Mass transfer rate axial dispersion

Other factors limiting the overall rate can be external or internal mass transfer, or axial dispersion in a fixed-bed reactor. Pertinent dimensionless numbers are the Biot number Bi, the Damkohler number of the second kind Dan, or the Bodenstein number Bo (Eqs. (5.46)—(5.48)]. 

A detailed model for a gas-liquid column with external recirculation loop has been published by Orejas [11]. The model takes into account the axial dispersion and mass transfer from bubbles. An important conclusion is that the mass-transfer rate is fast compared with the chemical reaction. As a result, a pseudohomoge-neous model for liquid-phase reaction may be applied for design purposes. 

Using the hodograph transform, Rhee and Amundson [3] have also shown that a plot of Q versus Q+i is a straight line (solid line in Figure 16.3), provided that these two components have the same axial dispersion coefficient (D ) and mass transfer rate constant (fcy), in addition to the competitive Langmuir isotherm behavior. The equation of this straight line is... 

Axial dispersion or backmixing31 36 can make a major contribution to the mass transfer rate in liquid-phase adsorption and cannot be ignored. This is especially true at low Reynolds numbers. 

Figure 3.8. Dispersion in coiled tubes according to Speberg s model [3.13]. (a) Equivelocity profiles in axial direction, showing the deformation of the parabolic profile by centrifugal forces, (b) Equivelocity profiles in radial direction showing the intense radial mass transfer rate close to the wall (De = 100 for other details see text).

Gas chromatography is a separation technique based on the fact that different components in the mixture exhibit different average residence times due to interactions with the porons packing material. These interactions can be classified as intrapellet diffusion and the column operates similar to a packed catalytic tubular reactor. The important mass transfer mechanisms are convection and diffusion. Hence, it is important to calculate the mass transfer Peclet number that represents an order-of-magnitude ratio of these two mass transfer rate processes. Intrapellet diffusion governs residence times, and interpellet axial dispersion affects the degree to which the output curve is broadened. For axial dispersion in packed columns and packed catalytic tubular reactors. 

Most of the research up to date was primarily directed toward the determination and prediction of major fluid dynamic properties, including heat and mass transfer rates. Although there is still lack of data and prediction models for some of the parameters, many favorable features have been identified low pressure drop, high heat and mass transfer rates, and low axial dispersion of gas and flowing solids. 

Usually the desorbent must be removed from the A and B product streams. Increasing the amount of desorbent will increase the cost for this removal and will also increase the diameters of the columns requiring more adsorbent. Thus, the ratio of desorbent to feed, D/F, often controls the cost of SMB systems. For an ideal system with no zone spreading (no axial dispersion and very fast mass transfer rates) the solute movement theory can be used to calculate D/F by solving Eqs. tl8-29al to tl8-29dl simultaneously with Eq. (18-15) and the mixing mass balances with constant density. 

In experiments tFigure 18-15A1 the oudet concentration profiles are not sharp as shown in Figures 18-17B and 18-18B. Instead the finite mass transfer rates and finite amounts of axial dispersion spread the wave while the isotherm effect (illustrated in Exanple 18-71 counteracts this spreading. The final result is a dynamic equilibrium where the wave spreads a certain amount and then stops spreading. Once formed, this constant pattern wave has a constant width regardless of the column length. 

The sensitivity of the process to mass transfer resistance and axial dispersion also means that in order to achieve an efficient practical system it is essential to minimize these effects. In most practical PSA systems the mass transfer rate is controlled by macropore diffusion and the mass transfer... 

It should be noted that for a given reactor configuration, different arrangements can be made for letting the phases flow through the reactor. In a continuous process this will result in different axial concentration profiles. This can have important consequences that will be discussed in Qiapter 7. In the next sections we shall look at dispersion and relative flows in two-phase systems, and mass transfer rates in two-phase systems. 

Models of BCR can be developed on the basis of various view points. The mathematical structure of the model equations is mainly determined by the residence time distribution of the phases, the reaction kinetics, the number of reactive species involved in the process, and the absorption-reaction regime (slow or fast reaction in comparison to mass transfer rate). One can anticipate that the gas phase as well as the liquid phase can be either completely backmixed (CSTR), partially mixed, as described by the axial dispersion model (ADM), or unmixed (PFR). Thus, it is possible to construct a model matrix as shown in Fig. 3. This matrix refers only to the gaseous key reactant (A) which is subjected to interphase mass transfer and undergoes chemical reaction in the liquid phase. The mass balances of the gaseous reactant A are the starting point of the model development. By solving the mass balances for A alone, it is often possible to calculate conversions and space-time-yields of the other reactive species which are only present in the liquid phase. Heat effects can be estimated, as well. It is, however, assumed that the temperature is constant throughout the reactor volume. Hence, isothermal models can be applied. 

The differential extractors are also subject to axial mixing (see Chap. 6) it severely reduces extraction rates because of the deterioration of the concentration differences between phases which is the driving force for mass transfer. This is illustrated in Fig. 10.50, where the real (axial-mixing) concentration profiles show a substantially smaller concentration difference than those for plug flow. If the flow ratio of the liquids is not unity, and it very rarely is, dispersing the liquid flowing at the lower rate will lead to small numbers of dispersed-phase drops, small interfacial areas, and small mass-transfer rates. On the other hand, if the majority liquid is dispersed, the axial-mixing problem is exacerbated. The... 

There are a number of models which do not ttssume the existence of equilibrium between the liquid and the solid phases. However, they do not incorporate the effect of axial dispersion. In such models of nonequilibrium nondispersive operation of the column, the mass-transfer rate between the liquid and the surface of the adsorbent (primarily in the pores) is not infinitely ftist rather it is finite. Further diffusion in the porous adsorbent particle is quite important One of the earlier models of this type is by Rosen (1952, 1954). 

A very high degree of separation can be achieved after multiple cycles, limited only by axial dispersion and finite mass-transfer rate between the mobile and fixed phases. 

Heat Transfer Heat-transfer rates are gener ly large despite severe axial dispersion, with Ua. frequently observed in the range 18.6 to 74.5 and even to 130 kW/(m K) [1000 to 4000 and even to 7000 Btu/(h fF °F)][see Bauerle and Ahlert, Ind. Eng. Chem. Process Des. Dev., 4, 225 (1965) and Greskovich et al.. Am. Tn.st. Chem. Eng. J., 13,1160 (1967) Sideman, in Drewet al. (eds.). Advances in Chemical Engineering, vol. 6, Academic, New York, 1966, p. 207, reviewed earlier work]. In the absence of specific heat-transfer correlations, it is suggested that rates be estimated from mass-transfer correlations via the heat-mass-transfer analogy. 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no-axial dispersion with external mass transfer and pore diffusion, an explicit time-domain solution, useful for the case of time-periodic injections, is also available [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. In most cases, however, when N > 50, use of Eq. (16-161), or (16-172) and (16-174) with N 2Np calculated from Eq. (16-181) provides an approximation sufficiently accurate for most practical purposes. 

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