Coefficient Method

Example 8 Calculation of Rate-Based Distillation The separation of 655 lb mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be earned out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed wiU enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate wiU be set at 167 lb mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 wiU be used. Plug flow of vapor and complete mixing of liquid wiU be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calciilate from the computed tray compositions, the component vapor-phase Miirphree-tray efficiencies. 

Practical applications of the influence coefficient method to multiplane, multispeed balancing are presented by Badgley and the author. The separate problem of choosing balancing planes is discussed at some length by Den Hartog, Kellenberger, and Miwa for the (N + 2)-plane method, and by Bishop and Parkinson in the V-plane method. 

The influence coefficient method is simple to apply, and data are now easily obtainable. Consider a rotor with n discs. The method of influence coefficients provides the means for measuring the compliance characteristics of the rotor. 

Tessarzik, J.M., Badgley, R.H., and Anderson, W.J., Flexible Rotor Balancing by the Exact-Point Speed Influence Coefficient Method, Transactions... 

Other important equations of state which can be related to fugacity and activity have been developed by Redlich-Kwong [56] with Chueh [10], which is an improvement over the original Redlich-Kwong, and Palmer s summary of activity coefficient methods [51]. 

An exchanger has heen examined, and the following individual coefficients and resistances determined. What is the overall coefficient of heat transfer referenced to outside coefficients (Methods for determining these film coefficients are given later). 

MHE can be used for substances of high volatility with a small partition coefficient. Method is based on a stepwise gas extraction at equal time intervals. Normal headspace chromatogram is run, a fraction of the gas phase exhausted, and a second headspace chromatogram is run. -The difference in petUc areas provides a measure of the total peak area of the analyte. 

Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

Alternatively, fundamental parameter methods (FPM) may be used to simulate analytical calibrations for homogeneous materials. From a theoretical point of view, there is a wide choice of equivalent fundamental algorithms for converting intensities to concentrations in quantitative XRF analysis. The fundamental parameters approach was originally proposed by Criss and Birks [239]. A number of assumptions underlie the application of theoretical methods, namely that the specimens be thick, flat and homogeneous, and that, for calibration purposes, the concentrations of all the elements in the reference material be known (having been determined by alternative methods). The classical formalism proposed by Criss and Birks [239] is equivalent to the fundamental influence coefficient formalisms (see ref. [232]). In contrast to empirical influence coefficient methods, in which the experimental intensities from reference materials are used to compute the values of the coefficients, the fundamental influence coefficient approach calculates... 

Unless liquid phase activity coefficients have been used, it is best to use the same equation of state for excess enthalpy that was selected for the vapour-liquid equilibria. If liquid-phase activity coefficients have been specified, then a correlation appropriate for the activity coefficient method should be used. 

Tanabe, K. and Saeki, S., Computer Retrieval of Infrared Spectra by a Correlation Coefficient Method , Analytical Chemistry 47, 118-122 (1975). 

Solubility modelling with activity coefficient methods is an under-utilized tool in the pharmaceutical sector. Within the last few years there have been several new developments that have increased the capabilities of these techniques. The NRTL-SAC model is a flexible new addition to the predictive armory and new software that facilitates local fitting of UNIFAC groups for Pharmaceutical molecules offers an interesting alternative. Quantum chemistry approaches like COSMO-RS [25] and COSMO-SAC [26] may allow realistic ab-initio calculations to be performed, although computational requirements are still restrictive in many corporate environments. Solubility modelling has an important role to play in the efficient development and fundamental understanding of pharmaceutical crystallization processes. The application of these methods to industrially relevant problems, and the development of new... 

The Pitzer virial coefficient method, see section 6.2.2. Methods 1 and 2 are equivalent and differ only in the form of the denominator in the Debye-Huckel term. Method 3 requires more parameters for the description of the activity factors. These parameters are not available in many cases. This is generally the case for complex formation reactions. 

One may sometimes have access to the parameters required for the Pitzer approaches, e.g., for some hydrolysis equilibria and for some solubility product data, cf. Baes and Mesmer [3] and Pitzer [4]. In this case, the reviewer should perform a calculation using both the B-G-S and the P-B equations and the full virial coefficient methods and compare the results. 

Activity coefficient methods work fairly well at temperatures well below the critical, at which the liquid phase is largely incompressible, and up to moderate pressures. 

The term (a, /xi) is called the activity coefficient. Methods of measuring the activity and activity coefficient are given in Chapter 5. Tabulations of Gibbs energies of mixing of numerous systems defined in various ways are available in standard reference books, such as Hultgren et al. (1973) and Kubaschewski et al. (1993). 

A somewhat more obviously empirical variation on the multilevel approach is the multi-coefficient method of Truhlar and co-workers. Although many different variations of this approach have now been described, it is simplest to illustrate the concept for the so-called multi-coefficient G3 (MCG3) model (Fast, Sanchez, and Truhlar 1999). In essence, the model assumes a G3-like energy expression, but each term has associated with it a coefficient that is not restricted to be unity, as is the case for G3. Specifically... 

The distribution coefficient method, often called the A -value method, was conceived by Wilcox et al. (1941) and finalized by Carson and Katz (1942). The best methane, ethane, and propane charts are from the latter reference. Updated charts are presented for carbon dioxide (Unruh and Katz, 1949), hydrogen sulfide (Noaker and Katz, 1954), nitrogen (Jhaveri and Robinson, 1965), isobutane (Wu et al., 1976), and n-butane (Poettmann, 1984), as well as for a method that is a function of hydrate structure (Mann et al., 1989). 

The UNIFAC (UNIQUAC functional group activity coefficient) method is an extension of the UNIQUAC (Universal quasi chemical) method, which has been used widely in chemical process engineering to describe partitioning in organic systems as occur in petroleum and chemical processing (Fredenslund et al., 1975,1977). It has been applied less frequently to aqueous systems. It expresses the activity coefficient as the sum of a "combinational" component, which quantifies the nature of the area "seen" by the solute molecule, and a "residual" component, which is deduced from group contributions. Arbuckle (1983,1986), Banerjee (1985), Banerjee and Howard (1988), and Campbell and Luthy (1985) have tested the applicability of the method to water solubility. 

Two methods are recommended for routine use for estimating solubility the correlation with octanol water partition coefficient method and the AQUAFAC group contribution method. When possible, use both approaches and compare the results. Insights can also be obtained if solubility data are available for structurally similar compounds. Both methods are illustrated below. 

There are many measurement techniques for activity coefficients. These include measuring the colligative property (osmotic coefficients) relationship, the junction potentials, the freezing point depression, or deviations from ideal solution theory of only one electrolyte. The osmotic coefficient method presented here can be used to determine activity coefficients of a 1 1 electrolyte in water. A vapor pressure osmometer (i.e., dew point osmometer) measures vapor pressure depression. 

One solution of the forced response xf is the undetermined coefficient method. Assuming the forced response has the same form of source function f(t) but a different coefficient, putting this trial forced response into the differential equation yields the coefficients in the forced response xf. 

Abstract. A recently developed new method to monitor reaction kinetics of intermolecular interaction is reviewed. This method is based on the measurement of the time-dependent diffusion coefficient using the pulsed-laser-induced transient grating technique. Using this method, conformation change, transient association, and transient dissociation on reactions are successfully detected. The principle and some applications to studies on changes in the intermolecular interactions of photosensor proteins (e.g., photoactive yellow protein, phototropins, AppA) in the time domain are described. In particular, unique features of this time-dependent diffusion coefficient method are discussed. 

Some valve manufacturers use the valve-flow coefficient C for valve sizing. This coefficient is defined as the flow rate, in gal/min, through a valve of given size when the pressure loss across the valve is 1 lb/in2. Tabulations such as Table 6.19 and flowcharts such as Fig. 6.9 incorporate this flow coefficient and are somewhat easier to use. Their accuracy equals that of the flow-coefficient method. 

Virial Coefficient Method.—One of the most valuable methods of studying intermolecidar forces consists in direct measurements of the deviations of real gases from the equation of state of the perfect gas. In 1873, van der... 

The virial coefficient method is applicable to arbitrary quantities Q describing the property of a real gas ... 

The alpha coefficients method based on simpler algorithms for calculating inter-element influence coefficients. 

All these methods for attenuation or compensation of matrix effects can be optimised by correcting the residual matrix effects using the alpha coefficient method. In this case, the mathematical equation becomes ... 

The addition of a second metal generally poses matrix problems which are resolved by creating numerous cross standards and using the alpha coefficient method to obtain a regression line. 

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