(22) where vi is the reciprocal of its volatility with respect to the least volatile component and x i 0 is the initial mole fraction of the i t h component, the summation on the right of ( 2 2 ) being over all components. Put in another way, if the vi are arranged in order of magnitude, the discriminant on the right of ( 2 2 ) divides the components into the two relevant classes. With
n components there are n - 2 intermediate components and n - 1types of behavior regarding the system as a whole. Department of Chemical Engineering Polytechnic of Wales Treforest,South wales,U,K, CF37 1DL
D. B. Scully
Receiued f o r reuiew September 18,1975 Accepted June 21,1976
Forces in Dispersive Mixing
Dispersive mixing in polymer processing is analyzed quantitatively by modelling agglomerates as dumbbells consisting of two unequal beads connected by a rigid connector. The force in the connector was calculated when the dumbbell is placed in a general homogeneous velocity field of a Newtonian fluid. Rupture occurs when the force in the connector exceeds a certain threshold value. In simple shearing flow and steady elongation flow the maximum force in the connector is proportional to the local shear stress and the product of beads radii. In the former the maximum value is obtained when dumbbell is 45’ to the direction of flow and in the latter when the dumbbell is aligned to the direction of flow.
Introduction Dispersive mixing in polymer processing involves the rupture of clumps and agglomerates of solid particles such as pigments and carbon black, or of liquid drops of polymer, usually in a deforming viscous liquid. It is accomplished by forcing the mixture to pass in high shear zones generated in narrow clearances such as the gap between the rolls in a roll mill or in the clearance between the blades and the shell in an internal mixer. A quantitative analysis of dispersive mixing would greatly aid the design of intensive mixers in the field of polymer processing. Such an analysis was proposed by McKelvey (1962). He considered the forces acting on a single dumbbell in a flow field. By following the path of one bead relative to the other and assuming breakup a t a certain critical separation, agglomerates rupture can be predicted. In this paper a similar approach is adopted but the agglomerates are assumed to behave as rigid dumbbells with unequal beads and following Bolen and Colwell (1958) it is assumed that the agglomerates break when internal stresses, induced by vis,cous drag on the particles, exceed a certain threshold value. Mathematical Formulation Consider the forces acting on a single agglomerate in the form of a rigid dumbbell, as shown in Figure 1,consisting of two unequal beads of radii r l and r2 connected by a rigid connector of length L , in a homogeneous velocity field of an incompressible Newtonian fluid. As a result of the viscous drag on each of the beads a certain force develops in the connector which depends on the magnitude of the viscous drag and on dumbbell orientation. When these forces exceed a certain critical value, which equals the attractive cohesive forces, the beads break apart. The mathematical formulation of this problem was proposed and solved in detail by Bird et al. (1971) in connection with molecular interpretation of macroscopic flow phenomena of polymer solutions. Their solution will be adopted here with two minor modifications: terms due to Brownian motion, which are irrelevant on the present scale, are neglected, and bead radii are assumed to be nonequal. 346
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
It is assumed that the presence of the dumbbell does not alter the flow field of the liquid in the neighbourhood of the dumbbell, that the dumbbell does not “drift”, and that the flow field is homogeneous by which it is meant that the rate of deformation is the same a t all points. Hence the velocity field is given by where K is a tensor which specifies the flow field and which may be time dependent, and p is a position vector as indicated in Figure 1. The rate of deformation tensor in terms of K is written as
+
’$ = K
+
K+
= VV
+ (VV)’
(2)
where K+ is the transpose of K. The nonvanishing velocity component for steady simple shearing flow is ux
=
KxyY
(3)
and the velocity components for steady elongational flow are Ux Uy
=
KxxX
= -1/2Kxxy
(4)
U z = -1/2KxxZ
For each bead of the dumbbell an equation of motion can be written indicating that mass-times-acceleration equals the sum of forces acting on it. Among the forces, viscous drag forces and the force through the connect0 are included
mlii = -3;(bi miiz = -l’zi(ba
- vi) - F - VZ) + F
(5)
(6)
where F is the force in the connector, vi is the local fluid velocity at Bead i, bi is the velocity of Bead i and Ti is the viscous drag on the Bead i which according to Stokes’ law is given by the following expression {i
= 63~wri
(7)
The acceleration term is small compared to the other terms
[K:RR]= Kzy(Y1 - y z ) (Xi - X 2 )
(14)
where x1 and y1 are the coordinates of Bead 1 and x i and y~ those of Bead 2. If the coordinate system is placed at the center of Bead 1 and recalling that for this flow situation according to eq 12, K~~ is simply the shear rate +; eq 10 reduces to
Equation 15 indicates that the force in the connector vanishes if the dumbbell is either parallel to the flow field (yz = 0) or if it is perpendicular to the flow field (x2 = 0). The length of the connector L is related to the position of Bead 2 by
X
L2 = x 2 2
+ y22 +
222
(16)
Inserting eq 16 into eq 15 results in Figure 1. Schematic representation of a rigid dumbbell.
and it can be neglected. Subtracting eq 6 from eq 5 gives
Substituting eq 1 into eq 8 and defining a vector R = p 1 - p2 which is a vector pointing from Bead 1to Bead 2, as shown in Figure 1, gives
R=
- R - F (-) + (1
Equation 17 indicates that the maximum force in the connector will be obtained when the dumbbell is placed in the x-y plane (i.e., t 2= 0) and it's orientation is a t a 45' angle to the direction of shear (i.e., x 2 = y~ = L / f i ) ; thus (18) Finally eq 18 for the special case of two beads in contact with each other where L = r l r 2 , gives
+
