Page 347 - Modelling in Transport Phenomena A Conceptual Approach

9.1. MOMENTUM TRANSPORT 327

Dividing Eq. (9.1-6) by W Ax Az and taking the limit as Ax -+ 0 gives

lim PIx - p[z+Ax +pg=O (9.1-7)
Ax-0 Ax
or,
dP
‘PS (9.1-8)

Note that Eq. (9.1-8) indicates the hydrostatic pressure distribution in the
x-direction.

z-component of the equation of motion

Over the differential volume element of thickness Ax, length Az and width W, Eq.
(9.1-2) takes the form

(Tzz~, WAz+ Tz,I,WAZ) - (~~z~Ix+~, WAX+ T ~ ~ I ~ + A ~
WAz)
+ (PI, - Pl,+A,)W AX = 0 (9.1-9)

Dividing Eq. (9.1-9) by Ax Az W and taking the limit as Ax --f 0 and Az -+ 0
gives

%zlz - %+I,+*, %rlz - ~zzIz+*x
lim + lim
AZ+O AZ Ax-& Ax

or,
an,, dr,, dP
- +-+-=o (9.1- 11)
dz dx
Substitution of Eqs. (9.1-3) and (9.1-5) into Eq. (9.1-11) and noting that dv,/dz =O
yields
(9.1-12)

Since the dependence of P on x is not known, integration of Eq. (9.1-12) with
respect to x is not possible at the moment. To circumvent this problem, the effects
of the static pressure and the gravitational force are combined in a single term
called the modified pressurn, P. According to Eq. (5.1-16), the modified pressure
for this problem is defined as
P=P-pgx (9.1-13)

so that
_- (9.1-14)
ap ap
--
ax ax - PS