convective heat transfer

momentum diffusivity :

themal diffusivity : p

k

c

Prpcv

k

0

s s

y

h T T k T Ty

tanNu=

tan

hL conductivethermal resis ce

k convectivethermal resis ce

0

/

/

s y

s

T T yhL

k T T L

2D PDt

v

v g

2 :pDT

c k T qDt

τ v

dimensional analysis

forced convection

in a closed conduit

# of variables; 7# of dimensions; 5# of ranks; 4# of independent dimensionless groups; 3

1a b c dD k v

2e f g h

pD k v c

3i j k lD k v h

1 ReDv

2 Pr pc

k

3 Nu

hD

k 1Nu Re, Prf

natural convection

0 1 T

buoyant 0F g

buoyant 0F g T

# of variables; 9# of dimensions; 5# of ranks; 5# of independent dimensionless groups; 4

1a b c d

pL k c

2f g h jL k g

3k l m n oL k g T

4p q r s tL k g h

1 Prpc

k

3 T

3 2

2 2

L g

2 3

2Gr

g L T

3Nu Gr, Prf

4 NuhL

k

Laminar boundary layer

No matter how turbulent the flow is far from the surface

0

y

u

x

u yx

2

2

2

2

y

u

x

u

y

p

y

uu

x

uu

yyy

y

y

x

2

2

2

2

y

u

x

u

x

p

y

uu

x

uu xxxy

xx

layerboundarythewithin0;0

y

pu y

Negligible viscous effect outside the boundary layer

0

y

u

x

u yx2

2

y

u

y

uu

x

uu xxy

xx

3

3

2

22

yyxyxy

Boundary conditions

0for,0on0 xyuu yx

0for xUux

yUux for0for,0on0

xy

xy

yxU

yand,0for

Blasius’ similarity transformation

1/2( )

( )f

U x

1/2

2

y U

x

''' '' 0f ff

' 0 at 0

' 2 as

f f

f

Ux becomes nearly Ualong the boundary layer

5

2/1

x

U

)(xy

Boundary layer thickness

Friction coeff.

1/2

0

2 1/212

( ) 0.6640.664

(Re )

x y

f

x

u yC

U xU

Shear force

L

y

x dxy

uWF

00

s s

2 1/2 1/21L2

/ 1.328 1.328

( / ) (Re )f

F LWC

U UL

RexxU

Local Reynolds

number1/2

0

0.332Rex x

yy x

laminar boundary layer; Blasius

2 2

2 2x y

T T T T Tv v

t x y x y

2

2x y

T T Tv v

x y y

2

2x x x

x y

v v vv v

x y y

0 at 0yx

vvy

v v

1 at xv

yv

0 at 0s

s

T Ty

T T

1 at s

s

T Ty

T T

1/2

0

0.332Rex x

yy x

1/2

0

0.332Res x

y

TT T

y x

1/2Nu 0.332Rex x xh x

k

for Pr=1

0

y

x s

y

q Th T T k

A y

effect of Pr; Pohlhausen 1/3Prt

1/2

0

0.332Res x

y

TT T

y x

1/2 1/3

0

0.332Re Prs x

y

TT T

y x

1/2Nu 0.332Rex x xh x

k

1/2 1/3Nu 0.332Re Prx x xh x

k

1/2 1/3Nu 0.664Re PrL LhL

k

2

sf

T TT

fluid properties are evaluated at the film temperature

; laminar boundary layer over a flat plate

1/2 1/3Nu 0.36Re Prx x(von Karman)

energy and momentum transfer analogy

Reynolds analogy ; for Pr=1

0 0

x s

sy y

v T Td d

dy v dy T T

0

p x

y

c dvh

v dy

02 2

0

2

/ 2

xf

y

dvC

dyv v

coefficient of skin friction

2

f

p

Ch v c

St2

f

p

Ch

v c

Colburn analogy ; 0.5

turbulent flow

xx x

y L y

dL

dy

x

x

dL

dy

turbyx x y

y L yy

dtT T L

dy dTT L

dy

turb

y

p y

qc v T

A

Prandtl analogy

/ 2St

1 5 / 2 Pr 1

f

f

C

C

/ 2St

51 5 / 2 Pr 1 ln 1 Pr 1

6

f

f

C

C

von Karman analogy

convective heat transfer correlations

natural convection;vertical plates, vertical cylinders, horizontal plates, horizontal cylinders,spheres, rectangular enclosures

forced convection for internal flow;laminar flow, turbulent flow

forced convection for external flow;flow parallel to plane surfaces, cylinders in crossflow,single spheres, tube banks in crossflow,

special considerations;whether to evaluate fluid properties at bulk or film temperaturewhat significant length is usedwhat is the allowable Pr, Re range for a given set of data

natural convection; vertical plates

12.5cm highTs=65CTinf=15C

2 3

2Gr

g L T

Pr

pc

k

2

1/6

8/279/16

0.387RaNu= 0.825

1 0.492 / Pr

L

PrRa Gr

forced convection for internal flow

laminar flow; Graetz solution

2

avg2 1xr

v vR

1x

T Tv r

x r r r

2

avg

12 1

r T Tv r

R x r r r

at 0 for 0eT T x r R

at 0, sT T x r R

0 at 0, 0T

x rr

2

0 avg

expe n nns e

T T r xc f

T T R Rv R

4 4 /

RePr / Pe

x D

D x Gz Pe

4

D

x

wallNu 3.658 for constantx T

wallNu 4.364 for / constantx q A

0.141/3

Nu 1.86 Pe bDw

D

L

2

0 avg

expe n nns e

T T r xc f

T T R Rv R

Sieder and Tate

turbulent flow; Dittus and Boelter

0.8Nu 0.023Re PrnD D

1. n=0.4 if the fluid is being heated, n=3 if the fluid is being cooled

2. all fluid properties are evaluated at the arithmetic-mean bulk temperature

3. Re>104

4. 0.7

forced convection for external flow

flow parallel to plane surfaces

laminar;

turbulent;

1/2 1/3Nu 0.332Re Prx x

1/2 1/3Nu 0.664Re PrL L

4/5 1/3Nu 0.0288Re Prx x

4/5 1/3Nu 0.036Re PrL L

2/3St Pr2

fx

x

C (Colburn analogy)

1/5

0.0576

Refx

x

C (von Karman)

cylinders in crossflow

1/2 1/3

1/42/3

0.62Re Pr ReNu 0.3 1

282,0001 0.4 / Pr

D DD