Peclet number defined

Naturally, there are two more Peclet numbers defined for the transverse direction dispersions. In these ranges of Reynolds number, the Peclet number for transverse mass transfer is 11, but the Peclet number for transverse heat transfer is not well agreed upon (121, 122). None of these dispersions numbers is known in the metal screen bed. A special problem is created in the monolith where transverse dispersion of mass must be zero, and the parallel dispersion of mass can be estimated by the Taylor axial dispersion theory (123). The dispersion of heat would depend principally on the properties of the monolith substrate. Often, these Peclet numbers for individual pellets are replaced by the Bodenstein numbers for the entire bed... 

For the study of flow stability in a heated capillary tube it is expedient to present the parameters P and q as a function of the Peclet number defined as Pe = (uLd) /ocl. We notice that the Peclet number in capillary flow, which results from liquid evaporation, is an unknown parameter, and is determined by solving the stationary problem (Yarin et al. 2002). Employing the Peclet number as a generalized parameter of the problem allows one to estimate the effect of physical properties of phases, micro-channel geometry, as well as wall heat flux, on the characteristics of the flow, in particular, its stability. 

The term Lu / D is known as the Peclet number, Pe, and its inverse as the dispersion number. The magnitude of the Peclet number defines the degree of axial mixing in the reactor. 

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. 

When Pe is less than about 46, heat loss effects on Umay be appreciable. If an analogy is made with turbulent flames using a turbulence Peclet number defined by Pe = Urd/aT, where ar 0.058RciZ/ [21, 22], it is readily shown that Pe is substantially above 46 for all of the present turbulent flame experiments. The influence of heat loss is therefore not likely a contributor to the bending effect exhibited in Fig. 15.4. 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

Equation (9) generalizes earlier porosity-Peclet number power-law correlations (Konstandopoulos et al., 2002) obtained at Pe > 0.3 down to the diffusion limited deposition limit. PeQ is a characteristic cross-over Peclet number defining the scale beyond which the convective mechanism will take over the diffusive mechanism of deposition and eK the large Peclet number asymptote of the porosity. K has a dependence on the aggregate size and it is described in a forthcoming publication (Konstandopoulos, 2007). Using Eq. (9) the experimental data of Fig. 9 can be collapsed on a single curve as shown in Fig. 10. 

Eqn. (3.4-76) has been solved for a number of initial- and boundary conditions using a variety of techniques [32]. Application to dispersion problems provides information on the axial Peclet number, defined as ... 

The parameter obtained after identification of the system as plug flow with axial dispersion, is the Peclet number defined from the packing height and the real fluid flow velocity ... 

The boundary conditions are dx/dz = Px at z =0 and dx/dz = 0 at z = 1. Here, P is the axial Peclet number, defined as the product of velocity and reactor length, divided by the effective axial diffusivity. 

Consider a Newtonian incompressible fluid containing a component A in high dilution (<0.05M) and moving under creeping flow conditions within a relatively high porosity porous medium. The solid surface adsorbs instantaneously the eomponent A. The mass transport regime (convection and/or diffusion) is expressed by the value of the Peclet number, defined... 

Notably, the Peclet number differs from the Bodenstein number (or axial Peclet number) defined in Eq. 6.106 by the ratio of particle diameter to column length ... 

Peclet number, Pe A dimensionless number used frequently by chemical engineers. Pe = (ux)/Dm, where x is a characteristic length, u the mobile phase velocity, and Dfn the molecular diffusivity. In chromatography, there are two Peclet numbers, defined by respect to the column length and to the particle size. The former must be large (i.e., 100 or more) for the simple models of chromatography to be valid. The second is known by chromatographers as the reduced mobile phase velocity. 

Equation (13-10) shows that the conversion depends on the dimensionless reaction-rate group TpPpdpICQU and the radial and axial Peclet numbers, defined by Eqs. (13-11) and (13-12). 

The coupled set of nonlinear differential equations (equations 1 and 4) are solved by the alternating direction implicit (ADI) method f9-10) on an evenly spaced grid. The advective transport of a solute species was solved using the Lax-Wendroff two-step method (10). To ensure that numerical dispersion is avoided, a grid spacing was chosen such that the grid Peclet number (defined by < 2 fll). The computational expense involved in using a... 

Pe Peclet number, defined by correlation Eq. 13.2.14 Pem modified Peclet number, defined by correlation Eq. 13.7.2 Pr Prandtl number... 

This is exactly the Peclet number defined by Eq. (5.3.25), which measures the characteristic rotational Brownian diffusion time to the time scale defined by the reciprocal of the shear rate. It is the same measure found for dilute polymer solutions with the particle radius here replacing the Flory radius for the polymer. 

The reduced shear stress is recognized to be essentially our Peclet number defined by Eq. (9.2.10). As can be seen from Fig. 9.2.3, as the Peclet number is increased, the viscosity reaches a stationary value and in this limit the suspension behaves as a Newtonian fluid. In the opposite limit, as the Peclet number tends to zero, the relative viscosity approaches a higher stationary value. The transition is seen to take place in the neighborhood of a Peclet number close to unity, consistent with our earlier discussion. 

Note that in the inequality of (5-33) that the quantity uR/Dm) can be considered a radial Peclet number defined analogously to the longitudinal quantity defined previously. It has been suggested that the right inequality is a little tight, and that iiR/Dm) > 50 is a more comfortable approximation [V. Ananthakrishnan, W.N. Gill and A.I. Barduhn, Amer. Inst. Chem. Eng. J., 11, 1063 (1965)). 

The subscript i refers to gas-phase reactants and products the superscript c refers to gas-phase cell threads. The accumulation term in (3.43) is equalized to the sum of the convective and reactive fluxes. The reactive term is computed only when a control volume has a face adjacent to the catalytic geometrical surface Fext), while the other terms are considered for internal faces Pint). Face variables, denoted by the subscript f, are computed with a first-order upwind scheme. If Peclet number, defined as shown in (3.44), is significantly greater than 1 (as happens for the LNT case study described later), it is possible to neglect the diffusion term vL... 

The following two models are frequently used to account for partial macromixing the dispersion model and the tanks-in-series model. In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. This model was anticipated in Chapter 12, and the governing equations for mass and heat are listed in Table 12.2 of that chapter. The derivation is similar to that for plug flow except that now a term is included for diffusive flow in addition to that for bulk flow. This term appears as -D ( d[A]/d ), where is the effective axial diffusion coefficient. When the equation is nondimensionalized, the diffusion coefficient appears as part of the Peclet number defined as = itd/D. A number of correlations for predicting the Peclet number for both liquids and gases in fixed and fluidized beds are available and have been reviewed by Wen and Fan (1975). 

The equation with axial mixing given in Table 12.2 for a fixed-bed reactor also holds for a homogeneous reactor, except that the effective diffusion coefficient is replaced by the real Df. This can be recast as /Pe )(d a/d ) da/dz) — (Z,/[v4]ou)(-rvA) = where Pe i is now the axial Peclet number defined as uL D(. This equation has been solved (see Danckwerts, 1953 Wehner and Wilhelm, 1956) to obtain an expression for concentration as a... 

Pe)d = pipe flow Peclet number defined in Eq. (114) Zi = vertical coordinate at point 1... 

Peclet number defined based on radial dispersion coefficient (= UD/Dr)... 

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