Calculated capillary pressure

One issue wifh fhis mefhod is fhaf, for larger pores shielded by smaller ones, the corresponding pressure necessary for the liquid to intrude them corresponds to the entry pressure for fhe smaller pores thus, the volume for the larger pores is incorrectly attributed to smaller ones. In addition, the assumption that the contact angle of fhe nonwetting liquid is the same on all solid surfaces is nof completely correct because diffusion layers with different treatments have pores with different wetting properties. For example, two pores of fhe same size may have different PTFE content and the entry pressure necessary for fhe liquid to penetrate them will be different for each pore, thus affecting the overall results and the calculated capillary pressures [196]. 

Figure 7. Calculated capillary pressures for aqueous solutions...

Figure 4.4 shows calculated capillary pressures for the typical pore size of each layer in typical ceramic membranes used for three-phase reactions. Vospernik et al. (2003b) have measured the displaced water by the application of an increasing transmembrane pressure. The importance was pointed out of a proper transmembrane pressure application when gas is fed from the support side. It must be underlined that the presence of defects in the top and intermediate layers will set a critical pressure that, if overcome, will result in the formation of gas bubbles. Therefore, the quality of the top-layer membrane is an important issue in the development of suitable catalytic membranes. 

To study the effect of phase separation on die pressures, we compared measurements of Pdie with the calculated capillary pressure drops (Pc) using Eq. 8 and 9. The calculations made use of the single-phase viscosity... 

Liquid between the surface of two solid bodies gives rise to boundary forces. A pressure difference arises and is known as the capillary pressure (Pc). This can be calculated from Laplace s equation. 

Figure 7 reports calculations of the effect of flow velocity on the critical capillary pressure for the constant-charge electrostatic model and for different initial film thicknesses. 

The most common cause of it is the neglect of 3-dimensional effects as compared with those in two dimensions. Thus, all stresses in a loaded wire or ribbon are disregarded in the shrinkage method, Section III. 1. The work of deformation leading to rupture is a bulk effect which does not receive its due consideration in the calculation of fracture energy, Section III.3. Bulk deformations associated with thermal etching, Section III.4, demand more attention than was alloted to them by many scientists. The method of bubbles, Section III.5, is invalid both because of the above neglect (that is, that of the volume stresses around the bubble) and because of another popular error, namely an erroneous treatment of capillary pressure Pc. 

If the dependence of Ar on saturation is known, then it can be used in eq 49 (via eq 50) directly. Nguyen and co-workers and Berning and Djilali assume a linear dependence of Ar on saturation, and most of the other models use a cubic dependence the model of Weber and Newman yields close to a cubic dependence. This last model differs from the others because it obtains an analytic expression for Ar as a function of the capillary pressure (the independent variable). Furthermore, they also calculated and used residual or irreducible saturations, which are known to but have only been... 

If the film is flat, (31) is automatically satisfied, as in the calculation of the viscous edge effect above, but, with surface tension, the capillary pressure in the curved part of the meniscus must be taken into account, and Eq. (31) provides the condition from which the shape of the free surface can be calculated. 

The size of the drop is varied by using a computer controlled microsyringe attached to the capillary and the output of the transducer is also fed into a computer. The volume and radius of the drop at any instant are determined by the position and speed of the microsyringe plunger. For the measurement of equilibrium IFT, a drop is formed at the capillary tip and maintained at that size. After sufficient time, chemical equilibrium is achieved and the equilibrium thermodynamic IFT can be calculated from the measured, steady state capillary pressure and drop radius by Eq. 1. 

An important factor of the surface s role in the water balance is infiltration of precipitation into the soil both during rainfall and in run-off. The rate of water take-up by soil wSH is described by the formula wsll = ksl, where ks is the coefficient of filtration, and l is the hydraulic slope. Let us denote the volume mass of the soil as k, which, on average, varies from 1.4g/cm3 to 1.5g/cm3, then for ks it is convenient to use the Azizov formula ks = 256.32k-7 28-1.27k1 14 (cm/da). The parameter / can be calculated using formula l = (z0 + zj + z2)/z0, where z0 is the depth of the column that leaches out, zx is the capillary pressure, and z2 is the height of the... 

When suction is generated within the sampling system, water is sucked inwards through the pores of the sampler until a corresponding capillary pressure occurs in the pores. If the capillary pressure in the sampler is lower than that in the soil, water flows from the soil into the sampler until the capillary pressure in the sampler and in the soil are equal. The maximum capillary pressure in a pore can be calculated by the following equation (Schubert, 1982) ... 

In spatially evolving multiphase media (e.g., during dissolution of a porous medium, or phase separation in a polymer blend), the mean curvature of the interface between two phases is of interest. Curvature is a sensitive indicator of morphological transitions such as the transition from spherical to rod-like micelles in an emulsion, or the degree of sintering in a porous ceramic material. Furthermore, important physicochemical parameters such as capillary pressure (from the Young-Laplace equation) are curvature-dependent. The local value of the mean curvature K — (1 /R + 1 /Ri) of an interface of phase i with principal radii of curvature Rx and R2 can be calculated as the divergence of the interface normal vector ,... 

Usually the estimation of structural parameters of foam formed by dispersion through gauzes is done on the basis of liquid and gas material balance [36,38]. Such calculations do not account for the properties of foaming solution and capillary pressures during the process of foam formation. That is why they cannot give reliable results. 

Eqs. (1.41) and (1.43) for the capillary pressure are derived assuming that foam is in equilibrium with the surrounding medium (air) under constant pressure. If an isolated foam with constant volume is submitted to drying , then in the calculation of pa the decrease in gas pressure should be considered. This pressure decrease results from the increase in gas volume caused by drainage of liquid from the foam. As far as changes in pressure and liquid volume usually are not large (Ap p0), the decrease in gas pressure can be derived from the equation of Boyle-Mariot. Then... 

When circular microscopic foam films (equilibrium or thinning) are studied it is necessary to know the pressure in the meniscus of the liquid being in contact with the film (see Fig. 2.2 A, B, C). In some cases it is very important to know the precise value of the capillary pressure, for example, in the calculation of low disjoining pressures n and the potential of the diffuse electric layer [Pg.50]

For films from solution of non-ionic surfactants within the concentration range corresponding to the plateau of the h(C) curves (curves 1 and 2, Fig. 3.20) the capillary pressure sharply decreases. Thus, ( -potential should also decrease. Calculations indicate a certain decrease of about 8 to 10 mV but the accuracy of (po evaluation for such kind of surfactants is 3 to 4 mV. As far as the change in (po within the plateau region of h(C) dependence is not large, the average value of (pq for the plateau of the (po(Q dependence could be used [189],... 

The course of h(Cci) dependence indicating the decrease in equilibrium thickness up to the transition to NBF as well as the course of n(Ii) isotherm with a distinct barrier transition, reveal the electrostatic character of the forces acting in the film. Thus, double electric layer can be estimated, knowing that n / = pc+T vw The capillary pressure pa was measured experimentally while Tlvw was calculated from Eq. (3.89). The potential was determined within the electrolyte concentration range of 5-10 4 to 10 3 mol dm 3 (Fig. 3.48) in which the films were relatively thick, yielding a value of (po = 36 3 mV. In this respect films stabilised with the zwitterionic lipid DMPC are very similar to those stabilised with non-ionic surfactants [e.g. 100,186,189] (see also Section 3.4.1.1). The low ( -potential leads to the low barrier in the FI(Ii) isotherm which can easily be overcome at relatively low electrolyte concentrations and low pressure values. 

The analysis of the above techniques (Section 3.4.2.2) developed to estimate the conditions under which stable CBF and NBF exist, and reveals the equilibrium character of the transition between them and the particular features of the two types of black films. Furthermore the difference between the techniques of investigation as well as the difference between their intrinsic characteristics proves to be a valuable source of information of these thinnest liquid formations. The transition theory of microscopic films evidences the existence of metastable black films. Due to the deformation of the diffuse electric layer of the CBF, the electrostatic component of disjoining pressure 1 L( appears and when it becomes equal to the capillary pressure plus Ylvw, the film is in equilibrium (in the case of DLVO-forces). As it is shown in Section 3.4.2.3, CBF exhibit several deviations from the DLVO-theory. The experimentally obtained value of ntheoretically calculated. This is valid also for the experimental dependence CeiiCr(r). Systematic divergences from the DLVO-theory are found also for the h(CeiXr) dependence of NaDoS microscopic films at thickness less than 20 nm. 

The data obtained indicate that Eqs. (4.24)-(4.26) can be used for the calculation of n of low expansion ratio foams involving the correcting function / = r rn n) and an equivalent by volume radius, evaluated from the capillary pressure. 

Values of experimenlal atxp and calculated acai cell sizes of a foam from NP-20. Foam dispersity is determined by the method for measuring local expansion ratio and capillary pressure. 

The radius of border curvature in a polyhedral foam which conforms the condition r/a 1 is the easiest to determine. If a micromanometer is used to measure the capillary pressure pa, then the radius of border curvature r can be calculated by combining Eqs. (1.40) and (1.45)... 

As confirmed by the experimental studies, the border profile corresponds to the cylindrical model only at the final stage of the drainage process when the capillary pressure in borders is close to the equilibrium value. That is why Eq. (5.37) can be used to calculate Cexp reached at the final drainage stage. However, it must be kept in mind that the pressure should be measured with a micromanometer at the foam column top. 

Recently a new method for formation of monodisperse emulsions that creates high capillary pressures, involving osmotic stress technique, has been introduced [73]. It proves to be most reliable for the purpose. Preliminary calculations showed that the emulsion films in such monodisperse systems rupture in a narrow range of critical disjoining pressure. For example, NaDoS emulsion films rupture in the range from 1 to 1.3-105 Pa, which is analogous to foam films from the same surfactant solution. Unfortunately, the foam film type has not been considered. 

The calculation of the average capillary pressure p and the disjoining pressure 77 in films (theses pressures are equal at equilibrium state) is according to the following formula, derived from Eqs. (1.40) and (4.10)... 

The maximum capillary pressure p max in a 1.5 cm layer of a NaDoS foam, calculated from Eq. (6.41) using the value of the maximum expansion ratio, is 13 kPa. It is slightly lower than the maximum pressure obtained earlier for a foam at the moment of its destruction, studied with the porous plate cells [84],... 

A qualitative evidence of the above are the data reported in [52]. It has been established that there is a correlation between the calculated rate of internal diffusion foam collapse and the experimentally determined rate. To obtain a stable foam from poor surfactants (alcohols, acids, etc.) under these conditions is hardly possible because of either insufficient dynamic elasticity of foam films or the lack of equilibrium elasticity (for films from insoluble surfactants). Furthermore, the n barrier for films from acid or alcohol solutions is low and the typical capillary pressures for a real foam are sufficient to induce disturbance of the film equilibrium and, respectively, foam collapse. 

---