Page 355 - Modelling in Transport Phenomena A Conceptual Approach

9.1. MOMENTUM TRANSPORT 335

Dividing Eq. (9.1-68) by 21rArAz and taking the limit as AT + 0 and Az -, 0
gives

+ Az-0 lim r ( -L'z+Az) = 0 (9.1-69)
+
rpg

(9.1-70)

Substitution of Eqs. (9.1-65) and (9.1-67) into Eq. (9.1-70) and noting that

(9.1-71)

The modified pressure is defined by
P = P - pgz (9.1-72)

so that
dP dP (9.1-73)
-- -
dz
dz - - PS
Substitution of Eq. (9.1-73) into Eq. (9.1-71) yields

(9.1-74)

Note that while the rightiside of Eq. (9.1-74) is a function of z only, the left-side
is dependent only on P. This is possible if and only if both sides of Eq. (9.1-74)
are equal to a constant, say A. Hence,

_- * A=- Po - PL (9.1-75)
dP
-
-A
da L
where Po and PL axe the values of P at z = 0 and z = L, respectively. Substitution
of Eq. (9.1-75) into h. (9.1-74) gives the governing equation for velocity as
I-~~[r(!2)]=po;pq (9.1-76)

Integration of Eq. (9.1-76) twice leads to

(9.1-77)