Ordinary differential equations computer simulation

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

The book explains how to solve coupled systems of ordinary differential equations of the kind that commonly arise in the quantitative description of the evolution of environmental properties. All of the computations that I shall describe can be performed on a personal computer, and all of the programs can be written in such familiar languages as BASIC, PASCAL, or FORTRAN. My goal is to teach the methods of computational simulation of environmental change, and so I do not favor the use of professionally developed black-box programs. 

In the computer simulation studies of the two preceding chapters, the systems and their describing equations could be quite complex and nonlinear. In the remaining parts of this book only systems described by linear ordinary differential equations will be considered (linearity is defined in Chap. 6). The reason we are limited to linear systems is that practically all the analytical mathematical techniques currently available are applicable only to linear equations. 

The Piecewise Linear Reasoner (PLR) [5] takes parameterized ordinary differential equations and produces maps with the global description of dynamic systems. Despite its present limitations, PLR is a typical example of a new approach that attempts to endow the computer with large amounts of analytical knowledge (dynamics of nonlinear systems, differential topology, asymptotic analysis, etc.) so that it can complement and expand the capabilities of numerical simulations. 

Here we focus on the issue of how to build computational models of biochemical reaction systems. The two foci of the chapter are on modeling chemical kinetics in well mixed systems using ordinary differential equations and on introducing the basic mathematics of the processes that transport material into and out of (and within) cells and tissues. The tools of chemical kinetics and mass transport are essential components in the toolbox for simulation and analysis of living biochemical systems. 

For large-scale problems, the most widely useful mathematical tool available is computational/numerical simulation. A great number of computer tools are available for simulation of ordinary differential equation (ODE) based models, such as Equations (3.27). Here we demonstrate how this system may be simulated using the ubiquitous Matlab software package. 

Unsteady-state or dynamic simulation accounts for process transients, from an initial state to a final state. Dynamic models for complex chemical processes typically consist of large systems of ordinary differential equations and algebraic equations. Therefore, dynamic process simulation is computationally intensive. Dynamic simulators typically contain three units (i) thermodynamic and physical properties packages, (ii) unit operation models, (hi) numerical solvers. Dynamic simulation is used for batch process design and development, control strategy development, control system check-out, the optimization of plant operations, process reliability/availability/safety studies, process improvement, process start-up and shutdown. There are countless dynamic process simulators available on the market. One of them has the commercial name Hysis [2.3]. 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

As mentioned above, the goal of MD is to compute the phase-space trajectories of a set of molecules. We shall just say a few words about numerical technicalities in MD simulations. One of the standard forms to solve these ordinary differential equations i.s by means of a finite difference approach and one typically uses a predictor-corrector algorithm of fourth order. The time step for integration must be below the vibrational frequency of the atoms, and therefore it is typically of the order of femtoseconds (fs). Consequently the simulation times achieved with MD are of the order of nanoseconds (ns). Processes related to collisions in solids are only of the order of a few picoseconds, and therefore ideal to be studied using this technique. 

For gaseous flames, the LES/FMDF can be implemented via two combustion models (1) a finite-rate, reduced-chemistry model for nonequilibrium flames and (2) a near-equilibrium model employing detailed kinetics. In (1), a system of nonlinear ordinary differential equations (ODEs) is solved together with the FMDF equation for all the scalars (mass fractions and enthalpy). Finite-rate chemistry effects are explicitly and exactly" included in this procedure since the chemistry is closed in the formulation. In (2). the LES/FMDF is employed in conjunction with the equilibrium fuel-oxidation model. This model is enacted via fiamelet simulations, which consider a laminar counterflow (opposed jet) flame configuration. At low strain rates, the flame is usually close to equilibrium. Thus, the thermochemical variables are determined completely by the mixture fraction variable. A fiamelet library is coupled with the LES/FMDF solver in which transport of the mixture fraction is considered. It is useful to emphasize here that the PDF of the mixture fraction is not assumed a priori (as done in almost all other flamelet-based models), but is calculated explicitly via the FMDF. The LES/FMDF/flamelet solver is computationally less expensive than that described in (1) thus, it can be used for more complex flow configurations. 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

The principal features of a mathematical model described for the enzymatic hydrolysis and fermentation of cellulose by Trichoderma reesei are the assumption of two forms of cellulose (crystalline and amorphous), two sugars (cellobiose and D-glucose), and two enzymes (cellulase and j3-D-glucosidase). An inducer-repressor-messenger RNA mechanism is used to predict enzyme formation, and pH effects are included. The model consists of 12 ordinary differential equations for 12 state variables and contains 38 parameters. The parameters were estimated from four sets of experimental data by optimization. The results appear satisfactory, and the computer programs permit simulation of a variety of system changes. 

To numerically simulate the above partial differential equation for n (Eq. 1), very fine grids should be utilized to discretize the computational domain (cell membrane). This is due to the exponential dependency of the creation rate to the pore energy and also the existence of disparate spatial and temporal scales. To avoid this costly computational effort, this PDE equation is asymptotically reduced to the system of ordinary differential equations [8]. 

The set of equations (10.5-22) is quite readily handled by the numerical method of orthogonal collocation. Basically, the coupled partial differential equations (eq. 10.5-22) are discretized in the sense that the spatial domain r is discretized into N collocation points, and the governing equation is valid at these points. In this way, the coupled partial differential equations will become coupled ordinary differential equations in terms of concentrations at those points. These resulting coupled ODEs are function of time and are solved by any standard ODE solver. Details of the orthogonal collocation analysis are given in Appendix 10.5, and a computer code ADSORB3A is provided with this book for the readers to learn interactively and explore the simulation of this model. 

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