Page 64 - Bird R.B. Transport phenomena

§2.3 Flow Through a Circular Tube 49

Ф | Fig. 2.3-1 Cylindrical shell of fluid
= о
2 2 г
of z-momentum over which the z-momentum bal-
in at 2 = 0 ance is made for axial flow in a cir-
cular tube (see Eqs. 2.3-1 to 5). The
z-momentum fluxes ф and ф гг are
гг
given in full in Eqs. 2.3-9a and 9b.

- of z-momentum
4> \ =uuxof_ out at r + Ar
rz r
z-momentum
in at r
Tube wall

Shell of
thickness
A rover
— which —i
momentum
balance
is made
|
т ^-—i—-^
ф,, |, _ i = flux
of z-momentum
out at z = L

The quantities ф and ф гг account for the momentum transport by all possible mecha-
22
nisms, convective and molecular. In Eq. 2.3-4, (r + Ar) and (г)| г+Дг are two ways of writ-
ing the same thing. Note that we take "in" and "out" to be in the positive directions of
the r- and z-axes.
We now add up the contributions to the momentum balance:

(2ттг1ф )\ - (2ттг1А )| + (2тггДг)(ф )| о " (2тггДг)(ф )| ^ + (2wrArL)pg = 0 (2.3-6)
Г2 г 2 Г+Дг 22 2= 22 2
When we divide Eq. (2.3-8) by lirL&r and take the limit as Дг —> 0, we get

.
ДГ L ( 2 3 7 )
The expression on the left side is the definition of the first derivative of rr with respect
rz
to r. Hence Eq. 2.3-7 may be written as
Pg ]r (2.3-8)

Now we have to evaluate the components ф and ф from Eq. 1.7-1 and Appendix B.I:
Г2 22
dv.
Фгг = + f*> V z = - / ! — + pV V z (2.3-9a)
r
r
= P-4.-g£ + W* (2.3-9Ы
Next we take into account the postulates made at the beginning of the problem—namely,
that v z = v (r), v r = 0, v e = 0, and p = p(z). Then we make the following simplifications:
z