Modeling Peclet number

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Estimate the Peclet number from Equation 8-141. If the value of Npg is 10 or greater, accept it. Otherwise, use a refinement of the theory, which accounts for the boundary conditions at the outlet, or formulate another model. 

The dispersed plug flow model has been successfully applied to describe the flow characteristics in the Kenics mixer. The complex flow behavior in the mixer is characterized by the one-parameter. The Peclet number, Np, is defined by ... 

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

Water at room temperature is flowing through a 1.0-in i.d. tubular reactor at Re= 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance What is the Peclet number at this minimum tube length Why would anyone build such a reactor ... 

When a number of competing reactions are involved in a process, and/or when the desired product is obtained at an intermediate stage of a reaction, it is important to keep the residence-time distribution in a reactor as narrow as possible. Usually, a broadening of the residence-time distribution results in a decrease in selectivity for the desired product. Hence, in addition to the pressure drop, the width of the residence-time distribution is an important figure characterizing the performance of a reactor. In order to estimate the axial dispersion in the fixed-bed reactor, the model of Doraiswamy and Sharma was used [117]. This model proposes a relationship between the dispersive Peclet number ... 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

To treat current distribution in through-holes, Hazlebeck and Talbot [140] formulated a model and established dimensionless criteria for ohmic and convective regimes. The most uniform distribution is obtained in the ohmically limited regime bounded by a numerical criterion based on a Peclet number and a Thiele modulus yT. In the notation of Takahashi and Gross,... 

Kondo and co-workers have studied through-mask deposition in the convective regime experimentally for both low [141] and high [142] Peclet numbers. They have also developed a numerical model which incorporates fluid flow [143], Their Peclet number is defined in terms of the cavity depth and has the same form as Eq. (6.12). They considered the effects of cavity geometry as determined by aspect ratio and sidewall angle. 

We consider two cases, one with a higher Peclet number than the other. Disper-sivity tt[, in the first case is set to 0.03 m in the second, it is 3 m. In both cases, the diffusion coefficient D is 10-6 cm2 s-1. Since Pe L/oti., the two cases on the scale of the aquifer correspond to Peclet numbers of 33 000 and 330. We could evaluate the model numerically, but Javandel el al. (1984) provide a closed form solution to Equation 20.25 that lets us calculate the solute distribution in the aquifer... 

Fig. 20.3. Transport model of the migration of a chemical species through an aquifer, calculated for two Peclet numbers, Pe. Species is not present initially, but from t = 0 to t = 2 years recharge at the left boundary contains the species at concentration C0. After this interval, concentration in recharge returns to zero. Fine line shows result for dispersivity aL of 0.03 m, corresponding to a P6clet number on the scale of the aquifer (1000 m) of 33 000 bold line shows results for oil = 3 m, or Pe = 330.

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some correlations of the Peclet number that have been achieved are cited in problem P5.08.14. It is related to the variance of an RTD, as discussed in problem P5.08.04. Consequently the dispersion model and the Gamma or Gaussian are interrelated. 

A significant merit of the dispersion model is some experimental correlations for the Peclet number. There are no such direct correlations for the parameters of the Gamma or Gaussian or other similar models. 

Writing the model in dimensionless form, the degree of axial dispersion of the liquid phase will be found to depend on a dimensionless group vL/D or Peclet number. This is completely analogous to the case of the tubular reactor with axial dispersion (Section 4.3.6). 

The equations have previously been derived in Section 4.4.4 in a form suitable for programming with MADONNA. Correlations for the column Peclet number are taken from the literature and used to calculate a suitable value for the dispersion coefficient for use in the model. 

The application of finite strains and stresses leads to a very wide range of responses. We have seen in Chapters 4 and 5 well-developed linear viscoelastic models, which were particularly important in the area of colloids and polymers, where unifying features are readily achievable in a manner not available to atomic fluids or solids. In Chapter 1 we introduced the Peclet number ... 

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