Peclet number, definition

Estimate die dispersion coefficient Dg[ from the definition of the Peclet number. 

Equations (8) are based on the assumption of plug flow in each phase but one may take account of any axial mixing in each liquid phase by replacing the molecular thermal conductivities fc, and ku with the effective thermal conductivities /c, eff and kn eff in the definition of the Peclet numbers. The evaluation of these conductivity terms is discussed in Section II,B,1. The wall heat-transfer terms may be defined as... 

The Peclet number defined for this case is unique to Regime II flows, due to the definition of kn eff given in Eq. (23). Therefore, Peu must be correlated with experimental data from Regime II flows. 

We might properly refer to this value as the apparent Peclet number, because by many formal definitions the Peclet number accounts for the relative importance of advection and molecular diffusion, without mention of hydrodynamic dispersion. 

Before leaving this discussion, it is important to note that other forms of Peclet numbers are also possible and may be more appropriate depending on the type of convective influence studied. For example, in the case of oscillatory flows (as in oscillatory viscometers), it is more useful to define the Peclet number as (Rfa/D), where co is the frequency of oscillation. Regardless of the particular definition, the general significance of the Peclet number remains the same, i.e., it compares the effect of convection relative to diffusion. 

Peclet Number and Damkdhler Number Box 22.2 Alternative Definitions of the Damkdhler Number Illustrative Example 22.1 Vertical Distribution of Dichlorodifluoromethane (CFC-12) in a Small Lake... 

Remembering the definition and meaning of the Peclet number (Eq. 22-11 a), this result is not really a big surprise. In fact, the left-hand side of Eq. 23-37 is the Peclet number for an advective (v /diffusive (Ez) flux over distance h. 

Figure 25.7 Nondimensional attenuation coefficient, a, and 10-1 specific phase shift, P, for sinusoidal input into a saturated groundwater system as a function 10 2 of the Peclet Number Pe. The numbers attributed to the different 1 q 3 lines give the nondimensional angular frequency 2. See text for definitions. 10-4...

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

The Peclet number gauges the magnitude of the departure from equilibrium configuration of the particles. (Note that the rotational Peclet number, Pe, for a sphere has nearly the same definition only 6 is replaced by 8.) As such the Peclet number can be used in the Cross equation to determine the value of > .[= 8], giving... 

To normalize the governing equations, we introduce a dimensionless position, z = x/a, and two dimensionless dependent variables,/ =/// and u = ua/DD. Note that the normalized velocity m is equivalent to a local Peclet number, indicating the relative magnitudes of the advective and diffusive fluxes of the reactive species. Applying these definitions to the transport equations yields the dimensionless governing equations... 

An important characteristic of a property distribution is encapsulated in the Peclet number, Pe = ULIk, which is the ratio of diffusive time-scale to advective timescale of the system. In this definition, U and L are the characteristic velocity and length scales of the flow. The Peclet number is a measure of the relative importance of advection versus diffusion, where a large number indicates an advectively dominated distribution, and a small number indicates a diffuse flow. Numerical modeling indicates that certain tracer distributions, in particular tracer-tracer relationships, are significantly affected by the Peclet number, and consequently can be used to determine the nature of the fluid flow (Jenkins, 1988 Musgrave, 1985, 1990). 

Chung and Wen (1968) and Wen and Fan (1975) have proposed a dimensionless equation using the dependency of the dispersion coefficient on the (particle) Reynolds number Re (Eq. 6.169) for fixed and expanded beds. It is an empirical correlation based on published experimental data and correlations from other authors that covers a wide range of Re. Owing to two different definitions of the Reynolds number, the actual appearance varies in the literature. Since the particle diameter dp, is the characteristic value of the packing, Eq. 6.168 based on the (particle) Peclet number Pe (Eq. 6.170) is used here ... 

We may note that the problem defined by (9-7) and (9-8) is identical mathematically to the corresponding single-solute mass transfer problem, provided the conditions at the body surface and at infinity are such that we can specify the solute concentration as known constants. In this case, we can substitute concentration c (measured as mass fraction of solute) for the temperature T in the definition (9-3) of 9. Then the boundary conditions are identical to (9-8), and the governing equation for solute transport is also the same as (9-7), with the exception that the Peclet number must now be defined in terms of the diffrisivity D for the solute in the solvent, rather than the thermal diffusivity k. Hence, in this case... 

Equation (9 274) can now be interpreted as a relationship between the Peclet number and the Sherwood number, and the constant c in (9 274) can be calculated from the definition... 

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). 

Size can refer to volume, area, or length, and therefore pore-size distribution may be defined in terms of any one of these properties. In practice, the definition of size adopted is highly dependent upon the method of measurement. For example, the area size distribution of pores is often measured by image analysis of soil thin sections, while water retention data are usually interpreted in terms of the distribution of pore diameters (Bullock Thomasson, 1979). For consistency with the definition of the Peclet number, we have chosen to define size in terms of length, L. Dullien (1991) has proposed the following interrelationships between the different definitions of size L = VIS in three-dimensions or L=AJP in two-dimensions, where V is volume, S is surface area, A is cross-sectional area and P is perimeter. These relations can be used to compare pore-size distributions measured using different methods. 

Values of the radial dispersion coefficient, or the corresponding radial Peclet number, udp/Dr), in packed beds have been determined for both liquids and gases by a number of researchers they are definitely not the same as those in the axial direction. These results are shown in Figure 5.10a and b for liquids and gases, respectively. The corresponding empirical equations fitting these data are... 

Convergence is obtained when the appropriate guess for d p./di at the reactor inlet predicts the correct Danckwerts condition in the exit stream, within acceptable tolerance. To determine the range of mass transfer Peclet numbers where residence-time distribution effects via interpellet axial dispersion are important, it is necessary to compare plug-flow tubular reactor simulations with and without axial dispersion. The solution to the non-ideal problem, described by equation (22-61) and the definition of Axial Grad, at the reactor outlet is I/a( = 1, RTD). The performance of the ideal plug-flow tubular reactor without interpellet axial dispersion is described by... 

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