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External Flow

External Flow. Chapter 7 Section 7.1 through 7.8. Lecture 12. Boundary Layer Analogies. To establish relationship between C f , Nu and Sh . 1. Empirical Method. Empirical Correlations. Empirical Correlation. Pr and Sc are properties of fluid at film temperature T f T f =(T s +T  )/2.

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External Flow

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  1. External Flow Chapter 7 Section 7.1 through 7.8 Lecture 12

  2. Boundary Layer Analogies To establish relationship between Cf, Nu and Sh

  3. 1. Empirical Method

  4. Empirical Correlations

  5. Empirical Correlation Pr and Sc are properties of fluid at film temperature Tf Tf =(Ts+T)/2

  6. 2. Flat Plate in Parallel Flow Laminar Flow:

  7. To determine the conditions, compute • and compare with the critical Reynolds number for transition to turbulence, Physical Features Physical Features • As with all external flows, the boundary layers develop freely without constraint. • Boundary layer conditions may be entirely laminar, laminar and turbulent, • or entirely turbulent.

  8. Value of depends on free stream turbulence and surface roughness. • Nominally, • If boundary layer istrippedat the leading edge • and the flow is turbulent throughout. • Surface thermal conditions are commonly idealized as being of uniform • temperature or uniform heat flux . Equivalent surface and free stream temperatures for and uniform (or ) for Physical Features (cont.) Is it possible for a surface to be concurrently characterized by uniform temperature and uniform heat flux? • Thermal boundary layer development may be delayed by an unheated • starting length.

  9. Based on premise that the dimensionless x-velocity component, , • and temperature, , can be represented exclusively in • terms of a dimensionless similarity parameter and Similarity Solution for Laminar, Constant-Property Flow over an Isothermal Plate Similarity Solution • Similarity permits transformation of the partial differential equations associated • with the transfer of x-momentum and thermal energy to ordinary differential • equations of the form

  10. Similarity Solution (cont.) • Subject to prescribed boundary conditions, numerical solutions to the momentum • and energy equations yield the following results for the x-component velocity • distributionand the temperature distribution in the boundary layer: What value of h corresponds to the edge of the velocity boundary layer? How thick is the thermal boundary layer, relative to the velocity boundary layer, for Pr = 1? Pr > 1? Pr < 1?

  11. Similarity Solution (cont.) and

  12. Similarity Solution (cont.) • How would you characterize relative laminar velocity and thermal boundary layer • growth for a gas? An oil? A liquid metal? • How do the local shear stress and convection coefficient vary with distance from • the leading edge? • Average Boundary Layer Parameters: • The effect of variable properties may be considered by evaluating all properties • at the film temperature.

  13. Empirical Correlations Substituting expressions for the local coefficients and assuming Turbulent Flow Turbulent Flow • Local Parameters: How do variations of the local shear stress and convection coefficient with distance from the leading edge for turbulent flow differ from those for laminar flow? • Average Parameters:

  14. Sketch the variation of hx versus for two conditions: What effect does an USL have on the local convection coefficient? Special Cases: Unheated Starting Length (USL) and/or Uniform Heat Flux Special Cases For both uniform surface temperature (UST) and uniform surface heat flux (USF), the effect of the USL on the local Nusselt number may be represented as follows:

  15. Special Cases (cont.) • UST: • USF: • Treatment of Non-Constant Property Effects: Evaluate properties at the film temperature.

  16. Flat Plate in Parallel Flow Laminar Flow: Boundary layer thickness: Local fraction coefficient: Local heat transfer: Pr ≥ 0.6 Local mass transfer: Sc ≥ 0.6

  17. Flat Plate in Parallel Flow Laminar Flow: Average fraction coefficient: Average heat transfer: Pr ≥ 0.6 Average mass transfer: Sc ≥ 0.6

  18. Flat Plate in Parallel Flow Turbulent Flow: Boundary layer thickness: Local fraction coefficient: Rex ≤ 107 Local heat transfer: 0.6<Pr<60 Local mass transfer: 0.6<Sc<3000

  19. Flat Plate in Parallel Flow Mixed Boundary Layer Conditions:

  20. Flat Plate in Parallel Flow Mixed Boundary Layer Conditions: Average fraction coefficient: 5x105<ReL≤108 Rex,c=5x105 Average heat transfer: 0.6<Pr<60 5x105<ReL≤108 Rex,c=5x105 Average mass transfer: 0.6<Sc<3000 5x105<ReL≤108 Rex,c=5x105

  21. 3. Cylinder in Cross Flow Boundary layer formation and separation ReD = VD/ν, CD≡ FD/(Af*(*V2/2)

  22. Cylinder in Cross Flow Boundary layer formation and separation

  23. Cylinder in Cross Flow Hilpert Correlation: (Pr>0.6) C and m depend on ReD and geometry of the cylinder, they can be found in Tables 7.2 & 7.3 (pages 458-459), where all properties are evaluated at film temperature Tf ReD = VD/ν

  24. Cylinder in Cross Flow Zhukauskas Correlation (circular cylinders): C and m depend on ReD, they can be found in Tables 7.4 (pages 459) where all properties are evaluated at T, except Prs at Ts n=0.37, Pr ≤10 n=0.36, Pr>10

  25. Cylinder in Cross Flow Chuchill-Bernstein Correlation: Pr > 0.2 and all ReD all properties are evaluated at film temperature Tf

  26. 4. Flow Cross Sphere ReD = VD/ν 0.71<Pr<380, 3.5<ReD<7.6x104 All properties, except μs, are evaluated at T , μs is at surface temperature Ts

  27. 5. Flow Across Banks of Tubes Aligned Staggered

  28. Flow Across Banks of Tubes Zhukauskas Correlation (Table 7.5, page 470): NL≥ 20 0.7 < Pr < 500 1000 < ReD,max < 2x106 where all properties, except Prs, are evaluated at average fluid T (Tf) ReD,max = VmaxD/ν Vmax = ST/(ST-D)*V for aligned or staggeredarrangement Vmax = ST/(SD-D)*V for staggered arrangement

  29. Flow Across Banks of Tubes Log-mean temperature difference: • Heat transfer rate per unit length:

  30. 6. Flow Through Packed Bed

  31. 6. Flow Through Packed Bed Pr ≈ 0.7 (gas) 90 ≤ ReD ≤4000 properties at average fluid T

  32. 7. Methodology for Convection Calculation • Identify the flow geometry • Specify reference T for fluid properties • Use dilute species to calculate Sc • Calculate Re • Local or average surface coefficient ? • Select an appropriate correlation

  33. Example 1 Air at pressure of 6 kN/m2 and a temperature of 300 ºC flows with a velocity of 10m/s over a flat plate of 0.5m long. Estimate the cooling rate per unit width of the plate needed to maintain it at a surface T of 27 ºC?

  34. Example 1 Known: Air flow over isothermal flat plate Find: Cooling rate per unit width, q’ (W/m) Schematic:

  35. Example 1 Assumptions: Steady-state, Negligible radiation Properties: At (300+27)/2 = 163.5 ºC, from Table A.4, for air v=30.84x10-6 m2/s, k=36.4x10-3W/mK, Pr=0.687 Analysis:

  36. Example 1 The flow is laminar over the entire plate, the corresponding correlation is:

  37. Example 2 The decorative plastic film on a copper sphere of 10-mm diameter is cured in an oven at 75ºC. Upon removal from the oven, the sphere is subjected to an air stream at 1 atm and 23 ºC having a velocity of 10m/s. Estimate how long it will take to cool the sphere to 35 ºC.

  38. Copper Sphere D=10 mm Air p=1 atm V=10 m/s T=23 C Ti =75 C T(t)=35 C Example 2 Known:Sphere cooling in air stream Find: Time required to cool from 75ºC to 35ºC Schematic:

  39. Example 2 Assumptions: 1. Negligible thermal resistance and capacitance for the plastic film; 2. Spatially isothermal sphere; 3. Negligible radiation effect. Properties: Copper:=8933 kg/m3, k=399 W/mK, cp=387 J/kgK, Air: At T (296K), μ=181.6x10-7Ns/m2, ν=15.36x10-6m2/s, k=0.0258 W/mK, Pr=0.709. At Ts, μ=197.8x10-7Ns/m2

  40. Example 2 Analysis: This is a transient problem, check Bi first Bi = hL/k=?, Assume Bi < 0.1, apply lumped capacitance method

  41. Example 2 Analysis: From Eqn. 7.59:

  42. Example 2 Analysis: Check if Bi < 0.1 ? Bi = hL/k= 122 W/m2K*(0.01m/6)/399W/mK = 5.1x10-4 <0.1 Where L = D/6

  43. Mass Transfer Mass Transfer • Correlations for convection mass transfer coefficients associated with evaporation • or sublimation from liquid or solid surfaces in external flow may be inferred from • the corresponding heat transfer correlations for an isothermal surface by invoking • the heat-and-mass transfer analogy: • Example: Flat Plate in Parallel Flow • Laminar Flow: • Laminar-to-Turbulent Flow: • Turbulent Flow:

  44. Hence, with , the properties of the mixture may be approximated by those of species B. • Evaluate the surface vapor concentration at the surface temperature (assuming saturated conditions). Mass Transfer (cont.) • Fluid Properties: • Assume a dilute, binary mixture of species A (the volatile species) and species B (the free stream fluid). • Evaluate the properties of species B at the reference temperature specified • for the correlation. If the correlation involves a property ratio, approximate the ratio as unity. How would you characterize the relative effectiveness of diffusion in gases, liquids and solids?

  45. Problem: Paper Drying Problem 7.130 Use of radiant energy to dry slurry in a paper production process. In a paper mill drying process, a sheet of paper slurry (water-fiber mixture) has a linear velocity of 5 m/s as it is rolled. Radiant heaters maintain a sheet temperature of Ts = 330 K, as evaporation occurs to dry, ambient air at 300K above and below the sheet. (a). What is the evaporation flux at a distance of x = 1m from the leading edge of the roll? What is the corresponding value of the radiant flux (irradiation, G) that must be supplied to the sheet to maintain its temperature at 330 K? The sheet has an absorptivity of α=1. (b). To accelerate the drying and paper production processes, the velocity and temperature of the strip are increased to 10 m/s and 340K, respectively. To maintain a uniform strip temperature, the irradiation G must be varied with x along the strip. For 0≤ x ≤1 m, compute and plot the variation hm,x(x), N”A(x) and G(x).

  46. Problem: Paper Drying Problem 7.130 Use of radiant energy to dry slurry in a paper production process.

  47. SCHEMATIC: Problem: Paper Drying (cont).

  48. Problem: Paper Drying (cont.)

  49. Problem: Paper Drying (cont.)

  50. Problem: Wet-and Dry-Bulb Thermometers Problem 7.139 Use of wet- and dry-bulb thermometers to determine temperature and relative humidity of air flow in a duct.

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