Fully developed, definition

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

The second equality in equation (6.28) is a definition of longitudinal dispersion coefficient, Dl- Taylor (1953) assumed that some of the terms in equation (6.28) would cancel and that longitudinal convective transport would achieve a balance with transverse diffusive transport. He then solved the second equality in equation (6.28), for a fully developed tubular fiow, resulting in the relation... 

Chemical metabolism can be described qualitatively or quantitatively. Many scientists can make qualitative predictions of the likely excretion products or blood plasma metabolites in mammals, or a particular animal including man, based on accumulated knowledge and experience. Such knowledge, in its raw form, generally consists of structure-metabolism relationships that are frequently expressible as qualitative structure-based rules that may be encoded into computer-based expert systems (see Chapter 9 for a full definition). Examples of such systems, in their more fully developed commercial forms, are discussed toward the end of this chapter. 

Because the area of the duct per unit width is 2w and because, by definition, the mass flow rate is equal to the (density x area X mean velocity), this shows that the mean velocity in fully developed laminar plane duct flow is 6/4 (= 1.5) times the center line velocity. 

Experimental work of Kalasz et al. resulted in the statement of the characteristics and basic rules of displacement chromatography. They conceived properties of the fully developed displacement train, factors affecting displacement development, efficacy of separation, analysis of displaced fractions, determination of displacement diagrams from Langmuirian isotherms, as well as selection of the column, carrier, and displacer for displacement chromatography. Concentration of the sample is a particular feature of displacement chromatography. However, the displacer in the carrier is also definitely concentrated through the development of the displacement train. 

The combination of all the points reported above seems to indicate versatile and efficient ab initio procedures as the best choice. However, there are other considerations to be added. Both continuum and discrete approaches suffer from limitations due to the separation of the whole liquid system into two parts, i.e. the primary part, or solute, and the secondary larger part, the solvent. These limitations cannot be eliminated until more holistic methods will be fully developed. We have already discussed some problems related to the shape of the cavity, which is the key point of this separation in continuum methods. We would like to remark that discrete methods suffer from similar problems of definition a tiny change in the non-boded interaction parameters in the solute-solvent interaction potential corresponds to a not so small change in the cavity shape. 

This relationship serves in many publications as the definition of a thermally fully developed flow. It is, as we have already seen, a result of the fact that the heat transfer coefficient reaches its asymptotic, constant end value downstream. 

For definiteness, we consider the transfer processes between a cylindrical wall and a turbulently flowing n-component fluid mixture. For condensation of vapor mixtures flowing inside a vertical tube, for example, the wall can be considered to be the surface of the liquid condensate film. We examine the phenomena occurring at any axial position in the tube, assuming that fully developed flow conditions are attained. For steady-state conditions, the equations of continuity of mass of component i (assuming no chemical reactions), Eqs. 1.3.7 take the form... 

The main caution is that these equations are only an approximation of reality, and the formulas and equations have been chosen to model experimental data. This means that they are only as good as the data they were derived from, and the data is usually found in relatively simple flows. The experiments are definitely not easy to carry out, but they are for idealized situations, such as fully developed flow in a pipe or past a flat plate or in a jet. Thus, the equations should be used with caution. Other methods in turbulence are much more time-consuming and may require banks of computers running for hours. That works for research, but is not suitable for day-to-day engineering work. 

Crystal engineering is a highly interdisciplinary area, which to some extent explains why unambiguous definitions and systematic nomenclatures have yet to be fully developed. The term co-crystal is not well-defined, and the existing literature contains terms such as molecular complexes, co-crystals, molecular adducts,... 

Heat exchange in fully developed laminar flow of fluids in tubes of various cross-sections was studied in many papers (e.g., see [80, 253, 341]). In what follows, we present some definitive results for the limit Nusselt numbers corresponding to the region of heat stabilization in the flow in the case of high Peclet numbers (when the molecular heat transfer can be neglected). 

Figure 4 presents the local Nusselt number variation along the microtube for the constant wall temperature boundary condition for cases where both viscous dissipation and axial conduction effects have been considered. A positive Br for this boundary condition refers to the fluid being cooled as it flows along the tube. Local Nu value first decreases due to temperature jump at the wall, then increases to its fully-developed value because of the heating due to the viscous dissipation effect. Before the increase, the values of local Nu match those for the Br = 0 case presented in Fig. 3 [10, 42]. However, because of the definition of Pe, local Nu curves deviate from those for Br = 0 as the minima are approached. This effect results in the overall increase in the average Nu in the tube, thus we can conclude that average Nu increases as the effect of axial conduction is more prominent. Also, the amount of viscous dissipation does not affect the fully developed Nu value.

With the example applications discussed here, it seems to us that it may not be fair at this stage to conclude definitively about the relative performance of SS-MRCEPA(O) and SS-MRCEPA(I) methods. At times CEPA(O) performs better than CEPA(l) and vice versa for the closed-shell case, as we found in more extensive applications, in the case of systems containing only closed-shell configurations [59]. More exhaustive calculations, in particular of the fully developed SS-MRCEPA(I), are needed to come to a definitive conclusion, which is on the way. For the SS-MRPT, we have more extensive applications, not all published yet, which definitely indicate the generally superior performance of the EN partitioning. 

It can be argued that any turbulent flow correlation should not be applied for Re <10,000. However, in current thin-channel ultrafiltration devices, the entrance geometry is such that fully developed turbulent flow occurs at much lower Reynold s numbers. Measurements of fluid velocity versus pressure drop show a definite transition from laminar to fully developed turbulent flow at Re = 2000. 

This use of different Reynolds numbers from one investigator to another makes the comparison of different sets of data quite difficult. The relative merits of the five definitions are discussed below. It was pointed out by Skelland [4] that for fully developed laminar circular-tube flow of nonnewtonian fluids, the wall shear stress xw is a unique function of 8Uld. This may be expressed as... 

K(o°) Incremental pressure drop number for fully developed flow, see Eq. 17.86 for definition, dimensionless... 

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