Ordinary differential equations numerical solution techniques

Batch and bio process modeling often results in a set of ordinary differential equations. Numerical solution techniques for solving these differential equations normally involve approximating the differential equations by difference equations that are solved in a step-by-step marching fashion. 

Thus 1 is the total heat energy required to bring a solid from an initial temperature Tso to Tm and to melt it at that temperature. Sundstrom and Young (33) solved this set of equations numerically after converting the partial differential equations into ordinary differential equations by similarity techniques. Pearson (35) used the same technique to obtain a number of useful solutions to simplified cases. He also used dimensionless variables, which aid in the physical interpretation of the results, as shown below ... 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

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. 

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

In this chapter, the numerical solution of ordinary differential equations (odes) will be described. There is a direct connection between this area and that of partial differential equations (pdes), as noted in, for example [558]. The ode field is large but here we restrict ourselves to those techniques that appear again in the pde field. Readers wishing greater depth than is presented here can find it in the great number of texts on the subject, such as the classics by Lapidus k Seinfeld [351], Gear [264] or Jain [314] there is a very clear chapter in Gerald [266]. 

The techniques just described have been extensively used in modeling reactive flow problems at NRL. Efficient solution of the coupled ordinary differential equations associated with these problems has enabled us to perform a wide variety of calculations on H2 °2 anC Ha/Oo mixtures which have greatly extended our understanding of tne combustion and detonation behavior of these systems. In addition numerous atmospheric problems have been studied. Details on these investigations are provided in references (7) and (9). 

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. 

In Chapters 2 and 3, various analytical techniques were given for solving ordinary differential equations. In this chapter, we develop an approximate solution technique called the perturbation method. This method is particularly useful for model equations that contain a small parameter, and the equation is analytically solvable when that small parameter is set to zero. We begin with a brief introduction into the technique. Following this, we teach the technique using a number of examples, from algebraic relations to differential equations. It is in the class of nonlinear differential equations that the perturbation method finds the most fruitful application, since numerical solutions to such equations are often mathematically intractable. 

In the previous chapter, we discussed analytical and approximate methods to obtain solutions to ordinary differential equations. When these approaches fail, the only remaining course of action is a numerical solution. This chapter and the next consider numerical methods for solving ODEs, which may not be obtainable by techniques presented in Chapters 2, 3, and 6. ODEs of initial value type (i.e., conditions are specified at one boundary in time or space) will be considered in this chapter, whereas ODEs of boundary value type (conditions are specified at two boundary points) will be considered in Chapter 8. 

The present chapter provides an overview of several numerical techniques that can be used to solve model equations of ordinary and partial differential type, both of which are frequently encountered in multiphase catalytic reactor analysis and design. Brief theories of the ordinary differential equation solution methods are provided. The techniques and software involved in the numerical solution of partial differential equation sets, which allow accurate prediction of nonreactive and reactive transport phenomena in conventional and nonconventional geometries, are explained briefly. The chapter is concluded with two case studies that demonstrate the application of numerical solution techniques in modeling and simulation of hydrocar-bon-to-hydrogen conversions in catalytic packed-bed and heat-exchange integrated microchannel reactors. 

Techniques for the numerical solution of ordinary differential equations... 

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