Mixing Peclet numbers

The value for the radial fluid-mixing Peclet number, Pej.f( >), is often given in the range 8.0-12.0, as determined by Fahien and Smith (28) and later workers. The radially-averaged Peclet numbers of such workers include mass transfer resistance all the way to the wall, however, whereas the extra resistance to fluid phase transport near the wall is covered in the present model by the use of the parameter Bif. Thus the appropriate value for present purposes would be nearer the bed-centre value of Perf( ) = 8.0, also reported by Fahien and Smith (28). 

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. 

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,... 

N = L/d, or that each row of pellets amounts to a mixing cell. This is intuitively reasonable. However, the parallel dispersion for heat has a Peclet number equal to 0.5, which would argue that four rows of pellets should be considered to be a mixing cell. The heat balance equation for cell i in a cascade of N cells is... 

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. 

Pe, coj, and fio are dimensionless parametas relating to the opoating conditions Pe is Peclet number denoting the inverse of axial mixing intensity, coj denotes the inverse of volumetric loading rate per mass of granules, and ySo daiotes the dimensionless inlet substrate concentration as respectively defined as follows ... 

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. 

Parameter estimation 112 Partial differential equation 578 Partial differentials 154 Peclet number 243, 579 Penetration distance 654 Perfect mixing 159 Perfectly mixed 142... 

The Murphree efficiency EmV is only equal to the point efficiency Emv if the liquid on the plate is perfectly mixed. On a real plate this will not be so, and to estimate the plate efficiency from the point efficiency some means of estimating the degree of mixing is needed. The dimensionless Peclet number characterises the degree of mixing in a system. For a plate the Peclet number is given by ... 

A Peclet number of zero indicates perfect mixing and a value of oo indicates plug flow. For bubble-cap and sieve plates the eddy diffusivity can be estimated from the equation ... 

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... 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

At high Reynolds numbers where molecular diffusion effects are negligible, experimental evidence confirms the general validity of equation 12.7.5. Figure 12.15 indicates how the Peclet number for radial mixing varies with the fluid Reynolds number. Above a Reynolds number of 40, the radial Peclet number is approximately 10. 

The mixing properties in a fluidized bed are a strong function of the fraction voids. Minimum values of radial Peclet numbers udp/Djt) are observed at e = 0.7, corresponding to a transition in the type of particle circulation in the bed. 

The main parameter in this model characterizing the quality of the flow is the axial dispersion coefficient. The term axial is used to distinguish mixing in the direction of flow from mixing in the radial direction. Then, based on this parameter, the particle Peclet number is introduced ... 

Concerning packed bubble bed reactors, the evaluation of the Peclet number of the liquid-phase is important in order to decide if we have to use a plug- or backmixed-flow model. The liquid-phase can be considered well mixed if (Ramachandran and Chaudhari, 1980)... 

It should be noted that since the solid phase is not stationary in a fluidized bed, the movement of the solid phase can also be described by a Peclet number. Thus, there are two Peclet numbers in fluidized beds for axial and radial mixing, i.e. one for the fluid side and one for the solid one. However, only the fluid-side Peclet numbers are presented here. 

According to Gunn (1968), the radial Peclet number in particulate fluidization (liquid-solid systems) ranges between 1 and 10 for values of Rep in the range 4—1000. Furthermore, the maximum mixing coefficient is found for sf = 0.7. Finally, the lateral (radial) mixing coefficients in gas-solid fluidized beds decrease constantly (for Rep > 10) from about 10 - 0.05 by increasing the expansion ratio from 0.01 - 0.2. 

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