3. Convective Heat Transfer 3.1 Fundamentals

( ∞−⋅α= TTq w& ) (1)

T

fluid~δλ

α

∫ ⋅α⋅=αL

0x dx

L1 (2)

λ⋅α

=xNu x

x (3)

λ⋅α

=LNu (4)

ν⋅

= ∞ xuRex (5)

ν⋅

= ∞ LuRe (6)

λ⋅ρ⋅ν

=ν

= pca

Pr (7)

⎟⎠⎞

⎜⎝⎛⋅⋅⋅=

LLifPrRecNu ba (8)

3.2 Forced Flow over Products

5crit 103Re ⋅≈ (9)

Form NUSSELTfunction validity range

plate laminar

33,0PrRe664,0Nu ⋅⋅= 33,0

xx PrRe332,0Nu ⋅⋅=

510Re <

5,0Pr >

(10)

(11)

plate turbulent

43,08,0 PrRe037,0Nu ⋅⋅= 43,08,0

xx PrRe030,0Nu ⋅⋅= 75 103Re10 ⋅<<

5,0Pr >

(12)

(13)

cylinder lam.-turb.

67,0d

33,0dd PrRe0012,0PrRe48,035,0Nu ⋅⋅+⋅⋅+=

und 5

d2 104Re10 ⋅<<−

5,0Pr >

(14)

sphere lam.-turb.

33,0dd PrRe6,02Nu ⋅⋅+= und 6

d 10Re <

5,0Pr >

(15)

Tab. 3.1: NUSSELTfunctions of forced convection over products

λα dNud⋅

= , ν

duRed⋅

= ∞ (16)

UA4Leg ⋅= (17)

Re → 0 stationary heat conduction

( ) ( )∞

∞

−−

λ⋅π⋅=→ TT

d1

d120ReQ wmin

&

( )∞−⋅⋅π⋅α= TTdQ w

2min&

2dNumin =λ⋅α

=

material properties for mean boundary temperature

( )∞+⋅= TT21T wm

Gases

n

000 TT⎟⎟⎠

⎞⎜⎜⎝

⎛≈

λλ

≈μμ ,

1

00 TT

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρρ ,

1n

000 TT

aa

+

⎟⎟⎠

⎞⎜⎜⎝

⎛≈

νν

≈

1n7.0 ≤≤

Pr independent on temperature

laminar flow 5.0RekNu ⋅=

( ) 5.01nn

00

05.0

5.0

5.0

TT

Lwk

Lwk

⋅+−

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

νλ⋅⎟

⎠⎞

⎜⎝⎛⋅=

νλ

⋅⎟⎠⎞

⎜⎝⎛⋅=α

( )00T/T~α for n = 1

( ) 15.0

0T/T~ −α for n = 0.7 turbulent flow 8.0RekNu ⋅=

8.0)1n(n

08.0 T

T~⋅+−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

νλ

α

( ) 6.00T/T~ −α for n = 1

( ) 66.0

0T/T~ −α for n = 0.7 Liquids

λρ ,c , nearly independ on temperature, ν strongly decreases with temperature,

25.0

w

42.08.0

PrPrPrRekNu ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅⋅=

3.3 Forced Flow in Tubes and Ducts Local heat transfer

( )[ ]xTTq wxx −⋅α=& T T

w

T(x=0)

0 L x

T(x)

Fig. 3.6: Axial temperature profile in a duct Mean heat transfer

( ) ( )( )( ) w

wln

TLTT0xTln

0xTLTTq

−−==−

⋅α=Δ⋅α=&

α mean heat transfer coefficient

λα dNud⋅

= (18)

νduRe ⋅

= (19)

Form NUSSELTfunction validity range laminar

3d

33d L

dPrRe61,166,3Nu ⋅⋅⋅+=

33,0dd Pr

LdRe664,0Nu ⋅⋅⋅=

2300Red <

Ld1,0 <

(20)

(21)

turbulent 48,08,0dd PrRe0235,0Nu ⋅⋅≈

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⋅⋅−⋅=

67,04,08,0

d Ld1Pr100Re0214,0Nu

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⋅⋅−⋅=

67,04,087,0

d Ld1Pr280Re012,0Nu

6d

4 10Re10 << Pr6,0 <

5,1Pr5,0 <<

500Pr5,1 <<

(22)

(23)

(24)

Tab. 3-2: NUSSELTfunctions of forced convection in ducts

materials properties for ( ))0x(T)L(T21Tm =+⋅=

for liquids ( ) ( )( )

11.0

w

mm TPr

TPrTNuNu ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=

(25)

UA4dh ⋅= (26)

a b

b

do

di

Tube

ddh =

rectangular duct

baba2dh +⋅⋅

=

Flat duct

b2dh ⋅= Annular duct

ioh ddd −=

d

Tab. 3.3: Hydraulic diameter of ducts

3.4 Impinging flows

λ⋅α

=dNu ,

ν⋅

=duRe

d diameter of nozzle u nozzle outlet velocity νλ, for gas temperature in nozzle

stagnation point

5.0st Re07.0Nu ⋅= 0 < r/d < 1

average for 5.3d/r1 <≤

65.0Re07.0Nu ⋅= 3.5 Packed Beds

void fraction V

VV S−=ψ

V volume of apparatus VS volume of solids d mean particle diameter

...d1

VV

d1

VV

1d

2S

2

1S

1 +⋅+⋅=

λ⋅α

=dNu ,

ν⋅

=dwRe

Superficial velocity

AVw&

=

V& volumne flow of gas A cross section area of empty apparatus

( ) ( )[ ]ψ−⋅+⋅⋅+= 15.11 PrRe6.02Nu 4.05.0

3.6 Natural convection

( )2

w3

T TTLgGrν

β ∞−⋅⋅= (27)

PrGrRa ⋅= (28)

910PrGr <⋅ laminar 910PrGr >⋅ turbulent

Shape Form NUSSELTfunction validity range vertical plate

laminar turbulent

25,0

L2

L Pr952,0GrPr677,0Nu ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⋅⋅=

( ) 33,0LL PrGr13,0Nu ⋅⋅=

9

L 10PrGr <⋅

PrGr10 L9 ⋅<

(29)

(30)horizontal cylinder

lam.-turb. ( ) 25,0dd PrGr40,0Nu ⋅⋅= 10

d2 10PrGr10 <⋅<

100Pr7,0 << (31)

horizontal plate

laminar turbulent

( ) 25,0LL PrGr54,0Nu ⋅⋅=

( ) 33,0LL PrGr14,0Nu ⋅⋅=

7L

5 102PrGr10 ⋅<⋅< 10

L7 103PrGr102 ⋅<⋅<⋅

(32)

(33)

Tab. 3.4: NUSSELTfunctions of free convection Forced and natural convection

Gr21ReRe 2

forcedeq ⋅+= (34)

Nu = f (Reeq) for forced convection

Fig. 3.1: Velocity boundary layer developing over a flat plate

Fig. 3.2: Thermal boundary layer

Fig. 3.3: Local heat transfer coefficient along a flat plate

Fig. 3.4a: Flow over a circular cylinder

Fig. 3-4b: Local Nusselt number (Nu0) for a flow normal to a circular cylinder

Fig. 3.4c: Local heat transfer coefficient across a cylinder

Fig. 3.5: Development of the boundary layer in a tube

Fig. 3.6: Axial temperature profile in a duct

wr δ1

δ2

Prallzone

Wandstrahl

unbeeinflusster Strahl

r

w D

Düse

Übergangsbereich

z

h

d

Fig. 3.7: Boundary layer and velocity profile for impingement flow

wD= 81 m/s

200

300

400

500

600

700

0 1 2 3r / d

αr

in W

/(m² K

)

4

h/d = 1h/d = 2h/d = 4h/d = 6h/d = 10

Fig. 3.8: Distribution of local heat transfer coefficient for several nozzle distances h/d

Re=91500

100

200

300

400

0 1 2 3 4 5r/d

Nu

h/d = 1h/d = 2h/d = 4h/d = 6h/d = 10

Fig. 3.9: Distribution of average Nusselt-number for several nozzle distances h/d

h/d=2Nu=k*Rem*Pr0,42

0,0

0,2

0,4

0,6

0,8

1,0

0 1 2 3 4 5 6r/d

m

gemitteltörtlich

Fig. 3.10a: Local Re-exponent of Nu-function for local and averaged heat transfer

at a nozzle distance of h/d=2

h/d=2Nu=k*Rem*Pr0,42

0,0

0,1

1,0

0 1 2 3 4 5 6r/d

k

gemitteltörtlich

Fig. 3.10b: Local values of factor k of Nu-function for local and averaged heat

transfer at a nozzle distance of h/d=2

1

10

100

1000

10 100 1000 10000 100000 1000000Re

Nu 0

/Pr0,

42

1 Bizzak 2 Brahma3 den Ouden 4 Garimella5 Hoogendoorn 6 Huang7 Hrycak 8 Ma9 Popiel 10 Sun11 Tawfek 12 Adler13 exakte Lösung

h/d = 2

1

2

3

45

6

7

8

9

10

11 12

13

8

11

Fig. 3.11: Comparison of several results in literature for heat transfer at stagnation Point

Fig. 3.12 –3.18: Heat transfer in an array of nozzles Fig. 3.19: Pached bed

Fig. 3.20: Development of the boundary layer on a heated vertical plate, left: interferogram of istoherms

Fig. 3-30: Approximate values of dynamic viscosity for various liquids and gases