Variables and Functions

A complex variable z = x + iy can be defined to represent a point in two-dimensional space, where the x-axis is taken to be real, and the y-axis is imaginary. A function of a complex variable f(z) can then be defined, where  

A variable is a quantity in mathematical relationships, represented by a symbol, that may take on any value firom a given set of values. Variables are related to one another by functions. If for every value of the variable x there corresponds at least one value of the variable y, then y is a function of x, written y = y x) or y = fix). 

Functions may also involve more than two variables, so that if values are assigned to all but one variable (the independent variables), the value of the remaining one (the dependent variable) is fixed, i.e., in w = w x, y, z), x, y, and z are the independent variables and w the dependent. Usually the choice is arbitrary, that is, we can usually solve for x and find x = xiw, y, z), making x the dependent variable, and so on. In cases where it is difficult or impossible to solve explicitly for the desired dependent variable, we may write an implicit function /(x, y, re, z) = 0 and treat it as described, for example, by Dence (1975, p. 53). 

A function having three variables, x, y, and z, therefore has two independent variables, and can be said to be divariant, or to have two degrees of freedom. Divariance and two degrees of freedom refer to the fact that we are free to choose the values of two of the variables (perhaps within certain ranges), the third then being fixed by the functional relationship. For example, for the function 

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These terms are analogous to those on p. 265 of [7], It will be noted that the symbol c has been reinstated as in Section VI.F, so as to facilitate the order of magnitude estimation in the nearly nonrelativistic limit. We now proceed based on Eq. (168) as it stands, since the transformation of Eq. (168) to modulus and phase variables and functional derivation gives rather involved expressions and will not be set out here. 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

Both the openloop and the closedloop frequency-response curves can be easily generated on a digital computer by using the complex variables and functions discussed in ( han. 12, Tlie freauencv-resnnnse curves for the closedloon... 

With these definitions, both differing from Eqs. (8), we can still transform variables and functions according to Eq. (7). (The reason for the change from Eq. (8) to Eq. (11) is that we want the Uhlenbeck-Ornstein weighting to come out in standard form). The result is 2... 

Most systems involve several interconnected feedback loops. Such systems cannot be analyzed seriously without a proper formalism, but their detailed description using differential equations is often too heavy. For these reasons we (as many others before) turned to a logical (or Boolean) description, that is, a description in which variables and functions can take only a limited number of values, typically two (1 and 0). Section II is an updated description of a logical method ( kinetic logic ) whose essential aspects were first presented by Thomas and Thomas and Van Ham.2 A less detailed version of this part can be found in Thomas.3 The present paper puts special emphasis on the fact that for each system the Boolean trajectories and final states can be obtained analytically (i.e.,... 

But real systems are usually not simple feedback loops. In a virus such as bacteriophage the decision to kill the infected bacterial cell or to establish a symbiotic association with it depends on complex interactions involving a number of interconnected feedback loops. Such systems (and even simpler ones) would need a formal description in view of their complexity but as a matter of fact this complexity is such that the classical methods are much too heavy. This was a reason for trying a logical description, that is, a description using variables and functions which can take only a limited number of values—typically two (1 and 0). 

In order to illuminate both the phase problem and its solution, I will represent structure factors as vectors on a two-dimensional plane of complex numbers of the form a + ib, where i is the imaginary number (—1)1/2. This allows me to show geometrically how to compute phases. I will begin by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Then I will show how to represent structure factors as vectors on the same plane. Because we will now start thinking of the structure factor as a vector, I will hereafter write it in boldface (FM,Z) instead of the italics used for simple variables and functions. Finally, I will use the vector representation of structure factors to explain a few common methods of obtaining phases. 

Either Qi or Q2 or both may be variable and functions of z, or one of the two may be either constant or zero. For example, if liquid is discharged through an orifice or a pipe of area A under a differential head z, Q2 = Cd (2gz)m, where Cd is a numerical discharge coefficient and z is a variable. If the liquid flows out over a weir or a spillway of length B, Q2 = CBz3/2, where C is the appropriate coefficient. (For steady flow, z would be the constant H.) In either case z is the variable height of the liquid surface above the appropriate datum. In like manner, Qi may be some function of z. 

Phenomenological approaches have been very successful in some areas, e.g. Miedema theory for predicting many quantities in metallurgy. The essential task in phenomenological theories is to identify a suitable set of physically meaningful variables, which are linearly independent , to characterize the materials. Experimental data are then correlated against this set of variables and functional relations are fitted. 

Complex columns were defined in Chap. 3 and illustrated by Figs. 3-1 and 3-4. To illustrate the application of the 2N Newton-Raphson method to the solution of problems involving complex columns, consider the simple case where the sidestream Wt is withdrawn in the liquid phase from some interior plate p. The withdrawal of the sidestream gives rise to one specification in addition to those stated for conventional columns, in items 1 through 4 of Table 4-2. When this additional specification is taken to be either the total-flow rate Wx or the ratio Wl/Lp, the sets of specifications, independent variables, and functions for this complex column are the same as those stated in Table 4-2 except that either Wt or Wx /Lp should be added to each set of specifications. 

Table 4-2 Specifications, independent variables, and functions for conventional distillation columns...

Table 7-2 Sets of specifications, variables, and functions for the capital 0 method for systems... 

Set System specifications specifications System variables and functions ... 

Instead of regarding all of the reflux and boilup ratios as fixed, let one of them be varied, say VNi/Bu as required to satisfy the condition that QRl = QC2. The new variable VNi/Bx is added to the set of variables and the variable QC1 is removed from the set of variables and functions by replacing QC2 by QRi. Thus, to solve this problem, the variables are taken to be... 

For the case where one or more reactions occur on each stage of an absorber or distillation column and the vapor and liquid phases form highly nonideal mixtures, a formulation of the Almost Band Algorithm is recommended. In the present formulation for the case where one or more chemical reactions occur on each stage of an absorber, the following choice of N(2c + 1 + r) independent variables and N(2c 4-1 + r) independent functions are made. In particular, for the case of one chemical reaction per stage, the independent variables and functions are taken to be... 

When the independent variables and functions are ordered as indicated by Eqs. (10-20) and (10-21), a banded jacobian matrix is obtained in which most of the elements lie along the principal diagonal as shown in Fig. 10-1. 

In the choice of the independent variables and functions, care must be exercised in the treatment of the stages adjacent to the limiting conditions (the pinches r and s) in order to avoid singularities in the jacobian matrix of the Newton-Raphson equations. In particular, the following limiting conditions which appear in the material balances must be observed... 

This Jacobian is of a block tridiagonal form like (15-12) because functions for stage / are only dependent on output variables for stages /-1, j, and j+l. Each A, B, or C block in (15-67) represents a (2C-I-1) by (2C+ 1) submatrix of partial derivatives, where the arrangements of output variables and functions are... 

Both the openloop and the closedloop frequency response curves can be easily generated on a digital computer by using the complex variables and functions in FORTRAN discussed in Chapter 10 or by using MATLAB software. The frequency response curves for the closedloop servo transfer function can also be fairly easily found graphically by using a Nichols chart. This chart was developed many years ago, before computers were available, and was widely used because it greatly facilitated the conversion of openloop frequency response to closedloop frequency response. 

We inspect now the thermodynamic variables and functions of the various energy forms by making the respective extensive variable dimensionless. 

We need to choose which variables and family of functions to utilize, and we make those choices correctly if we think that the variables and function hierarchy selected somehow map onto the basic physical reality at hand (Norton 2003 656). 

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