Empirical expressions

In the case of nitrogen on iron, the experimental desorption activation energies are also shown in Fig. XVIII-13 the desorption rate was given by the empirical expression... 

In the original study (61), NTU /NTU was calculated for thousands of hypothetical design cases as a function of both Pe and Pep -pu- The results were correlated and empirical expressions were given that can be evaluated on a handheld calculator, just as equations 76 and 77, but which approximate the computer calculation much better, to within about 5%. 

An estimate of AR ia alloys can be made usiug the empirical expression (12) of equation 10 where the average alloy reduced energy, is defined by equation 11, where C (i = 1, 2,... , n) is the elemental atomic fraction of the /th element, and is the elemental reduced energy defined iu equation 3. Using this formulation, the projected range straggling iu compounds can be calculated to within 20%. 

Evaluation of the integrals requires an empirical expression for the temperature dependence of the ideal gas heat capacity, (3p (8). The residual Gibbs energy is related to and by equation 138 ... 

Size-dependent Crystal Growth. A number of empirical expressions correlate the apparent effect of crystal size on growth rate (30). The most commonly used correlation uses three empirical parameters to correlate growth rate with crystal size ... 

The standard Gibbs-energy change of reaction AG° is used in the calculation of equilibrium compositions. The standard heat of reaclion AH° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating AH° and AG° with T. Numerical values for AH° and AG° are computed from tabulated formation data, and AC° is determined from empirical expressions for the T dependence of the C° (see, e.g., Eq. [4-142]). 

The following empirical expression (Treybal, Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963) has been found to represent all the available data reasonably well, considering the great variety of circumstances and the considerable scatter in many of the original data ... 

For many years, it was thought that the macro solute forms a new phase near the membrane—that of a gel or gel-like layer. The model provided good correlations of experimental data and has been widely used. It does not fit known experimental facts. An explanation that fits the known data well is based on osmotic pressure. The van t Hoff equation [Eq. (22-75)] is hopelessly inadequate to predict the osmotic pressure of a macromolecular solution. Using the empirical expression... 

Using the Stokes-Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals... 

Above a concentration of 2 percent, additional terms must be added to the Dubinin expression, or an empirical expression can be used. 

Ahn and Shull [118] have shown that the relationship between G and v can be adequately described by the following empirical expression ... 

Estimation of the heat transfer coefficients for forced convection of a fluid in pipes is usually based on empirical expressions. The most well known expression for this purpose is ... 

Another empirical expression that can be used to predict pressure drop, again based upon application of dimensional analysis is given by the following formula. This relationship is applicable to fabrics having porosities over the range of 0.88 to 0.96,... 

A final empirical expression that can be used is given below. This equation relates pressure drop to fabric properties which are readily obtainable from equipment suppliers. 

But the following empirical expression can be applied to describing washing ... 

The hold-up and bubble diameter in mechanically agitated vessels are given by the following empirical expressions ... 

In the second approach, a value for is not assumed but a relationship for dip/dT is determined from semi-empirical expressions for the amount of cooling air that is required in an (elementary) turbine blade row. One such relationship, derived in Ref. [5], gives... 

Evidence for secondary nucleation has came from the early continuous MSMPR studies. MSMPR crystallization kinetics are usually correlated with supersaturation using empirical expressions of the form... 

Rowe, P.N., 1987. A convenient empirical expression for the estimation of the Richard-son-Zaki exponent. Chemical Engineering Science, 42, 2795-2796. 

Corrections due to remaining correlation effects are estimated by an empirical expression. 

For mixed homogeneous/heterogeneous primary nucleation, the following empirical expression can be used to describe its rate Kirk-Othmer Encyclopedia of Technology, 1978-1984) ... 

The next step is to find suitable expressions for r and r. Considerable simplifications are in any case needed. Because of these reasons, the empirical expressions for rj and o listed below were used as a starting approach (L=ligand),... 

Above a Reynolds number of around 2, Equation 8.5 will underestimate the drag coefficient and hence overestimate the settling velocity. Also, for Re > 2, an empirical expression must be used7 ... 

Equation B.21 assumes the compression to be adiabatic ideal gas (isentropic) compression. In practice, the compression will be neither perfectly adiabatic nor ideal. To allow for this, the gas compression can be assumed to follow a polytropic compression represented by the empirical expression ... 

The model is a significant improvement over the Lockhart and Martinelli correlations for pressure drop and holdup (discussed in Sec. 3.5.3). A severe limitation of the model, however, is the dependency on the empirical expression for Cf j [Eq. 3-126]. This expression is based on air-water data only, and has not been shown to apply to other systems. 

A procedure for characterizing the rates of the volume change of gels has not been uniformly adopted. Often, the kinetics are simply presented as empirical sorption/desorption curves without quantitative analysis. In other cases, only the time required for a sample of given dimensions to reach a certain percentage of equilibrium is cited. One means of reducing sorption/desorption curves to empirical parameters is to fit the first 60% of the sorption curve to the empirical expression [119,141]... 

Such empirical expressions have been also verified by Kulkarni et A1.(21) and Kheradmand(7) but they have been established for room temperature. In fact, we have to know the temperature... 

For the Bingham plastic, there is no abrupt transition from laminar to turbulent flow as is observed for Newtonian fluids. Instead, there is a gradual deviation from purely laminar flow to fully turbulent flow. For turbulent flow, the friction factor can be represented by the empirical expression of Darby and Melson (1981) [as modified by Darby et al. (1992)] ... 

A widely quoted empirical expression for the function in Eq. (14-22) is that of Richardson and Zaki (1954) ... 

It has been found that this relationship can be represented by the empirical expression (Coulson et al., 1991)... 

In the opposite limit, go 1, r(oo) can be described by formula (6.16), with the potential (6.12) averaged over the period of the driving signal. Therefore we can obtain the following empirical expressions for the amplitude of the resonant activation effect for co = 0, = oo ... 

We have been able to derive an approximate analytical expression for A as a function of which describes the curves shown in Figure 3 quite accurately over nearly the whole concentration range, from dilute solutions up to 0.6. The derivation is based upon the greatest eigenvalue of the matrix which, in the case of nonadsorbing long chains at low concentrations between the plates, is easily computed (19). We will report the derivation elsewhere and give here only the result, which in the present context may be considered as an empirical expression ... 

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