External Applied Pressure

Increasing the external pressure (Pjj) leads to an increase in both the cavitation threshold and the intensity of bubble collapse. Qualitatively it can be assumed that there will no longer be a resultant negative pressure phase of the sound wave (since Pj, — 0) and so cavitation bubbles cannot be created. Clearly, a sufficiently large 

Pj = P sin 27tft) making Pj — P 0. In that P (the pressure in the bubble at the moment of collapse) is approximately Ph + Pa increasing the value of Pj will lead to a more rapid (Eq. 2.27) and violent (Eq. 2.36) collapse. 

In general an increase in intensity (I) will provide for an increase in the sonochemical effects. Cavitation bubbles, initially difficult to create at the higher frequencies (due to the shorter time periods involved in the rarefaction cycles) will now be possible, and since both the collapse time (Eq. 2.27), the temperature (Eq. 2.35) and the pressure (Eq. 2.36) on collapse are dependent on P i(=Ph + PA) bubble collapse will be more violent. However it must be realised that intensity cannot be increased indefinitely, since (the maximum bubble size) is also dependent upon the pressure amplitude (Eq. 2.38). With increase in the pressure amplitude (P ) the bubble may grow so large on rarefaction (R g, ) that the time available for collapse is insufficient. 

As an example to illustrate this point consider the effect of applying an acoustic wave of 20 kHz and pressure amplitude 2 atm (P ) to a reaction in water. According to Eq. 2.38, this amplitude will produce a bubble of maximum radius, R ng 1.27 X 10 cm, which if it can be assumed that Pm = Pa + Ph collapses in approx. 6.8 ps, (Eq. 2.27). This is less than l/5th of a cycle (10 ps), the assumption often 

Cavitation, McGraw Hill, Maidenhead, England, 1989, 

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Here we have expressed the stress as the sum of the (external) applied pressure PappUed together with a static pressure Pstatic- which arises from the internal forces acting on the uiiit cell. 

For a given reaction studied in a series of solvents, (8r- 8 f) is essentially constant, and most of the change in In k will come from the term — AV (8j — 8s)". If AV is positive, an increase in 8s (increase in solvent internal pressure) results in a rate decrease. If AV is negative, the reverse effect is predicted. Thus reactivity is predicted by regular solution theory to respond to internal pressure just as it does to externally applied pressure (Section 6.2). This connection between reactivity and internal pressure was noted long ago," and it has been systematized by Dack. -" ... 

Fluids may be classified in two different ways either according to their behaviour under the action of externally applied pressure, or according to the effects produced by the... 

Water s internal pressure acts on the volume of activation (AV ) of a reaction in the same way as an externally applied pressure does. Thus, the internal pressure of water influences the rates of nonpolar reactions in water in the same direction as external pressures. Nonpolar reactions with a negative volume of activation will thus be accelerated by the internal pressure of water, whereas nonpolar reactions with a positive volume of activation will be slowed by the internal pressure. For example, at 20° C the rate of Diels-Alder reaction between cyclopentadiene and butenone, which is known to have a negative volume of activation, in a 4.86 M LiCl solution is about twice as that of the reaction in water alone (Eq. 1.1).4... 

Werner (1980) has studied devolatilization in corotating twin-screw extruders when the volatile component was stripped from the polymeric solution by applying a vacuum to the system. Rough estimates of the equilibrium partial pressure of the volatile component in the feedstream for each of the systems studied by Werner indicate that this pressure was less than the applied pressure, which means that bubbles could have been formed. Figure 17 shows the influence of the externally applied pressure on the exit concentration for a methyl methacrylate-poly(methyl methacrylate) system of fixed concentration. Note that the exit concentration decreases as the pressure is decreased, but seems to approach an asymptotic value at the lowest pressures studied. Werner also reported that at a fixed flow rate and feed concentration the exit concentration did not vary with screw speed (over the range 150-300 min" ), which also suggests that ky alay, is independent of screw speed. Figure 18 is a plot of data obtained by Werner on an ethylene-low-density poly(ethylene) system and also shows that decreases in the applied pressure result in decreases in the exit concentration, but here a lower asymptote is not observed. 

The ultrasonic degradation of polystyrene in toluene increases vnth an increase in the externally applied pressure. 

Johannes van der Waals developed his famous equation of state by the introduction of both the attractive and the repulsive forces between the molecules. First he postulated that the gas behaves as if there is an additional internal pressure to augment the external applied pressure, which is based on the mutual attraction of molecules since the density of molecules is proportional to 1/V, the intensity of the binary attractive force would be proportional to 1/V. Then he postulated that when the measured total volume begins to approach the volume occupied by the real gaseous molecules, the free volume is obtained by subtracting the molecular volume from the measured volume. Then he introduced the parameter a, which represents an attractive force responsible for the internal pressure, and the parameter b, which represents the volume taken by the molecules. He arrived at... 

For an externally applied pressure, a compression-compression stress field is obtained with a thick-walled sphere. If the sphere wall is thin, a biaxial field is produced. If the pressure is applied internally, a triaxial tension-tension-compression state is generated. A nearly uniform stress field is produced over the entire specimen. The supporting tube is surrounded with a low modulus material to avoid stress concentration and... 

Equation (26) represents the intersection of two surfaces in p(P, T) space. The intersection of two surfaces is a curve in the three-dimensional space. The projection of this curve on the PT plane is given by P(T). Because P is a function of T, at equilibrium between two phases, the system has been reduced to one degree of freedom by the requirement of Eq. (26). If one of the phases is a gas, P(T) is the vapor pressure curve of the condensed phase. If both phases are condensed, P is the externally applied pressure. Alternatively, we could consider T(P), which gives the temperature at which two phases are at equilibrium as a function of pressure. 

To stop the osmosis occurring, the pressure P, in Figure 8.3, can be applied to the left-hand side. This pressure will be equal to the osmotic pressure exerted by the solution in the opposite direction. If the external applied pressure, P, is greater than the osmotic pressure then reverse osmosis occurs and molecules can be forced to pass from the stronger to the weaker solution. In this process, the semi-permeable membrane acts as a molecular filter to remove the solute particles. In some areas of the world this process is used for desalination of sea water, i.e. getting rid of salts from water. It is also used in emergency life raft survival kits to enable drinking water to be made from sea water. 

The driving force behind the fluid flow can be the capillary suction pressure of the support or an external applied pressure. In the former case the process is equivalent to the slip casting process in pljister moulds well-known in ceramics. 

Mercury, which has a contact angle on glass of approximately 140°, is most commonly used as the intrusion fluid. The mercury is forced into the pores of the sample using an externally applied pressure the smallest pores require the highest pressures to effect filling. The Washburn equation, as applied to circular pore openings, is used to relate the applied pressure and the pore size opening. 

Little is known about the mechanism of inflammatory pain. Kelly has discussed the question of pressure on nerves. The accumulation of exudate in an abscess leads to pain which is immediately relieved when the abscess is incised. On the other hand, greater tension exists in tissues affected by angioneurotic oedema although no pain is produced. It is known that externally-applied pressure, insufficient to elicit pain in normal tissue, causes pain in an inflamed area. Randall and Selitto used this hyperalgesic state as the basis for their now widely used test for analgesic drugs. 

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