Nonlinear ode

The gravity-flow tank that we considered in Chap. 1 and later in Example 2.9 makes a nice simple system to start our simulation examples. The force balance on the outlet line gave us the nonlinear ODE... 

The jacketed exothermic CSTR discussed in Sec. 3.6 provides a good example of the simulation of very nonlinear ODEs. Both flow rates and holdups will be... 

Sometimes useful information and insight can be obtained about the dynamics of a system from just the steadystate equations of the system. Van Heerden Ind. Eng. Chem. Vol. 45, 1953, p. 1242) proposed the application of the following steadystate analysis to a continuous perfectly mixed chemical reactor. Consider a nonisothermal CSTR described by the two nonlinear ODEs... 

Example 11.3. The nonlinear ODEs describing the constant holdup, nonisothemial CSTR system are... 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

The LCF model allows another (trivial) simplification. Because the bulk fluid phase is assumed to be perfectly mixed, it will also be axisymmetric consequently, the concentration profile in the catalyst particles will also be axisynrunetric. As a result, the three-dimensional PDE of Eq. (15), which could be reduced to a two-dimensional PDE due to the long cylinder approximation, can be further reduced to the second-order linear or nonlinear ODE of Eq. (21), since the angular coordinate drops out. 

Solving Nonlinear ODEs Using Maple s dsolve Command... 

Series solutions for nonlinear ODEs can be obtained using Maple s dsolve command. The syntax is ... 

In this chapter, nonlinear IVPs were solved numerically. In section 2.2.2 a noniinear IVP was solved analytically using Mapie s dsolve command. This approach is limited to very few nonlinear ODEs. In section 2.2.3, series solutions were obtained using Maple s dsolve command. This approach is valid for all... 

When the boundary conditions are nonlinear, the procedure described in section 5.2.2 cannot be used because the boundary values cannot be eliminated because of the nonlinear boundary conditions. This is handled by differentiating the finite difference form of the boundary condition with respect to t. This yields two additional nonlinear ODEs in time (see section 2.2.6 on DAEs), which are then solved simultaneously with N nonlinear ODEs arising from the discretization of... 

The gravity-flow tank discussed in Chapter 1 is described by two nonlinear ODEs ... 

Derive the nonlinear ODE dynamic mathematical model of this simplified process. (Hint Your model should consist of two ODES.)... 

The numerical solution of second-order nonlinear ODEs with split boundary conditions requires trial and error integration of two coupled first-order ODEs. If one defines d p./di = Axial Grad, then the one-dimensional plug-flow mass balance with axial dispersion,... 

An example from population dynamics is presented in Fig. 7.2 that shows the variation with time of the number of furs from hares and lynxes delivered to the Hudson s Bay Company [18]. The oscillatory populations of both species are obviously coupled to each other with a certain phase shift. There is a plausible explanation When the l5mxes find enough food (i.e., hares) then-population increases, while that of the hares decays as soon as the birth rate cannot compensate the growing loss anymore. When the supply of hares drops, the lynxes begin to starve and their population decreases, so the population of the hares recover again. An approximate mathematical description of this effect can be achieved in terms of two coupled, nonlinear ODEs (Lotka-Volterra model) [19] for the concentrations of hares, x, and lynxes, y, as presented in Fig. 7.3, together with the solution for properly chosen parameters a and )8. 

Equation 7.1 includes all first order nonlinear ODEs, provided the independent... 

For instance, suppose that the degradation trend i (t) can be approximated by the simple nonlinear ODE... 

Fig. 4.18 Numerical solution of the nonlinear ODE (4.38) (a) Charge density (b) electrical field. Reprinted from [Chen et al. (2009)].

The dynamic equations describing this fluidized bed after the above-described manipulations are the following nonlinear ODEs in terms of dimensionless variables and parameters ... 

For the particular test cases, we have simulated a digester that produces pulp out of softwood. The digester was split into 50 zones, thus producing a set of 950 nonlinear ODEs. The simulation was performed using the ode23 solver of MATLAB. The simulation time corresponding to one-hour real operation time was 30s in a Pentium IV 1400 MHz processor, which, given the complexity of the system, is a rather short time. 

The evolution of the method of solution is as follows The first solution to porous channel flow was the similarity solution from Berman (1953), from which a nonlinear ODE was reported for channel flow with two porous walls and uniform velocity injection. Berman also reported a perturbation solution to this ODE for small Rcinj. A more mathematically challenging problem was the perturbation solution to the ODE for large RCinj which was solved for by both Yuan (1956) and Terrill (1965a). By changing one boundary condition... 

Only two BCs are required to solve the second order nonlinear ODE. To enforce zero injection velocity at the channel dead end, and thus impose the no-slip condition, the derivative of Equation (12.34) is taken ... 

The collocation method makes the residuals and equal to zero at each collocation point x. The system of nonlinear ODEs is thus transformed into a system of 2N nonlinear algebraic equations. This can be solved by some iterative procedure.The effectiveness factor is calculated by ... 

This is exactly Equation 20-66, with kjf x) replaced by kg(p /j) and (dkj-/dx) replaced by (dkj,/dp)(dp/dx) Q/j. Thus, an equation analogous to Equation 20-69 is easily obtained. The function k(.(p) and the function ( t(p) in Equation 20-64 could be hard-coded into the main program, or declared as subroutines or statement functions, as desired. Finally, mudcake compressibility transients are easily modeled using the ideas developed in Example 20-4. For such problems, instead of the nonlinear ODE d(k(, dp/dx)/dx = 0, we would instead solve the nonlinear parabolic equation... 

Notice that n, the reaction order, can be any real number, including zero. Solve (10.1071 via transforming the second-order nonlinear ODE into a set of two first-order ODEs (see Sec.7.51. 

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