ME 150 – Heat and Mass Transfer Chap. 14.5: Emperical Correlations – Internal Flow

Forced Convection in Internal Geometries No free flow condition (u∞ and T∞) exists in this case: Flow is determined by two boundary layers

Thermal entrance length (laminar): Prof. Nico Hotz

⎛ X fd ,t ⎜⎜ ⎝ D

⎞ ⎟⎟ ≈ 0.05 ⋅ Re D ⋅ Pr ⎠lam 1

ME 150 – Heat and Mass Transfer

Chap. 14.6: Emperical Correlations – Pipe Flow

For pipe flow: r0

Mean velocity:

m um = = ρ⋅A

2π ⋅ ρ ⋅ ∫ u (x, r ) ⋅ r ⋅ dr 0

ρ ⋅ π ⋅ r02

2 r0 = 2 ⋅ ∫ u ( x, r ) ⋅ r ⋅ dr r0 0

r

Mean temperature:

2 0 Tm ( x) = u (r ) ⋅ T ( x, r ) ⋅ r ⋅dr 2 ∫ um ⋅ r0 0

Note: For pipe flow, um und Tm are used instead of u∞ and T∞

Prof. Nico Hotz

2

ME 150 – Heat and Mass Transfer

Chap. 14.6: Emperical Correlations – Pipe Flow

Valid for laminar pipe flow in fully developed region.

Constant surface heat flux: Nu D =

h⋅ D = 4.36 k

Constant wall temperature: NuD = 3.66

Prof. Nico Hotz

3

ME 150 – Heat and Mass Transfer

Chap. 14.6: Emperical Correlations – Pipe Flow

Heat transfer within the entrance region Empirical solution for the case of identical hydrodynamic and thermal entrance length (Pr ≈ 1): 1.33 ⎡ D ⎞ ⎤ ⎛ 0.14 0.067 ⋅ ⎜ Re D ⋅ Pr⋅ ⎟ ⎥ ⎢ ⎛ ⎞ µ L ⎠ ⎥ ⎝ Nu D = ⎢3.66 + ⋅ ⎜⎜ ⎟⎟ 0.83 ⎢ ⎥ D ⎞ ⎛ ⎝ µ w ⎠ 1 + 0 . 1 Pr ⋅ Re ⋅ ⎜ D ⎟ ⎥ ⎢ L ⎠ ⎦ ⎝ ⎣

Valid for laminar pipe flow: ReD < 2300 !

Prof. Nico Hotz

4

ME 150 – Heat and Mass Transfer

Chap. 14.7: Correlations – Turbulent Pipe Flow

Turbulent pipe flow: Only experimental / emperical solutions possible: 23 ⎡ ⎤ ⎛ µ ⎞ D ⎛ ⎞ 0.8 0.3 Nu D = 0.0235 ⋅ (Re D − 230) ⋅ (1.8 ⋅ Pr − 0.8) ⋅ ⎢1 + ⎜ ⎟ ⎥ ⋅ ⎜⎜ ⎟⎟ ⎢⎣ ⎝ L ⎠ ⎥⎦ ⎝ µ w ⎠

0.14

Valid for thermal entrance length = 10-40 times diameter D

Simple solution for limited parameter range: 3000 0.6

< <

ReD Pr L/D

< < >

⎛ µ ⎞ Nu D = 0.027 ⋅ Re 0D.8 ⋅ Pr1 3 ⋅ ⎜⎜ ⎟⎟ ⎝ µ w ⎠ Prof. Nico Hotz

105 500 10 0.14

5

ME 150 – Heat and Mass Transfer

Chap. 14.8: Correlations – Non-Circular Channels

Laminar Flow in NonCircular Channels Nusselt number and friction coefficient comparable to laminar flow in circular pipes But: ReD calculated with hydraulic diameter Dh Dh = 4 ⋅

Geometry a a a a

b b b b

a a

b b

c f Re D

b a

q ʹ′wʹ′ = const

T w = const

-

4.36

3.66

64

1.0

3.61

2.98

57

1.43

3.73

3.08

50

2.0

4.12

3.39

62

3.0

4.79

3.96

69

4.0

5.33

4.44

73

8.0

6.49

5.60

82

∞ -

8.23 3.11

7.54 2.47

96 53

cross − sectional area wetted perimeter

Turbulent case: same equation as in laminar case, with Dh instead of D Prof. Nico Hotz

6

ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer

Prof. Nico Hotz

7

ME150_Lect13-1_Empirical Correlations for Internal Convection-1
ME150_Lect13-1_Empirical Correlations for Internal Convection-1

⋅ ⋅ ≈ ⎟⎟ ⎠ D ME 150 – Heat and Mass Transfer X ⎞ ⎜⎜ ⎝ Chap. 14.5: Emperical Correlations – Internal Flow Prof. Nico Hotz 1 tfd ⎛ , M...