Page 206 - Bird R.B. Transport phenomena

190 Chapter 6 Interphase Transport in Isothermal Systems

in which the friction factor for a single tube, / tube , is a function of the Reynolds number
Re /; = 4R h(v)p/jjL. When this pressure difference is substituted into Eq. 6.4-1, we get

f-lEL<*Zf - L D ' f (6 43)
/ / (643)
In the second expression, we have introduced the void fraction, E, the fraction of space in
the column not occupied by the packing. Then v 0 = (V)E, which results from the defini-
tion of the superficial velocity. We now need an expression for R h.
The hydraulic radius can be expressed in terms of the void fraction E and the wetted
surface a per unit volume of bed as follows:

R _ [cross section available for flow \
h
\ wetted perimeter /
_ (volume available for flow
V total wetted surface
/ volume of voids ]
V volume of bed / P
= i f = I (6 4 4)
- -
[ wetted surface \
\ volume of bed/
The quantity a is related to the "specific surface" a v (total particle surface per volume of
particles) by
a v = T-^— (6.4-5)
1 E
The quantity a v is in turn used to define the mean particle diameter D p as follows:

D p = %- (6.4-6)

This definition is chosen because, for spheres of uniform diameter, D p is exactly the di-
ameter of a sphere. From the last three expressions we find that the hydraulic radius is
= D p£/6(l — E). When this is substituted into Eq. 6.4-3, we get
R h
о Л Л
e (6.4-7)
[b
We now adapt this result to laminar and turbulent flows by inserting appropriate ex-
pressions for/ .
tube
(a) For laminar flow in tubes, / = 16/Re This is exact for circular tubes only. To
tub e /r
account for the fact that the fluid is flowing through tubes that are roncircular and that
its path is quite tortuous, it has been found that replacing 16 by 100/3 allows the tube
bundle model to describe the packed-column data. When this modified expression for
the tube friction factor is used, Eq. 6.4-7 becomes

s* (D G /n)
p
0
in which G = pv 0 is the mass flux through the system. When this expression for / is sub-
o
stituted into Eq. 6.4-1 we get
" " ' 0 1 \ A W / S A fW
v 5— (6.4-9)