MANIPULATING AN ALGEBRAIC EQUATION

Tabular (C,t) data usually are easier to manipulate when put in the form of an algebraic equation. Then necessary integrals and derivatives can be formed readily and most accurately. The calculation of chemical conversions by such mechanisms as segregation, maximum mixing or dispersion also is easier with data in the form of equations. 

The model involves four variables and three independent nonlinear algebraic equations, hence one degree of freedom exists. The equality constraints can be manipulated using direct substitution to eliminate all variables except one, say the diameter, which would then represent the independent variables. The other three variables would be dependent. Of course, we could select the velocity as the single independent variable of any of the four variables. See Example 13.1 for use of this model in an optimization problem. 

We have found that dynamics can be more conveniently handled in the Russian transfer-function language than in the English ODE language. However, the manipulation of the algebraic equations becomes more and more difficult as the system becomes more complex and higher in order, if the system is th-order, an Afth-order polynomial in s must be factored into its N roots. For N greater than 2, we usually abandon analytical methods and turn to numerical... 

As an alternative to algebraic manipulation of these equations, graphic solutions, such as shown in Fig. 2-f5, have been developed for determining the pH of water as a function of Aik and Cx, where Cx is the sum of the concentration of carbonic acid and the concentrations of the two anions produced when it ionizes (i.e., HCOj and CO -). 

An algebraic manipulation has been done in arriving at equation (2.22) by first expanding the summation from m>l and then subtracting m= term in equation 2.21. It is then easy to see... 

Using operator algebra, we manipulate this equation as though it were an ordinary equation. We factor it to obtain ... 

Before proceeding further, it will perhaps make things plainer to put the meaning of this differential equation into words. The manipulation of the equations so far introduced, involves little more than an application of common algebraic principles. Dexterity in solving comes by practice. Of even greater importance than quick manipulation is the ability to form a clear concept of the physical process symbolized by the differential equation. Some of the most... 

The calculations for the nonconstant-density case may be greatly simplified by using a differential-algebraic equation (DAE) solver, All three cases enumerated above can be handled by modifying the residual equations provided to the DAE solver. We do not have to differentiate the equation of state or perform other algebraic manipulations that are required if one uses an ordinary differential equation (ODE) solver. 

Figure 5.26 shows a flow diagram for the Laplace transformation and inverse transformation. It is clear that the main function of the Laplace transformation is to put the differential equation (in the time domain) into an algebraic form (in the 5-domain). These 5-domain algebraic equations can be easily manipulated as input-output relations. 

The LFERs given in Eqs. 8.57 and 8.59 are not those routinely applied to experiments, such as a Hammett plot. Instead, a form that relates the sensitivity factor Q and the sukstitu-ent constant Cx to a number that can be experimentally determined is used. To derive this form of the LFER, we perform some algebraic manipulations (left as an Exercise at the end of the chapter) to derive Eq. 8.61. This final form of an LFER is the one used in most such analyses (see Sections 8.3-8.5). An identical equation can be written for rate constants (Eq. 8.62). 

Let us consider Eqs. (5.1.3a), (5.1.3b) and (5.1.5b). This set of equations was obtained by twice differentiating the position constraint with respect to time and some algebraic manipulations of the equations. A further time differentiation of the algebraic equation leads to an ODE in all variables p, v and A ... 

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