Dispersion crossed axial

Local equihbrium theory also pertains to adsorption with axial dispersion, since this mechanism does not disallow existence of equilibrium between stationary and fluid phases across the cross section of the bed [Rhee et al., Chem. Eng. ScL, 26, 1571 (1971)]. It is discussed below in further detail from the standpoint of the constant pattern. 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

Equation (9.27) defines the so-called axial dispersion coefficient Dax as a model parameter of mixing. Nd is the dispersion flow rate, c the concentration of the tracer mentioned earlier, and S the cross-sectional area of the column. The complete mole flow rate of the tracer consists of an axial convection flow and the axial dispersion flow. The balance of the tracer amount at a cross section of the extractor leads to second-order partial differential equations for both phase flows at steady state. For example, for continuous liquids ... 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity (m s ) as an indicator of the degree of mixing in the flow direction. If the flow in a tube is plug flow, the axial dispersion is zero. On the other hand, if the fluid in a tube is perfectly mixed, the axial dispersion is infinity. For turbulent flow in a tube, the dimensionless Peclet number (Pe) deflned by the tube diameter (v dlE-Q is correlated as a function of the Reynolds number, as shown in Figure 10.3 [3] dz is the axial eddy diffusivity, d is the tube diameter, and v is the velocity of liquid averaged over the cross section of the flow channel. 

Fig. 56. Dispersion of optic axes in orthorhombic crystals, a. p > u. b-d. Crossed axial plane dispersion.

Ex Effective axial dispersion coefficient, based upon total cross-sectional area of bed, cm.2/sec. 

By At we denote the cross sectional area of the reactor tube. Then the steady-state mass balance with axial dispersion gives us the equation... 

Three-parameter PDE model (Van Swaaij et aL106) This model is largely used to correlate the RTD curves from a trickle-bed reactor. The model is based on the same concept as the crossflow or modified mixing-cell model, except that axial dispersion in the mobile phase is also considered. The model, therefore, contains three arbitrary parameters, two of which are the same as those used in the cross-flow model and the third one is the axial dispersion coefficient (or the Peclet number in dimensionless form) in the mobile phase (see Fig. 3-11). 

The third and fourth condition are fulfilled by Tarhan [25]. Axial dispersion is fundamentally local backmixing of reactants and products in the axial, or longitudinal direction in the small interstices of the packed bed, which is due to molecular diffusion, convection, and turbulence. Axial dispersion has been shown to be negligible in fixed-bed gas reactors. The fourth condition (no radial dispersion) can be met if the flow pattern through the bed already meets the second condition. If the flow velocity in the axial direction is constant through the entire cross section and if the reactor is well insulated (first condition), there can be no radial dispersion to speak of in gas reactors. Thus, the one-dimensional adiabatic reactor model may be actualized without great difficulties. ... 

Descriptions for the molar flow rate of species i in a PFR and an axially-dispersed PPTl. Ac, cross-sectional diameter of tube, u linear velocity. Da, axial dispersion coefficient. 

The axial dispersion in a single channel is low due to the very thin film surrounding the bubbles. For the low conversion that is usually obtained in a single pass through the monolith reactor, the residence-time distribution within the channels will have an insignificant effect on conversion. However, the difference between the channels can be important. In downflow where the velocity is controlled by gravity, the linear velocity will be almost the same in all channels, but the gas hold-up will be different in the channels due to uneven liquid distribution over the cross section. 

The dispersion model is also used to describe nonideal tubular reactors. In this model, there is an axial dispersion of the material, which is governed by an analogy to Pick s law of diffusion, superimposed on the flow. So in addition to transport by bulk flow, UAqC, every component in the mixture is transported through any cross section of the reactor at a rate equal to [—DaAddCldz)] resulting from molecular and convective diffusion. By convective diffusion we mean either Aris-Taylor dispersion in laminar flow reactors or turbulent diffusion resulting from turbulent eddies. 

The Flow Equation. Consider a differential cross-sectional slice, dx, at distance x from the feed end of the devolatilizer. A volatile component material balance across this slice will include net inputs due to mean axial flow and axial dispersion (the latter arising from the nip mixing action), and depletion through the regenerated surface films. In addition to the three assumptions made above, it is assumed that uniform conditions prevail throughout the length—i.e., constant Uy p, S, Wy D y etc.-and that the effect of axial dispersion may be characterized by a constant axial eddy diffusivity, E. The steady-state material balance for a volatile component across dx reduces to ... 

The dispersion coefficient and the parameter pipe length" in Eq. 6.119 are mainly adjusted parameters, especially since the actual cross section of the pipe is not known exactly and therefore a representative value has to be chosen. For the axial dispersion coefficient, initial guesses using a very low value (e.g. 10 5 cm2 s 1) should be used and the pipe length is set according to Eq. 6.119 and the specified cross section. 

It can be seen from Fig. 7 that in curved tubes, there is a drastic reduction in axial dispersion with an increase in Dean number. These studies reveal that very high Dean numbers are required to induce significant mixing in the cross-sectional plane. Saxena proposed a simple and more effective alternative to helical coils. Helical coils are very efficient in inverting the flow... 

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