ME 150 – Heat and Mass Transfer
Chap. 13: Emperical Correlations – External Flow
Forced Convection: External Flow Flate Plate: Development of turbulence over a certain length: transition
x crit crit
crit
Prof. Nico Hotz
1
ME 150 – Heat and Mass Transfer
Chap. 13: Emperical Correlations – External Flow
Laminar Flow: Exact solution δ
Boundary layer thickness:
x
Friction coefficient:
Heat transfer
c fx =
= 4.92 ⋅ Re
1 δ = Pr 3 δT
−1 2 x
−1 0.664 ⋅ Re x 2
−1 c f x = 1.328 ⋅ Re x 2
Pr > 0.6
Nu x =
1 1 h⋅ x = 0.332 ⋅ Re x 2 ⋅ Pr 3 k
Pr < 0.6
Nu x =
1 h⋅ x = 0.565 ⋅ (Re x ⋅ Pr) 2 k 1
Pe x = Re x ⋅ Pr > 100
Prof. Nico Hotz
Nu x =
0.3387 ⋅ Re x ⋅ Pr
(
2
⎡ 0.0468 ⎢⎣1 + Pr
2
)
3
1
⎤ ⎥⎦
3 1
4
2
ME 150 – Heat and Mass Transfer
Chap. 13: Emperical Correlations – External Flow
Turbulent Flow: Semi-empirical solution − 15
Re x ≤ 10 7
c f x = 0.0592 ⋅ Re x
4
Nu x = 0.0296 ⋅ Re x ⋅ Pr Re x =
U∞ ⋅ x ν
5
1
3
Nu x =
Prof. Nico Hotz
h ⋅x α λk
3
ME 150 – Heat and Mass Transfer
Chap. 13.1: Emperical Correlations – Cylinders
Circular Cylinder in Cross-Flow Cross-Flow with U∞ und T∞ Wall temperature TW = constant Nu D =
h ⋅D = C ⋅ Re mD ⋅ Pr k
D 1
3
For temperature-dependent values ρ(T), µ(T): Use mean film temperature: Tfilm = (TW + T∞)/2
Prof. Nico Hotz
4
ME 150 – Heat and Mass Transfer
Chap. 13.1: Emperical Correlations – Cylinders
Non-Circular Cylinder in Cross-Flow
Square
Square
Hexagon
Hexagon
Vertical plate
U
U
Re D
C
m
D
5.103 – 105
0.246
0.588
D
5.103 – 105
0.102
0.675
5.103 – 1.95.104
0.160
0.638
1.95.104 - 105
0.0385
0.782
5.103 – 105
0.153
0.638
0.228
0.731
U
D
U
D
U
D 4.103 – 1.5.104
Prof. Nico Hotz
5
ME 150 – Heat and Mass Transfer
Chap. 13.2: Emperical Correlations – Spheres
Flow around Spheres ⎛ µ ⎞ Nu D = 2 + (0.4 ⋅ Re + 0.06 ⋅ Re ) ⋅ Pr ⎜⎜ ⎟⎟ ⎝ µ w ⎠ 1
2 D
2
3 D
0.4
1
4
Valid for: 0.71
<
Pr
<
380
3.5
<
Re D
<
7.6 ⋅ 10 4
1.0
<
µ
µW <
3.2
Example for an important application: evaporating small droplets in sprays Prof. Nico Hotz
Nu ≈ 2 ≈ const .
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ME 150 – Heat and Mass Transfer Chap. 13.3: Emperical Correlations – Multi-Structures
Bank of tubes Aligned
Nu D = C ⋅ Re
m D , max
⋅ Pr
ReD,max
C
m
10 - 102 103 - 2.105 2.105 - 2.106
0.80 0.27 0.021
0.40 0.63 0.84
0.36
⎛ Pr ⋅ ⎜⎜ ⎝ PrW
⎞ ⎟⎟ ⎠
1
ReD,max: based on maximum velocity (i.e. minimum cross section A). All properties evaluated for mean temperature (inlet/outlet).
4
ReD,max
C
m
10 - 102 103 - 2.105 2.105 - 2.106
0.90 0.40 0.022
0.40 0.60 0.84
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A
Staggered A1 A2
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ME 150 – Heat and Mass Transfer Chap. 13.3: Emperical Correlations – Multi-Structures
Packed Bed of Spheres
Nu D =
2.06
ε
⋅ Re
0.425 D
⋅ Pr
1
3
U
90 ≤ Re D ≤ 4000 Pr ≈ 0.7
ReD,max: based on undisturbed inlet velocity and particle diameter ε: Porosity or void fraction Pr: valid for gases
Prof. Nico Hotz
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ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer
Chap. 13.4: Emperical Correlations â&#x20AC;&#x201C; Methodology
Methodology (1)â&#x20AC;Ż Identify the flow geometry (configuration, wetted area, etc.) (2) Specify the appropriate reference temperature and determine the flow properties (density, viscosity, etc) at that temperature. Appropriate reference temperature: often the free-stream temperature. Some correlations use other reference temperatures! (3) Calculate the Reynolds number using the appropriate reference dimension (length for plates / wings, diameter for spheres, cylinders, etc.) (4) Decide whether you want an average heat transfer coefficient (often the case) or a local heat transfer. (5) Select the appropriate correlation (often: Nusselt correlations)
Prof. Nico Hotz
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ME 150 â&#x20AC;&#x201C; Heat and Mass Transfer
Prof. Nico Hotz
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