Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process

Various particulate systems were modeled by the population balance equation (PBE). However, only few cases of analytical solutions for the breakage process do exist, with most solutions being valid for the batch stirred vessel. The analytical solutions of the PBE for particulate processes under the influence of particle breakage in batch and continuous processes were investigated. Such solutions are obtained from the integro‐differential PBE governing the particle size distribution density function by two analytical approaches: the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). ADM generates an infinite series which converges uniformly to the exact solution of the problem, while HPM transforms a difficult problem into a simple one which can be easily handled. The results indicate that the two methods can avoid numerical stability problems which often characterize general numerical techniques in this area.     

Simulation of rapid pressure swing adsorption and reaction processes

A general model for non-isothermal adsorption and reaction in a rapid pressure swing process is described. Several numerical discretisation methods for the solution of the model are compared. These include the methods of orthogonal collocation, orthogonal collocation on finite elements, double orthogonal collocation on finite elements, and cells-in-series. Computationally, orthogonal collocation on finite elements is found to be the most efficient of these. The model is applied to air separation for oxygen production. Calculations confirm the formation of a concentration shock when an adsorbent bed is pressurised with air. The form and propagation of the shock over short times is found to be in excellent agreement with the exact similarity transformation solutions derived for an infinitely long bed. For air separation, novel experimental measurements, showing an optimum particle size for maximum product oxygen purity, are accurately described by the model. Calculations indicate that a poor separation results from ineffective pressure swing for beds containing very small particles, and from intraparticle diffusional limitations for beds containing very large particles. For adsorption coupled with reaction, finite rate and reversible reactions are considered. These include both competitive and non-competitive reaction schemes. For the test case of a dilute reaction A &.rlhar2; B + 3C, with B the only adsorbing species, bed pressurisation calculations are found to be in excellent agreement with the solutions obtained by the method of characteristics.     

Micromixing effects on series–parallel and autocatalytic reactions in a turbulent single-phase gas flow

A hybrid solution algorithm is implemented to simulate turbulent reactive single-phase gas flow in an isothermal tubular reactor. This algorithm is a combination of a Finite Volume (FV) and a Probability Density Function (PDF) method. The FV method is used to solve the mass and momentum conservation equations combined with a standardized k–ε model for single-phase gas flow. The PDF method is applied to solve the species continuity equations. The advantage of using the PDF method is the fact that there is no need of any closures for chemical reaction source terms in a turbulent flow. The mesomixing and the micromixing contributions in the PDF equation are closed using the gradient-diffusion model and the Interaction-by-Exchange-with-the-Mean (IEM) model, respectively. This hybrid solution algorithm is applied to simulate a series–parallel and an autocatalytic reactive single-phase gas flow. The mechanical-to-scalar time-scale ratio, i.e. the IEM model parameter, is found to have an influence on the simulation results that cannot be neglected. As expected, in series–parallel reactions, the desired product selectivity increases when the reaction rate coefficient, corresponding to its formation, increases. Moreover multiple solutions are observed in the autocatalytic reaction for a given feed ratio, Damköhler (Da^I) and Péclet (Pe) numbers. To validate the hybrid FV–PDF solution algorithm, the calculated results are compared with the results obtained when using the Reynolds-averaged species continuity equation model. A good agreement is observed when infinite-rate mixing is applied.     

General analytical solutions for first and second order,homogeneous, irreversible reactions,in ideal tubular reactors

The two particular exponential integrals involved in the conventional solutions for ideal adiabatic tubular reactors are represented by a modified series for computation purposes. For arguments larger than ten in absolute value, the method of asymptotic expansion removes the inconveniences that may be encountered in the numerical evaluation of the solutions. The expressions obtained with the modified series are not restricted by zero heat of reaction as is the case with the conventional solutions. General solutions, easily amenable to computer evaluation, are presented for first and second order irreversible reactions, under adiabatic or isothermal conditions.     

Isothermal nth order reaction in catalytic pellets: Effect of external mass transfer resistance

The effectiveness factor for the isothermal power-law decomposition of a reactant within symmetrical catalytic pellets is obtained from the solution of a single first order differential equation. The effects of external mass transfer are taken into account, as well as the effect of simple types of non-uniform distribution of the catalyst within the particle.For fractional and negative reaction orders there are solutions with zero concentration in a central region of the particle. For negative reaction orders intervals of multiplicity of solutions are found.     

Description of osmotic dehydration of apple using two-dimensional diffusion models considering shrinkage and variations in process parameters

The main purpose of this paper is to study osmotic dehydration of apple in water and sucrose solutions. The kinetics of water quantity and solid gain are described by two mathematical models using numerical solutions of the two-dimensional diffusion equation in Cartesian coordinates with boundary condition of third kind. For the first model, both the process parameters and the product dimensions have been considered constant. For the second model these quantities have been taken as variable. The numerical solutions have been obtained via the finite volume method with fully implicit formulation. Process parameter estimation, using experimental data, was obtained by an optimizer based on the inverse method. The third-kind boundary condition was shown to be appropriate in view of the fact that the resistances to mass flows on the product surface are not at all negligible. The temperature and concentration of the osmotic solution used in the experiments influenced significantly the kinetics of solid uptake and water quantity as well as the values of estimated parameters. Mathematical model which considers the variations in the process parameters and the product shrinkage presents good physical consistency and well describes the mass migrations.     

Mass transfer with Nth order chemical reaction around spheres in the presence of surfactants

A numerical technique has been used to solve the nonlinear equation describing the mass transfer taking place for single and multiple particles for Nth order chemical reactions in the presence of a surfactant. A generalized form is given for the Sherwood number as a function of the Peclet number for reaction orders of 1/2, 1 and 2 in the presence of surfactants, (Sh = aPe^b). The parameters (a) and (b) are dependent on the reaction orders and the amount of surfactant present. The coefficient (b) had an average value of 0.46 with the parameter (a) increasing with larger reaction rate constants. The numerical technique involved DISPL, a computer software package for the solution of partial differential equations by the method of lines.     

Numerical modelling and simulation of pulverized solid‐fuel combustion in swirl burners

A finite‐volume numerical model for computer simulation of pulverized solid‐fuel combustion in furnaces with axisymmetric‐geometry swirl burner is presented. The simulation model is based on the k ? ε single phase turbulence model, considering the presence of the dispersed solid phase via additional source terms in the gas phase equations. The dispersed phase is treated by the particle source in cell (PSIC) method. Solid fuel particle devolatilization, homogenous and heterogeneous chemical reaction processes are modelled via a global combustion model. The radiative heat transfer equation is also resolved using the finite volume method. The numerical simulation code is validated by comparing computational and experimental results of pulverized coal in an experimental furnace equipped with a swirl burner. It is shown that the developed numerical code can successfully predict the flow field and flame structure including swirl effects and can therefore be used for the design and optimization of pulverized solid‐fuel swirl burners.     

Intermediate storage in batch/continuous processing systems under stochastic operation conditions

A mathematical model and model-based method is presented to design the intermediate storages aiming to buffer the operational differences between the batch and continuous subsystems in processing systems. The occurrence times of the inputs are assumed to be described by a Poisson process, while the amounts of the material transferred by the batch units allowed changing according to general probability distributions. Based on the stochastic differential equation model of operation, integral equations for determining the overflow and underflow probabilities of a finite storage are formulated for both infinite and finite operation horizons that provide the basis for the rational design of such intermediate storages. Analytical solutions to the integral equations for infinite horizons are derived in the cases of constant and exponentially distributed inputs. For the batch sizes described by general distribution functions, solutions to the integral equations are obtained in the form of approximating functions generated by stochastic simulation. A number of numerical experiments with exponential, normal and lognormal distributions of the batch sizes are presented and analyzed. The effects of process parameters on the design are also investigated.     

A numerical analysis of rate data for packed bed reactors in gas-solid reaction systems

A numerical solution of the pseudo-steady state governing equations on the basis of the Langmuir-Hinshelwood type rate equation was obtained by the approximate finite difference method in packed bed reactors for gas-solid reaction system. It was proved that the numerical method has good accuracy compared with the strict solution in the special case that the reaction rate can be represented by the first-order kinetics in terms of gaseous reactant and the effectiveness factor is unity druing the reaction. The numerical method is proposed to predict the transient of exit-gas compositions of a packed bed reactor used for gas-solid reaction systems. The exit-gas composition can be predicted from the conversion data of a single particle with varying reaction time. The present method can be easily applied to the systems involving adsorptive gaseous reactants and complex reaction behavior with structural changes of particles.     

Velocity,temperature and concentration profiles in a vertical flow reactor

A vertical, adiabatic, steady-state, flow reactor in which a homogeneous chemical reaction takes place has been studied. Release of heat by the reaction sets up density gradients, which cause free convective flow to be superimposed on the laminar upflow. The coupled differential equations of momentum, energy, and mass transfer have been solved numerically by two different finite difference methods. In the first numerical method, the momentum transfer equation formulated in terms of the stream function required large amounts of computer time and storage for solution. In the second method, a simplified form of the momentum transfer equation was solved with a significant reduction of the computer requirements. Velocity, temperature, and concentration profiles have been obtained from the two numerical methods and the results analyzed in terms of dimensionless groups. The excellent agreement which has been found between the two methods confirms the accuracy and usefulness of the simplified method.     

Sorption and migration of aliphatic organic esters into VITON® fluoroelastomer membranes

Sorption and migration of six aliphatic esters into four VITON^® fluoroelastomers were studied by a gravimetric sorption method in the temperature interval of 30–60°C. Fick's equation was used to obtain diffusion coefficients. The dependence of fluorine contents and the polymer morphology on the sorption and diffusion characteristics of esters was investigated. The permeability coefficients were obtained from the sorption and diffusion data. Fick's equation was solved to compute the concentration profiles of liquids at various locations within the membrane materials using initial and boundary conditions. These profiles were compared with those obtained from the numerical method based on finite difference technique. Activation parameters for diffusion and sorption were calculated using the Arrhenius relationship. These results were discussed in terms of molecular size and shapes of the esters. For higher esters, namely, n- and iso-amyl acetates, a concentration dependency of the diffusion coefficient was investigated. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 63: 1223–1235, 1997     

Unsteady viscometric flow

Poncin's Navier-Stokes solution for the falling head viscometer is obtained more directly by a Laplace transform method. Approximations which facilitate numerical evaluation are derived and Grumbach's result for an infinite capillary is obtained as a limiting case. A systematic series of approximations to the general, unsteady Euler duct equation are solved analytically. All solutions are compared with a numerical integration of the complete Euler duct equation. It is shown that all results become indistinguishable for sufficiently small Reynold's number and that the kinetic energy effect is clearly the major inertial effort.     

Numerical simulation of resin transfer molding using linear boundary element method

Using mass conservation and Darcy's law to describe flow through isotropic porous media leads to a Laplace equation, which may be solved numerically at each time step using the boundary element method. For anisotropic porous media in which the permeability matrix is symmetric, the problem can be solved in the same way by rotating and stretching the coordinates. The numerical model has been compared with analytical solutions and experimental data in the case of radial front flows and with finite element for a frontal injection.     

On the Solution of Non-Isothermal Reaction-Diffusion Model Equations in a Spherical Catalyst by the Modified Adomian Method

We investigate the reactant diffusion within a porous catalytic pellet including the heat of reaction. The Lane–Emden boundary value problem with an Arrhenius reaction rate is used to model the reactant concentration. We combine the Volterra integral form with the Adomian decomposition method to solve the equivalent Fredholm–Volterra integral equation. We then estimate the reactant concentration at the center of the catalytic pellet and the iterated Shanks transform is used to improve the accuracy of the approximations. The objective error analysis formulas are used to demonstrate a high accuracy and rapid rate of convergence, which does not depend on a priori knowledge of the exact solution or comparison with an alternate approximation method. Thus low-stage approximations by the Adomian decomposition method are validated for parametric simulations of the reactant concentration profiles and the effectiveness factor profiles. Our approach demonstrates enhancements over previous investigations, and is readily extensible to more general diffusion reaction models in catalytic reactor engineering.     

An Efficient Numerical Scheme to Solve a Quintic Equation of State for Supercritical Fluids

In this study, an efficient iterative algorithm is devised to handle a nonlinear equation arising in estimation of thermodynamic properties at supercritical conditions. The approach is based on a synergistic combination of the classic Newton-Raphshon algorithm and the Adomian decomposition method. We demonstrate that the proposed method enjoys a higher degree of accuracy while requiring fewer iterations to reach a specific solution compared to that by the Newton-Raphson algorithm. To illustrate the efficiency of the aforementioned solution technique, several numerical examples are provided. The proposed method has been easily implemented in computer codes to provide parametric, not just numeric, solutions to the model equations. Consequently, one can derive other thermodynamic properties, which have not been treated parametrically to date, based on our new combined approach.     

ANALYTIC SOLUTIONS FOR THE UNSTEADY LONGITUDINAL FLOW OF AN OLDROYD-B FLUID WITH FRACTIONAL MODEL

The unsteady flow of an Oldroyd-B fluid with fractional derivative model, between two infinite coaxial circular cylinders, is studied by using finite Hankel and Laplace transforms. The motion is produced by the inner cylinder that, at time t = 0⁺, is subject to a time-dependent longitudinal shear stress. The solutions that have been obtained, presented under series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B and generalized and ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Concentration dependence of viscometric properties of model short chain polymer solutions

We study the concentration dependence of the conformational and viscometric behaviour of short-chain polymer solutions in shear flow by conducting a series of non-equilibrium molecular dynamics simulations, covering the entire concentration range. Our model explicitly incorporates all of the important generic features of real polymer solutions—excluded volume, hydrodynamic interactions and finite chain extensibility. Hydrodynamic interactions are included exactly by treating the solvent explicitly as an atomic fluid. The polymer molecules studied consist of 20-site bead-rod model molecules, which correspond approximately to 12 Kuhn steps in the melt. For polyethylene, this represents a molar mass of 1800 g mol⁻¹. In some respects, our results are consistent with experimental and theoretical results obtained for long-chain polymer solutions. We calculate the Flory-Fox constant and find a value that agrees reasonably well with results for long chain polymer solutions. Due to the short chain length of the molecules investigated, no semidilute region exists for these solutions. However, the radius of gyration and viscosity still exhibit strong concentration dependence, which is well described by power series, rather than power law expressions, in contrast to the behaviour usually observed in long-chain polymer solutions.     

Numerical methods for the simulation of the settling of flocculated suspensions

For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic–parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e. conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.     

Numerical solution of population balance equations for nucleation, growth and aggregation processes

This article focuses on the derivation of numerical schemes for solving population balance models (PBMs) with simultaneous nucleation, growth and aggregation processes. Two numerical methods are proposed for this purpose. The first method combines a method of characteristics (MOC) for growth process with a finite volume scheme (FVS) for aggregation process. For handling nucleation terms, a cell of nuclei size is added at a given time level. The second method purely uses a semi-discrete finite volume scheme for nucleation, growth and aggregation of particles. Note that both schemes use the same finite volume scheme for aggregation process. On one hand, the method of characteristics offers a technique which is in general a powerful tool for solving linear growth processes, has the capability to overcome numerical diffusion and dispersion, is computationally efficient, as well as give highly resolved solutions. On the other hand, the finite volume schemes which were derived for a general system in divergence form, are applicable to any grid to control resolution, and are also computationally not expensive. In the first method a combination of finite volume scheme and the method of characteristics gives a highly accurate and efficient scheme for simultaneous nucleation, growth and aggregation processes. The second method demonstrates the applicability, generality, robustness and efficiency of high-resolution schemes. The proposed techniques are tested for pure growth, simultaneous growth and aggregation, nucleation and growth, as well as simultaneous nucleation, growth and aggregation processes. The numerical results of both schemes are compared with each other and are also validated against available analytical solutions. The numerical results of the schemes are in good agreement with the analytical solutions.