Empirical, equation factor

The practice of estabHshing empirical equations has provided useflil information, but also exhibits some deficiencies. Eor example, a single spray parameter, such as may not be the only parameter that characterizes the performance of a spray system. The effect of cross-correlations or interactions between variables has received scant attention. Using the approach of varying one parameter at a time to develop correlations cannot completely reveal the tme physics of compHcated spray phenomena. Hence, methods employing the statistical design of experiments must be utilized to investigate multiple factors simultaneously. 

How these factors affect erosion can be determined from the empirical equation shown in Table 4-14. The equation shows erosion rate proportional to a constant, the mass of the particle, its velocity, and angle of attack. 

In practical terms the above analysis is tcx) simplistic, particularly in regard to the assumption that the stresses in the fibre and matrix are equal. Generally the fibres are dispersed at random on any cross-section of the composite (see Fig. 3.8) and so the applied force will be shared by the fibres and matrix but not necessarily equally. Other inaccuracies also arise due to the mis-match of the Poisson s ratios for the fibres and matrix. Several other empirical equations have been suggested to take these factors into account. One of these is the Halpin-Tsai equation which has the following form... 

It is gratifying that no empirical calibrating factor was needed with the Fe-55 source, which means that the results were predictable from Equation 5-6 by insertion of accepted values for the mass absorption coefficients. The deviation corrected by the introduction of this empirical factor (Equation 5-7) was of the kind produced by the filtering of polychromatic beams. About all that can be said about such empirical factors and about background corrections is this Always unwelcome, not to be introduced unless necessary, the need for them does not in itself make a method less desirable, but it does usually indicate that something is incompletely understood. 

These differences have been attributed to various factors caused by the introduction of new structural features. Thus isopentane has a tertiary carbon whose C—H bond does not have exactly the same amount of s character as the C—H bond in pentane, which for that matter contains secondary carbons not possessed by methane. It is known that D values, which can be measured, are not the same for primary, secondary, and tertiary C—H bonds (see Table 5.3). There is also the steric factor. Hence, it is certainly not correct to use the value of 99.5 kcal mol (416 kJ mol ) from methane as the E value for all C—H bonds. Several empirical equations have been devised that account for these factors the total energy can be computed if the proper set of parameters (one for each structural feature) is inserted. Of course these parameters are originally calculated from the known total energies of some molecules that contain the structural feature. 

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. 

The equations for estimating nucleate boiling coefficients given in Section 12.11.1 can be used for close boiling mixtures, say less than 5°C, but will overestimate the coefficient if used for mixtures with a wide boiling range. Palen and Small (1964) give an empirical correction factor for mixtures which can be used to estimate the heat-transfer coefficient in the absence of experimental data ... 

S3) reported that the size distributions of teconite pellets are self-preserving over a wide pelletizing interval. The scaling factor Dw, weight median diameter, is related to the agglomeration time by the following empirical equation ... 

Many studies investigating one or more of these potential rate-determining steps have been carried out over the years. These studies have shown that the rate of reaction depends upon many factors such as temperature [15, 27-29], pellet size [27-29], crystallinity [28], additive types and concentrations [30], process gas type and quantity [31, 32], molecular weight [22, 31] and end group concentrations [16, 33] - all of which will be addressed individually later in this section. Various models have also been proposed involving kinetics [33] and/or by-product diffusion [11, 16, 21, 27-29, 34, 35] through to empirical Equations [15]. The variety of models used and the wide range of kinetic and physical data published demonstrate the complexity of the mechanisms involved. 

A number of empirical equations have been obtained for the rate of sedimentation of suspensions, as a result of tests carried out in vertical tubes. For a given solid and liquid, the main factors which affect the process are the height of the suspension, the diameter of the containing vessel, and the volumetric concentration. An attempt at co-ordinating the results obtained under a variety of conditions has been made by Wallis 8 . ... 

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 ... 

Another approach developed on the basis of an empirical compliance calibration, which was designed originally for isotropic brittle materials (Berry 1963), appears to avoid certain problems associated with correction factors. The compliance is given in the form of empirical equation... 

Nagaoka s equation (Nagaoka et al, 1955) is an extension of Plank s model and takes into account the time required to reduce the temperature from an initial temperature T, above the freezing poinf. The lafenf heat of fusion in equafion 3.3 is replaced by the total enthalpy change A/i which includes the sensible heat which must be removed in reducing the temperature from an initial T and in addition an empirical correction factor is included. Thus... 

Approximate derivation o Tobwg)- Given values of and the basic BWG treatment also leads to explicit equations for the ordering temperature, T ", but the omission of sro inevitably leads to calculated values that are appreciably higher than shown by experiment. If the simplicity of the BWG method is to be retained, an empirical correction factor (x) has to be included, where X = Typical equations for various structural transitions are given... 

The effect of a given salt on vapor composition in a given system is, of course, a function of the relative proportions of the two volatile components in the liquid as well as of salt concentration, and an equation for correlation of salt effect at other than fixed liquid composition should contain liquid composition as a factor. Hashitani and Hirata (18) reported some success with a purely empirical equation which related the improvement factor of Equation 1 both to salt concentration and to liquid composition. Guyer, Guyer, and Johnsen (19) proposed an empirical relationship between vapor composition change and the concentration... 

An extensive system of metallic radii has been formulated on the basis of this equation. It is evident that there is some uncertainty about this empirical equation in particular, the value 0.60 A for the factor of the logarithmic term is somewhat uncertain but, in fact, the conclusions about electronic structure, bond numbers, and valence in metals and intermetallic compounds that have been reached through use of the equation would not be significantly changed by some change in the value of this factor. <... 

In granular solids or in analysis of liquids or slurries in which a considerable amount of particulate material exists, the scattering effect attenuates the optical signal in addition to the absorption. Scattering back from the body of the sample toward the surface produces the intensity to be measured as diffuse reflectance. Scattering also controls the depth of penetration of the sample as well as does its absorptivity (10). The complexity of these two factors acting at once is difficult to predict a priori. This is another reason why the empirical method and the empirical equation coefficients produced by a training set are essential. 

The maximum velocity at the axis of the contactor, Wj(0), is a function of the geometric factors and operating conditions. The experimental results have been fitted to an empirical equation for calculation of the interstitial velocity at the contactor axis in jet spouted beds. The fitting has a... 

Tooker [2] then used this approach to formulate a means of calculating the extract air requirements for enclosures, such as conveyor transfer stations. The equations of Hemeon [1] and Tooker [2], and variations thereon, are used widely in industry. However, they generally grossly over-predict the rate of air entrainment unless empirical correction factors... 

If we express EM in terms of free energies we obtain Equation 10.13 and we can recognise that RTIn (EM) is an empirical correction factor when an intermolecular process is replaced by an intramolecular one. If we substitute into Equation 10.13 the separate enthalpy and entropy terms and recognise that in a strain free ring AE/ nter = AHintm we get Equation 10.14. 

Substituting the known factors into the expression above and rearranging the result in the following empirical equation ... 

Other models are presented in references 9-14. Gates et al. (9) also considered piston flow and d = 1. Van der Baan (70) considered piston flow but with d undetermined. One patent (14) is based on piston flow, sv = 1, d = 2.3 (0 = 0.76), and some corrective scale factors/ In our previous work on the subject (77), we considered piston flow, sv = 1, and d = 3. The recent work of Bernard et al. (72) points to the necessity of taking into account radial velocity profiles. Finally, in an earlier work, Wollaston et al. (73) used only an empirical equation relating conversion at the riser exit and process severity. 

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