Single, First-Order Ordinary Differential Equation

APPENDIX 7-A NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 7-A.l Single, First-Order Ordinary Differential Equation 

The value of y at x = xq is jo- We are interested in calculating the value of y at some value of X, designated Xf, that is greater than xq. 

Begin by dividing the interval between xq and Xf into N equal segments, so that 

The value of h is called the step size. The change in y when x is changed by Ax = /i will be denoted Ay. 

Calculating an accurate value off is the most challenging part of solving differential equations numerically. Textbooks on numerical analysis discuss this issue in detail. For many problems, the fourth-order Runge-Kutta method can be used. In this method. 

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In this family of helices, k = 0, r = 2ro = 1/6 is a straight filament with a — O, the case k — 0,t = 0 occurs when a = 00. Although the height of the helix does not change, its radius does a = a t). The evolution of the helix is governed by a single first-order ordinary differential equation, which we choose to express in terms of r(f) ... 

Note that all the conditions are known at one time, t = 0. Thus it is possible to calculate the function on the right-hand side at f = 0 to obtain the derivative there. This makes the set of equations initial value problems. The equations are ordinary differential equations because there is only one independent variable. Any higher-order ordinary differential equation can be turned into a set of first-order ordinary differential equations they are initial value problems if all the conditions are known at the same value of the independent variable [Finlayson, 1980, 1997 (p. 3-54), 1990 (Vol. BI, p. 1-55)]. The methods for initial value problems are explained here for a single equation extension to multiple equations is straightforward. These methods are used when solving plug-flow reactors (Chapter 8) as well as time-dependent transport problems (Chapters 9-11). 

Once the effective rate forms at the particle/bubble level are established, and flow patterns as assumed in Figure 6.2 are available, one simply uses these effective rate expressions to write down the corresponding steady-state material balances for the reactor for the assumed flow patterns. Under steady-state conditions, this involves either first-order ordinary differential equations for the phases in which plug flow is assumed or simple difference equations in species concentration in phases in which completely mixed flow is assumed. The treatment in all these cases is very similar to what will be in a single-phase reactor (see, e.g.. Ref [48]), except that one has a separate differential equation balancing for each species concentration and they are coupled through the effective reaction rate term. 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

A transformation of the dependent variables Cjt, and Cs allowed DelBorghi, Dunn, and Bischoff [9] and E>udukovic [25] to reduce the coupled set of partial differential equations for reactions first-order in the fluid concentration and with constant porosity and diffusivity, into a single partial differential equation. With the pseudo-steady-state approximation, this latter equation is further reduced to an ordinary differential equation of the form considered in Chapter 3 on diffusion and reaction (sk Problem 4.2). An extensive collection of solutions of such equations has been presented by Aris [7]. 

Situations may arise when there is a need to simultaneously solve more than one ordinary differential equation (ODE). In a more general case, one could have n dependent variables yi yi. > 3 n with each related to a single independent variable x by the following system of n simultaneous first-order ODEs ... 

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