Axial Diffusion Coefficient

The axial diffusion (or dispersion) coefficient is a measure of mixing in a vertical bioreactor like a bubble column or airlift bioreactor. It has also been used to quantify 

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Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

Figure 8. Effective axial diffusivity coefficients for solids mixing, (From DeGroot, 1967.)...

L.Dq/u.Rq dimensionless number Mark-Houwink-Sakurada constant Axial diffusion coefficient... 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

G. I. Taylor s concept of the effective axial diffusion coefficient, which has proved so useful in combining variable axial advection with radial transfer into one parameter, works best when there is no exchange of a passive tracer with the pipe walls. An analogue of his method, which should be applicable when development lengths are large and there is exchange at the wall, has yet to be provided. It would be of great value. 

Assuming that radial and axial diffusion coefficients are equal, that is, Da - Dr, Eq. (1) then becomes... 

In the bubble column the velocity profile of recirculating liquid is shown in Fig. 27, where the momentum of the mixed gas and liquid phases diffuses radially, controlled by the turbulent kinematic viscosity Pf When I/l = 0 (essentially no liquid feed), there is still an intense recirculation flow inside the column. If a tracer solution is introduced at a given cross section of the column, the solution diffuses radially with the radial diffusion coefficient Er and axially with the axial diffusion coefficient E. At the same time the tracer solution is transported axially Iby the recirculating liquid flow. Thus, the tracer material disperses axially by virtue of both the axial diffusivity and the combined effect of radial diffusion and the radial velocity profile. 

As for axial diflfusivity included in Eq. (4-2), the situation is not much different from that for radial diflfusivity. Pozin et al. (P6) report that the mean value of axial diflfusivity is 2.5 times the value of Er, when radial and axial diffusion are assumed to be homogeneous and nonisotropic throughout the bubble column. Hence it is reasonable to assume for the local axial diffusion coefficient Ez = Ipr, with i of order unity. On the other hand, one has the relations that Ey = Vm/cL [see Eq. (4-4)] and = au so that E equals the product (aC/eJvf Accordingly, the cross-sectionally averaged axial diflfusivity E is expressed by the relation Ez = where the overbar shows an effective mean value. 

Increasing surfactant concentrations in the aeration cell has been found to decrease bubble diameter, bubble velocity, axial diffusion coefficient, but increase bubble s surface-to-volume ratio, and total bubble surface area in the system. The effect of a surface-active agent on the total surface area of the bubbles is also a function of its operating conditions. The surfactant s effect is pronounced in the case of a coarse gas diffuser where the chances of coalescence are great and the effectiveness of a surface-active solute in preventing coalescence increases with the length of its carbon chain. 

The axial Peclet number, Pe = uL/Di, where L is the column length and Di the axial diffusion coefficient,... 

How docM the FEMLAB result compare with the solution to Example 14-2 Repeat (a) for a second-order reaction with k = 0.5 dm-Vraol min. Repeat (a) but a,ssume laminar (low and consider radial gradient. in concentration. Use for both the radial and axial diffusion coefficients. Plot the axial and radial profiles. Compare your results with pan (a). 

Where Dax is the axial diffusion coefficient, x is the position of the profile along the flow direction, v is the linear flow velocity, t is the time variable, and c and q are the concentrations of the solute in solution and the adsorbed state, respectively (19). 

The pulse spreading has been characterized by an axial diffusion coefficient and an axial Peclet number ... 

In this equation, z is the coordinate along the reactor axis, Dax is the axial diffusion coefficient and vint the interstitial velocity, which can be calculated from the gas flow speed Vsup using vint = Vsup/e. The axial diffusivity can be calculated from the molecular diffusion coefficient of the component. For Rc, the radius of the crystals, the equivalent spherical particle radius is taken, defined as the radius of the sphere having the same external surface area to volume ratio [3]. We have estimated a value of 25 pm for the zeolite crystals in the current study. 

The following two models are frequently used to account for partial macromixing the dispersion model and the tanks-in-series model. In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. This model was anticipated in Chapter 12, and the governing equations for mass and heat are listed in Table 12.2 of that chapter. The derivation is similar to that for plug flow except that now a term is included for diffusive flow in addition to that for bulk flow. This term appears as -D ( d[A]/d ), where is the effective axial diffusion coefficient. When the equation is nondimensionalized, the diffusion coefficient appears as part of the Peclet number defined as = itd/D. A number of correlations for predicting the Peclet number for both liquids and gases in fixed and fluidized beds are available and have been reviewed by Wen and Fan (1975). 

Unlike in MASRs, where liquid mixing is always considered complete, in this case allowance must be made for partial mixing. Thus it may often be necessary to use the dispersion model given by Equation 17.25. The liquid-phase axial diffusion coefficient for estimating the Peclet number in this equation may be calculated from the correlations of Hikita and Kikukawa (1975) or Mangartz and Pilhofer (1981). 

Hydrodynamics in gas-liquid systems have been studied extensively in the past due to their wide range of applications. Characteristics of interest include flow regimes, local pressure drop, gas residence time, axial diffusion coefficients, bubble size, bubble rise velocity, gas holdup, and power consumption. This section will summarize various experimental techniques to quantify some of these characteristics. 

Axial dispersion model In the dispersion model, deviation from plug flow is expressed in terms of a dispersion or effective axial diffusion coefficient. The mathematical derivation is similar to that for plug flow except that a term is now included for diffusive flow in addition to that for convective flow. This term appears as (d[A]ldz), where is the effective axial dispersion coefficient. The continuity equation in the absence of radial variations takes the form... 

Solids mixing in beds with an assembly of vertical tubes has not been widely investigated. Ramamoorthy and Subramanian (1981), applying a one-dimensional diffusion model, determined an empirical correlation to predict the axial diffusion coefficient at given fluidizing velocity and packing density of vertical rods. The diffusivity is substantially decreased in the presence of internals and further decreased with a reduction in the spacing of internals. 

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