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CHAPTER 10

CHAPTER 10. CORRELATION EQUATIONS:. FORCED AND FREE CONVECTION. 10.1 Introduction. Correlation equations: Based on experimental data. Chapter outline: Correlation equations for:. (1) External forced convection over:. Plates. Cylinders. Spheres.

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CHAPTER 10

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  1. CHAPTER 10 CORRELATION EQUATIONS: FORCED AND FREE CONVECTION 10.1 Introduction • Correlation equations: Based on experimental data Chapter outline: Correlation equations for: (1) External forced convection over: Plates Cylinders Spheres (2) Internal forced convection through channels (3) External free convection over: Plates Cylinders Spheres

  2. D V ¢ ¢ q - s · · T + s · T ¥ V ¥ Fig. 10.1 10.2 Experimental Determination of Heat Transfer Coefficient h Newton's law of cooling defines h: (10.1) = surface flux = surface temperature = ambient temperature Example: Electric heating , Measure: Electric power, Use (10.1) to calculate h • Form of correlation equations: • Dimensionless: Nusselt number Is a dimensionless heat transfer • coefficient.

  3. Example: Forced convection with no dissipation (2.52) = Use (2.52) to plan experiments and correlate data 10.3 Limitations and Accuracy of Correlation Equations • Limitations on: • Geometry (2) Range of parameters: Reynolds, Prandtl, Grashof, etc. (3) Surface condition: Uniform flux, uniform temperature, etc. • Accuracy: Errors as high as 25% are not uncommon! 10.4 Procedure for Selecting and Applying Correlation Equations (1) Identify the geometry

  4. (2) Identify problem classification: Forced convection Free convection External flow Internal flow Entrance region Fully developed region Boiling Condensation Etc. (3) Define objective: Finding local or average heat transfer coefficient (4) Check the Reynolds number: (a) Laminar (b) Turbulent (c) Mixed (5) Identify surface boundary condition: (a) Uniform temperature

  5. (b) Uniform flux (6) Note limitations on correlation equation (7) Determine properties at the specified temperature: (a) External flow: at the film temperature (10.2) (b) Internal flow: at the mean temperature (c)However, there are exceptions (8) Use a consistent set of units (9) Compare calculated values of h with Table 1.1 10.5 External Forced Convection Correlations 10.5.1 Uniform Flow over a Flat Plate: Transition to Turbulent Flow • Boundary layer flow over a semi-infinite flat plate

  6. V ¥ T x ¥ t · x laminar turbulent transition Fig. 10.2 Three regions: • Laminar (2) Transition (3) Turbulent = Transition or criticalReynolds number: depends on: Geometry, surface finish, pressure gradient, etc. For flow over a flat plate: (10.3) • Examples of correlation equations for plates: Laminar region, x < xt :

  7. Use (4.72a) or (4.72b) for localNusselt number to obtain local h Turbulent region,x > xt: Local h: (10.4a) Limitations: (10.4b) Average (10.5)

  8. d V ¥ t x T t x ¥ · · · x 0 insulation o T s Fig. 10.3 = local laminar heat transfer coefficient = local turbulent heat transfer coefficient (4.72b) and (10.4a) into (10.5): (10.6) Integrate (10.7a) Dimensionless form: (10.7b) (2) Plate at uniform surface temperature with an insulated leading section x0=Length of insulated section

  9. V ¥ x T t ¥ x · 0 ¢ ¢ q s Fig. 10.4 Two cases: • Laminar flow, : Use (5.21) for the local Nusselt numberto obtain > local h • Turbulent flow, : The local Nusselt number is < (10.8) (3) Plate with uniform surface flux Two regions: • Laminar flow, 0 < x < xt Use (5.36) or (5.37) for the local Nusselt numberto obtain local h • Turbulent flow, : (10.9)

  10. V q ¥ T ¥ Fig. 10.5 and is the average surface temperature Properties at 10.5 External Flow Normal to a Cylinder • For uniform surface temperature or uniform surface flux (10.10a) Limitations: (10.10b) Pe = Peclet number = ReDPr

  11. For Pe < 0.2, use: (10.11a) Limitations (10.11b) 10.5.3 External Flow over a Sphere (10.12a) Limitations: (10.12b)

  12. 10.6 Internal Forced Convection Correlations • Transition or critical Reynolds number for smooth tubes: (10.13)

  13. T T s · u developing u x 0 FDV temperatur e d t insulation Fig. 10.6 10.6.1 Entrance Region: Laminar Flow Through Tubes at Uniform Surface Temperature • Two cases: • Fully Developed Velocity, Developing Temperature: Laminar Flow • Solution: Analytic Correlation of analytic results: (10.14a)

  14. Limitations: (10.14b) (2) Developing Velocity and Temperature: Laminar flow (10.15a)

  15. Limitations: 10.6.2 Fully Developed Velocity and Temperature in Tubes: Turbulent Flow • Entrance region is short: 10-20 diameters • Surface B.C. have minor effect on h for Pr > 1 • Several correlation equations for h: (1) The Colburn Equation: Simple but not very accurate (10.16a) Limitations:

  16. (10.16b) • Accuracy: Errors can be as high as 25% (2) The Gnielinski Equation: Provides best correlation of experimental data (10.17a) • Valid for: developing or fully developed turbulent flow

  17. Limitations: (10.17b) • The D/L factor in equation accounts for entrance effects • For fully developed flow set D/L = 0 The Darcy friction factorf is defined as (10.18) For smooth tubes f is approximated by (10.19)

  18. x u · T T s ¥ g y Fig. 10.7 10.6.3 Non-circular Channels: Turbulent Flow Use equations for tubes. Set (equivalent diameter) (10.20) = flow area = wet perimeter 10.7 Free ConvectionCorrelations 10.7.1 External Free Convection Correlations • Vertical plate: Laminar Flow, Uniform Surface • Temperature • Local Nusselt number:

  19. (10.21a) • Average Nusselt number: (10.21b) (10.21a) and (10.21b) are valid for: Limitations: (10.21c)

  20. (2) Vertical plates: Laminar and Turbulent, Uniform Surface Temperature (10.22a) Limitations: (10.22b) (3) Vertical Plates: Laminar Flow, Uniform Heat Flux • Local Nusselt number:

  21. (10.24) and (10.25) into (10.23) and solve for (10.23) Determine surface temperature: Apply Newton’s law: (10.24) where is defined as (10.25) (10.26a) (10.23) and (10.26a) are valid for:

  22. > T T ¥ s q < T T q ¥ s g T ¥ (a) (b) Fig. 10.9 (10.26b) (x) which is not • Properties in (10.26a) depend on surface temperature known. Solution is by iteration (4) Inclined plates: Constant surface temperature • Use equations for vertical plates • Modify Rayleigh number as: (10.27)

  23. Limitations: (10.28) (5) Horizontal plates: Uniform surface temperature: (i) Heated upper surface or cooled lower surface , (10.29a) , (10.29b) Limitations: (10.29c)

  24. (ii) Heated lower surface or cooled upper surface , (10.30a) Limitations: horizontal platehot surface down or cold surface up all properties, except, β, at Tfβ at Tffor liquids, Ts for gases (10.30b) Characteristic length L: (10.31) (6) Vertical Cylinders. Use vertical plate correlations for: for Pr 1(10.32) (7) Horizontal Cylinders:

  25. (10.33a) Limitations: (10.33b) (8) Spheres (10.34a) Limitations: (10.34b)

  26. 10.7.2 Free Convection in Enclosures Examples: • Double-glazed windows • Solar collectors • Building walls • Concentric cryogenic tubes • Electronic packages Fluid Circulation: • Driving force: Gravity and unequal surface temperatures Heat flux: Newton’s law: (10.35) Heat transfer coefficient h: Nusselt number correlations depend on:

  27. d T T c c L g Fig. 10.10 • Configuration • Orientation • Aspect ratio • Prandtl number • Rayleigh number (1) Vertical Rectangular Enclosures Rayleigh number (10.36) Several equations:

  28. (10.37a) Valid for (10.37b) (10.38a) Valid for (10.38b)

  29. (10.39a) Valid for (10.39b) (10.40a) Valid for (10.40b)

  30. T L c g d T h Fig. 10.11 (2) Horizontal Rectangular Enclosures • Enclosure heated from below Cellular flow pattern develops at critical Rayleigh number Nusselt number: (10.41a) Valid for (10.41b)

  31. d T c T g L h q Fig. 10.12 Table 10.1 critical tilt angle > 12 d 6 1 12 L / 3 o o o o o q 25 5 3 6 0 6 7 7 0 c (3) Inclined Rectangular Enclosures • Applications: Solar collectors • Nusselt number: correlations depend on: • Inclination angle • Aspect ratio • Prandtl number • Rayleigh number For: : heated lower surface, cooled upper surface cooled lower surface, heated upper surface • Nusselt number is minimum at

  32. a critical angle : Table 10.1 (10.42a) Valid for (10.42b)

  33. (10.43a) Valid for (10.43b) (10.44a) Valid for (10.44b)

  34. D o T o 5 T i D i Fig. 10..13 (10.45a) Valid for (10.45b) (4) Horizontal Concentric Cylinders • Flow circulation for • Flow direction is reversed for • Circulation enhances thermal conductivity (10.46)

  35. : Correlation equation for the effective conductivity (10.47a) (10.47b) (10.47c) Valid for (10.47d)

  36. 10.8 Other Correlations • Boiling • Condensation • Jet impingement • High speed flow • Dissipation • Liquid metals • Enhancements • Finned geometries • Irregular geometries • Non-Newtonian fluids • Etc.

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