1. FREE CONVECTION 7.1 Introduction Solar collectors Pipes Ducts Electronic packages Walls and windows 7.2 Features and Parameters of Free Convection (1) Driving Force:Natural 1

2. Requirements: (i) Acceleration field, (ii) Density gradient (2) Governing Parameters: Two: 2

3. Criterion for vertical plates: Transition Rayleigh number 3

4. (5) External vs. Enclosure free convection: (i) External: over: Vertical surfaces Inclined surfaces Horizontal cylinders Spheres (ii) Enclosure: in: 4

5. Rectangular confines Concentric cylinders Concentric spheres (6) Analytic Solution 7.3 Governing Equations Approximations: (1) Density is constant except in gravity forces 5

6. (2) Boussinesq approximation: relate density change to temperature change Assume: Steady state Two-dimensional Laminar Continuity: x-momentum: 6 (3) Negligible dissipation

7. y-momentum: Energy: NOTE: (1) Gravity points in the negative x-direction (2) Flow and temperature fields are coupled 7

8. 7.3.1 Boundary Layer Equations Velocity and temperature boundary layers Apply approximation used in forced convection y-component of the Navier-Stokes equations reduces to External flow: Neglect ambient pressure variation in x Furthermore, for boundary layer flow 8

9. x-momentum: Neglect axial conduction Energy: (7.4) becomes 9

10. 7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature Determine: velocity and temperature distribution 7.4.1 Assumptions (1) Steady state (2)Laminar flow (3) Two-dimensional 10

11. (1) Constant properties (2) Boussinesq approximation (3) Uniform surface temperature (4) Uniform ambient temperature (5) Vertical plate (9) Negligible dissipation 7.4.2 Governing Equations Continuity: x-momentum: 11

12. Energy: 7.4.3 Boundary Conditions Velocity: where is a dimensionless temperature defined as 12

13. Temperature: 7.4.4 Similarity Transformation 13

14. Stream function satisfies continuity Let Local Grashof number: (7.9) into (7.8)7 14

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16. Combining (7.5), (7.7), (7.8), (7.12), (7.16), (7.17) NOTE Velocity: Transformation of boundary conditions: 16

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18. NOTE: 7.4.5 Solution (1) Three PDE are transformed into two ODE (3) Five BC are needed (4) Seven BC are transformed into five (5) One parameter: Prandtl number. (7.18) and (7.19) are solved numerically Solution is presented graphically Figs. 7.2 gives u(x,y) Fig. 7.3 gives T(x,y) 18

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21. 7.4.5 Heat Transfer Coefficient and Nusselt Number Start with Use (7.8) and (7.10) 21

22. Define: local Nusselt number: Average heat transfer coefficient Average Nusselt number (7.21) into (2.50), integrate 22

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24. Special Cases Example 7.1: Vertical Plate at Uniform Surface Temperature (3) , (4) t  24

25. Solution (2) Problem Definition (1) Observations  External free convection  Vertical plate  Uniform surface temperature Check Rayleigh number for laminar flow 25

26. Determine flow and heat transfer characteristics for free convection over a vertical flat plate at uniform surface temperature. (3) Solution Plan (4) Plan Execution (i) Assumptions  Laminar flow? Compute Rayleigh number  If laminar, use Figs. 7.2 and 7.3.  Use solution for Nu and h (1)Newtonian fluid (2)Steady state (3)Boussinesq approximations (4)Two-dimensional 26

27. (6) Flat plate (7) Uniform surface temperature (8) No dissipation (9) No radiation. (ii) Analysis and Computation Compute the Rayleigh number: 27

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29. (1) Axial velocity u: Substituting into (7.2) Thus the flow is laminar 29

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31. Solve for u (1) Temperature T: 31

32. Solve for 32

33. Nusselt Number: Use (7.22), evaluate at x = L = 0.08 m (5) Local Nusselt number: 33

34. (6) Local heat transfer coefficient:at x = L = 0.08 m Newton’s law gives(7) Heat flux: (8) Total heat transfer: 34

35. (iii) Checking Dimensional check: Quantitative check: 35

36. 7.5 Laminar Free Convection over a Vertical Plate: Uniform Surface Heat Flux (5) Comments Infinite fluid at temperature Assumptions: Same as Section 7.4 Vertical Plate Uniform surface heat flux 36

37. Governing equations: Same as Section 7.4 Boundary conditions: Replace uniform surface temperature with uniform surface flux Solution: Similarity transformation (Appendix F) Results: (1) Surface temperature 37

38. (2) Nusselt number 38

39. Example 7.2: Vertical Plate at Uniform Surface Flux Solution (1) Observations External free convection Vertical plate (1) Surface temperature (2) Nusselt number (3) Heat transfer coefficient 39

40. Uniform surface heat flux Check Rayleigh number for laminar flow If laminar: Laminar flow? Compute Rayleigh number (2) Problem Definition (3) Solution Plan 40

41. (10) Newtonian fluid (11) Steady state (12) Boussinesq approximations (13) Two-dimensional (15) Flat plate (i) Assumptions (4) Plan Execution 41

42. (16) Uniform surface heat flux (17) No dissipation (18) No radiation (ii) Analysis Rayleigh number: Surface temperature: Nusselt number: 42

43. Heat transfer coefficient (iii) Computations 43

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46. Substitute into (7.2) Flow is laminar 46

47. Result: 47

48. Thus, 48

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50. Use (d)-(f) to tabulate results at x = 0.02, 0.04, 0.06 and 0.08 m (iii) Checking 50

51. Dimensional check: Quantitative check: (5) Comments (i) Magnitude of h is relatively small (ii) Surface temperature increases with distance along plate 7.6 Inclined Plates Two cases: 51

52. Hot side is facing downward, Fig. 7.5a Cold side is facing upward, Fig. 7.5b Flow field: Same for both Solution: Limitations 52 7.7 Integral Method

53. Obtain approximate solutions to free convection problems Example: Vertical plate at uniform surface temperature (a) Simplifying assumption 7.7.1 Integral Formulation of Conservation of Momentum 53

54. Wall stress Pressure Gravity Simplify Wall stress: 54 (c)

55. Weight: x-momentum: Substitute(e), (f) and (g) into (d) Constant density Variable density 55

56. Combine pressure and gravity terms (B.L. flow) (j) into (h) Density change: Rewrite 56

57. NOTE: Assume: 7.7.2 Integral Formulation of Conservation of Energy (1)no shearing force on the slanted surface (2)(7.30) applies to laminar and turbulent flow (3)(7.30) is a first order O.D.E. 57

58. No changes in kinetic and potential energy Neglect axial conduction Neglect dissipation Properties are constant NOTE: 7.7.3 Integral Solution 58

59. Solution Procedure for Forced Convection: Vertical plate 59

60. Solution Procedure for Free Convection: 60

61. Assumed Velocity Profile Boundary conditions: 61

62. (a) gives Rewrite as Introduce a second unknown in (b). Define (b) becomes 62

63. (2) (3) (1) Assumed Temperature Profile Boundary Conditions: 63

64. (d) becomes Heat Transfer Coefficient and Nusselt Number (7.33) into (7.20) Local Nusselt number: 64

65. Solution Use momentum: substitute (7.32) and (7.33) into (7.30) Evaluating the integrals Use energy: substitute(7.32) and (7.33) into (7.31) 65

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68. Solve (g) and (h) for m and n gives Introduce (i) into (7.40) and (7.41) Solve (j) and (k) for A and B and 68

69. Substitute (i) and (m) into (7.39) Introduce(7.42) into (7.35) 7.7.4 Comparison with Exact Solution for Nusselt Number Exact solution: 69

70. Rewrite above as Rewrite integral solution (7.43) as 70

71. Limiting Cases 71

72. NOTE: 72