1. Heat Transfer Su Yongkang School of Mechanical Engineering # 1 HEAT TRANSFER CHAPTER 9 Free Convection

2. Heat Transfer Su Yongkang School of Mechanical Engineering # 2 Natural Convection Where we’ve been …… Up to now, have considered forced convection, that is an external driving force causes the flow. Where we’re going: Consider the case where fluid movement is by buoyancy effects caused by temperature differential

3. Heat Transfer Su Yongkang School of Mechanical Engineering # 3 When natural convection is important Weather events such as a thunderstorm Glider planes Radiator heaters Hot air balloon Heat transfer with pipes and electrical lines Heat flow through and on outside of a double pane window Just sitting there Oceanic and atmospheric motions Coffee cup example …. Small velocity

4. Heat Transfer Su Yongkang School of Mechanical Engineering # 4 Natural Convection KEY POINTS THIS LECTURE New terms –Volumetric thermal expansion coefficient –Grashof number –Rayleigh number Buoyancy is the driving force –Stable versus unstable conditions Nusselt number relationship for laminar free convection on vertical surface Boundary layer impacts: laminar  turbulent Text book sections: §9.1 – 9.5

5. Heat Transfer Su Yongkang School of Mechanical Engineering # 5 Buoyancy is the driving force Buoyancy is due to combination of –Differences in fluid density –Body force proportional to density –Body forces gravity, also Coriolis force in atmosphere and oceans Convection flow is driven by buoyancy in unstable conditions Fluid motion may be (no constraining surface) or along a surface

6. Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Buoyancy is the driving force (Cont’d) Free boundary layer flows Heated wire or hot pipe

7. Heat Transfer Su Yongkang School of Mechanical Engineering # 7 A heated vertical plate We focus on free convection flows bounded by a surface. The classic example is u(x,y) y g x v u Extensive, quiescent fluid

8. Heat Transfer Su Yongkang School of Mechanical Engineering # 8 Governing Equations The difference between the two flows (forced flow and free flow) is that, in free convection, a major role is played by buoyancy forces. Consider the x-momentum equation. As we know,, hence the x-pressure gradient in the boundary layer must equal that in the quiescent region outside the boundary layer. Very important Buoyancy force

9. Heat Transfer Su Yongkang School of Mechanical Engineering # 9 Governing Equations (Cont’d) Define , the volumetric thermal expansion coefficient. In general, For liquids and non-ideal gases, see appendix A Density gradient is due to the temperature gradient

10. Heat Transfer Su Yongkang School of Mechanical Engineering # 10 Governing Equations (cont’d) Now, we can see buoyancy effects replace pressure gradient in the momentum equation The buoyancy effects are confined to the momentum equation, so the mass and energy equations are the same. Strongly coupled and must be solved simultaneously

11. Heat Transfer Su Yongkang School of Mechanical Engineering # 11 Dimensionless Similarity Parameter The x-momentum and energy equations are

12. Heat Transfer Su Yongkang School of Mechanical Engineering # 12 Dimensionless Similarity Parameter (cont’d) Define new dimensionless parameter, Grashof number in natural convection is analogous to the Reynolds number in forced convection. Grashof number indicates the ratio of the buoyancy force to the viscous force. –Higher Gr number means increased natural convection flow Eq 9.12 forced natural

13. Heat Transfer Su Yongkang School of Mechanical Engineering # 13 Example #1: Consider a object having a characteristic length of 0.25m and a situation for which the temperature difference is 25 ℃. Using thermophysical properties evaluated at 350K, calculate the Grashof number for air, hydrogen, water, and ethylene glycol. Assume a pressure of 1 atm.

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15. Heat Transfer Su Yongkang School of Mechanical Engineering # 15 u(x,y) y g x v u Laminar Free Convection on Vertical Surface As y   : u = 0, T = T  As y  0 : u = 0, T = T s With little or no external driving flow, Re  0 and forced convection effects can be safely neglects –

16. Heat Transfer Su Yongkang School of Mechanical Engineering # 16 Analytical similarity solution for the local Nusselt number in laminar free convection Average Nusselt # = Eq 9.21

17. Heat Transfer Su Yongkang School of Mechanical Engineering # 17 Effects of Turbulence Just like in forced convection flow, hydrodynamic instabilities may result in the flow. For example, illustrated for a heated vertical surface: Define the Rayleigh number for relative magnitude of buoyancy and viscous forces For vertical surface, transition to turbulence at

18. Heat Transfer Su Yongkang School of Mechanical Engineering # 18 Effects of Turbulence (cont’d) Transition to turbulent flow greatly effects heat transfer rate.

19. Heat Transfer Su Yongkang School of Mechanical Engineering # 19 Example #2: Consider a large vertical plate with a uniform surface temperature of 130 ℃ suspended in quiescent air at 25 ℃ and atmospheric pressure. a)Estimate the boundary layer thickness at a location 0.25m measured from the lower edge. b)What is the maximum velocity in the boundary layer at this location and at what position in the boundary layer does the maximum occur?

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21. Heat Transfer Su Yongkang School of Mechanical Engineering # 21 Example #3: A number of thin plates are to be cooled by vertically suspending them in a water bath at a temperature of 20 ℃. If the plates are initially at 54 ℃ and 0.15m long, what minimum spacing would prevent interference between their free convection boundary layers?

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23. Heat Transfer Su Yongkang School of Mechanical Engineering # 23 Have a good time!