Simplified Expressions

In isothermal conditions, G = const., and if the nucleation is either purely instantaneous dNa t)ldx = DS x), S being the Dirac delta function), or purely sporadic (or spontaneous according to Chapter 7) in time (dNai ) /dx = B), Equation (15.8) assumes the form  

For cooling at a constant rate t and the same types of nucleation as above (instantaneous or sporadic in time), Ozawa [62] has proposed the following expression  

Nevertheless, from an experimental point of view, Nakamura s approach utilizes results of isothermal crystallization, while Ozawa s involves experiments performed at constant cooling rates. The kinetic parameters xi T) and n can be modified to take into account flow effects [64,65]  

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Equations n82)-(185) are valid when all three masses are different. In case two masses are equal, namely, = me, we get for the simplified expression... 

Linear Driving Force Approximation Simplified expressions can also be used for an approximate description of adsorption in terms of rate coefficients for both extrapai ticle and intraparticle mass transfer controlling. As an approximation, the rate of adsorption on a particle can be written as ... 

He analyzed dispersion data from 11 field experiments in order to determine the form of the functions fy and f, including release height effects. Irwin (9) has used simplified expressions for these functions where both fy and /, have the form... 

Hence, a simplified expression may be written to describe the filtration process as follows ... 

This expression is difficult to use, as iteration is required. A simplified expression can be used with sufficient accuracy ... 

More recently, Rao and Yanseen (1986) gave a simplified expression ... 

A simplified expression for the desorption of a molecule from an immobile adsorption state is ... 

Discharge Current and Discharge Power The discharge current of a battery depends on the external circuit resistance, and is given by Ohm s law as when using the simplified expression (19.5), as... 

This result was taken as an experimental eonfirmation of the model developed by Sehmiekler [7]. However, it appeared somehow eontradictory with other results obtained with SECM. It was also suggested that eoneentration polarization phenomena occurring at the aqueous side are negligible as the whole potential drop is presumably developed in the benzene phase. This assumption can be qualitatively verified by evaluating a simplified expression for the potential distribution based on a back-to-back diffuse double layer [40,113],... 

A scoring system was fixed to simplify expression of the test results (table 3). 

Simplified expressions were developed for estimating the magnitude of each force. 

Inasmuch as the rheological properties are very difficult to measure for very dilute solutions (100 ppm or less), a simplified expression was developed by Darby and Pivsa-Art (1991) in which these rheological parameters are contained within two constants, /q and k2 ... 

Table 2.5, together with the subsequent worked examples, illustrates the application of the statistical tests to real laboratory situations. Equation (2.10) is a simplified expression derived on the assumption that the precisions of the two sets of data are not significantly different. Thus the application of the F-test (equation (2.8)) is a prerequisite for its use. The evaluation of t in more general circumstances is of course possible, but from a much more complex expression requiring tedious calculations. Recent and rapid developments in desk top computers are removing the tedium and making use of the general expression more acceptable. The references at the end of the chapter will serve to amplify this point. 

Robinson and Walsh have reviewed earlier cell designs. The performance of a 500 A pilot plant reactor for copper ion removal is described. Simplified expressions were derived for mass transport both in single pass [243] and batch recirculation [244]. For a detailed discussion of the principle and the role of the rotating cylinder electrode reactor in metal ion removal the reader is referred to Refs. [13] and [241] (46 references). 

Figure 1. Potential profile ij/(x) across the cell wall (x < 0) according to the simplified expression (6) from the Ohshima and Kondo model [15]. The profile within the solution (x > 0) is assumed to follow the usual GC decay expression ij/ = ij/Dm exp ( — kx) (equation (2)). Parameters T — 298 K, z—+1, = 78.5, thickness of the Donnan layer = 10 nm, p — 1447Cdm Curves correspond with c — 0.2, 0.1, 0.05, 0.02 and...

Throughout the preceding discussion, simplified expressions of stream-specific enthalpy as a function of temperature are used. They have to be updated during process operation to consider changes in steady-state compositions. 

A prime on Z denotes the configurational integral without the factorial coefficients. This notation is introduced to simplify expressions which appear later. 

Thus, in Chapter 6, the transport equations for /++ x, t) and the one-point joint velocity, composition PDF /u+V, + x. / ) are derived and discussed in detail. Nevertheless, the computational effort required to solve the PDF transport equations is often considered to be too large for practical applications. Therefore, in Chapter 5, we will look at alternative closures that attempt to replace /++ x, t) in (3.24) by a simplified expression that can be evaluated based on one-point scalar statistics that are easier to compute. 

The effect of pressure P on the rate constant k for a reaction can be summarized in the simplified expression... 

This maximum velocity depends on the molecular mass Mg, the specific heat y, and the stagnation temperature Tg. The velocity increases as y and Mg decrease, and as To increases. Based on Eq. (1.52), a simplified expression for mass flow rate in terms of the nozzle throat area Aj (= A ) and the chamber pressure (= pg) is given by... 

Equation (1.68) can be represented by a simplified expression for thrust in terms of nozzle throat area and chamber pressure ... 

Table 1.1 shows form functions for several types of propellant grains. Substituting Eqs. (1.80), (1.86), and (1.88) into Eq. (1.87), one obtains a simplified expression for the volumetric burning rate change ... 

When the term in the exponent Xg/6g >> 1, a simplified expression for the heat flux feedback from the gas phase to the solid phase is obtained ... 

Though the theoretical detonation velocity and pressure at the CJ point are expressed by very simplified expressions, the computed results obtained by means of Eq. (9.7) are confirmed by measured data for RDX- and TNT-based explosives, as shown in Table 9.110 (Cp-B indicates Composition B , with the two columns relating to different particle sizes). 

We will explicitly ignore entropy and any other temperature-related phenomena and use the simplified expression given by equation 13,... 

Thus our unimolecular isomerization reaction actually occurs by a sequence of steps and is therefore a multiple-reaction system We need a simplified expression for the overall rate of this rate in terms of Ca alone because zl is an intermediate whose concentration is always very small just as for the free-radical intermediates and dimers in the previous examples. 

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