Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

1

Forced Convection: External Flow

Chap. 13: Emperical Correlations – External Flow

Development of turbulence over a certain length:

crit crit crit x

transition

Flate Plate:

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

2

Chap. 13: Emperical Correlations – External Flow

124.92 Rexx

δ −= ⋅

21

Re664.0 −⋅= xfxc

31

Pr=Tδδ

21

Re328.1 −⋅= xfxc

Heat transfer

6.0Pr >

100PrRePex >⋅= x ( ) 41

32

31

21

Pr0468.01

PrRe3387.0Nu

⎥⎦

⎤⎢⎣

⎡ +

⋅⋅= x

x

Laminar Flow: Exact solution

Boundary layer thickness:

Friction coefficient:

6.0Pr < 21

Pr)(Re565.0Nu ⋅⋅=⋅

= xx kxh

31

21PrRe332.0Nu ⋅⋅=

⋅= xx k

xh

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

3

Chap. 13: Emperical Correlations – External Flow

Turbulent Flow: Semi-empirical solution

λα xxU

c

x

x

xxf x

⋅=

⋅=

⋅⋅=

≤⋅=

∞

−

x

31

54

x

751

NuRe

PrRe0296.0Nu

10ReRe0592.0

ν k h

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

4

Circular Cylinder in Cross-Flow

D

Cross-Flow with U∞ und T∞ Wall temperature TW = constant

For temperature-dependent values ρ(T), µ(T):

Use mean film temperature:

Tfilm = (TW + T∞)/2

Chap. 13.1: Emperical Correlations – Cylinders

31

PrReNu ⋅⋅=⋅

= mDD C

kDh

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

5

Chap. 13.1: Emperical Correlations – Cylinders

Non-Circular Cylinder in Cross-Flow

U

U

U

U

U

D

D

D

D

D

Square

Square

Hexagon

Hexagon

Vertical plate

Re C m

5.103 – 105

5.103 – 105

5.103 – 1.95.104

1.95.104 - 105

5.103 – 105

4.103 – 1.5.104

0.246

0.102

0.160

0.0385

0.153

0.228

0.588

0.675

0.638

0.731

0.638

0.782

D

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

6

41

4.032

21

Pr)Re06.0Re4.0(2Nu ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅+⋅+=

wDDD

µµ

Flow around Spheres

Example for an important application: evaporating small droplets in sprays

.2 constNu ≈≈

2.30.1

106.7Re5.3

380Pr71.04

<<

⋅<<

<<

W

D

µµ

Valid for:

Chap. 13.2: Emperical Correlations – Spheres

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

7

Chap. 13.3: Emperical Correlations – Multi-Structures

Bank of tubes

41

36.0max, Pr

PrPrReNu ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅⋅=

W

mDD C

ReD,max C m

10 - 102 0.80 0.40 103 - 2.105 0.27 0.63 2.105 - 2.106 0.021 0.84

ReD,max C m

10 - 102 0.90 0.40 103 - 2.105 0.40 0.60 2.105 - 2.106 0.022 0.84

Aligned

Staggered

A

A1

A2

ReD,max: based on maximum velocity (i.e. minimum cross section A). All properties evaluated for mean temperature (inlet/outlet).

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

8

Packed Bed of Spheres

Chap. 13.3: Emperical Correlations – Multi-Structures

31

PrRe06.2Nu 425.0 ⋅⋅= DDε

ReD,max: based on undisturbed inlet velocity and particle diameter

ε: Porosity or void fraction

Pr: valid for gases

7.0Pr4000Re90

≈

≤≤ D

U

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

9

Chap. 13.4: Emperical Correlations – Methodology

(1)  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)

Methodology

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

10