Pressure difference

Inside the capillary tube, the capillary pressure (P ) is the pressure difference between the oil phase pressure (PJ and the water phase pressure (P ) at the interface between the oil and the water. 

Finally, it is worth remembering the sequence of events which occur during hydrocarbon accumulation. Initially, the pores in the structure are filled with water. As oil migrates into the structure, it displaces water downwards, and starts with the larger pore throats where lower pressures are required to curve the oil-water interface sufficiently for oil to enter the pore throats. As the process of accumulation continues the pressure difference between the oil and water phases increases above the free water level because of the density difference between the two fluids. As this happens the narrower pore throats begin to fill with oil and the smallest pore throats are the last to be filled. 

AP such that the work against this pressure difference APAvr dr is just equal to the decrease in surface free energy. Thus... 

If the first plane is rotated through a full circle, the first radius of curvature will go through a minimum, and its value at this minimum is called the principal radius of curvature. The second principal radius of curvature is then that in the second plane, kept at right angles to the first. Because Fig. II-3 and Eq. II-7 are obtained by quite arbitrary orientation of the first plane, the radii R and R2 are not necessarily the principal radii of curvature. The pressure difference AP, cannot depend upon the manner in which and R2 are chosen, however, and it follows that the sum ( /R + l/f 2) is independent of how the first plane is oriented (although, of course, the second plane is always at right angles to it). 

There will be a pressure difference AP across the surface it acts on the area xy and through a distance dz. The corresponding work is thus... 

It is apparent Aat Eq. II-7 reduces to Eq. II-3 for the case of both radii being equal, as is true for a sphere. For a plane surface, the two radii are each infinite and AP is therefore zero thus there is no pressure difference across a plane surface. 

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

In Fig. III-7 we show a molecular dynamics computation for the density profile and pressure difference P - p across the interface of an argonlike system [66] (see also Refs. 67, 68 and citations therein). Similar calculations have been made of 5 in Eq. III-20 [69, 70]. Monte Carlo calculations of the density profile of the vapor-liquid interface of magnesium how stratification penetrating about three atomic diameters into the liquid [71]. Experimental measurement of the transverse structure of the vapor-liquid interface of mercury and gallium showed structures that were indistinguishable from that of the bulk fluids [72, 73]. 

It was pointed out at the beginning of this section that ir could be viewed as arising from an osmotic pressure difference between a surface region comprising an adsorbed film and that of the pure solvent. It is instructive to develop... 

The surface viscosity can be measured in a manner entirely analogous to the Poiseuille method for liquids, by determining the rate of flow of a film through a narrow canal under a two-dimensional pressure difference Ay. The apparatus is illustrated schematically in Fig. IV-7, and the corresponding equation for calculating rj is analogous to the Poiseuille equation [99,100]... 

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,... 

In the example of pressure-volume work in die previous section, the adiabatic reversible process consisted simply of the sufficiently slow motion of an adiabatic wall as a result of an infinitesimal pressure difference. The work done on the system during an infinitesimal reversible change in volume is then -pdVand one can write equation (A2.1.11) in the fomi... 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

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. 

The Laplace equation, which defines tire pressure difference, AP, across a curved surface of radius, r. 

Chambers A and B are separated by an elastic rubber membrane whose motion in response to pressure differences is transmitted to the needle N. Variations in the pressure difference can then be followed by observing the displacement of the point of the needle. Initially chamber A and the capillary are filled to the desired pressure by opening cocks and C, ... 

The conditloci to be satisfied by the pressure difference, if (10,18) is to be a good approximation, is not so demanding as that needed to ensure that (10.14) is a good approximation, and it does not depend on the permeability of the medium. Thus (10.18), applied at sufficiently high pressure, should provide a method of determining which is more robust than (10.14)... 

In this case the pressure difference across the porous medium no longer vanishes (if it did, (10.33) would be replaced by Graham s relation) but. 

To prevent this flow, the pressure on the hotter side must be larger than the pressure on the colder side. The required pressure difference depends on the nature of the gas, its mean pressure and absolute temperature, the relation between its density and the pore size, and the temperature difference. However, it does not depend on the thickness of the plate. 

With a given gas, plate and temperature difference, the pressure difference required to prevent flow is approximately inversely proportional to the density of the gas when the density is sufficiently high. However,... 

When the density Is sufficiently low that the pressure difference Is proportional to density, then the ratio of the absolute pressures on the two sides of the plate Is equal to the square root of Che ratio of tha absolute temperatures. 

In the second part of hla memoir Reynolds gave a theoretical account of thermal transpiration, based on the kinetic theory of gases, and was able CO account for Che above "laws", Chough he was not able to calculate Che actual value of the pressure difference required Co prevent flow over Che whole range of densities. ... 

Reynolds also discussed transpiration under the Influence of a pressure difference alone and gave an account of the phenomenon of Impulsion In a Crookes radiometer, an effect of great Interest to 19ch century scientists. 

A mixture of helium and oxygen is used as an artificial atmosphere for divers and others working under pressure. Different ratios of He/02 are used for different depths at which the diver is operating. 

Chapter 3, there is often a region immediately preceding the lower closure point, in which increased adsorption is brought about by reversible capillary condensation. The meniscus now tends to be somewhat ill defined owing to its small dimensions (p. 153), but the mechanism can still be thought of in Kelvin terms, where the driving force is the pressure difference across an interface. 

A hydrodynamic injection is made by applying a pressure difference of 2.5 X 10 Pa (approximately 0.02 atm) for 2 s to a 75-cm long capillary tube with an internal diameter of 50 Jtm. Assuming that the buffer solution s viscosity is 10 kg m s what volume of sample is injected ... 

This predicts the rate of flow of liquid through a smooth tube under the effect of a pressure difference between the ends of the tube (equation 1). 

---