Peclet number diffusion

The concentration part of the problem is described by the convective diffusion equation (4.4.3) (the subscript 1 is introduced for the stream function) and by the boundary conditions (4.4.4) and (4.4.5) specifying the concentration on the interface and remote from the drop. The diffusion Peclet number Pe is assumed to be small. 

Pe diffusion Peclet number, Pe = aU/D Per heat Peclet number Pr( turbulent Prandtl number Q volume rate of flow (through tube cross-section)... 

Besides the Knudsen number, some other dimensionless parameters become important in microscale flow and heat transfer problems. The first such number is the Peclet number (Pe), which is the product of Reynolds (Re) and Prandtl (Pr) numbers (Pe = Re Pr), and signifies the ratio of rates of advection to diffusion. Peclet number enumerates the axial conduction effect in flow. In macro-sized conduits, Pe is generally large and the effect of axial conduction may be neglected. However as the channel dimensions get smaller, it may become important. 

McHenry and Wilhelm (7), extended the previous work of Deisler and Wilhelm 5) and determined axial diffusion (Peclet numbers) for turbulent flow of gases. Other references in this general area of investigation include Wehner and Wilhelm 8). 

Both UL/a and UL/D are termed the Peclet number and usually given the same symbol. When there is no reason for confusion, we shall do the same otherwise we distinguish between the two by reference to the thermal Peclet number or the diffusion Peclet number. The Peclet number plays a similar role in heat and mass transport as the Reynolds number in momentum transport. The thermal and diffusion Peclet numbers may be written somewhat differently to bring out their relation to the Reynolds number in particular. 

We have already observed that for dilute solutions the Schmidt number is very large, as a consequence of which the diffusion Peclet number... 

We treat first the capture by a collector of submicrometer-size particles undergoing Brownian motion in a low-speed flow of velocity U. The collector is taken to be a sphere of radius a and is assumed to be ideal in that all of the particles that impinge on its surface stick to it (Fig. 8.3.1). Because the Brownian particle diffusivities D - kTIGiruap, where is the particle radius, are typically about a thousand times smaller than the molecular diffusivities, the diffusion Peclet number (Ua/D) is generally very large compared with unity. The diffusive flux of the particles to the surface is therefore governed by the steady, convective, diffusion boundary layer equation, with the particles treated as diffusing points. ... 

Pcpg diffusion Peclet number for the vapor-gas phase, Pe, = tp /to... 

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. 

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. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

Figure 3.2.1 illustrates the mixing in packed beds (Wilhelm 1962). As Reynolds number approaches the industrial range Rep > 100, the Peclet numbers approach a constant value. This means that dispersion is influenced by turbulence and the effect of molecular diffusion is negligible. 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

Equations 8-148 and 8-149 give the fraction unreacted C /C o for a first order reaction in a closed axial dispersion system. The solution contains the two dimensionless parameters, Np and kf. The Peclet number controls the level of mixing in the system. If Np —> 0 (either small u or large [), diffusion becomes so important that the system acts as a perfect mixer. Therefore,... 

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 parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

In this expression, w is a typical velocity scale and d a typical length scale, for example the diameter of a micro charmel. The Peclet number represents the ratio of the diffusive and the convective time-scales, i.e. flows with large Peclet numbers are dominated by convection. 

Study the effect of varying mass transfer and heat transfer diffusivities (D and X, respectively) and hence Peclet numbers (Pi and P2) on the resulting dimensionless concentration and temperature reactor profiles. 

Strictly speaking, in this formulation the effective diffusion coefficient, is replaced by an empirical dispersion coefficient, D, to account for the effect of water flow on diffusion. However, in practice, the rate of transpirational water flow is sufficiently slow that dispersion effects are minimal and Eq. (8) can be used without error. This is because the Peclet number (see Sect. F.2) is small. For the same reason, in almost all cases diffusion is the most important process in moving nutrients to the root and the convection term can be omitted entirely. 

These substitutions replace the eight dimensional parameters in the original equations by the four nondimensional parameters above. The parameter Pe is the Peclet number (37) and shows the relative importance of convection compared to diffusion. The advantage of this formulation becomes obvious when typical parameter values are substituted into the equations. 

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]

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

Fig. 2.3.5 Profiles recorded from a drying alkyd left. The Peclet number is defined as HE/D emulsion layer are shown on the right. At low where H is the film height, the evaporation Peclet number (upper set of profiles), drying is rate and D the particle diffusivity. The upper set...

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