I
W. D. MOHR, R. L. SAXTON, and C. H. JEPSON Polychemicals Department, Du Pont Experimental Station, E. I. du Pont d e Nemours & Co., Inc. Wilmington, Del.
Mixing in laminar-Flow Systems A theoretical treatment guides the design of compounding processes in the plastics industry, and permits thoroughness of mixing to be related to the amount and orientation of the initial interfacial surface and net shear strain imposed
M I X I N G O F LIQUIDS is a common chemical engineering problem. However, only over the range of viscosity in which the production of turbulence is feasible do there exist proved design principles to guide the specification of equipment, and these are largely empirical. If the production of turbulence is costly, or impossible-e.g.. in the viscosity range including paints, sirups, molten plastics, and glasses-the engineer must rely mainly on his experience with similar materials, or run small scale tests. The goodness of mixing achieved in a laminar-flow process is believed to depend, other conditions held constant, on the net amount of shear supplied to the material. The mixing action of fluid shear consists in generating new interfacial surface, and so reducing the size of the regions occupied exclusively by one component. In principle, the velocity distribution, and hence both rate of shear and time under that rate of shear, can be specified for any location in a laminar-flow system. So the path of an element of fluid can be traced, and
INCREASING INTENSITY
I
INCREASING SCALE
Figure 1. gation
Scale and intensity of segre-
the amounts of shear received can be calculated and summed. An equation has been derived to relate the goodness of mixing achieved to the initial state of the system, and the amount of net shear.
Fundamental Concepts The term “mixture” is defined by Webster as ‘‘q complex of two or more ingredients which do not bear a fixed proportion to one another and which, however thoroughly commingled, are conceived as retaining a separate existence.” In this discussion a “mixing process” is considered one in which a mixture is produced by the subdivision and distribution of the components throughout the entire volume of the system without any intrinsic change in the ultimate elements of the components. If intrinsic changes such as solution, molecular weight reduction, or deagglomeration occur, processes other than mixing are involved. An inevitable problem in mixing work is to evaluate the quality of mixtures, or in Danckwerts’ phrase (7), the “goodness of mixing.” Danckwerts has defined two properties that are useful in evaluating goodness of mixing: the “scale of segregation” and the “intensity of segregation.” The scale of segregation is a measure of the size of undistributed portions of the components, while the intensity refers to the difference between the compositions of undistributed portions and the desired mean composition. The natures of these properties are illustrated in Figure 1 ; scale increases to the right, and intensity increases upward. I n general, reduction of the scale of segregation is accomplished by mechanical energy supplied by the mixing device, while reduction of intensity can be effected only by diffusion. The effect of reduced intensity can, of course, be achieved by carrying the scale of seg-
regation down to the dimensions of the ultimate units of the components, as by emulsifying immiscible liquids to a molecular scale. But mixing by itself cannot reduce the intensity of segregation, although if the components are mutually soluble the rate of reduction of intensity -Le., the rate of solution--will be increased because of the shortened average path length for diffusion. Some mechanisms by which the scale of segregation can be reduced include: turbulence, breaking apart and recombination, fluid deformation as in stretching or kneading, and shear. In the following discussion of laminar-flow systems only shear is considered as a mixing mechanism, and diffusion is assumed to be negligible. These conditions are closely approached in the processing of molten plastics. The mixing action of fluid shear consists in drawing out the components into progressively thinner striations. Figure 2,A, shows a system of two liquids of equal viscosity enclosed between parallel plates. Initially, the minor, darkcolored component exists as discrete
Figure 2.
Mixing action of fluid shear
VOL. 49, NO. 1 1
NOVEMBER 1957
1855
cubes. randomly distributed. Under shear induced by movement of the upper boundary these particles would be stretched out, and eventually this view of the system would show only thin striations of light and dark material. as in Figure 2,B. If enough shear were imposed, the combined thickness of each pair of light and dark layers could be brought below the limit of resolution, and the eye would see only uniform gray. The average combined thickness of a pair of layers, or the separation of a pair of like interfaces, is a measure of the scale of segregation, and hence of goodness of mixing. This characteristic property of mixtures prepared in laminar-flow systems is called the striation thickness, 7.
oriented with edges parallel to the coordinate axes, the direction cosines of the surfaces are : xg-plane. cos y,z-plane.
0 1 ~=
0, cos a y = 0, cos
cy8
= 1
cos a, = 1, cos a y =
0, cos olr. = 0
x,a-plane. cos az = 0, cos a y = 1, cos a$ = 0 From Equation 2 the area ratios for the three types of sides are : x,y-plane.
=
y,t-plane.
(i)' + =
1 1
($)2
Calculation of Striation Thickness
The striation thickness is related to the amount of interfacial area developed by the mixing operation. Spencer and Wiley (2) have considered how this interfacial area can be calculated, and the following treatment stems from their work. Consider a striated mixture of dark with clear liquid. If the radii of curvature of all interfacial surfaces are large relative to the striation thickness, the total volume may be expressed in terms of interfacial area
and since
s = 2(sz.,
+ + sy.s
SLIE)
and SO= 6 so s
so
1 =
3+
;dl + (2y+ 1
(3)
In any practical mixing operation it is apparent that
Nomenclature (4)
2
(iy
=
(2)
1 - 2 - cos Lyz cos a y
+
as
(2)
The initial ratio of interfacial surface to volume is So
v
-
6Y2
a7
simple shear in the x-direction; a measure of the shearing strain imposed
Equation 2 shows that the new is proportional to the original surface, and depends on the orientation of the original surface to the direction of shear. This relation has been applied to the case in which the minor component exists originally as randomly distributed, discrete cubes. If the cubes are assumed
1 856
I M
If M be substituted for du,/by, the amount of resultant shear, the striation thickness is expressed by (7) If the components differ in viscosity, a correction must be applied :
=
In the derivation of this correction it is assumed that the boundary surfaces of the system are wet by the major component, Relative motion of portions of these surfaces imposes a shear rate on this component. The shear stress induced is the same in both components. The shear rate of the minor component, however, depends on its viscosity:
Rearranging and multiplying both shear rates by the residence time give
a direction cosine of normal to original interface a t point
( x , Y? 2) = cube side length = net amount of shear supplied;
s
= striation thickness = cube side area
S
=
u
=
V
=
x,y,z =
Y
=
7 ,M
= =
total interfacial area between components of mixture displacement vector of a point on original interface total volume of system coordinate axes volume fraction , shear stress viscosity
SUBSCRIPTS 0 = referring to original statr of 1 2
INDUSTRIAL AND ENGINEERING CHEMISTRY
cy
shearing strain
r
2)
= a constant for the case of
cos
(5
12
Combining this with Equation 4 gives
where cos LY = a direction cosine of the normal to the original surface a t point (x,y , z ) = x-component of the displaceuz ment vector of point ( x , y ,
The amount of shear, is calculated as the product of the rate of shear imposed on the major component and the residence time. The ratio of viscosities is taken as unity for (#~/,MI)< 1.0. because the minor component can never be deformed more rapidly than the matrix of major component. Stated in words, Equation 11 says that the final striation thickness equals the initial striation thickness divided by the product of the amount of shear supplied and the relative fluidity. An analogous statement for distillation might be that the mole fraction of undesired component retained in the bottoms equals the mole fraction in the feed, divided by the product of heat supplied and relative volatility. The assumption of orientation of the cube faces parallel to the coordinate axes should not seriously restrict application of Equation 11, as for large amounts of shear the initial orientation of symmetrical particles should have little effect on the final striation thickness.
so Equation 3 can be reduced to
v = rS This neglects interfacial area at ends and edges of striations of the minor component. The factor of 2 enters because all but two striations have two interfaces. For a system in unidirectional shear the ratio of final to initial. interfacial area is given by ( 2 ):
from which
= =
system major component minor component
Literature Cited
(1) Danckwerts, P. V., Appl. Sci. Research A3, 279-96 (1951-3). ( 2 ) Spencer, R. S.:Wiley, R. h4., J . Colloid Sci.6 , 133-43 (1951). RECEIVED for review May 29, 1957 ACCEPTED September 27, 1957 Division of Industrial and Engineering Chemistry, Symposium on Engineering Aspects of Polymer Processes and Applications, Joint with Divisions of Paint, Plastics, and Printing Ink and Polymer Chemistry, 131st Meeting, XCS, Miami, Fla., April 1957.
