Equality independent variables

Sinee E and A are independent variables, tlieir variations are arbitrary. Henee, for the above equality to be satisfied, eaeh of the two braeketed expressions must vanish when the (E, N) partition is most probable. The vanishing of the eoeffieient of dE implies the equality of temperatures of I and II, eonsistent with thennal equilibrium ... 

The isotherm under test is then re-drawn as a t-plot, i.e. a curve of the amount adsorbed plotted against t rather than against p/p° the change of independent variable from p/p° to t is effected by reference to the standard t-curve. If the isotherm under test is identical in shape with the standard, the t-plot must be a straight line passing through the origin its slope = b say) must be equal to nja, since the number of molecular layers is equal to both t/ff and n/n ... 

Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 18.5) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points ... 

Having phases together in equilibrium restricts the number of thermodynamic variables that can be varied independently and still maintain equilibrium. An expression known as the Gibbs phase rule relates the number of independent components C and number of phases P to the number of variables that can be changed independently. This number, known as the degrees of freedom f is equal to the number of independent variables present in the system minus the number of equations of constraint between the variables. 

Thus the problem involves the two independent variables, time t and length Z. The distance variable can be eliminated by finite-differencing the reactor length into N equal-sized segments of length AZ such that N AZ equals L, where L is the total reactor length. 

The left-hand side of equation (2.28) is a function only of t, while the right-hand side is a function only of x. Since x and t are independent variables, each side of equation (2.28) must equal a constant. If this were not true, then the left-hand side could be changed by varying t while the right-hand side remained fixed and so the equality would no longer apply. For reasons that will soon be apparent, we designate this separation constant by E and assume that it is a real number. 

To a first approximation the three terms in equation (1.46) and (1.47) can be treated as independent variables. For a fixed value of n Figure 1.8 Indicates the influence of the separation factor and capacity factor on the observed resolution, when the separation factor equals 1.0 there is no possibility of any separation. The separation factor is a function of the distribution coefficients of the solutes, that is the thermodynamic properties of the system, and without some... 

If the equality constraint involves independent variables and parameters in an algebraic model, i.e.. it is of the form, model equations reduces the number of unknown parameters by one. 

Each term on the left-hand side of Eq. (30) is a function of only a single independent variable. Each term is, therefore, equal to a constant, such that sx + y+ez = e. The first term is identified as... 

The left-hand side of Eq. (7) does not depend on the time r it is only a function of the coordinate x. On the other hand, the right-hand side of this equation depends only on the time. As t and x are independent variables, each side of Eq. (7) must be equal to a constant Furthermore, it must be the same constant, if Eq. (7) is to hold. This argument which will be employed often in subsequent examples, is the basis of the method of the separation of variables. Clearly, the constant in question can be chosen at will. For convenience in this example, it will be set equal to —ca1. 

Because each of the terms in Eq. (19) involves a different independent variable but all terms are equal, each term must be equal to the same constant. Thus, we may write... 

An independent variable. The number of degrees of freedom possessed by a replicate set of results equals the total number of results in the set. When another quantity such as the mean is derived from the set, the degrees of freedom are reduced by one, and by one again for each subsequent derivation made. 

Note that two functions of independent variables can be equal only when they are constant. Later developments show that the constant must be negative. Accordingly, the solution of the partial differential equation is reduced to that of the pair of ordinary ones,... 

The two functions of separate independent variables can equal each other only when they equal the same constant. The assumed negativity is substantiated later. The solutions of the two ODEs making up (8) are,... 

For each of the following six problems, formulate the objective function, the equality constraints (if any), and the inequality constraints (if any). Specify and list the independent variables, the number of degrees of freedom, and the coefficients in the optimization problem. 

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. 

Geometry of a quadratic objective function of two independent variables—elliptical contours. If the eigenvalues are equal, then the contours are circles. 

Various rules of thumb and empirical correlations exist to assist in making initial guesses for the values of the independent variables. All the values of the feeds here can be assumed to be equal initially. If the reflux ratio is selected as an independent variable, a value of 1 to 1.5 times the minimum reflux ratio is generally appropriate. 

To this point we isolated four variables D, v, Ap, and/, and have introduced three equality constraints—Equations (d (e), and (/)—leaving 1 degree of freedom (one independent variable). To facilitate the solution of the optimization problem, we eliminate three of the four unknown variables (Ap, v, and/) from the objective function using the three equality constraints, leaving D as the single independent variable. Direct substitution yields the cost equation... 

Equality constraints. Only 1 degree of freedom exists in the problem because there are three constraints x is designated to be the independent variable. 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

The program starts from the known concentration and temperature conditions at the top of the column and integrates down the column until the condition Yin (calculated) is equal to the known inlet value Yin. A variable Yheigh, is defined in the program to locate the YIN position. Note the use of HEIGHT, instead of TIME, as the independent variable in this steady state example. 

At steady state, die left-hand side of this expression has independent variables ip. For fixed ip = ip, the integral on the right-hand side sweeps over all fluid elements in search of those whose concentrations ),atcll are equal to ip Af these fluid elements have the same age (say, a = o ), then the joint PDF reduces to Xoutiet) = E(a ),... 

We have considered in some detail in Section 4.2 the case where the random vector Y of n ancillary or dependent variables relates linearly to those of a vector X of n principal or independent variables (e.g., raw data) with covariance matrix L through the matrix equality... 

Since the two members in the last equation cannot be functions of the independent variables x and t, they must be equal to a same constant, which suggests using an exponential form for/(f) and a trigonometric form for g(x). The diffusion equation is indeed identically verified for... 

Eqs. (2), (3) are appropriate for the canonical ensemble, with T and 8 the fixed independent variables. However, if the adsorbed layer is in thermal equilibrium with an external adsorbate gas reservoir, it is the pressure of this gas which is controlled experimentally, rather than 8. Since the chemical potential of the gas equals the chemical potential /i of the layer, ft rather than 8 is... 

One other alternative for obtaming derivatives from experimental data is to fit the data to a function by the method of least squares, either linear or nonlinear, and then to obtain the derivative analytically. We carried out both procedures for Exercise 18.4(c), and the different procedures agreed very well. Another alternative is to use a software package for numerical differentiation that does not require equal intervals in the independent variable. In any case, it is preferable to use more than one method. 

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