Foam model

In general the foam density reduces as the amount of blowing agent is increased, with a lower limit set by foam stability. It is possible to model the factors which affect the final density Mahapatro and co-workers (206) used a regular Kelvin foam model to analyse the expansion of PE foams. The foam has uniform sized cells, each with eight hexagonal faces and four square... 

The faces in low density LDPE foams are partly buckled or wrinkled, as a result of processing (a.l7). This affects both the bulk modulus and the Young s modulus. The foam bulk modulus Kp is predicted, using the Kelvin closed cell foam model, to be ... 

Mills and Zhu (a. 15) used a Kelvin foam model, in which face tensions restrain the bending of cell edges... 

A Kelvin foam model with planar cell faces was used (a. 17) to predict the thermal expansion coefficient of LDPE foams as a function of density. The expansion of the heated gas is resisted by biaxial elastic stresses in the cell faces. However SEM shows that the cell faces are slightly wrinkled or buckled as a result of processing. This decreases the bulk modulus of the... 

In a subsequent theoretical study, Stamenovic [60] obtained an expression for the shear modulus independent of foam geometry or deformation model. The value of G was reported to depend only on the capillary pressure, which is the difference between the gas pressure in the foam cells and the external pressure, again for the case of <)> ca 1. Budiansky et al. [61] employed a foam model consisting of 3D dodecahedral cells, and found that the ratio of shear modulus to capillary pressure was close to that obtained by Princen, but within the experimental limits given by Stamenovic and Wilson. 

Open and closed cell foams modeled by tetrakaidecahedra. 

Descriptive Foam Model Depending on the Microstructure of the (PPE/PS)/SAN Blend... 

For establishing a descriptive foam model of such blend systems, the (PPE/PS)/SAN blend with a PPE/PS ratio of 75/25 is exemplarily used (Fig. 30), as it further reveals pronounced differences in microstructure ... 

Fig. 30 Schematic foam model of (PPE/PS)/SAN blends with a PPE to PS ratio of 75/25...

Descriptive Foam Model of the Microstructured and Nanostructured Blend Systems... 

Geometrical parameters of films and Plateau borders in the dodecahedral foam model... 

In order to compare the structural parameters of the foam model studied by Kruglyakov et al. [18] with the respective parameters of a real polydisperse foam (individual bubbles of different degree of polyherdisity) Kachalova et. al. [19] performed measurements of the average border radius of curvature of foams with variable expansion ratio. The foam studied, generated by the set-up shown in Fig. 1.4, was obtained from a nonionic surfactant solution of Triton-X-100 (a commercial product) to which NaCl (0.4 mol dm 3) was added. The expansion ratio was determined conductometrically with correction of the change in electrolyte concentration due to the internal foam destruction. The electrolyte concentration... 

Important are the equations derived on the basis of various capillary foam models. For instance, Eqs. (5.57) and (5.58) [65] are obtained if the model of liquid flow through the foam films is assumed... 

This expression can be used to derive Eq. (5.46) which can be applied for a cylindrical border foam model in the following form... 

A number of physicochemical constants are derived and used to characterise the foaming agents on the basis of the properties of microscopic foam films being the most appropriate foam model. The first place belongs to the surfactant concentration Cm at which black spots are formed in the film [43,67]. As already shown, this concentration is related to the sharp increase in foam stability. Each foaming agent is characterised by its Cu value which... 

Several other more complex dependences accounting for the shape of grains and pores have been proposed. However, it should be pointed out that in most of these models the disperse phase consists of solid particles, while the dispersion phase is gas in the form of pores. Such models do not correspond to the process for they do not account for the effect of the liquid phase on heat transfer. Considering a cubic foam model Manegold [5] has suggested a relation between the thermal conductivity of a foam and the liquid content in it... 

For the border foam model (10.35ah 1.16r2a) and at the assumption that the foam expansion ratio is expressed by Eq. (4.10)... 

Figure 9.29 Two-dimensional hexagonal foam model. In (a), the foam structure changes from hexagonally close-packed cylinders to regular hexagons as the volume fraction increases from 0.9069 to unity. In (b), the three films and...

Foams usually possess a finite low-frequency elastic modulus, along with static and dynamic yield stresses. These and other aspects of foam flow and rheology can be captured qualitatively and even semiquantitatively by cellular foam models. 

Figure 9-33a shows the predicted shear stress as a function of strain for the initial foam orientation depicted in Fig. 9-32. The stress grows continuously until at y = 1.15 a T1 reorganization occurs which brings the cell structure back to its starting state, and the stress jumps back to zero. Thereafter, the stress history repeats itself. Similar periodic stress patterns and stress jumps have been predicted for the three-dimensional tetrakaidecahedron foam model (Reinelt 1993). If the initial orientation is rotated through an angle of r/12 with respect to that shown in Fig. 9-32, the stress history also has jumps, but is aperiodic (see Fig. 9-33b). Aperiodic behavior is the norm, and periodic stress histories occur only for special initial orientations (Kraynik and Hansen 1986). These unsteady, discontinuous stress...

Nevertheless, some of the predictions of simple regular foam models are relevant to real foams. One such property is the linear modulus Go, which is the slope of the stress-strain curve at zero strain. From Eq. (9-57a), we obtain... 

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. 

Here p and p are the liquid and gas densities, respectively, g is the vector of the gravitational acceleration, and AP is the capillary rarefaction given by (7.1.10) and (7.1.15). The kinetic coefficient H was called the coefficient of hydroconductivity and calculated for polyhedral foam models [245, 246]. Generally speaking, the variable H is a tensor, but usually the isotropic approximation is used, where this parameter is a scalar. Various expressions for the coefficient H were proposed and made more precise in [125, 214, 245]. Thus, different approaches used to calculate the coefficient of hydroconductivity were analyzed in [488]. For example, the structure of spherical and cellular foam was studied under the assumption that liquid flows through a porous layer according... 

Yield stress and plastic viscosity. The most important rheological characteristic determining the foam behavior ( solid-shaped or fluid-shaped ) is the yield stress To. This variable was calculated in [379] for a two-dimensional foam model ... 

Another steam-foam model was developed at the Alberta Research Council (46, 47). It considers surfactant transport and flow resistance to foam. Thermal degradation is assumed to be first-order. The rate constant is dependent on both temperature and pH. Surfactant adsorption is... 

Shell (48) used a simple foam model (49) for their Bishop Fee pilot. The foam generation rate was matched by using an effective surfactant partition coefficient that took into account surfactant losses and foam generation inefficiencies. The value of this coefficient was selected so that the numerical surfactant propagation rate was equal to the actual growth rate. Foam was considered to exist in grid blocks where steam was present and the surfactant concentration was at least 0.1 wt%. The foam mobility was assumed to be the gas-phase relative permeability divided by the steam viscosity and the MRF. The MRF increased with increasing surfactant concentration. The predicted incremental oil production [5.5% of the... 

Foam is a disperse system in which the dispersed phase is a gas (most commonly air) and the dispersion medium is a liquid (for aqueous foams, it is water). Foam structure and foam properties have been a subject of a number of comprehensive reviews [6, 17, 18]. From the viewpoint of practical applications, aqueous foams can be, provisionally, divided into two big classes dynamic (bubble) foams which are stable only when gas is constantly being dispersed in the liquid 2) medium and high-expansion foams capable of maintaining the volume during several hours or even days. In general, the basic surface science rules are established in foam models foam films, monodisperse foams in which the dispersed phase is in the form of spheres (bubble foams) or polyhedral (high-expansion foams). Meanwhile, real foams are considerably different from these models. First of all, the main foam structure parameters (dispersity, expansion, foam film thickness, pressure in the Plateau-Gibbs boarders) depend... 

The polyurethane foam model of a shark used in the Jaws films became dirty with both use and degradation and appears discoloured as a result (upper image). Mechanical cleaning of selected areas improved its appearance (lower image). 

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