Numbers of measurement

The basic data gathering methods are direct methods which allow visual inspection or at least direct measurement of properties, and indirect methods whereby we infer reservoir parameters from a number of measurements taken in a borehole. The main techniques available within these categories are summarised in the following table ... 

The 3D inspection system has a number of measuring and report utilities that enables the user to easily find, analyse and report possible indications in the test object. As an example, a moveable 2D projection view plane can be moved along e.g, the welding geometry dynamically updating the content of the 2D projection view window. Indications can be measured using any referenee co-ordinate system and the results and screen dumps can automatically be dumped in report files suited for later import into a word processing application. 

A major advance in force measurement was the development by Tabor, Win-terton and Israelachvili of a surface force apparatus (SFA) involving crossed cylinders coated with molecularly smooth cleaved mica sheets [11, 28]. A current version of an apparatus is shown in Fig. VI-4 from Ref. 29. The separation between surfaces is measured interferometrically to a precision of 0.1 nm the surfaces are driven together with piezoelectric transducers. The combination of a stiff double-cantilever spring with one of a number of measuring leaf springs provides force resolution down to 10 dyn (10 N). Since its development, several groups have used the SFA to measure the retarded and unretarded dispersion forces, electrostatic repulsions in a variety of electrolytes, structural and solvation forces (see below), and numerous studies of polymeric and biological systems. 

These equations apply when an entire population is available for measurement. The most common situation in practical problems is one in which the number of measurements is smaller than the entire population. A group of selected measurements smaller than the population is called a sample. Sample statistics are slightly different from population statistics but, for large samples, the equations of sample statistics approach those of population statistics. 

Inserting the full expression for /(x,y) and integrating over x and y from 0 to Lx and Ly, respeetively, this integral is seen to vanish. This means that the result of a large number of measurements of px on eleetrons eaeh deseribed by the same / will yield zero net momentum along the x-axis. half of the measurements will yield positive momenta and half will yield negative momenta of the same magnitude. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

Mean The mean, X, is the numerical average obtained by dividing the sum of the individual measurements by the number of measurements... 

Median The median, Xmed) is the middle value when data are ordered from the smallest to the largest value. When the data include an odd number of measurements, the median is the middle value. For an even number of measurements, the median is the average of the n/2 and the nil) + 1 measurements, where n is the number of measurements. 

The Chauvenet tables relate the number of measurements to p values and assess the probability that the unusual value may be real or may be ignored. In the present example, the tables show that p = 1.73 for 6 measurements. Since the calculated value (1.80) is greater than 1.73, the result may be rejected. 

This process of rejecting values should not to be repeated on any one set of measurements, unless the number of measurements is large. 

The definition of polymer thermal stabiUty is not simple owing to the number of measurement techniques, desired properties, and factors that affect each (time, heating rate, atmosphere, etc). The easiest evaluation of thermal stabiUty is by the temperature at which a certain weight loss occurs as observed by thermogravimetric analysis (tga). Early work assigned a 7% loss as the point of stabiUty more recentiy a 10% value or the extrapolated break in the tga curve has been used. A more reaUstic view is to compare weight loss vs time at constant temperature, and better yet is to evaluate property retention time at temperature one set of criteria has been 177°C for 30,000 h, or 240°C for 1000 h, or 538°C for 1 h, or 816°C for 5 min (1). 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

The successful appHcation of pattern recognition methods depends on a number of assumptions (14). Obviously, there must be multiple samples from a system with multiple measurements consistendy made on each sample. For many techniques the system should be overdeterrnined the ratio of number of samples to number of measurements should be at least three. These techniques assume that the nearness of points in hyperspace faithfully redects the similarity of the properties of the samples. The data should be arranged in a data matrix with one row per sample, and the entries of each row should be the measurements made on the sample, as shown in Figure 1. The information needed to answer the questions must be implicitly contained in that data matrix, and the data representation must be conformable with the pattern recognition algorithms used. 

For compounds with three or four heteroatoms in the ring the number of measurements is much fewer, and these are summarized in Table 1. 

Continuous-Pliase Coefficients There have been a large number of measurements of kc for sohd particles and gas bubbles suspended in agitated hquids [for review, see Miller, Jnd. Fng. Chem., 56(10), 18 (1964)]. A typical correlation of these data is that of Calder-bankand Moo-Young [Chem. Fng. Sci., 16, 39 (1961)] ... 

Ferefs diameter (Fig. 20-5) is the perpendicular projection, in a fixed direction, of the tangents to the extremities of the particle profile. Martin s diameter is a hne, parallel to a fixed direction, which divides the particle profile into two equal areas. Since the magnitude of these statistical diameters varies with particle orientation, these diameters have meaning only when a sufficient number of measurements are averaged. 

Sample Containers More saiTiple containers will be required for a complex test than are typically used for normal operation. The number and type of sample containers must be gathered in advance, recognizing the number of measurements that 1 be required. The sample containers should be tagged for the sample location, type, and conditions. 

Motivation Unit tests require a substantial investment in time and resources to complete successfully. This is the case whether the test is a straightforward analysis of pump performance or a complex analysis of an integrated reactor and separation train. The uncertainties in the measurements, the likelihood that different underlying problems lead to the same symptoms, and the multiple interpretations of unit performance are barriers against accurate understanding of the unit operation. The goal of any unit test should be to maximize the success (i.e., to describe accurately unit performance) while minimizing the resources necessary to arrive at the description and the subsequent recommendations. The number of measurements and the number of trials should be selected so that they are minimized. 

Modern petro/chemical processes provide the opportunity for gathering a large number of measurements automatically and frequently. Most are redundant and provide little additional insight into unit performance. The difficulties in handling a large amount of information with httle intimate knowledge of the operation increases the hkeli-hood that some of the conclusions drawn will be erroneous. 

Identifying the minimum number of specific measurements containing the most information such that the model parameters are uniquely estimated requires that the model and parameter estimates be known in advance. Repeated unit tests and model building exercises will ultimately lead to the appropriate measurements. However, for the first unit test in absence of a model, the identification of the minimum number of measurements is not possible. 

The methodology of identifying the optimum test and number of measurements has received httle attention in analysis of plant performance and design literature. 

Increa.se the number of operating conditions in the mea.surement. set. Measurement sets from different operating conditions have the same effect as increasing the number of measurements. They have the added benefit identifying weaknesses in the model when it cannot accurately describe all of the conditions. 

Periodic fault detection is readily done by analysts without extensive software support. Process monitoring such as the examination of the traces discussed above are one exam e. However, the number of measurements in a single set have such complex interactions that it is... 

When the number of measurement sets is substantially less than that indicated Fig. 30-25, the interpretation becomes problematic. One option is to use the parameter v ues from one period to describe the measurements from another. If the description is within measurement error, the operation has not changed. If there is a substantial difference between the predictions and the measurements, it is hkely that the operation has changed. Methods such as those developed by Narasimhan et al. (1986) can be used when the number of measurements are large. When implementing automatic methods to treat a large number of measurements, analysts should ensure that the unit is at steady state for each time period. 

For rectangular ducts, the area is evenly divided into the necessary number of measurement points. For circular ducts. Table 32-3 can be used to... 

The number of measurable layers of a stack is limited only by the optical contrast between the different layers. In practice stacks of ten layers and more can be analyzed by ellipsometry. Further advantages of ellipsometry compared with other metrological methods are the non-invasive and non-destructive character of the optical method, the low energy entry into the sample, the direct measurement of the dielectric function of materials, and the possibility of making the measurement in any kind of optical transparent environment. 

Two measured ellipsometric angles T and d at a fixed wavelength and a fixed angle of incidence enable calculation of a maximum of two other properties, e. g. the film thickness and refractive index of a transparent layer. Multiple angle measurements increase the number of measured quantities and hence the number of properties which can be determined for a specific sample, although even under these condi-... 

State estimation is the process of extracting a best estimate of a variable from a number of measurements that contain noise. 

Measurement noise covariance matrix R The main problem with the instrumentation system was the randomness of the infrared absorption moisture eontent analyser. A number of measurements were taken from the analyser and eompared with samples taken simultaneously by work laboratory staff. The errors eould be approximated to a normal distribution with a standard deviation of 2.73%, or a varianee of 7.46. 

FIGURE 6.8 Plant maximum concentrations of xylene (number of plants = 139, number of measurements = 865). 

Random error varies in an unpredictable manner in absolute value and in sign when a large number of measurements of the same value of a quantity are made under essentially identical conditions. 

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