Naphtali

King, 1971 Naphtali and Sandholm, 1971 Newman, 1963 and Tomich, 1970). Moreover the choice of appropriate computation procedures for distillation, absorption, and extraction is highly dependent on the system being separated, the conditions of separation, and the specifications to be satisfied (Friday and Smith, 1964 Seppala and Luus, 1972). The thermodynamic methods presented in Chapters 3, 4, and 5, particularly when combined to... 

Such step-limiting is often helpful because the direction of correction provided by the Newton-Raphson procedure, that is, the relative magnitudes of the elements of the vector J G, is very frequently more reliable than the magnitude of the correction (Naphtali, 1964). In application, t is initially set to 1, and remains at this value as long as the Newton-Raphson correotions serve to decrease the norm (magnitude) of G, that is, for... 

Use the Naphtali-Sandholm SC method to compute stage temperatures and interstage vapor and liquid flow rates and compositions for the rehoiled-stripper specifications shown in Fig. 13-53. The specified hottoms rate is equivalent to removing most of the nCs and nCe and some of the nC in the hottoms. 

FIG. 13-53 Specifications for the calculation of a rebelled stripper by the Naphtali-Sandholm method. 

Dynamic Methods for Characterization of Adsorptive Properties of Solid Catalysts L. POLINSKI AND L. NaPHTALI Enhanced Reactivity at Dislocations in Solids... 

This approach has been further developed by other workers notably Wang and Henke (1966) and Naphtali and Sandholm (1971). 

The linearisation method of Naphtali and Sandholm has been used by Fredenslund et al. (1977) for the multicomponent distillation program given in their book. Included in then-book, and coupled to the distillation program, are methods for estimation of the liquid-vapour relationships (activity coefficients) using the UNIFAC method (see Chapter 8, Section 16.3). This makes the program particularly useful for the design of columns for... 

Leo Digestion, the tribe of Judah, Strength Virgo Distillation, the tribe of Naphtali, The Hermit Libra Sublimation, the tribe of Asher, Justice Scorpio Putrefaction, the tribe of Dan, Death Sagittarius Incineration, the tribe of Benjamin, Temperance Capricorn Fermentation, the tribe of Issachar, The Devil Aquarius Dissolution, the tribe of Manasseh, The Star Pisces Multiplication, the tribe of Reuben, The Moon... 

L. Polanski, L. Naphtali, AIChE J., Chemical engineering research and development, American Institute of Chemical Engineers, New York, ISSN 0001-1541, 1969. 

Naphtali (10) solved Equations (7) by obtaining corrections to extents of reaction from ... 

He showed that positive dX give d that reduce Gibbs free energy. This method is analogous to that of steepest descent, a first-order method for minimization of Gibbs free energy. Ma and Ship-man (11) used Naphtali s method to estimate compositions at equilibrium and the Newton-Raphson method to achieve convergence. 

In one approach, T is assumed and Gibbs free energy minimized to give compositions. Then T is adjusted to satisfy the specification e.g., H or S. Naphtali (10) suggests this procedure, but does not present an algorithm for adjusting T. 

Naphtali, L.M., "Computing Complex Chemical Equilibria by Minimizing Free Energy," First Conference on Kinetics, Equilibria and Performance of High Temperature Systems, Butterworth, London, 1959. 

With analytical derivatives, the TAYLOR method should handle all problems. If not, composition derivatives will be added to the matrix solution in the manner of Naphtali and Sandholm (16). 

The Naphtali-Sandholm (42) method. This method chooses the stage temperatures and component vapor and liquid rates from the MESH variables as the independent variables of the Newton-Raphson calcu-... 

The equations are normalised to keep the numerical values in the same order of magnitude. Since Naphtali-Sandholm equations are grouped by stage and for a system with C components, the independent functions for any stage) are... 

The functions and variables are salved together using a large -Jacobian of size N(2C + 1) x N(2C + 1). When originally presented, The Naphtali-Sandholm method used derivatives of IT-values and enthalpies with respect to composition and temperature, but it was not stated whether these are analytical or numerical derivatives. 

Christiansen et al. (54) applied the Naphtali-Sandholm method to natural gas mixtures. They replaced the equilibrium relationships and component vapor rates with the bubble-point equation and total liquid rate to get practically half the number of functions and variables [to iV(C + 2)]. By exclusively using the Soave-Redlich-Kwong equation of state, they were able to use analytical derivatives of revalues and enthalpies with respect to composition and temperature. To improve stability in the calculation, they limited the changes in the independent variables between trials to where each change did not exceed a preset maximum. There is a Naphtali-Sandholm method in the FraChem program of OLI Systems, Florham Park, New Jersey CHEMCAD of Coade Inc, of Houston, Texas PRO/II of Simulation Sciences of Fullerton, California and Distil-R of TECS Software, Houston, Texas. Variations of the Naphtali-Sandholm method are used in other methods such as the homotopy methods (Sec. 4,2.12) and the nonequilibrium methods (Sec. 4.2.13). 

The matrix solution techniques of the block-banded formulations of Naphtali and Sandholm 42) and of Holland (6) are generally simpler than that of the other global Newton methods. Also, the Naphtali-Sandhoha and almost hend methods are better suited for nonideal mixtures than other global Newton methods. 

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