Pressure, internal

9 Internal Pressure The internal pressure, TOp /kt., of water increases as the temperature is raised, /MPa=63+0.352(7 /K contrary to most other liquids as shown in the review by Marcus [58]. This behavior was related to the breakdown of the large hydrogen bonded aggregates in water, and similar effects on the internal pressure of water in the presence of salts were ascribed by Dack [59] to their water structure-making and structure-breaking effects. The experimental values for IM salt solutions at 25°C, accurate to 4 to 8 MPa, were compared with those of certain non-electrolytes (urea, formamide, acetonitrile, dioxane, and piperidine) that established the size (volume) effect of the solutes as  

TABLE S3 The internal Pressure Differences At 25°C of Aqueous 1M (or 1 m) Electrolytes (-values in parentheses corrected for valency, see the text), their Structural Temperature Differences AT= T, and their Osmotic Coefficient Differences cp(0.4m)-(p(0.2m) 

A multitude of additive terms make up the iouic values accordiug to Leyeudekkers [65], only some of which are straight forward the rest depeudiug ou ad-hoc assumed values of their parameters. Her work was discussed iu detail iu the review by Marcus [58]. It is rather questiouable whether this treatmeut provides iudepeudeut estimates of the effects of the ious ou the structure of water. 

The internal pressure n is closely related to the cohesive energy density ecoh. Therefore we pay some attention to this subject. Spencer and Gilmore (1949,1950) showed that the p-v-T behaviour of polymer melts can be represented reasonably well by the following modified Van der Waals equation of state4  

4 A comparison between some empirical equations of state for polymers with regard to their standard deviations was made by Kamal and Levan (1973). 

At atmospheric conditions the internal pressure n is much greater than the external pressure p, so that for the liquid polymer  

In Table 7.14 the results calculated by means of Eq. (7.28) are compared with the experimental data. The figures obtained are of the right order of magnitude. 

Smith (1970) derived an equation of state for liquid polymers based on the hole theory of liquids. For higher temperatures, this equation can be reduced to a form equivalent to that of Eq. (7.26). 

The water structure-making and -breaking effects of salts, as obtainable from the internal pressure of their aqueous solutions, was discussed by Dack (1976). The internal pressure, Pi, is defined by the first equality  

These values were compared with those of certain non-electrolytes (urea, for-mamide, acetonitrile, dioxane, and piperidine) that establish the size effect of the solutes as  

3 Effects of Ions on Water Stiucture and Vice Versa 

Leyendekkers (1983) obtained some additional internal pressure values, estimated according to several models of the solutions, and her Pi values are also included in Table 3.5, from which APi and APi con- values are derived as described above. In a following paper Leyendekkers (1983a) expressed the internal pressure as a sum of 6 terms  

Individual ionic values of the terms in Eq. (3.11) are calculated as follows. Pvoi +Avw = 1.173 (VrTGi/cm mol ) MPa, where Vttg i = 2520(ri + Ag + Aeii) is the Tamman-Tait-Gibson ionic volume (Leyendekkers 1982) with the ionic radius n and the additives specified in nm. The geometrical factor Ag = 0.055 nm is due to the packing and Aei i is an electric deformation term. Pei + P2c = — Ck(Ae i + A2c) 

The second total derivative along the path is positive. This means that if we are moving on a tangent away from the path, the energy will increase. Thus, the isodynamic path is a minimum with respect to a deviation in direction of any tangent touching the path. The total differential of third order is 

For the reasons explained in Sect. 1.11.3, the energy function along the isodynamic path flattens out, as both the entropy and the volume increase. 

The internal pressure is reflected by the dependence of the internal energy now as function of temperature and volume with the volume as n = dU(T, V)/dV. The ordinary pressure is p = -dU(S, V)/dV. We can access the internal pressure from Eq. (4.33)  

There are two ways of approach. The common approach is thinking about a process running at constant temperature dT = 0. Then we have 

The last equality follows from Maxwell s relations. The other approach is to think about an isentropic process with d5 = 0. Then we have 

To find how p varies with T in a closed system kept at constant volume, we set dV equal to zero in Eq. 7.1.6 0 = aVdT — ktVdp, or dp/dT = a/Kj. Since dp/dT under the condition of constant volume is the partial derivative (dp/dT)v, we have the general 

The partial derivative (dU/dV)T applied to a fluid phase in a closed system is called the internal pressure. (Note that U and pV have dimensions of energy therefore, U/V has dimensions of pressure.) 

This equation is sometimes called the thermodynamic equation of state of the fluid. For an ideal-gas phase, we can write p = nRT/V and then 

In Sec. 3.5.1, an ideal gas was defined as a gas (1) that obeys the ideal gas equation, and (2) for which f/ in a closed system depends only on T. Equation 7.2.3, derived from the first part of this definition, expresses the second part. It thus appears that the second part of the definition is redundant, and that we could define an ideal gas simply as a gas obeying the ideal gas equation. This argument is valid only if we assume the ideal-gas temperature is the same as the thermodynamic temperature (Secs. 2.3.5 and 4.3.4) since this assumption is required to derive Eq. 7.2.3. Without this assumption, we can t define an ideal gas solely by pV = nRT, where T is the ideal gas temperature. 

The internal pressure of a liquid at / = 1 bar is t5q)ically much larger than 1 bar (see Prob. 7.6). Equation 7.2.4 shows that, in this situation, the internal pressure is approximately equal to oiTIkt- 

---

Burst load This is the internal pressure the casing will be exposed to during operations... 

The calculation was carried out using the ANSYS F.E.M. code. The pressure vessel was meshed with a 4 nodes shell element. Fig. 18 shows a view of the results of calculation of the sum of principal stresses on the vessel surface represented on the undeformed shape. For the calculation it was assumed an internal pressure equal to 5 bar and the same mechanical characteristics for the test material. 

In a 250 ml. separatory funnel place 25 g. of anhydrous feri.-butyl alcohol (b.p. 82-83°, m.p. 25°) (1) and 85 ml. of concentrated hydrochloric acid (2) and shake the mixture from time to time during 20 minutes. After each shaking, loosen the stopper to relieve any internal pressure. Allow the mixture to stand for a few minutes until the layers have separated sharply draw off and discard the lower acid layer. Wash the halide with 20 ml. of 5 per cent, sodium bicarbonate solution and then with 20 ml. of water. Dry the preparation with 5 g. of anhydrous calcium chloride or anhydrous calcium, sulphate. Decant the dried liquid through a funnel supporting a fluted Alter paper or a small plug of cotton wool into a 100 ml. distilling flask, add 2-3 chips of porous porcelain, and distil. Collect the fraction boiling at 49-51°. The yield of feri.-butyl chloride is 28 g. 

The yield of iso-propylbenzene is influenced considerably by the quality of the anhydrous aluminium chloride employed. It Is recommended that a good grade of technical material be purchase in small bottles containing not more than 100 g. each undue exposure to the atmosphere, which results in some hydrolysis, is thus avoided. Sealed bottles containing the reagent sometimes have a high internal pressure they should be wrapped in a dry cloth and opened with care. 

Alternatively, authors have repeatedly invoked the internal pressure of water as an explanation of the rate enhancements of Diels-Alder reactions in this solvent ". They were probably inspired by the well known large effects of the external pressure " on rates of cycloadditions. However, the internal pressure of water is very low and offers no valid explanation for its effect on the Diels-Alder reaction. The internal pressure is defined as the energy required to bring about an infinitesimal change in the volume of the solvents at constant temperature pi = (r)E / Due to the open and... 

Mechanistic studies have tried to unravel the origin of the special effect of water. Some authors erroneously have held aggregation phenomena responsible for the observed acceleration, whereas others have hinted at effects due to the internal pressure. However, detailed studies have identified two other effects that govern the rate of Diels-Alder reactions in water. 

The use of indium in acpieous solution has been reported by Li and co-workers as a new tool in org nometallic chemistry. Recently Loh reported catalysis of the Mukaiyama-aldol reaction by indium trichloride in aqueous solution". Fie attributed the beneficial effect of water to a eg tion phenomena in connection with the high internal pressure of this solvenf This woric has been severely criticised by... 

Loh most likely mixed up the internal pressure with the cohesive eneigy density. See (a) Dack, M. 

The problem has been discussed in terms of chemical potential by Everett and Haynes, who emphasize that the condition of diffusional equilibrium throughout the adsorbed phase requires that the chemical potential shall be the same at all points within the phase and since, as already noted, the interaction energy varies wtih distance from the wall, the internal pressure must vary in sympathy, so as to enable the chemical potential to remain constant. 

Fig. 1. Solution-type aerosol system in which internal pressure is typically 240 kPa at 21°C. To convert kPa to psi, multiply by 0.145.

Elastic Behavior. In the following discussion of the equations relevant to the design of thick-walled hoUow cylinders, it should be assumed that the material of which the cylinder is made is isotropic and that the cylinder is long and initially free from stress. It may be shown (1,2) that if a cylinder of inner radius, and outer radius, is subjected to a uniform internal pressure, the principal stresses in the radial and tangential directions, and <7, at any radius r, such that > r > are given by... 

Fig. 1. Principal stresses acting on small element of cylinder waU at radius, r, when the cylinder is stressed elastically by internal pressure, (3).

Equations 1 to 3 enable the stresses which exist at any point across the wall thickness of a cylindrical shell to be calculated when the material is stressed elastically by applying an internal pressure. The principal stresses cannot be used to determine how thick a shell must be to withstand a particular pressure until a criterion of elastic failure is defined in terms of some limiting combination of the principal stresses. 

The state of stress in a cylinder subjected to an internal pressure has been shown to be equivalent to a simple shear stress, T, which varies across the wall thickness in accordance with equation 5 together with a superimposed uniform (triaxial) tensile stress (6). 

If it is assumed that uniform tensile stress, like uniform compressive stress (7), has no significant effect on yield, then the yield pressure of a cylinder subjected solely to an internal pressure may be calculated from... 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

Assume pressure, needed to take the elastic—plastic boundary to radius r corresponds to point B (see Fig. 3). Then provided the cylinder unloads elasticady when the internal pressure is removed, ie, unloading path BE is paradel to OA, the residual shear stress distribution is as fodows. 

Fig. 5. Internal pressure expansion curves for cylinders of EN25 (16,17) k = T jr. To convert MPa to psi, multiply by 145.

A more important effect of prestressiag is its effect on the mean stress at the bore of the cylinder when an internal pressure is appHed. It may be seen from Figure 6 that when an initially stress-free cylinder is subjected to an internal pressure, the shear stress at the bore of the cylinder increases from O to A. On the other hand, when a prestressed cylinder of the same dimensions is subjected to the same internal pressure, the shear stress at the bore changes from C to E. Although the range of shear stress is the same ia the two cases (distance OA = CE), the mean shear stress ia the prestressed cylinder, represented by point G, is smaller than that for the initially stress-free cylinder represented by point H. This reduction in the mean shear stress increases the fatigue strength of components subjected to repeated internal pressure. 

The residual shear stress distribution in the assembled cylinders, prior to the appHcation of internal pressure, may be calculated, from pressure P, generated across the interface. The resulting shear stress distribution in the compound cylinder, when subjected to an internal pressure may be calculated from the sum of the residual stress distribution and that which would have been generated elastically in a simple cylinder of the same overall radius ratio as that of the compound cylinder. 

It may be shown (33) that when the inner surface of a cylinder made of components of the same material is subjected to an internal pressure, the bote of each component experiences the same shear stress provided all components have the same diameter ratio. For these optimum conditions,... 

The maximum internal pressure, subject to the avoidanceof reversed yielding, is then given by... 

Fig. 12. Pressure and temperature stresses in a cylinder, k = 2 subjected to a steady temperature gradient of 100°C and an internal pressure of 138 MPa...

Creep Rupture. The results from creep mpture tests on tubes under internal pressure at elevated temperatures (71,72) may be correlated by equation 16, in which is replaced by the tensile creep mpture stress after time t at temperature T. 

---