Mass concentration profiles

Fig. 25. Predicted mass concentration profiles for S02 and 02 in the experimental trickle bed used by Haure et al. (1989) at vL = 1.65 mm/s. (Figure from Stegasov et at, 1992, with permission, 1992 Elsevier Science Publishers.)...

Another two-dimensional, discrete element model was applied by Cartaxo and Rocha [43]. In this work, only the dynamic phenomena were investigated, that is, heat and mass transfer between the phases were not considered. Thns, the inflnence of the momentum coupling between the discrete particles and the conveying air on the air radial velocity and the mass concentration profiles was presented. An object-oriented numerical model was developed to simulate the conveying of large spherical particles (3 mm) through 9.14 m vertical tube with 7.62 cm bore size. 

In the presence of flocculating cells or pellets a considerable longitudinal cell mass concentration profile usually prevails. This can cause very significant nonuniformity of dissolved oxygen concentration profiles. It is unlikely that under these conditions the longitudinal profile of product concentration is uniform in tower... 

The radial mass concentration profiles follow a Gaussian contur perpendicular to the jet axis, where the concentration decays inversely proportional ro distance . 

Dietz (1981) ctssumed (i) t r(z) to be a constant, i/ ), and (ii) that, due to turbulent mixing, the particle mass concentration profile in each region was radially uniform. However, each such profile, />pi(z), Ppzi ) snd Ppsi ) regions 1, 2 and 3, respectively, changes with the z-coordinate due to particle deposition at the cyclone wall and/or particle exchange between regions 2 and 3 ... 

We now consider the modification of the Leith and Licht (1972) model by CUft et al. (1991) the result is a relatively simple procedure for estimating for a cyclone. It assumes that the particle mass concentration profile, is uniform radially, but varies axially it is represented by Ppg z) for the whole cyclone. Consider now a length dz of the cylindrical part of the cyclone. The rate at which particle masses are being deposited from this volume to the cyclone wall at r = is (2ntc dz)Ppg z)Upruj. However, this must equal the rate of change of particle mass in that volume ... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Fig. 6. Concentration profiles through an idealized biporous adsorbent particle showing some of the possible regimes. (1) + (a) rapid mass transfer, equihbrium throughout particle (1) + (b) micropore diffusion control with no significant macropore or external resistance (1) + (c) controlling resistance at the surface of the microparticles (2) + (a) macropore diffusion control with some external resistance and no resistance within the microparticle (2) + (b) all three resistances (micropore, macropore, and film) significant (2) + (c) diffusional resistance within the macroparticle and resistance at the surface of the...

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

Fig. 3. Concentration profiles near an iaterface where the arrow represents the direction of mass transfer, = concentration of C in A-rich phase,...

The equations of combiaed diffusion and reaction, and their solutions, are analogous to those for gas absorption (qv) (47). It has been shown how the concentration profiles and rate-controlling steps change as the rate constant iacreases (48). When the reaction is very slow and the B-rich phase is essentially saturated with C, the mass-transfer rate is governed by the kinetics within the bulk of the B-rich phase. This is defined as regime 1. 

Fig. 5. Concentration profiles for different numbers N of 2one passes for k = 0.5 and a rod containing 10 2one masses. Obtained by numerical methods of...

M. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant mass flux at wall... 

Figure 14-10 illustrates the gas-film and liquid-film concentration profiles one might find in an extremely fast (gas-phase mass-transfer limited) second-order irreversible reaction system. The solid curve for reagent B represents the case in which there is a large excess of bulk-liquid reagent B. The dashed curve in Fig. 14-10 represents the case in which the bulk concentration B is not sufficiently large to prevent the depletion of B near the liquid interface and for which the equation ( ) = I -t- B /vCj is applicable. 

FIG. 14-13 Gas-phase and liquid-phase solute-concentration profiles for a liquid-phase mass-transfer limited reaction system in which is larger than 3. 

Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The cui ves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The cui ve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about cf= nf = 0.5, N — Ti) = 0]. The hnear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. 

In neutron reflectivity, neutrons strike the surface of a specimen at small angles and the percentage of neutrons reflected at the corresponding angle are measured. The an jular dependence of the reflectivity is related to the variation in concentration of a labeled component as a function of distance from the surface. Typically the component of interest is labeled with deuterium to provide mass contrast against hydrogen. Use of polarized neutrons permits the determination of the variation in the magnetic moment as a function of depth. In all cases the optical transform of the concentration profiles is obtained experimentally. 

The influence of the mass transfer resistance on the purity and on the steady state internal concentration profiles are shown in Figs. 9-11 and 9-12. A higher value for the mass transfer coefficient corresponds to a situation where mass transfer resistance is less important, and a better performance of the SMB will be obtained with sharper internal concentration profiles. 

Fig. 9-12. Effect of the mass transfer resistance on the internal concentration profiles ...

Thus t]X s 1, corresponding to the region where mass transfer effects dominate. The concentration profile is given by equation 10.198 as ... 

X-ray scattering studies at a renewed pc-Ag/electrolyte interface366,823 provide evidence for assuming that fast relaxation and diffu-sional processes are probable at a renewed Sn + Pb alloy surface. Investigations by secondary-ion mass spectroscopy (SIMS) of the Pb concentration profile in a thin Sn + Pb alloy surface layer show that the concentration penetration depth in the solid phase is on the order of 0.2 pm, which leads to an estimate of a surface diffusion coefficient for Pb atoms in the Sn + Pb alloy surface layer on the order of 10"13 to lCT12 cm2 s i 820 ( p,emicai analysis by electron spectroscopy for chemical analysis (ESCA) and Auger ofjust-renewed Sn + Pb alloy surfaces in a vacuum confirms that enrichment with Pb of the surface layer is probable.810... 

Figure 8.8 Internal mass transfer resistance and catalyst deactivation concentration profiles inside a catalyst particle-lactose hydrogenation to lactitol and by-products (sponge Ni).

The volumetric gas-liquid mass transfer coefficient ki a) has been obtained by fitting the concentration profile of dissolved oxygen to the axial dispersion model [8, 18]. The value of... 

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

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