Algebraic Example

To illustrate this we consider first the simultaneous equations 

In Table 17.1 we have calculated values of m and n for various values of p, and you will note that the constant of proportionality between p and (which is 1 /fcifc2) 

All we have done, really, is to retain the Henry s Law relation 

These solutions are a little too concentrated to illustrate the dilute solution properties we are discussing, but they serve as a basis for calculating the fugacity of HCl in more dilute solutions. This is done using values of 7 (to be defined shortly see 17.2.4), which are measured electrochemically.  

These were divided by 7 at each concentration to obtain a value of k. The average value of k in this range is 0.000704, and this value was used to calculate /hci at each concentration as (0.000704 7 mHCi) - 

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The intensity setting of an input is called a level. It is possible for an input to be at different levels at different times. Thus, if x designates the x in the previous algebraic example, it might have had the value 0 when we were first interested in the system now we might want x to have the value 5. To designate these different sets of conditions, a second subscript is added. Thus, x, = 0 means that in the first instance, x, = 0 and x,2 = 5 means that in the second instance, x, = 5. 

TABLE 9.1 Dirac Enclosure Symbols for Different Types of Mathematical Object, with Matrix-Algebraic Examples... 

In the class of methods proposed by Broyden, the partial derivatives df/dxj in the jacobian matrix Jk of Eq. (4-29) are generally evaluated only once. In each successive trial, the elements of the inverse of the jacobian matrix are corrected by use of the computed values of the functions. An algebraic example will be given after the calculational procedure proposed by Broyden has been presented. 

Example 4-8 consists of a simple algebraic example which illustrates the application of this method. 

As originally proposed, the sparsity of the jacobian matrix is destroyed by Broyden s method. Two procedures (or modifications) which preserves the sparsity of the jacobian matrices are presented. The procedures are demonstrated by use of simple algebraic examples and applied to the solution of distillation problems whose jacobian matrices are sparse. 

To demonstrate the application of this algorithm, the following algebraic example is used. In order to reduce the arithmetic required to demonstrate the application of the algorithm, a very simple example was selected whose solution is seen by inspection to be Xj = 1, x2 = yjl, and x3 = y/3. 

Motivation R-matrix theory Electronic model Basis set expansion Matrix algebra Examples Further comments Acknowledgment Appendix... 

This algebraic example illustrates the salient features of the perturbation method. In the next example, we apply the method to an elementary differential equation. 

One can write acid-base equilibrium constants for the species in the inner compact layer and ion pair association constants for the outer compact layer. In these constants, the concentration or activity of an ion is related to that in the bulk by a term e p(-erp/kT), where yp is the potential appropriate to the layer [25]. The charge density in both layers is given by the algebraic sum of the ions present per unit area, which is related to the number of ions removed from solution by, for example, a pH titration. If the capacity of the layers can be estimated, one has a relationship between the charge density and potential and thence to the experimentally measurable zeta potential [26]. 

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

The algebraic form of the expression (9.24) for the enhancement factor is specific to the particular reaction rate expression we have considered, and corresponding results can easily be obtained for other reactions in binary mixtures, for example the irreversible cracking A—2B. ... 

Note that in contrast to the example shown in Section 2.2,2 the element stiffness equation obtained for this problem is not symmetric. After the substitution for the shape functions and algebraic manipulations... 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

Some systems can give quantitative results from known pieces of data complete with proper units. For example, these systems can take all the starting information and then determine a set of equations from the available list that can yield the desired result. The program could subsequently convert units or algebraically solve the equations if necessary. 

Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as... 

The generalizations demonstrated in the preceding example can be proved without resorting to numbers, but we have looked at enough algebra in this chapter already without adding these manipulations as well ... 

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. 

Differential-Algebraic Systems Sometimes models involve ordinary differentia equations subject to some algebraic constraints. For example, the equations governing one equihbrium stage (as in a distillation column) are... 

Comparison with Eq. (4-20) provides an example of the parallelism that exists between the eqnaOons for a constant-composition sohiOon and those for the corresponding partial properties. This parallelism exists whenever the sohidon properties in the parent equation are related hnearly (in the algebraic sense). Thus, in view of Eqs. (4-17), (4-18), and (4-19) ... 

Simultaneous reactions. The overall rate is the algebraic sum of the rates of the individual reactions. For example, take the three reactions ... 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

We need this speeial algebra to operate on the engineering equations as part of probabilistie design, for example the bending stress equation, beeause the parameters are random variables of a distributional nature rather than unique values. When these random variables are mathematieally manipulated, the result of the operation is another random variable. The algebra has been almost entirely developed with the applieation of the Normal distribution, beeause numerous funetions of random variables are normally distributed or are approximately normally distributed in engineering (Haugen, 1980). 

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