A numerical solution for the drainage of two-dimensional packed beds

A numerical procedure which allows the calculation of the position of the gas-liquid interface during the drainage of a two-dimensional packed bed is described. The calculations are compared with drainage experiments conducted on a laboratory scale packed bed and the agreement between the two results is shown to be excellent.The residual volume of liquid which remains undrained when the gas-liquid interface first reaches the drain point increases with increasing drainage rate and decreasing hydraulic conductivity of the bed.Computational difficulties which can arise at very high and very low drainage rates are discussed and a smoothing procedure which overcomes this problem is described.     

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.     

A numerical study of the Jacobson-Oksman algorithm

The Jacobson-Oksman algorithm is a method for finding the unconstrained minimum of homogeneous functions. The optimal solution is obtained by solving a set of (n + 2) linear equation by a method based on Householder updating. The method has some very attractive features including robustness and rapid convergence, and it does not require accurate one dimensional searches. However, to date, little if any use has been recorded in the literature.We have studied the properties of the algorithm and also compared the method with the Fletcher-Reeves and Davidon-Fletcher-Powell methods. We have found the Jacobson-Oksman method to be superior to the former and comparable to the latter on general nonlinear functions. Therefore we recommend the Jacobson-Oksman algorithm as a competitive alternative to the celebrated class of quasi-Newton methods for unconstrained minimization.     

Reactions of solid particles with nonuinform distribution of solid reactant: The volume reaction model

The effect of non-uniform solid reactant distribution on conversion of solid particles in gas-solid reactions is analyzed based on the volume reaction model. Certain special features of such systems are pointed out. The possibility of ash layer formation in the kinetically controlled regime is discussed. Conditions leading to single or double ash layer formation, both at the center and surface of the particle, in the intermediate regime of diffusion with simultaneous reaction are described. Detailed mathematical equations which are useful for calculation of the conversion-time relationship for particles with non-uniform solid reactant distribution are presented. Comparison is made to reaction of uniform particles and differences in required reaction time for desired conversion are outlined.     

Steady state multiplicity in surface reactions with coverage dependent parameters

Surface reaction rates may not be unique even when gas pressure and composition above an isothermal catalytic surface are constant. In this paper we examine general ways of obtaining necessary and sufficient conditions for multiple reaction rates for mechanisms that include coverage dependent rate parameters, surface heterogeneity or formation of a complex compound between two reactants or between a reactant and a product. It is shown that variations of the reaction activation energy or heat of adsorption with coverage can enlarge regions of rate multiplicity. Several numerical examples are studied in detail.     

An analytical solution for dispersion in packed beds with low bed-to-particle diameter ratios

An analytic solution to the unsteady advective-diffusion equation is obtained which correctly describes the dispersion from an instantaneous point source in a flow field having a velocity distribution generally found in packed beds of low bed-to-particle diameter ratios. The non-uniform but symmetrical velocity profile shows a great influence on the local distribution of concentrations, C(τ,?,ξ), but this effect disappears when only C_m(τ,ξ), known as the breakthrough curve, is of interest.The effective dispersion coefficient (K₂(τ) - (1/Pe²)) approaches a constant of 1/1196 for the particular velocity profile used as τ → ∞; it reduces itself to 1/192 as expected or laminar flow in empty tubes.     

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.     

Suspending solid particles in an agitated tank with axial-type impellers

Experimental investigation of the critical impeller speed in systems solids-liquid agitated by axial-type turbine impellers was drawn to more general relations as regards the impact of two more important variables: angle of blade inclination and volume fraction of solids. Both variables manifest existence of separate hydrodynamic regions which are reflected in the respective shape of the theoretically derived correlation function.     

A generalized corresponding states method for the prediction of the thermal conductivity of liquids and liquid mixtures

We have recently developed an extension of the three parameter corresponding states principle based on the properties of two non-spherical reference fluids for the viscosity of liquids and liquid mixtures. We extend the method here to the thermal conductivity. We have tested the method for a large number of binary liquid mixtures using the two pure components in each case as our reference fluids. Good agreement between predicted and experimental thermal conductivities was obtained using only the data for the pure components. This agreement becomes excellent if a single binary interaction consta independent of temperature and composition, is used in the mixture calculations. If the pure component reference properties are not available, then the may he obtained from the properties of two similar fluids.     

Optimal temperature profiles for heterogeneous catalytic parallel reactions

The problem of determination of the optimal temperature profiles in a fixed-bed catalytic reactor for the catalytic parallel reactions including transport phenomena (external or internal diffusion) has been worked out.Special attention was given to predict the shapes of the optimal temperature profiles. The general considerations are illustrated by some numerical examples. It has been found that for every studied case the optimal temperature profile need not to end by the isotherm T = T_max.     

Solution of models for gassolid noncatalytic reactions by orthogonal collocation on finite elements with moving boundary

A variable size grain model is used to describe gassolid non-catalytic reactions which are accompanied by change in solid structure. A finite element collocation method with a moving boundary is developed to describe gas and solid concentration profiles in the reacting pellet and to compute conversion-time relationships. Such a method is necessary in case of structural changes which lead to complete pore mouth blockage and gives good results until such blockage occurs. A modified shrinking core model is used as an approximation after the time at which complete pore mouth closure occurs. The developed computational algorithms show that the effect of structural changes on time-conversion relationship is significant.     

Analytical solution of the breakthrough curve for rectangular isotherm systems

A general solution including both film and intraparticle diffusion resistances is derived for the breakthrough curve for an adsorption or ion exchange system with irreversible equilibrium. Both fluid and solid resistances are represented by linearized rate expressions but the solution is obtained without recourse to the constant pattern assumption. A numerical solution has also been obtained for intraparticle diffusion with external fluid film resistance. The deviation between the predictions of the two models increases with the magnitude of the intraparticle resistance. Numerical solutions were also obtained for Langmuir and Freundlich isotherms in order to establish the range where the simple rectangular models can be considered as an acceptable approximation.Experimental data for the ion exchange systems of RNaZn(NO₃)₂ and RNaCe(NO₃)₃ and for the adsor     

The implementation of an optimal control policy for a batch thermal system

A system comprising a steam heated exchanger through which fluid is pumped into a tank containing a steam heating coil has been constructed. This system has been modelled and the optimal charge rate to bring the tank to known conditions of mass and temperature in minimum time has been calculated by use of Pontryagin's maximum principle.The optimal policies calculated in this way have been implemented on the experimental equipment and comparisons with an equivalent constant charge rate made. Time advantages up to nearly 20% have been achieved over the constant charge rate by use of the optimal policy.Four control schemes have been compared to implement the optimal policy. These have been the control of mass alone, the control of temperature alone an two policies involving switching between mass and temperature. The two policies involving switching have given much closer agreement with predictions t those involving a single control variable.     

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.     

Automation of group-contribution techniques for estimation of thermophysical properties

A computer algorithm is presented which predicts thermophysical properties of mixtures using only two-dimensional chemical structures. It applies a combination of techniques for the analysis of group contributions and other characteristics of a chemical structure in a table-driven format. These techniques are used to examine the error behavior for predictions of critical properties and saturated properties. UNIFAC predictions of activity coefficients are combined with pure-component properties to demonstrate that accurate estimates of azeotrope behavior and K-values can be obtained.     

A unified approach to transport phenomena in polymeric media—II Diffusion in solid structured polymers

The altered free volume state model proposed in the earlier work for polymer solutions and melts is extended to cover solid polymeric systems. Within this framework, model equations have been developed to correlate diffusion data in semicrystalline polymers, crosslinked polymers, filled polymers, block and graft copolymers, polymeric blends and drawn fibres. Extensive data from the literature have been analysed to test the validity of the model. Excellent agreement between model predictions and the data has been obtained.     

Dynamic behaviour of an adiabatic adsorption column—I Analytic solution for irreversible adsorption

A simple theoretical model, based on the rectangular isotherm approximation, is shown to provide a good representation of the dynamic behaviour of an adiabatic adsorption column. The model provides an analytic expression for the dynamic temperature response in terms of the usual kinetic and equilibrium parameters and it is shown that within the constant pattern regime the relationship between the concentration and temperature profiles assumes a very simple form. The magnitude of the temperature rise depends on the relative velocities of thermal and concentration waves as well as the heat of adsorption and sorbate concentration. The maximum temperature rise occurs when the velocities of the temperature and concentration fronts are indentical. Under these conditions there is no temperature plateau and the magnitude of the thermal wave increases continuously as the adsorption front progresses through the bed.Experimental data for the adiabatic adsorption of moisture from an air stream are presented and discussed in relation to the theory.