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Free Convection

Free Convection. Chapter 9 Section 9.1 through 9.9. Lecture 14. General Considerations. General Considerations. Free convection refers to fluid motion induced by buoyancy forces . Buoyancy forces may arise in a fluid for which there are density gradients

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Free Convection

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  1. Free Convection Chapter 9 Section 9.1 through 9.9 Lecture 14

  2. General Considerations General Considerations • Free convection refers to fluid motion induced by buoyancy forces. • Buoyancy forces may arise in a fluid for which there are density gradients • and a body force that is proportional to density. • In heat transfer, density gradients are due to temperature gradients and the • body force is gravitational. • Stable and Unstable Temperature Gradients

  3. Boundary layer flow on a hot or cold surface induced by buoyancy forces. General Considerations (cont.) • Free Boundary Flows • Occur in an extensive (in principle, infinite), quiescent (motionless at locations far from the source of buoyancy) fluid. • Plumes and Buoyant Jets: • Free Convection Boundary Layers

  4. General Considerations (cont.) • Pertinent Dimensionless Parameters • Grashof Number: [K-1] • Rayleigh Number:

  5. General Considerations (cont.) • Mixed Convection • A condition for which forced and free convection effects are comparable. • Exists if • Heat Transfer Correlations for Mixed Convection: NuFC → Nusselt number for forced convection NuNC → Nusselt number for natural (free) convection

  6. How do conditions differ for a cold plate Vertical Plates Vertical Plates • Free Convection Boundary Layer Development on a Hot Plate: x-component velocity temperature • Ascending flow with the maximum velocity occurring in the boundary layer and zero velocity at both the surface and outer edge. • How do conditions differ from those associated with forced convection?

  7. Net Momentum Fluxes ( Inertia Forces) Buoyancy Force Viscous Force Vertical Plates (cont.) • Form of the x-Momentum Equation for Laminar Flow • Temperature dependence requires that solution for u (x,y) be obtained concurrently with solution of the boundary layer energy equation for T (x,y). • The solutions are said to be coupled.

  8. Based on existence of a similarity variable, through which the x-momentum equation may be transformed from a partial differential equation with two- independent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of . Vertical Plates (cont.) • Similarity Solution • Transformed momentum and energy equations:

  9. Numerical integration of the equations yields the following results for • Velocity boundary layer thickness Vertical Plates (cont.) dimensionless x-component velocity dimensionless temperature

  10. Nusselt Numbers Vertical Plates (cont.) • Transition to Turbulence • Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces. • Transition occurs at a critical Rayleigh Number.

  11. Laminar Flow Vertical Plates (cont.) • Empirical Heat Transfer Correlations (9.27) • All Conditions: (9.26)

  12. Hot Surface Facing Upward or Cold Surface Facing Downward where L = As/P. How doesdepend on L when Horizontal Plates Horizontal Plates • Buoyancy force is normal, instead of parallel, to the plate. • Flow and heat transfer depend on whether the plate is hot or cold and • whether it is facing upward or downward. (9.30) (9.31)

  13. Horizontal Plates (cont.) • Hot Surface Facing Downward or Cold Surface Facing Upward (9.32) • Why do these conditions yield smaller heat transfer rates than those for a hot upper surface or cold lower surface?

  14. Similarity Solutions (Laminar Flow)

  15. 2. Laminar Free Convection

  16. 3. Effects of Turbulence Rayleigh Number Rax≡ Grx * Pr Rax,c≈ 109 Rax,c< 109, Laminar free convection Rax,c > 109, Turbulent free convection

  17. 4. External Free Convections n=1/4 for laminar flow; n=1/3 for turbulent flow Fluid properties evaluated at film temperature Tf Tf =(Ts+T)/2

  18. The Vertical Plate Churchill-Chu Equation (valid for all RaL & isothermal plate): For laminar free convection only (RaL < 109, isothermal plate):

  19. The Vertical Long Cylinder The above correlations may also be applied to long cylinders when the free convection boundary layer thickness is much smaller Than the cylinder diameter.

  20. Heated Surface Facing Upward or Cooled Surface Facing Downward Inclined and Horizontal Plates • Heated Surface Facing Downward or Cooled Surface Facing Upward

  21. Inclined and Horizontal Plates

  22. Upper Surface of Heated Plate or Lower Surface of Cooled Plate 104≤ RaL ≤ 107 107≤ RaL ≤ 1011 Inclined and Horizontal Plates Characteristic Length: L≡As/P=Plate Surface Area/Plate Perimeter Lower Surface of Heated Plate or Upper Surface of Cooled Plate 104≤ RaL ≤ 109 g = g*cosθ, using correlations for vertical plates For Inclined Heated or Cooled Plate

  23. Long Horizontal Cylinders C and n can be found in table 9.1 in page 613 Churchill-Chu Equation (RaL ≤ 1012 ):

  24. Spheres Churchill Correlation (RaL ≤ 1011, Pr≥0.7 ):

  25. 5. Internal Free Convection Symmetrical Vertical Channels in Isothermal Plates: As (S/L) → 0

  26. Internal Free Convection Concentric Cylinders

  27. Summary of Correlations

  28. Example 9.1 Consider a 0.25m long vertical plate that is at 70C. The plate is suspended in air that is at 25C. Estimate the boundary layer thickness at the trailing edge of the plate if the air is quiescent. How does it compare this thickness with that which would exist if the air were over the plate at a free stream velocity of 5 m/s?

  29. Example 9.1 Known:Vertical plate in quiescent air at a lower T Find: 1. Boundary layer thickness at trailing edge. 2. Compare it with thickness corresponding to an air speed of 5m/s Schematic:

  30. Example 9.1 Assumptions: 1.Constant properties, 2. Negligible buoyancy force when V=5 m/s Properties: At (70+25)/2 = 47.5 ºC, from Table A.4, for air v=17.95x10-6 m2/s, =Tf-1 =3.12x10-3 K-1, Pr=0.7 Analysis: RaL=GrL * Pr = 4.68x107

  31. Example 9.1 RaL < 109, Laminar free convection exists. From Figure 9.4, (page 567) For Pr=0.7, η≈ 6 at the edge of the boundary layer, that is, at y=δ. For airflow at u=5 m/s, ReL=(uL/v)=6.97x104, it isLaminar flow,

  32. Example 2 (Problem 9.56) Beverage in cans of 150mm long and 60 mm in diameter is initially at 27ºC and is to be cooled by placing in a refrigerator at 4ºC. In the interest of maximizing the cooling rate, should the cans be laid horizontally or vertically in the compartment? As a first estimation, neglect heat transfer from the ends.

  33. Example 2 Known:Dimensions and temperature of beer can in refrigerator compartment Find: Orientation which maximizes cooling rate Schematic:

  34. Example 2 Assumptions: • End effects are negligible, • Compartment air is quiescent, • Conduction is negligible (4) Constant properties. Properties: Air: At Tf (288.5K), ν=14.87x10-6m2/s, k=0.0254 W/mK, Pr=0.709, =21.0x10-6 m2/s, =1/Tf=3.47x10-3 K-1

  35. Example 2 Analysis: The ratio of cooling rate may be expressed as:

  36. Example 2 Lecture 14

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