Drop profile

A set of pendant drop profiles is shown in Fig. 11-17 as an illustration of the range of shapes that may be observed. It has been pointed out that for practical reasons, the size of the tip from which the drop is suspended should be such that rja is about 0.5 or less [66]. 

A modem alternative procedure involves computer matching of the entire drop profile to a best fitting theoretical curve in this way the entire profile is used, rather than just d and de, so that precision is increased. Also, drops whose ds is not measurable (how does this happen ) can be used. References 61 and 71-74 provide examples of this type of approach. 

Usually one varies the head of mercury or applied gas pressure so as to bring the meniscus to a fixed reference point [118], Grahame and co-workers [119], Hansen and co-workers [120] (see also Ref. 121), and Hills and Payne [122] have given more or less elaborate descriptions of the capillary electrometer apparatus. Nowadays, the capillary electrometer is customarily used in conjunction with capacitance measurements (see below). Vos and Vos [111] describe the use of sessile drop profiles (Section II-7B) for interfacial tension measurements, thus avoiding an assumption as to the solution-Hg-glass contact angle. 

For an operating tray the pressure drop profile is shown in Figure 8-148 [201]. The valves are closed at low hole vapor velocities, although, due to the design of the valves (see Figures 8-72 and 8-74), the metal tabs keep some styles of valves open sufficiently to allow some vapor and some liquid through, even at low flow rates. 

Figure 8-148. Typical operating valve tray pressure drop profile. Valves start to open at A, the closed balance point. Used by permission, Klein, G. F. Chem. Eng. V. 89, No. 9 (1982) p. 81 all rights reserved.

Figure 6 shows droplets of KOH solution on mica produced by similar methods. In both cases the drop profiles are very close to a spherical cap. In Figure 7 we have plotted the effective contact angle as a function of droplet height. The deviation from the macroscopic contact angle with decreasing droplet volume can clearly be seen. 

The compact can either be saturated with the fluid for static measurements or dynamic measurements may be made through a computer imaging goniometer which takes successive images of the drop profile. 

The Bashforth-Adams tables provide an alternate way of evaluating 7 by observing the profile of a sessile drop of the liquid under investigation. If, after all, the drop profiles of Figure 6.15 can be drawn using 0 as a parameter, then it should also be possible to match an experimental drop profile with the (3 value that characterizes it. Equation (85) then relates 7 to 0 and other measurable quantities. This method is claimed to have an error of only 0.1%, but it is slow and tedious and hence not often the method of choice in practice. 

Figure 6.15a may be used as a numerical example to illustrate this procedure. Suppose an actual experimental drop profile is matched with theoretical profiles and is shown to correspond to a j8 value of 10.0. Then b is evaluated as follows ... 

Suppose the drop profile shown in Figure 6.15b describes an actual drop for which the radius at the widest point equals 0.135 cm. Use the value of (x/b) from the figure to calculate y for each of the following situations ... 

While the measurement protocol is fairly simple, there are a number of important factors that need to be considered when using the drop shape analysis method. First, a sufficient visual contrast between the drop and the surrounding liquid is required to be able to extract the drop profile. If the external phase is slightly turbid, or if the refractive indices of the two phases match each other, it may be difficult or even impossible to extract the drop profile. If possible, the more turbid phase should be chosen as the internal drop-forming... 

A set of experimental drop profiles is usually nonlinearly fitted to numerically generated... 

Merte, H., Jr. and Son, S., Further Consideration of Two-Dimensional Condensation Drop Profiles and Departure Sizes, Warmeund Stoffubertragung, Vol. 21, pp. 163-168. 

Figure 3.5. Video images of sessile drops of two Ni alloys on Ali03 at 1500" C with (left) and without (right) background illumination (Labrousse 1998). On the left image, one can distinguish the reflection of the resistance heater on the droplet as well as the capillary used to dispense the droplet. By tilting slightly the substrate, it is possible to determine accurately the position of the triple line (dashed line) by the intersection of the drop profile and the drop shadow (or reflection) on the substrate surface.

To have unreserved confidence in such 0 data it is advisable to measure several geometric and dimensional characteristics of the sessile drop and to test for their internal consistency. Thus the 0 values assumed by the left hand and right hand sides of the drop profile should be measured and compared for consistency. Additionally, it is prudent to measure the height of the drop apex, H, and the radius R of the substrate contact area. If the drop has the profile of a spherical cap then a value can be calculated for the contact angle, 0Caic. by substitution in... 

Sessile drop experiments are also used to measure the effects of temperature on liquid surface energies. Because the temperature coefficient dliquid metals and oxides is usually a very small, negative, value (—0.05 to —0.5 mJ.m-2.K-1), a temperature rise of several hundred degrees is necessary to produce decreases in the surface energy that can be reliably detected by measurements of drop profiles. Even in this case, the error on the temperature coefficient lies between 30% and 100% (see Section 4.1.1). 

Liquid density and other characteristics. Profile measurements for a symmetrical sessile drop can be used to calculate its volume and hence its density if the original mass is known and there has been no significant loss due to evaporation. As for the measurement of numerical integration by using image analysis techniques (Naidich and Grigorenko 1992). Alternatively, the... 

Thus, measurements of solidified sessile drops can be and are used to derive estimated values of contact angles, but these estimates are less precise than those derived from direct observation of liquid drop profiles except for very well wetting systems. The estimates for solidified drops tend to be smaller, typically by up to 5°, and more variable than those derived from observation of liquid profiles. Larger differences can be observed when, during cooling, dewetting of the liquid drop occurs before the resolidification temperature is reached. In this case, traces of the location of the triple line before dewetting are easily seen on the substrate surface. 

Maze and Burnet [61,62] developed a more satisfactory scheme for the determination of contact angle and interfacial tension from the shape of sessile drops both above and below 90. They utilized a numerical nonlinear regression procedure in which a calculated drop shape is made to fit a number of arbitrary selected and measured points on the drop profile. In other words, the measured drop shape (one-half or the meridian section) is described by a set of coordinate... 

Two similar techniques to probe hydrodynamic behaviour have been recently developed by Horn et al. [33] and Gee and co-workers [34-37]. The measurements by Horn et al. employ interferometry to measure the shape of a mercury drop at the end of a microcapillary as it approaches a flat interface. The capillaries are much larger than those used in an LSFA, where the drop profile is recorded under hydrodynamic forces. The work by Gee [34-37] is similar but uses hydrocarbon drops and determines the separation using imaging ellipsometry/reflectometry. This method is useful for systems... 

An expression for the cumulative pressure drop profile in a spouted bed was derived by Mamuro and Hattori (M6) from a consideration of the balance of forces acting on a differential height dz of the annular solids (Fig. 10) ... 

Using the vertical profile of v and Ua (estimated from the measured pressure drop profile as explained in Section IV,A), the authors evaluated the right-hand side of Eq. (74) for each 2-in. increment of the spout, assuming constant pba over this increment. This value, on substitution in Eq. (70) along with the measured total pressure drop for the increment, gave the pressure drop per foot due to the solids bulk density in the spout, from which the voidage profile was calculated. The results agreed well with those from method (a). 

Figure 3.9 Pressure drop profile across the membrane/support structure during dip coating process [Leenaars and Burggraaf t985]...

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