Algebra simplifying

After substituting the species-continuity equation for the mass-fraction derivative and algebraically simplifying we have an energy equation with temperature as the dependent... 

Relate equilibrium mole fractions to the equilibrium constants. By definition, K = YUf1, where the a,- are the activities of the components with the mixture, and the v, are the stoichiometric numbers for the reaction (see Example 4.4). The present example is at relatively large reduced temperatures and relatively low reduced pressures, so the activities can be represented by the equilibrium mole fractions y,-. For Reaction 4.11, Kk = yco,Vch4 A> co)2( Vh, )2- Substituting the value for K from step 2 and the values for the y, from the last column of the table in step 3 and algebraically simplifying,... 

Problems like this are often algebraically simplified if they are written in terms of dimensionless variables. This is standard practice in many advanced textbooks. I often find this procedure confusing, because for me it produces only a small gain in algebra at the expense of a large loss in physical insight. Nonetheless, we shall follow this procedure here to illustrate the simplification possible. We first define three new variables ... 

The derivation of the flux relations given here is a simplified treatment which yields the correct algebraic form but does not provide estimates of the various diffusion parameters. A more complete discussion is given by Mason et al. [21]. 

The existence of these aimple algebraic relations enormously simplifies the problem of solving the implicit flux equations, since (11.3) permit all the flux vectors to be expressed In terms of any one of them. From equations (11.1), clearly... 

To simplify the algebra, the error in the a variable will be considered negligible relative to the error in the y variable, although this is not a necessar y condition. Equation (3-54) describes a plane passing through the origin. Let us restrict it to positive values of Xi, a 2, and y. 

Combine your equations to solve for one unknown (usually the one you are most interested in knowing). Whenever possible, simplify the algebra by making appropriate assumptions. 

We now have four equations (6.35, 6.36, 6.37, and 6.38) and four unknowns ([HE], [E-], [H3O4], and [OH-]) and are ready to solve the problem. Before doing so, however, we will simplify the algebra by making two reasonable assumptions. Eirst, since HE is a weak acid, we expect the solution to be acidic thus it is reasonable to assume that... 

Our problem looks challenging, but several assumptions greatly simplify the algebra. First, since the formation of the Ag(NlT3)27" complex is favorable, we will assume that... 

For algebraic convenience will be substituted for (v) in functions other than in ( ) which to simplify the algebra will be replaced later. It follows that,. 

If the letter symbols for sets are replaced by numbers, tlie commutative and associative laws become familiar laws of aritlimetic. In Boolean algebra tlie first of tlie two distributive laws, Eq. (19.3.5), lias an analogous counterpart in arithmetic. Tlie second, Eq. (19.3.6), does not. In risk analysis. Boolean algebra is used to simplify e. pressions for complicated events. For example, consider tlie event... 

It is now useful to show how steady-state solutions can be obtained in several other cases. We shall also present two shortcuts to simplify the algebraic steps in the derivations for complicated situations. The first example is the S>j 1 mechanism for the substitution reaction of an organic halide RX by a nucleophile Y ... 

The starting point in more a detailed exploration is the simplest systems of linear algebraic equations, namely, difference equations with special matrices in simplified form, for example, with tridiagonal matrices. 

Eq. (122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161],... 

Let us now consider the situation where [/] [E], We have here a situation that is analogous to our discussion of pseudo-first-order kinetics in Appendix 1. When [/] E in equilibrium binding studies, the diminution of [/]f due to formation of El is so insignificant that we can ignore it and therefore make the simplifying assumption that [/]f = [/]T. Combining this with the mass balance Equations (A2.1) and (A2.2), and a little algebra, we obtain... 

In our examples, we will take Gm = Ga = 1, and use a servo system with L = 0 to highlight the basic ideas. The algebra tends to be more tractable in this simplified unity feedback system with only Gc and Gp (Fig. 5.6), and the closed-loop transfer function is... 

To arrive at the last term, we have used a simple Taylor expansion (e s 1 — 9s) of the exponential term. This is purposely done to simplify the algebra as shown in the next example. (We could have used the Pade approximation in Eq. (6-26), but the result will not be the simple PI controller.)... 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

The rather fearsome algebra can be somewhat simplified by defining ... 

Thus matrix operations provide a simplified method for solving equation systems as compared to elementary algebraic operations for linear equations. 

This is, incidentally, the first example of a Lie algebra that we encounter. We will return to it later. Also we have set h = 1 in (1.19) to simplify the notation. 

This celebrated factorization lemma (Wyboume, 1974 Biedenham and Louck, 1981) allows one to simplify considerably the calculations in the molecular case. The algebras of interest are... 

Equation (28) can be simplified at the cost of a certain amount of accuracy. The second term on its right-hand side denotes the difference in the radii of the final bubble and the force-balance bubble, and its value is normally small. Similarly, the third term on the right-hand side is a multiplication of two small numbers and is hence very small. As these two terms have algebraically opposite signs, their difference can be neglected when compared with the first term on the right-hand side. With all these simplifications and by expressing rfb in terms of Vfb, Eq. (28) reduces to... 

The digital simulation of a distillation column is fairly straightforward. The main complication is the large number of ODEs and algebraic equations that must be solved. We will illustrate the procedure first with the simplified binary distillation column for which we developed the equations in Chap. 3 (Sec. 3.11). Equimolal overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODEs per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship). 

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