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.     

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.     

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.     

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.     

Monotone iteration methods with adaptive collocation for solving coupled systems of nonlinear boundary value problems

Monotone iteration procedures are developed for the solution of a class of coupled and highly nonlinear boundary value problems. These procedures are of guaranteed convergence and can be used for a numerical investigation of the multiplicity structure of steady-state solutions. The monotone iterations provide also a basis for an automatic scheme for adapting and refining the collocation point grid between iterations. The local error of approximation is controlled very accurately and problems exhibiting extremely steep multiple concentration profiles of significantly different scales can be treated.     

Finite element collocation solution of tubular and packed-bed reactor two point boundary value problems

Nonlinear two point boundary value problems (TPBVP's) are frequently encountered in chemical engineering. TPBVP's describing tubular and packed-bed reactors are important and have received much attention in the literature. For certain ranges of parameter values of the reactors, the TPBVP's become stiff and pose problems in numerical solution. These problems have been efficiently solved by the finite element collocation method using Hermite and B-spline functions in conjunction with quasilinearization. Numerical results have been presented and discussed. The problems have been solved for a wide range of parameter values without encountering any numerical difficulty.     

The use of a computer algebra system in the analysis of chemical engineering problems

Computer algebra systems are powerful processors designed to treat formal mathematical expressions. Routine features include many operations that occur frequently in the analysis of engineering problems, and provide the analyst with a powerful and reliable tool for manipulating algebraic, differential and integral equations. The power and general utility of one particular computer algebra system is first demonstrated on two elementary examples—the first involving the solution of an ordinary differential equation by weighted residual techniques, and the second, a perturbation solution of Duffing's equation. Ideas stemming from these examples are then applied to a current research problem involving the determination of the amplitude and frequency of self-sustained density wave oscillations in a once-through evaporator. All examples show the power of this system in handling large amounts of tedious algebra with relative ease, thus eliminating the potential for manual error.     

Conjugated Graetz problems—I: General formalism and a class of solid-fluid problems

Conjugated Graetz problems arise where two (or more) phases, with at least one phase in fully developed flow, exchange energy or mass across an intervening surface. In these problems the temperature or concentration fields are coupled through the conjugating conditions which express the continuity of fluxes and the rate of transfer. A general formalism is presented first for the analysis of these problems, employing a matrix differential operator with respect to the radial variable and following the decomposition technique of [1–3]. The aforementioned operator is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues and a complete set of eigenvectors. A class of solid-fluid problems, involving the removal of heat from a heated solid cylinder by a surrounding annular-flow fluid, is then discussed in detail; analytical solutions are obtained by expansion in terms of the above eigenvectors. These solid-fluid problems may be viewed as extensions of the extended, one-phase, Graetz problem with prescribed wall flux [2].     

Conjugated Graetz problems—II: Fluid-fluid problems

Conjugated Graetz problems [1], involving two co- or counter-currently flowing phases are discussed following the general formalism of [1]. Although the difficulty, which arises from the changing sign of the velocity profile over the total cross-section of the two-fluid conduit of the counter-current problems, may be readily handled by the general formalism, the inclusion of axial heat conduction in the analysis presents special difficulties in obtaining analytical and/or computationally efficient solutions. The present analysis shows that analytical and computationally efficient solutions may be obtained only for these problems where the temperature profile at the entrance of the heating section is known for at least one of the fluids. The solution of a class of problems with long heating sections is obtained untilizing the general formalism together with the Gram-Schmidt orthonormalization process in the spirit of [5]. Problems with low Peclet numbers for both fluids and with an adiabatic section preceding the heating section or problems with very short heating sections are also briefly discussed.     

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.     

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.     

Computer-generation of reaction paths and rate equations in the thermal cracking of normal and branched paraffins

Reaction networks for the thermal cracking of hydrocarbons and the associated rate equations are generated by computer using an approach based upon Boolean relation matrices.     

Exact solutions of the Newman,and of the handlos-baron,model equations for countercurrent flow extraction

The objective of this study was to develop numerical algorithms for the simultaneous solution of the mass transfer and hydrodynamic equations for swarms of liquid drops in a countercurrent flow liquidliquid extraction column. Two drop diffusion models were used—the Newman rigid drop, and the Handlos-Baron turbulent-circulating drop. Algorithms were developed which gave accurate and rapid solutions of the model equations. Mass transfer rates in a Pilot Plant Rotating Disc Contactor were predicted for comparison with experimental data and with rates predicted by approximate calculation methods. The latter were found to be 20–30% below the exact solution values.     

Evaluation of Hopf bifurcation points in parabolic equations describing heat and mass transfer in chemical reactors

Direct iteration algorithm for evaluation of Hopf bifurcation points in partial differential equations describing heat and mass transfer in chemical re     

Shifted Legendre function approximation of differential equations; application to crystallization processes

An effective new method solves either the stretched functional ordinary differential equation describing a crystallization process using breakage models, or the partial differential equations for a population balance which represent the transient crystallization process. The population balance density function is first expanded into a series of shifted Legendre polynomial functions. The partial differential equation (or the ordinary differential equation) is first transformed into a series of ordinary differential equations (or of algebraic equations) for the expansion coefficients which are solved by numerical computation. The computational time is greatly reduced through a recursive algorithm for the integration of the triple product of the shifted Legendre polynomial functions. Illustrative examples are given and the results are compared with data available in the literature. The proposed method is powerful, accurate and more precise than previously documented ones.     

Prediction of the solubility of hydrogen in hydrocarbon solvents through cubic equations of state

Soave—Redlich—Kwong and the Peng—Robinson equations of state have been applied to the calculation of hydrogen solubility in hydrocarbon solvents. The interaction parameter can be obtained from experimental data: vapour liquid equilibria or Henry's constants. It has been shown that the so obtained value can be correlated as a simple function of temperature only, regardless of the solvent nature, and a predictive procedure has been developed which results in a rather good calculation of hydrogen-solvents vapour liquid equilibria.     

On efficient solution of large-scale Newton-Raphson-based flowsheeting problems in limited core

A strategy is presented for solving in limited core the very large sparse linear equation systems that arise in the Newton-Raphson approach to large-scale flowsheeting problems. Such systems may be too large to handle in core all at once and thus must be decomposed. The strategy employs a bordered-block-triangular decomposition and solves the system by block elimination. This prevents loss of sparsity in off-diagonal blocks and facilitiates the efficient use of core and mass storage. Results indicate the strategy to be more effective in this regard than strategies based on a bordered-block-diagonal decomposition.     

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.