Concentration radial profiles

In a sedimentation equilibrium run, the stationary radial concentration profile, which is established after a few hours for a 1-mm column, is analyzed according to Equation (72) or, in case of polydisperse samples, Equations (74) or (75). Contrary to the sedimentation velocity experiment, the diffusion coefficient D is not required. 

The lumped pore model (often referred to as the POR model) was derived from the general rate model by ignoring two details of this model [5]. The first assumption made is that the adsorption-desorption process is very fast. The second assumption is that diffusion in the stagnant mobile phase is also very fast. This latter assumption leads to the consequence that there is no radial concentration gradient within a particle. Instead of the actual radial concentration profile across the porous particle, the model considers simply its average value. 

However, flow tube systems for use at much higher pressures, up to several hundred Torr, have also been designed and applied to reactions of atmospheric interest (e.g., see Keyser, 1984 Abbatt et al., 1990, 1992 Seeley et al., 1993 and Donahue et al., 1996a). At these higher pressures, the velocity and radical axial and radial concentration profiles are experimentally determined and the full continuity equation describing the concentration profiles is solved. 

For radial concentration profiles, a quadratic representation may not be adequate since application of the zero flux boundary conditions at r, = cp0 and r, = 1.0 leads to d2 = d3 = 0. Thus a quadratic representation for the concentration profiles reduces to the assumption of uniform radial concentrations, which for a highly exothermic system may be significantly inaccurate. 

Although additional radial collocation points increase the dimensionality of the resulting model, they may be necessary to accurately express the radial concentration profiles. Preliminary analysis in this section considers only one interior radial concentration collocation point, although a detailed analysis of this assumption is presented in Section VI,E. 

However, a quadratic representation of the radial concentration profile may not be adequate since application of zero flux conditions at the inner thermal well and outer cooling wall with a quadratic profile reduces to an assumption of uniform radial concentrations. Although additional radial... 

Simulations show that the radial and axial temperature and bulk concentration profiles are effectively not influenced by these modeling differences. Figure 9 shows the radial concentration profiles at = 0.38 and at the reactor outlet. Even with very high Peclet numbers, the differences between the radial concentration profile across the relatively small bed and the assumed uniform profile are minimal. Under typical operating conditions with small Peclet numbers, there is no benefit to increasing the number of radial collocation points, especially in light of the increased dimensionality of the resulting system. 

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

Figure 13.7 Radial concentration profiles for an A —> B —> C reaction in the case of convection-diffusion-reaction in a catalytic hollow-fiber membrane.

When the Pedet number equals about R3, or when Ad < 1 and Brownian motion is negligible, the radial concentration profile displays a maximum value greater than unity. Then a generalized form of transformation (11a)... 

Use of Ficks law to describe the diffusion process requires the solute particle to be small compared with the diffusion boundary layer. The analysis presented above suggests that, for Peelet numbers greater than 100, the ratio 8o/0p is proportional to (Pe)ua/R. The solid curves in Figure 3 are truncated at the value of the Peelet number corresponding to Pe/R3 10 "2, where an inspection of the radial concentration profile revealed that the ratio 8d/Op is about ten. 

Figure 9. The radial concentration profiles for the impurities detected in two adjacent breakdown trees 5A-1 and 5A-2. Note the breaks in the logarithmic concentration scale and also the essentially constant concentration of sulphur in comparison with the large variations in the Na, Cl, K and Ca concentrations.

Dry, R. J. Radial concentration profiles in a fast fluidized bed, Powder Technol. 49,37-44(1987). Dry. R. J., and White, C. C. Gas residence time characteristics in a high-velocity circulating fluidized bed of FCC catalyst, Powder Techn. 58, 17 (1989). 

A full analysis of the efficiency of particle separation in CHDF gives the appropriate criteria for the development of a steady state radial concentration profile [68]. Particle transit time is a logarithmic function of particle size. Pressures of up to 30,000 Pa are required and give a separating range from 0.2 to 200 nm [69]. 

Equation 2 expresses whether radial diffusion, which in the case of laminar flow is due to molecular diffusion, is fast enough to outlevel radial concentration profiles. This approximation usually holds for monolithic reactors because of the rather small channel diameter. The corresponding axial dispersion coefficient can be calculated [1] from the following ... 

Dry R.J, Radial concentration profiles in a fast fluidised bed, Powder Technology. 49. 37(1986)... 

According to Eq. 6.4 the mass flow wmt is generally equal to the overall accumulation in the adsorbent. If concentrations and loadings change inside the particles, the overall accumulation (Eq. 6.14) must be calculated by integration (Eq. 6.19) after the radial concentration profiles are obtained. In this case it is reasonable to replace Eq. 6.4 with the continuity equation around the particles, as the mass flow through the outer liquid film is equal to the mass flow entering the adsorbent particle (Eqs. 6.25 and 6.26). 

Models lor Nonideai Reciciars Chap. 14 Radial Concentration Profiles... 

The at erage oullei conversion becomes 68.8 55 . not much differem from the one in pan (a) in agreement with the Aris-Taylor analysis. However. due to the laminar flow assumption in the reactor, the radial concentration profiles are very different throughout the reactor,... 

FIGURE 13.1-12 Radial concentration profiles in particle-diffusion-conlrolled ion exchange with ions of differing mobility. Figure reproduced from F. Helfferich, Ion Exchange. McGraw-Hill, New York. 1962. Available ftom Univemity Microfilms International, Ann Arbor. Ml. 

FIGURE 12.12 Steady-state radial concentration profiles of concentration of reactant species C(r) relative to the concentration at the surface of the drop Cs for a drop radius Rp as a function of the dimensionless parameter q (Schwartz and Freiberg 1981). 

Two proximations allow the solution of this balance. First, the dependency of the reaction rate on temperature shall follow the approximation already used in the discussion of the heat explosion theory. The second step neglects the development of radial concentration profiles. 

Figure 5.14 Effect of radial concentration profiles on the chain length and chemical composition distributions of an ethylene/1-hexene copolymer (four microparticle layers were used for simplicity sake many more layers are required in an actual MGM simulation) [131],...

See Hunt [40] for a recent exposition.) The step from Eq. (b) to Eq. (c) is only possible under conditions that the radial concentration profile has a chance to stabilize somewhat, or, in practical terms, that the length to diameter ratio be sufficiently large ... 

In this type of reactor, as the name implies, the flow is laminar. In other words, the radial concentration profile is parabolic and not uniform as in a PFR. This is because there is hardly any mixing and the reactant concentration profile closely matches the velocity profile (lowest near the wall to highest at the center). Thus each element of fluid flowing through the reactor is completely independent of the other elements, so that the fluid as a whole tends to behave as a macrofluid. In essence, therefore. Equation 13.10 would be valid for this case also. Integrated... 

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