Porosity profile

Fio. 10. Illustration of the porosity profile through a rod at two times, h and /j, when reaction is occurring at a constant rate in temperature Zone II. 

Fio. 13. Porosity profiles through spectroscopic carbon rods before and after ca. 11% burnoff at different temperatures. 

At 1200°, no decrease in external radius occurred, with the surface porosity reaching a value of only 0.56. At this temperature, there was a significant increase in porosity even near the center hole in the rod consequently, it may be assumed that the carbon dioxide concentration was not zero in this part of the rod. Therefore, reaction was in the transition region between Zones I and II. The reaction should be in Zone II when = (R/CBDe(i)dw/dt > 4, as previously discussed. Since R is ca. 0.48 cm., and at 1200°, Cr is 1 X 10 g. of carbon per cc., dw/dt is 0.22 X 10 g. of carbon/min./cm. and the mean Deti (as discussed shortly) is ca. 0.1 cm.Vsec., = 1.7. Thus, the reaction should be near, but not in. Zone II, in agreement with the interpretation of the porosity profile. 

A very important form of such disturbances is caused by the presence of the wall of the tube containing the packed bed. Vortmeyer and Schuster (1983) have used a variational approach to evaluate the steady two-dimensional velocity profiles for isothermal incompressible flow in rectangular and circular packed beds. They used the continuity equation, Brinkman s equation (1947), and a semiempirical expression for the radial porosity profile in the packed bed to compute these profiles. They were able to show that significant preferential wall flow occurs when the ratio of the channel diameter to the particle diameter becomes sufficiently small. Although their study was done for an idealized situation it has laid the foundation for more detailed studies. Here CFD has definitely contributed to the improvements of theoretical prediction of reactor performance. 

Kuipers, J. A. M., Tammes, H., Prins, W., and van Swaaij, W. P M., Expermental and theoretical porosity profiles in a two-dimensional gas-fluidized bed with a central jet. Powder Technol. 71, 87 (1992b). 

Fig. 12.4 Effects of the depth resolution in pore water concentration profiles on calculating the rates of diffusive transport. Three samples drawn from surface sediments are shown to possess different resolutions (intervals 0.5 cm - dots, 1.0 cm diamonds, 2.0 cm - squares). All values are sufficient to plot the idealized concentration profile within the hounds of analytical error, yet very different flux rates are calculated in dependence on the depth resolution values. In the demonstrated example, the smallest sample distance indicates the highest diffusion (2.98 mmol cmA f ). As soon as the vertical distance between single values increases, or, when the sediment segments under study grows in thickness, the calculated export across the sediment-water boundary diminishes (2.34-t.64mmol cm yr ). In our example, this error which is due to the coarse depth resolution can be reduced by applying a mathematical Fit-function. A truncation of 0.05 cm yields a flux rate of 2.84 mmol cm yr. (The indicated values were calculated under the assumption of the presented porosity profile according to Pick s first law of diffusion - see Chapter 3. A diffusion coefficient of 1 cmA f was assumed. Adaptation to the resolution interval of 2.0 cm was accomplished by using a simple exponential equation).

Table 2. Ratio of uptake in the matrix with porosity profile to that in the low porosity matrix. 2% 14%...

The present model could be refined by introducing a velocity profile. This was done by Valstar [128], who used the velocity profiles published by Schwartz and Smith [8S] that exhibit a maximum at 1.5 dp of the wall, and also by Lerou and Froment [144]. The latter authors concluded from a simulation of experimental radial temperature profiles that a radial velocity profile inversely proportional to the radial porosity profile led to the b t fit. Such a radial velocity profile exhibits more than one peak. It follows from these studies that the influence of radial nonuniformities in the velocity profile are worthwhile accounting for in the simulation of severe operating conditions. Progress in this field will require more extensive basic knowledge of the packing pattern and hydrodynamics of fixed beds. 

Haidegger et al, 1989, have studied the total oxidation of ethane in a fixed-bed reactor and found a better agreement between experiment and simulation with the A (r)-model, compared to the model using a wall resistance (l/ w)-Simulation results for models with and without the radial porosity profile will be compared below. In the model neglecting the influence of the voidage profile, constant mass and heat dispersion and a wall resistance (I/Kw 0) will be applied. The effects of the radial voidage profile will be studied using the same correlation as used in Kiirten, 2003, for the mass and the A(r)-model for the heat dispersion. 

The analogy of these results to that in fixed-bed simulations without sidestream is pronounced and can be explained with the predominance of the bed friction-force term in the extended Navier-Stokes equation (Eq. (5.6)). HydrodynamicaUy developed flow is achieved after a distance of just about one particle diameter in the axial direction. However, the developed profile in a PBMR is characterized by a radial velocity different from zero. One can prove analytically that for reactive flow problems with negligible change in the physical properties (density, viscosity) the superficial radial velocity decreases linearly towards the core (Kiirten, 2003). In Fig. 5.17b the superficial radial velocities are compared. Using the radial porosity profile, smaller absolute values of the local superficial velocity are calcu-... 

Figure 5.17 (a) Axial and (b) radial velocity profile at z/L = 0.5 in the PBMR - effect of porosity profile. 

Reactor models accounting for radial porosity profile were compared with models using the averaged bed-porosity value. Isothermal conditions were applied in order to rule out thermal effects on the concentration profiles. To check the need for two-dimensional models the results were compared with that obtained by using the pseudohomogeneous one-dimensional reactor model (Eqs. (5.1)-(5.4)). 

Figure 5.18 Effect of porosity profile on the local oxygen concentration, (a) 2D oxygen profiles (b) radially averaged axial O2 profile. ID simulation with Temperature 600°C,...

Figure 5.19 Effect of porosity profile on the ethylene selectivity - Temperature 600°C, = 250kgs/m ...

Summarizing, the effect of the porosity profile on the integral reactor performance is rather small for the conditions studied. This can, however, change for systems with kinetics more sensitive to the educt concentration (higher reaction orders). In comparison with the ID model results it was found that the simple model overpredicts the achievable intermediate yields in the PBMR. Consequently, radial mass-transfer limitations can not be neglected if more precise predictions are required. 

Figure 5.28 Radial porosity profile averaged over interparticle space along angular coordinate at two different axial positions and over the entire packing.

Figure 11.10 Porous medium (a), averaging volume, (c) radial porosity profile.

In this chapter it will be shown how 2D models can be used to predict the extent of external mass transfer limitations and their effect on the reactor performance. Also the effect of a radial porosity profile (important where the ratio of the tube diameter over the particle diameter is smaller than about 10) can be included in the model. 

The definition of porosity profile and the way it was computed in the 2D model is also reported. 

When catalytic particles (spheres) are packed in tubes as in the packed bed membrane reactors, a porosity profile occurs which influences the performance of the reactor. The porosity profile in a packed bed of uniform spheres was studied by several research groups as a function of the distance from the wall (Benenati and Brosilow," and Schuster and Vortmeyer ). 

In Figure lO.Al(a) the different porosity profiles are compared with the data points of Benenati and Brosilow, while Figure 10.A 1(b) shows the resulting dimensionless axial velocity profiles, calculated with the PBMR model with the membrane flux set to zero. 

U. Kiirten, M. van Sint Annaland and J. A. M. Kuipers, Oxygen distribution in packed-bed membrane reactors for partial oxidations Effect of the radial porosity profiles on the product selectivity, Ind. Eng. Chem. Res., 2004, 43, 4753 760. 

J. G. H. Borkink, K. R. Westererp, Significant of the radial porosity profile for the description of heat transport in wall-cooled packed beds, Chem. Eng. Sci, 49, S63-ST6 (1994). 

The porosity profile of the PS multilayer stmcture - five low/high porosity bilayers Torres-Costa et al. (2004)... 

TEM has also been used to analyze the in-depth porosity profile of PS-based multilayer stacks. An... 

Torres-Costa V, Paszti F, Climent-Font A, Martin-Palma RJ, Martinez-Duart JM (2005) Porosity profile determination of porous silieon interference filters by RBS. Phys Stat Solid ... 

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