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

Chap. 13: Emperical Correlations â&#x20AC;&#x201C; 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 .

6

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

Prof. Nico Hotz

A

Staggered A1 A2

7

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

8

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

9

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

Prof. Nico Hotz

10

ME150_Lect11-2_External Convection
ME150_Lect11-2_External Convection

Flate Plate: ME 150 – Heat and Mass Transfer Chap. 13: Emperical Correlations – External Flow transition Prof. Nico Hotz 1 x crit xh xh Re66...