Internal Forced Convection Internal Forced Convection ::Heat Transfer CorrelationsHeat Transfer Correlations

Chapter 8Chapter 8Sections 8.4 through 8.6Sections 8.4 through 8.6

8.4 Laminar Flow in Circular Tubes

Laminar Flow in a Circular Tube:The local Nusselt number is a constant throughout the fully developed region, but its value depends on the surface thermal condition.

Uniform Surface Heat Flux : ( )sq 4.36D hDNu k= =

Uniform Surface Temperature : ( )sT 3.66DhDNuk

= =

sT =

= sq

Temperature Profile and the Temperature Profile and the NusseltNusselt NumberNumber

The steady flow energy balance for a cylindricalThe steady flow energy balance for a cylindricalshell element can be expressed as:shell element can be expressed as:

SubstitutingSubstitutingand dividing by and dividing by 22r r drdr dxdx gives, after rearranginggives, after rearranging

0p x p x dx r r drmc T mc T q q+ + + =

( )2cm uA u rdr = = =

1 ( ) ( )( )2

x dx x r dr rp

T T q qc udx rdx dr

+ + =

OrOr

Since Since Fouriers Law

( Energy Equation)

1( )2 p

T qux c rdx r

=

(2 ) 2q T Tk rdx kdx rr r r r r

= =

; p

T T ku rx r dr r c

= = (8-48)

Constant Surface Heat Flux ()

'' '' ''

2(2 ) (2 ) 2= =constant

( )s m s o s o s

p m o p m o p

dT dT q r q r qTx dx dx mc u r c u r c

= = = =

2

1( ) 0 , ( ) ( ) 0( )

,

(i) For constant surface hea

( ) (

t fl

)

s s s s m

s m s m s m

s s s s m

s m s m

T T T T T T T THencex T T T T x x T T x x

T dT T T dT T T dTorx dx T T dx T T dx

= =

= +

'' ( ) constant

( ) must also be a constant. So, , substitute it back,

ux

it

case:

yields

s m s

m sm s

s

q h T TdT dTT Tdx dx

T dTx dx

= =

=

=

Constant Surface Heat Flux ()

Substituting u and into Eq. 8.48

"2 constantsm p o

qTx u c r

= =

T Tu rx r dr r

=

( )2

22 1mo

ru r ur

=

" 2

2

4 11so o

q r d dTrkr r r dr dr

=

( / )T x

Separating the variables and integrating twice

Boundary conditionsBoundary conditions

Symmetry at r = 0::

At r = ro: (ii)(ii) T(r = R) = Ts

" 42

1 224s

o

q rT r C r CkR r

= + +

( )0(i) 0T rr

==

" 2 4

2 2

34 4

s os

o o

q r r rT Tk r r

= +

(8-57)

C1 = 0

C2

"1124

s om s

q rT Tk

= ( )" s s mq h T T

24 48 4.3611 11o

k k khr D D

= = =

4.36hDNuk

= = = (8-53)

Constant heat flux (circular tube, laminar)

( ) ( )20

2 R

mm o

T T r u r rdru r

= =

3.66hDNuk

= = (8-55)

()

Constant Surface Temperature(circular tube, laminar)

Thermal Entry Length:

Average Nusselt Number for Laminar Flow in a Circular Tube with Uniform Surface Temperature:

Combined Entry Length (Sieder and Tate ):

( ) ( )1/ 3 0.14Re Pr/ / / 2 :D sL D > 0.141/ 3Re Pr1.86

/D

Ds

NuL D

=

( ) ( )1/ 3 0.14Re Pr/ / / 2 :D sL D

Effect of the Entry Region The manner in which the Nusselt decays from inlet to fully developed conditions

for laminar flow depends on the nature of thermal and velocity boundary layerdevelopment in the entry region, as well as the surface thermal condition.

Laminar flow in a circular tube

Combined Entry Length:Thermal and velocity boundary layers develop concurrently from uniform profiles at the inlet.

8.5 Turbulent Flow in Circular Tubes

Turbulent Flow in a Circular Tube:

For a smooth surface and fully turbulent conditions ,the DittusBoelter equation may be used as a first approximation:

( )Re 10,000D >

4 / 50.023Re PrnD DNu =

( )( )( ) ( )1/ 2 2 / 3

/8 Re 1000 Pr1 12.7 /8 Pr 1

DD

fNu

f

=+

The effects of wall roughness and transitional flow conditions may be considered by using the Gnielinski correlation:

( )Re 3000D >

( )( )

0.3 0.4

s m

s m

n T Tn T T=

( ) 20.790 1n Re 1.64Df= Smooth surface (e = 0):

Surface of roughness : 0e > Figure 8.3f

Average Nusselt Number for Turbulent Flow in a Circular Tube :

Effects of entry and surface thermal conditions are less pronounced for turbulent flow and can be neglected.

For long tubes :( )/ 60L D>

,D D fdNu Nu

For short tubes : ( )/ 60L D