Useful Definitions

This section presents a number of mathematical definitions considered useful to conduct various types of transportation system reliability and safety studies. 

Advanced process control (APC) applications involve the use of control algorithms to provide improved process control when compared to regulatory PID controllers in loops or cascades. 

Traditional Advanced Control (TAC) employs the use of advanced control algorithms combined with regulatory control functions (i.e., lead/lag, ratio, high/low selectors, etc) to implement a control strategy. 

A programmable logic controller (PLC) is a small computer used to automate real-world processes. A PLC receives input from various sensors 

In a distributed control system (DCS), process measurements and control functions such as multiple PID loops are connected to application processors, which are networked throughout the plant. A graphical user interface (GUI) makes it easier for operators to view data, create plots, change setpoints, and respond to alarms. In addition to process control, modem DCS software includes sophisticated trending and data storage. 

Multivariable process control (MVPC) applications use one or more independent variables to control two or more dependant variables. If a variable is tmly independent, its value is not affected by other process variables. There are two t) es of independent variables. Manipulated variables (MV) can be changed by an operator to control the process. These include setpoints for regulatory PID controllers and valve positions. Feedforward (FF) and disturbance variables (DV) affect the process but can t be manipulated. These include ambient temperature, the quality of an external feed, etc. 

The probability that two events Ei and E2 occur is called the joint probability and is denoted by P(Ei, Ef). 

Two events 1 and E2, with probabilities PfEi) and P E2), respectively, are called mutually exclusive, when the probability that one of these events occurs is 

Consider a deck of cards. The probability of drawing the ace of spades is ( ace of spades ) = 1/52. The probability of drawing a ten is ( ten ) = 4/52. The two events, drawing the ace of spades or a ten, are mutually exclusive, because they cannot both occur in a single draw. The probability of either one occurring is equal to the sum of event probabilities, P = P( 2tx, of spades ) + P( ten ) = 5/52. 

Two events E and E2, with respective probabilities P E ) and ( 2). are called independent if the joint probability of both of them occurring is equal to the product of the individual event 

The conditional probability that an event 2 occurs provided that event 1 has occurred is denoted by ( 2 i). It is related to the joint probability through 

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The flmetion 5(co), ealled the Dirae delta flmetion, is the eontinuous analog to 5nm-It is zero unless co = o. If co = o, 5(co) is infinite, but it is infinite in sueh a way that the area under the eurve is preeisely unity. Its most useful definition is that 5(co) is the funetion whieh, for arbitrary f(co), the following identity holds ... 

In the 1960s Gillespie suggested calling protic acids stronger than 100% sulfuric acid superacids. This arbitrary but most useful definition is now generally accepted. It should be mentioned, however, that... 

Finally, a number of useful definitions of quantities directly or indirectly involved in the study of the surface area and porosity of both particulate and massive solids are given in Table 1.6. 

A useful definition of acids and bases is that independently introduced by Johannes Bronsted (1879-1947) and Thomas Lowry (1874-1936) in 1923. In the Bronsted-Lowry definition, acids are proton donors, and bases are proton acceptors. Note that these definitions are interrelated. Defining a base as a proton acceptor means an acid must be available to provide the proton. For example, in reaction 6.7 acetic acid, CH3COOH, donates a proton to ammonia, NH3, which serves as the base. 

Capacity Definitions In any analysis, the capacity per unit time of dynamic equipment (such as conveyors and bagging machines), as well as the rates at which they ac tuaUy perform, must be defined more precisely and realistically than by a mere statement of kilograms or pounds per hour. Some useful definitions employed by the equipment industi y are the following ... 

Originally confined to the shipment of crude raw materials and fuels, the term transportation of bulk sohds now apphes also to manufactured produc ts, which often become raw materials for other industries. In recent years, increasing tonnages of highly processed, finished chemical products have moved to customers in large bulk units. A useful definition of a bulk shipment is any unit greater than 2000 kg (4000 lb) or 2 m (70 fF). The containers available range from small portable hoppers of 2-m (70-fF) capacity to railroad cars of 255-m (9000-fF) capacity. 

This chapter has given an overview of the structure and dynamics of lipid and water molecules in membrane systems, viewed with atomic resolution by molecular dynamics simulations of fully hydrated phospholipid bilayers. The calculations have permitted a detailed picture of the solvation of the lipid polar groups to be developed, and this picture has been used to elucidate the molecular origins of the dipole potential. The solvation structure has been discussed in terms of a somewhat arbitrary, but useful, definition of bound and bulk water molecules. 

The title of the book requires that a definition of plastics materials be given. This is however very difficult. For the purpose of this book I eventually used as a working definition Those materials which are considered to be plastics materials by common acceptance . Not a positive definition but one which is probably less capable of being criticised than any other definition I have seen. Perhaps a rather more useful definition but one which requires clarification is... 

Note In general, property mean size is not equal to mean property size, (i) property mean size is the more commonly used definition. 

The rate of a chemical reaction can be described in any of several different ways. The most commonly used definition involves the time rate of change in tlie amount of one of the components participating in tlie reaction tliis rate is usually based on some arbitrary factor related to tlie reacting system size or geometry, such as volume, mass, or interfacial area. Tlie definition shown in Eq. (4.6.7), wliich applies to homogeneous reactions, is a convenient one from an engineering point of view. 

In this paper, asymmetric carbon atoms are considered only in paraffins and substituted paraffins, and the following definition will be retained (cf. Sec. 36(b)) A carbon atom is called asymmetric if the four bonded radicals arc pairwise structurally different. (Thus, it is not sufficient to require that the four radicals are not stereoisomers in order to declare a carbon atom asymmetric. One could envisage other, possibly useful, definitions.)... 

A further important concept related to electronegativity and polarity is that of acidity and basicity. We ll see, in fact, that much of the chemistry of organic molecules can be explained by their acid-base behavior. You may recall from a course in general chemistry that there are two frequently used definitions of acidity the Brtfnsted-Lowry definition and the Lewis definition. We ll look at the... 

Thomas and Deemer, in a well-known paper,48 have set down useful definitions and thoughts on simulation, Monte Carlo, and operational gaming methods. 

Chemists debated the concepts of acid and base for many years before precise definitions emerged. Among the first useful definitions was the one proposed by the Swedish chemist Svante Arrhenius in about 1884 ... 

Before setting out to discuss the vertical structure of the atmosphere, we note that it is useful to have access to conventional nomenclature. Figure 7-1, based on the thermal profile of the atmosphere, includes a number of commonly used definitions. 

GP 2] [R 2] The definition of space-time yield in a micro reactor depends on the definition of the reactor volume . Owing to the large amoimt of construction material relative to the reaction channels and the neglect of some reactor parts ( abstraction to the real reaction zone ), several more or less useful definitions can be made. In the following, two definitions concerning the time yield divided by the pure reaction channel volume and the platelet volume were used. 

Where is the dividing line between bulk and fine chemicals Here again there is no universally accepted definition. Nonetheless, a useful definition of a fine chemical is one with a price of more than 10 US dollars/kg and a volume of less than 10,000 tons per annum. The latter is rather a large volume for most fine chemicals, e.g. most pharmaceutical intermediates are produced in quantities of less than 100 tons per annum. In practice, the type of technology used is probably more dictated by volume than by product application. 

The formalism of graph theory lends itself to a number of very useful definitions. One useful concept is the degree d(v) of a vertex v, which is defined as the number of edges with which the vertex is incident. Another is a... 

Ans. Choice (a). This is a useful definition of empirical formula. The molecular formula gives the ratio of moles of each element to moles of the compound, plus the information given by the empirical formula. The percent composition does not deal with moles, but is a ratio of masses. 

In analytical practice, some methods using definitive measurements, in principle, are also caUbrated in an experimental way (e.g. spectrophotometry, polarography) to provide reliable estimates of S. 

The second observable is the reaction order, which has two eommonly used definitions. The simplest definition to use, though not necessarily to measure,... 

Some of the commonly used definitions related to fires and explosions are given in what follows. These definitions are discussed in greater detail in later sections. 

A reaction can be thought of as a multi-step process first the reactants approach and then they collide. Only after touching do they react. One of the more useful definitions... 

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