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ITER

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. [Pg.4]

The algorithm employed in the estimation process linearizes the constraint equations at each iterative step at current estimates of the true values for the variables and parameters. [Pg.99]

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. [Pg.102]

Many different manipulations of these equations have been used to obtain solutions. As discussed by King (1971), many of the older approaches work in terms of V/L, which has the disadvantage of being unbounded and which, in the classical implementation, leads to poorly convergent iterative calculations. A preferable arrangement of this equation system for solution is based on the ratio V/F, which must lie between 0 and 1. If we substitute in Equation (7-1) for L from Equation (7-2) and for y from Equation (7-4), and then divide by F, we obtain... [Pg.113]

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. [Pg.113]

These systems are solved by a step-limited Newton-Raphson iteration, which, because of its second-order convergence characteristic, avoids the problem of "creeping" often encountered with first-order methods (Law and Bailey, 1967) ... [Pg.116]

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

The scalar t is in the range 0 to 1 and provides step-limiting, or damping, when required to obtain a convergent iteration. [Pg.116]

There is justification for allowing t to increase beyond 1, and in many particular applications this may be desirable. Here a more conservative approach is used to reduce the chance of unstable iterations. [Pg.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. [Pg.117]

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. [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]

At low or moderate pressures,a Newton-Raphson iteration is not required, and the bubble and dew-point pressure iteration can be, respectively. [Pg.119]

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. [Pg.120]

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]

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]

Again, Equations (7-8) and (7-9) are then used to calculate new compositions. These compositions, normalized, and the new value for T are utilized in thermodynamic subroutine calls to find equilibrium ratios and enthalpies for use in the next iteration. [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]

The convergence rate depends somewhat on the problem and on the initial estimates used. For mixtures that are not extremely wide-boiling, convergence is usually accomplished in three or four iterations,t even in the presence of relatively strong liquid-phase nonidealities. For example, cases 1 through 4 in Table 1 are typical of relatively close-boiling mixtures the latter three exhibit significant liquid-phase nonidealities. [Pg.122]

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

Flash calculations for these mixtures usually require four to eight iterations. Cases 5 and 6 in Table 1 have feeds of this type, including noncondensable components in Case 6. Within the limits of the thermodynamic framework used here, no case has been encountered where FLASH has required more than 12 iterations for satisfactory convergence. [Pg.124]

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]

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... [Pg.125]

Each iteration requires only one call of the thermodynamic liquid-liquid subroutine LILIK. The inner iteration loop requires no thermodynamic subroutine calls thus is uses extremely little computation effort. [Pg.125]

As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. [Pg.127]

Outside the two-phase region, ELIPS yields a value of 0 for E/F on the R-phase side and 1 for E/F on the E-phase side. Con-, vergence to these values again requires about eight or fewer iterations, except near the plait-point region where convergence is somewhat slower. [Pg.127]

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. [Pg.128]

The special estimates used in BLIPS, which are essentially pure phases of the "solvent components" (98%, with 2% of the other solvent) are chosen to avoid these problems in this iterative procedure. [Pg.128]

From Equation (35), an iteration function can be developed in the form... [Pg.135]

Convergence of this iteration is influenced by initial estimates for the true mole fractions, zThe following rules have been found to lead to rapid convergence in all cases. [Pg.135]

These initial estimates are used in the iteration function. Equation (37), to obtain values of the 2 s that do not change significantly from one iteration to the next. These true mole fractions, with Equation (3-13), yield the desired fugacity... [Pg.135]

The subsequent representations are probably reliable within the range of data used (always less broad than 200° to 600°K), but they are only approximations outside that range. The functions are, however, always monotonic in temperature, to provide appropriate corrections when iterative programs choose temperature excursions outside the range of data. [Pg.138]

Subroutine MULLER. MULLER iteratively solves the equilibrium relations and computes the equilibrium vapor composition when organic acids are present. These compositions are used by subroutine PHIS2 to calculate fugacity coefficients by the chemical theory. [Pg.220]

Second card FORMAT(8F10.2), control variables for the regression. This program uses a Newton-Raphson type iteration which is susceptible to convergence problems with poor initial parameter estimates. Therefore, several features are implemented which help control oscillations, prevent divergence, and determine when convergence has been achieved. These features are controlled by the parameters on this card. The default values are the result of considerable experience and are adequate for the majority of situations. However, convergence may be enhanced in some cases with user supplied values. [Pg.222]


See other pages where ITER is mentioned: [Pg.4]    [Pg.25]    [Pg.100]    [Pg.114]    [Pg.115]    [Pg.116]    [Pg.116]    [Pg.117]    [Pg.118]    [Pg.120]    [Pg.122]    [Pg.125]    [Pg.125]    [Pg.125]    [Pg.135]    [Pg.218]    [Pg.222]   
See also in sourсe #XX -- [ Pg.1194 ]




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A Note on In-Cell Iteration

A-Tocopherol via iterative Claisen rearrangement

Accuracy iterative method

Agile and Iterative Development

Agile iterative development process

Alkylation iterative

Allocation iterative/constructive

An Iteration Method of Unfolding

An Iterative Algorithm

Analyzing iterative incremental scheduling

Applying Gershgorins theorem to study the convergence of iterative linear solvers

Arm-First Convergent Iterative Methodology

Arm-First Divergent Iterative Methodology

B-factor iterative test

Biological rhythms limit cycles or discontinuous iterative behaviour

Born inversion iterative

Calculation of the Activation Energy by Iterative Procedure

Chebyshev filtered subspace iteration

Chebyshev iteration procedure

Chebyshev semi-iterative method

Chemistry iterative approach

Claisen condensations, iterative

Claisen iterative

Command iteration

Comparing Nonlinear Systems to Other Iterative Problems

Constraints generic iteration

Contracted Schrodinger equation iterative solution

Conventional iterative synthesis

Convergence by Dynamic Iteration

Convergence iteration schemes

Core-First Divergent Iterative Methodology

Coupled-cluster theory iterative schemes

Cross-coupling strategy, iterative

DFT Total Energies—An Iterative Optimization Problem

Damped iteration

Data-dependent iterations

Databases iterative multiple sequence method

Deconvolution iterative

Dendrimers iterative preparation

Derivatives iterative averaging method

Design iterative

Design-analysis iterations

Development iterating

Difference scheme explicit iteration

Difference scheme implicit iteration

Direct Inversion in the Iterative Subspace

Direct Inversion in the Iterative Subspace DIIS)

Direct inversion of iterative subspace

Direct inversion of the iterative subspace

Dynamic programming iterative

Electrophilic iterative

Explosion in the Number of Iterations

FINITE ELEMENT ITERATIVE

FINITE ELEMENT ITERATIVE METHOD

Factor iterative target transformation

Factoring Case Iterated Squaring and Doubling (Or A Useful Homomorphism on an Ugly Group)

Feedback iteration plan

Finite difference iterative schemes

Finite-difference methods iteration

Fixed point iteration

Fixed-Point Iteration (Direct Substitution)

Fractals iterative averaging method

Functional iteration

GDIIS iterative subspace

Gauss-Newton iteration

Gauss-Seidel Iteration Method

Gauss-Seidel iteration

Gauss-Seidel point iteration method

Generic iteration

Generic iteration function

Geometry Direct Inversion in the Iterative

Glycals iterative assembly

Gradient method iterative procedure

Hamiltonian second-order iterated

INDEX iteration

INDEX iterative

ITER (International Thermonuclear

ITER Design

ITER Tritium Retention Estimates and Uncertainties

ITER tokamak devices

Image reconstruction iterative

Inner and outer iterations

Inner iteration

International Thermonuclear Experimental Reactor ITER)

Inverse iteration

Inverse iteration method

Inversion of the Iterative Subspace

Iter functions

Iter representations

Iter simulation

Iterated

Iterated

Iterated Dirac Equation

Iterated function

Iterated function systems

Iterated kernels

Iterated map

Iterated permutation

Iterated similarity

Iterated squaring and doubling

Iterated waterfall model

Iteration

Iteration

Iteration filling model

Iteration iterator

Iteration iterator

Iteration limit

Iteration method

Iteration pattern

Iteration penalty method

Iteration penalty method in Hilbert spaces

Iteration planning

Iteration procedure

Iteration solutions

Iteration space

Iteration strategies

Iteration variables

Iteration, direct

Iteration, monotone

Iteration-perturbation method

Iteration/iterative

Iteration/iterative algorithms

Iteration/iterative geminal

Iteration/iterative purification

Iteration/iterative purification procedures

Iteration/iterative self-consistent

Iteration/iterative solutions

Iterations Converge

Iterative

Iterative

Iterative Boltzmann inversion

Iterative Boltzmann inversion methods

Iterative Boltzmann method

Iterative Born inversions of the wavefield

Iterative Chemical Hardness of AIM

Iterative Component-Wise Solution of the Nonlinear Equations

Iterative Convergence Methods

Iterative Extended Huckel

Iterative Fock Matrix Construction

Iterative Methodology with Regeneration of DPE Function

Iterative Methodology with Regeneration of Two or More DPE Functions

Iterative Minimization Technique for Total Energy Calculations

Iterative Problem Solving Strategy

Iterative Search

Iterative Search Subject

Iterative Searching

Iterative Solution of Nonlinear Algebraic Equations

Iterative Suzuki-cross

Iterative Suzuki-cross couplings

Iterative Target Transform Factor Analysis

Iterative Weighted

Iterative Wittig olefination

Iterative algorithm

Iterative alternating direction methods

Iterative analysis

Iterative approach

Iterative assay

Iterative averaging, fractal structures

Iterative binding refinement

Iterative bisection method

Iterative block synthesis

Iterative calculation, example

Iterative calculations

Iterative characteristics

Iterative closest point

Iterative computations

Iterative cycle

Iterative cyclic approaches

Iterative cyclic voltammograms

Iterative deconvolution process

Iterative definition

Iterative design process

Iterative development method

Iterative diagonalization

Iterative evaluations

Iterative extended Hiickel theory

Iterative extended Huckel theory

Iterative formula

Iterative formula spreadsheet

Iterative gravity migration

Iterative hybrid methods

Iterative improvement

Iterative incremental scheduling

Iterative key set factor analysis

Iterative large linear system solution

Iterative learning

Iterative least-squares methods

Iterative linear solvers

Iterative linear solvers Conjugate Gradient

Iterative linear solvers Conjugate Gradient method

Iterative linearized density matrix

Iterative localization

Iterative localization Reliability

Iterative localization accuracy

Iterative localization density functions

Iterative loops

Iterative method and stability

Iterative method direct

Iterative method kinetic data analysis

Iterative method, description

Iterative methods

Iterative methods problems

Iterative methods to solve the linear system

Iterative migration

Iterative migration in the time domain

Iterative modules

Iterative multi-dimensional

Iterative mutual interactions

Iterative natural orbitals

Iterative natural-orbital method

Iterative numerical integration

Iterative oligosaccharide synthesis

Iterative optimization approach

Iterative optimization methods —

Iterative optimization technique

Iterative optimization, tissue

Iterative play

Iterative problems solving

Iterative procedure

Iterative process

Iterative protein crystallographic analysis

Iterative reactions

Iterative rearrangements

Iterative reconstruction

Iterative reconvolution

Iterative refinement

Iterative refinement process, structure

Iterative regression strategy

Iterative renormalization of polymers on a lattice

Iterative resynthesis

Iterative reweighting

Iterative saturated mutagenesis

Iterative saturation mutagenesis

Iterative scheme of the symmetry adapted perturbation theory

Iterative self-assembly

Iterative simulation modules

Iterative solution method

Iterative solution strategy

Iterative solution technique

Iterative solutions

Iterative solutions of the linear inverse problem

Iterative solutions, positive function

Iterative solver

Iterative statement

Iterative strategy

Iterative structural coarse-graining

Iterative subspaces

Iterative synthesis

Iterative synthesis, oligomeric isoprenoids

Iterative synthetic sequence

Iterative tandem catalysis

Iterative target transform factor analysis ITTFA)

Iterative target transformation factor analysi

Iterative technique

Iterative threshold

Iterative update of the Hessian matrix

Iterative updates

Iteratively reweighted least

Iteratively reweighted least squares

Iteratively weighted least squares

Jacobi iteration

Jacobi iterative algorithm

Jacobi point iteration method

Jacobi-Newton iteration

Jacobian changes with iterations

Juvenile hormone via iterative rearrangements

Lactones iterative tandem catalysis

Lanczos iteration

Large linear system solution, with iterative

Large linear system solution, with iterative methods

Least squares iterative

Linear System Solution with Iterative Methods

Linear operator equations and their solution by iterative methods

Linearize Iterate

Linearize and iterate

Lineshape iterative

Loop iteration

Matrix Newton iteration

Maximum Iterations parameter

Method simple iteration

Methods variation-iteration

Methynolide via iterative rearrangements

Micro-iterations

Micro-iterative method

Model iterative simulation

Modified Newton-Raphson iteration

Molecular function iterative methods

Multi-scale modeling, iterative

Multigrid iteration

Multiple iteration

Natural iteration method

Nest iteration method

Nested iterations

Newton iteration

Newton iteration damping

Newton iteration method

Newton-Raphson iteration

Newton-Raphson iteration procedure

Newton-Raphson iteration technique

Newton-Raphson iterative method

Newton-Raphson iterative technique

Newton-type iteration around stationary flame equations

Newton’s iteration method

Non-Newtonian Dynamics-Based Iterations for Molecular Sampling

Non-linear iterative partial least squares

Non-linear iterative partial least squares NIPALS)

Nonlinear Iterative Partial Least Squares

Nonlinear Iterative Partial Least Squares NIPALS)

Nonlinear comparing, other iterative problems

Nonlinear equations Newton-Raphson iteration

Nonlinear iterative

Nonlinear iterative least squares algorithm (NIPALS

Nonlinear iterative partial least

Nonlinear iterative partial least squares NIPALS) algorithm

Nonlinear terms Newton iteration

Number of iterations

Numerical Iterative Methods of Solution

Numerical methods iterative method

Oligosaccharides iterative reactions

Optimal control theory iterative methods

Optimization characteristics iteration

Optimization iterative

Optimization self-consistent iteration

Orbitals, iterative natural spin

Other iterative methods

Outer iteration

Outer iteration convergence

Partial least squares nonlinear iterative algorithm

Peaceman-Rachford iterative method

Point-Iterative Methods

Polarizable continuum model iterative polarization

Position-specific Iterated BLAST

Power method iteration

Protecting-group-free iterative

Protecting-group-free iterative synthesis

Reconstruction, image iterative method

Regression, parameter estimation iterations

Regular iterative algorithms

Reich Stivala iterative method

Residual function Newton-Raphson iteration

SCF Iteration

Sampling Iterations

Screening iterations

Secant iterative method

Self consistent field iteration

Self-consistent field method iterative minimization

Self-consistent iteration

Sequence-controlled polymers iterative synthesis

Sequential iterative procedure

Short-iterative Lanczos method

Short-time iterative Lanczos

Simple iteration

Simulation iterative

Size extensivity iterative corrections

Solution , algebraic iterative

Solving nonlinear simultaneous equations in a process model iterative method

Sparse iterative methods

Squalene via iterative rearrangements

Stepwise iterative methodology

Stiff equations Newton iteration

Strategy for solving flow networks using iterative methods

Structure-based drug design iterative cycles

Subject iterative

Successive overrelaxation iterative method

Successive substitution iterative method

Synthetic biology iterative cycle

The Agile and Iterative Development Process

The Iteration Method

Two-layer iteration schemes

Vitamin via iterative Claisen rearrangement

Wall procedure iteration

Wegstein iterative method

Zero-iteration technique

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