Peclet number simplified

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

Because the term r(CA T) is exponentially dependent on T and can be nonlinear as well, a numerical solution or piecewise linearization must be used. To simplify the numerical manipulations, equations in Table IX are normalized by = z/L, r = ut/L, and jc = 1 - C,/(C,)0, where i is normally S02. y also is a normalized quantity. The Peclet numbers for mass and heat are written PeM = 2Rpu/D) and PeH = 2Rpcpul t for a spherical particle. They are also written in terms of bed length as Bodenstein numbers. It is... 

The presentation of numerical solutions of Eq. 2.18 and their discussion are greatly simplified if some reduced variables are introduced at this stage. These variables are the reduced axial position (x)> the reduced radial position (p), the reduced time (t), the axial (Pea) and the radial (Per) column Peclet numbers (note that these two Peclet numbers are different from the conventional particle Peclet number or reduced velocity, v = udp / D ), and the column aspect ratio (0), which are defined as follows... 

These relations can be used as rough estimates of steric rejection, if the solute and membrane pore dimensions are known. The derivation is based on a strictly model situation (see Figure 1) and a long list of necessary assumptions can be written. Apart from the simplified geometry (hard sphere in a cylindrical pore), it was also assumed that the solute travels at the same velocity as the surrounding liquid, that the solute concentration in the accessible parts of the pore is uniform and equal to the concentration in the feed, that the flow pattern is laminar, the liquid is Newtonian, diffusional contribution to solute transport is negligible (pore Peclet number is sufficiently high), concentration polarization and membrane-solute interactions are absent, etc. 

If the Peclet number is high compared to 1 (Pe 1), Eq. 37 can be simplified by omitting the diffusion term in the x direction and by linearizing the flow velocity dependence ony, because we can limit calculations of the analyte concentration to a region close to the sensor surface (y < h) [20-22]. We find ... 

If the Peclet number is small, Pe l, two simplifying approximations are possible (Dukhin, 1965, 1981). As it will be shown in the following, if Pe l the surface concentrations differ negligibly from their equilibrium values,... 

Order-of-magnitude analysis indicates that diffusion is neghgible relative to convective mass transfer in the primary flow direction within the concentration boundary layer at large values of the Peclet number. Typically, liquid-phase Schmidt numbers are at least 10 because momentum diffusivities (i.e., i/p) are on the order of 10 cm /s and the Stokes-Einstein equation predicts diffusion coefficients on the order of 10 cm /s. Hence, the Peclet number should be large for liquids even under slow-flow conditions. Now, the partial differential mass balance for Cji,(r,0) is simplified for axisynunetric flow (i.e., = 0), angu-... 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

Answer At large Prandtl and heat transfer Peclet numbers, the fluid temperature must satisfy the following simplified thermal energy balance ... 

If there is only one chemical reaction on the internal catalytic surface, then vai = — 1 and subscript j is not required for all quantities that are specific to the yth chemical reaction. When the mass transfer Peclet number which accounts for interpellet axial dispersion in packed beds is large, residence-time distribution effects are insignificant and axial diffusion can be neglected in the plug-flow mass balance given by equation (22-11). Under these conditions, reactor performance can be predicted from a simplified one-dimensional model. The differential design equation is... 

The following strategy should be used to calculate the interpellet axial dispersion coefficient and the mass transfer Peclet number in packed catalytic tubular reactors (see Dullien, 1992, Chap. 6). Initially, one should calculate a simplified mass transfer Peclet number (i.e., Pesimpie) based on the equivalent diameter of the catalytic pellets, equivalent, the average interstitial fluid velocity through the packed bed, (Uj>intetstitiai, and the ordinary molecular diffusion coefficient of reactant A, a, ordinary-... 

Empirical and theoretical correlations for T are summarized in Table 22-6, and documented for the following range of simplified mass transfer Peclet numbers 10 < Pcsimpie < 3 X 10 . Theoretical correlation (4) in Table 22-6, which has been developed in detail by Been (1998, pp. 398-404) and Bird et al. (2002,... 

If T ss 2 Pesimpie when the simplified Peclet number is greater... 

When the simplified mass transfer Peclet number is very small (i.e., <1), T 0.67 instead of unity because the numerator of T (i.e., lSA.eff. axial disp.) is based on unsteady-state pore diffusion without convection, whereas the denominator of T (i.e., a, ordinary) is measured in an unrestricted bulk fluid phase. In other words, the diffusivity in the numerator of T is reduced by porosity and tortuosity factors. 

Step 5. Use your results from step 4 together with the following assumptions the mass transfer Peclet number is large, and the tube diameter is much smaller than the overall length of the reactor to simplify the mass transfer equation. 

Hlavacek and Hofmann also defined necessary and sufficient conditions for multiplicity, for a simplified rate law of the type Barkelew used (Eq. 11.5-c) and equality of the Peclet numbers for heat and mass transfer. The necessary and sufficient conditions for multiplicity, which have to be fulfilled simultaneously are... 

In addition, the Peclet number Pe = QUAD, and c = NJ2Q. More simplifieations are possible. Where double layers overlap, the solution to the Poisson-Boltzmann equation... 

Peclet number mixing, Hardt, Schbnfeld and co-workers [69, 70] have developed some theoretical models that are based on the solution of the diffusion equation with simplifying assumptions. Although these models are rather simple, they enable quantitative information on mixing to be obtained that is both free of numerical diffusion and in good agreement with experimental data. 

In liquids, the Schmidt number is much higher than 1, of the order of 1000 in water. As a result, even at low flow velocity, the Peclet number is much higher than 1 (for a velocity of 1 mm s in a channel of 100 pm diameter, Pe= 100). Inthis case, the arcsinh function can be simplified by a logarithmic function and the mixing time is... 

The axial Peclet number for heat has about the same value as the corresponding value for dispersion of mass, that is, a value of 2 (Section 4.11.4), and so Eq. (4.10.151) simplifies to ... 

As the value of the Peclet number increases, the behavior of the axial dispersion model in fact approaches that of a plug flow model. As a result of this, it is possible to simplify... 

The ratio of reaction and permeation rates is critical in designing an MR. Dimensionless numbers are important in parametric analysis of engineering problems. They allow comparison of two systems that are vastly different by combining the parameters of interest. Dimensionless numbers are used to simplify the meaning of the information in scaUng-up the reactor for real flow conditions and to determine the relative significance of the terms in the equations. The Damkohler number (Da) is the ratio of characteristic fluid motion or residence time to the reaction time, and the Peclet number (Pe) defines the ratio of transport rate by convection to diffusion or dispersion (Basile et al, 2008a Battersby et al., 2006 Moon and Park, 2000 Tosti et al., 2009). In the case of an MR, Da and Pe are defined in Equations [11.1] and [11.2]. 

Knowing properties of liquids, it is possible to find Pr, Sc, Le, and consequently Peclet s number for various values of Re. Such estimations are of great importance, because they allow us to simplify the system of equations (5.106) to (5.109). 

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