Calculated capillary

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...

Equation (3.27) can be applied to both adsorption and desorption branches of the isotherm. For the model of a bundle of capillary tubes, it is more appropriate to use the desorption branch of the isotherm for the determination of the pore size distribution. The basic idea is that the effective meniscus radius is the difference between the capillary radius and the thickness of the multilayer adsorption at p/p°, which can be obtained from de Boer s t-plot. In practice, at each desorption pressure, P, the capillary radius can be calculated from Eq. (3.27). The actual pore radius is then the sum of the calculated capillary radius and the estimated thickness of the multilayer. The exposed pore volume and surface area can be obtained from the volume desorbed at that specific desorption pressure. This step can be repeated at different desorption pressures. Except for the first desorption step, the desorbed volume should be corrected for the multilayer thinning on the sum of the area of the previously exposed pores. The pore size distribution can then be determined from the slope of the cumulative volume versus r curve. 

This section discusses the definitions of capillary number and how to calculate capillary numbers. 

Note that in this definition, the porosity term is included. If the velocity is used, the preceding equation becomes Eq. 7.82. It is expected that for a group of rocks with different porosities, if the porosities are included, the calculated capillary numbers should be closer to their average. However, the data that is shown in Table 7.7 do not consistently support this expectation. The ratio of the average to the standard deviation decreases for the data from Chatzis and Morrow (1984) if the porosity is included, but it increases for data from Taber et al. (1973). From these two data sets, it seems as though the capillary numbers that do and do not include porosity are equally good. 

This section discusses how to select the parameters to calculate capillary number. Initially, capillary number was proposed to correlate the residual saturation of the fluid (oil) displaced by another fluid (water) in the two-phase system. In surfactant-related flooding, there is multiphase flow (water, oil, and microemulsion), especially at the displacing front. If we use up/a to define the relationship between capillary number and residual oil saturation, which phase u and p and which o should be used then To the best of the author s knowledge, this issue has not been discussed in the literature. The following is what we propose. 

The reader may ask is (Nc)ow for the water phase displacing the oil phase the same as (Nc) o for the oil phase displacing the water phase, for example In principle, they are different from the preceding discussion. However, the formulas to calculate capillary number are empirical. In most practical cases. 

Nc)ow and (Nc)ow are not differentiated, simply calculating the single form of Nc. This is more obvious when the definition Nc = k(Ap/L)/a is used to calculate capillary number then we use the same absolute permeability (k), the pressure drop (Ap) along the core with the length L, and the interfacial tension... 

Classically, the approach used to calculate capillary flow has been to determine the curvature of liquid interfaces in the system and calculate Pcap from Equation (6.1). Those values could then be used to calculate the direction and magnitude of the driving forces. In systems of simple geometry such as liquids which form spherical interfaces and smooth cylindrical solid surfaces, the technique works out very well. Perhaps the best known example of such a system is the capillary rise method for determining the surface tension of a hquid, illustrated in Figure 6.10. In this system, capillary forces cause the hquid to rise in the tube due to differences in curvature of the liquid-air interface within the tube (a small radius of curvature) and that in the reservoir... 

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. 

For equilibrated systems, there is an excellent correlation between the capillary number and oil recovery efficiency. However, in calculating capillary number for nonequilibrated systems, care should be exercised because the IFT measured in vitro may not be achieved in situ and, in certain cases, the interfacial viscosity and not interfacial tension, may be a predominant factor influencing the oil displacement efficiency. 

Figure 10.12 The calculated capillary force acting between two tungsten spheres separated a liquid copper bridge as a function of the interparticle distance for several values of the liquid volume V normalized to the volume of a tungsten sphere Vq at two extreme values of the contact angle, (a) 9 = 8° (b) 9 = 85°. (From Ref. 24.)...

Applying the circular approximation allowed us to calculate capillary forces between axisymmetric objects analytically. What are the limits and errors involved. One limit... 

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... 

In a reservoir at initial conditions, an equilibrium exists between buoyancy forces and capillary forces. These forces determine the initial distribution of fluids, and hence the volumes of fluid in place. An understanding of the relationship between these forces is useful in calculating volumetries, and in explaining the difference between free water level (FWL) and oil-water contact (OWC) introduced in the last section. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

As in the case of capillary rise, Sugden [27] has made use of Bashforth s and Adams tables to calculate correction factors for this method. Because the figure is again one of revolution, the equation h = a lb + z is exact, where b is the value of / i = R2 at the origin and z is the distance of OC. The equation simply states that AP, expressed as height of a column of liquid, equals the sum of the hydrostatic head and the pressure... 

The table is used in much the same manner as are Eqs. 11-19 and 11-20 in the case of capillary rise. As a first approximation, one assumes the simple Eq. II-10 to apply, that is, that X=r, this gives (he first approximation ai to the capillary constant. From this, one obtains r/ai and reads the corresponding value of X/r from Table II-2. From the derivation of X(X = a /h), a second approximation a to the capillary constant is obtained, and so on. Some mote recent calculations have been made by Johnson and Lane [28]. 

Calculate to 1% accuracy the capillary rise for water at 20°C in a 1.2-cm-diameter capillary. 

A liquid of density 2.0 g/cm forms a meniscus of shape corresponding to /3 = 80 in a metal capillary tube with which the contact angle is 30°. The capillary rise is 0.063 cm. Calculate the surface tension of the liquid and the radius of the capillary, using Table II-l. 

Calculate the vapor pressure of water when present in a capillary of 0.1 m radius (assume zero contact angle). Express your result as percent change from the normal value at 25°C. Suppose now that the effective radius of the capillary is reduced because of the presence of an adsorbed film of water 100 A thick. Show what the percent reduction in vapor pressure should now be. 

The reports were that water condensed from the vapor phase into 10-100-/im quartz or pyrex capillaries had physical properties distinctly different from those of bulk liquid water. Confirmations came from a variety of laboratories around the world (see the August 1971 issue of Journal of Colloid Interface Science), and it was proposed that a new phase of water had been found many called this water polywater rather than the original Deijaguin term, anomalous water. There were confirming theoretical calculations (see Refs. 121, 122) Eventually, however, it was determined that the micro-amoimts of water that could be isolated from small capillaries was always contaminated by salts and other impurities leached from the walls. The nonexistence of anomalous or poly water as a new, pure phase of water was acknowledged in 1974 by Deijaguin and co-workers [123]. There is a mass of fascinating anecdotal history omitted here for lack of space but told very well by Frank [124]. 

As illustrated in Fig. XU-13, a drop of water is placed between two large parallel plates it wets both surfaces. Both the capillary constant a and d in the figure are much greater than the plate separation x. Derive an equation for the force between the two plates and calculate the value for a 1-cm drop of water at 20°C, with x = 0.5, 1, and 2 mm. 

Templeton obtained data of the following type for the rate of displacement of water in a 30-/im capillary by oil (n-cetane) (the capillary having previously been wet by water). The capillary was 10 cm long, and the driving pressure was 45 cm of water. When the meniscus was 2 cm from the oil end of the capillary, the velocity of motion of the meniscus was 3.6 x 10 cm/sec, and when the meniscus was 8 cm from the oil end, its velocity was 1 x 10 cm/sec. Water wet the capillary, and the water-oil interfacial tension was 30 dyn/cm. Calculate the apparent viscosities of the oil and the water. Assuming that both come out to be 0.9 of the actual bulk viscosities, calculate the thickness of the stagnant annular film of liquid in the capillary. 

Thus D(r) is given by the slope of the V versus P plot. The same distribution function can be calculated from an analysis of vapor adsorption data showing hysteresis due to capillary condensation (see Section XVII-16). Joyner and co-woikers [38] found that the two methods gave very similar results in the case of charcoal, as illustrated in Fig. XVI-2. See Refs. 36 and 39 for more recent such comparisons. There can be some question as to what the local contact angle is [31,40] an error here would shift the distribution curve. 

This equation describes the additional amount of gas adsorbed into the pores due to capillary action. In this case, V is the molar volume of the gas, y its surface tension, R the gas constant, T absolute temperature and r the Kelvin radius. The distribution in the sizes of micropores may be detenninated using the Horvath-Kawazoe method [19]. If the sample has both micropores and mesopores, then the J-plot calculation may be used [20]. The J-plot is obtained by plotting the volume adsorbed against the statistical thickness of adsorbate. This thickness is derived from the surface area of a non-porous sample, and the volume of the liquified gas. 

The creatmenc of the boundary conditions given here ts a generali2a-tion to multicomponent mixtures of a result originally obtained for a binary mixture by Kramers and Kistecnaker (25].These authors also obtained results equivalent to the binary special case of our equations (4.21) and (4.25), and integrated their equations to calculate the p.ressure drop which accompanies equimolar counterdiffusion in a capillary. Their results, and the important accompanying experimental measurements, will be discussed in Chapter 6 ... 

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

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