Capacity curves

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

We have seen in previous sections that the two-dimensional Ising model yields a syimnetrical heat capacity curve tliat is divergent, but with no discontinuity, and that the experimental heat capacity at the k-transition of helium is finite without a discontinuity. Thus, according to the Elirenfest-Pippard criterion these transitions might be called third-order. 

Fig. 2. Head—capacity curve, where A represents the minimum allowable capacity B, the best efficiency point (BEP) and C, the maximum allowable...

Probably the most widely used capacity control for the centrifugal compressor is speed control. The capacity curve when used with speed control covers a wide range. While electric variable speed motors offer a continuation to the speed control practice, there are some other alternatives available. Suction throttling has been widely used and offers a r sonable control range for a relatively low cost. 

A speed controller is needed in conjunction with the surge control sy tern. A new head-capacity curve is established for each speed, as shown in Figure 10-14. 

The performance curve can also be shifted to match the process requirements by variable inlet guide vanes. Located at the compressor inlet, these vanes change the direction of the velocity entering the first-stage impeller. By changing the angle at which these vanes direct the flow at the impeller, the shape of the head capacity curve can be changed. 

If you do not select a non-overloading motor, and variations in head and/or flow occur, the motor could overheat and stop operating. Study the pump-capacity curve shape to recognize the possible variations. 

The backpressure is represented by the straight lines labeled minimum, normal and maximum. Only one capacity curve is shown since the increase in capacity resuldng from the lower steam pressure is negligible [4]. 

The sucdon pressure of an ejector is expressed in absolute units. If it is given as inches of vacuum it must be converted to absolute units by using the local or reference barometer. The suction pressure follows the ejector capacity curve, varying with the non-condensable and vapor load to the unit. 

Performance capacity curves are based on standard dry air with 60°F water as the liquid compressajit or seal liquid. The pumps operate on a displacement or volumetric basis therefore, the CFM capacities are about the same for any particular pump for any dry gas mixture. To calculate pounds/hours of air or gas mixture, the appropriate calculation must be made. 

Figure 9-54. Typical pressure drop—capacity curves for countercurrent gas-liquid operation, for Koch Flexipac . Used by permission of Koch Engineering Co., Inc., Bull. KFP-4. Parameter lines on charts are gpm/ft. ...

Figure 13.3 Capacity curves for AS75 condensing unit...

Curves 1, 2 and 3 represent the maximum safe discharge pressure, as the system will operate along the capacity curve as long as the system discharge pressure from the ejector is less than the maximum value of the curve, all for a given suction pressure [4]. The slopes of the curves are a function of the type of ejector, its physical design and relative pressure conditions. Whenever the discharge backpressure exceeds the maximum safe dis-... 

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

Figure 10-16 The Schottky heat capacity curve for a single excited state with energies of 30, 300 and 1000 cm-1 and a degeneracy of one.

The electronic contribution is generally only a relatively small part of the total heat capacity in solids. In a few compounds like PrfOHE with excited electronic states just a few wavenumbers above the ground state, the Schottky anomaly occurs at such a low temperature that other contributions to the total heat capacity are still small, and hence, the Schottky anomaly shows up. Even in compounds like Eu(OH)i where the excited electronic states are only several hundred wavenumbers above the ground state, the Schottky maximum occurs at temperatures where the total heat capacity curve is dominated by the vibrational modes of the solid, and a peak is not apparent in the measured heat capacity. In compounds where the electronic and lattice heat capacity contributions can be separated, calorimetric measurements of the heat capacity can provide a useful check on the accuracy of spectroscopic measurements of electronic energy levels. 

A very small hump in the heat capacity curve at —30° was also found for ammonium fluoride the interpretation of this is uncertain (the structure of this crystal is not the same as that of the other ammonium halides). 

The value of the index n is traditionally taken as 0.6 the well-known six-tenths rule. This value can be used to get a rough estimate of the capital cost if there are not sufficient data available to calculate the index for the particular process. Estrup (1972) gives a critical review of the six-tenths rule. Equation 6.2 is only an approximation, and if sufficient data are available the relationship is best represented on a log-log plot. Garrett (1989) has published capital cost-plant capacity curves for over 250 processes. 

The charge density on the electrode a(m) is mostly found from Eq. (4.2.24) or (4.2.26) or measured directly (see Section 4.4). The differential capacity of the compact layer Cc can be calculated from Eq. (4.3.1) for known values of C and Cd. It follows from experiments that the quantity Cc for surface inactive electrolytes is a function of the potential applied to the electrode, but is not a function of the concentration of the electrolyte. Thus, if the value of Cc is known for a single concentration, it can be used to calculate the total differential capacity C at an arbitrary concentration of the surface-inactive electrolyte and the calculated values can be compared with experiment. This comparison is a test of the validity of the diffuse layer theory. Figure 4.5 provides examples of theoretical and experimental capacity curves for the non-adsorbing electrolyte NaF. Even at a concentration of 0.916 mol dm-3, the Cd value is not sufficient to permit us to set C Cc. 

Electroneutral substances that are less polar than the solvent and also those that exhibit a tendency to interact chemically with the electrode surface, e.g. substances containing sulphur (thiourea, etc.), are adsorbed on the electrode. During adsorption, solvent molecules in the compact layer are replaced by molecules of the adsorbed substance, called surface-active substance (surfactant).t The effect of adsorption on the individual electrocapillary terms can best be expressed in terms of the difference of these quantities for the original (base) electrolyte and for the same electrolyte in the presence of surfactants. Figure 4.7 schematically depicts this dependence for the interfacial tension, surface electrode charge and differential capacity and also the dependence of the surface excess on the potential. It can be seen that, at sufficiently positive or negative potentials, the surfactant is completely desorbed from the electrode. The strong electric field leads to replacement of the less polar particles of the surface-active substance by polar solvent molecules. The desorption potentials are characterized by sharp peaks on the differential capacity curves. 

The appearance of peaks on the differential capacity curves can be derived from this potential dependence in the following manner. The Gibbs-Lippmann equation (see Eq. 4.2.23) gives... 

This contribution involves the positive-ion and electron density profiles of the metal, and the former is often assumed not to change with charging of the interface. In 1983 and 1984, several workers30-32,79 showed how certain features of the interfacial capacity curves should depend on the metal. 

Kornyshev et al.76 proposed several models of the interface, including both orienting solvent dipoles and polarizable metal electrons, to calculate the position of the capacitance hump. Although it had been shown32,79 101 that this was one of the features of the interfacial capacity curves that should depend on the nature of the metal, available calculations did not give the proper position of the hump. The solvent molecules in the surface layer were modeled as charged layers, associated with the protons and the oxygen atoms of molecules oriented either toward or away from the surface. These layers also carried Harrison-type pseudopoten-... 

Figure 6. Potential vs. capacity curves obtainedfrom cycling tests of synthetic graphite flakes in EC-PC/LiCIO4 solutions in different discharge rates. Notice that as the discharge rate decreases - the irreversible capacity decreases accordingly.

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