Intraparticle velocity

Only the component of the intraparticle velocity that is parallel to the column axis is significant. 

This result shows that increasing the flowrate (or the Reynolds number Re) the intraparticle velocity v increases also (and so does X) and the apparent effective diffusivity also increases. 

In Figure 4 the intraparticle velocity v is shown as a function of the superficial velocity u. The agreement between experimental results and model calculations is quite satisfactory. 

Figure 4- Intraparticle velocity v versus superficial velocity u (tracer gas H2 cata yst BM 329)...

The reduced velocity compares the mobile phase velocity with the velocity of the solute diffusion through the pores of the particle. In fact, the mobile phase velocity is measured in units of the intraparticle diffusion velocity. As the reduced velocity is a ratio of velocities then, like the reduced plate height, it also is dimensionless. Employing the reduced parameters, the equation of Knox takes the following form... 

This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using matliematical expressions show how physical... 

Here uR is the velocity at which the solute band moves along the column and u is the velocity of the mobile phase that is, u = (superficial velocity)/e, where superficial velocity is volumetric flow rate divided by cross-sectional area of column and s is the fractional volume of column occupied by mobile phase. Most column packings are porous, in which case s includes both interstitial and pore (intraparticle) voidage, as defined in the note to Table 19.1, and here u is less than the interstitial velocity. 

The lumped kinetic model can be obtained with further simplifications from the lumped pore model. We now ignore the presence of the intraparticle pores in which the mobile phase is stagnant. Thus, p = 0 and the external porosity becomes identical to the total bed porosity e. The mobile phase velocity in this model is the linear mobile phase velocity rather than the interstitial velocity u = L/Iq. There is now a single mass balance equation that is written in the same form as Equation 10.8. 

Brusseau, M. L., The influence of solute size, pore water velocity, and intraparticle porosity on solute dispersion and transport in soil , Water Resour. Res., 29,1071-1080 (1993). 

If we express the free cross-section of the column bed by the interparticle porosity (knowing total porosity of packed beds with porous particles is larger because of intraparticle space), we can obtain the true average fluid velocity, v,... 

Figure 1.13 illustrates flow through a bed packed with porous particles and ui and U2 are the EOF velocities through the small intraparticle pores and the larger interstitial channels, respectively. For pressure driven flow, assuming that the path length and the pressure drops are the same for the two cases the flow velocity varies as square of... 

Since electroosmotic flow can exist in both the interparticle and intraparticle spaces, numerous studies have focused on the existence of intraparticle flow in CEC. Several groups have investigated the existence of electroosmotic flow in wide-pore materials [41-44], A model was developed to estimate the extent of perfusive flow in CEC packed with macroporous particles [41] by employing the Rice and Whitehead relationship. Results showed the presence of intraparticle EOF in large-pore packings (> 1000 A) at buffer concentrations as low as 1.0 mM. Additional parameters had been investigated [43,44] to control intraparticle flow by the application of pressure to electro-driven flow. Enhancement in mass transfer processes was obtained at low pore flow velocities under the application of pressure. The authors pointed out that macroporous particles could be used as an alternative to very small particles, as smaller particles were difficult to pack uniformly into capillary columns. 

Another approach is to scale the A-term contribution to a so-called effective particle diameter as has been proposed by Vallano and Remcho [19], They defined an effective particle diameter from the perfusive EOF velocity within each volume fraction of the pore size distribution of the particles. Their model allows inclusion of both the intraparticle EOF and a pore size distribution. However, both this approach as well as Eq. (10) are more or less empirical. 

Here, the parameter F = Uo]dJ2De( — t) considers the effect of intraparticle diffusion, Pe = V dJlEzi. takes into account the effect of axial dispersion, S = 3(1 — e)Kt/U0L considers the effect of total external mass-transfer resistance, and A0 = /j (l — )k dp/2UoL considers the effect of surface reaction on the conversion. In these reactions L/0l, s the superficial liquid velocity, dp is the particle... 

Equation (11) is valid for small fractional conversions (F < 0.5). Its main advantage is that the effective diffiisivity, D f, is obtained from the individual diffusivities Dq and D,. The linear dependence of the fractional conversion F versus (Cot) values in the initial stage is typical for intraparticle kinetic processes. Within the approximate solution, the moving boundary velocity is independent of the diffusivity Dy. There is dependence on charge, Zy, and this aspect is considered. 

The gas flow velocity through the emulsion phase is close to the minimum fluidization velocity When the particles are spherical and have diameters of several tens of microns, this flow condition gives a quite small particle Peclet number, dpUmf/Dc. For example, the Peclet number is estimated as 0.1-0.01 when 122-/Lim-diam. cracking catalyst is fluidized by gas, with Umt = 0.73 cm/sec and Dq = 0.09 cmVsec and it is estimated as 0.001-0.01 for 58-/u.m-diam. particles, with Umt = 0.16 cm/sec. The mechanism of mass transfer between fluid and particles in packed beds is controlled by molecular diffusion under such low Peclet numbers, and the particle Sherwood number kfdp/Dc, is well over 10 (M24). Consequently with intraparticle diffusion shown to be negligible (M21), instantaneous equilibrium is established to be a good approximation [see Eq. (6-24)]. 

Considering the effects of the radial and the angular components of the velocity vector of intraparticle convective flow in the pores of the spherical particles. 

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