First-order ODEs

NUMERICAL SOLUTIONS TO SETS OF FIRST-ORDER ODEs... 

The extension to multiple reactions is done by writing Equation (3.1) (or the more complicated versions of Equation (3.1) that will soon be developed) for each of the N components. The component reaction rates are found from Equation (2.7) in exactly the same ways as in a batch reactor. The result is an initial value problem consisting of N simultaneous, first-order ODEs that can be solved using your favorite ODE solver. The same kind of problem was solved in Chapter 2, but the independent variable is now z rather than t. 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

The numerical solution of Equations (9.14) and (9.24) is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs that result from applying the method of lines to PDEs. The reason for the complication is the second derivative in the axial direction, Sajdz. ... 

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

Then Equations (9.14) and (9.24) can be written as a set of four, first-order ODEs with boundary conditions as indicated below ... 

Equations (11.20) and (11.21) are linear, first-order ODEs with coefficients that are assumed constant. The equations can be combined to give a second-order ODE in af. 

A phase space is established for a typical particle, whose coordinates specify the location of the particle as well as its quality. Then, ordinary differential equations describe how these phase coordinates evolve in time. In other words, the state of a particle in a processing system is specified by the values of a number of phase coordinates z. The only requirement on z is that they describe the state of the particle fully enough to permit one to write a set of first order ode s of the form ... 

For numerical processing, a first order ODE is arranged explicitly for the derivative,... 

The available software for numerical integration of first order ODEs is applicable only when dC/dt is available explicitly. Here a "root solver" is used to find the relation between C and r. Then the relation to t is obtained by integration with the trapezoidal rule,... 

We need to study the numerical integration of only first-order ODEs. Any higher-order equations, say with Mth-order derivatives, can be reduced to N first-order ODEs. For example, suppose we have a third-order ODE ... 

Thus wc have three first-order ODEs to solve ... 

The solution of a second-order ODE can be deduced from the solution of a first-order ODE. Equation (6.45) can be broken up into two parts ... 

A. COMPLEMENTARY SOLUTION, Since the complementary solution of the first-order ODE is an exponential, it is reasonable to guess that the complementary solution of the second-order ODE will also be of exponential form. Let us guess that... 

Example 6l9. If two CSTRs like the one considered in Example 6.6 are run in series, two first-order ODEs describe the system ... 

Combine the three first-order ODEs describing the three-CSTR system of Sec. 3.2 into one third-order ODE in terms of Then solve for the response of to a unit step change in C 0 assuming all Jt s and t s are identical. 

The fourth-order Runge-Kutta method is applicable for a set of N first-order ODEs, where the functional form of the first derivative for each equation i is known ... 

Example (Transforming a third order ODE into a system of three first-order ODEs)... 

This is a fairly simple set of first-order ODEs. The set is difficult to solve analytically, but numerical solutions are easy. 

Equations (2.4.10a) to (2.4.101) are six first order ODEs for the six unknown variables y to ye- Note that the order of system is increased from four to six in CMM, while the governing equation is transformed from a boundary value problem to an initial value problem. To solve these six equations, we therefore need to generate initial conditions for the unknowns. As we know the property of the fundamental solutions in the free stream, we can use that information to generate the initial conditions for yi to r/g. As at r/ —> 00 and (f>s, therefore we can estimate the... 

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. 

In Example 2.1, Maple was used to solve two simultaneous first order ODEs. The same methodology can be used to solve more than two simultaneous ODEs. Eor example, the material balance equations for the time dependent concentration of each species (A, B, and C) in an isothermal batch reactor with reversible series... 

Higher order linear ODEs can also be solved by changing them into a system of first order ODEs and using the exponential matrix approach discussed earlier. The most general form of a linear ODE of n order is[l]... 

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