Axial mixing coefficient

At any level in the transition region, there will be a balance between the mixing effects attributable to (a) axial dispersion and to (b) the segregating effect which will depend on the difference between the interstitial velocity of the liquid and that interstitial velocity which would be required to produce a bed of the same voidage for particles of that size on their own. On this basis a model may be set up to give the vertical concentration profile of each component in terms of the axial mixing coefficients for the large and the small particles. 

Figure 1 shows an example of the time development of the spatial profiles of Ethanol concentration, temperature and fluid velocity for a 10m reactor tube with Ea=340 kJ/mol and ko=6.46E21/sec [6], Heat transfer is kept constant at 4kJ/s/m2/K. Steady state is reached at about lOOsec. The bath temperature of 500°C is too low for complete oxidation under these conditions. Pivot spacing increases with reactor length and the axial mixing coefficients D are 0.5m2/sec. The fluid velocity is constant beyond z=2m. 

The liquid flowing inside a MWPB can be described with a one-parameter dispersion flow model. As we show in Section 3.3, the axial mixing coefficient or, more correctly, the axial dispersion coefficient is the specific parameter for this model. Relation (3.112) contains the link between the variance of the residence time of liquid elements and the Peclet number. We can rewrite this relation so as to particularize it to the case of a MWPB. Here, we have the possibility to compute the variance of the residence time of the liquid through the stochastic model for the liquid flow developed previously in order to obtain the value of the axial dispersion coefficient ... 

Considering relations (4.220) and (4.221), we can observe that we have all the required elements to compute the axial dispersion coefficient. The theoretical computed values for the axial mixing coefficient for the case where the bed height has a practical importance are shown in Table 4.5. For the cases when the selection Vx = 1 dm/s is not justified by the operational conditions, we replace FIj by H. We can introduce Hj through equation (4.210) ... 

Figure 4.28 State of the axial mixing coefficient versus gas velocity.

E Axial mixing coefficient mVs cmVs K Stripping section partition Mass ratio/ Mass ratio/... 

Axial mixing effects commonly are taken into account by using a diffusion analogy and an axial mixing coefficient E, also called the longitudinal dispersion coefficient or eddy diffusivity, to account for the spreading of the concentration profiles. At steady state, the conservation equation has the general form... 

HDUr+ HDUe Xf The axial mixing coefficient is correlated by... 

Since the reactor is usually cylindrical and partially filled, generally it possesses two dispersion mechanisms, one in the axial direction that is characterized by an axial mixing coefficient, and the other in... 

Carlos CR, Richardson JF. Solids movement in liquid fluidised beds—II. Measurement of axial mixing coefficients. Chem Eng Sci 23 825-831, 1968b. 

This flux in the z-direction is superimposed on the net fluid flow. This is, however, not the only possible way to model axial mixing. Recently other models have been developed, fliat consider the total flow through the tube to consist of two or more parallel flows with different flow rates. This model can be extended by adding radial mixing (Westerterp, 1994). This more sophisticated model appears to describe the residence time distribution in tubes more accurately. However, one needs two or more parameters. In this text we shall restrict ourselves to the use of the concept of the axial mixing coefficient. 

When the reaction rate is known as a function of the -concentration, and when the axial mixing coefficient is known, this equation can be integrated, giving the A-concentration as a function of the length coordinate z. A relatively simple solution is found for first order reactions (for constant density of the reaction mixture), compare eqs. (3.28) - (3.32). We first substitute the first order rate equation, eq. (3.1) ... 

In order to calculate the conversion of a reaction the axial mixing coefficient has to be known. There are ample experimental data available. In principle, the axial mixing coefficient of a component A is determined by flow conditions and by the diffusivity of A, Generally speaking, the Bodenstein number for axial mixing is a function of the Reynolds and Schmidt numbers (for definitions see eq. 4.23) ... 

It is of course possible that the flow of the dispersed phase shows some residence time distribution. When the RTD can be quantified in terms of an axial mixing coefficient, the conversion of reactant A can be described in the manner shown in section 7.2.2.2. Instead of eq. (7.37) one would then find, for high Peclet numbers... 

The correlation proposed by Joshi [71] was tested using about 80% of the liquid axial mixing coefficients reported in the literature by Rato et al,[46], Michelsen and Ostergaard [64], Ostergaard [67], and Vail et al. [68], For the selected data points, the model conditions of 20% dissipation of input energy, was fulfilled, A 16% deviation between the experimental values and the model predictions was observed. 

Because the axial mixing coefficients are depending on die packing dim sions, they are determined in the dimensionless equaticms as Peclet numbers defined for die two phases by ... 

The axial mixing coefficient Z) is determined fitmi die experimentally obtained value of the Bodenstein number defined as follows ... 

Since the gas velocity is sigoificanfly higher than the liquid one, foe influence of foe residence time in foe measurement cell in this case is more important. That is why foe lusual analytical methods for measuring foe tracer concentration are not appropriate, and foe best possibility is to use as a tracer a radioactive substance, fed in foe gas plmse before entering foe column. The data of foe end concentration are processed in the same way as in foe measurement of foe axial mixing coefficient in foe liquid phase. 

As already mentioned foe mass transfer coefficients used for calculations in chemical engineeting are rather parameters in foe calculation model than physical values. That is diy speaking of methods for determination of these coefficients we should distinguish methods for determination of foe mass transfer coefficients for foe piston flow model and for foe diffusion model. Since for foe latter model foese coefficients are connected with foe axial mixing coefficients in gas and liquid phase, foe methods for their determination are discussed after foe mefoods for determination of foe axial mixing coefficients. 

Equation for calculation ofthe axial mixing coefficients... 

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