7/24/2019 Convective Heat Transfer With Multiflow in an Annular Pipe of Circular Cross Section

1/4

JANUARY 1989

TECHNICAL NOTES

89

If

we substitute the value of K from Eq. (13) into Eq. (12),

the following expression for the terminal velocity of the

bubble

is

obtained:

(14)

With

order-of-magnitude

analysis,

it may be

shown from

the results that th e radial pressure gradient resulting

from

liquid acceleration near th e ends of the slug is negligible

compared to theaxialpressure gradient in the liquidfilmwhen

r

{

is

sufficiently

close to

1,

fo r

both

small and large Reynolds

numbers. Also, analysis of the normal stress balance at the

interface reveals

that

the surface deformation is negligible if

(

a

T

)AL/a

7/24/2019 Convective Heat Transfer With Multiflow in an Annular Pipe of Circular Cross Section

2/4

90

J.THERMOPHYSICS

VOL.

3, NO. 1

The

energyequations

to be satisfied in the inner and

outer

flow regions ar e

cz

~

T

cw

=

l

d

R

dT