Ordinary differential equation solver, ODE

One particular class of ordinary differential equation solvers (ODE-solvers) handles stiff ODEs and these are widely known as stiff solvers. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps or relatively high and low concentrations. A typical example would be an oscillating reaction. Here, a highly sophisticated step-size control is required to achieve a reasonable compromise between accuracy and computation time. It is well outside the scope of this chapter to expand on the intricacies of modem numerical... 

For the ordinary differential equation solver (ODE solver), contact ... 

The design Equation 14-2 and the rate law r(X) are now combined and solved using an ordinary differential equation solver ODE. The POLYMATH-program and the result is shown as follows ... 

A commercial stiff ordinary differential equation solver subroutine, DVOGER, is available in the IMSL Library (3). This subroutine uses Gear s method for the solution of stiff ODE s with analytic or numerical Jacobians. The pyrolysis model was solved using DVOGER and the analytical Jacobians of Eqs. (14) and (15). For a residence time of 0.0511 in dimensionless time, defined as t/t where 9... 

A large number of ordinary differential equation solver software packages (i.e., ODE solvers) which are extremely user friendly have become available. We shall use POLYMATH S to solve the examples in the printed text. However, the CD-ROM contains an example that uses ASPEN, as well as all the MATLAB and POLYMATH solution programs to the example programs. With POLYMATH one simply enters Equations (E4-7.3) and (E4-7.4) and the corresponding parameter value into the computer Table E4-7.1) with the initial (rather, boundary) conditions and they are solved and displayed as shown in Figure E4-7.1,... 

ODE (Ordinary Differential Equation) solver option config.ODEMethod = 23 ... 

A large number of ordinary differential equation solver software packages (i.e., ODE solvers), which are extremely user friendly, have become available. We... 

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... 

Develop two method-of-lines simulations to solve this problem. In the first, formulate the problem as standard-form ordinary differential equations, y7 = ff(f, y). In the second, formulate the problem in differential-algebraic (DAE) form, 0 = g(t, y, y ). Standard-form stiff, ordinary-differential-equation (ODE) solvers are readily avalaible. DAE solvers are less readily available, but Dassl is a good choice. The Fortran source code for Dassl is available at http //wwwjietlib.org. 

Most commonly used ordinary differential equation (ODE) solvers provide options of several different integration techniques. Most solvers also automatically vary the integration step size during the simulation to allow the best trade-off between accuracy and solution time, based on user-specified numerical tolerances. There is no single best integration technique—different methods work better for various problems. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

When using an ordinary differential equation (ODE) solver such as POLYMATH or MATLAB, it is usually easier to leave the mole balances, rate laws, and concentrations as separate equations rather than combining them into a single equation as we did to obtain an analytical solution. Writing She equations separately leaves it to the computer to combine them and produce a solution. The formulations for a packed-bed reactor with pressure drop and a semibatch reactor are given below for two elementary reactions. 

We see that we have j coupled ordinary differential equations that imist be solved simultaneously with either a numerical package or by writing an ODE solver. In fact, this procedure has been developed to take advantage of the vast number of computation techniques now available on mainframe (e.g,... 

For the case of isothermal operafion with no pressure drop, we were able to obtain an analytical solution, given by equation B, which gives the reactor volume necessary to achieve a conversion X for a gas-phase reaction carried out isothermaliy in a PFR, However, in the majority of situations, analytical solutions to the ordinary differential equations appearing in (he combine step are not possible. Consequently, we include POLYMATH, or some other ODE solver such as MATLAB, in our menu in that it makes obtaining solutions to the differential equations much more palatable, ... 

These formulas are useful in illustrating how the reaction engineering integrals and coupled DDEs (ordinary differential equation(s)) can be solved and also when there is an ODE solver power failure or some other malfunction. 

An orthogonal co-location method can be used to convert the above partial differential equahons (PDE) into the ordinary differential equations (ODEs). An ODE solver, EPISODE can be used to solve the (ODEs) (25). In the model, the diffusivity is obtained from a batch kinetic study while the external mass transfer coefficient can be calculated from empirical equations or is available in the literature. The longitudinal dispersion coefficient (D ) is determined by matching the model output with the experimental data. Readers may like to refer to Chen and Wang s work (9) for detailed information. 

To calculate dispersed phase particle trajectories it will be necessary to solve a set of coupled ordinary differential equations (Eqs. 4.1 and 4.9). Any standard initial value ODE solvers can be used for this purpose. These methods are not discussed here. Necessary details may be found in texts such as Numerical Recipes (Press et al., 1992). When calculating the trajectories of dispersed phase particles, any other auxiliary equations to account for heat transfer or chemical reactions can also be solved following similar procedures. Care must be taken to ensure that the time steps used for integration are sufficiently small and the trajectory integration is adequately time accurate. It is often necessary to use different time steps to simulate transients in the continuous flow field and trajectories of dispersed phase particles. 

Simultaneous to the graph creation, kinetic properties in each vRxn are used to create the appropriate reaction rate equations (ordinary differential equations, ODE). These properties include rate constants (e.g., Michaelis constant, Km, and maximum velocity, Vmax, for enzyme-catalyzed reactions, and k for nonenzymatic reactions), inhibitor constants, A) and modes of inhibition or allosterism. The total set of rate equations and specified initial conditions forms an initial value problem that is solved by a stiff ODE equation solver for the concentrations of all species as a function of time. The constituent transforms for the each virtual enzyme are compiled by carefully culling the literature for data on enzymes known to act on the chemicals and chemical metabolites of interest. 

Given the initial conditions (concentrations of the 22 chemical species at t = 0), the concentrations of the chemical species with time are found by numerically solving the set of the 22 stiff ordinary differential equations (ODE). An ordinary differential equation system solver, EPISODE (17) is used. The method chosen for the numerical solution of the system includes variable step size, variable-order backward differentiation, and a chord or semistationary Newton method with an internally computed finite difference approximation to the Jacobian equation. 

Global configuration variables are used to share Information between this % function and derivative code, which is Invoked indirectly via the % Ordinary Differential Equation (ODE) solver... 

When using an ordinary differential equation (ODE) solver such as Polymath or ... 

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. 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

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