Independent variable, defined

By the components of the system we are to understand the least number of independently variable constituents, in terms of which the composition of every phase in the system can be completely specified. The number of components will therefore contribute to the total number of independent variables defining the state of chemical and physical equilibrium of the system. It is not necessary that the components shall be actual constituents of the system all that is required is that they shall be independently variable, Le.y the least number has been chosen. Thus, in systems composed of solid fuming sulphuric acid in presence... 

A matrix of independent variables, defined for a linear model in Eq. (29) and for nonlinear models in Eq. (43) Transpose of matrix X Inverse of the matrix X Conversion in reactor A generalized independent variable... 

The values of these parameters can be described as functions of the feed. In order to collapse the data from the cumene-decane mixtures and the cumene-naphthalene mixtures into a single curve, a single variable has been defined to describe the nature of the two different types of feeds. Note that decane, cumene and naphthalene contain respectively 0, 1 and 2 aromatic rings per molecule, and the different aromatic rings appear to be related to the differences between the coking abilities of these three species. Hence, we suggest that the aromatic ring percent in the feed, A, may be a possible independent variable, defined by ... 

Suppose we take T, p and Xg as the independent variables defining the state of the system, and compare two states of the system in which the variables have the following values ... 

The column operable range is determined in part by the requirement that no tray be allowed to dry up, that is, liquid and vapor must exist on each tray to maintain phase equilibrium. This range may be defined by the limits over which the condenser and reboiler duties may vary. As such, the condenser and reboiler duties are considered the two independent variables required to define the column performance (Sections 3.2.3 and 5.2.1). Alternatively, other pairs of variables may be chosen as the independent variables defining the column performance and each set can vary... 

Degree of freedom Independent variable defining the position. A free rigid body has six degrees of freedom, a ball joint three, and a hinge joint one. 

The shear plane is the one which maximises the value of sin a cos a +ksin a, and it is easily shown that this plane makes an angle = itan" fc with the plane of maximum shear stress. On the Coulomb theory a is uniquely defined by a material constant, whereas in (10) 0 is an independent variable defining the specimen orientation. 

As an example let us discuss the triple of independent variables defining the Gibbs function G. Temperature , the coordinates of the electric field vector and... 

Let us consider once more the set of constitutive equations (4.11), (4.12), and (4.13). It was mentioned in this context that partial derivatives of dependent variables with respect to independent variables define the material coefficients involved. The terms appearing in the leading diagonal of the array on the right hand side of this system of equations represent the so called principal effects. Writing... 

Consider an equilibrium stage with a feed stream at fixed rate, composition, and thermal conditions (temperature and pressure). Let the two independent variables defining the system be the pressure and duty, making it an adiabatic flash. If now it is desired that another variable such as the temperature or the vapor or liquid rate or composition be specified, then at least one of the independent variables—the pressure or duty— must be allowed to vary in order for the new specification to be satisfied. Certain parameters are either inherently or by necessity fixed for a given system and may not be varied. In an equilibrium stage, for instance, the fact that it is a single stage is a fixed parameter. 

The change in composition of the system can also be described in terms of variables based on extents of advancement of various chemical processes. Because there are constraints such as conservation of the number of atoms of each element, the number of independent variables defined in terms of extents of reaction will always be less than the number of species undergoing reaction, and the state of even a very complex system can often be described in terms of only one or two such variables. Description of dynamic behavior will usually require more variables, but even this number will often be much less than the number of reactive species. 

Let us begin by considering a prismatic piece of rubber with orthogonal edges Lz,Lx = Ly, and inquire about the force produced by an extension ALz in the 2 -direction. We will use the extension ratio A as an independent variable, defined as... 

Degree of freedom An independent variable defining the position. 

---