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Laminar Free Convection: Vertical Plate Heat Transfer

This guide explores free convection over vertical plates, covering governing equations, solutions, and heat transfer coefficients. Learn about boundary layer equations and compute Nusselt numbers for different scenarios.

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Laminar Free Convection: Vertical Plate Heat Transfer

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  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 (3) Negligible dissipation Assume: Steady state Two-dimensional Laminar Continuity: x-momentum: 6

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

  13. Temperature: 7.4.4 Similarity Transformation 13

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

  15. 15

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

  17. 17

  18. NOTE: (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. 7.4.5 Solution • (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

  19. 19

  20. 20

  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 (7.21) into (2.50), integrate Average Nusselt number 22

  23. 23

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

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

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

  28. 28

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

  30. 30

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

  32. Solve for 32

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

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

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

  36. Infinite fluid at temperature (5) Comments 7.5 Laminar Free Convection over a Vertical Plate: Uniform Surface Heat Flux • Vertical Plate • Uniform surface heat flux • Assumptions: Same as Section 7.4 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 (1) Surface temperature (2) Nusselt number (3) Heat transfer coefficient Solution (1) Observations • External free convection • Vertical plate 39

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

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

  44. 44

  45. 45

  46. Substitute into (7.2) • Flow is laminar 46

  47. Result: 47

  48. Thus, 48

  49. 49

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

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