Pressure and Curved Surfaces

The Young-Laplace equation (3.4/3.5) shows that, pA pB, the pressure inside a bubble or drop exceeds that outside. For a sphere, Ap=pA - Pb = 2y/R, so that Ap varies with the radius, R. Thus the vapour pressure of a drop should be higher, the smaller the drop. This is shown by a related equation, the Kelvin equation [13,26], which is described here. 

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A related phenomenon is the difference in vapour pressure between the flat and curved surfaces of a given liquid. The application of classical thermodynamics (cf. Defay and Prigogine, 1951) allows us to replace the difference in mechanical pressure Ap=pg -p1, by a function of the relative vapour pressure p/p°. The condition for physicochemical equilibrium is ... 

Table 4.1 Relative vapor pressures of curved surfaces in equilibrium at 25°C, for water drops in air and air bubbles in water...

For a liquid surface, if no force acts normal to a tensioned surface, the surface must remain flat. However, if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. The surface tension forces must cancel the force due to pressure and the surface must be curved. Figure 2.18 shows how the curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the result is known as the Young-Laplace equation ... 

Polyethylene and PP have a surface tension of 30 dynes/cm, which has to be raised to a minimum of 38 dynes/cm, but preferably raised to 42 dynes/cm, in order to render their surfaces adherable. Flame treatment is the most widely used method of pretreatment. It is flexible and reliable if carefully controlled. It enables uneven and curved surfaces to be treated. It uses a mixture of air at 138—345 kPa and a fuel gas at a (low) pressure of 1.7 kPa. The gas can be butane, propane, natural gas (methane), and coal gas. 

The thermographic activity on the pressure vessel was carried out considering a part of it because of the axial symmetry. Three different partially overlapping area were inspected since it was optically impossible to scan the curved surface of the pressure vessel by a single sweep. The selected areas are shown in fig.7 and the correspondent positions of the thermographic scan unit are also illustrated. The tests were performed with a load frequency of 2, 5 and 10 Hz. 

The Kelvin equation (Eq. HI-18), which gives the increase in vapor pressure for a curved surface and hence of small liquid drops, should also apply to crystals. Thus... 

For some types of wetting more than just the contact angle is involved in the basic mechanism of the action. This is true in the laying of dust and the wetting of a fabric since in these situations the liquid is required to penetrate between dust particles or between the fibers of the fabric. TTie phenomenon is related to that of capillary rise, where the driving force is the pressure difference across the curved surface of the meniscus. The relevant equation is then Eq. X-36,... 

Ultimately, the surface energy is used to produce a cohesive body during sintering. As such, surface energy, which is also referred to as surface tension, y, is obviously very important in ceramic powder processing. Surface tension causes liquids to fonn spherical drops, and allows solids to preferentially adsorb atoms to lower tire free energy of tire system. Also, surface tension creates pressure differences and chemical potential differences across curved surfaces tlrat cause matter to move. 

Equations (5,61) and (5.62) can be used to derive a pressure potential equation applicable to thin-layer flow between curved surfaces using the following procedure. In a thin-layer flow, the following velocity boundary conditions are prescribed ... 

More often, however, microporosity is associated with an appreciable external surface, or with mesoporosity, or with both. The effect of microporosity on the isotherm will be seen from Fig. 4.11(a) and Fig. 4.12(a). In Fig. 4.11(a) curve (i) refers to a powder made up of nonporous particles and curve (ii) to a solid which is wholly microporous. However, if the particles of the powder are microporous (the total micropore volume being given by the plateau of curve (ii)), the isotherm will assume the form of curve (iii), obtained by summing curves (i) and (ii). Like isotherm (i), the composite isotherm is of Type II, but because of the contribution from the Type 1 isotherm, it has a steep initial portion the relative enhancement of adsorption in the low-pressure region will be reflected in a significantly increased value of the BET c-constant and a shortened linear branch of the BET plot. 

The force exerted on a submerged planar surface of area A is given by F = p A where p is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example. Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. 

Insulants for pipework and curved pressure vessels can be obtained ready shaped, so that they fit tightly to the surface. All surfaces should be quite dry before the material is applied, even if the adhesive is a water-based emulsion, and the water or other solvent must be given ample time to dry or set before any outer wrapping is applied. 

The pitot tube is used to measure the difference between the impact and static pressures in a fluid. It normally consists of two concentric tubes arranged parallel to the direction of flow the impact pressure is measured on the open end of the inner tube. The end of the outer concentric tube is sealed and a series of orifices on the curved surface give an accurate indication of the static pressure. The position of these orifices must be carefully chosen because there are two disturbances which may cause an incorrect reading of the static pressure. These are due to ... 

Capillary phenomena are due to the curvature of liquid surfaces. To maintain a curved surface, a force is needed. In Eq. (8), this force is related to the second term in the integral. The so-called Laplace pressure due to the force to maintain the curved surface can be expressed as, Pl = 2-yIr, where r is the curvature radius. The combination of the capillary effects and disjoining pressure can make a liquid film climb a wall. 

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