Elasticity modulus magnitude

For most hydrardic pressure-driven processes (eg, reverse osmosis), dense membranes in hoUow-fiber configuration can be employed only if the internal diameters of the fibers are kept within the order of magnitude of the fiber-wall thickness. The asymmetric hoUow fiber has to have a high elastic modulus to prevent catastrophic coUapse of the filament. The yield-stress CJy of the fiber material, operating under hydrardic pressure, can be related to the fiber coUapse pressure to yield a more reaUstic estimate of plastic coUapse ... 

With the Interparticle Force Theory, interparticle forces (van der Waals, etc.) are what cause the bed to be elastic. Bed elasticity is characterized by an Elasticity Modulus, M.. The criterion which determines when the fluidized bed starts to bubble is determined by the relative magnitudes of the two sides of Eq. (7). 

Figure 9. The plot of the elasic modulus (GO as a function of the extent of reaction shows more clearly that once the elastic modulus is detected, it increases at a rapid rate. In fact, G increases by more than three orders of magnitude for less than 20% change in conversion of the thiol functional groups.

The polyethenes prepared with catalyst 2 (Fig. 3a) have greatly elevated elastic modulus G values due to LCB compared to the linear polymers shown in Fig. 3b. LCB also shifts the crossover point to lower frequencies and modulus values. The measured complex viscosities of branched polymers (see also Table 2) are more than an order of magnitude higher than calculated zero shear viscosities of polymers having the same molecular weight but a linear structure. The linear polymers have, in turn, t] (0.02 radvs)... 

The behaviour and magnitude of the storage and loss moduli and yield stress as a function of applied stress or oscillatory frequency and concentration can be modelled mathematically and leads to conclusions about the structure of the material.3 For supramolecular gels, for example, their structure is not simple and may be described as cellular solids, fractal/colloidal systems or soft glassy materials. In order to be considered as gels (which are solid-like) the elastic modulus (O ) should be invariant with frequency up to a particular yield point, and should exceed G" by at least an order of magnitude (Figure 14.2). 

Numerical simulations produce force-deformation data whose shape and magnitude is dependent on the initial parameters defined within the model, including the elastic modulus (E), the uninflated cell radius (rQ) and the initial stretch ratio (ls). Experimental data are fitted to these numerical simulations allowing intrinsic material properties to be derived. 

However, the measured values of elasticity modulus are several orders of magnitude lower than the values predicted by these equations. Perhaps the bed elasticity may have a role to play for fine cohesive particles at high solid hold-ups. In that case, this can be included in future in the equation of motion for the particle phase as an extra force arising out of particle-particle interaction, without affecting the fundamental approach of linear stability analysis. 

As expected, magnitudes of instantaneous elastic modulus. Go = (l/./o), and of storage modulus G at 1 rad s (Yoo and Rao, 1996). Also, as noted by Giboreau et al. (1994) for a gel-like modified starch paste, the two parameters were of the same order of magnitude. Further, as can be inferred from data in Table 5-E, the other moduli from creep-compliance data (Gi = 1/Ji, G2 = 1//2) and the Newtonian viscosity increased with increase in °Brix. 

Emulsions with a high volume fraction of droplets (0 > 0.64) and foams show solidlike properties such as a yield stress and a low-frequency plateau value of G. The magnitudes of the yield stress and elastic modulus can be estimated using simple cellular foam models. These and related models show that at low shear rates where the shear stress is close to the yield value, the flow occurs by way of intermittent bubble-reorganization events. The dissipative processes that occur during foam and emulsion flows are still under active investigation. 

---