Representational

Figure 4-4. Representation of vapor-liquid equilibria for a binary system showing moderate positive deviations from Raoult s law.

Figure 5 shows the isothermal data of Edwards (1962) for n-hexane and nitroethane. This system also exhibits positive deviations from Raoult s law however, these deviations are much larger than those shown in Figure 4. At 45°C the mixture shown in Figure 5 is only 15° above its critical solution temperature. Again, representation with the UNIQUAC equation is excellent.

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

The continuous line in Figure 16 shows results from fitting a single tie line in addition to the binary data. Only slight improvement is obtained in prediction of the two-phase region more important, however, prediction of solute distribution is improved. Incorporation of the single ternary tie line into the method of data reduction produces only a small loss of accuracy in the representation of VLE for the two binary systems. 

Figure 4-16. Representation of ternary liquid-liquid equilibria using the UNIQUAC equation is improved by incorporating ternary tie-line data into binary-parameter estimation. Representation of binary VLB shows small loss of accuracy. ---- Binary...

Figure 17 shows results for the acetonitrile-n-heptane-benzene system. Here, however, the two-phase region is somewhat smaller ternary equilibrium calculations using binary data alone considerably overestimate the two-phase region. Upon including a single ternary tie line, satisfactory ternary representation is obtained. Unfortunately, there is some loss of accuracy in the representation of the binary VLB (particularly for the acetonitrile-benzene system where the shift of the aceotrope is evident) but the loss is not severe. 

Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. 

For multicomponent VLE, the method used for obtaining binary parameters is of some, but not crucial, importance. Provided that the binary parameters give good representation of the binary data, good multicomponent results are usually obtained... 

Many well-known models can predict ternary LLE for type-II systems, using parameters estimated from binary data alone. Unfortunately, similar predictions for type-I LLE systems are not nearly as good. In most cases, representation of type-I systems requires that some ternary information be used in determining optimum binary parameter. 

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

At low pressures, it is often permissible to neglect nonidealities of the vapor phase. If these nonidealities are not negligible, they can have the effect of introducing a nonrandom trend into the plotted residuals similar to that introduced by systematic error. Experience here has shown that application of vapor-phase corrections for nonidealities gives a better representation of the data by the model, oven when these corrections... 

The subsequent representations are probably reliable within the range of data used (always less broad than 200° to 600°K), but they are only approximations outside that range. The functions are, however, always monotonic in temperature, to provide appropriate corrections when iterative programs choose temperature excursions outside the range of data. 

Waste from steam systems. If steam is used as a hot utility, then inefficiencies in the steam system itself cause utility waste. Figure 10.9 shows a schematic representation of a steam system. Raw water from a river or other source is fed to the steam system. This is... 

Let us now consider a few examples for the use of this simple representation. A grand composite curve is shown in Fig. 14.2. The distillation column reboiler and condenser duties are shown separately and are matched against it. Neither of the distillation columns in Fig. 14.2 fits. The column in Fig. 14.2a is clearly across the pinch. The distillation column in Fig. 14.26 does not fit, despite the fact that both reboiler and condenser temperatures are above the pinch. Strictly speaking, it is not appropriately placed, and yet some energy can be saved. By contrast, the distillation shown in Fig. 14.3a fits. The reboiler duty can be supplied by the hot utility. The condenser duty must be integrated with the rest of the process. Another example is shown in Fig. 14.36. This distillation also fits. The reboiler duty must be supplied by integration with the process. Part of the condenser duty must be integrated, but the remainder of the condenser duty can be rejected to the cold utility. 

Figure 15.1 The representation of evaporators in shifted temperatures. (Repnnted from Smith, R., and Jams, P. S., The Optimal Design of Integrated Evaporation Systems, Heat Recovery Systems and CHP, 10 341, 19, with permission from Elsevier Science Ltd.)...

Structurally benzene is the simplest stable compound having aromatic character, but a satisfactory graphical representation of its formula proved to be a perplexing problem for chemists. Kekule is usually credited with description of two resonating structures which. 

In a mass spectrometer, the molecules, in the gaseous state, are ionized and fragmented. The fragments are detected as a function of their mass-to-charge ratio, m/e. The graphical representation of the ion intensity as a function of m/e makes up the mass spectrogram as illustrated In Figure 3.1. 

The constants k- enable the improved representation of binary equilibria and should be carefully determined starting from experimental results. The API Technical Data Book has published the values of constants k j for a number of binary systems. The use of these binary interaction coefficients is necessary for obtaining accurate calculation results for mixtures containing light components such as ... 

The range of uncertainty in the UR may be too large to commit to a particular development plan, and field appraisal may be required to reduce the uncertainty and allow a more suitable development plan to be formed. Unless the range of uncertainty is quantified using statistical techniques and representations, the need for appraisal cannot be determined. Statistical methods are used to express ranges of values of STOMP, GIIP, UR, and reserves. 

An alternative and commonly used representation of the range of reserves is the proven, proven plus probable, and proven plus probable plus possible definition. The exact cumulative probability which these definitions correspond to on the expectation curve... 

The most informative method of expressing uncertainty in HCIIP or ultimate recovery (UR) is by use of the expectation curve, as introduced in Section 6.2. The high (H) medium (M) and low (L) values can be read from the expectation curve. A mathematical representation of the uncertainty n a parameter (e.g. STOMP) can be defined as... 

Reservoir simulation is a technique in which a computer-based mathematical representation of the reservoir is constructed and then used to predict its dynamic behaviour. The reservoir is gridded up into a number of grid blocks. The reservoir rock properties (porosity, saturation, and permeability), and the fluid properties (viscosity and the PVT properties) are specified for each grid block. 

The bar chart below is a representation of the network shown in Figure 12.4. In addition the chart has been used to display the resource loading. 

Fig.5 Example of histogram presentatioa The lower picture is a schematic representation of the monitored component with Uie position of the transducers (1-4). The higher window is the total AE counts vs linear location representation of the located AE sources...

Fig.6 Example of trend vs time parameter representation the view is Epical of a paper recorder, with time in Y coordinate decreasing from the top to the bottom. The eight top buttons represent the different choice of plant parameters...

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