Page 117 - Bird R.B. Transport phenomena

102 Chapter 3 The Equations of Change for Isothermal Systems

Equations 3.7-26 to 28 are the no-slip and impermeability conditions; the surface S (r,
[mp
в - 2тгМ, z) = 0 describes the location of the impeller after Nt rotations. Equation 3.7-29 is the
condition of no mass flow through the gas-liquid interface, described by S (r, 0, z, t) = 0,
mt
which has a local unit normal vector n. Equation 3.7-30 is a force balance on an element of this
interface (or a statement of the continuity of the normal component of the momentum flux ten-
sor IT) in which the viscous contributions from the gas side are neglected. This interface is ini-
tially stationary in the plane z = H, and its motion thereafter is best obtained by measurement,
though it is also predictable in principle by numerical solution of this equation system, which
describes the initial conditions and subsequent acceleration Dv/Dt of every fluid element.
Next we nondimensionalize the equations using the characteristic quantities v = ND,
0
/ = D, and ^ = p along with dimensionless polar coordinates r = r/D, 6, and z = z/D.
0 0 atm
2
Then the equations of continuity and motion appear as in Eqs. 3.7-8 and 9, with Re = D Np/fx.
The initial condition takes the form
[ j?1 II w l
^ a n d O < z < Ш, v = 0 (3.7-31)
and the boundary conditions become: I
[ j^l, v = 0 (3.7-32)
R

tank wall a t f = M v = 0 (3.7-33)
impeller surface at S (r, в - 2тг}, z) = 0, v = 2тггЬ (3.7-34)
[mp 0
gas-liquid interface at S (f, 6, z, 0 = 0, (n • v) = 0 (3.7-35)
mt
] (з 7 зб)
=° -"
In going from Eq. 3.7-30 to 3.7-36 we have used Newton's law of viscosity in the form of Eq.
1.2-7 (but with the last term omitted, as is appropriate for incompressible liquids). We have
also used the abbreviation 7 = Vv + (Vv) + for the rate-of-deformation tensor, whose dimen-
sionless Cartesian components are у = dv^/dx) + (dv-Jdx^.
J
The quantities in double brackets are known dimensionless quantities. be function
Sim (b 0 ~ 2?rf, z) is known for a given impeller design. The unknown function S (r, 9, z, t) is
P int
measurable photographically, or in principle is computable from the problem statement.
By inspection of the dimensionless equations, we find that the velocity and pressure pro-
files must have the form
1? и \
^
v = v( r, в, z, t; , ^ , Re, Fr I (3.7-37)
g
r, 0, z, t; ^, , Re, Fr J (3.7-38)

for a given impeller shape and location. The corresponding locus of the free surface is given by
§
S = S (b в, z, U , , Re, Fr j = 0 (3.7-39)
g
int int
2
2
in which Re = D Np/ JX and Fr = DN /g. For time-smoothed observations at large t, the depen-
dence on t will disappear, as will the dependence on в for this axisymmetric tank geometry.
These results provide the necessary conditions for the proposed model experiment: the
two systems must be (i) geometrically similar (same values of R/D and H/D, same impeller
geometry and location), and (ii) operated at the same values of the Reynolds and Froude
numbers. Condition (ii) requires that
^ ^ (3.7-40)

^ (3.7-41)