Variables, dependent independent

A single experiment consists of the measurement of each of the g observed variables for a given set of state variables (dependent, independent). Now if the independent state variables are error-free (explicit models), the optimization need only be performed in the parameter space, which is usually small. 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

Observation number Parameter number Independent variables Dependent variable Calculated variable... 

The 0]/0 and 02/0 ratios also do not depend upon 9, the empirical temperature. However, in the case of this variable, the independence does not arise because both the numerator and denominator are independent of 9, but rather because the dependence cancels out in the ratio. This arises most generally when both numerator and denominator in each ratio are of the following form ... 

The original intensity of the radiation is defined as t. A part of the intensity is absorbed, another part is transmitted, still another part is scattered, and a part of the total intensity is refiected. The components, S and T, are processes which are independent of the wavelength (firequency) of the incident photons, whereeis R and A are primarily wavelength dependent. It is here that the factor of "color" arises. The exact amount of energy extracted from lo by each process is a complex set of variables depending upon the type and arrangement of atoms composing the solid. 

In optimization using a process simulator to represent the model of the process, the degrees of freedom are the number of decision variables (independent variables) whose values are to be determined by the optimization, hence the results of an optimization yield a fully determined set of variables, both independent and dependent. Chapter 2 discussed the concept of the degrees of freedom. Example 15.1 demonstrates the identification of the degrees of freedom in a small process. 

Most graphs typically plot two variables the independent variable (or cause ) and the dependent variable (or effect ). 

The variables that are to be set must of course be independent. The first one set is naturally independent, but all other variables that are assigned values could be dependent on those previously set. Thus, care must be exercised in choosing the variables to be set fortunately, in most problems which can be solved directly, it is relatively easy to determine which variables are independent. 

A different classification scheme for DEs, short for differential equations, separates those DEs with a single independent variable dependence, such as only time or only 1-dimensional position, from those depending on several variables, such as time and spatial position. DEs involving a single independent variable are routinely called ODEs, or ordinary differential equations. DEs involving several independent variables such as space and time are called PDEs, or partial differential equations because they involve partial derivatives. 

The state of the system is defined in terms of certain state variables. The state of the system is then fixed by assigning definite values to sufficient variables, chosen to be independent, so that the values of all other variables are fixed. The number of independent variables depends in general upon the problem at hand and upon the system with which we are dealing. The... 

We have implied in Section 1.1 that certain properties of a thermodynamic system can be used as mathematical variables. Several independent and different classifications of these variables may be made. In the first place there are many variables that can be evaluated by experimental measurement. Such quantities are the temperature, pressure, volume, the amount of substance of the components (i.e., the mole numbers), and the position of the system in some potential field. There are other properties or variables of a thermodynamic system that can be evaluated only by means of mathematical calculations in terms of the measurable variables. Such quantities may be called derived quantities. Of the many variables, those that can be measured experimentally as well as those that must be calculated, some will be considered as independent and the others are dependent. The choice of which variables are independent for a given thermodynamic problem is rather arbitrary and a matter of convenience, dictated somewhat by the system itself. 

We see from these equations that we deal with a large number of variables in thermodynamics. These variables are E, S, H, A, G, P, V, T, each nh and two variables for each additional work term. In any given thermodynamic problem only some of these variables are independent, and the rest are dependent. The question of how many independent variables are necessary to define completely the state of a system is discussed in Chapter 5. For the present we take the number of moles f each component, the generalized coordinate associated with each additional work term, and two of the four variables S, T, P, and V as the independent variables. Moreover, not all of... 

A significant part of developing a model used for other than determining static sets of heat and material balances (which are sufficient for some model objectives, such as providing the basis for new plant design) is specifying which variables are independent and which are dependent. Far more variables are dependent variables than are independent in essentially all models. For simulation and optimization... 

In most multiparameter instrumental techniques, the parameters can be classified into two types independent and dependent. Independent parameters can be optimized independently from all other parameters and can therefore be subjected to a univariate approach i.e., the variable can be adjusted until the largest signal-to-noise ratio (SNR) is obtained and set at that value for the best instrumental performance. This is the simplest situation and can be handled in a very straightforward manner. 

Partial Least Squares Regression is one of the many available regression techniques. Regression techniques are used to model the relation between 2 blocks of variables, called independent or x variables and dependent or y variables (figure 12.15 a). The general regression equation is (figure 12.15 b) ... 

A graph pictorially shows the relationship between two variables, the independent variable and the dependent variable. The independent variable is the variable you control. The dependent variable is the one that changes in response to the independent variable. A graph is set up so the values for the independent variable are found along the x- axis (horizontal) and the dependent variable values are found along the y- axis (vertical). 

In this reaction we cannot alter any one of the mole fractions (xn2, xh2 or xnh3) without automatically altering the other two, thus of the three variables each one of them is dependent on the other two, so that we have one dependent variable. Put another way only two of the variables are independent. Thus in this example ... 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

The presence of both mutually dependent (mixture) and independent (process) variables calls for a new type of regression model that can accommodate these peculiarities. The models, which serve quite satisfactorily, are combined canonical models. They are derived from the usual polynomials by a transformation on the mixture-related terms. To construct these types of models, one must keep in mind some simple rules these models do not have an intercept term, and for second-order models, only the terms corresponding to the process variables can be squared. Also, despite the external similarity to the polynomials for process variables only, it is not possible to make any conclusions about the importance of the terms by inspecting the values of the regression coefficients. Because the process variables depend on one another, the coefficients are correlated. Basically, the regression model for mixture and process variables can be divided into three main parts mixture terms, process terms, and mixture-process interaction terms that describe the interaction between both types of variables. To clearly understand these kinds of models, the order of the mixture and process parts of the model must be specified. Below are listed some widely used structures of combined canonical models. The number of the mixture variables is designated by q, the number of the process variables is designated by p, and the total number of variables is n = q + p. 

The Clausius-Duhem equation is the fundamental inequality for a single-component system. The selection of the independent constitutive variables depends on the type of system being considered. A process is then described by solving the balance equations with the constitutive relations and the Clausius-Duhem inequality. 

---