Convection in Flow Past Cylinders : Cross Flows

1. Convection in Flow Past Cylinders : Cross Flows P M V Subbarao Professor Mechanical Engineering Department IIT Delhi Most Popular Thermal Equipment in Industry…

2. Cylinder in Cross Flow

6. Generally the overall average Nusselt number for heat transfer with the entire object is important. As with a flat plate, correlations developed from experimental data to compute Nu as a f(Rem,Prn) Overall Average Nusselt number • All properties are evaluated at the freestream temperature, except Prs which is evaluated at the surface temperature.

7. Values for C and m Expect an accuracy within  20% with these correlations

8. Cylinder in Cross Flow The empirical correlation due to Hilpert • All properties are evaluated at the bulk mean temperature ,Tb.

9. D D Square Cylinder in Cross Flow Valid for 5 X 103 < ReD < 105 Valid for 5 X 103 < ReD < 105

10. D D Hexagonal Cylinder in Cross Flow Valid for 5 X 103 < ReD <1.95X104 Valid for 1.95X104 < ReD < 105 Valid for 5 X 103 < ReD < 105

11. Vertical Plate in Cross Flow Valid for 4 X 103 < ReD < 1.5 X104 D

12. Convection heat transfer with a sphere External flow and heat transfer relations are similar to those around a cylinder. Numerous correlations proposed from lab experiments, one being: All properties except ms are evaluated at T∞.

13. Special case: Free falling liquid drops

14. Convection heat transfer with banks of tubes • Typically, one fluid moves over the tubes, while a second fluid at a different temperature passes through the tubes. (cross flow) • The tube rows of a bank are staggered or aligned. • The configuration is characterized by the tube diameter D, • the transverse pitch ST and longitudinal pitch SL.

15. Internal or External Flow !?!?! Inline Arrangement Zig-Zag Arrangement Square Pitch Triangular Pitch

16. For tube bundles composed of 10 or more rows

17. For Reynolds number or If staggered and

18. All properties are evaluated at the bulk mean temperature.

19. If number of tubes are less than 10, a correction factor is applied as: And values for C2 are from table

21. CONVECTION IN INTERNAL FLOWS P M V Subbarao Professor Mechanical Engineering Department IIT Delhi An Essential Part of Exchanging Heat……..

22. Development of Flow

23. Hydrodynamic Vs Thermal Development of FLow • There are several fundamental problems in laminar internal flow that can be considered. • The following problems arise as a result of considering the thermal entrance length (Lt)in proportion to the hydrodynamic entrance length(Lh): • L >> Lh, L >> Lt, i.e. thermally and hydrodynamically fully developed flow. • This rarely occurs in practice, but it affords many theoretical solutions. • L >> Lh, L << Lt, i.e. hydrodynamically fully developed, but thermally developing flow, sometimes called the thermal entrance problem. • This type of flow is characteristic of high Prandtl number fluids Pr >> 1, e.g. oils. • L << Lh, L << Lt, i.e. hydrodynamically and thermally developing flow, sometimes called the combined entrance problem • L << Lh, L >> Lt, i.e. hydrodynamically developing flow and thermally fully developed. • This type of flow occurs with low Prandtl number fluids Pr << 1, e.g. liquid metals.

24. Nature of Convection • In general, heat transfer is always higher in developing flows, since the thermal resistance of the boundary layer is lower. • In the thermal entrance region, heat is being transferred from a warmer wall temperature (in the case of heating) to the lowest temperature which is the inlet fluid temperature. • However, when the thermal boundary layers merge, the lowest temperature is at the centre and bulk fluid temperature rises quickly. • The local heat transfer rate is:

25. Temperature Profile in Internal Flow Hot Wall & Cold Fluid q’’ Ts(x) Ti Cold Wall & Hot Fluid q’’ Ti Ts(x)

26. An universally acceptable definition of Newton's Law of cooling for internal flows : • The local heat transfer rate is: We also often define a Nusselt number as:

27. Mean Velocity and Bulk Temperature Two important parameters in internal forced convection are the mean flow velocity u and the bulk or mixed mean fluid temperature Tm(z). The mass flow rate is defined as: For Incompressible Flows: while the bulk or mixed mean temperature is defined as: For Incompressible Flows:

28. Mean Temperature (Tm) • We characterise the local fluid temperature by using the mean temperature of the fluid at a given cross-section. • Heat addition to the fluid leads to increase in mean temperature and vice versa. • For the existence of convection heat transfer, the mean temperature of the fluid should monotonically vary. • Computation of local convection heat transfer coefficient involves: • Computation of temperature & velocity profiles. • Computation of local Mean Temperature.

29. First Law for A CV : SSSF qz Tm,in Tm,exit dx No work transfer, change in kinetic and potential energies are negligible

31. THERMALLY FULLY DEVELOPED FLOW • There should be heat transfer from wall to fluid or vice versa, over the entire length of the pipe being used. • Then What does fully developed flow signify in Thermal view?

32. FULLY DEVELOPED CONDITIONS (THERMALLY) What does this signify? How to model this limit? Use a dimensionless temperature difference to characterise the profile, i.e. use This ratio is independent of x in the fully developed region, i.e.

33. This is condition for Thermal Development Internal Flows

34. Temperature Profile in Fully Developed Region Uniform Wall Temperature (UWT)  axial temp. gradient is not independent of r and shape of temperature profile is changing.

35. The shape of the temperature profile is changing, but the relative shape is unchanged (for UWT conditions). At the tube surface:

36. i.e. the Nusselt number is independent of x in the thermally fully developed region. Assuming const. fluid properties:- This is the real significance of thermally fully developed

37. Evolution of Macro Flow Parameters

38. Uniform Wall Heat flux : Fully Developed Region

39. For a thermally developed flow with constant wall flux: Temperature profile shape is unchanging.

40. Constant Surface Heat Flux : Heating of Fluid Integrating from x=0 (Tm = T m,i) to x = L (Tm = Tm,o):

41. Thermal Considerations – Internal Flow • T fluid Tsurface • a thermal boundary layer develops • The growth of dthdepends on whether the flow is laminar or turbulent Extent of Thermal Entrance Region: Laminar Flow: Turbulent Flow: