Concave

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... 

The area under a parabolic arc concave upward is bh, where b is the base of the figure and h is its height. The area of a parabolic arc concave downward is jh/t. The areas of parts of the figure diagrammed for Simpson s rule integration are shown in Fig. 1-3. The area A under the parabolic arc in Fig. 1-3 is given by the sum of four terms ... 

From the Kelvin equation it follows that the vapour pressure p over a concave meniscus must be less than the saturation vapour pressure p°. Consequently capillary condensation of a vapour to a liquid should occur within a pore at some pressure p determined by the value of r for the pore, and less than the saturation vapour pressure—always provided that the meniscus is concave (i.e. angle of contact <90°). 

Frequently, however, the DR plot deviates from linearity, and in a number of ways. Sometimes the plot is convex to the log (p°/p) axis, as in Fig. 4.19(a), and sometimes concave, as in Fig. 4.19(b). The question then arises as to whether one should extropolate from the low-pressure. 

An example from Dubinin s paper illustrates the application of the new equation. For benzene at 293 K, the plot of Equation (4.18) with m = 2 gave a line concave to the log (p°/p) axis. A revised value of m was therefore required, and was obtained as follows. A provisional value of IVq (= 408 mm g ) was estimated from the plateau of the isotherm the relative pressure (p/p°), corresponding to IV/tV = 0-368 was then read off, and a first value of S calculated from... 

I (curve D). Thus the micropores had been able to enhance the adsorbent-adsorbate interaction sufficiently to replace monolayer-multilayer formation by micropore filling and thereby change the isotherm from being convex to being concave to the pressure axis. 

When Fj = 1/f2, the copolymer composition curve will be either convex or concave when viewed from the Fj axis, depending on whether Fj is greater or less than unity. The further removed from unity rj is, the farther the composition curve will be displaced from the 45° line. This situation is called ideal copolymerization. The example below explores the origin of this terminology. 

The dispersing element is a diffraction grating preferably used under conditions of grazing incidence (6 in Equation 3.9 about 89°) to improve the reflectance. The grating may also be concave to avoid the use of a focusing mirror. 

Fig. 8. Correlation of the effective average slopes m of the equilibrium line (4). (a) Equilibrium line curved concave upward (b) equilibrium line curved...

Fig. 14. Coiielation of effective average slope m of equilibrium line (dilute part of absorber) equilibrium line concave upward. gas mole fraction at...

Several recent patents describe improvements in the basic belt process. In one case a higher soHds polymerization is achieved by cooling the starting monomer until some monomer crystallizes and then introducing the resulting monomer slurry onto the belt as above. The latent heat of fusion of the monomer crystals absorbs some of the heat of polymerization, which otherwise limits the soHds content of the polymerization (87). In another patent a concave belt is described which becomes flat near the end. This change leads to improved release of polymer (88). 

The 2eohte and the resin adsorbents show different adsorption isotherm characteristics, particularly at higher concentration (51). The resin adsorbent isotherm is slightly concave upward, whereas the 2eohte isotherm is linear, or even slightly concave downward. Resins, therefore, have an advantage in a UOP Sarex operation that involves high feed-soHds concentration. 

McCabe-Thiele diagrams for nonlinear and more practical systems with pertinent inequaUty constraints are illustrated in Figures 11 and 12. The convex isotherms are generally observed for 2eohtic adsorbents, particularly in hydrocarbon separation systems, whereas the concave isotherms are observed for ion-exchange resins used in sugar separations. 

Steel. The steel container s most usual form is cylindrical with a concave (or flat) bottom and a convex top dome with a circular opening finished to receive a valve with a standard 2.54-cm opening. The three pieces (body, bottom, and top) are produced separately and joined by high speed manufacturing. The size of the container is described by its diameter and height to top seam, in that order. Hence a 202 x 509 container is 54.0 mm (2 /jg in.) in diameter by 141.3 mm (5 /jg in.) high. Tables of available sizes and overflow volumes and suggested fill levels can be readily obtained from manufacturers. 

A perforated plate can be flat, concave, convex, or double-dished. The main advantages of the perforated plate are that it is simple, inexpensive, easy to modify, and easy to clean. The disadvantages of a perforated plate are the possibiUty of soflds leaking, ie, weeping through it into the plenum lower turndown capabiUty than other distributors the requirement of a peripheral seal and a relatively high pressure drop requited for good distribution. 

Fig. 13. Sketch of Kelvin-Helmholtz instabiHty, where C, the convex side of fluid M, is at a lower pressure than D, the concave side of fluid B.

Minimum thickness requirements are 11/2 times the indentation diagonal measurement, and there should be no visible marking or bulge visible on the side opposite the indentation.The Vickers test is based on a plane surface however, correction tables are available for both convex and concave surfaces (see ASTM E92) (2). 

The first of these reactions takes place at temperatures of about 150°C, the second reaction proceeds at about 550—660°C. Typical furnaces used to carry out the reaction include cast-iron retorts the Mannheim mechanical furnace, which consists of an enclosed stationary circular muffle having a concave bottom pan and a domed cover and the Laury furnace, which employs a horizontal two-chambered rotating cylinder for the reaction vessel. The most recent design is the Cannon fluid-bed reactor in which the sulfuric acid vapor is injected with the combustion gases into a fluidized bed of salts. The Mannaheim furnace has also been used with potassium chloride as the feed. 

Fig. 1. The principle of formation of an inclusion compound, (a) concave host (b) convex guest component (c) host—guest compound.

Uses. Sound-absorbing materials are frequendy used to reduce reverberation, or the persistence of sound in a space after generation of the sound ceases to reduce focused reflections from concave surfaces to prevent echoes, or delayed sound reflections from distant surfaces and to prevent the buildup of sound by multiple reflections within rooms and other enclosures. Sound-absorbing materials also are used to reduce the transmission of noise from one location to another by multiple reflections from sound-reflecting surfaces. 

In plotting on WeibuU paper, a downward concave plot implies a non2ero minimum life. Values for S < can be selected by trial and error. When they are subtracted from each /, a relatively straight line is produced. This essentially translates the three-parameter WeibuU distribution back to a two-parameter distribution. 

RoUer-top cards have five to seven sets of workers and strippers to mix and card the fibers carried on the cylinder. The multiple transferring action and re-introduction of new groupings of fibers to the carding zones provides a doubling effect which enhances web uniformity. Stationary-top cards have strips of metallic clothing mounted on plates positioned concavely around the upper periphery of the cylinder. The additional carding surfaces thus estabhshed provide expanded fiber alignment with minimum fiber extraction. 

Etch Profiles. The final profile of a wet etch can be strongly influenced by the crystalline orientation of the semiconductor sample. Many wet etches have different etch rates for various exposed crystal planes. In contrast, several etches are available for specific materials which show Httle dependence on the crystal plane, resulting in a nearly perfect isotropic profile. The different profiles that can be achieved in GaAs etching, as well as InP-based materials, have been discussed (130—132). Similar behavior can be expected for other crystalline semiconductors. It can be important to control the etch profile if a subsequent metallisation step has to pass over the etched step. For reflable metal step coverage it is desirable to have a sloped etched step or at worst a vertical profile. If the profile is re-entrant (concave) then it is possible to have a break in the metal film, causing an open defect. 

When the void space in an agglomerate is completely filled with a Hquid (Fig. Ic), the capillary state of wetting is reached, and the tensile strength of the wet particle matrix arises from the pressure deficiency in the Hquid network owing to the concave Hquid interfaces at the agglomerate surface. This pressure deficiency can be calculated from the Laplace equation for chcular capillaries to yield, for Hquids which completely wet the particles ... 

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