Diffusion coefficient intraparticle

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

Surface diffusion considerably influences the mass transfer inside the intraparticle space of a porus adsorbent. The surface diffusion coefficient Ds can be expressed by... 

Aguwa, A.A.. Patterson, J.W., Haas. C.N., and Noll, K.E. Estimation of effective intraparticle diffusion coefficients with differential reactor columns./. Water PoMut Control Fed., 56(5) 442-448,1984. 

The intraparticle phenomena The next step is the evaluation of the internal effectiveness factor. The unknown parameter is the effective solid-phase diffusion coefficient, which is (eq. (3.602))... 

The viscosity of most gases at atmospheric pressure is of the order of 10"7 Ns/m2, so for pores of about 1 /mi radius DP is approximately 10"5 m2/s. Molecular diffusion coefficients are of similar magnitude so that in small pores forced flow will compete with molecular diffusion. For fast reactions accompanied by an increase in the number of moles an excess pressure is developed in the interior recesses of the porous particle which results in the forced flow of excess product and reactant molecules to the particle exterior. Conversely, for pores greater than about 100/im radius, DP is as high as 10"3 m2/s and the coefficient of diffusion which will determine the rate of intraparticle transport will be the coefficient of molecular diffusion. 

The question remains as to when the various diffusion effects really influence the conversion rate in fluid-solid reactions. Many criteria have been developed in the past for the determination of the absence of diffusion resistance. In using the many criteria no more information is required than the diffusion coefficient DA for fluid phase diffusion and for internal diffusion in a porous pellet, the heat of reaction and the physical properties of the gas and the solid or catalyst, together with an experimental value of the observed global reaction rate (R ) per unit volume or weight of solid or catalyst. For the time being the following criteria are recommended. Note that intraparticle criteria are discussed in much greater detail in Chapter 6. 

Using the dusty gas model [5] analytical solutions are derived to describe the internal pressure gradients and the dependence of the effective diffusion coefficient on the gas composition. Use of the binary flow model (BFM, Chapter 3) would also have yielded almost similar results to those discussed below. After discussion of the dusty gas model, results are then implemented in the Aris numbers. Finally, negligibility criteria are derived, this time for intraparticle pressure gradients. Calculations are given in appendices here we focus on the results. 

Now that we have derived the intraparticle pressure gradients, we can also determine the effective diffusion coefficient as a function of the gas composition. 

The diffusion coefficient of S02 through the product layer has also been measured, the value of which is about 10 12 m2/s. Figure 9 shows comparison of intraparticle gas diffusivity between the sulfurated sorbent and its calcined sample, indicating calcination can enhance gas diffusivity. The inert matter in the sorbent, however, is beneficial for improving gas diffusion. Measurements (Zhang, 1992) indicated that pores greater than 700 A in diameter for different limestones possess the same distribution function /i(r), as can be expressed by the following correlation ... 

The intraparticle (pore) diffusion coefficient defined in Section 6.2.2.4 may estimated by the Mackie-Meares correlation (Mackie and Meares, 1955) ... 

A plot of H/(2mq) versus 1/Ug is a straight line with a slope equal to Di and an ordinate equal to 3f + S )/Sq. The coefficient of external mass transfer is estimated using one of the several correlations available for it (see Chapter 5, subsection 5.2.5, correlation of Wilson and Geankoplis [62], Kataoka et al. [87], or the penetration theory [88]). Correcting for the contribution due to the external mass transfer resistance gives the last term in the plate height equation, 5, hence the intraparticle diffusion coefficient, Dg. 

In industrial practice, the most convenient way of accounting for mass-transfer effects is to view the penetrable catalyst particle as a pseudo-homogeneous phase. Obstruction of mass transfer by the solid material in the particle then is reflected by an "effective" intraparticle mass-transfer or diffusion coefficient that is appropriately lower than in the contacting fluid. If this approach is taken, two fundamentally different mass-transfer situations appear Mass transfer to and from the particle across an adherent boundary layer is affected by the reaction only in that the latter sets the boundary condition at the particle. Here, mass transfer and reaction are sequential and occur in different parts of the system, and the slower of the two is the bottleneck and dictates the overall rate and its temperature dependence. Within the particle, however, mass transfer and reaction occur simultaneously and in the same volume element. Here, the reaction introduces a source-or-sink term into the basic differential material balance. If the reaction is slow, it alone controls the overall rate and its temperature dependence. If mass transfer is slow, both reaction and mass transfer affect the rate, and the apparent reaction order and activation energy are the arithmetic means of those of reaction and mass transfer. 

Consequently, diffusion coefficients were calculated for all runs assuming that intraparticle diffusion predominates and that the characteristic length value is 2 microns. Typical values of the diffusion coefficients as a function of the SO2 partial pressures are shown in Figure 11. The order of magnitude of the diffusion coefficients, -10 to -12, is within the usual range of intraparticle diffusion coefficients in molecular sieves. The diffusion coefficient decreased both with the flow rate and temperature (not... 

Mass transfer within the catalyst particles occurs by diffusion only which may be expressed by means of a constant effective diffusion coefficient Dg, and the rate of intraparticle diffusion is described by Pick s law. 

Under these circumstances, the interparticle transport resistances can be neglected. What are left are the intraparticle resistances, i.e. the heat and mass transfer effects inside the catalyst particles. Since the current case reflects the situation that few reactant and product molecules exist in an environment of solvent molecules, the simplest Fick s law approach with effective diffusion coefficients can be considered as sufficient for the description of molecular diffusion. 

It can be seen that the first breakthrough for the sandstone column (run 2) occurs at later times than for the limestone columns. This is due to the higher apparent intraparticle diffusion coefficient D ppj of the sandstone grains where D ppj is defined by ... 

In (5.51), r stands for the intrinsic reaction rate at liquid bulk conditions. For worst-case-estimations, one should use a highest rate value possible in the considered RD column. In this respect it should be kept in mind that the reaction rates under RD conditions strongly depends on the operating pressure that influences the boiling temperatures, that is the reaction temperature. D g/ represents the effective diffusion coefficient of a selected reaction component inside the catalyst particles. One should use the component with the lowest mole fraction Xj in the liquid bulk mixture as key component [35]. Its effective diffusion coefficient can be estimated from the diffusion coefficient at infinite dilution Dg((/ = (sfr)D with the total porosity e and the tortuosity r of the applied catalyst. Based on (5.51) one can say that intraparticle diffusion resistances will be negligible, if 1. 

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