Simultaneous, First-Order, Ordinary Differential Equations

2 Simultaneous, First-Order, Ordinary Differential Equations 

The following spreadsheet shows how the Runge-Kutta technique for simultaneous differential equations can be used to solve Example 7-4. 

we note that this problem is more complex than Example 7-2. The value of t was known in that problem, and the only challenge was to find a value of h that was small enough. In this problem, the value of r is not known, so no guidance is available in selecting ft. 

Let s begin by finding an approximate value of r. Better yet, let s find values of t that bracket the true value. Equation (7-24) can be rearranged and integrated from the reactor inlet to the reactor outlet to give 

The last integral in the above equation may be evaluated by taking an average value of po and substituting the value of kiRT. 

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In Chapter 3, the analytieal method of solving kinetie sehemes in a bateh system was eonsidered. Generally, industrial realistie sehemes are eomplex and obtaining analytieal solutions ean be very diffieult. Beeause this is often the ease for sueh systems as isothermal, eonstant volume bateh reaetors and semibateh systems, the designer must review an alternative to the analytieal teehnique, namely a numerieal method, to obtain a solution. Eor systems sueh as the bateh, semibateh, and plug flow reaetors, sets of simultaneous, first order ordinary differential equations are often neeessary to obtain die required solutions. Transient situations often arise in die ease of eontinuous flow stirred tank reaetors, and die use of numerieal teehniques is die most eonvenient and appropriate mediod. 

For this case, Eqns. (3-5) and (8-35) are a pair of simultaneous, first-order, ordinary differential equations. They are subject to the initial conditions T = To, Na = t = 0. These equations can be solved for T and xa as a function of time, using the numerical techniques discussed in Chapter 7, Section 7.4.3.2 and Appendix 7-A.2. 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

Numerical integration of ordinary differential equations is most conveniently performed when the system consists of a set of n simultaneous first-order ordinary differential equations of the form ... 

The above formulation of solution for a two-equation boundary-value problem can be extended to the solution of m simultaneous first-order ordinary differential equations. For this purpose, we define the following matrices ... 

Steady state heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear elliptic partial differential equation. For linear parabolic partial differential equations, finite differences can be used to convert to any given partial differential equation to system of linear first order ordinary differential equations in time. In chapter 5.1, we showed how an exponential matrix method [3] [4] [5] could be used to integrate these simultaneous equations... 

Under these conditions, the describing equations for both mass and energy transfer are first order ordinary differential equations. These must be solved simultaneously using a suitable analytical or numerical procedure. 

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 ... 

This is a set of simultaneous first-order nonlinear ordinary differential equations. The solution of these equations first requites the determination of the constants a, p, y, and 6, and the specification... 

The mathematical problem posed is the solution of the simultaneous differential equations which arise from the mass-action treatment of the chemistry. For the homogeneous, well-mixed reactor, this becomes a set of ordinary, non-linear, first-order differential equations. For systems that are not... 

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