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Correlation Equations for Heat Transfer: Forced and Free Convection

Learn about experimental data, limitations, and applications of correlation equations for forced and free convection. Understand the process for selecting and applying these equations with examples and calculations provided.

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Correlation Equations for Heat Transfer: Forced and Free Convection

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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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