Measured variable

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

For each experiment, the true values of the measured variables are related by one or more constraints. Because the number of data points exceeds the number of parameters to be estimated, all constraint equations are not exactly satisfied for all experimental measurements. Exact agreement between theory and experiment is not achieved due to random and systematic errors in the data and to "lack of fit" of the model to the data. Optimum parameters and true values corresponding to the experimental measurements must be found by satisfaction of an appropriate statistical criterion. 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

Measured Variables and Estimates of Their True Values for Acetone(1)/Methanol(2) System (Othmer, 1928)... 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

Subroutine VLDTA2. VLDTA2 loads the binary vapor-liquid equilibrium data to be correlated. If the data are in units other than those used internally, the correct conversions are made here. This subroutine also reads the estimated standard deviations for the measured variables and the initial parameter estimates. All input data are printed for verification. 

Subroutine REGRES. REGRES is the main subroutine responsible for performing the regression. It solves for the parameters in nonlinear models where all the measured variables are subject to error and are related by one or two constraints. It uses subroutines FUNG, FUNDR, SUMSQ, and SYMINV. 

SOLVES FOR THE PARAMETERS IN NON-LINEAR MEASURED VARIABLES ARE SUBJECT TC ERROR ONE OR TWO CONSTRAINTS. 

CALCULATES THE SUM OF THE SQUARES OF THE DEVIATIONS OF ALL MEASURED VARIABLES FROM THEIR TRUE VALUES FOR REGRES. 

CALCULATES THE CONSTRAINT FUNCTIONS FOP BINARY VAaOR-LIOUIO EQUILIBRIUM OATA PEOUCTION. THE CCNSTRAINT FUNCTIONS RELATING THE TRUE VALUES QP THE MEASURED VARIABLES ARE (1) PRESS = F(LI0 COMP,TEMP,PARAMETERS)... 

A simple decision-making problem is I measure variable x of a population A and the same variable xof a population B. I get (slightly) different results. Is there areal difference between populations A and B based on the difference in measurements, or am I only seeing different parts of the distributions of identical populations ... 

Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a hnear combination of the measured variables [yj], Y = CL y + Cioyo + Let the random variables yi, yo,. . . have means E yi), E y, . . . and variances G yi), G y, . The variable Y has mean... 

Process measurements encompass the apphcation of the principles of metrology to the process in question. The objective is to obtain values for the current conditions within the process and make this information available in a form usable by either the control system, process operators, or any other entity that needs to know The term measured variable or process variable designates the process condition that is being determined. 

Accuracy Accuracy refers to the difference between the measured value and the true value of the measured variable. Unfortunately, the true value is never known, so in practice accuracy refers to the difference between the measured value and an accepted standard value for the measured variable. 

In practice, most attention is given to accuracy when the measured variable is the basis for billing, such as in custody transfer applications. However, whenever a measurement device provides data to any type of optimization strategy, accuracy is veiy impoi tant. 

For regulatory control, repeatability is of major interest. The basic-objective of regulatory control is to maintain uniform process operation. Suppose that on two different occasions, it is desired that the temperature in a vessel be 80°C. The regulatoiy control system takes appropriate actions to bring the measured variable to 80°C. The difference between the process conditions at these two times is determined by the repeatability of the measurement device. 

Measurement span. The measurement span required for the measured variable must lie entirely within the instrument s envelope of performance. 

Modern control systems permit the measurement device, the control unit, and the final actuator to be physically separated by several hundred meters, if necessary. This requires the transmission of the measured variable from the measurement device to the control unit, and the transmission of the controller output from the control unit to the final ac tuator. 

In each case, transmission of a single value in only one direction is required. Such requirements can be met by analog signal transmission. A span is defined for the value to be transmitted, and the value is basically transmitted as a percent of this span. For the measured variable, the logical span is the measurement span. For the controller output, the logical span is the range of the final actuator (e.g., valve fully closed to valve fully open). 

Transmission of more than one value from a transmitter. Information beyond the measured variable is available from the smart transmitter. For example, a smart pressure transmitter can also report the temperature within its housing. Knowing that this temperature is above normal values permits corrective action to be taken before the device fails. Such information is especially important during the initial commissioning of a plant. 

Analog alarms can be defined on measured variables, calculated variables, controller outputs, and the like. For analog alarms, the following possibilities exist ... 

Coupling digital controls with networking technology permits information to be passed from level-to-level within a corporation at high rates of speed. This technology is capable of presenting the measured variable brom a flow transmitter installed in a plant in a remote location anywhere in the world to the company headquarters in less than a second. 

Figure 6-45 shows a simple eompressor system with surge proteetion. The eharaeteristie eurve of this eompressor at a eonstant rotation speed and eonstant inlet eonditions is illustrated in Figure 6-46. The two measured variables in Figure 6-47 are the eoordinates for the eompressor eurve in Figure 6-45 and are defined as ...

In experimental load studies, the measurable variables are often surface strain, acceleration, weight, pressure or temperature (Haugen, 1980). A discussion of the techniques on how to measure the different types of load parameters can be found in Figliola and Beasley (1995). The measurement of stress directly would be advantageous, you would assume, for use in subsequent calculations to predict reliability. However, no translation of the dimensional variability of the part could then be accounted for in the probabilistic model to give the stress distribution. A better test would be to output the load directly as shown and then use the appropriate probabilistic model to determine the stress distribution. 

A weighted least-squares analysis is used for a better estimate of rate law parameters where the variance is not constant throughout the range of measured variables. If the error in measurement is corrected, then the relative error in the dependent variable will increase as the independent variable increases or decreases. 

Equation 11-18 is referred to as the Eadie-Hofstee equation, where V is plotted against v/C . However, both of these equations are sub-jeeted to large eiTors. Equation 11-18 in partieular eontains tlie measured variable v in both eoordinates, whieh is subjeeted to the largest errors. Eigures 11-2 and 11-3 show plots of Lineweaver-Burk and Eadie-Hofstee equations, respeetively. 

Often we will have data rather than a function and we wish to plot it, so that we can find a function that describes the data by analysis. In such cases we can manipulate the data by bringing it into a matrix form and then plotting it with ListPlot. We also can compare it to the behavior of functions that are meant to represent the data. The following is a typical set of data obtained from an experiment, appropriately named "data." (This could have been imported to Mathematica by any number of different means.) The first column is time and the second is the value of the measured variable in the system ... 

Compensating control The process of au tomatically adjusting the control point of a controller to compensate for changes in a second measured variable. 

Measured variable A variable that is measured, and may be controlled. 

X = temperature, concentration or some other measurable variable... 

Concentration of measurable variable = Dimension of model = Dimension of scale-up unit = Ratio of dimensions on scale-up = Overall liquid t ertical height erf mixing vessel, from top liquid level to bottom (flat or dished or elliptical), ft or in., consistent with other components of equations, see Figure 5-34 = Empirical constant... 

By mathematical manipulation, numerous additional relationships can be derived from those given in Table 2-19. Of particular significance are expressions that relate enthalpy H and internal energy U to the measurable variables, P, V, and T. Thus, choosing the basis as one pound mass,... 

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