Diffusivity axial eddy

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

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. 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find Din = Dout = 0, which is the definition of a closed system. See Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ain. The flux just inside the reactor at location z = 0+ has two components. One component, Qina(0+), is due to convection. The other component, —DAc[da/dz 0+, is due to diffusion (albeit eddy diffusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,... 

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 ... 

Dm — molecular diffusivity of volatile component, D/B E = axial eddy diffusivity, L /B... 

Equation 6.84 shows that the column HETP is the sum of the independent contributions of the axial dispersion (molecular diffusion and eddy diffusion), the film mass transfer resistance, the pore diffusion, and slow kinetics of adsorption-desorption. By comparing Eqs. 6.84 and 6.57a, we obtain ... 

Apparent dispersion coefficient, Dapi The apparent dispersion coefficient lumps all the contributions to axial dispersion arising from axial molecular diffusion, tortuosity, eddy diffusion, and from a finite rate of mass transfer, adsorption-desorption, or other phenomena, such as reactions, in which the eluites may be involved. It is used in the equilibrium-dispersive model of chromatography to ac-coimt for the finite efficiency of the column (Eq. 2.53 and 10.11). See equilibrium-dispersive model. 

The Peclet number of radial dispersion was found to be between 8 and 15. as we have noted above. However, the axial Peclet number is about 2. which shows that the axial eddy diffusion coefficient is anywhere from four to seven times the radial coefficient Er. A very simple model gives some indication why this should be so. The flow through a packed bed has been described... 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity f Dz (m2s-1) as an indicator of the degree of mixing in the flow direction. If a 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... 

Deisler and Wilhelm (<5) were the first to introduce frequency response as an experimental technique for investigating the simultaneous diffusion phenomena of gases in a packed bed. A single value of axial eddy diffusivity was found in the gaseous flow through a bed of fused alumina spheres. 

The axial dispersion coefficient can be estimated from the concentration profile of a nonpenetrating tracer (Tl). An approximation regarding its velocity dependence goes back to Van Deemter, Zuiderweg, and Klinkenberg (1956). The axial dispersion coefficient is seen as the sum of the contributions of molecular diffusion and eddy diffusion (Section 2.3.4 Ruthven, 1984) ... 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

In PF, the transport of material through a vessel is by convective or bulk flow. All elements of fluid, at a particular axial position in the direction of flow, have the same concentration and axial velocity (no radial variation). We can imagine this ideal flow being blurred or dispersed by backmixing of material as a result of local disturbances (eddies, vortices, etc.). This can be treated as a diffusive flow superimposed on the convective flow. If the disturbances are essentially axial in direction and not radial, we refer to this as axial dispersion, and the flow as dispersed plug flow (DPF). (Radial dispersion may also be significant, but we consider only axial dispersion here.)... 

The quantities in the numerator and denominator were directly calculated from the experimental concentration profile data. Some results for Dr R) are shown in Fig. 13. It is seen that the eddy diffusivity is relatively constant except near the walls. The values seem quite sensitive to the axial position where the concentration profile was measured in other words, the system is not axially homogeneous. These results have been... 

---