State variable, definition

The state variables define a point on the diagram the "constitution point". If this point is given, then the equilibrium number of phases can be read off. So, too, can their composition and the quantity of each phase - but that comes later. So the diagram tells you the entire constitution of any given alloy, at equilibrium. Refer back to the definition of eonstitution (p. 311) and check that this is so. 

In these cases there is no well defined notion of a looser constraint, the choice is then either to force those variables to be equal in x and y, or to find some path from their value to a constraint on another inter- or intrasituational variable and thus be able to show that their values in jc, y should obey some ordering based on these other constraints. This topic is the subject of current research, but is not limiting in the flowshop example, since no such constraints exist. Lastly, it is not enough to assert conditions on the state variables in x and y, since we have made no reference to the discrete space of alternatives that the two solutions admit. Our definition of equivalence and dominance constrains us to have the same set of possible completions. For equivalence relationships the previous statement requires that the partial solutions, x and y, contain the same set of alphabet symbols, and for dominance relations the symbols of JC have to be equal to, or a subset of those of y. Thus our sufficient theory can be informally stated as follows ... 

These implications types correspond closely to the definitions of state variable types we gave in Section II, E. Those variables that appear in the... 

Definition of parameters and their values are listed in Table 4. For simplicity in the notation, the time dependence of state variables as well as the dilution... 

According to the definitions given in Fig. 12.5, the input/is not influenced by what happens in the system it is a so-called external variable. This is not true for the two elimination processes, O and RtoV Thus, in order to solve Eqs. 12-42 or 12-43 we have to determine how these terms depend on the system state. The only quantities which characterize the system state are (M or C. In fact, there is only one independent state variable since M and C are proportional to each other. 

Let us discuss the meaning of some of the expressions which appear in the above definition (see Box 21.1). A subunit of the environment can be any spatial compartment of the world, from the whole planet to a single algal cell floating in the ocean. The term state variable refers to those properties which are used to characterize the... 

Figure 21.8 Two-box model of trwo completely mixed environmental compartments. Definition of subscripts first subscript (7 or j) designs the compound, the second (1 or 2) the box. Transfer fluxes Tcarry three subscripts. For instance, T ll describes the interbox flux of variable i from box 1 to box 2. X and Y denote other chemicals which are not state variables.

The foregoing chapters mark a long and not yet finished journey through the special field of the chemical kinetics of solids. It differs from the more common textbooks on kinetics not only because of the immense variety of crystalline phases, but even more in view of the ambiguity in the definition of the correct number of independent thermodynamic state variables. This is the source of many difficulties and particularly with solids containing one or more immobile components or multiphase systems composed of coherent or semicoherent crystals. In coping with this inherent complexity in the foregoing chapters, we chose to restrict ourselves mainly to the fundamental aspects rather than to present many uncorrelated details. 

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... 

Definition of variables relevant to the process state variables, such as cells, substrates, and product concentrations, that characterize the system studied and operational variables, that represent particular conditions of the system, that may be initial or fixed conditions such as initial concentrations, feeding rates, etc. ... 

Loosely speaking, a manifold of dimension (n-k) is a set of points in an n-dimensional space defined by k < n equations. Suppose that, after some algebraic manipulations, the plant model given by Eqs. (9.11) to (9.15) and the sensitivity definition (9.4) are reduced to one equation with one unknown aP/A. Then, the dependence of the state variable aP/A versus one parameter (the volume V) as defined by the model equation can be graphically depicted, for example as in Figure 9.6. The plot is a one-dimensional manifold in the two-dimensional space... 

Biomass concentration is of paramount importance to scientists as well as engineers. It is a simple measure of the available quantity of a biocatalyst and is definitely an important key variable because it determines - simplifying - the rates of growth and/or product formation. Almost all mathematical models used to describe growth or product formation contain biomass as a most important state variable. Many control strategies involve the objective of maximizing biomass concentration it remains to be decided whether this is always wise. 

In other cases, the data trajectories are translated into trend-qualities via shape descriptors such as glucose uptake rate is decreasing (concave down) while RQ is increasing (linear up) and. .. These combinations of trends of the trajectories of various state variables or derived variables define a certain physiological state the advantage of this definition is that the association is no longer dependent on time and on the actual numerical values of variables and rates [413]. 

The zeroth law of thermodynamics involves some simple definition of thermodynamic equilibrium. Thermodynamic equilibrium leads to the large-scale definition of temperature, as opposed to the small-scale definition related to the kinetic energy of the molecules. The first law of thermodynamics relates the various forms of kinetic and potential energy in a system to the work which a system can perform and to the transfer of heat. This law is sometimes taken as the definition of internal energy, and introduces an additional state variable, enthalpy. 

The first law of thermodynamics allows for many possible states of a system to exist. But experience indicates that only certain states occur. This leads to the second law of thermodynamics and the definition of another state variable called entropy. The second law stipulates that the total entropy of a system plus its environment can not decrease it can remain constant for a reversible process but must always increase for an irreversible process. 

These occurrences are formulated using big-M constraints, which require the definition of a tolerance to identify each state (within the given tolerance between the desired value and the variable value, the state variable is activated). Moreover, each variable that occurs in these constraints is now modeled as an integer variable (tank inventory, input and output volumes), enabling the definition of a tolerance lower than 1. 

The definition of a full or empty tank is applied for specific cases (tank inventory is at its limits). The remaining states are exclusive at any moment, i.e., each tanks is always either in a filling up cycle or in an emptying cycle (idle time intervals may occur in both cycles). Finally, whenever the tank is full and the corresponding state variable is activated, it also controls the settling period accomplishment. 

As follows from the Thom theorem, each structurally stable function of one state variable, dependent on at most five control parameters, must be equivalent to one of the functions listed in Table 2.2. Recall that functions are considered to be equivalent if they have identical sets of critical points. Another, equivalent definition, which will help to understand better the meaning of local equivalence of a function near to a critical point, is given below. Before that, however, let us examine two examples. 

We could now institute an extensive search of possible thermodynamic functions in the hope of finding a function that is a state variable and also has the property that its rate of internal generation is a positive quantity. Instead, we will just introduce this new thermodynamic property by its definition and then show that the property so defined has the desired characteristics. 

Since state variables have fixed values in equilibrium states and have changes between equilibrium states that do not depend on how the change is carried out, it follows that the differentials of state variables will always be exact differentials, according to our definitions in Chapter 2. 

Systems at equilibrium have literally dozens of properties. In addition to those state variables applicable to the system as a whole such as T and V, each phase within the system has a host of properties such as heat capacity, cell size, optic angle, refractive index, and so on, and all can be considered properties of the system according to our definition. 

Since A is defined in terms of state variables, it is itself a state variable, and its differential is exact. From the definition, the differential dA is... 

Expressions for Growth Rate. At this stage, a model must be introduced if further progress is to be made, and we choose a deterministic model. All deterministic models which have been introduced (with two exceptions) have been autonomous they assume that the growth rates are explicit functions only of the state of the system, and not of time. But state in an unstructured model can only refer to population density or to concentration of protoplasm, since, by definition, a model is unstructured if only one state variable appears. Thus, the model is... 

State Variable,. A state variable is one that has a definite value when the state of a system is specified—... 

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