrying out the integration leads to
suppose
Appling boundary conditions obtains
Separation of variables leads to

17. , so that
In terms of Renolds number
Exact solution

18. 5-5 ENERGY EQUATION OF THE BOUNDARY LAYER
Assumptions:
incompressible steady flow
Constant viscosity ,thermal conductivity, and specific heat
Negligible heat conduction in the direction of flow
Energy convected in left face
+ energy convected in bottom face
+ energy conducted in bottom face
+net viscous work done on element
= energy convected out right face
+ energy out top face
+heat conducted out top face

19. ［5-12］ ［5-22］ The viscous shear force over dx
The distance through which the force moves in respect to the control volume dxdy is
The net viscous energy delivered to the element is
［5-12］
Energy balance corresponding to the quantities shown in figure 5-6 is
［5-22］
Using
And dividing by gives
——Energy equation of the laminar boundary layer.

20. Order-of-magnitude analysis
［5-25］
［5-23］
［5-26］
A striking similarity between [5-25] and [5-26]
［5-24］

21. 5-6 THE THERMAL BOUNDARY LAYER
2. Definition of h
［5-27］
［5-28］
［5-29］
3. Temperature distribution in the thermal boundary layer
Boundary conditions
At y=0
［a］ At
［c］ At
［b］
At y=0
［d］

22. Conditions (a) to (d) may be fitted to a cubic polynomial
［5-30］
4. Integral energy equation of the boundary layer
Energy convected in +viscous work within element +heat transfer at wall=energy convected out
The energy convected through plane 1 is
The energy convected out through plane 2 is
The net viscous work done within element is
The mass flow through plane A-A
Heat transfer at wall
The energy carried with is

23. ［5-32］ Combining the above energy quantities gives
——integral energy equation of the boundary layer.
5. Thermal boundary layer thickness
Inserting (5-30) and (5-19) into (5-32) gives
［5-30］

24. Assume thermal boundary layer is thinner than the hydrodynamic boundary layer
Making substitution
5-33
5-34
Neglecting gives
Performing the differentiation gives or
But according to page 217
and

25. so that we have
5-35
Noting that
Solution is
When the boundary condition at at
is applied , the final solution become
5-36
where
5-37
When the plate is heated over the entire length
5-38

26. 6. Prandtl number(see page 225)
5-39
7. Nusselt number
5-40
Substituting (5-21) and (5-36) gives
5-41
Nusselt number
5-42
Finally,
5-43
For the plate heated over its entire length
5-44

27. 9. Average heat transfer coefficient and Nusselt number
5-45 or
5-46a
5-46b
where
Film temperature
5-47

28. 10. Constant heat flux 11. Other relations or
5-48
5-49
5-50 or
11. Other relations
For laminar flow on an isothermal flat plate
For Rex Pr＞100
5-51
For the constant-heat-flux case, is changed to , and is replaced by

29. Basic laws for inviscous flow
Velocity and temperature distributions
Mass continuity equation
Results:
Momentum equation
Integral momentum equation
Energy equation
A striking similarity
Integral energy equation

30. 5-7 THE RELATION BETWEEN FLUID FRICTION AND HEAT TRANSFER
[5-52]
The shear stress is
Using the velocity distribution given by equation(5-19), we have
Making use of the relation for the boundary-layer thickness gives
Combining (5-52) and (5-53) leads to

31. The exact solution is
[5-44] may be rewritten as
By introduction of Stanton number
——Reynolds-Colburn analogy

32. 5-8 TURBULENT-BOUNDARY-LAYER HEAT TRANSFER
Structure of turbulent flow:
Laminar sublayer
Buffer layer
Turbulent
The physical mechanism of heat transfer in turbulent flow is similar to that in laminar flow.
Difficulty: there is no completely adequate theory to predict turbulent-flow behavior
velocity fluctuation in a turbulent flow

33. Shear stress giving rise to velocity fluctuations in turbulent flow
Eddy Viscosity and Mixing Length ( 湍流粘度与混合长度 )
Mean free path and Prandtl mixing length
Prandtl postulated:

34. Nondimensional coordinates
Prandtal’s hypothesis
In the near-wall region
For laminar sublayer
Universal velocity profile

35. Universal velocity profile
For fully turbulent region
From equation [5-65], we have
Substituting this relation along with equation (5-64) into equation (5-63) gives or
Substituting this relation into Eq (5-69) for and integrating gives
Universal velocity profile
Laminar sublayer : 0<y+<5
Buffer layer : 5<y+<30
Turbulent layer : 30<y+<400
For fully turbulent region
For regions where both molecular and turbulent energy transport are important

36. Turbulent Heat Transfer Based on Fluid-Friction Analogy
3. Average-friction coefficient for a flat plate:
A simpler formula for lower Reynolds number is
2. The local skin-friction coefficient over a flat plate:
Table 5-1
4. Local turbulent heat transfer coefficient

37. 5. Average heat transfer coefficient over the entire laminar-turbulent boundary layer
For higher Reynolds number,using equation (5-79), one obtains
From , the above equation can be rewritten as
6. Equation suggested by Whitaker
Alternatively,
for the laminar portion
for the turbulent portion
Constant Heat Flux
One obtains

38. TURBULENT BOUNDARY LAYER THICKNESS
1. Velocity profile in a turbulent boundary layer
The first case: The boundary layer is fully turbulent from the leading edge of the plate:
2. Shear stress at wall
The second case: The boundary layer follows a laminar growth pattern up to
and a turbulent growth thereafter
So that
3. Integrating the integral momentum equation
Integrating [5-90] leads to
Combining the above various relations gives
Integrating and clearing terms gives

39. (a) Semilog scale
Boundary-layer thickness for atmospheric air at u=30m/s.
(b) log scale

40. 5-10 HEAT TRANSFER IN LAMINAR TUBE FLOW
1. Velocity distribution
5-98

41. 2. Energy balance analysis and temperature distribution
Net energy convected out = net heat conducted in
which may be rewritten

42. 5-98
assume
B.C:
Inserting Eq (5-98) into Eq (5-99)

43. Bulk temperature 4. Convection heat transfer coefficient
1. Definition of convection heat transfer coefficient in tube flow
Local heat flux =
2. Bulk temperature ( 整体温度 )
3. Wall temperature

44. 5-11 TURBULENT FLOW IN A TUBE
For laminar flow
assume
Integrating (a)
For turbulent flow
Heat transfer at wall is

45. Substituting ( B ) and ( C ) into ( A ) gives
Reynolds analogy for tube flow
Heat transfer at wall is
( B )
Shear stress at wall is
The pressure drop can be expressed in terms of friction factor
( D ) is modified by Pr
So that
A more correct relation
( C )