Capillary liquid flow through

In many processes (such as oil recovery, blood flow, underground water), one encounters liquid flow through thin (micrometer diameter), noncircular-shaped tubes, or pores. In the literature, one finds studies that address these latter systems. In another context of liquid drop formation, for example, in an inkjet nozzle, this technique falls under a class of scientifically challenging technology. The inkjet printer demands such quality that this branch of drop-on-demand technology is much in the realm of industrial research. All combustion engines are controlled by oil drop formation and evaporation characteristics. The important role of capillary forces is obvious in such systems. 

Although reverse osmosis, ultrafiltration and microfiltration are conceptually similar processes, the difference in pore diameter (or apparent pore diameter) produces dramatic differences in the way the membranes are used. A simple model of liquid flow through these membranes is to describe the membranes as a series of cylindrical capillary pores of diameter d. The liquid flow through a pore (q) is given by Poiseuille s law as ... 

A stable and sensitive conductivity cell has been described that measures the DC current that passes through a small volume of solution confined in a small capillary tube. The very simple apparatus is illustrated in Figure 6.19. When filled with 0.1 M KC1, the cell exhibits an Ohm s law behavior with an applied voltage from 5 to 100 V (approximately 5-100 mA of current). The cell also can be used for conductimetric titrations, with the titration carried out in vessel B. Because some liquid flows through the capillary during a titration, a correction factor is used to obtain the most accurate results (although corrections usually are <0.2%). 

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... 

The maximum electroosmotic pressure of the liquid flowing through a capillary or a membrane equals... 

When there is a difference of pressure between two points of a capillary tube, the fluid flows from the high pressure side to the low pressure side of the tube. Let us assume that a Newtonian liquid flows through a capillary tube of radius r and length AL (Fig. 13.5). Once steady-state conditions are reached, that is, the applied energy is totally dissipated into friction energy, a simple balance of energy gives... 

Many flow patterns have been described for two-phase, gas-liquid flow through narrow tubes. Figure 1 shows different types of gas-liquid flow in vertical capillaries. 

V. Hatziantoniou and B. Andersson, Solid-Liquid Mass Transfer in Segmented Gas-Liquid Flow Through a Capillary, Ind. Eng. Chem. Fundam. 27 451 (1982). 

Combining (9.21) with the Kelvin equation (9.18) and using Darcy s law for liquid flow through porous media (as similarly done before for gas flow, see Eq. (9.1)) yields an expression for the gas flux Jd3 of capillary condensate in case 3 of Fig. 9.9 ... 

In foams with charged gas/liquid interface, various electrokinetic parameters can be obtained, such as streaming potential and zeta-potential. For example, the relationship between the volumetric flow of a liquid flowing through a capillary or membrane and the zeta-potential can be given by the Smoluchowski equation. 

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... 

Figure 4-3. Force balance on a colum.n of liquid flowing through a capillary. ...

When the liquid flows through the tube T with a constant flow rate the interface moves downwards with the velocity v. Under the assumption that the liquid film adheres to the wall of the capillary new surface is created by the receding liquid meniscus. Assuming a hemispherical meniscus, the dilation rate is given by... 

Characterizing the porous bed by means of a capillary model of the interstitial space, the physical basis of the size separation procedure can be demonstrated through examination of the convection and Brownian diffusion of the colloidal particles in a liquid flow through a circular capillary. Figure 5.7.1 shows two freely-rotating spherical Brownian particles of different size sampling a nonuniform Poiseuille flow. The center of the larger particle in its travel... 

In capillary electrochromatography (CEC) a liquid flow through a packed capillary is created by appHcation of an electric field. Several interfaces have been designed for coupHng to ESI-MS, and current appHcations have been reviewed [93, 94]. CEC is a good alternative for neutral analytes in combination with mass spectrometric detection, since no interferences with miceUular matrix can occur [95]. The combination with LC, resulting in an electrically and a pneumatically driven flow of solvent, has been called pressurised CEC. This combination offers new opportunities for the separation of complex mixtures. 

When a liquid flows through a capillary with radius R and length L under a pressure/ , a force nr p is exerted on the liquid column at a radius r. It is counteracted by a frictional force iTrrLon. At equilibrium, therefore,... 

At a constant flow rate, the pressure drop across a capillary tube P is proportional to the viscosity of the liquid flowing through the tube. For a polymer solution, the ratio of this pressure to the pressure for the pure solvent is equal to the relative viscosity of the solution. 

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