Simplified Equations

Boron, being chemically a non-metal, is resistant to attack by nonoxidising acids but the other members of the group react as typical metals and evolve hydrogen. Aluminium, gallium and indium are oxidised to the + 3 oxidation state, the simplified equation being... 

In the context of our semi-implicit methods, we typically consider the special case Vi = V2 = 0 which leads to the simplified equations of motion... 

It is helpful to simplify Equation (2.157) by introducing the charge density matrix, F, whose elements are defined as ... 

In applying the BET procedure to Type III isotherms, c = 1 constitutes a special case insertion of c = 1 into the standard BET equation (2.12) leads to the simplified equation... 

NSP is produced by the reaction of phosphate rock and sulfuric acid. This reaction quickly yields a soHd mass containing monocalcium phosphate monohydrate and gypsum, CaSO 2H20, according to the simplified equation... 

The focus herein is a survey of contemporary experimental approaches to determining the form of equation 3 and quantifying the parameters. In general, the differential equation could be very compHcated, eg, the concentrations maybe functions of spatial coordinates as well as time. Experimental measurements are arranged to ensure that simplified equations apply. 

For many modeling purposes, Nhas been assumed to be 1 (42), resulting in a simplified equation, S = C, where is the linear distribution coefficient. This assumption usually works for hydrophobic polycycHc aromatic compounds sorbed on sediments, if the equdibrium solution concentration is <10 M (43). For many pesticides, the error introduced by the assumption of linearity depends on the deviation from linearity. 

Electrochemistry and Kinetics. The electrochemistry of the nickel—iron battery and the crystal stmctures of the active materials depends on the method of preparation of the material, degree of discharge, the age (Life cycle), concentration of electrolyte, and type and degree of additives, particularly the presence of lithium and cobalt. A simplified equation representing the charge—discharge cycle can be given as ... 

Starches. In the United States, all potable alcohol, most fermentation industrial alcohol, and most fuel alcohol is currendy made principally from grains com is the principal feedstock for fuel alcohol. Fermentation of starch from grain is somewhat more complex than fermentation of sugars because starch must first be converted to sugar and then to ethanol. This process was known to the ancient Egyptians and Mesopotamians who brewed beer almost 5000 years ago (202). The simplified equations for the conversion of starch to ethanol are... 

Calvert (R-15) critiqued the many pressure-drop equations and suggested the following simplified equation as accurate to 10 percent ... 

It is a common procedure to assume certain conditions for the chromatographic system and operating conditions and, as a result, simplify equations (20) and (21). However, in many cases the assumptions can easily be over-optimistic, to say the least. It is necessary, therefore, to carefully consider the conditions that may allow such simplifying procedures and take steps to ensure that such conditions are carefully met when such expressions are used in practice. Now, the relative magnitudes of the resistance to mass transfer terms will vary with the type of columns (packed or capillary), the type of chromatography (GC or LC) and the type of particle, i.e., porous or microporous (diatomaceous support or silica gel). 

It is clear that, in LC, the resistance to mass transfer in the mobile phase (albeit within the pores of the particle) is much greater than the resistance to mass transfer in the stationary phase and, thus, simplifying equations (23) and (24), by ignoring the dispersion contribution from the resistance to mass transfer in the stationary phase, will be quite valid. 

Simplifying equation (24) using numerical data from Table 2, then equation (2) becomes... 

Constants C and K can be determined from several measurements of filtrate volumes taken at different time intervals. There are some doubts as to the actual constancy of C and K during constant pressure filtration. Constants C and K depend on r (specific volumetric cake resistance), which, in turn, depends on the pressure drop across the cake. This AP causes some changes in the cake, especially during the initial stages of filtration. When the cake is very thin, the main portion of the total pressure drop is exerted on the filter medium. As the cake becomes thicker, the pressure drop through the cake increases rapidly but then levels off to a constant value. Isobaric filtration shows insignificant deviation from the expressions developed. For approximate calculations, it is possible to neglect the resistance of the filter plate, provided the cake is not too thin. Then the filter plate resistance, Rf, is equal to zero, C = 0, and r = 0. Hence, a simplified equation is = Kr. 

Another detailed method of determining pressures is computational fluid dynamics (CFD), which uses a numerical solution of simplified equations of motion over a dense grid of points around the building. Murakami et al. and Zhoy and Stathopoulos found less agreement with computational fluid dynamics methods using the k-e turbulence model typically used in current commercial codes. More advanced turbulence models such as large eddy simulation were more successful but much more costly. ... 

Equation (9.50) is a simplified equation that can be used at normal conditions. For practical purposes,... 

This equation was then solved numerieally with known values of erystal growth rate (G) and solubility dc jdO) for seeded solutions. It was further assumed that if G f G(L, 6), dc /d6 = eonstant and dAc/d6 = 0 then the simplified equation for ealeulating transient temperature results (Mullin and Nyvlt, 1971)... 

Larson and Garside (1973) extended the eontrolled eooling approaeh to provide simplified equations to prediet programmed evaporative erystallization polieies based on the population balanee. 

The equations provide reasonable estimates per Rohsenow, who suggests using with caution, only when performance on the system is not available. Ganapathy offers simplified equations and nomographs to solve these relations. 

In this equation is the standard electrode potential of the water/oxygen reaction, i.e. —AG% o/nF. Simplifying, equation 12.4 at 298K becomes... 

Azeotropic compositions are rare for terpolymerization and Ham 14 has shown that it follows from the simplified eqs. 38-40 that ternary azeotropes should not exist. Nonetheless, a few systems for which a ternary azeotrope exists have now been described (this is perhaps a proof of the limitations of the simplified equations) and equations for predicting whether an azeotropic composition will exist for copolymerizations of three or more monomers have been formulated.20113 This work also shows that a ternary azeotrope can, in principle, exist even in circumstances where there is no azeotropic composition for any of the three possible binary copolymerizations of tire monomers involved. 

Another class of hydrocarbon binders used in propints are the carboxy-terminated polybutadiene polymers which are cross-linked with either tris[l-(2-methyl)aziridinyl] phosphine oxide (MAPO) or combinations with phenyl bis [l -(2-methyl)aziridinyl] phosphine oxide (Phenyl MAPO). Phenyl MAPO is a difunctional counterpart of MAPO which makes possible chain extension of polymers with two carboxylic acid groups. A typical propint formulation with ballistic properties is in Table 11 (Ref 83) Another class of composites includes those using hydroxy-terminated polybutadienes cross-linked with toluene diisocyanate as binders. The following simplified equations illustrate typical reactions involved in binder formation... 

The simplified equation (for the general equations, see Section IV, L) in the case of unsteady-state diffusion with a simultaneous chemical reaction in isothermal, incompressible dilute binary solutions with constant p and D and with coupled phenomena neglected is... 

Water is very frequently used as the cooling medium and the effect of the variation of physical properties with temperature may be included in equation 9.64 to give a simplified equation which is useful for design purposes (Section 9.9.4). 

The mechanism of this type of instability can be elucidated as follows First, at the portion where the anion concentration happens to be higher than other portions of the surface, according to the first reaction equation [Eq. (76a)], the dissolution current density also becomes higher. From Eq. (78b), the current density fluctuation is expressed by the following simplified equation,... 

Also, the zeroth moment of the differential molecular weight distribution, DMWD, may be obtained by integration of the simplified equation ... 

Assuming perfect gas behavior a simplified equation can be used... 

Uncertainties in Photochemical Models. The ability of photochemical models to accurately predict HO concentrations is undoubtedly more reliable in clean vs. polluted air, since the number of processes that affect [HO ] and [H02 ] is much greater in the presence of NMHC. Logan et al (58) have obtained simplified equations for [HO ] and [HO2 ] for conditions where NMHC chemistry can be ignored. The equation for HO concentration is given in Equation E6. The first term in the numerator refers to the fraction of excited oxygen atoms formed in R1 that react to form HO J refers to the photodissociation of hydrogen peroxide to form 2 HO molecules other rate constants refer to numbered reactions above. 

Very complex systems of equations can be handled this allows interactions to be studied that elude those who simplify equations to make them manageable at the paper and pencil level." ... 

When combining Eqs. (A.l) and (A.4), we obtain a second-order nonlinear differential equation for /o(r) which is mathematically very difficult to solve. Therefore, in DH theory a simplified equation is used The exponential terms of Eq. (A.4) are expanded into series and only the first two terms of each series are retained [exp(y) 1 +y]. When we include the condition of electroneutrality and use the ionic strength we can write this equation as... 

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