Qualitative equations

Qualitatively equation (7.15) is adequate to describe tiM f influence of layer quality, selectivity, and zone position in the 1 chromatogram upon resolution for a single unidimensional development under capillary flow controlled conditions. The variation of R, with Rf is not a simple function as can be seen from Figure 7.6. The resolution increases with the layer efficiency in a manner that depends linearly on the R, value. — Relatively small changes in selectivity have an enormous impact on... 

There is a need for advanced discrete-continuous optimization tools that can handle mixed-integer, discrete-logic, and quantitative-qualitative equations to model synthesis and planning and scheduling problems. 

The conditions of the experiment discussed here are different than the restrictions imposed to obtain the Mott-Gurney equation. However, at least qualitatively, Equation 8.38 can describe the space charge-limited current effect. To test this hypothesis, Equation 8.38 was experimentally tested, and it was shown that it is approximately satisfied (see Figures 8.13 and 8.14) [112]. 

Qualitative description of physical behaviors require that each continuous variable space be quantized. Quantization is typically based on landmark values that are boundary points separating qualitatively distinct regions of continuous values. By using these qualitative quantity descriptions, dynamic relations between variables can be modeled as qualitative equations that represent the structure of the system. The... 

When model parameters are unknown or uncertain, but the model structure is well defined, qualitative equations may be used in place of differential equations for fault diagnosis. The quahtative state of a parameter or variable is defined as... 

Qualitative equations are written in terms of functions. As an example, relation /6 = (1 /R ) (ee) is written in functional terms as U = R(, ( e) where corresponds to the inverse function for valve resistance. Likewise, expression like C4 Q4) returns K4Q4 for linear springs. These functions are often termed as constitutive relations or characteristic functions, and they may be nonlinear. The qualitative equation for the thermo-fluid process may be written in function form as... 

In terms of human responses to toxic chemicals, methods of low LOD and long response times can be qualitatively equated to higher LOD and shorter response times (the relationship between toxicity and time of exposure is not necessarily linear). As an illustration, human exposure of 10 mg-min/m of GB produces different results depending upon the concentration and time of exposure. If the exposure was to 10 mg/m for one minute, a person would probably go into convulsions but, if the exposure were to 0.08 mg/m for 2 hours (still 10 mg-min/m ) symptoms would probably be miosis and headache. It is a reasonable statement, therefore, that instruments with lowest LOD and the shortest response time commensurate with the task at hand provide the greatest protection. This is illustrated in Figure 2 for some chemical warfare materials. 

Qualitatively the equation shows that solutes which lower the surface tension have a positive surface concentration, e.g. soaps in water or amyl alcohol in water. Conversely solutes which increase the surface tension have a negative surface concentration. 

To first order, the dispersion (a-a) interaction is independent of the structure in a condensed medium and should be approximately pairwise additive. Qualitatively, this is because the dispersion interaction results from a small perturbation of electronic motions so that many such perturbations can add without serious mutual interaction. Because of this simplification and its ubiquity in colloid and surface science, dispersion forces have received the most significant attention in the past half-century. The way dispersion forces lead to long-range interactions is discussed in Section VI-3 below. Before we present this discussion, it is useful to recast the key equations in cgs/esu units and SI units in Tables VI-2 and VI-3. 

The final equation obtained by Becker and Doting may be written down immediately by means of the following qualitative argument. Since the flux I is taken to be the same for any size nucleus, it follows that it is related to the rate of formation of a cluster of two molecules, that is, to Z, the gas kinetic collision frequency (collisions per cubic centimeter-second). 

Equation X-17 was stated in qualitative form by Young in 1805 [30], and we will follow its designation as Young s equation. The equivalent equation, Eq. X-19, was stated in algebraic form by Dupre in 1869 [31], along with the definition of work of adhesion. An alternative designation for both equations, which are really the same, is that of the Young and Dupre equation (see Ref. 32 for an emphatic dissent). 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

The simplest extension to the DH equation that does at least allow the qualitative trends at higher concentrations to be examined is to treat the excluded volume rationally. This model, in which the ion of charge z-Cq is given an ionic radius d- is temied the primitive model. If we assume an essentially spherical equation for the u. . 

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

The qualitative solution behavior becomes more apparent when going to local coordinates, i.e., we rewrite the equations of motion in terms of the center of mass... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

As a result of these assumptions, qualitative moleeular orbital models ean be developed in whieh one assumes that eaeh mo (jti obeys a one-eleetron Sehrodinger equation... 

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