Differential Equations In Time Research Articles

THE DEVELOPMENT AND VALIDATION OF A SYSTEM MODEL FOR A COUNTERCURRENT CASCADING ROTARY DRYER

ABSTRACT An overall system model for a countercurrent rotary dryer has been developed with the ullimale aim of assessing controller pairings in these dryers. This model is based on heat and mass balances within dryer regions combined with two subsidiary models, one describing the equipment (which determines particle transport and heat transfer)and the other describing the behaviour of the material (the drying kinetics). Six partial differential equations have been set up to evaluate six state variables: solids moisture content, solids temperature, gas humidity, gas temperature, solids holdup and gas holdup as functions of time and rotary dryer length. A control-volume method has been used to reduce the six partial differential equations with respect to time and the length of the rotary dryer to six ordinary differential equations in time. The drying model has been implemented in the SPEEDUP flowsheeting package (with FORTRAN subroutines) The model has been validated by fifteen experiments-in a pilot scale countercurrent-flow rotary dryer (0.2m in diameter and 2m in length)

Modelling the dynamics of supply chains

A summary of the important issues arising in supply chain research is presented. A review of the various mathematical methods used to model and analyse supply chains is then undertaken. They are categorized as: continuous time differential equation models, discrete time difference equation models, discrete event simulation models and operational research techniques. Each method is then appraised from a system dynamics perspective. To conclude, a judgement is made on whether the merits of operational research techniques justify their predominance in industry over the other methods presented here.

Sheared salt fingers: Instability in a truncated system

We derive a model for fingering doubly diffusive convection from a truncated expansion in horizontal planform functions with the inclusion of a large-scale shearing mode. This produces nonlinear partial differential equations in time and in vertical coordinate. At a high enough Rayleigh number, both convection and shear modes are sustained and their interaction produces rich cyclic dynamics with the fingering layer dividing into two distinct finger layers that engender steps in the mean salinity before being disrupted by the beginning of a new cycle.

Heating kinetics of a plane-parallel light-scattering layer

A layer heated from both sides by plates as a result of radiative energy transport and a heat-conduction mechanism is considered. A system of differential equations describing radiation transport and heat transfer in the layer is formulated. Its solution is given on the basis of a fixed finite-difference approximation of the spatial derivatives that reduces the problem to a system of ordinary differential equations in time for the temperature and the radiation fluence in the process of heating of both the entire layer and separate portions of it.

Solution of inverse heat conduction problems using control volume approach

A new method for solving multidimensional inverse heat conduction problems is presented. Using control volume methods, the partial heat conduction equation is replaced by a system of ordinary differential equations in time, which are then solved sequentially. The procedure is started at a spatial node where temperature sensor is located and sequentially marches through space to the surface node. The accuracy of the method was demonstrated by comparison of the calculated surface heat flux and temperature with the known exact solution. Two numerical experiments, one- and two-dimensional inverse heat conduction problems, are solved to illustrate the effectiveness, computational efficiency and good accuracy of the presented method. In the third example, the inverse formulation is applied to a set of experimental data. The method can easily be extended to three-dimensional cases.

A Simple Model for a Fluid-Filled Open-Cell Foam

A simple microstructure model is used to describe a fluid-filled open-cell foam. In the simplest case it consists of parallel elastic plates with gaps between them, which are filled with a Newtonian fluid. We assume that the load applied to this model material is uniaxial. The constitutive equation is formulated with the pressure of the fluid as an inner variable. The model yields an evolutional equation for the fluid pressure which itself is a field equation, that is a partial differential equation in time and space coordinates. This differential equation is solved for an instantaneously applied constant load and for a harmonically oscillating load. The solution of the differential equation, in combination with the constitutive equation leads to a relation between mean applied load and global strain of the test specimen. Finally, we obtain the creep compliance and the complex modulus of the foam material, respectively. The influence of different geometries of the foam and of different material behaviour of the matrix and fluid on the creep compliance and the complex modulus is discussed.

Transient spheroidal elements for unbounded wave problems

A wave-envelope element numerical scheme is applied to the solution of unbounded wave problems. The scheme is based on a Fourier transformation of a discrete model formulated in the frequency domain. This yields a discrete system of ordinary differential equations in time which are local in space. Oblate and prolate spheroidal element geometries are used. The accuracy of the scheme is demonstrated by a comparison of computed and analytic solutions for axisymmetric test cases. Time-harmonic and transient solution are presented. An indirect solution procedure is also presented which permits the sparsity of the transient equations to be utilised more effectively.

Unsteady flow of an axisymmetric annular film under gravity

The unsteady flow of an annular and axisymmetric film under gravity is examined. This moving boundary problem is solved by mapping the inner and the outer interface of the film in the radial direction onto fixed ones and by transforming the governing equations accordingly. The ratio of the film thickness to its inner radius at the exit of the die is small in relevant processes with polymer melts. This ratio, ε, is used as the small parameter in a perturbation expansion of the general Navier–Stokes equations. Forces applied on the film include gravity, surface tension, inertia, and viscous forces. Their ratios give rise to three dimensionless numbers, St, Ca, and Re. When these dimensionless numbers are up to order one, the base state is quite deformed and it is calculated numerically by simultaneously solving three nonlinear partial differential equations in time and the axial direction. Intuitively it is expected that when the dimensionless numbers are small the base state in the perturbation scheme is a uniformly falling film. This is confirmed by analysis and the two next orders in the perturbation scheme are calculated analytically. In both cases, it was found that increasing the St number (i) accelerates the downward motion of the film, (ii) deflects its inner and outer surfaces towards its axis of symmetry, and (iii) decreases its thickness around the middle of its length. The latter effect may lead to breakup of the film in two parts. It was also found that increasing the Ca number deflects these two interfaces towards its axis of symmetry and increases its thickness monotonically with time and the axial distance. Increasing the Re number from zero, but to not very large values, generally decelerates the film and decreases its deflection from the vertical. Given typical fluid properties and process conditions the St number is up to O(ε0), i.e., much larger than the other two dimensionless numbers, and affects the film shape more significantly.

Relative Income and Investment Comparisons among OECD Nations

Cross-section and time series studies of convergence generally have led to opposite conclusions about the convergence hypothesis. The methodologies employed in these studies suffer certain conceptual or statistical weaknesses. In this study, an entirely different approach is taken. The analytics of convergence is modeled using differential equations and the necessary and sufficient condition for absolute (steady state) convergence is derived. While our results reject absolute convergence for the OECD nations in a differential equation-time only model and in a differential equation model with capital accumulation and time as arguments, we find evidence of relative convergence with a weaker differential equation model.

Electromechanical wave propagation in large electric power systems

An electrical power network consisting of generators and transmission lines is treated as a continuum system. The application of the limit of zero generator spacing, with finite rotor inertia and transmission line impedance per unit length, yields a nonlinear partial differential equation in time and two spatial dimensions for the rotor phase angle. The equation is a nonlinear version of the standard second-order wave equation which exhibits an explicit expression for the finite wave phase velocity. The electromechanical wave propagation characteristics, equilibrium solutions, and linear stability are investigated and some potentially important results are presented. Numerical simulations of the usual discrete generator model, based upon the swing equation, are presented and demonstrate the electromechanical wave propagation as having interesting properties. Numerical solutions of the analogous continuum model are compared to the discrete model and are found to be in excellent agreement. A numerical estimate of the wave phase velocity for the U.S. power grid is consistent with observations of the transient wave phenomena during staged fault events. The continuum model enables an array of alternative analytic and simulation methods to be applied to the study of global power system characteristics, such as stability and transient dynamics.

Linearization methods for reaction-diffusion equations: 1-D problems

Three types of linearized finite difference methods are presented. The first type is based on a θ-formulation which provides discrete solutions in both space and time. It is shown that this type is a linearly-implicit Rosenbrock's or a W-method depending on whether the full or an appropriate Jacobian is employed. The second one requires the solution of two-point, linear, ordinary differential equations and provides either piecewise continuous or piecewise differentiable solutions in space and discrete in time. The third type is based on the discretization of the spatial coordinate and provides a system of linear, ordinary differential equations in time which can be integrated analytically. Therefore, the third type of linearized finite difference methods provides continuous solutions in time and discrete in space, and may be transformed into one of the first type by discretizing the time variable.

The DAE-aspect of the charge cycle in a combustion engine

The gas exchange in the cylinders is essential for the overall performance of an internal combustion engine. A CAE approach leads to technically improved and less polluting engines. Its base is a network formulation of the engine. The mathematical model describes the gas flow in the various parts of the engine. A system of ordinary differential equations in time represents conservation of mass and energy in the chambers. Considering the gas flow in the pipes yields an hyperbolic system of partial differential equaitons, the unsteady one-dimensional Euler equations. In contrast to classical difference schemes the numerical treatment with an ENO scheme prevents spurious numerical oscillations. The coupling of the pipes with the cylinders and other elements of the network leads to differential-algebraic equations (DAEs), which are solved by a predictor-corrector method. The implementation in an industrial simulation package shows that even large-scale problems can be solved with a very good congruence of measured and computed data.

On direct methods for the discretization of a heat-conduction equation using spline functions

The finite element collocation method is sometimes equivalent to the Galerkin method. The two methods are therefore considered together here. The nonhomogeneous heat-conduction equation with nonhomogeneous boundary condition is chosen for consideration. The equation is transformed into a variational form by continuous time Galerkin method and expressed in terms of linear spline basis function. On the other hand, the collocation method is applied on the heat equation with cubic splines as the basis functions. The two methods of discretization transform the heat equation to systems of ordinary differential equations in time, respectively. The corresponding stiffness matrices for the two systems of equations turn out to be the same. These matrices are sparse, banded and symmetric, and are therefore convenient for computer application.

Images in linearly conducting dielectrics

Methods of images for electrostatic fields and steady conduction fields are fairly well known. Whenever applicable, they have made the solution of field problems much simpler. However, when dealing with lossy dielectrics both permittivity and nonzero conductivity are to be considered. A generalised method of images is developed which can deal with such linearly lossy (conducting) dielectrics. For linearly conducting dielectrics, a point charge is equivalent to a point current source and vice versa. At t = 0+, only the dielectric image will be seen. Subsequently, because of the finite nonzero conductivities of the associated media, a current will flow and surface charges will accumulate at the interface due to mismatch in the material properties. The equation governing this surface-charge accumulatioin is derived. Linearity of the media permits fields in either medium to be obtained by superposing the fields due to the dielectric images and that due to the interfacial charges. The field can be obtained in either medium by replacing these surface charges by an equivalent point charge kept ait the appropriate image point. This equivalent point charge satisfies a similar differential equation in time to that of the surface charge. The cases of time-varying chargeicurrent sources and the source-current requirement for keeping any point very close to the source at a specified potential are also discussed.

Low collisionality transport in a torus

Low collisionality transport analysis methods developed earlier [Phys. Plasmas 1, 3942 (1994)] are applied to several problems. Slow time dependence is added to the earlier work on low collisionality steady states. It is shown that two types of time evolution are possible. In one case the energy balance relations adjust instantaneously to time dependent changes, while the flux surface averaged density satisfies an ordinary differential equation in time. Another possible type of solution, which requires self-consistent external energy sources and/or sinks is a flux conserving time evolution in which the safety factor Q moves with the plasma and the magnetic field. The low collisionality solution methods are then applied to the usual neoclassical parameter ordering. Without the use of moment equations to close the system of drift kinetic equations, solvability conditions on higher order drift kinetic equations complete the specification of the solution. The electron energy balance relation drops from the system, so that no electron energy transport anomaly appears. The results are not significantly different from the low collisionality case. Finally, to exhibit the nature of the constraints implicit in the low collisionality analysis procedure, it is shown that the magnetic field in a steady state, low collisionality stellarator must satisfy the condition that the magnetic field must have approximately a tokamak-like symmetry. Specifically, on a flux surface B⋅∇B must be approximately a function of B alone. A more direct solution technique is employed here, compared with the previous work, although certain ambiguities occur. The technique used here greatly simplifies the previous analysis but a solution accurate to N-th order in the expansion in some parameter admits arbitrary N+1-st and higher order additions.

Geriatric-patient flow-rate modelling

We consider the application of a three-compartment mathematical model using difference equations in discrete time to model the flow of patients through departments of geriatric medicine. It has been shown empirically that the time spent in hospital since admission may be described by a two-term mixed-exponential distribution. Previous work has explained this empirical finding in terms of a two-compartment model of geriatric in-patient behaviour where the two compartments are acute/rehabilitative and long-stay care. Our model extends this approach by considering patients released from geriatric departments and their subsequent length of stay in the community. We have therefore, two states of patient behaviour while in hospital: (i) patients can be admitted to the acute-rehabilitative state, from which they may die or be released back into the community; (ii) patients can be admitted to the long-stay state, from which they eventually die. The community state currently has only one state, from which patients may be readmitted back into the geriatric department or they may die. The model may then be used to estimate the average numbers and lengths of stay for short-term and long-term patients and the average number and length of stay in the community for released patients, allowing for a significant improvement in the forecasting of future bed requirements to aid the planning of geriatric departments.

Doubly asymptotic approximations for transient poroelastodynamics

A doubly asymptotic approximation (DAA) is an approximate temporal impedance relation at the boundary of a continuous medium. Here, first- and second-order DAAs are formulated for an infinite external poroelastic medium described by Biot’s equations. As with their acoustic, elastodynamic, and electromagnetic predecessors, the poroelastodynamic DAAs approach exactness in both the early-time (high-frequency) and late-time (low-frequency) limits, effecting a smooth transition between. They also lend themselves to straightforward boundary-element discretization, producing matrix ordinary differential equations in time that are readily solved by numerical integration. An initial examination of poroelastodynamic DAA accuracy is presented for two problems with spherical symmetry.

The use of Hamilton's principle to derive time-advance algorithms for ordinary differential equations

Hamilton's principle is applied to derive a class of numerical algorithms for systems of ordinary differential equations when the equations are derivable from a Lagrangian. This is an important extension into the time domain of an earlier use of Hamilton's principle to derive algorithms for the spatial operators in Maxwell's equations. In that work, given a set of expansion functions for spatial dependences, the Vlasov-Maxwell equations were replaced by a system of ordinary differential equations in time, but the question of solving the ordinary differential equations was not addressed. Advantageous properties of the new time-advance algorithms have been identified analytically and by numerical comparison with other methods, such as Runge-Kutta and symplectic algorithms. This approach to time advance can be extended to include partial differential equations and the Vlasov-Maxwell equations. An interesting issue that could be studied is whether a collisionless plasma simulation completely based on Hamilton's principle can be used to obtain a convergent computation of average properties, such as the electric energy, even when the underlying particle motion is characterized by sensitive dependence on initial conditions.

THE VAN KAMPEN EXPANSION FOR LINKED DUFFING-LINEAR OSCILLATORS EXCITED BY COLORED NOISE

This paper expands on previous work by Rodriguez and van Kampen, and by Weinstein and Benaroya. The original work by Rodriguez and van Kampen outlined a method of extracting information from the Fokker-Planck equation without having to solve the equation itself. In the van Kampen expansion, the Fokker-Planck equation is expanded about the random component of the response. This expansion is used to derive a series of first order differential equations in time for the moments of the response. These can then be solved for moments as a function of time. In the original work, this method was demonstrated on the case of a Duffing oscillator excited by white noise. Weinstein and Benaroya developed this method for the case of a Duffing ocsillator excited by colored noise, performed parametric studies on the system parameters, and verified their results by comparing them to Monte Carlo experiments. In this work, the van Kampen expansion is applied to systems of linked Duffing-linear oscillators excited by colored noise. Again, parametric studies are performed on the system parameters and the results compared to those of Monte Carlo experiments. It is found that the analytical results compare closely with the numerical results as long as the initial assumptions of the expansion are not violated. The closeness of the comparisons indicates that the van Kampen expansion is a valid technique for the study of this class of problem and is a useful tool in modelling offshore structures, plates in turbulent flows, and other fluid-structure interaction phenomena.

Algorithm for optimal filtering of random Markov signals with pulse-disturbance-amplitude estimation

Optimal filtering problem of random Markov signals with simultaneous estimation of pulse disturbance amplitudes is considered. Linear models of stochastic difference equations in discrete time are used to describe signals, observed processes, and pulse disturbances. Pulse disturbances occur at random times with random amplitudes. A real-time calculation procedure is obtained for the joint a posteriori probability density function of random signals and pulse disturbance amplitudes. A quasioptimal filtering algorithm is derived in the case of scalar signals and scalar observed processes by a partition method. Computer simulation results are presented.