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Solution methods

A shooting method was used to solve the steady-state model. The wall temperature at 2 = 0 is guessed and the boundary conditions of / = 0 and l2=o are applied. The solution is marched forward using the wall equation (7.8.11) and the two-phase equation (7.8.13) until / = 1.0. The one-phase equation (7.8.12) is applied until z = 2g d. This procedure is repeated until the boundary conditions at 2 = Zgnd are met. [Pg.344]

The first derivative of the wall temperature is discontinuous at the points where the pipe wall thickness changes. An energy balance around the point of discontinuity and temperature continuity lead to the following equations  [Pg.344]

Choose a fluid outlet temperature. Use a macroscopic energy balance to determine qh- [Pg.345]

Step the solution forward in z until the thickness changes. [Pg.345]

Restart the solution. (Repeat steps 4, 5, and 6 when changes in the wall thickness occur.) [Pg.345]


Gilson, M. K., Sharp, K. A., Honig, B. H. Calculating the electrostatic potential of molecules in solution Method and error assessment. J. Comp. Chem. 9 (1988) 327-335. [Pg.195]

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. [Pg.199]

The relativistic Schrodinger equation is very difficult to solve because it requires that electrons be described by four component vectors, called spinnors. When this equation is used, numerical solution methods must be chosen. [Pg.262]

The major disadvantage of solid-phase peptide synthesis is the fact that ail the by-products attached to the resin can only be removed at the final stages of synthesis. Another problem is the relatively low local concentration of peptide which can be obtained on the polymer, and this limits the turnover of all other educts. Preparation of large quantities (> 1 g) is therefore difficult. Thirdly, the racemization-safe methods for acid activation, e.g. with azides, are too mild (= slow) for solid-phase synthesis. For these reasons the convenient Menifield procedures are quite generally used for syntheses of small peptides, whereas for larger polypeptides many research groups adhere to classic solution methods and purification after each condensation step (F.M. Finn, 1976). [Pg.237]

Because almost any diacid can be leaddy converted to the acid chloride, this reaction is quite versatile and several variations have been developed. In the interfacial polymerization method the reaction occurs at the boundary of two phases one contains a solution of the acid chloride in a water-immiscible solvent and the other is a solution of the diamine in water with an inorganic base and a surfactant (48). In the solution method, only one phase is present, which contains a solution of the diamine and diacid chloride. An organic base is added as an acceptor for the hydrogen chloride produced in the reaction (49). Following any of these methods of preparation, the polymer is exposed to water and the acid chloride end is converted to a carboxyhc acid end. However, it is very difficult to remove all traces of chloride from the polymer, even with repeated washings with a strong base. [Pg.224]

P. W. Morgan, Condensation Polymers by Interfacial and Solution Methods olm. Wiley Sons, Inc., New York, 1965. [Pg.242]

Several derivatives of cellulose, including cellulose acetate, can be prepared in solution in dimethylacetamide—lithium chloride (65). Reportedly, this combination does not react with the hydroxy groups, thus leaving them free for esterification or etherification reactions. In another homogeneous-solution method, cellulose is treated with dinitrogen tetroxide in DMF to form the soluble cellulose nitrite ester this is then ester-interchanged with acetic anhydride (66). With pyridine as the catalyst, this method yields cellulose acetate with DS < 2.0. [Pg.253]

Fault Tree Solution. Solving the fault tree means obtaining the minimal cut sets. The minimal cut sets are all the combinations of equipment failures that can result in the fault tree TOP event. Computer programs are requked to determine the minimal cut sets for large fault trees (72). The solution method has four steps ... [Pg.84]

This linear quadratic program will have a unique solution if B i) is kept positive definite. Efncient solution methods exist for solving it (Refs. 119 and 123). [Pg.486]

Single-Effect Evaporators The heat requirements of a singleeffect continuous evaporator can be calculated by the usual methods of stoichiometry. If enthalpy data or specific heat and heat-of-solution data are not available, the heat requirement can be estimated as the sum of the heat needed to raise the feed from feed to product temperature and the heat required to evaporate the water. The latent heat of water is taken at the vapor-head pressure instead of at the product temperature in order to compensate partiaUv for any heat of solution. If sufficient vapor-pressure data are available for the solution, methods are available to calculate the true latent heat from the slope of the Diihriugliue [Othmer, Ind. Eng. Chem., 32, 841 (1940)]. [Pg.1145]

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... [Pg.1277]

Using the isotherm to calculate loadings in equilibrium with the feed gives rii = 3.87 mol/kg and ri2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written... [Pg.1523]

Succinyl coenzyme A trisodium salt [108347-97-3] M 933.5. If it should be purified further then it should be dissolved in H2O (0.05g/mL) adjusted to pH 1 with 2M H2SO4 and extracted several times with Et20. Excess Et20 is removed from the aqueous layer by bubbling N2 through it and stored frozen at pH 1. When required the pH should be adjusted to 7 with dilute NaOH and used within 2 weeks (samples should be frozen). Succinyl coenzyme A is estimated by the hydroxamic acid method [J Biol Chem 242 3468 1967]. It is more stable in acidic than in neutral aqueous solutions. [Methods Enzymol 128 435 7956.]... [Pg.568]

Polymers produced by methods as described above have thermal stabilities many times greater than those obtained by the earlier bulk and solution methods of Staudinger. Staudinger had, however, shown that the diacetates of low molecular weight polyoxymethylenes (I) (polyformaldehydes) were more stable than the simple polyoxymethylene glycols (II) (Figure 19.2). [Pg.534]

Alkyl fluorides have been prepared by reaction between elementary fluorine and the paraffins, by the addition of hydrogen fluoride to olefins, by the reaction of alkyl halides with mercurous fluoride, with mercuric fluoride, with silver fluoride, or with potassium fluoride under pressure. The procedure used is based on that of Hoffmann involving interaction at atmospheric pressure of anhydrous potassium fluoride with an alkyl halide in the presence of ethylene glycol as a solvent for the inorganic fluoride a small amount of olefin accompanies the alkyl fluoride produced and is readily removed by treatment with bromine-potassium bromide solution. Methods for the preparation of alkyl monofluorides have been reviewed. ... [Pg.43]

The solution method using the Plate Constitutive Equation is therefore straightforward and very powerful. Generally a computer is needed to handle... [Pg.214]

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. [Pg.47]

The SIMPLE solution method, including compressibility (Hjertager 1982a)... [Pg.367]

Naherung, /. approach approximation. Naherungs-formel, /. approximation formula. -Idsung, /. approximate solution, -method , /, -verfahren, n. method of approximation, -wert, m. approximate value, nahe-stehend, p.a. closely related or connected ... [Pg.311]

The presence of ether linkages in the polymer molecule imparts chain flexibility, lowers glass transition temperature, and enhances solubility while maintaining the desired high temperature characteristics [192]. Recently, polyether imines were prepared by the reaction of different diamines with 4,4 -[l,4-phenylene bis(oxy)] bisbenzaldehyde [184]. The polymers synthesized by the solution method were yellow to white in color and had inherent viscosities up to 0.59 dl/g in concentrated H2SO4. Some of these polyimines can be considered as... [Pg.50]

The direct-solution method of Akers and Wade [1] is among several which attempt to reduce the amount of trial-and-error solutions. This has been accomplished and has proven quite versatile in application. The adaptation outlined modifies the symbols and rearranges some terms for convenient use by the designer [3]. Dew point and bubble point compositions and the plate temperatures can be determined directly. Constant molal overflow is assumed, and relative volatility is held constant over sections of the column. [Pg.87]


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See also in sourсe #XX -- [ Pg.27 , Pg.40 , Pg.49 , Pg.50 , Pg.51 , Pg.52 , Pg.53 , Pg.54 , Pg.55 , Pg.65 , Pg.66 ]

See also in sourсe #XX -- [ Pg.321 ]

See also in sourсe #XX -- [ Pg.877 ]




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APPROXIMATE ANALYTICAL SOLUTIONS FOR MODELS OF THREE-DIMENSIONAL ELECTRODES BY ADOMIANS ECOMPOSITION METHOD Keith Scott and Yan-Ping Sun ntroduction

Acidic solution oxidation number method

Acidic solutions half-reaction method

Additional Methods Liquids and Solutions

Advanced Solution Methods

Algebraic equations linear, matrix method solution

Algebraic equations numerical solution methods

Analysis Using Slurry Solution Method

Analytical closed-form solution methods

Approximate Method for Sufficiently Dilute Solutions

Approximate and Numerical Solution Methods for PDEs

Aqueous Solution-Gel Methods

Basic solutions half-reaction method

Binary distillation solution methods

Bromleys Method for Multicomponent Solutions

Building Blocks of a Numerical Solution Method

Characterisation dilute solution methods

Characterisation of Gum Rubber Using Dilute Solution Methods

Chemical relaxation methods micellar solution studies

Chemical relaxation methods oxide-solution

Chemical solution deposition characterization methods used

Coating methods solution-phase

Comparison of Euler and backward Eulers methods with exact solution

Comparison of Eulers method with exact solution

Continuum-solvation methods solution acidity

Differential equation solution method

Dilute solution method

Dilute solution method properties

Dilute solution method viscosity

Direct methods of structure solution

Direct methods recognizing solutions

Direct solution methods

Dried-down solution method

Error distribution solutions method

Experimental methods solution

Explicit solution method

Extraction methods solution

Finite-difference solution by the explicit method

Flow methods in solution

Fokker-Planck equation solution methods

Formulations and Methods for the Preparation of Solution Acrylic Resins

Free-energy functional method, interface solutions

Full solution methods

Gas flow-solution trap method

General Solution for Two Degrees of Freedom and Relationship to Jacobian Methods

General solution methods

Groups method, analytical solution

Guggenheims Method for Multicomponent Solutions

Homogenous solution methods, functional

Hydrodynamic methods convective-diffusion equation, solution

Ill-posed problems and the methods of their solution

Initial value problem, solutions NUMERICAL COMPUTER METHODS

Instrumental Methods for Analyzing Polymer Solution Interfaces

Integral equations solution, methods

Isopotential solutions method

Iterative solution method

Large linear system solution, with iterative methods

Linear System Solution with Iterative Methods

Linear operator equations and their solution by iterative methods

Linear solution methods

Liquid solutions group contribution methods

Manufacturing methods solution mixing

Mathematical modeling solution method specification

Membrane transport solution methods

Method development sample solutions

Method for Multicomponent Solutions

Method of Isopotential Solutions

Method of Micro-Phase Separation by Blending Polymer Solutions

Method of Solution for Homogeneous Equations

Method of Solution for Inhomogeneous Equations

Method of manufactured solutions

Method of solution

Method ophthalmic solutions

Method, analytical solutions

Methods of the Thirty-six Aqueous Solutions

Mitigation solution evaluation method

Mixed-solution method

Modeling advanced solution methods

Monte Carlo methods criticality problem solution

Multicomponent systems rigorous solution procedures (computer methods)

Nanocomposites solution casting method

Numerical Iterative Methods of Solution

Numerical Methods for Solution of Partial Differential Equations

Numerical Solution Methods (Initial Value Problems)

Numerical Solutions Finite Element Method

Numerical solution methods

Nutrient solution culture methods

ODE solution methods

Oligonucleotide solutions, synthesis, phosphotriester method

Optimal periodic control solution methods

POLYMER-ASSISTED SOLUTION-PHASE METHODS FOR CHEMICAL LIBRARY SYNTHESIS

Particle fabrication solution methods

Particle size reduction solution method

Particular Solution Methods for Forced Equations

Particular solution method

Peptide synthesis classical solution method

Peptides: synthesis solution methods

Polymerization methods solution

Potential Step in an Infinite Solution—Explicit Method

Process Model and the Solution Method

Pyrolysis solution method

Recycling methods contamination problem, solution

Regularized gradient-type methods in the solution of nonlinear inverse problems

Regularized solution in the Backus-Gilbert method

SEDS (Solution Enhanced method

SIMULTANEOUS METHODS WITHOUT SOLUTE RECOGNITION

Sampling methods solution techniques

Separate solution method

Separate solution method, selectivity

Separation Solution Methods

Sequential solution method

Series Solution Methods and Special Functions

Series solution method

Series solution method for harmonic oscillator

Series solution method for orbital angular momentum

Series solution method for radial equation

Silver nitrate solutions Argentometric methods)

Simultaneous solution method

Size methods spinning solution additive

Skill 4.1e-Describe various methods for separation of solutions (e.g., chromatography, distillation)

Soil solution isolation methods

Solid solutions testing method

Solution ( Liquid-Phase) Methods for Peptide Synthesis

Solution - precipitation methods

Solution Based Methods

Solution Methods for Linear Algebraic Systems

Solution Methods for Linear Finite Difference Equations

Solution Methods for Second Order Nonlinear Equations

Solution Spectroscopic Methods

Solution algorithms based on the Gaussian elimination method

Solution blending method

Solution casting method

Solution crystallization method

Solution depletion method

Solution distillation and special methods

Solution method specification

Solution methods and

Solution methods background

Solution methods ligand substitution reactions

Solution phase synthesis methods

Solution-Based Oxidation Method

Solution-Induced Intercalation Method

Solution-based growth method

Solution-evaporation method

Solution-intercalation/mixing method

Solution-state methods

Solution/dispersion method

Solutions contribution method, UNIQUAC

Spectrophotometric Methods as Solutions to Pharmaceutical Analysis of p-Lactam Antibiotics

Spectroturbidimetric titration of polymer solutions as a method for analytical fractionation

Statistical Methods for Studying Solutions

Steady-State Solution Methods

Structure solution, Patterson heavy-atom method

Structure solution, direct method

Summary of Particular Solution Methods

Summary of Solution Methods for ODE

Systems Rigorous Solution Procedures (Computer Methods)

Test methods solution preparation

Test solute method

The Newton-Raphson method applied to solutions

The Solution X-Ray Diffraction Method

The Solution of Stochastic Models with Analytical Methods

The quasi-chemical method of modeling solutions

The regularization method in a linear inverse problem solution

Thymosin by solution method

Titration with alkaline solutions method

Toxicity test methods extraction solution

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