Two Independent Variables

We will determine the dependence of the P.O.P. stack loss (to be denoted by y) upon the cooling water temperature, Xw, and the air temperature, Xa. 

C ) A full discussion of various aspects of correlation is to be found in Chapters 11 to 16 of Yule and Kendall s An Introduction to the Theory of Statistics (Charles Grilfin). 

It was assumed that the stack loss was to a first approximation related linearly to the tw o independent variables., so that it could be represented by an equation of the form 

If we had considered it more appropriate, we could, of course have used an equation of the type 

Taking the straightforward linear equation we will determine the values of these three coefficients which give the best general fit to the data. We proceed as follows. 

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This is the method used by the commercial software packages Crystal Ball and RISK . The method is ideally suited to computers as the description of the method will reveal. Suppose we are trying to combine two independent variables, say gross reservoir thickness and net-to-gross ratio (the ratio of the net sand thickness to the gross thickness of the reservoir section) which need to be multiplied to produce a net sand thickness. We have described the two variables as follows ... 

We have already seen the normal equations in matrix form. In the multivariate case, there are as many slope parameters as there are independent variables and there is one intercept. The simplest multivariate problem is that in which there are only two independent variables and the intercept is zero... 

What is the most meaningful way to express the controllable or independent variables For example, should current density and time be taken as the experimental variables, or are time and the product of current density and time the real variables affecting response Judicious selection of the independent variables often reduces or eliminates interactions between variables, thereby leading to a simpler experiment and analysis. Also inter-relationships among variables need be recognized. For example, in an atomic absorption analysis, there are four possible variables air-flow rate, fuel-flow rate, gas-flow rate, and air/fuel ratio, but there are really only two independent variables. 

Self-similarity applies to one-dimensional, time-dependent problems in which dependence on one of two independent variables can be eliminated by nondimen-... 

The double integral of a function with two independent variables is of the form... 

There are three basic classes of second-order partial differential equations involving two independent variables ... 

We can now proceed exactly as we did when there were only two independent variables, v and T, and find ... 

All the coefficients will, in general, be functions of both independent variables, and since we know that the heat absorbed depends on the path of change, it follows that the coefficients are not, in general, partial derivatives of a function of the two independent variables, for SQ would then be a perfect differential (cf. H. M., 115). 

We start by noting that any dependent thermodynamic variable Z is completely specified by two — and only two — independent variables X and Y (if n held constant). As an example, the molar volume of the ideal gas depends upon the pressure and temperature. Setting p and T fixes the value of Vm through the equation... 

Let us define the system as being characterized by an empirical temperature,aa 9 and any other number of variables, at, aa,. .., v . Initially, we will change only two independent variables, 9 and x. Our generalized example will be set up in such a way as to be consistent with the behavior of gas isotherms. That is, 9 will behave in a way analogous to temperature T, and the other variable, at, will be the analogue of pressure p. Then, we will consider the implications when 9 and another variable are allowed to change. 

The quantities dX and d Y are called differentials, the coefficients in front of dX and dT are called partial derivatives,11 and dZ is referred to as a total differential because it gives the total change in Z arising from changes in both X and Y. If Z were to depend upon additional variables, additional terms would be included in equation (A 1.1) to represent the changes in Z arising from changes in those variables. For much of our discussion, two variables describe the processes of interest, and therefore, we will limit our discussion to two independent variables, with the exception of the description of Pfaffian differentials in... 

This equation cannot be integrated directly since the temperature 9 is expressed as a function of two independent variables, distance jc and time t. The method of solution involves transforming the equation so that the Laplace transform of 6 with respect to time is used in place of 9. The equation then involves only the Laplace transform 0 and the distance jc. The Laplace transform of 9 is defined by the relation ... 

It follows that for a special value of one parameter, the observed value of y is independent of the second parameter. This happens at Ii= a2/ai2 or I2 = -ai/ai2 any of these values determines y= a -aia2/ai2, the so called isoparametrical point. The argument can evidently be extended to more than two independently variable parameters. Experimental evidence is scarce. In the field of extrathermodynamic relationships, i.e., when j and 2 are kinds of a constants, eq. (84) was derived by Miller (237) and the isoparametrical point was called the isokinetic point (170). Most of the available examples originate from this area (9), but it is difficult to attribute to the isoparametrical point a definite value and even to obtain a significant proof that a is different from zero (9, 170). It can happen—probably still more frequently than with the isokinetic temperature—that it is merely a product of extrapolation without any immediate physical meaning. 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

An alternating direction scheme. Further developments are concerned with the heat conduction equation of two independent variables that can serve as test vehicles for the difference schemes to be presented ... 

Given the absence of correlation between the sensitivity and range descriptors, we also examined whether a two-variable equation would improve on Eq. (2). As shown by Eq. (3), the inclusion of two independent variables in the same equations improved their predictive capacity ... 

The main process variables in differential contacting devices vary continuously with respect to distance. Dynamic simulations therefore involve variations with respect to both time and position. Thus two independent variables, time and position, are now involved. Although the basic principles remain the same, the mathematical formulation, for the dynamic system, now results in the form of partial differential equations. As most digital simulation languages permit the use of only one independent variable, the second independent variable, either time or distance is normally eliminated by the use of a finite-differencing procedure. In this chapter, the approach is based very largely on that of Franks (1967), and the distance coordinate is treated by finite differencing. 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

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. 

Equation (2.28) is now separable into two independent differential equations, one for each of the two independent variables x and t. The time-dependent equation is... 

The two independent variables, the pH and the potential, are of major concern in the construction of Pourbaix diagrams. The method and details of construction of a potential-pH diagram are available in many standard texts hence, this presentation focuses attention on the conditions and the principles behind leaching in order to determine the likely products of leaching reactions. 

For example, if we have two independent variables (x, and x2) the following four (22) experiments can be readily designed ... 

When the relationship for the response Y is given as a function of two independent variables, Xi and X2,... 

The two independent variables (the axes) show the pump speeds for the two reagents required in the analysis reaction. The initial simplex is represented by the lowest triangle the vertices represent the spectro-photometrie response. The strategy is to move toward a better response by moving away from the worst response. Since the worst response is 0.25, conditions are selected at the vortex, 0.6, and, indeed, improvement is obtained. One can follow the experimental path to the optimum, 0.721. 

This optimization method, which represents the mathematical techniques, is an extension of the classic method and was the first, to our knowledge, to be applied to a pharmaceutical formulation and processing problem. Fonner et al. [15] chose to apply this method to a tablet formulation and to consider two independent variables. The active ingredient, phenylpropanolamine HC1, was kept at a constant level, and the levels of disintegrant (corn starch) and lubricant (stearic acid) were selected as the independent variables, X and Xj. The dependent variables include tablet hardness, friability, volume, in vitro release rate, and urinary excretion rate in human subjects. 

Although the Lagrangian method was able to handle several responses or dependent variables, it was generally limited to two independent variables. A search method of optimization was also applied to a pharmaceutical system and was reported by Schwartz et al. [17], It takes five independent variables into... 

For this last stage, the one-at-a-time procedure may be a very poor choice. At Union Carbide, use of the one-at-a-time method increased the yield in one plant from 80 to 83% in 3 years. When one of the techniques, to be discussed later, was used in just 15 runs the yield was increased to 94%. To see why this might happen, consider a plug flow reactor where the only variables that can be manipulated are temperature and pressure. A possible response surface for this reactor is given in Figure 14-1. The response is the yield, which is also the objective function. It is plotted as a function of the two independent variables, temperature and pressure. The designer does not know the response surface. Often all he knows is the yield at point A. He wants to determine the optimum yield. The only way he usually has to obtain more information is to pick some combinations of temperature and pressure and then have a laboratory or pilot plant experimentally determine the yields at those conditions. 

In the geometric method1,11 experimental results are used to minimize the region in which the optimum exists. The response is obtained for a number of points that are located very near one another. The number of points should be one greater than the number of independent variables. From the results a surface (this is aline when there are two independent variables) representing a constant value of the response is constructed. This method hypothesizes that on one side of this surface will be all the pointsthatyieldabetterresponse,andthereforetheoptimummustlieonthatsideofthe surface. 

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