Myxin


Numerous empirical equations of state have been proposed but the theoretically based virial equation (Mason and Spurling, 1969) is most useful for our purposes. We use this equation for systems which do not contain carboxylic acids.  [c.27]

Mason, E. A., Spurling, T. H., The International Encyclopedia of Phys. Chem. and Chem. Phys., Topic 10, The Fluid State, Vol.  [c.38]

For all calculations reported here, binary parameters from VLE data were obtained using the principle of maximum likelihood as discussed in Chapter 6, Binary parameters for partially miscible pairs were obtained from mutual-solubility data alone.  [c.64]

As indicated in Chapter 6, and discussed in detail by Anderson et al. (1978), optimum parameters, based on the maximum-likelihood principle, are those which minimize the objective function  [c.67]

The method used here is based on a general application of the maximum-likelihood principle. A rigorous discussion is given by Bard (1974) on nonlinear-parameter estimation based on the maximum-likelihood principle. The most important feature of this method is that it attempts properly to account for all measurement errors. A discussion of the background of this method and details of its implementation are given by Anderson et al. (1978).  [c.97]

Maximum-Likelihood Principle  [c.97]

In the maximum-likelihood analysis, it is assumed that all  [c.97]

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors.  [c.98]

When there is significant random error in all the variables, as in this example, the maximum-likelihood method can lead to better parameter estimates than those obtained by other methods. When Barker s method was used to estimate the van Laar parameters for the acetone-methanol system from these data, it was estimated that = 0.960 and A j = 0.633, compared with A 2 0.857 and A2- = 0.681 using the method of maximum likelihood. Barker s method uses only the P-T-x data and assumes that the T and x measurements are error free.  [c.100]

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation.  [c.105]

The maximum-likelihood method is not limited to phase equilibrium data. It is applicable to any type of data for which a model can be postulated and for which there are known random measurement errors in the variables. P-V-T data, enthalpy data, solid-liquid adsorption data, etc., can all be reduced by this method. The advantages indicated here for vapor-liquid equilibrium data apply also to other data.  [c.108]

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand.  [c.108]

However, each of these forms possesses a spurious root and has other characteristics (maxima or minima) that often give rise to convergence problems with common iterative-solution techniques.  [c.113]

However, because the differences are not large, there are some cases where a four-parameter fit was used instead of a five-parameter fit, to avoid maxima or minima with respect to temperature.  [c.141]

Because the precision assigned to the upper and lower extrapolated points is relatively poor, it is possible to obtain a maximum or minimum in the curve, even when fitting all real and extrapolated data from 200 to 600°C. Extrema can occur anywhere, but generally they occur very close to either the lower or the upper end. A check of the sign of the slope at 200°C and 600°C easily indicated the presence of an extremum. When an extremum occurred, a new fit was established to avoid it.  [c.142]

CHU, J.C./VAPOR-LIQUID EQUILIBRIUM DATA, ANN ARBOR, MICHIGAN (1956)  [c.203]

VLE data are correlated by any one of thirteen equations representing the excess Gibbs energy in the liquid phase. These equations contain from two to five adjustable binary parameters these are estimated by a nonlinear regression method based on the maximum-likelihood principle (Anderson et al., 1978).  [c.211]

MINIMUM PRINTED OUTPUT 6 - MAXIMUM PRINTED OUTPUT ITERATION L MIT  [c.229]

MAXIMUM ABSOLUTE VALUE OF PARAMETER CHANGES WHEN LMP EQUAL 2  [c.229]

MINIMUM PRINTED OUTPUT 6 - MAXIMUM PRINTED OUTPUT EXECUTION CODE FOR TYPE OF REGRESSION  [c.240]

MAXIMUM ABSOLUTE CHANGE IS LIMITED TO VALUE  [c.240]

MAXIMUM ABSOLUTE VALUE OF PARAMETER CHANGES WHEN LMP equal 2  [c.240]

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM VLE DATA  [c.278]

The parameters characterizing pure components and their binary interactions are stored in labeled common blocks /PURE/ and /BINARY/ for a maximum of 100 components (see Appendix E).  [c.340]

PARIN loads values of pure component and binary parameters from formatted card images into labeled common blocks /PURE/ and /BINARY/ for a maximum of 100 components.  [c.341]

If a reaction is reversible, there is a maximum conversion that can be achieved, the equilibrium conversion, which is less than 1.0. Fixing the mole ratio of reactants, temperature, and pressure fixes the equilibrium conversion.  [c.25]

Single reversible reactions. The maximum conversion in reversible reactions is limited by the equilibrium conversion, and conditions in the reactor are usually chosen to increase the equilibrium conversion. Le Chatelier s principle dictates the changes required to increase equilibrium conversion  [c.35]

The choice of reactor temperature depends on many factors. Generally, the higher the rate of reaction, the smaller the reactor volume. Practical upper limits are set by safety considerations, materials-of-construction limitations, or maximum operating temperature for the catalyst. Whether the reaction system involves single or multiple reactions, and whether the reactions are reversible, also affects the choice of reactor temperature, as we shall now discuss.  [c.41]

Allow vaporization of one of the components in a reversible reaction in order that removal increases maximum conversion.  [c.45]

Laboratory studies indicate that the reactor yield is a maximum when the concentration of sulfuric acid is maintained at 63 percent.  [c.52]

Figure 3.9a shows the temperature-composition diagram for a maximum-boiling azeotrope that is sensitive to changes in pressure. Again, this can be separated using two columns operating at different pressures, as shown in Fig. 3.96. Feed with, say, rpA = 0.8 is fed to the high-pressure column. This produces relatively pure A in the overheads and an azeotrope with xba = 0.2, Xbb = 0.8 in the bottoms. This azeotrope is then fed to a low-pressure column, which produces relatively pure B in the overhead and an azeotrope with 3 ba = 0.5, BB = 0.5 in the bottoms. This azeotrope is added to the feed to the high-pressure column. Figure 3.9a shows the temperature-composition diagram for a maximum-boiling azeotrope that is sensitive to changes in pressure. Again, this can be separated using two columns operating at different pressures, as shown in Fig. 3.96. Feed with, say, rpA = 0.8 is fed to the high-pressure column. This produces relatively pure A in the overheads and an azeotrope with xba = 0.2, Xbb = 0.8 in the bottoms. This azeotrope is then fed to a low-pressure column, which produces relatively pure B in the overhead and an azeotrope with 3 ba = 0.5, BB = 0.5 in the bottoms. This azeotrope is added to the feed to the high-pressure column.
Figure 3.9 Separation of a maximum boiling azeotrope by pressure change. (From Holland, Gallun, and Lockett, Chemical Engineering, March 23, 1981, 88 185-200 reproduced by permission.) Figure 3.9 Separation of a maximum boiling azeotrope by pressure change. (From Holland, Gallun, and Lockett, Chemical Engineering, March 23, 1981, 88 185-200 reproduced by permission.)
As illustrated here with the UNIQDAC equation, an optimum set of binary parameters can be obtained using simultaneously binary VLE data, binary LLE data, and one (or more) ternary tieline data. The maximum-likelihood principle described in Chapter 6 provides the basis for parameter estimation. The parameters obtained give good representation of ternary data for a wide variety of systems. More important, however, as outlined here, calculations based on a model for the excess Gibbs energy provide a systematic procedure for predicting VLE and LLE for systems containing more than three components.  [c.79]

PRCG cols 21-30 the maximum allowable change in any of the parameters when LMP = 1, default value is 1000. Limiting the change in the parameters prevents totally unreasonable values from being attained in the first several iterations when poor initial estimates are used. A value of PRCG equal to the magnitude of that anticipated for the parameters is usually appropriate.  [c.223]

MAIN PROGRAM AND DRIVER FOR FITTING BINARY VLF DATA USING METHOD EASED ON THE MAXIMUM LIKELIHCOO PRINCIPLE ONLY CONTROL VARIABLES APE READ IN THIS ROUTINE.  [c.229]

MAXIMUM LlKeLIHQOO ESTIMATION OF PARAMETERS FROM VLE OATA CONTROL PARAMETERS WERE SET AS FOLLOWS -  [c.272]

Unwanted byproducts usually cannot be converted back to useful products or raw materials. The reaction to unwanted byproducts creates both raw materials costs due to the raw materials which are wasted in their formation and environmental costs for their disposal. Thus maximum selectivity is wanted for the chosen reactor conversion. The objectives at this stage can be summarized as follows  [c.25]

Maximum selectivity requires a minimum ratio rjr in Eq. (2.17). A high conversion in the reactor tends to decrease Cfeed- Thus  [c.26]


See pages that mention the term Myxin : [c.67]    [c.80]    [c.204]    [c.205]    [c.221]    [c.222]    [c.222]    [c.229]    [c.241]    [c.284]    [c.11]    [c.41]    [c.41]   
Advances in heterocyclic chemistry Vol.85 (2003) -- [ c.85 , c.122 ]