Primary dimensions

There are nine variables and three primary dimensions, and therefore by Buekingham s theorem. Equation 7-1 ean be expressed by (9-3) dimensionless groups. Employing dimensional analysis. Equation 7-1 in terms of the three basie dimensions (mass M, length L, and time T) yields Power = ML T. ... 

The ultimate extension of two-dimensional GC was introduced in the early 1990s (4) and involves the reanalysis of all components from the primary dimension on a secondary column. To enable this, the peak capacity of the secondary column is often very much smaller than the primary thus allowing completion of the separation in a time that may be considered insignificant as a fraction of the time required to complete the primary stage. This is discussed in much further detail in Chapter 4. 

For the most part, in this book we use SI dimensions and units (SI stands for le systeme international d unites). A dimension is a name given to a measurable quantity (e.g., length), and a unit is a standard measure of a dimension (e.g., meter (for length)). SI specifies certain quantities as primary dimensions, together with their units. A primary dimension is one of a set, the members of which, in an absolute system, cannot be related to each other by definitions or laws. All other dimensions are secondary, and each can be related to the primary dimensions by a dimensional formula. The choice of primary dimensions is, to a certain extent, arbitrary, but their minimum number, determined as a matter of experience, is not. The number of primary dimensions chosen may be increased above the minimum number, but for each one added, a dimensional constant is required to relate two (or more) of them. 

The SI primary dimensions and their units are given in Table 1.1, together with their dimensional formulas, denoted by square brackets, and symbols of the units. The number of primary dimensions (7) is one more than required for an absolute system, since... 

The secondary units are derived from the primary ones according to physical laws, e.g., velocity = length/time. (The borderline separating both types of quantities is largely arbitrary for example, 50 years ago a measuring system was used in which force was a primary dimension instead of mass.)... 

Some of the many kinds of impellers are shown in Figure 7.10. For clear liquids, some form of closed impeller [Figure 7.10(c)] is favored. They may differ in width and number and curvature of the vanes, and of course in the primary dimension, the diameter. Various extents of openness of impellers, [Figs. 7.10(a) and (b)] are desirable when there is a possibility of clogging as with slurries or pulps. The impeller of Figure 7.10(e) has both axial propeller and... 

A flow field is best cliaraclerizcd by the velocity distribution, and thus a flow is said to be one-, two-, or three-dimensional if Ihe flow velocity varies in one, two, or three primary dimensions, respectively. A typical fluid flow involves a three-dimensional geometry, and the velocity may vary in all three dimensions, rendering the flow three-dimensional [V (.r. y, z) in rectangular or V (r, 0, z) in cylindrical coordinates]. However, the variation of velocity in certain directions can be small relative to the variation in oUicr directions and can be ignored with negligible error, In such cases, the flow can be modeled conveniently as being one- or two-dimensional, which is easier to analyze. 

Comprehensive two-dimensional GC (GC X GC) was introduced in the early nineties by the late John Phillips. This technique differs from GC-GC in that the entire sample injected to the column is subjected to separation in both dimensions. In GC X GC, the sample injected into the system is first subjected to chromatographic separation in the first column (primary dimension), as... 

Process reliability is the ability of the process to consistently produce required results as measured through the primary dimensions of uptime, dependability, and first run yield [97]. Uptime is the time the process is in operation compared with the available time the process is scheduled to be in operation. Dependability is the repeatability of a process actual run rate compared with its scheduled run rate. First run yield is the ability to produce quality outputs the first time through production run without any rework. Reliability losses are measured in terms of unscheduled downtime, run rate losses, yield losses, and reprocessing [97]. 

Accuracy of movement The primary dimension of performance achieved by the sensory-motor control performance resource. 

Vowel sounds are described with the primary dimensions of high/low and front/back, and secondary dimensions of rounding and length. 

Engineered nanomaterials are materials intentionally produced that have one primary dimension from 1-100 nm. By comparison, a sheet of newspaper is about 100,000 nm thick and a human hair is about 80,000 nm thick. There are more than 25 million nm in an inch. If a marble were a nm, the Earth would be one meter in diameter. 

In the SI system, where length, mass, and time are defined as primary dimensions and force a secondary dimension, is equal to 1. In the English Engineering system mass and force are defined... 

Symptom Checklist-90-Revised (SCL-90-R). The SCU-90-R is a 90-item, self-administered, inventory developed by Derogatis in 1977 and is derived from the Hopkins Symptom Checklist (17). The inventory was developed to screen for psychopathology and assesses symptomatic distress in nine primary dimensions and three global indices of distress. The dimensions include somatization, obsessive-compulsive features, interpersonal sensitivity, depression, anxiety, hostility, phobic anxiety, paranoid ideation, and psychoti-cism. The measure takes 15 to 20 minutes to complete. It has been used widely with community adolescents and adults, inpatients, and outpatients. It has been shown to have good reliability and validity (18). 

Before considering the method of proceeding from Eq.(2.24) to Eq. (3.2) in this chapter, a few quantities will be defined. Fundamental or primary dimensions are properties of a system under study that may be considered independent of the other properties of interest. For example, there is one fundamental dimension in any geometry problem and this is length (L). The fundamental dimensions involved in different classes of mechanical problems are listed in Table 3.1 where Z stands for length, F for force, and T for time. Dimensions other than F, L, and T, which are considered fundamental in areas other than mechanics, include temperature... 

The dimensionless form is also suitable for developing models or correlations that have a less theoretical basis and cannot be deduced from simple balance equations. The advantage of using dimensionless variables is that, if there are any n variables containing m primary dimensions, tiie correlation can be formulated using n — m dimensionless groups. However, we need experience or some preliminary experiments to identify the variables that are important. 

By this time, you are probably experienced in working with units. Most science and engineering texts have a section in the first chapter on this topic. In this text, we will mainly use the Systeme International, or SI units. The SI unit system uses the primary dimensions m, s, kg, mol, and K. Details of different unit systems can be found in Appendix D. One of the easiest ways to tell that an equation is wrong is that the units on one side do not match the units on the other side. Probably the most common errors in solving problems result from dimensional inconsistencies. The upshot is Pay close attention to units Try not to write a number down without the associated units. You should be able to convert between unit systems. It is often easiest to put all variables into the same unit system before solving a problem. 

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