Sarthit Toolthaisong

1. Heat Transfer In Channels Flow Sarthit Toolthaisong

2. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong We wish to determine the following:

3. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong Applying conservation of energy to the element dx Eq. (a) = Eq. (b), we get

4. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong From the average heat transfer coefficient over the length x We get (d)

5. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong Introducing (d) into (6.11) and solving the resulting equation for Tm(x) Application of conservation of energy between the inlet of the channel and a section x gives Application of Newton’s law of cooling gives the heat flux q”s(x) at location x gives

6. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong

7. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong Solution For flow through a tube at uniform surface temperature, applying Eq.(6.13) At the outlet of the heat section (x=L) and solving for L Where

8. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong The properties of air using at the mean temperature Tm(x) Check the flow is laminar or turbulent

9. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong Since the Reynolds number is smaller than 2300, the flow is laminar. Thus The mass flow rate.

10. 6.5 Channels with Uniform Surface Temperature Sarthit Toolthaisong The perimeter. Finally, the length of tube

11. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong 6.6.1 Scale Analysis Equating Fourier’s law with Newton’s law A scale for r is

12. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong

13. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong From Eq. (6.18) applying thermal thickness of external flow

14. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong

15. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong 6.6.2 Basic Considerations for the Analytical Determination of Heat Flux, Heat Transfer Coefficient and Nusselt Number (1) Fourier’s law and Newton’s law. (6.21)

16. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong

17. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong Substituting into (a) (6.22) We define h using Newton’s law of cooling (6.23) Combining (6.22) and (6.23)

18. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong Where

19. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong (2) The Energy Equation The last term in Eq.(6.28) can be neglected for where

20. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong Thus, under such conditions, Eq.(6.28) becomes 3) Mean (Bulk) Temperature, Tm Where

21. 6.6 Determination of Heat Transfer Coefficient h(x) and Nusselt Number NuD Sarthit Toolthaisong

22. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong This section focuses on the fully developed region. 6.7.1 Definition of Fully Developed Temperature Profile Far away from the entrance of a channel We introduce a dimensionless temperature defined as For fully developed is independent of x. That is

23. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Thus.

24. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong 6.7.2 Heat Transfer Coefficient and Nusselt Number Equating Fourier’s with Newton’s law Using Eq.(6.37) in the definition of the Nusselt number, give

25. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong For scale analysis of temperature gradient Compared Eq.(6.19)

26. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong 6.7.3 Fully Developed Region for Tubes at Uniform Surface flux Application of Newton’s law of cooling gives

27. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong

28. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Using energy balance on element dx for detemine eq.(6.41)

29. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Assume Cp and m constant Substituting eq.(6.42) into (6.41)

30. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong For determine fluid temperature distribution T(r,x) and surface temperature Ts(x), from energy equation

31. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong The axial velocity for fully developed flow is

32. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong

33. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Substituting eq.(6.46) and (6.49) into (6.32a) gives

34. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Substituting T(r,x), Tm(x) and Ts(x) into eq.(6.33) gives Differentiating (6.54) and substituting into (6.38) gives the Nusselt number From scaling analysis

35. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong From eq.(6.44) and (6.50), we get Substituting (6.51) into (6.49) Surface temperature, by setting r=ro in (6.52)

36. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong 6.7.4 Fully Developed Region for Tubes at Uniform Surface Temperature By energy equation - Neglecting axial conduction and dissipation - vr = 0 Simplifies to Boundary conditions

37. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Using equation (6.36a) to eliminate

38. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong Applied boundary condition to Eq.(6.58)

39. 6.7 Heat Transfer Coefficient in the Fully Developed Temperature Region Sarthit Toolthaisong 6.7.5 Nusselt Number for Laminar Fully Developed Velocity and Temperature in Channels of Various Cross-Sections

40. Example 6.4: Maximum Surface Temperature in an Air Duct Sarthit Toolthaisong Solution Temperature distribution for uniform heat flux, given by eq.(6.10)

41. Example 6.4: Maximum Surface Temperature in an Air Duct Sarthit Toolthaisong

42. Example 6.4: Maximum Surface Temperature in an Air Duct Sarthit Toolthaisong Using Energy conservation to determine L

43. Example 6.4: Maximum Surface Temperature in an Air Duct Sarthit Toolthaisong

44. Example 6.4: Maximum Surface Temperature in an Air Duct Sarthit Toolthaisong Laminar flow From Table.6.2 for uniform heat flux

45. Example 6.4: Maximum Surface Temperature in an Air Duct Sarthit Toolthaisong

46. 6.8 Thermal Entrance Region: Laminar Flow through Tubes Sarthit Toolthaisong 6.8.1 Uniform Surface Temperature: Graetz Solution Consider laminar flow in Fig. 6.8 Fluid enters a heated or cooled section with a fully developed velocity We neglect axial conduction (Pe >100)

47. 6.8 Thermal Entrance Region: Laminar Flow through Tubes Sarthit Toolthaisong

48. 6.8 Thermal Entrance Region: Laminar Flow through Tubes Sarthit Toolthaisong Assume product solution as the form

49. 6.8 Thermal Entrance Region: Laminar Flow through Tubes Sarthit Toolthaisong Substitution the solution of (b) and (c) into (a) Where Cn is constant

50. 6.8 Thermal Entrance Region: Laminar Flow through Tubes Sarthit Toolthaisong The surface heat flux is given by