Gear's procedure for the simultaneous solution of differential and algebraic equations with application to unsteady state distillation problems

The dynamic equations modeling a sieve plate at unsteady state are developed. Gear's procedure for the simultaneous solution of systems of stiff differential and algebraic equations is presented and demonstrated for the solution of unsteady state distillation problems. It is shown that the basic stage model can be modified by the addition of one variable and one equation such that Gear's procedures are readily applied. The proposed model and solution procedure is contrasted to recently published procedures. Numerical results are given for the solution of a problem involving an extractive distillation column at unsteady state.     

Sensitivity analysis of systems of differential and algebraic equations

Formulae are derived for parametric sensitivity analysis of mathematical mo dels consisting of sets of differential and algebraic equations. Such equations often arise in dynamic modeling of equilibrium stage processes, and in solution of partial differential equations via the numerical method of lines. These formulae can be used to efficiently produce the model sensitivity coefficients, simultaneously with the solution of the model.     

Development and comparison of a generalized semi-implicit Runge-Kutta method with Gear's method for systems of coupled differential and algebraic equations

A generalized form of the semi-implicit Runge-Kutta method proposed by Michelsen[1, 2] is developed and its performance is compared with Michelsen's method and Gear's method[3, 4] in the solution of a dynamic model for an absorber. The necessity for the definition of new variables in the application of Michelsen's method in order to place the set of differential equations in state variable form [equations of the form ? = f(y)] is removed by the generalized semi-implicit Runge-Kutta method.     

Simulation of batch crystallization

The closed mathematical model of a well-mixed batch crystallizer has been presented. This model takes into consideration crystals growth rate fluctuations producing an increased spread of crystal sizes. The numerical methods for solving partial integro-differential system of this model equations are proposed for the cases of when growth rate is expressible as product of supersaturation and size functions.Simultaneous population and mass balance equations have been calculated together with power law kinetics of nucleation and growth to determine crystal-size distribution functions (CSD), variation coefficients (CV), produced crystal sizes and numbers. The case of size-dependent crystal growth determined by sequential processes of diffusion and surface reaction is also investigated.The computer results are presented in the dimensionless form as functions on both diffusion and fluctuation Peclet numbers and dimensionless cooling rate. In particular, it has been established that the CV of weight CSD increases approximately twice due to fluctuations and becomes close to 40%; the CV also increases owing to the decrease of cooling rate and approaches 28%. Some formulae are suggested for predicting the kinetics of crystallization (or precipitation) and determining the kinetic parameters of crystals growth and nucleation.     

The application of a modified damped newton method for the simulation of vapor-liquid stagewise processes

In this paper we describe the application of a modified damped Newton method for simulating the steady-state operation of vapor-liquid stagewise processes. The modifications introduced in Newton's method are such that a monotonic convergence toward the solution was always achieved. In addition to this, we analyzed the influence of the various parameters within the method upon its performance; the results indicate that the method is very robust mainly due to its convergence properties. Finally, we compared the method presented here with several others previously reported in the literature, observing that the present method compares favorably.     

A modification of Powell's dogleg method for solving systems of nonlinear equations

We present in this paper an algorithm for solving nonlinear equation systems that is a modification of Powell's dogleg method. The modifications are designed to make the technique more efficient and reliable and to reduce storage requirements. The performance of the new algorithm on a set of standard test problems demonstrates its effectiveness.     

Some characteristics of chemical crystallization in water purification process

Technological scheme for water purification using coprecipitation was analysed. Crystallization of BaCO₃ under the conditions of rapid mixing of reagent solution was investigated with special emphasis on: size distribution of crystals and aggregates by states in time, intensities of nucleation and growth of crystals, mean frequency of molecule binding to the surface during growth, etc. This analysis and results show that the above characteristics must be taken into account in optimization of the technological scheme.     

A note on approximation techniques used for process optimization

The need for increasingly complex models for process simulation has led several researchers to develop more efficient steady-state simulation strategies. In a common approach, rigorous physical property calculations are replaced by simplified ones which approximate locally the behavior of the rigorous ones. Periodically the rigorous models are executed to check if the simplified ones are still accurate enough, and, if not, parameters for the simplified ones are adjusted so both the rigorous and simplified models again predict the same values locally. Convergence is assumed when no adjustments are needed for these parameters and the simplified model is converged.This approach works well for phase equilibrium and even general process simulation problems. Some researchers have extended this approach to optimization problems as well. In this note we discuss problems which arise when adapting this approach for process optimization problems. We also present three examples where the use of simplified models can lead to detection of false optima or convergence failures.     

A simple approximation of the error function

A very simple approximation formula of the error function, with sufficient accuracy for engineering calculations, is proposed in this investigation. The presented form is compared with some of the less sophisticated approximations available in the literature. Aspects such as mnemonic form, computation time, accuracy and ease of inversion are considered.     

A modification of broyden's method for the solution of sparse systems—with application to distillation problems described by nonideal thermodynamic functions

An algorithm that permits Broyden's method to be used for the solution of large systems of algebraic equations with sparse Jacobians is presented. The new procedure is compared to Schubert's modification of Broyden's method and to the Newton-Raphson method by solving an extractive distillation problem. It is demonstrated that the new procedure is competitive with Schubert's method when it is necessary to evaluate Jacobian matrices numerically.     

Application of the method of weighted residuals to mixed (split) boundary value problems. Triple series equations

The method of weighted residuals is applied as a means of providing approximate solutions to mixed (split) boundary value problems which are described by triple-series relations. This approach relies upon using the triple series kernel as the basis function and results in a linear system of equations with the series coefficients as the unknowns. Any available method such as LU decomposition or Gaussian elimination is then used to solve this system. The approach developed in this study is illustrated by numerical examples for diffusion and reaction which are solved using a special software package which implements the proposed algorithms.     

Simulation of packed-bed separation processes using orthogonal collocation

The two-point boundary value problem resulting from the heat and material balance equations of a packed separation column are solved using polynomial approximation techniques. The model equations are based on the two-film theory of mass transfer. The resulting partial differential equations are first reduced to ordinary differential equations and then integrated using semi-implicit Runge-Kutta method of integration. Application of orthogonal collocation simplifies the solution of the two-point boundary value problem. For the examples studied, the algorithm is found to converge rapidly with respect to the number of collocation points used in the polynomial approximation.     

Heat integration of distillation columns into overall processes

There are many well-known schemes for better energy efficiency in distillation. Examples are thermal coupling, multiple effect, heat pumping, etc. Usually these schemes are discussed for individual columns in isolation, independently from the overall process they are a part of.This paper puts the design of individual distillation columns into context with the heat integration for the overall process. An insight is discussed wGenerally, the paper defines good integration as a column not crossing the heat recovery pinch of the process and either the reboiler or the condenser     

A Newton type linearization method for solution of nonlinear equations

Convergence properties of a new iterative method for solution of non-linear equations are investigated. It is shown that for a system of equations which contain mixed linear equations and homogeneous functions of degree n, the convergence of this method is equivalent to the convergence of Newton's method. In contrast to the latter, this new method does not require evaluation of the vector of function values at every iteration.     

Methane-based equations of state for a corresponding states reference substance

Two equations of state have been developed, one valid over the reduced temperature range 0.2–26 and the other over 0.35–26. Both are intended for use as reference equations of state in the calculation of thermodynamic properties via the principle of corresponding states. The equations are essentially equations of state for methane in that they reproduce the experimentally measured properties of the fluid phase over the whole region for which they exist (reduced temperatures of 0.47 to 3.3) but the extension to higher temperatures was made by utilizing experimental measurements made on nitrogen and hydrogen. An empirical scheme was used for temperatures below 0.47.     

New criteria for application of the well-mixed model to gas-liquid mass transfer studies

The well-mixed model presents several advantages for the evaluation of interphase mass transfer rates in gas-liquid contactors: it simplifies the mathematical analysis and eliminates the need for measuring numerous mixing parameters. Although the well-mixed model corresponds to a theoretical concept, it is shown in this paper that, under certain conditions, it can be safely applied to the liquid phase of contactors for the purpose of determining the overall volumetric mass transfer coefficient K_La_L. Based on a computer simulation of absorption and mixing processes, simple criteria were developed that specify the conditions for which the well-mixed model is applicable in practice, for either a semi-batch gas-liquid contactor or a continuous flow absorber at steady-state. The dimensionless group K_La_Lτ_c, where τ_c is the average circulation time in the liquid phase of the contactor, was found to be a key parameter in establishing the validity of the well-mixed model.     

Periodical operation of chemical processes and evaluation of conversion improvements

The periodical operation of chemical processes and the possibility of conversion improvements have been studied. Especially, methods are discussed which allow to evaluate the result of dynamic control of inlet-concentrations. A tank reaktor with CSTR behaviours has been regarded. The isothermal balance of the reaction encloses dynamic steps at the surface of the catalyst. The derivated evaluations have been checked by numerical simulations and confirmed in many cases.     

Incipient fault diagnosis of nonlinear processes with multiple causes of faults

We describe how to detect and diagnose the causes of faults for stochastic processes. Causes of the faults are presumed to be deterioration of variousWe illustrate the proposed strategy for fault detection and diagnosis for a chemical reactor with heat exchange via simulation, and compare the results     

Lateral migration of spherical particles in porous flow channels: application to membrane filtration

Lateral migration of spherical rigid neutrally buoyant particles moving in a laminar flow field in a porous channel is induced by an inertial lift force (tubular-pinch effect) and by a permeation drag force due to convection into the porous walls. The analysis of Cox and Brenner [7], for the particle motion in a nonporous duct is extended to include the effect of the wall porosity. Criteria are established under which the inertial and permeation drag force in the lateral direction can be vectorially added. Particle trajectories and concentrations profiles are calculated for a plane Poiseuille flow with one porous wall. For particles with radius of 1 μm, inertial and permeation drag forces are of comparable size under flow conditions often met in ultra- and hyperfiltration of dilute suspensions. For smaller particles the permeation drag force dominates.