Equation-based calculations

Definition A.12 Residual) Fault indicator based on deviations between measurements and model equation based calculations. 

For dilute solutions, the Debye-Huckel equation by calculations based on these Coulombic interactions is... 

The effective saturation depth,, represents the depth of water under which the total pressure (hydrostatic plus atmospheric) would produce a saturation concentration equal to for water ia contact with air at 100% relative humidity. This can be calculated usiag the above equation, based on a spatial average value of T, measured by a clean water test. For design purposes,, can be estimated from clean water test results on similar systems, and it can range from 5 to 50% of tank Hquid depth. Effective depth values for coarse bubble diffused air, fine bubble diffused air, and low speed surface aerators are 26 to 34%, 21 to 44%, and 5 to 7%, of the Hquid depth, respectively. 

An equation for calculating the partial pressure of chlorine dioxide above specified chlorine dioxide solutions at various temperatures based on the data from reference 24 has been developed (25) ... 

The concept of a mass-transfer unit was developed many years ago to represent more rigorously what happens in a differential contactor rather than a stagewise contactor. For a straight operating line and a straight equilibrium line with an intercept of zero, the equation for calculating the number of mass-transfer units based on the overall raffinate phase N r is identical to the Kremser equation except for the denominator when the extraction factor is not equal to 1.0 [Eq. (15-23)]. 

Equations to calculate size distributions from sedimentation data are based on the assumption that the particles fall freely in the suspension. In order to ensure that particle-particle interactton does not prevent free fall, an upper-volume concentration hmit of around 0.2 percent is recommended. 

Centerline velocity in Zone 3 of the supply jet can be calculated from the equations based on the principle of momentum conservation along the jet ... 

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

Ward and Sommerfeld [130] present an equation based on the curves shown in Figure 9-21C, D and referenced to Eckert [125] and Leva [43] for calculating the gas and liquid flooding rates. There have been numerous other equations targeted for this pmpose, but many are too awkward for easy general use. The proposed equations have been tested by the authors. 

Filtration of 300 ml of fermentation broth was carried out in a laboratory-sized filter with a pressure drop of lOpsi. The filtration took 20 min. Based on previous studies, the filter cake obtained from Penicillium chrysogenum was compressible with the exponent 5 in the equation for calculation of filter area equal to 0.5. 

Since most equations of state are pressure-explicit, Eqs. (6) and (99) are often more convenient than Eqs. (5) and (98). With these equations, basing his calculations on van der Waals equation of state, Temkin (Tl) showed that gas-gas immiscibility may occur if the van der Waals constants a and b... 

A simulation was performed for an experiment with [O3] = 3.16 X 10-5 M and [OH-] = 7.17 X 10-3 M. The kinetic curve calculated by the kinsim program form this model is depicted in Fig. 5-4. Also shown is an experimental curve calculated from an empirical equation, based on the equation that applies at this OH- concentration ... 

Step 5 Use an equilibrium table to find the H.O concentration in a weak acid or the OH concentration in a weak base. Alternatively, if the concentrations of conjugate acid and base calculated in step 4 are both large relative to the concentration of hydronium ions, use them in the expression for /<, or the Henderson—Hasselbalch equation to determine the pH. In each case, if the pH is less than 6 or greater than 8, assume that the autoprotolysis of water does not significantly affect the pH. If necessary, convert between Ka and Kh by using Kw = KA X Kb. 

Values of e are correlated by the Op constants in accord with eq. (61) (119, 120). Kawabata, Tsuruta, and Furukawa (121) have proposed a revised form of the Price-Alfrey equation based on the definition e = 0 for styrene. On the basis of this redefinition, they have calculated a new set of Q values. These Q values are linear in the Hammett 0 constants. It is not clear from their paper whether this linear relationship is for substituents directly bonded to the carbon-carbon double bond or whether it is applicable only to Q values for substituted styrenes. Charton and Capato (119) were unable to obtain significant correlations between any a constants and the q values of Schwan and Price (117). Zutty and Burkhart (122) have proposed a redefinition of the Price-Alfrey equation based on ethylene as the reference system with Qo and e defined as 1 and 0, respectively. 

Amotf was the first to develop a set of equations for acetone to simultaneously calculate chlorophyll a and chlorophyll b in 1949. Several authors later proposed different new equations based on more adjusted and accurate extinction coefficients due to the development of higher resolution spectrophotometers adapted to each special condition. Moreover, besides 80% acetone, coefficients for diethyl ether and ethanol were also established and their respective equations developed, as reviewed by Schwartz and Lorenzo and Eder. Solvents chosen should be those for which specific absorbance coefficients have been published to derive equations and updates should be carefully tracked for new values. 

Chlorophylls and their derivatives fluoresce independently from one another and their concentrations can be calculated by equations based on the total fluorescence — the sum of individual fluorescences when together or in mixtures. Equations for 80% acetone, diethyl ether, DMF, methanol and ethanol can be found in literature. 

Equation (9.2) can be used to calculate the metal s surface potential. The value of the electron work function X can be determined experimentally. The chemical potential of the electrons in the metal can be calculated approximately from equations based on the models in modem theories of metals. The accuracy of such calculations is not very high. The surface potential of mercury determined in this way is roughly -F2.2V. 

A theoretical approach based on the electrical double layer correction has been proposed to explain the observed enhancement of the rate of ion transfer across zwitter-ionic phospholipid monolayers at ITIES [17]. If the orientation of the headgroups is such that the phosphonic group remains closer to the ITIES than the ammonium groups, the local concentration of cations is increased at the ITIES and hence the current observed due to cation transfer is larger than in the absence of phospholipids at the interface. This enhancement is evaluated from the solution of the PB equation, and calculations have been carried out for the conditions of the experiments presented in the literature. The theoretical results turn out to be in good agreement with those experimental studies, thus showing the importance of the electrostatic correction on the rate of ion transfer across an ITIES with adsorbed phospholipids. 

We have only just started to explore empirical access to FMO parameters based on these similarities 52). Recently, others have reported empirical equations for calculating IP s and EA s for a variety of Ji-bonded systems 64). This approach used a large number of parameters for the underlying rc-system, heteroatom substitution, and the substituents on the 7t-system. However, we aim at calculating FMO parameters from fundamental atomic data while taking due account of the bond structure of a molecule. 

The term "semi-empirical" has been reserved commonly for electronic-based calculations which also starts with the Schrodinger equation.9-31 Due to the mathematical complexity, which involve the calculation of many integrals, certain families of integrals have been eliminated or approximated. Unlike ab initio methods, the semi-empirical approach adds terms and parameters to fit experimental data (e.g., heats of formation). The level of approximations define the different semi-empirical methods. The original semi-empirical methods can be traced back to the CNDO,12 13 NDDO, and INDO.15 The success of the MINDO,16 MINDO/3,17-21 and MNDO22-27 level of theory ultimately led to the development of AMI28 and a reparameterized variant known as PM3.29 30 In 1993, Dewar et al. introduced SAMI.31 Semi-empirical calculations have provided a wealth of information for practical applications. 

The relief sizing calculation procedure involves, first, using an appropriate source model to determine the rate of material release through the relief device (see chapter 4) and, second, using an appropriate equation based on fundamental hydrodynamic principles to determine the relief device vent area. 

When the modules are connected to one another as represented in a flowsheet, a long train of units may become coupled together for calculations. Thus, a set of modules may require a fixed precedence order of solution so that convergence of the calculations may be slower than in equation-based codes. 

For the delivery of atomization gas, different types of nozzles have been employed, such as straight, converging, and converging-diverging nozzles. Two major types of atomizers, i.e., free-fall and close-coupled atomizers, have been used, in which gas flows may be subsonic, sonic, or supersonic, depending on process parameters and gas nozzle designs. In sonic or supersonic flows, the mass flow rate of atomization gas can be calculated with the following equation based on the compressible fluid dynamics ... 

The dose modifications based on body surface area provide a better approximation of dose than the dose based on body weight. The general equation to calculate the child s dose by the surface area method is as follows ... 

Figure 27.2 (a) Symmetry properties for core-shell structures where r,/r2 < 1.20 (b) sterically induced stoichiometry (SIS) based on respective radii (r,) and (r2) core and shell dendrimers respectively (c) Mansfield-Tomalia-Rakesh equation for calculation of maximum shell filling when r,/r2 < 1.20... 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

Miller has shown that TBTO will prevent fouling attachment at leaching rates as low as 1.25 yg/cm2/day (18). It is thus reasonable to assume that fouling commences when the rate of release falls below 0.5 yg Sn/cm2/day. Based on this, the effective dif-fusivities are calculated, using Crank s rate equation. The calculated effective diffusivities are then substituted in the integral form of Crank s equation to estimate the amount of Sn lost. 

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