Dispersions axial

It is assumed that axial dispersion (Equation 6.22) in the liquid phase can be defined in analogy to Pick s first law of diffusion  

The axial dispersion coefficient depends essentially on the quality of the packing and captures deviations of the fluid dynamics from plug flow. In preparative chromatography contributions of molecular diffusion are generally negligible (Section 6.5.6.2). 

Diffusion inside the adsorbent particles can also be quantified using Pick s law. As for the accumulation terms, size and number of particles per unit volume (Equation 6.20) are taken into account  

In Equation 6.23 the transport in the pore fluid is modeled as free diffusion in the macropores and mesopores, but the diffusion coefficient Dpo. i is usually smaller than the molecular diffusivity characterizing transport in the liquid mobile phase due to the random orientations and variations in the diameter of the pores (tortuosity) (Section 6.5.8). 

In Equation 6.24 transport is assumed to occur by micropore or surface diffusion, where the molecules are under the influence of a force fields of the inner adsorbent surface. The surface diffusion concept is applied to quantify the transport in the adsorbed phase. 

In a sparged reactor, the behavior of the liquid and gas phase deviates significantly from plug flow, particularly at high gas and low liquid velocities. This deviation is generally accounted for by the axial dispersion coefficient, D. Deckwer (1992) has discussed this matter in detail outlining the various approaches used to quantify axial dispersion. Tables 10.3 and 10.4 list some of the correlations available in the hterature for estimating the liquid- and gas-phase axial dispersion coefficient, and D, respectively. 

Liquid-phase back mixing is a serious issue for reactions that have nonzero-order kinetics with respect to the liquid-phase reactant. Sectionalization of bubble column using sieve plates of relatively low free area is an attractive choice in such a case. Although this choice has been mentioned in the literature, its application in solid-catalyzed reactions has not attracted any attention. The sieve plate design must be such that it prevents weeping (Prince 1960). The free area in such sieve plates is 

Joshi and Sharma D =03lxTV Equation derived from liquid 

Garcia-Calvo and Leton (1994) Correlation valid for Newtonian as well as non-Newtonian liquids over a wide range of conditions 

Empirical correlation based on authors own and literature data (CGS units) Empirical correlation based on authors own and literature data (CGS units) Empirical correlation based on authors own and literature data (CGS units) Introduction of sieve plates (free area 44%) in the 0.5 m dia. column gave lower dispersion for V- 0.1m/s 

We evaluate D or D/wL by recording the shape of the tracer curve as it passes the exit of the vessel. In particular, we measure 

These measures, t and cr, are directly linked by theory to D and D/wL. The mean, for continuous or discrete data, is defined as 

Since the mixing process involves a shuffling or redistribution of material either by slippage or eddies, and since this is repeated many, many times during the flow of fluid through the vessel we can consider these disturbances to be statistical in nature, somewhat as in molecular diffusion. For molecular diffusion in the x-direction the governing differential equation is given by Fick s law  

Fluctuations due to different flow -y velocities and due to molecular / and turbulent diffusion / 

In dimensionless form where z = ut x)IL and 6 - tl t = tu/L, the basic differential equation representing this dispersion model becomes 

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Due to a limited axial dispersion, the first passages are sufficiently separated to calculated the leakage size directly, as the ratio between ... 

Effect of Axial Dispersion on Column Performance. Another assumption underlying standard design methods is that the gas and the Hquid phases move in plug-flow fashion through the column. In reaHty, considerable departure from this ideal flow assumption exists (4) and different fluid... 

Fig. 17. Effect of axial dispersion in both phases on solute distribution through countercurrent mass transfer equipment. A, piston or plug flow B, axial...

Determination of separation efficiencies from pilot-plant data is also affected by axial dispersion. Neglecting it yields high or values. Literature data for this parameter have usually not been corrected for this effect. 

Rapid Approximate Design Procedure. Several simplified approximations to the rigorous solutions have been developed over the years (57—60), but they aU. remain too compHcated for practical use. A simple method proposed in 1989 (61,62) uses a correction factor accounting for the effect of axial dispersion, which is defined as (57)... 

NTU p is the "exterior apparent" overall gas-phase number of transfer units calculated neglecting axial dispersion simply on the basis of equation 56, whereas NTU stands for the higher real number of transfer units (Nq ) which is actually required under the influence of axial dispersion. The correction factor ratio can be represented as a function of those parameters that are actually known at the outset of the calculation... 

Fig. 19. Correction factor for axial dispersion as a function of NTU. SoHd lines are rigorous calculations broken lines, approximate formulas according to hterature (61). (a) Numbers on lines represent Pe values Pe = 20 /Lj = 0.8. (b) For design calculations. Numbers on lines represent Pep u ...

The recommended rapid design procedure consists of the following steps (/) The apparent is calculated using equation 56. (2) The extent of axial dispersion is estimated from Hterature correlations for each phase in terms of Pe numbers and transformed into values. (3) The correction... 

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

Two alternative approaches are used ia axial mixing calculations. For differential contactors, the axial dispersion model is used, based on an equation analogous to equation 13 ... 

Pulsed Columns. The efficiency of sieve-plate or packed columns is increased by the appHcation of sinusoidal pulsation to the contents of the column. The weU-distributed turbulence promotes dispersion and mass transfer while tending to reduce axial dispersion in comparison with the unpulsed column. This leads to a substantial reduction in HETS or HTU values. 

Chemical Reaction Measurements. Experimental studies of incineration kinetics have been described (37—39), where the waste species is generally introduced as a gas in a large excess of oxidant so that the oxidant concentration is constant, and the heat of reaction is negligible compared to the heat flux required to maintain the reacting mixture at temperature. The reaction is conducted in an externally heated reactor so that the temperature can be controlled to a known value and both oxidant concentration and temperature can be easily varied. The experimental reactor is generally a long tube of small diameter so that the residence time is well defined and axial dispersion may be neglected as a source of variation. Off-gas analysis is used to track both the disappearance of the feed material and the appearance and disappearance of any products of incomplete combustion. 

Axial Dispersion Backmixing in bubble columns has been extensively studied. An excellent review article by Shah et al. [AIChE... 

The recommended correlation for the gas-phase axial-dispersion coefficient is given by Field and Davidson (loc. cit.) ... 

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 Vermeiilen et al. [Chem. Eng. Prog., 62(9), 95 (1966)] summarized many of the data for packings. Their correlation for the continuous phase is shown in Fig. 15-36. For the dispersed phase, their correlation is given by... 

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