For bubble and dew-point calculations we have, respectively, the objective functions [Pg.118]

In the case of the adiabatic flash, application of a two-dimensional Newton-Raphson iteration to the objective functions represented by Equations (7-13) and (7-14), with Q/F = 0, is used to provide new estimates of a and T simultaneously. The derivatives with respect to a in the Jacobian matrix are found analytically while those with respect to T are found by finite-difference approximation [Pg.121]

DFA Partial derivative of the Rachford-Rice objective function (7-13) with respect to the vapor-feed ratio. [Pg.321]

FIND NORM OF OBJECTIVE FUNCTION AND CHECK FOR DECREASE 260 FV ABS(F) [Pg.325]

Value of the objective function [(7-23) or (7-24)] at T + AT used for finite difference approximation of the derivative. [Pg.327]

F Rachford-Rice objective function for liquid-liquid separa- [Pg.335]

When the main object of the absorption is to remove impurities these columns are often referred to as scrubbers. [Pg.9]

For binary vapor-liquid equilibrium measurements, the parameters sought are those that minimize the objective function [Pg.98]

The equation systems representing equilibrium separation calculations can be considered multidimensional, nonlinear objective functions [Pg.115]

For liquid-liquid systems, the separations are isothermal and the objective function is one-dimensional, consisting of Equation (7-17). However, the composition dependence of the [Pg.117]

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function [Pg.118]

In application of the Newton-Raphson iteration to these objective functions [Equations (7-23) through (7-26)], the near linear nature of the functions makes the use of step-limiting unnecessary. [Pg.119]

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be [Pg.121]

Convergence of the iteration requires the norm of the objective vector 1g to be less than the convergence criterion, e. The initial estimates used, if not provided externally, are, in addition to Equation (7-28) [Pg.122]

Liquid-liquid equilibrium separation calculations are superficially similar to isothermal vapor-liquid flash calculations. They also use the objective function. Equation (7-13), in a step-limited Newton-Raphson iteration for a, which is here E/F. However, because of the very strong dependence of equilibrium ratios on phase compositions, a computation as described for isothermal flash processes can converge very slowly, especially near the plait point. (Sometimes 50 or more iterations are required. ) [Pg.124]

Subroutine FUNDR. This subroutine calculates the required derivatives for REGRES by central difference, using EVAL to calculate the objective functions. [Pg.218]

A step-limited Newton-Raphson iteration, applied to the Rachford-Rice objective function, is used to solve for A, the vapor to feed mole ratio, for an isothermal flash. For an adiabatic flash, an enthalpy balance is included in a two-dimensional Newton-Raphson iteration to yield both A and T. Details are given in Chapter 7. [Pg.319]

FIND EQUILIBRIUM OBJECTIVE FUNCTION F AND UNNORMALIZEO COMPOSITIONS [Pg.324]

FIND ENTHALPY OBJECTIVE FUNCTION G lAOIABATICI 255 CALL ENTH(N lOfKEEfOtX TNyPfHLfER) [Pg.325]

APPLY STEP-LIMITING PROCEDURE TO DECREASE OBJECTIVE FUNCTION 265 KD=l [Pg.325]

Bubble-point temperature or dew-point temperatures are calculated iteratively by applying the Newton-Raphson iteration to the objective functions given by Equations (7-23) or (7-24) respectively. [Pg.326]

Current value of the objective function given by either (7-23) or (7-24) [Pg.327]

Change in extract-feed ratio from one iteration to the next. Partial derivative of Rachford-Rice objective function with respect to extract-feed ratio. [Pg.335]

Given that the objective is to manufacture a certain product, there are often a number of alternative reaction paths to that product. Reaction paths which use the cheapest raw materials and produce the smallest quantities of byproducts are to be preferred. Reaction paths which produce significant quantities of unwanted byproducts should especially be avoided, since they create significant environmental problems. [Pg.16]

Example 2.1 Given that the objective is to manufacture vinyl chloride, there are at least three reaction paths which can be readily exploited. [Pg.16]

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 [Pg.25]

Centrifugal separators make use of the common principle that an object whirled about an axis at a constant radial distance from the point is acted on by a force. Use of centrifugal forces increases the force acting on the particles. Particles that do not settle readily in gravity settlers often can be separated from fluids by centrifugal force. [Pg.71]

More than 7.5 MW could be added from a hot utility to the first interval, but the objective is to find the minimum hot and cold utility. Thus from Fig. 6.186, QHmin = 7.5MW and Qcmm = 10MW. This corresponds with the values obtained from the composite curves in Fig. 6.5a. One further important piece of information can be deduced from the cascade in Fig. 6.186. The point where the heat flow goes to zero at T = 145°C corresponds to the pinch. Thus the actual hot and cold stream pinch temperatures are 150 and 140°C. Again, this agrees with the result from the composite curves in Fig. 6.5a. [Pg.179]

In batch process optimization, one of the principal objectives is to improve equipment utilization through reduction in dead time. This requires both structural and parameter optimization, with many options available. [Pg.252]

In Chap. 2 the objective set was to maximize selectivity for a given conversion. This also will minimize waste generation in reactors for a given conversion. [Pg.276]

Life-cycle analysis, in principle, allows an objective and complete view of the impact of processes and products on the environment. For a manufacturer, life-cycle analysis requires an acceptance of responsibility for the impact of manufacturing in total. This means not just the manufacturers operations and the disposal of waste created by those operations but also those of raw materials suppliers and product users. [Pg.296]

Inertial collectors. In inertial collectors, an object is placed in the path of the gas. An example is shown in Fig. 11.1. While the gas passes around the shutters, particles with sufficiently high inertia impinge on them and are removed from the stream. Only particles in excess of 50/um can reasonably be removed. Like gravity settlers, inertial collectors are widely used as prefilters. [Pg.302]

Overall strategy, 399-404 final design, 403-404 hierarchy, 400-403 objectives, 399-400 Oxidation, 283 [Pg.456]

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