1. Sarthit Toolthaisong FREE CONVECTION

2. Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection 1) Driving Force In general, two conditions are required for fluids to be set in motion in free convection. - acceleration field (gravity) - density gradient (temperature gradient) 2) Governing Parameters. Two parameters play a key role in the determination of the Nusselt number in free convection: - the Grashof number - the Prandtl number

3. Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection Grashof number Coefficient of thermal expansion or Compressibility factor For ideal gases it is given by

4. Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection Rayleigh number

5. Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection 3) Boundary Layer Boundary layer approximations for free convection are valid for 4) Transition from Laminar to Turbulent Flow For vertical plates the transition Rayleigh number is given by

6. Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection 5) External vs. Enclosure Free Convection. In external free convection a surface is immersed in a fluid of infinite extent. Enclosure free convection takes place inside closed volumetric regions.

7. Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection 6) Analytic Solutions. Analytic solutions require the simultaneous integration of the continuity, momentum and energy equations.

8. Sarthit Toolthaisong 7.3 Governing Equations Analysis of free convection is usually based on following approxi-mations: (1)Density is assumed constant except in evaluating gravity forces. (2) The Boussinesq approximation which relates density change to temperature change is used in formulating buoyancy force in the momentum equation. (3) Dissipation effect is neglected in the energy equation.

9. Sarthit Toolthaisong 7.3 Governing Equations

10. Sarthit Toolthaisong 7.3 Governing Equations 7.3.1 Boundary Layer Equations From Eq.(7.6) reduce to

11. Sarthit Toolthaisong 7.3 Governing Equations Applied (c) for Eq.(7.5), thus the x-component of the Navier-Stokes equations simplifies to

12. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 7.4.1 Assumptions. (1) Continuum (2) Newtonian (3) Steady state (4) Laminar flow (5) Two-dimensional (6) Constant properties (7) Boussinesq approximation (8) Uniform surface (9) Uniform ambient temperature (10) Vertical plate (11) Negligible dissipation

13. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 7.4.2 Governing Equations. Based on the above assumptions we get the governing equations

14. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 7.4.3 Boundary Conditions No slip No dissipation No density change T= Ts T= T 

15. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 7.4.4 Similarity Transformation

16. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature By stream function 

17. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature Using stream function  of Blasius solution

18. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

19. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

20. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 7.4.5 Solution Solution by numerical

21. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature 7.4.6 Heat Transfer Coefficient and Nusselt Number. Based on Fourier’s law and Newton’s law

22. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

23. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

24. Sarthit Toolthaisong 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

25. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

26. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature Solution. Assumptions - Continuum - Newtonian fluid - Steady state - Boussinesq approximations - Two – dimensional - Laminar flow - Flat plate - Uniform surface temperature - No dissipation - No radiation

27. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature The properties are evaluated at the temperature

28. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature Check laminar or turbulent by Rayleigh number Thus the flow is laminar. Axial velocity u is given by (7.20).

29. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

30. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

31. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

32. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

33. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

34. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature

35. Sarthit Toolthaisong Example 7.1 Vertical at Uniform Surface Temperature