Mathematical representation

Additionally, Equations 3.1 and 3.2 used for cascaded stages in Section 3.2.2 are 

Two more equations may be derived to represent energy balances in each stage. The energy balance equations provide functional relationships between stage temperatures T and T2 and heat duties Q and Q - 

and 22- Hence, in order to solve the column, two of the variables must be specifled. The distribution coefficients may be supplied as additional data or may be calculated from appropriate correlations. The column equations may be manipulated in different ways, depending on which of the variables are specifled and which are to be solved. For instance, a chain of eliminations and substitutions may be carried out to develop an expression for LJV, the ratio of reflux to vapor from the top tray, or reflux ratio  

The application of these equations to speciflc cases is demonstrated in Example 3.2. The number and complexity of the equations for multistage, multi-component columns are much higher, which rules out the elimination and substitution methods. General systematic techniques for solving systems of column equations are discussed in Chapter 13. 

If the process is operated at a specified liquid rate from stage 2 (A2), the vapor rate from stage 1 (Vj) is fixed by virtue of overall material balance. A second independent variable, the liquid from stage 1, or the reflux rate Lj, is now specified. For a given bottoms rate L2 or overhead rate V, the effect of reflux rate on stage temperatures and separation is examined next. As Lj is increased, the fraction of feed vaporized on stage 2 increases because of what follows. The vapor fraction on stage 2 is 

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In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The most informative method of expressing uncertainty in HCIIP or ultimate recovery (UR) is by use of the expectation curve, as introduced in Section 6.2. The high (H) medium (M) and low (L) values can be read from the expectation curve. A mathematical representation of the uncertainty n a parameter (e.g. STOMP) can be defined as... 

Reservoir simulation is a technique in which a computer-based mathematical representation of the reservoir is constructed and then used to predict its dynamic behaviour. The reservoir is gridded up into a number of grid blocks. The reservoir rock properties (porosity, saturation, and permeability), and the fluid properties (viscosity and the PVT properties) are specified for each grid block. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

In the complex mathematical representation, quadrature means that, at the (s + 1) wave mixing level, the product of. s input fields constituting the. sth order generator and the signal field can be organized as a product of (s + l)/2 conjugately paired fields. Such a pair for field is given by = ,One sees that the exponent... 

To see the contributions to the molecular mechanics potential energy function and their mathematical representation... 

A structure descriptor is a mathematical representation of a molecule resulting from a procedure transforming the structural information encoded within a symbolic representation of a molecule. This mathematical representation has to be invariant to the molecule s size and number of atoms, to allow model building with statistical methods and artificial neural networks. 

Descriptor Molecular representation Mathematical representation Inuariance properties ... 

A structure descriptor is a mathematical representation of a molecule resulting from a procedure transforming the structural information encoded within a symbolic representation of a molecule. 

Conventional computers initially were not conceived to handle vague data. Human reasoning, however, uses vague information and uncertainty to come to a decision. In the mid-1960 this discrepancy led to the conception of fuzzy theory [14]. In fuzzy logic the strict scheme of Boolean logic, which has only two statements true and false), is extended to handle information about partial truth, i.e., truth values between "absolutely true" and absolutely false". It thus gives a mathematical representation of uncertainty and vagueness and provides a tool to treat them. 

Pearson, J. R. A. and Petrie, C. J. S., 1970a. The flow of a tabular film, part 1 formal mathematical representation. J. Fluid Mech. 40, 1-19. 

In formulating a mathematical representation of molecules, it is necessary to define a reference system that is defined as having zero energy. This zero of energy is different from one approximation to the next. For ah initio or density functional theory (DFT) methods, which model all the electrons in a system, zero energy corresponds to having all nuclei and electrons at an infinite distance from one another. Most semiempirical methods use a valence energy that cor-... 

CSTRs and other devices that require flow control are more expensive and difficult to operate. Particularly in steady operation, however, the great merit of CSTRs is their isothermicity and the fact that their mathematical representation is algebraic, involving no differential equations, thus maldng data analysis simpler. 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Focus For the purposes of this discussion, a model is a mathematical representation of the unit. The purpose of the model is to tie operating specifications and unit input to the products. A model can be used for troubleshooting, fault detection, control, and design. Development and refinement of the unit model is one of the principal results of analysis of plant performance. There are two broad model classifications. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

We need a mathematical representation of our prior knowledge and a likelihood function to establish a model for any system to be analyzed. The calculation of the posterior distribution can be perfonned analytically in some cases or by simulation, which I... 

As a final comment on inadequacies of mathematical representations of risk, those who bear the risk are not necessarily those who receive the benefit. While unequal distribution of risk and benefit may not be fair, it is difficult to redress die incquily. 

A basis set is a mathematical representation of the molecular orbitals within a molecule. The basis set can be interpreted as restricting each electron to a particular region of space. Larger basis sets impose fewer constraints on electrons and more accurately approximate exact molecular orbitals. They require correspondingly more computational resources. Available basis sets and their characteristics are discussed in Chapter 5. 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

Before we enter into a mathematical representation of gravitational waves, it is useful to summarize their main characteristics ... 

The mathematical representation of a lag takes the form of a first order differential equation ... 

Based on Fick s laws, the mathematical representation for the illustration taken is as follows ... 

Figure A.l Mathematical representation of flow redistribution. (From Tong and Weisman, 1979. Copyright 1979 by American Nuclear Society, LaGrange Park, IL. Reprinted with permission.)...

Because the dependence of kinetic parameters k and r on temperature as described by Equations 5 and 6 is constrained by the factorial design, it is possible that these equations and therefore Equation 4 may not give the best overall representation of the kinetic data. We have therefore carried out four more experiments and analyzed these results along with those mentioned earlier to obtain ( 2) the best overall mathematical representation that is summarized by the following equations ... 

The prospect of intelligent life anywhere in the Universe has been puzzling astronomers and recently astrobiologists, and there have been some attempts to estimate probabilities. This led Drake to construct a now famous equation that collects the ideas together the Drake equation. It is a mathematical representation of factors relating the probability of finding life and, in particular, an intelligent civilisation elsewhere in the Universe. This is an extreme example of hypothesis multiplication and should be treated with caution. The equation is written ... 

A molecular orbital (MO) is an orbital resulting from the overlap and combination of atomic orbitals on different atoms. An MO and the electrons in it belong to the molecule as a whole. Molecular orbitals calculations are used to develop (1) mathematical representations of the orbital shapes, and (2) energy level diagrams for the molecules. 

For PF, the F function requires another type of special mathematical representation. For this, however, consider a sudden change in a property of the fluid flowing that is maintained (and not pulsed) (e.g., a sudden change from pure water to a salt solution). If the change occurs at the inlet at t = 0, it is not observed at the outlet until t = t. For the exit stream, F(t) = 0 from ( = 0 to t = t, since the fraction of the exit stream of age less than ( is 0 for t < f in other words, the exit stream is pure water. For t > t, F(t) = 1, since all the exit stream (composed of the salt solution) is of age less than t. This behavior is represented by the unit step function S(t - b) (sometimes called the Heaviside unit function), and is illustrated in Figure 13.7, in which the arbitrary constant b = t. With this change, the unit step function is... 

Similar equations may be developed for other geometries such as spheres and cylinders. To complete the mathematical representation of a problem, initial and boundary conditions are specified. 

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