Steady-state breakup

Atomization. A gas or Hquid may be dispersed into another Hquid by the action of shearing or turbulent impact forces that are present in the flow field. The steady-state drop si2e represents a balance between the fluid forces tending to dismpt the drop and the forces of interfacial tension tending to oppose distortion and breakup. When the flow field is laminar the abiHty to disperse is strongly affected by the ratio of viscosities of the two phases. Dispersion, in the sense of droplet formation, does not occur when the viscosity of the dispersed phase significantly exceeds that of the dispersing medium (13). 

The use of a wetted spherical model affords the opportunity of studying combustion under steady-state conditions. Forced convection of the ambient gas may be employed without distortion of the object. Sufficiently large models may be employed when it is desired to probe the gas zones surrounding the burning sphere. It is apparent that the method is restricted to conditions where polymerization products and carbonaceous residues are not formed. In the application of such models, the conditions of internal circulation, radiant heat transmission, and thermal conductivity of the interior are somewhat altered from those encountered in a liquid droplet. Thus the problem of breakup of the droplet as a result of internal temperature rise cannot be investigated by this method. 

To determine the existence of stable steady state, a model [109] was studied of the destmction of clusters in the case vp = 700. At the initial instant of time 10 uniformly distributed clusters of 300 vacancies each, were put into the crystal , and interstitial atoms in the intervals between them (Uq 10). Then pairs of randomly distributed defects of different types were created in the crystal . The newly generated defects break up the orginally existing clusters and the concentration of defects declines to a steady-state value. The values of Uo were obtained by averaging a region of the curve of length 2.5 x 104 events of defect creation. The result unambiguously implies the existence of stable steady state in the problem of accumulation of point defects and in the problem of breakup of clusters. 

A unique steady state mobile ganglion size is established for each Ca when there is a homogeneous pore geometry. Smaller ganglia trap momentarily, and larger ganglia break up by either snap-off or dynamic breakup. 

Various studies about processes where the Smoluchowski equation is applicable have recognized that fragmentation of particles could be important because the possibility of breakup increases as clusters grow [14-17]. Under these conditions a steady-state particle size distribution as t-+Qo is possible, in agreement with the observations by Fuentes [5-7]. 

The most efficient mechanism of drop breakup involves its deformation into a fiber followed by the thread disintegration under the influence of capillary forces. Fibrillation occurs in both steady state shear and uniaxial extension. In shear (= rotation + extension) the process is less efficient and limited to low-X region, e.g. X < 2. In irrotatlonal uniaxial extension (in absence of the interphase slip) the phases codeform into threadlike structures. 

The idea that, under steady state flow condition, the morphology is fully defined by the dynamic breakup and coalescence processes is an attractive one — one may say, a natural one. 

A more recent theory for the dynamic equilibrium drop diameter also started from separate calculations of the drop breakup and coalescence during the steady state shearing. The rate of particle generation was taken to be determined by microrheology, viz. Eq 7.52, [Huneault et al, 1995] ... 

Predictive model for the morphology variation during simple shear flow under steady state uniform shear field was developed. The model considers the balance between the rate of breakup and the rate of drop coalescence. The theory makes it possible to compute the drop aspect ratio, p a parameter that was directly measured for PS/PMMA =1 9 blends. 

BCs tend to be tall vessels with a large aspect ratio (///Dr). The height is an important design variable because of its influence on the process and residence times, especially for batch and semibatch operations (Roy and Joshi, 2008). Biochemical processes require an aspect ratio between 2 and 5 even for experimental work. Industrial applications require much taller vessels with an aspect ratio of at least 5 (Kantarci et al., 2005), and it is fairly conunon to have vessels with an aspect ratio greater than 10 (Bellgardt, 2000b). An aspect ratio greater than 5 is also preferred because it does not influence BC hydrodynamics (Ribeiro Jr., 2008). It also allows for the breakup and coalescence mechanism to stabilize and reach steady state. 

Under steady-state flow conditions, the morphology is fully defined by the dynamic breakup and coalescence processes. However, behind is an implicit assumption that the flow conditions are strong enough to erase the initial morphology. The presence of the critical value of shear rate, Yct, has been documented (Minale et al. 1997). The authors reported that the unique morphology was observed only above ycr- Below this limit, multiple pseudo-steady-state structures were observed for the model PDMS/PIB system. No attempt was made to generalize this observation. In principle, the phenomenmi should be related to the critical value of the capillary number, Kcr, and a ratio of the polymer(s) relaxation time to... 

Predictive model for the morphology variation during simple shear flow under steady-state uniform shear field was developed. The model considers the balance between the rate of breakup and the rate of drop coalescence. The theory makes it possible to ctunpule the drop aspect ratio (p = 01/02), a parameter that was directly measured for PS/PMMA =1 9 blends. Theoretically and exptaimtaitally, p vs. shear stress shows a sharp peak at the stresses cturesponding to a transition from the Newtonian plateau to the power-law flow, i.e., to the onset of the elastic behavior (see Fig. 9.9) Lyngaae-J0rgensen et al. 1993, 1999... 

The steady-state deformation of isolated droplets decreases with increasing dispersed phase elasticity for the same imposed capillary number. A linear relationship between critical capillary number for droplet breakup (Kn-i,) and dispersed-phase Weissenberg number (Wi[Pg.934]

The influence of a block copolymer on the droplet breakup and coalescence in model immiscible PEP/PPO polymer blends was investigated by Ramie et fd. [18], who found that the addition of 0.1 wt% or 1.0 wt% of PEO-b-PPO-b-PEO [poly (ethylene oxide)-poly(propylene oxide) copolymer] triblock copolymers facilitated breakup and inhibited coalescence. The steady-state droplet size resulting from breakup was reduced only slightly by the addition of 0.1 wt% copolymer, but more substantially by addition of lwt%. However, the kinetics of coalescence were suppressed effectively even when 0.1 wt% of copolymer was added. In these systems, the copolymer seems to reduce the efficiency of both droplet collision and film drainage and/or rupture. 

Generally, the distribution of droplet sizes in flow can be obtained as a solution of the generalized Smoluchowski (balance population) equation describing the competition between the droplet breakup and coalescence. Various approximate approaches to the solution of the equation with various expressions for breakup and coalescence frequencies have been used in the hteratnre (101,105-115). For rather long mixing in batch mixers, achievement of a steady state in the droplet size distribution is assumed. For mixing in extruders, development of the droplet... 

---