Use of empirical equation

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

W. Krzyzanski and W. J. Jusko, Note Caution in the use of empirical equations for pharmacodynamic indirect response models. J Pharmacokinet Biopharm 26 735-741 (1998). 

S OLID-LIQUID FLOWS are encountered in a variety of applications ranging from food to mining industries (I). Unlike single fluid flow in pipes, slurry flow in pipelines is complex. The complexity of these flows has necessitated the use of empirical equations in the design of slurry handling equipment, often leading to expensive systems. This complexity depends on the physical properties of the solid particles, for example, particle density, shape, and mean diameter. It also depends on the viscosity and density of the carrier fluid and, finally, on the operating con-... 

The performance of the titration can be controlled In a variety of ways (see Table 13.1) by use of empirical equations for the calculation of AV from preceding titration data points by use of microprocessors to control volumetric equipment (e.g. in photometric, potentlometrlc, coulometrlc titrations) or expand the scope of a given technique by use of robot stations In Implementing laborious manual methods or In handling toxic or hazardous substances etc. End-point detection Is usually based on E/A.V maxima and on first or second derivatives In the case of microprocessor- and microcomputer-controlled processes, respectively. Table 13.2 lists a chronological selection of calculation methods applied to titration curves [46]. 

There were three themes of the previous chapter. First was the infrastructure for describing systems via potentials, state variables, and differentials. Second was the use of empirical equations to model systems at equilibrium. Third, was that fluctuations impose a nonzero width on every state point. A point anticipated by the ideal gas, van der Waals, or other equations of state is not infinitely sharp as in analytic geometry. Rather, a system demonstrates a range of pressure, density, and other properties. The fluctuations are as integral to the thermodynamic behavior as the average values. 

A third alternative for U-cai is the use of empirical equations other than Eq. (17), which have usually been developed for more restrictive conditions. Thus, for liquid fluidization of spheres, based on an empirical... 

The direct measurement of rjg is often practically impossible, especially for polydisperse samples. This is because standard melt rheometers are often unable to provide reliable data at sufficiently low shear rates to reach the region of Newtonian behavior. While this issue is discussed in Section 10.8, it is important to note here that the use of empirical equations for the viscosity function rj y) to extrapolate data is an unreliable procedure. Sometimes it is found that within a given family of polymers (same structure and shape of the MWD) a... 

Caution is advised in the use of empirical equations such as those presented above, because each is based on data for one type of branching structure. Vega et al [ 1 ] evaluated several such correlations and concluded that simple correlations for the number of branches based on a few rheological and molecular parameters are approximately valid only for one specific type of branching structure, e.g randomly branched polymers, metallocene polymers, stars, combs, etc. 

Many empirical correlations have been pubHshed in the Hterature for various types of Hquid atomizers, eg, one book (2) provides an extensive coUection of empirical equations. Unfortunately, most of the correlations share some common problems. Eor example, they are only vaHd for a specific type of atomizer, thereby imposing strict limitations on thein use. They do not represent any specific physical processes and seldom relate to the design of the atomizer. More important, they do not reveal the effect of interactions among key variables. This indicates the difficulty of finding a universal expression that can cover a wide range of operating conditions and atomizer designs. 

Calvert et al. []. Air Pollut. Control Assoc., 22, 529 (1972)] obtained an explicit equation by making some simplifying assumptions and incorporating an empirical constant that must be evaluated experimentally the constant may absorb some of the deficiencies in the model. Although other models avoid direct incorporation of empirical constants, use of empirical relationships is necessary to obtain specific-estimates of scrubber collec tion efficiency. One of the areas of greatest uncertainty is the estimation of droplet size. 

Most of the investigators have assumed the effective drop size of the spray to be the Sauter (surface-mean) diameter and have used the empirical equation of Nuldyama and Tanasawa [Trons. Soc. Mech. Eng., Japan, 5, 63 (1939)] to estimate the Sauter diameter ... 

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

The only problem with the foregoing approach to molecular interactions is that the accurate solution of Schrddinger s equation is possible only for very small systems, due to the limitations in current algorithms and computer power. Eor systems of biological interest, molecular interactions must be approximated by the use of empirical force fields made up of parametrized tenns, most of which bear no recognizable relation to Coulomb s law. Nonetheless the force fields in use today all include tenns describing electrostatic interactions. This is due at least in part to the following facts. 

For other die geometries it is necessary to use the appropriate form of equation (4.12). The equations for a capillaiy and a slit die are derived in Chapter 5. For other geometries it is possible to use the empirical equation which was developed by Boussinesq. This has the form... 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

The experimental results obtained in the laboratory by the researchers can be monitored using computer programs with help of empirical equations or models. Most of the computer-assisted procedures have been developed for HPLC separations and mainly for RPLC, and some of them are commercially available. 

The equation of motion given above may be solved for maximum displacement response using Figure B.2. Transformation factors K, K, and Xlm are provided for a variety of structural elements in References 7, 73, 75, 92, and 93. Solutions in terms of maximum displacement response of the nonlinear SDOF model to transient loads are also provided by these references in graphical form or in the form of empirical equations. 

A limitation of the Erbar-Maddox, and similar empirical methods, is that they do not give the feed-point location. An estimate can be made by using the Fenske equation to calculate the number of stages in the rectifying and stripping sections separately, but this requires an estimate of the feed-point temperature. An alternative approach is to use the empirical equation given by Kirkbride (1944) ... 

These assumptions are akin to those taken in account in the mixed adsorption model of Trogus (12). The difference between the two models lies in the relationship linking CMCs of single and mixed surfactants and monomer molar fractions Trogus used the empirical equation proposed by Mysels and Otter (13) in our model, the application of RST leads to an equation of the same type. 

Extensive testing with dusts and vapors has resulted in a detailed set of empirical equations for the relief vent area (published as NFPA 68).18 The length-to-diameter ratio LID of the enclosure determines the equation(s) used for calculating the necessary vent area. For noncircular enclosures the value used for the diameter is the equivalent diameter given by D = 2VAhr, where A is the cross-sectional area normal to the longitudinal axis of the space. 

Osmotic concentration kinetics were also studied for the purpose of manufacturing carrot preserves (Singh et al., 1999). The preserve quality was assessed as a function of sugar solution concentration and sample-to-syrup ratio, and the kinetics of preserve manufacture were described using an empirical equation. 

Various attempts have been made to increase the valid range of the Debye- Huckel equation to regions of high ionic strength by the use of empirically fitted parameters. Several of these equations are listed in table I. 

It is not a problem either for the protonation constant of S (i.e. the reciprocal of the second dissociation constant of H2S) some estimates of which are shown in Fig. 4. Neither Cobble s estimate (68), using the correspondence principle (curve a) nor Pohl s (69) extrapolation (curve b) using an empirical equation due to Harned and Embree (70) is showing any indication of the expected minimum in K. The extrapolation used by Khodakovskii et al (71) (curve c) is based on the more frequently used expression of Harned and Robinson (72) and a different selection of low temperature data. While their result looks more reasonable it is difficult to have much confidence in any of the results even up to 200 C. The apparent failure of the correspondence principle may arise as much from the choice of low temperature data as a failure of the relationship itself. 

It can be seen from Equation (7.70) that to calculate AG at any temperature and pressure we need to know values of AH and AS at standard conditions (P= 100 kPa, T = 298 K), the value of ACp as a function of temperature at the standard pressure, and the value of AEj- as a function of pressure at each temperature T. Thus, the temperature dependence of AC/> and the temperature and pressure dependence of AVj-are needed. If such data are available in the form of empirical equations, the required integrations can be carried out analytically. If the data are available in tabular form, graphical or numerical integration can be used. If the data are not available, an approximate result can be obtained by assuming ACp and AVp are constant... 

In order to use the correction factors in a generalized Newtonian code, the factors need to be functionalized using an empirical equation. A total of 160 numerical experiments were performed to determine the effect of the design parameters on the correction factors. A random sample of 95 numerical data points were used to evaluate the correction factor fitting function. The equation for the correction factors is as follows ... 

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