# Number

The possible number of liquid and vapor mixtures in technological processes is incredibly large, and it is unreasonable to expect that experimental vapor-liquid and liquid-liquid equilibria will ever be available for a significant fraction of this number. Further, obtaining good experimental data requires appreciable experimental skill, experience, and patience. It is, therefore, an economic necessity to consider techniques for calculating phase equilibria for multicomponent mixtures from few experimental data. Such techniques should require only a [c.1]

Presents vapor-pressure data for a large number of substances. [c.12]

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture. [c.26]

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. [c.44]

Sum of squared, weighted residuals °F Number of degrees of freedom [c.46]

Equations (4) and (5) are not limited to binary systems they are applicable to systems containing any number of components. [c.51]

The total number of experimental data points is N. Data points 1 through L and L+1 through M refer to VLB measurements (P, T, [c.68]

Two generally accepted models for the vapor phase were discussed in Chapter 3 and one particular model for the liquid phase (UNIQUAC) was discussed in Chapter 4. Unfortunately, these, and all other presently available models, are only approximate when used to calculate equilibrium properties of dense fluid mixtures. Therefore, any such model must contain a number of adjustable parameters, which can only be obtained from experimental measurements. The predictions of the model may be sensitive to the values selected for model parameters, and the data available may contain significant measurement errors. Thus, it is of major importance that serious consideration be given to the proper treatment of experimental measurements for mixtures to obtain the most appropriate values for parameters in models such as UNIQUAC. [c.96]

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. [c.96]

For each experiment, the true values of the measured variables are related by one or more constraints. Because the number of data points exceeds the number of parameters to be estimated, all constraint equations are not exactly satisfied for all experimental measurements. Exact agreement between theory and experiment is not achieved due to random and systematic errors in the data and to "lack of fit" of the model to the data. Optimum parameters and true values corresponding to the experimental measurements must be found by satisfaction of an appropriate statistical criterion. [c.98]

Large confidence regions are obtained for the parameters because of the random error in the data. For a "correct" model, the regions become vanishingly small as the random error becomes very small or as the number of experimental measurements becomes very large. [c.104]

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. [c.106]

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). [c.107]

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). [c.108]

In this chapter we present efficient calculation procedures for single-stage equilibrium separations subroutines implementing these procedures are given in Appendices F and G. While we recognize the great importance of multistage separations, it must be realized that the efficient computation of such processes requires very careful resolution of the large number of simultaneous equilibrium stages involved in a countercurrent cascade. The dominant consideration in such multistage computation procedures is usually the technique used to achieve this simultaneous solution rather than the equilibrium treatment of the stages themselves. (Goldstein and Stanfield, 1970 Holland, [c.110]

The single-stage separations for which we present computational procedures are the incipient separations (one product phase present in very small amount) represented by bubble and dew-point calculations, vapor-liquid equilibrium separations at fixed pressure under isothermal or adiabatic conditions, and liquid-liquid equilibrium separations at fixed pressure and temperature. These calculations are implemented by FORTRAN IV subroutines designed to minimize the number of vapor and liquid-phase fugacity evaluations necessary to achieve satisfactory solutions. This criterion for efficiency of the algorithms is based on the recognition that, with relatively rigorous thermodynamic methods such as those used here, most of the computation effort in any separation calculation is devoted to evaluation of thermodynamic equilibrium functions. It is important to avoid unnecessary calculations of fugacities or fugacity (activity) coefficients in computer programs used in chemical engineering practice. [c.111]

Here the superscript (r) represents the iteration number and g is the Jacobian derivative matrix whose elements are [c.116]

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. [c.117]

Case Flash Number Type Components Mole Fraction Pressure (bar) Temperature (K) Pressure (bar) Temperature V (K) F Mole Fractions Liquid Vapor No. of Iterations [c.123]

The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives [c.136]

For a large number of liquids, the quantity Z is tabulated [c.139]

Appendix C-6 gives parameters for all the condensable binary systems we have here investigated literature references are also given for experimental data. Parameters given are for each set of data analyzed they often reflect in temperature (or pressure) range, number of data points, and experimental accuracy. Best calculated results are usually obtained when the parameters are obtained from experimental data at conditions of temperature, pressure, and composition close to those where the calculations are performed. However, sometimes, if the experimental data at these conditions are of low quality, better calculated results may be obtained with parameters obtained from good experimental data measured at other conditions. [c.144]

VARIANCE OF PIT = (sum of squared, weighted residuals)/(number of degrees of freedom) [c.144]

A number of correlation options are possible [c.211]

NN cols 21-22 number of experimental VLE data points [c.225]

ND cols 41-42 number of times this set of standard devia- [c.227]

INDICATOR FOR NUMBER OF REFERENCE LINES, COLS 0-60 [c.234]

READ STANDARD DEVIATIONS OF VLE MEASUREMENTS- ND IS THE NUMBER OF STANDARD DEVIATIONS WHICH ARE DUPLICATED (1.UE.NO.LE-NN). [c.235]

NUMBER OF DATA POINTS [c.240]

NUMBER OF INDEPENDENT VARIABLES [c.240]

NN - NUMBER OF EXPERIMENTAL POINTS. [c.251]

NUMBER OF EXPERIMENTAL POINTS [c.252]

Vapor-liquid and liquid-liquid equilibria depend on the nature of the components present, on their concentrations in both phases, and on the temperature and pressure of the system. Because of the large number of variables which determine multi-component equilibria, it is essential to utilize an efficient organizational tool which reduces available experimental data to a small number of theoretically significant functions and parameters these functions and parcimeters may then be called upon to form the building blocks upon which to construct the desired equilibria. Such an organizational tool is provided by thermodynamic analysis and synthesis. First, limited pure-component and binary data are analyzed to yield fundamental thermodynamic quantities. Second, these quantities are reduced to obtain parameters in a molecular model. That model, by synthesis, may be used to calculate the phase behavior of multicomponent liquids and vapors. In this way, it is possible to "scale up" data on binary and pure-component systems to obtain good estimates of the properties of multicomponent mixtures of a large variety of components including water, polar organic solvents such as ketones, alcohols, nitriles, etc., and paraffinic, naphthenic, and aromatic hydrocarbons. [c.2]

The total number of stoichiometric (nonreacted) components is designated by m there are m(m+l)/2 equations of the form given by Equations (33) and (34). [c.134]

G. The next NN cards supply the VLB data. NN equals the number of experimental points. Bach card has one set of data. FORMAT(8F10.2). [c.226]

NUMBER OF PARAMETEPSt DETERMINED Y CHOICE OF LIQUID-PHASE MODEL [c.229]

See pages that mention the term

**Number**:

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Advanced control engineering (2001) -- [ c.0 ]