Page 624 - Bird R.B. Transport phenomena

604 Chapter 19 Equations of Change for Multicomponent Systems

с Fig. 19.5-3. Blending of miscible fluids. At zero time, the
z = H upper half of this tank is solute free, and the lower half
contains a uniform distribution of solute at a dimen-
sionless concentration of unity, and the fluid is motion-
« . . . less. The impeller is caused to turn at a constant rate of
rotation N for all time greater than zero. Positions in the
И tank are given by the coordinates г, 0, z, with r measured
и
radially from the impeller axis, and z upward from the
1—\- J 1 . bottom of the tank.

г
- в -

EXAMPLE 19.5-3 Develop by dimensional analysis the general form of a correlation for the time required to
blend two miscible fluids in an agitated tank. Consider a tank of the type described in Fig.
Blending of Miscible 19.5-3, and assume that the two fluids and their mixtures have essentially the same physical
Fluids properties.

SOLUTION It will be assumed that the achievement of "equal degrees of blending" in any two mixing op-
erations means obtaining the same dimensionless concentration profile in each. That is, the
dimensionless solute concentration co is the same function of suitable dimensionless coordi-
A
nates (г, 0, z) of the two systems when the degrees of blending are equal. These concentration
profiles will depend on suitably defined dimensionless groups appearing in the pertinent
conservation equations and their boundary conditions, and on a dimensionless time.
In this problem we select the following definitions for the dimensionless variables:

: = — v = = Nt P = (19.5-21)
ND N D 2
2
P
Here D is the impeller diameter, N is the rate of rotation of the impeller in revolutions per
unit time, and p is the prevailing atmospheric pressure. The dimensionless pressure p is used
0
here rather than the quantity Ф defined in §3.7; the formulation with p is simpler and gives
equivalent results. Note that t is equal to the total number of turns of the impeller since the
start of mixing.
The conservation equations describing this system are Eqs. 19.5-8, 9, and 11 with zero
Grashof numbers. The dimensionless groups arising in these equations are Re, Fr, and Sc. The
boundary conditions include the vanishing of v on the tank wall and of p on the free liquid
surface. In addition we have to specify the initial conditions

C.I: atf<0, (19.5-22)

С 2: at} < 0, for 0 < z < ~ (19.5-23)

C.3: at t < 0, v = 0 for 0 < z < ^ and 0 < f < ~ (19.5-24)
and the requirement of no slip on the impeller (see Eq. 3.7-34).
We find then that the concentration profiles are functions of Re, Sc, Fr, the dimensionless
time t, the tank geometry (via H/D and B/D), and the relative proportions of the two fluids.
That is,
й) = /(Re, Fr, Sc, t, geometry, initial conditions) (19.5-25)
А
It is frequently possible to reduce the number of variables to be investigated.