Nonlinear First-Order Differential Equation

For ideal batch and plug-flow reactors, we obtain a set of nonlinear, first-order, differential equations of the form... 

The design formulation of nonisothermal batch reactors consists of + 1 nonlinear first-order differential equations whose initial values are specified. The solutions of these equations provide Z s and 6 as functions of t. The examples below illustrate the design of nonisothermal ideal batch reactors. 

The design formulation of nonisothermal plug-flow reactors consists of /+2 nonlinear first-order differential equations. Note that usually the inlet temperature of the heating/cooling fluid, Tp, is known. Hence, the case of co-current... 

It is straightforward to see that the superpotential [Pg.46]

For those first-order equations that cannot be expressed in polynomial form, there is no single analytical method to produce a solution as seen earlier in Section 2.1. This difficulty increases the importance of the issues of existence and uniqueness of a solution. For a very lucid discussion on the existence and uniqueness theorem for nonlinear first-order differential equations, many excellent texts are available [1,2]. 

To find the condition for the occurrence of undamped oscillations during a chemical reaction or another process described by nonlinear first order differential equations in some variables A2, we use the fact that if the oscillations are around some stationary point the equations can always be linearized near that point to gi e... 

Using the Carreau-Yasuda constitutive equation, this yields a nonlinear first-order differential equation for the velocity field... 

Eq. (3.34) cannot be solved analytically because it is a nonlinear differential equation. It can be solved by various numerical techniques. Again Advanced Continuous Simulation Language (ACSL, 1975) can be used to solve the problem. Since Eq. (3. 34) is a second-order differential equation, it has to be changed to two simultaneous first-order differential equations to be solved by ACSL as... 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

The spurious or satellite term in the solution is introduced by using a second-order difference equation to approximate a first-order differential equation. An extra condition is needed to fix the solution of the second-order equation, and this condition must be that the coefficient of the spurious part of the solution is zero. In the general case of a nonlinear difference equation, no method is available for meeting this condition exactly. 

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

Students require knowledge of solving (numerically) simultaneous first-order differential equations (initial value problems) and multiple nonlinear algebraie equations. The use of mathematical software that provides numerieal solutions to those types of equations (e.g., Matlab, Mathematica, Maple, Matbead, Polymatb, HiQ, etc.) is required. Numerical solutions of all the examples in the text are posted on the book web page. 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

Note that this last equation would be of the standard form of a first-order differential equation, with an exponential function solution, except for the time dependencies of the terms a(f) and P(t). These terms make this a nonlinear differential equation that most likely must be solved numerically. 

The substitutions translate the second order, nonlinear Poisson-Boltzmann equation into two coupled first order differential equations. The boundary conditions are ... 

For the numerical solution of this nonlinear equation, the first step is to put the second-order differential equation in the form of two first-order differential equations. Let... 

As can be seen by inspection of the set of three first-order differential equations, Eqs. (6) to (8), the model profile depends nonlinearly on the set of 2 unknown parameters, namely the apex radius b and the shape factor p. Moreover, as an additional third parameter, the apex correction error, e, is also taken into account. 

The Poisson-Boltzmann equation (23.5) is a nonlinear second-order differential equation from which you can compute ip if you know the charge density on P and the bulk salt concentration, rioo- This equation can be solved numerically by a computer. However, a linear approximation, which is easy to solve without a computer, applies when the electrostatic potential is small. For small potentials, zeip/kT 1, you can use the approximation sinh(x) s [(I + x) - (I - x)] 12 = X (which is the first term of the Taylor series expansion for the two exponentials in sinh(x) (see Appendix C, Equation (C.l)). Then Equation (23.5) becomes... 

In the simulation, the first step is to develop mathematical modeling. The modeling, based on first principles, is done by applying a standard input/ output approach for time-dependent systems with one or multiple inputs x t) and one or multiple outputs y t). The mathematical descriptions of components and hardware are formulated in the form of ordinary differential equations with the time t as the independent variable. The system description is represented mathematically by a system of coupled, nonlinear, first-order differential (or integral) equations ... 

Such an expansion reduces the partial differential equation, Eq. (9.37), into a set of nonlinear, ordinary first-order differential equations for the time-dependent coefficient (Doi and Edwards 1978b) ... 

The ability to solve nonlinear differential equations as readily as linear equations is one of the major advantages of the numerical solution of differential equations. For one such example, the Van der Pol equation is a classical nonlinear equation that has been extensively studied in the literature. It is defined in second order form and first order differential equation form as ... 

The second line gives mixed boundaiy conditions at bofli boundaries involving in this case a linear combination of the function value and the first derivative. A more general case would be some nonlinear combination of the function value and first derivative. Expressing this in terms of two first order differential equations gives ... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

These equations are a set of nonlinear first-order ordinary differential equations that describe the evolution of the n species as a function of time starting from a set of initial conditions... 

The problem in obtaining a state space model for the dynamics of the CSD from this physical model is that the population balance is a (nonlinear) first-order partial differential equation. Consequently, to obtain a state space model the population balance must be transformed into a set of ordinary differential equations. After this transformation, the state space model is easily obtained by substitution of the algebraic relations and linearization of the ordinary differential equations. 

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