A modified “inside-out” algorithm for simulation of multistage multicomponent separation processes using the UNIFAC group-contribution method

A computational procedure using a modification of Boston and Sullivan's “inside-out” multistage multicomponent separation algorithm (1974) is developed. In order to improve convergence behavior for problems involving mixtures with highly nonideal liquid phases, a two-parameter model is used to describe liquid-phase compositional effects upon the K-factor. The quasi-Newton methods of Mehra et al. (1983) and Nghiem (1983) are applied to solving various sets of solution variables in the proposed algorithm. Activity coefficients are calculated using the UNIQUAC activity-coefficient model (1975) with parameters obtained from the UNIFAC group-contribution method (1975). The computational procedure is applicable to distillation, absorption and reboiled-absorption configurations. The proposed algorithm was implemented in a FORTRAN 77 program and tested on the Honeywell DPS 8/70M computer at the University of Calgary. Inclusion of the liquid-phase model resulted in improved convergence behavior for nonideal systems in which the original “inside-out” method failed to converge.     

An improved algorithm for solving the mass balance equations in multistage separation processes

The reliability and efficiency of any method of solving the equations that describe multicomponent, multistage, counter-current separation processes depend on the accuracy and efficiency of the algorithm that is employed to solve the component mass conservation (CMC) equations. Previous investigators have shown that ordinary Gaussian elimination is very efficient for solving the CMC equations when they are expressed in tridiagonal form. The present work, however, demonstrates that this procedure is subject to excessive round-off error under certain conditions. In such cases, the error can be reduced to insignificant levels by using a newly developed algorithm based on recursion formulas derived specifically to treat the tridiagonal system of CMC equations.     

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.     

A method for steady-state simulation of multistage separation columns involving nonideal systems

A new algorithm is developed for steady-state simulation of multistage separation columns. The algorithm decouples the model equations into two groups. The component-material balance and summation equations are solved simultaneously by the Newton-Raphson method for temperatures and component flow rates in the inner loop. The energy-balance equations are solved in the outer loop to obtain total flow rates. The K-factor is separated into three parts, each representing separately the effect of temperature, composition and apparent volatility. This makes the analytical calculation of the partial derivatives of the K-factors in the Jacobian matrix a simple task. The stability and efficiency of the algorithm is illustrated by solving a variety of problems for distillation, absorber and reboiled absorber columns. The algorithm does not require large storage space.     

The Extent of Separation: Applications to Multistage Systems

Abstract

The fundamental equations for calculating the extent of separation are given for countercurrent multistage systems operated at either zero reflux, partial reflux, or total reflux. With the component distribution coefficients assumed constant, simple expressions for the maximum extent of separation are derived. The calculations clearly show how multistaging and reflux influence component separation.     

LEGENDRE FUNCTION APPROXIMATIONS OF ORDINARY DIFFERENTIAL EQUATION AND APPLICATION TO CONTINUOUS CRYSTALLIZATION PROCESSES

The technique of the shifted Legendre integral transformation associated with its series expansion is applied to solve population balance equations describing the continuous crystallization processes. The ordinary differential equation of population balance is transformed into a series of algebraic equations of the expansion coefficients which can be easily solved. Illustrative examples for realistic cases are given to simulate the systems. Satisfactory computational results are obtained when are compared with those of the exact solutions. The method is straightforward and the computational algorithm is effective.     

LEGENDRE FUNCTION APPROXIMATIONS OF ORDINARY DIFFERENTIAL EQUATION AND APPLICATION TO CONTINUOUS CRYSTALLIZATION PROCESSES

The technique of the shifted Legendre integral transformation associated with its series expansion is applied to solve population balance equations describing the continuous crystallization processes. The ordinary differential equation of population balance is transformed into a series of algebraic equations of the expansion coefficients which can be easily solved. Illustrative examples for realistic cases are given to simulate the systems. Satisfactory computational results are obtained when are compared with those of the exact solutions. The method is straightforward and the computational algorithm is effective.     

A solution to the distributed parameter model of a continuous grinding mill at steady state

A distributed parameter model of an open-circuit size reduction device at steady state is considered. The model is applicable to tumbling or vibratory mills. Material transport is described in terms of a size-dependent axial convective velocity and a size-dependent axial dispersion coefficient. For n size fractions, the model consists of a set of n coupled ordinary differential equations of second order with two-point boundary conditions. An explicit solution to the model equations is derived for the case in which these equations are linear with constant coefficients. This corresponds to the circumstance in which the axial convective velocities, axial dispersion coefficients and the quantities used to describe the size reduction process are independent of the axial position or of the dependence of the material holdup or size distribution on axial position in the device. These assumptions are not too restrictive and the solution appears to be a useful approximation for a broad range of material-mill combinations. Applications to other process unit operations are suggested.     

Mass transport corrected Tafel analysis of voltammetric waves: When can it be applied?

The derivations of mass transport corrected Tafel equations are presented for a heterogeneous electron transfer. It is shown that the equations are valid for an electrode which is uniformly accessible to the electroactive species in solution. It is also shown that the equations are valid for any electrode geometry when the electron transfer is fast and reversible and the chemical species involved have equal diffusion coefficients. The microdisc and tubular flow electrodes are used as examples of non-uniformly accessible electrodes. The case is considered when the electron transfer is fast and reversible, the electrode is non-uniformly accessible and the species involved have unequal diffusion coefficients. Under these conditions it is shown by analytical solution that the Tafel equation is valid at the tubular flow electrode when axial and radial diffusion are insignificant. It is shown by numerical solution that at the same electrode the Tafel equation is not valid when axial and radial diffusion are significant.     

LABORATORY CATALYTIC REACTORS

A basic requirement in the analysis of catalytic reactors is a rate expression for the reaction concerned. In the design of any equipment the equations to be used contain certain basic physical-chemical coefficients whose values must be known. In the case of physical operations, these coefficients can usually be predicted from available correlations and the design of equipment is often carried out from first principles. But there is no method available by which the kinetic rate parameters of a chemical reaction can be predicted; thus experimental determination of these parameters is unavoidable.     

LABORATORY CATALYTIC REACTORS

A basic requirement in the analysis of catalytic reactors is a rate expression for the reaction concerned. In the design of any equipment the equations to be used contain certain basic physical-chemical coefficients whose values must be known. In the case of physical operations, these coefficients can usually be predicted from available correlations and the design of equipment is often carried out from first principles. But there is no method available by which the kinetic rate parameters of a chemical reaction can be predicted; thus experimental determination of these parameters is unavoidable.     

A new class of solution methods for multicomponent,multistage separation processes

A new algorithm was developed for the solution of the equations that describe multicomponent, multistage separation processes operating at steady state. The algorithm is based on the use of newly defined energy and volatility parameters as the primary successive approximation variables. A third parameter was defined for each stage as a unique combination of the liquid and vapor phase rates and the temperature, and the quasi-Newton method of Broyden was employed to iterate on these parameters. The exceptional stability of the new algorithm in very difficult cases, as well as its efficiency in easy cases, are demonstrated using a variety of example problems.     

The dynamical behavior of catalyst pellets in the limiting cases of finely porous and coarsely porous structures

The pair of partial differential equations conventionally used to describe the dynamical behavior of a porous catalyst pellet is appropiate only in the limiting case when all pore diameters are small compared with mean free path lengths in the reaction mixture. It is here shown that, at the opposite limit, when all pores are large compared with mean free path lengths, this pair of equations must be replaced by three simultaneous partial differential equations. Quantitatively significant differences are found between the stability limits for steady states obtained from the two sets of equations.     

NUMERICAL ANALYSIS OF THE EFFECTS OF LOCAL HYDRODYNAMICS ON MASS TRANSFER IN HETEROGENEOUS POROUS MEDIA

Many engineering problems require the estimation of mass transfer coefficients in porous materials. In heterogeneous materials or in cases where mass transfer sites are not spatially uniform, empirical equations for mass transfer coefficients vary widely, and the origin of these differences is not well understood. In this article, we use a stochastic algorithm to model mass transfer from single particles in a two-dimensional heterogeneous packed bed. The computed mass transfer coefficients are used to generate a distribution of local Peclet numbers in the bed. Detailed hydrodynamics are then used to interpret variations in the local Peclet number. The results show clear relationships between pore structure, streamline patterns, and mass transfer rates.     

A tridiagonal matrix model for multistage flash desalination plants

An efficient and reliable technique has been developed to solve the large system of non-linear equations which describe the behaviour of multistage flash (MSF) desalination plants. The new method depends on decomposition of the equations, after linearization, into subsets which are grouped by type rather than by stage, followed by formulation of the enthalpy balance equations into a tridiagonal matrix (TDM) form which is solved by the Thomas algorithm.The computer program developed in this work converged rapidly over a wide range of conditions, showing significant reduction in run times relative to those required by previous stage-by-stage approaches. Both recirculatory and once-through type plants have been simulated for different practical operating situations.The development of the linearized equations is discussed in detail, all physical property correlation are documented and full details of a numerical example given.     

SELECTION OF EMC/ERH ISOTHERM EQUATIONS FOR SHELLED CORN BASED ON FITTING TO AVAILABLE DATA

Five hundred and ninety-one experimental EMC/ERH data points for shelled corn are extracted from twenty publications identified in literature. These data are grouped into adsorption, desorption, average, yellow dent, dent and yellow flint data sets according to varieties and sorption processes, Thirteen out of the twenty publications contain data over a wide range of temperatures, thus producing nineteen data sub-sets suitable for comparing EMC/ERH isotherm equations. Three commonly cited shelled com EMC/ERH equations (Modified-Henderson, Modified-Chung-Pfost and Modified-Oswin equations) are evaluated for their ability to fit these nineteen data sub-sets. The evaluation, based on the residual sum of squares (RSS), standard error of estimate (SEE) and mean relative deviation (MRD), shows that the Modified-Oswin and Modified-Chung-Pfost equations have a better ability to describe the EMC/ERH isotherms of shelled com, with the Modified-Oswin equation slightly better than the Modified-Chung-Pfost equation. Both equations are then used to fit the adsorption, desorption, average, yellow dent, dent and yellow flint data sets and the best fitted coefficients are given. These fitted equations are based on a large data set and therefore provide a sound basis for future work on drying and storage of shelled com.     

Parameter estimation in models with hidden variables : An application to a biotech process

Biological processes are often characterised by significant nonlinearities, noisy measurements and hidden process variables. The dynamic behaviour of such processes can be represented by stochastic differential equations obtained from physical laws. We propose a Bayesian algorithm for parameter estimation in stochastic nonlinear biological processes with unmeasured (or hidden) variables. The proposed algorithm, involves drawing random samples iteratively from a posterior density functions of the parameters and the hidden variables. A Bayesian sampling techniques is used to approximate these posterior density functions. Both Metropolis–Hastings algorithm and Gibbs sampling are used for sample generation. The algorithm is extended to handle multiple data sets and missing observations. The algorithm is applied to an experimental data set collected from an algal bioreactor system. © 2011 Canadian Society for Chemical Engineering     

Simulation of air-cooled heat exchangers

An algorithm is presented for the simulation of air-cooled finned tubes heat exchangers with topological configuration which preclude the use of method based on overall coefficients and mean temperatures differences. It is a generalization of the step by step method arising when the transfer area of each tube is divided into a finite number of elements. The heat transfer rate equations and the energy balances corresponding to all the elements of all tubes constitute a system of linear equations which is solved iteratively.     

Reaction steady states and stationary states in modelling semi-batch enzyme hydrolysis

Dimensionless material balance equations describing an uninhibited enzyme hydrolysis process in a semi-batch reactor (i.e. fed-batch reactor) are formulated; numerical solution of these equations provided concentration profiles of the enzyme-substrate complex by using published kinetic parameters. The unrestricted values obtained are compared with estimates based separately on the reaction steady state and stationary state assumptions. Results are discussed in terms of the enzyme/substrate inventory used and it is found that the reaction steady state is a satisfactory approximation only when this ratio is sufficiently small. The stationary state may be a better approximation at other values, particularly when enzyme is added to substrate or when an empty tank is being filled. Reaction yields from semi-batch and batch operations are compared. Processing takes longer in the semi-batch operations and complete conversions are only practical in this mode when enzyme is added to substrate.