Ideal flows, deviation from dispersion

Much of Fig. 18 refers to conditions under which the dispersion model should only be used with caution. Levenspiel [26] suggests that, if DluL is greater than unity, then other models may be more appropriate and Dudukovic and Felder [59] comment that the dispersion model should only be used with confidence when the dispersion number is less than 0.05. As is clear from both Figs. 16 and 18, these conditions represent relatively minor deviations from the plug flow ideal. 

Both the dispersion and tanks in series models can be used to represent the non-ideal flow behavior of fluids in packed bed and tubular reactors. As mentioned in the previous sections dealing with these models, they are both good for the slight deviations from plug flow encountered in the above systems. 

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. 

The deviation from ideal plug flow due to the axial mixing can be described by the dispersion model (Levenspiel, 1972). Let s look at the differential element with a thickness dx in a holding tube as shown in Figure 8.1. The basic material balance for the microorganisms suspended in the medium is... 

Tubular flow reactors (TFR) deviate from the idealized PFR, since the applied pressure drop creates with viscous fluids a laminar shear flow field. As discussed in Section 7.1, shear flow leads to mixing. This is shown schematically in Fig. 11.9(a) and 11.9(b). In the former, we show laminar distributive mixing whereby a thin disk of a miscible reactive component is deformed and distributed (somewhat) over the volume whereas, in the latter we show laminar dispersive mixing whereby a thin disk of immiscible fluid, subsequent to being deformed and stretched, breaks up into droplets. In either case, diffusion mixing is superimposed on convective distributive mixing. Figure 11.9(c) shows schematically the... 

Tubular Reactor with Dispersion An alternative approach to describe deviation from ideal plug flow due to backmixing is to include a term that allows for axial dispersion De in the plug flow reactor equations. The reactor mass balance equation now becomes... 

Causes for deviations from ideal plug flow are molecular diffusion in the gas and dispersion caused by flow in the interstitial channels of the bed, and uneveness of flow over the cross section of the bed. 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

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... 

Here D, the axial dispersion (or diffusion) coefficient, is the parameter used to describe the deviations from ideal flow. If u is taken to be constant in the radial direction, the rightmost terms in equation (5-20) constitute the plug-flow mixing model and D (f Cld ) is a Fickian form of a diflusional correction term. 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

The dispersion model (diffusion model) explains the deviation of the real flow profile from the ideal plug flow profile due to dispersion analogously to molecular diffusion. For example, in the continuous phase, the axial distribution of the key component is... 

The picture involved in the one-dimensional dispersion model is the onedimensional process of flow in a tube. There is a flow velocity in direction z, which, in the ideal case, is constant over the reactor cross section %. Because of molecular diffusion, turbulent convection, and the parabolic velocity profile that results from boundary friction (roughness, s), there are large deviations from a uniform flow front. The effective longitudinal dispersion coefficient... 

The quantification of real reactors, those with deviations from ideal plug flow, can now also be undertaken. In agreement with Equ. 3.3a, in addition to the convection term operating via the flow velocity the term with the dispersion coefficient J9l is also operative. 

For a plug-flow fluidized bed dryer, the residence time will deviate from idealized plug-flow because of backmixing. This can be accounted for by employing the axial dispersion number, B = where D is... 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

To inspect the deviation from an ideal PFR, we use the axially dispersed plug flow model (Sections 4.10.6.1-4.10.6.3) and recall the equation for the term of axial dispersion ... 

Hence we need respective criteria for the design and operation of a laboratory reactor to ensure negligible deviations from the ideal. Subsequently, we repeat these criteria, which were already derived in Sections 4.7, 4.10.6.5, and 4.10.7.2, and specify them for laboratory-scale experiments. In the next subsection, the criteria for ideal plug flow behavior (exclusion of an influence of axial and radial dispersion of mass and heat are covered), and in the subsequent subsection, the criteria for gradientless deal particle behavior (exclusion of an influence of interphase and intraparticle transport of mass and heat) are outlined. 

Table 4.11.2 Comparison of critical values according to the criteria for negligible influence of axial and radial dispersion of mass and heat (deviation from ideal plug flow behavior of the experimental reactor for 1-hexene hydrogenation forthe conditions listed in Table 4.11.1).

The concept of dispersion is used to describe the degree of backmixing in continuous flow systems. Dispersion models have been developed to correct experimentally recorded deviations from the ideal plug flow model. As described in previous sections, the residence time functions E(t) mdF(t) can be used to establish whether a real reactor can be described by the ideal flow models (CSTR, PFR, or laminar flow) or not. In cases where none of the models fits the residence time distribution (RTD), the tanks-in-series model can be used, as discussed in Section 4.4. However, the use of a tanks-in-series model might be somewhat artificial for cases in which tanks do not exists in reality but only form a mathematical abstraction. The concept of a dispersion model thus becomes actual. 

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