Capillary number equation defining

As defined by Radke and Ransohoff (Equation 7), the "snap-off" capillary number, C, contains the effective grain radius, R the permeability, K anS the relative permeability of the nonwetting phase, k ( ). In field applications, the values of all of these parameters are set by the reservoir. Also contained in C are the total superficial velocity, U, and the distance between injection and production wells, L. Within narrow limits, L can be changed by... 

He solved the equations and matched the asymptotic expansions for each of the regions at velocities of the TCL small enough to neglect inertial and viscous effects in comparison with capillary ones. This last condition is insured if the nondimensional capillary number defined as Ca = /U.(7/y, is much smaller than 1. 

The coefficients A = (1 -Aq)IBq (3 Cay, Aq, and Bq as functions of the capillary number according to Equation 3.237 and Equation 3.236 obtained from the numerical calculation and are presented in Figure 3.18. The isotherm of the disjoining pressure is defined by Equation 3.240, that is, the case of complete wetting is under consideration. Figure 3.18 shows that in this case at Ca 0, B 3/2, which corresponds to the prediction (3.249). At the same lime, Aq 1 - (3/2) He as shown above, and A . 

Table 10.1 lists several equations that apply to CE. Equation 1 states that the velocity (cm/s) with which an ion moves through the capillary is a function of its mobility and field strength. Field strength (V/cm) is defined in Eq. 2. Ionic mobility (cm /V s) is defined in terms of column length, migration time and applied voltage in Eq. 3. Equation 4 states that ionic mobility (cmW s) is made up of electrophoretic and elec-troosmotic mobility. In CE as in chromatography the separation power is often stated by the plate number, N (also called the number of theoretical plates). 

With the introduction of capillary open tube columns (Chapter 4) it became possible to obtain chromatograms corresponding to hundreds of thousands (even >10 ) of theoretical plates as evaluated via Equation [3.20] or [3.21]. However, as a consequence of the issues discussed in Section 3.4.6, such enormous efficiencies could be obtained only for compounds with very low k values, i.e., those that eluted very close to the column dead volume. To provide a more realistic measure of column efficiency in such cases, the effective plate number (Ng) was defined by replacing t, by (L—tg) in Equations [3.20-3.21], where tg is the elution time after injection for unretained solutes ... 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

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