Dependent and Independent Variables

Before concluding our discussion, remember the distinction that has to be made between dependent and independent variables. This is necessary in the context of differential equations, which for ODEs generally lead to solutions of the following form  

The dependent variable u — also termed a state variable — is, for mass transfer operations, usually represented by the concentration of a system, or its total mass. Concentration can be expressed in a variety of ways, the most common being kilogram per cubic meter (kg/m ) or mole per cubic meter (mol/m ), or in terms of mole and mass fractions and ratios. Remember that the number of dependent variables equals the number of unknowns. For a system to be fully specified, the number of equations must therefore equal the number of unknowns, in other words, the number of dependent variables. The model is then said to be complete. 

The independent variables are usually represented by time t and distance X, y, z or, in the case of radial coordinates, by the radial distance variable r. Occasionally, distance may depend on time and then becomes the dependent variable. This is the case, for example, with spherical particles that undergo a change in size due to reaction, dissolution, or deposition of material. The attendant change in mass is then expressed by the derivative 

Another departure from the normal definition of variables occurs when two first-order differential equations are combined by division into a single ODE. Consider, for example, the system 

Mass Transfer and Separation Processes Principles and Applications 

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Equation 11-17 is refeiTed to as a Lineweaver-Burk equation involving separate dependent and independent variables 1/v and 1/Cg, respeetively. Equation 11-17 ean be further rearranged to give... 

As we will soon see, the nature of the work makes it extremely convenient to organize our data into matrices. (If you are not familiar with data matrices, please see the explanation of matrices in Appendix A before continuing.) In particular, it is useful to organize the dependent and independent variables into separate matrices. In the case of spectroscopy, if we measure the absorbance spectra of a number of samples of known composition, we assemble all of these spectra into one matrix which we will call the absorbance matrix. We also assemble all of the concentration values for the sample s components into a separate matrix called the concentration matrix. For those who are keeping score, the absorbance matrix contains the independent variables (also known as the x-data or the x-block), and the concentration matrix contains the dependent variables (also called the y-data or the y-block). 

Whenever a linear relationship between dependent and independent variables (ordinate-resp. abscissa-values) is obtained, the straightforward linear regression technique is used the equations make for a simple implementation, even on programmable calculators. 

In many applications the goal of predictive modelling is not a detailed understanding of the relation between dependent and independent variables. Ability to interpret the model, therefore, is not a requirement perse. This should not preclude the exploitation of available background knowledge on the problem at hand during calibration modelling. A model that can be sensibly interpreted certainly adds value and confidence to the calibration result. 

In certain circumstances, the model equations may not have an explicit expression for the measured variables. Namely, the model can only be represented implicitly. In such cases, the distinction between dependent and independent variables becomes rather fuzzy, particularly when all the variables are subject to experimental error. As a result, it is preferable to consider an augmented vector of measured variables, y, that contains both regressor and response variables (Box. 

The dimensionality of a functional relationship will be defined here (ax-iomatically) by the number of dependent and independent variables in such a function. Therefore, functions of the type a = f(b) are two-dimensional, of the type a = f(bi,b2) three-dimensional, and of the type a = /(fri, b2y... bn) are (n + 1 )-dimensional. The representation of various realizations of only one variable (either y or z) is one-dimensional (Danzer et al. [2002]). 

Alternative characterization of the dimensionality. The SIMS example demonstrates that the dimensionality of analytical information and of signal functions occasionally follow other principles than those given above, where the dimensionality of a functional relationship is determined by the number of dependent and independent variables in such a function. 

The situation is illustrated in Fig. 6.4 from which can be seen that the reversion of the dependent and independent variables gives different estimates y and x. 

In order to compute error propagation, we must evaluate the partial derivatives with respect to both the dependent and independent variables. This will be more clearly seen by differentiating equation (4.3.24)... 

Unfortunately, many transformations purported to linearize the model also interchange the role of dependent and independent variables. Important examplE are the various linearization transformations of the simple steady-state Michaelis-Menten model... 

If the assumption of neglecting errors in independent variables cannot tie justified, there is no statistical distinction between dependent and independent variables. Then we rather use the vector z = (z ,Z2,..., znz)7 to denote the variables of the model written in the more general implicit form... 

By casting the governing equation in nondimensional form, important insights about relative scales and the contributions various terms can be revealed. One has choices in the establishment of reference scales on which the nondimensional. Generally, the objective is to select scales such that the nondimensional dependent and independent variables are order one. Thus the selection of reference scales requires some understanding of the class of problems for which nondimensionalization is sought. 

The scaling factors for the nondimensionalization have been chosen so that the dependent and independent variables are roughly order one. There are two parameters that remain in the equations the Reynolds number, Re, and the wedge angle, a. Depending on the magnitudes of the parameters, there can be further simplifications of the system. 

While Scatchard plots may be useful to display the binding data, they are not a useful way to analyze them. It is usually preferred to fit the raw binding data with a nonlinear regression. Selection of the dependent and independent variables... 

Aris (3) more formally defined a mathematical model thus a system of equations, S, is said to be a model of prototypical system, S, if it is formulated to express the laws of S and its solution is intended to represent some aspect of the behavior of S. Seinfeld and Lapidus (4) gave a more specific definition Mathematical model is taken to mean the formulation of mathematical relationships, which describe the behavior of actual systems such that the dependent and independent variables and parameters of the model are directly related to physical and chemical quantities in the real system. ... 

And the following are functions of the dependent and independent variables of stage j ... 

Figure 6. Variations of dependent and independent variables, pyrolysis reactor...

Himmelblau [32] and Himmelblau and Bischoff [33] have considered three types of model which are useful in process analysis, i.e. empirical models, population balance models and transport phenomena models. Empirical models involve mathematical relationships between dependent and independent variables, which are postulated either entirely a priori, or by considering the nature of the experimental data, or by analogies, etc. On the other hand, transport phenomena models are based on the laws of... 

The procedure above seems rather cumbersome. Would it not be possible simply to carry out the regression, reversing the dependent and independent variables and get directly to an equation that predicts PABA concentration from absorption We could do this, but the fitted line would be slightly different and would not be properly optimized. The correct procedure is the one shown above. 

Integral PFR data cannot be used directly in this way, since one has a differential equation that describes the conversion profile along the catalyst bed (eq 2). For simple cases this can be integrated analytically, yielding an implicit expression in the observed variable (eq S). Sometimes the independent variable W/Ff is now used as observed variable and its SSR minimized [9], but this interchange of dependent and independent variable destroys the error properties and the parameter error limits are not correct. Often the parameter estimates correspond well [9] and can be used as starting guesses for more robust minimization to determine the real error bounds. 

Mathematical models that represented the relationships between the dependent and independent variables were developed. Once the model... 

A comparison of this equation with Equation 1) demonstrates that functional relationships between dependent and independent variables can be quite different for CSTR s and batch reactors, even with the same reaction system and the same kinetic mechanisms. 

A plant model is a complex mathematical relationship between the dependent and independent variables of the process in a real unit. These are obtained by the assembly of one or more process models. 

When the preliminary steps of the statistical model have been accomplished, the researchers must focus their attention on the problem of correlation between dependent and independent variables (see Fig. 5.1). At this stage, they must use the description and the statistical selections of the process, so as to propose a model state with a mathematical expression showing the relation between each of the dependent variables and all independent variables (relation (5.3)). During this selection, the researchers might erroneously use two restrictions Firstly, they may tend to introduce a limitation concerning the degree of the polynomial that describes the relation between the dependent variable y( and the independent variables Xj, j = l,n Secondly, they may tend to extract some independent variables or terms which show the effect of the interactions between two or more independent variables on the dependent variable from the above mentioned relationship. 

The problem of simplifying the regression relationship can be omitted if, before establishing those simplifications, the specific procedure that defines the type of the correlations between the dependent and independent variables of the process, is applied on the basis of a statistical process analysis. 

The extreme values = 1 ryx = —1) for the correlation coefficient show that a linear relationship exists between the dependent and independent variables. 

For a general discussion of characteristics and for the definition of characteristic surfaces with an arbitrary number of dependent and independent variables, see, for example, [56]. The use of the method of characteristics in solving equations is considered in [57]. 

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