Controlled-independent-variable

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... 

Having selected the system response, we start choosing factors, levels of the factors and center point of the design (basic level or the null point). By factor we understand the controllable independent variable that corresponds to one possibility of influence on the object of research. A factor is considered defined if its name and domain of factors are determined. A factor may take several values in this field. The chosen factor values, both qualitative and quantitative, are called factor variation levels. Factor variation levels in the design of experiments are coded values. Under factor interval of variation we understand the difference between two factor levels, which in their coded form have value one. When selecting the factors one should pay attention to the conditions they must meet. 

The intercept (b) mathematically describes the value of the function where it crosses the y-axis. In a physical sense, however, the intercept represents a correcting or offset value that relates the dependent variable to the independent variable. That is, when your controlled (independent) variable is zero, the experimental (dependent) variable may have an inherently nonzero value. 

In this formulation, x t) are state (dependent) variables, u(t) are control (independent) variables and z(t) are algebraic variables, while p are time-independent parameters. 

The second approach is called the controlled-independent-variable approach. In this approach, the experimenter decides before the experiment, what values of d will be observed (hence the name controlled-independent-variable). For example, the experimenter may choose to obtain responses at d=5, 10, and 20... 

A prominent measure of overall success is the profit obtained with a given set of resources in a given time span. Basically, the profit depends on costs, revenues and assets employed, whereby those depend on system-internal and -external influences. In SC simulation the aim is to investigate the behaviour of the system and explain the relations between its components. Here, the focus is on studying internal relations under external disturbances. Let X 6 R denote the vector of controllable independent variables and y 6 R" denote the vector of dependent response variables defined in the model building phase. Furthermore, e 6 R° denotes the vector of uncontrollable variables which are not considered in detail in the model but are captured as environmental effects. The function H -) describes the real relation between dependent and independent as well as environmental variables... 

Buonaiuto and Kraus S analyzed the bathymetry of 18 inlets around the coast of the USA to determine a predictive expression for the minimum depth over crest her of the ebb shoal. It was reasoned that, because both incident waves and the tidal prism are expected to be controlling independent variables for ebb shoal development, a parameter combining both average annual significant wave height Hs and... 

Ratio and Multiplicative Feedforward Control. In many physical and chemical processes and portions thereof, it is important to maintain a desired ratio between certain input (independent) variables in order to control certain output (dependent) variables (1,3,6). For example, it is important to maintain the ratio of reactants in certain chemical reactors to control conversion and selectivity the ratio of energy input to material input in a distillation column to control separation the ratio of energy input to material flow in a process heater to control the outlet temperature the fuel—air ratio to ensure proper combustion in a furnace and the ratio of blending components in a blending process. Indeed, the value of maintaining the ratio of independent variables in order more easily to control an output variable occurs in virtually every class of unit operation. 

In the context of chemometrics, optimization refers to the use of estimated parameters to control and optimize the outcome of experiments. Given a model that relates input variables to the output of a system, it is possible to find the set of inputs that optimizes the output. The system to be optimized may pertain to any type of analytical process, such as increasing resolution in hplc separations, increasing sensitivity in atomic emission spectrometry by controlling fuel and oxidant flow rates (14), or even in industrial processes, to optimize yield of a reaction as a function of input variables, temperature, pressure, and reactant concentration. The outputs ate the dependent variables, usually quantities such as instmment response, yield of a reaction, and resolution, and the input, or independent, variables are typically quantities like instmment settings, reaction conditions, or experimental media. 

An important part of planning an experimental program is the identification of the variables that affect the response and deciding what to do about them. The decision as to how to deal with each of the candidate variables can be made jointiy by the experimenter and the statistician. However, identifying the variables is the experimenter s responsibiUty. Controllable or independent variables in a statistical experiment can be dealt with in four different ways. The assignment of a particular variable to a category often involves a trade-off among information, cost, and time. 

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. 

The equilibrium constitution of an alloy can be determined experimentally by metallography and thermal analysis (described later). If the pressure is held constant at 1 atm., then the independent variables which control the constitution of a binary alloy are T and Xr or Wg. 

To recast the thermodynamic description in terms of independent variables that can be controlled in actual laboratory experiments (i.e., T, /i, and the set of strains or their conjugate stresses), it is sensible to introduce certain auxiliary thermodynamic potentials via Legendre transformations. This chapter is primarily concerned with... 

The use of a polymeric support also affords a unique opportunity to control independently the variables that may affect the chiral recognition process, which is hard to achieve with silica. For example, the type and number of reactive sites can be easily adjusted with a polymer support. We recently reported an extensive study of the... 

For a system such as discussed here, the Gibb s Phase Rule [59] applies and establishes the degrees of freedom for control and operation of the system at equilibrium. The number of independent variables that can be defined for a system are ... 

Variable, independent An experimental factor that can be controlled (temperature, pressure, order of test, etc.) or independently measured (hours of sunshine, specimen thickness, etc.). Independent variables may be qualitative (such as a qualitative difference in operating technique) or quantitative (such as temperature, pressure, or duration). Thus, if variable A is a function of variable B, than B is the independent variable. 

Most often, we will choose the independent variables to be those quantities we control in the laboratory. The usual thermodynamic choices are (p and T) or (Vand T), Then, we measure changes in the thermodynamic properties of the system as these variables are altered. Thus, for a pure substance, writing... 

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives... 

The experimental basis for the model should span a broader range of the independent variables than will be encountered in the use of the model. To develop a comprehensive model, it is often necessary to add components to the feed in amounts that would not normally be present. For A —> B, the concentration of B is correlated to that of A aj — a = b — bi . Varying bin wiU lessen the correlation and will help distinguish between rate expressions such 2i 01 = ka ox 01 = kfa — krb ox 01 = ka/ + ksb). Books and courses on the design of experiments can provide guidance, although our need for formalized techniques is less than that in the social and biological sciences, where experiments are much more difficult to control and reproduce. 

After the dominant independent variables have been brought under control, many small and poorly characterized ones remain that limit further improvement in modeling the response surface when going to full-scale production, control of experimental conditions drops behind what is possible in laboratory-scale work (e.g., temperature gradients across vessels), but this is where, in the long term, the real data is acquired. Chemistry abounds with examples of complex interactions among the many compounds found in a simple synthesis step,... 

The development of a pharmaceutical formulation and the associated process usually involves several variables. Mathematically, they can be divided into two groups. The independent variables are the formulation and process variables directly under the control of the formulator. These might include the level of a given... 

A reported application of canonical analysis involved a novel combination of the canonical form of the regression equation with a computer-aided grid search technique to optimize controlled drug release from a pellet system prepared by extrusion and spheronization [28,29]. Formulation factors were used as independent variables, and in vitro dissolution was the main response, or dependent variable. Both a minimum and a maximum drug release rate was predicted and verified by preparation and testing of the predicted formulations. Excellent agreement between the predicted values and the actual values was evident for the four-component pellet system in this study. 

The emphasis, once again, is that appropriate statistical design is an important consideration. For a formulator planning such a study, it should be noted that the independent variables can be anything that he or she can quantitate and control and the dependent variables can be anything that he or she can quantitate. 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

As we stressed earlier, the reactor space time is the independent variable at the control of the reactor designer. This parameter is more meaningful than the mean residence time in the reactor. 

Such internal thermodynamic equilibria where A is a protein are found for non-metal components, including free coenzymes and substrates where B is a small molecule, or where free M is an ion of either a non-metal, e.g. Cl" or HCOj, or a metal, e.g. K+ or Mg2+, or is H+, and they are involved in, even necessary for, catalysis, pumping and cooperative controls of many metabolic paths. All such combinations reach equilibrium, as long as exchange is fast, where a fast rate can be taken as, say, 10-3 s for dissociation in cells. Note that equilibria with defined binding constants for AB or AM formation in any system reduce the number of variables and hence AB and AM concentrations are defined by those of free A, B and M, leaving two independent variables for each equilibrium. In some cases, the... 

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