Equation syneresis

The continuous liquid phase in a foam layer is a surfactant solution whose volume and interface vary in the course of evolution. The system evolves, its volume varies because of syneresis, and the interface area varies because of the diffusion gas exchange between the system and the ambient medium. For a system with variable volume V(t) and surface area S(t), we can write out the balance equation for the surfactant mass in the system in the form... 

Equation of syneresis. Syneresis coefficient. Together with the liquid flux (7.4.1) relative to the skeleton phase, there also exists translational transfer VU determined by the local velocity field U of the entire foam (or of its skeleton phase). In this case, the law of conservation of liquid mass has the form... 

Transient problems of syneresis are of great interest. For example, the transient syneresis in a stagnant foam layer (U = 0) under the action of constant mass forces is governed by a complex nonlinear parabolic equation. Some self-similar solutions and traveling wave type solutions were found in [152] for some special forms of this equation. For one-dimensional barosyneresis (g = 0), Eq. (7.4.4) has the form... 

Krotov, V. V., The theory ofsyneresis offoams and concentrated emulsions. 3. Local syneresis equation and setting of boundary conditions, Colloid Journal, Vol. 43, No. 1, 1981. 

Consequently, a particle network in which junctions can occasionally be broken will build up a pressure on the liquid, called endogenous syneresis pressure, psyn. If the liquid can indeed flow out, the Darcy equation can be used to obtain the rate of the process. We write it in the form... 

As is mentioned above, the syneresis rate increases with increasing temperature and decreasing pH. A look at Eq. (17.17) shows that the total outflow of whey is proportional to the surface area of the gel and inversely proportional to the distance over which the whey has to flow. Cutting the gel into small pieces thus very much enhances the syneresis rate (in fact, syneresis is almost imperceptible before the gel is cut). The equation contains the endogenous syneresis pressure psyn. However, for a system that can show structural rearrangement leading to syneresis, any externally... 

Two more features are needed to complete the constitutive equation for a gel allowance for the syneresis strain rate (which occurs in the absence of applied stress) and for the presence of the liquid phase. When a force F is applied to the face of the sample cube (see Fig. 11), a portion of that force is borne by the solid phase and a portion F is borne by the liquid (and = Fxs + F i). The pressure in the liquid phase, is equal to the force on the liquid divided by the area of the liquid, The negative... 

From Eq. 43 we see that a = 3pP = per, so the total force on the solid phase is p(/ i + Fy + Fj) that is, the load is borne by the solid phase in proportion to the volume fraction that it occupies. Finally, we identify the linear syneresis strain rate as and modify the constitutive equation by adding that quantity to the right side of each line in Eq. 46. That is, we assume that the stress-induced strain rate and the inherent syneresis strain rate are linearly additive. Thus the constitutive equations for an isotropic viscous gel are ... 

The maximum volume decrease (AV/Vo)max and the characteristic time constant T are modeled as a function of pressure difference Ap and temperature to correlate natural and enforced syneresis, and the predictive model equations for syneresis Eqs. (12) and (13) are proposed. 

The values of the fitting parameters are given in Table 3. They sufficiently reflect the progression of enforced syneresis [19]. Besides that reflection, the predictive model equations allow for extrapolation to natural syneresis (dotted lines for Ap < 0.8 bar shown in Figs. 16 and 17). 

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