The Deviation Function

The deviation function AW Too) is obtained as a function of r90 for various temperature intervals by calibration of the platinum resistance thermometer, using specified fixed points from Table A2.1. The form of the AW(Too) function is dependent on the temperature range in which the thermometer is being calibrated. For example, in the temperature subrange from 234.3156 to 302.9146 K, the form of the deviation function is 

The coefficients a5 and b5 are obtained by calibrating the thermometer at the triple points of mercury (234.3156 K) and water (273,16 K.) and the melting point of gallium (302.9146 K). 

Calibration (Fixed) points used to determine coefficients in the deviation function 

In summary, to obtain 7% from a platinum resistance thermometer, one selects the range of interest, calibrates the thermometer at the fixed points specified for those ranges, and uses the appropriate function to calculate AW(Tw) to be used in equation (A2.5). Companies are available that perform these calibrations and provide tables of W T )0) versus 790 that can be interpolated to give 7% for a measured W T90). 

At temperatures above the melting point of silver (1234.93 K), radiation thermometry is used. The equation that applies is 

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For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

Each range has a specific deviation function to be used in equation 8. For example, over the range 0—961.78°C, the deviation function is... 

In general, a thermometer is called primary if a theoretical reliable relation exists between a measured quantity (e.g. p in constant volume gas thermometer) and the temperature T. The realization and use of a primary thermometer are extremely difficult tasks reserved to metrological institutes. These difficulties have led to the definition of a practical temperature scale, mainly based on reference fixed points, which mimics, as well as possible, the thermodynamic temperature scale, but is easier to realize and disseminate. The main characteristics of a practical temperature scale are both a good reproducibility and a deviation from the thermodynamic temperature T which can be represented by a smooth function of T. In fact, if the deviation function is not smooth, the use of the practical scale would produce steps in the measured quantities as function of T, using the practical scale. The latter is based on ... 

It is noted that an independent value for D is obtained from every fringe numbered / whose deflection Yj is measured. For nonideal diffusion, Ct actually will show a drift with fringe number, and the value Ct corresponding approximately to the middle of the diffusing boundary is obtained by an extrapolation procedure (27). Akeley and Gosting (27) developed the deviation function defined by... 

Estimate the deviation functions. For n-octane, Tr = (273 + 27)/568.8 = 0.527. From Table 1.4, using linear interpolation and the nonpolar terms,... 

One simple universal equation applies to all substances, requiring no substance-specific parameters. However, for most real states, the ideal-gas equation is inadequate, and real-fluid properties are obtained by adding to the ideal-gas equation the contribution of intermolecular potential in the form of deviation functions, also called residual functions. A major objective of Section 4.2 is to derive the deviation functions from the equation of state of the substance. Because the ideal-gas properties are known, to And the deviation function is as good as finding the state function of a real substance. In this way the ideal-gas equation is used universally in all equation-of-state calculations of thermodynamic functions. 

The component-material balances are formulated in a manner analogous to that shown in Chap. 2 except for the fact that Boston and Sullivan3 stated these balances in terms of the liquid rates /,-, rather than the vapor rates t ,. Temperatures were computed use of a variation of the method wherein a different base component is used for each plate as suggested by Billingsley.2 Partial molar enthalpies were also expressed in terms of the deviation function Q. 

In the definition of ITS-90, interpolation formulas are provided for the calibration of SPRTs. These formulas are rather involved, including reference functions and deviation functions. For temperature above 0°C, the reference function is a 9th-order polynomial with fixed coefficients and the deviation function is a cubic polynomial with four constants, determined by calibration at the triple point of water (0.01°C) and the freezing points of tin (231.928°C), zinc (419.527°C), aluminum (660.323°C), and silver (961.78°C). These equations are complex and usually of interest only to... 

Almost all definitions of molar properties for mixtures lack an unambiguous definition in the sense that they can be related directly to measurable properties. Therefore, it is common practice to compare an actual mixture property with its corresponding value obtained from an arbitrary model, for instance, an equation of state. This approach leads to the introduction of deviation functions. For a general mixture molar property Mm, the deviation function is defined by ... 

As pointed out in the previous section, the calculation of deviation functions requires a choice of an appropriate model. If the model system is chosen to be an ideal gas mixture, which is an obvious choice for fluid mixtures, then the deviation functions are called residual functions. With temperature, volume and composition as independent variables eq 2.33 becomes ... 

Approximate solutions to Eq. 11-12 have been obtained in two forms. The first, given by Lord Rayleigh [13], is that of a series approximation. The derivation is not repeated here, but for the case of a nearly spherical meniscus, that is, r h, expansion around a deviation function led to the equation... 

The second application is to temperature fluctuations in an equilibrium fluid [18]. Using (A3.2.321 and (A3.2.331 the correlation function for temperature deviations is found to be... 

Many of the molecular modelling force fields in use today for molecular systems can be interpreted in terms of a relatively simple four-component picture of the intra- and inter-molecular forces within the system. Energetic penalties are associated with the deviation of bonds and angles away from their reference or equilibrium values, there is a function... 

By combining Equations (8.4) and (8.6) we can see that the partition function for a re system has a contribution due to ideal gas behaviour (the momenta) and a contributii due to the interactions between the particles. Any deviations from ideal gas behaviour a due to interactions within the system as a consequence of these interactions. This enabl us to write the partition function as ... 

We have already found that the probability function governing observation of a single event x from among a continuous random distribution of possible events x having a population mean p and a population standard deviation a is... 

The potential energy of a molecular system in a force field is the sum of individual components of the potential, such as bond, angle, and van der Waals potentials (equation 8). The energies of the individual bonding components (bonds, angles, and dihedrals) are functions of the deviation of a molecule from a hypothetical compound that has bonded interactions at minimum values. 

To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subjec t to the initial condition that = 0 at t = 0, and Cj is constant. If were not initially zero, one would define a deviation variable between and its initial value (c — Cq). Then the transfer function would be developed using this deviation variable. Taking the Laplace transform of both sides of Eq. (8-2) gives ... 

The safe operation of a chemical process requires continuous monitoring of the operation to stabilize the system, prevent deviations, and optimize system performance. This can be accomplished through the use of instrumentation/control systems, and through human intervention. The human element is discussed in Chapter 6. Proper operation requires a close interaction between the operators and the instrumentation/control system. To a large extent, batch operations have simple control systems and are frequently operated in the manual mode. The instrumentation system is the main source of information about the state of the process. Some of the typical functions of the instrumentation/control system are... 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

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