The most plausible mechanisms of aggregate disruption to the present date appear to be those based on the turbulent dynamic forces acting across an aggregate and turbulent drag forces acting on the surface of the aggregates.
Considering whether disruption occurs by dynamic shear forces acting across the aggregates or by turbulent surface drag depends on aggregate size and
aggregate-liquid density difference. For relatively large density differences between the liquid and the aggregates it has been reported (Ayazi Shamlou et al., 1990) that disruption by turbulent drag forces can be significant. Experimental data were obtained on the disruption rate of model aggregates and crystals of potassium sulphate in aqueous suspension under mechanical agitation in a stirred vessel. The results appear to support the theory which states that disruption is caused by turbulent drag forces on the surface of the aggregates. In addition to a large density difference, the sizes of both the crystals and the model agglomerates were much larger than the Kolmogoroff microscale of turbulence. This results in significant velocity differences between the particles and the local liquid motion. Consequently, disruption by surface drag forces resulting from these relative velocities is to be expected. No comprehensive experimental data exist, however, for comparison and verification of this mechanism.
For small density differences and small aggregate sizes as in the case of the protein precipitate in the present investigation, disruption by surface drag forces will be negligible. The forces which are believed to be primarily responsible for the breakage of the protein precipitates therefore are the turbulent dynamic forces acting across the aggregates and for this reason it is these forces which will be considered here in greater detail.
Turbulent dynamic forces originate from the instantaneous turbulent velocity differences acting on the opposite sides of the aggregates (Hinze, 1975). These stresses may be either shear or normal stresses depending on the direction of the fluctuating velocities influencing the aggregates.
For instantaneous velocities acting parallel to the surface of the aggregates the local shear stress can be written as
= ^ l 0
1/2
assuming that the fluctuating velocities are separated by a distance equal to the aggregate diameter.
Multiplying Equation 2.32 by the aggregate surface area ^df^), the shear force across the aggregate can be shown as
i ê
C . 3 3 ). v > '"
The dynamic pressure fluctuations acting on opposite sides of an aggregate result in normal stresses. These pressure fluctuations are related to the instantaneous eddy velocities by the following equation assuming that the eddy scales influencing the aggregates are equal to the diameter of the aggregates:
Ap = p [ Au(df) f (2.34)
Substituting for the fluctuation velocity, Au(df), using Equations 3.14 and 3.18 (Chapter 3), and multiplying by the surface area of the aggregate the instantaneous normal forces acting across the aggregates can be written as
FnO^df^ P ( “ ) viscous (2.35)
and
The simplest correlation for break-up usually relates to the maximum stable size of an aggregate, (d^)max' which exists in a shear field. It is now possible to examine this critical situation in which an entrained aggregate can just withstand hydrodynamic break-up effects under conditions of fully developed turbulence. The maximum stable aggregate size may be obtained from a force balance on an aggregate at the point when the instantaneous turbulent forces accelerating the aggregate are just equal to the mechanical strength of the aggregate (Parker et a i, 1972). This term represents a hydrodynamically controlled limit of performance and is therefore of interest in design. Assuming the aggregate to be spherical and uniform in its structure, the force balance on the aggregate may be written as
d [Au(df)]
^ (^f)max o<^(^f)max Pf (2.37)
where a is the characteristic mechanical strength of the aggregates; d [Au(df)] / dt is the aggregate acceleration which is assumed to be the same as the eddy acceleration influencing the aggregates and is given by
d [Au(df)]
^ --- = [ Au(df) ] [ fe(df) ] (2.38)
For eddies in the viscous dissipation subrange, substituting into Equation 2.38 for the fluctuating velocity and eddy frequency using the expressions in Equations 3.14 and 3.17, the aggregate acceleration becomes
d [Au(df)]
For eddies in the inertial subrange, the acceleration term is obtained by substituting into Equation 2.38 for the fluctuating velocity and eddy frequency using the expressions in Equations 3.18 and 3.21. Thus
[ûu(df)] = e2/3 inertial (2.40)
Substituting Equations 2.39 or 2.40 into Equation 2.37 and rearranging gives
(df)max“ y ( - ) (2.41)
(df)max“ © G ) inertial (2.42)
•p r 'E-
However, these relationships are by no means conclusive. Tambo and Hozumi (1979) describe successfully the breakage of clay-aluminium floes in a mechanically stirred batch flocculator using the condition of viscous dissipation subrange. The data are produced in a recently submitted article (Appendix D3), where plots are shown of maximum floe diameter as a function of the energy dissipation rate with pH as a parameter. In this case the lines have a constant slope of approximately -0.3 which tends to disagree with the value of -0.5 given in Equation 2.41. In addition to this, similar information has been obtained only recently (written in the communication of Appendix D3) regarding the breakage of the protein precipitate aggregates studied in this work.
For isoelectrically prepared Soya protein aggregates, the slope of the line of best fit through the data points obtained for all three impeller-vessel configurations has a value of -0.24. This value compares well with the experimental findings of Tambo and Hozumi (1979) and therefore deviates
somewhat from the imphcations of Equation 2.41. Consequently, it is clear that further investigation into this field is necessary, for better understanding of this otherwise equivocal concept in particle breakage.