Page 109 - Bird R.B. Transport phenomena

94 Chapter 3 The Equations of Change for Isothermal Systems

Fig. 3.6-6. Rotating liquid with a free surface, the
shape of which is a paraboloid of revolution.

V = Pati
surface /

p = p(r, z)
-и ~ within fluid

SOLUTION Cylindrical coordinates are appropriate for this problem, and the equations of change are
given in Tables B.2 and B.6. At steady state we postulate that v and v are both zero and that
r z
v depends only on r. We also postulate that p depends on z because of the gravitational force
0
and on r because of the centrifugal force but not on 0.
These postulates give 0 = 0 for the equation of continuity, and the equation of motion gives:
Vl dp
(3.6-33)
r-component —p — = ——
(3.6-34)
0-component 0 = /x -
(3.6-35)
z-component 0 = —-^- — pg
The 0-component of the equation of motion can be integrated to give

(3.6-36)
in which C l and C are constants of integration. Because v cannot be infinite at r = 0, the con-
0
2
stant C must be zero. At r = R the velocity v is RCl. Hence C } = 2П and
2
0
v* = (3.6-37)
This states that each element of the rotating liquid moves as an element of a rigid body (we
could have actually postulated that the liquid would rotate as a rigid body and written down
Eq. 3.6-37 directly). When the result in Eq. 3.6-37 is substituted into Eq. 3.6-33, we then have
these two equations for the pressure gradients:

dp dp
2
= pu r and = ~Pg (3.6-38,39)
dr T z
Each of these equations can be integrated, as follows:
p = Iptfr 2 + / (0, z) and p = -pgz + / (r, в) (3.6-40,41)
:
2
where fa and / are arbitrary functions of integration. Since we have postulated that p does not
2
2
depend on в, we can choose fa = -pgz + С and / 2 = \pCl rp + C, where С is a constant, and
satisfy Eqs. 3.6-38 and 39. Thus the solution to those equations has the form
2 2
p = -pgz + \рп г + С (3.6-42)
The constant С may be determined by requiring that p = p at r = 0 and z = z , the latter
atm 0
being the elevation of the liquid surface at r = 0. When С is obtained in this way, we get
V ~ Patm = ~ Z 0 ) (3.6-43)