Pseudo-values

Figure 12-15 is a compressibility chart for natural gas based on pseudo-reduced pressure and temperature. The reduced pressure is the ratio of the absolute operating pressure to the critical pressure, P and the reduced temperature is the ratio of the absolute operating temperature to the critical temperature, T, for a pure gas or vapor. The pseudo value is the reduced value for a mixture calculated as the sum of the mol percentages of the reduced values of the pure constituents. 

At 800°C with excess carbon, Kshift is approximately 1.0. Invariably the oxygen exchange reactions never reached equilibrium. The apparent or pseudo values for K8hift were generally less than 0.2. Ergun and Menster found similar low values when steam conversion was low. 

The alternative to the leave-one-out approach to the JKK is the grouped or blocked JKK (18). Here there are g blocks of size s. The grouped JKK can save time by executing the PM procedure on the g blocks. Here again, one has an estimator of 6, t, which is the estimate of 6 with the block eliminated. Next, a pseudo value is calculated as follows ... 

For simplicity 9 will be assumed to be one dimensional, but what will be discussed below applies to multivariate 9 as well. First, define 9 and 9, as the statistic of interest with and without the ith observation, respectively. Then, define the ith pseudo value for 9 as... 

The main reason for the deficiencies of the present state of the art is, however, the maldistribution of gas and liquid in packed beds. Many studies reveal that there exist large deviations from plug flow of gas and in particular of hquid within the bed. (Hoek et al. 1986 Kammermaier 2008). The degree of maldistribution as well as its effect on mass transfer are unknown and, in turn, not accounted for in existing mass transfer models. Thus the pubhshed data for interfacial area and mass transfer coefficients comprise the maldistribution in an undefined manner. The data are not true but pseudo values which are not predictable within the plug flow model. 

Pseudo-values do not correspond to the real peak spectral velocity or acceleration. For... 

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the pseudo-density method described in Chapter 3, Section 5.1. 

The value for the pseudo-first-order rate constant is determined by solving equation 13.6 for k and making appropriate substitutions thus... 

Over 25 years ago the coking factor of the radiant coil was empirically correlated to operating conditions (48). It has been assumed that the mass transfer of coke precursors from the bulk of the gas to the walls was controlling the rate of deposition (39). Kinetic models (24,49,50) were developed based on the chemical reaction at the wall as a controlling step. Bench-scale data (51—53) appear to indicate that a chemical reaction controls. However, flow regimes of bench-scale reactors are so different from the commercial furnaces that scale-up of bench-scale results caimot be confidently appHed to commercial furnaces. For example. Figure 3 shows the coke deposited on a controlled cylindrical specimen in a continuous stirred tank reactor (CSTR) and the rate of coke deposition. The deposition rate decreases with time and attains a pseudo steady value. Though this is achieved in a matter of rninutes in bench-scale reactors, it takes a few days in a commercial furnace. 

Feed analyses in terms of component concentrations are usually not available for complex hydrocarbon mixtures with a final normal boihng point above about 38°C (100°F) (/i-pentane). One method of haudhug such a feed is to break it down into pseudo components (narrow-boihng fractions) and then estimate the mole fraction and value for each such component. Edmister [2nd. Eng. Chem., 47,1685 (1955)] and Maxwell (Data Book on Hydrocarbons, Van Nostrand, Princeton, N.J., 1958) give charts that are useful for this estimation. Once values are available, the calculation proceeds as described above for multicomponent mixtures. Another approach to complex mixtures is to obtain an American Society for Testing and Materials (ASTM) or true-boihng point (TBP) cui ve for the mixture and then use empirical correlations to con-strucl the atmospheric-pressure eqiiihbrium-flash cui ve (EF 0, which can then be corrected to the desired operating pressure. A discussion of this method and the necessary charts are presented in a later subsection entitled Tetroleum and Complex-Mixture Distillation. ... 

The application of a 50 percent Murphree vapor-phase efficiency on a y-x magram is illustrated in Fig. 13-40. A pseudo-equilibrium cui ve is drawn halfway (on a vertical line) between the operating hnes and the true-equilibrium cui ve. The true-equilibrium cui ve is used for the first stage (the partial reboiler is assumed to be an equilibrium stage), but for 1 other stages the vapor leaving each stage is assumed to approach the equilibrium value only 50 percent of me way Consequently, the steps in Fig. 13-40 represent actual trays. 

Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickeiy and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to ac tual values so as to solve the MESH equations when veiy nonideal hquid solutions were involved. Taylor, Wayburn, and Vickeiy [I. Chem. E. Symp. Sen No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where httle separation occurs, to a higher and more reasonable efficiency. 

The laboratory studies utilized small-scale (1-5-L) reactors. These are satisfactoiy because the reaction rates observed are independent of reac tor size. Several reac tors are operated in parallel on the waste, each at a different BSRT When steady state is reached after several weeks, data on the biomass level (X) in the system and the untreated waste level in the effluent (usually in terms of BOD or COD) are collected. These data can be plotted for equation forms that will yield linear plots on rec tangular coordinates. From the intercepts and the slope or the hnes, it is possible to determine values of the four pseudo constants. Table 25-42 presents some available data from the literature on these pseudo constants. Figure 25-53 illustrates the procedure for their determination from the laboratory studies discussed previously. 

An anaerobic digester is a no-recycle complete mix reactor. Thus, its performance is independent of organic loading but is controlled by hydraulic retention time (HRT). Based on kinetic theoiy and values of the pseudo constants for methane bac teria, a minimum HRT of 3 to 4 days is required. To provide a safety factor and compensate for load variation as indicated earlier, HRT is kept in the range 10 to 30 days. Thickening of feed sludge is used to reduce the tank volume required... 

Distribution of benzodiazepines in system micellar pseudophase - water was investigated in micellar solutions of sodium dodecylsulfate. The protonization constants of benzodiazepines were determined by the UV-spectophotometry. Values of protonization constants increase with increasing of sodium dodecylsulfate concentration. The binding constants of two protolytic forms of benzodiazepines with a micellar pseudo-phase and P, values were evaluated from obtained dependence. 

From this equation it is seen that parameters have been introduced into the QM-MM method, with K, L, M, and corresponding to the pseudo s orbital on the classical atom. These parameters can be optimized to reproduce experimental or high level theoretical data. Field et al. [9] perfonned extensive investigations of the values of these extra parameters and suggested that the parameters K, L, and M (i = 1,. . . , 4) can be set to zero and that should take a value of 5.0. These are generally the values used in most current QM-MM implementations that employ semiempirical methods in the quantum region. 

The silica dispersion showed the smallest retention volume. It should be noted, however, that the authors reported that the silica dispersion required sonicating for 5 hours before the silica was sufficiently dispersed to be used as "pseudo-solute". The retention volume of the silica dispersion gave the value of the kinetic dead volume, /.e., the volume of the moving portion of the mobile phase. It is clear that the difference between the retention volume of sodium nitroprusside and that of the silica dispersion is very small, and so the sodium nitroprusside can be used to measure the kinetic dead volume of a packed column. From such data, the mean kinetic linear velocity and the kinetic capacity ratio can be calculated for use with the Van Deemter equation [12] or the Golay equation [13]. 

The unit of the veloeity eonstant k is see Many reaetions follow first order kineties or pseudo-first order kineties over eertain ranges of experimental eonditions. Examples are the eraeking of butane, many pyrolysis reaetions, the deeomposition of nitrogen pentoxide (NjOj), and the radioaetive disintegration of unstable nuelei. Instead of the veloeity eonstant, a quantity referred to as the half-life iyj is often used. The half-life is the time required for the eoneentration of the reaetant to drop to one-half of its initial value. Substitution of the appropriate numerieal values into Equation 3-33 gives... 

To illustrate the relationship between the microscopic structure and experimentally accessible information, we compute pseudo-experimental solvation-force curves F h)/R [see Eq. (22)] as they would be determined in SEA experiments from computer-simulation data for T z [see Eqs. (93), (94), (97)]. Numerical values indicated by an asterisk are given in the customary dimensionless (i.e., reduced) units (see [33,75,78] for definitions in various model systems). Results are correlated with the microscopic structure of a thin film confined between plane parallel substrates separated by a distance = h. Here the focus is specifically on a simple fluid in which the interaction between a pair of film molecules is governed by the Lennard-Jones (12,6) potential [33,58,59,77,79-84]. A confined simple fluid serves as a suitable model for approximately spherical OMCTS molecules confined... 

It is sometimes useful to recast the equation as the expectation value of a sum of one-electron and pseudo one-electron operators... 

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