Capillary tube geometry

For the study of flow stability in a heated capillary tube it is expedient to present the parameters P and q as a function of the Peclet number defined as Pe = (uLd) /ocl. We notice that the Peclet number in capillary flow, which results from liquid evaporation, is an unknown parameter, and is determined by solving the stationary problem (Yarin et al. 2002). Employing the Peclet number as a generalized parameter of the problem allows one to estimate the effect of physical properties of phases, micro-channel geometry, as well as wall heat flux, on the characteristics of the flow, in particular, its stability. 

In bringing the models to a non-dimensional form, the presence of dominant Peclet and Damkohler numbers in reactive flows is observed. The problems of interest arise in complex geometries-like porous media or systems of capillary tubes. 

Several experimental arrangements are used to measure and analyze the wetting of liquids on solid surfaces. Typical geometries are a spreading drop on a solid surface, liquid-fluid displacement through a capillary tube, steady immersion or withdrawal of fibers, plates or tapes from a pool of liquid, and the rotation of a horizontal cylinder in a liquid (Fig. 7.12). 

Some rotational viscometers employ a rotating disc, bar, paddle or pin at a constant speed (or series of constant speeds). It is extremely difficult to obtain tme shear stress, and the shear rate usually varies from point to point in the rotating member. In particular, the velocity field of a rotating disc geometry can be considerably distorted in viscoelastic fluids. Nevertheless, because they are simple to operate and give results easily, and their cost is low, they are widely used in the food industry. While they may be useful for quality control purposes, especially Newtonian foods, the reliability of their values should be verified by comparison with data obtained with well defined geometries (capillary/tube, concentric cylinder, and cone-plate). 

Although the capillary tube is a simple representation of fluid in a pore, several valid comments can be made. The capillary pressure increases as pore diameter decreases, or as interfacial tension (IFT) or contact angle increases. Capillary forces are therefore influenced by pore geometry, interfacial tension, and surface wettability. 

This equation can be used to explain that stable adsorption on the inner wall of a capillary tube is possible up to a certain critical thickness. Capillary condensation starts from this critical thickness. It appears that capillary condensation in the cylindrical pores with radii between 2 and 7 run is well described by the corrected Kelvin equation. The deviation from the ideal behavior can amount to up to +20%. Nevertheless, it is difficult to estimate the limits of the applicability of the Kelvin equation. It seems likely that distortions of the meniscus in small pores may occur and that the local pore geometry may have a marked influence [1,15]... 

Figure 4.7 Concave liquid meniscus in a capillary tube during a liquid surface tension measurement 6 is the angle of contact between the liquid and the capillary wall c is the point at the liquid level, a is the point just above, and b is just below, the meniscus level and h shows the height of the liquid column. The crown of the concave meniscus is the liquid between the top and the lower end of the meniscus. The term cp is the inclination angle, where (cp = 90° - 6) from plane geometry.

Since Oq is very small (ca. 0.5-3°, or 0.0087-0.0523 radians), sin (jt/2 -So) will close to one, and t wiU be nearly independent of position [see Eq. (59)] that is, the tested material between the gap will experience uniform shear stress. This is the advantage of cone-plate compared to other geometries (i.e., capillary tube and parallel disk). For example, at an angle of 1°, the pereentage difference in shear stress between cone and plate is 0.1218% (Fredrickson, 1964). This is within the precision of measurements that must be made therefore, one can assume that shear stress, and, hence, shear rate and apparent viscosity, are uniform throughout the fluid. 

Even the measurement of the steady-state characteristics of shear-dependent fluids is more complex than the determination of viscosities for Newtonian fluids. In simple geometries, such as capillary tubes, the shear stress and shear rate vary over the cross-section and consequently, at a given operating condition, the apparent viscosity will vary with location. Rheological measurements are therefore usually made with instraments in which the sample to be sheared is subjected to the same rate of shear throughout its whole mass. This condition is achieved in concentric cylinder geometry (Fi re 3.37) where the fluid is sheared in the annular space between a fixed and a rotating cylinder if the gap is small compared with the dimneters of the cylinders, the shear rate is approximately... 

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