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REUEL SHINNAR’ and JAMES M. CHURCH Department of Chemical Engineering, Columbia University, New York, N. Y.
Statistical Theories of Turbulence in
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Predicting Particle Size in Agitated Dispersions Behavior of turbulent flow and particle size distribution in stirred tanks can be predicted by using the concepts of local isotropy
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SCALING up stirred reactors from the pilot plant to commercial scale, two methods are commonly used-one uses constant energy per unit mass and the other uses dimensional analysis. The main difficulty in using energy input is lack of theoretical derivation to explain its limited applicability. And, when dispersions of small drops or particles are involved and only pilot plant data are available, dimensional analysis also presents difficulties. In the work described here, the applicability of Kolmogoroffs theory (74, 77) of local isotropy, which describes the behavior of dispersions in stirred tanks, is demonstrated and supported by experimental results obtained from the literature.
Theory of local Isotropy In turbulent flow, instability of the main flobv amplifies existing disturbances and produces primary eddies which have a wave length or scale similar to that of the main flow. The large primary eddies are also unstable and disintegrate into smaller and smaller eddies until all their energy is dissipated by viscous flow. “hen the Reynolds number of the main flow is high, most of all the kinetic energy is contained in the large eddies, but nearly all dissipation occurs in the smallest eddies. If the scale of the main flow is large compared to that of the energy-dissipating eddies, a wide
Present address, Dept. of Chemical Engineering, Technion, Haifa, Israel.
spectrum of intermediate oscillations or eddies exist: which contain and therefore dissipate little of the total energy. These eddies transfer kinetic energy from the large to small eddies, and because this transfer occurs in different directions, directional information of the large eddies is gradually lost. Kolmogoroff concludes therefore that all eddies which are much smaller than the primary eddies are completely independent from them statistically. T h e only remaining information received by these small eddies from the primary eddies is the amount of kinetic energy transferred by them to smaller eddies. If the main flow is time-independent, the statistical properties of any oscillation of a scale much smaller than the main flow, should therefore be determined by the local energy dissipation rate per unit mass, E . The velocity of a point in a turbulent fluid is normally defined relative to a fixed system of coordinates. The magnitude of the fluctuating components of the velocity vector is _ defined _ _ by the root mean square values, u2, u2,-w2. Isotropic turbulence is defined by uz = v2 = w 2 but. most practical cases are unisotropic, which complicates the mathematical treatment considerably. Kolmogoroff overcame this difficulty by assuming isotropy if the volume under consideration is small enough compared to the scale of the main flow, L . If rl and r2 are two points in the small volume of the fluid, and r is the radius vector r l y 2 , a relative velocity, u(T) may be defined by :
Now, all eddies much larger than 7 will contribute little to uz(@ ; therefore, u2(r) is determined mainly by the small eddies which are statistically independent of the main flow. For r > r >> 7, _ .
u2(r)
-
~
4 7 )
= C1e
7
3
(2)
For r > 7. Both conditions apply frequently to processes in stirred reactors, where R e numbers above 100,000 are common, and L is often very large. L is given approximately by the width of the fluid ejected by the agitator. For water at an energy input of 1 hp. per 100 gallon, the average value o f f is 25 microns; thus, for a 5cm. wide agitator blade, L/7 is 2000 which is large enough to assume local isotropy. Application to Agitated Dispersions Agitation influences both dispersions and processes involving dispersions in several ways:
1. Agitation increases heat transfer through the vessel wall. The heat transfer coefficient as a function of agitation can be estimated from Rushton's equations and data (27). He demonstrated that the principle of equal energy input per unit volume cannot be applied to the scale-up of heat transfer problems in stirred tanks. 2. Agitation affects over-all mixing. The mixing time, defined as the time needed to ensure uniformity, should be a function of the ratio of the volume of liquid pumped by the agitator per unit time to the volume of the vessel. I n geometrically similar agitators, mixing time increases with tank size if energtinput per unit volume is kept constant (28). 3. Turbulent flow in the immediate vicinity of the particle affects the particle and the processes around it. For example, the turbulent pressure fluctuations in this area may cause a droplet to break up. The rate of coalescence of liquid-liquid dispersions depends on the relative velocity of two adjacent drops. Mass transfer from a small liquid or solid particle is sometimes controlled by the thickness of the boundary layer around the particle. I n such cases the influence of the main flow is small and the process is mainly determined by what occurs in a small volume around the particle. The theory of local isotropy is especially useful here, because it allows an approximate statistical definition of the flow field in a small volume of a turbulently agitated fluid. For very small particle diameters, the average statistical properties of the turbulent flow in their immediate neighborhood are determined entirely by the local energy dissipation per unit mass. Variations in the agitator's dimensions a n d rotational speed affect particles only by changing local energy dissipation. E can therefore be used to describe the effect of agitation on the system. Alter-
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natively, u2(d) as calculated from Equations 2 and 3, may replace E , where d, is used instead of distance, r . u(d), defined as d u ( d ) , s h o u l d not be taken as the real velocity of the particle in the common meaning. It is a statistical parameter describing the flow fluid around the particle. - Use of dimensionless groups on u2(d) simplifies the dimensionless analysis of mixing problems considerably. The concept of local isotropy may be applied to mixing problems where principal interest is in what happens in the immediate neighborhood of a particle, and where the dispersions have highly turbulent flow (Re + m , or at least > 50,000) and particle diameters much smaller than the scale of the main flow (a > 7. should not be used to predict performance of agitators in processes where d
