Dispersion coefficient experimental determination

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for moleciilar diffusion and convec tive mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contac tor. 

Parameter estimation problems result when we attempt to match a model of known form to experimental data by an optimal determination of unknown model parameters. The exact nature of the parameter estimation problem will depend on the mathematical model. An important distinction has to be made at this point. A model will contain both state variables (concentrations, temperatures, pressures, etc.) and parameters (rate constants, dispersion coefficients, activation energies, etc.). 

Lipophilicity is a molecular property experimentally determined as the logarithm of the partition coefficient (log P) of a solute between two non-miscible solvent phases, typically n-octanol and water. An experimental log P is valid for only a single chemical species, while a mixture of chemical species is defined by a distribution, log D. Because log P is a ratio of two concentrations at saturation, it is essentially the net result of all intermolecular forces between a solute and the two phases into which it partitions (1) and is generally pH-dependent. According to Testa et al. (1) lipophilicity can be represented (Fig. 1) as the difference between the hydrophobicity, which accounts for hydrophobic interactions, and dispersion forces and polarity, which account for hydrogen bonds, orientation, and induction forces ... 

We have discussed methods for experimentally finding dispersion coefficients for the various classes of dispersion models. Although the models were treated completely separately, there are interrelations between them such that the simpler plug-fiow models may be derived from the more complicated general models. Naturally, we would like to use the simplest possible model whenever possible. Conditions will be developed here for determining when it is justifiable to use a simpler plug-flow model rather than the more cumbersome general model. 

Whereas there is little doubt that the method of moments, as the procedure is called, is basically sound, it is obvious that for reliable results high-quality experimental data over a broad range of frequencies and temperatures are desirable. As importantly, reliable models of the interaction potential must be known. Since these requirements have rarely been met, ambiguous dipole models have sometimes been reported, especially if for the determination of the spectral moments substantial extrapolations to high or to low frequencies were involved. Furthermore, since for most works of the kind only two moments have been determined, refined dipole models that attempt to combine overlap and dispersion contributions cannot be obtained, because more than two parameters need to be determined in such case. As a consequence, empirical dipole models based on moments do not attempt to specify a dispersion component, or test theoretical values of the dispersion coefficient B(7) (Hunt 1985). 

It is important to remember that often, especially for the He-Ar system, the empirical B(7> coefficients given in Table 4.1 are not necessarily the same as the lowest-order theoretical dispersion coefficients, Dy. An accurate determination of the true dispersion coefficient would require the inclusion of higher-order dispersion terms in the analysis and, moreover, experimental data of a quality that is presently not available. We note that for He-Ar the ab initio dipole specified below is probably superior to the empirical model of Table 4.1. 

Experimental Determination of Dispersion Coefficient from a Pulse Input... 

Fig. 2.16. Experimental determination of dispersion coefficient-ideal-pulse injection. Expected symmetrical distribution of concentration measurements. Small DJuL value...

Fig. 2.17. Experimental determination of dispersion coefficient. (a) Treatment of data by linear interpolation (b) Treatment of mixing-cup data...

The above example demonstrates that treatment of the basic data by different numerical methods can produce distinctly different results. The discrepancy between the results in this case is, in part, due to the inadequacy of the data provided the data points are too few in number and their precision is poor. A lesson to be drawn from this example is that tracer experiments set up with the intention of measuring dispersion coefficients accurately need to be very carefully designed. As an alternative to the pulse injection method considered here, it is possible to introduce the tracer as a continuous sinusoidal concentration wave (Fig. 2.2c), the amplitude and frequency of which can be adjusted. Also there is a variety of different ways of numerically treating the data from either pulse or sinusoidal injection so that more weight is given to the most accurate and reliable of the data points. There has been extensive research to determine the best experimental method to adopt in particular circumstances 7 " . 

Fig. 2.20. Dimensionless axial-dispersion coefficients for fluids flowing in circular pipes. In the turbulent region, graph shows upper and lower limits of a band of experimentally determined values. In the laminar region the lines are based on the theoretical equation 2.37...

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

The model is referred to as a dispersion model, and the value of the dispersion coefficient De is determined empirically based on correlations or experimental data. In a case where Eq. (19-21) is converted to dimensionless variables, the coefficient of the second derivative is referred to as the Peclet number (Pe = uL/De), where L is the reactor length and u is the linear velocity. For plug flow, De = 0 (Pe ) while for a CSTR, De = oo (Pe = 0). To solve Eq. (19-21), one initial condition and two boundary conditions are needed. The closed-ends boundary conditions are uC0 = (uC — DedC/dL)L=o and (dC/BL)i = i = 0 (e.g., see Wen and Fan, Models for Flow Systems in Chemical Reactors, Marcel Dekker, 1975). Figure 19-2 shows the performance of a tubular reactor with dispersion compared to that of a plug flow reactor. 

Determinations of Peclet number were carried out by comparison between experimental residence time distribution curves and the plug flow model with axial dispersion. Hold-up and axial dispersion coefficient, for the gas and liquid phases are then obtained as a function of pressure. In the range from 0.1-1.3 MPa, the obtained results show that the hydrodynamic behaviour of the liquid phase is independant of pressure. The influence of pressure on the axial dispersion coefficient in the gas phase is demonstrated for a constant gas flow velocity maintained at 0.037 m s. 

Empirical parameters governing atmospheric dispersal pervade the literature on this subject. Like most cases of turbulent transport, elimination of a disposable coefficient in one place leads to a reappearance of one somewhere else. The present work uses an experimentally determined turbulent diffusion coefficient, D, in Equation 19. Near the ground and near the inversion base we must assign a height (z) dependence to the diffusion coefficient. 

The first 3D model of FFF was developed in Ref. 2. The 3D diffusion-convection equation was solved with the help of generalized dispersion theory, resulting in the equations for the cross-sectional average concentration of the solute and dispersion coefficients and K2, representing the normalized solute zone velocity and the velocity of the corresponding peak width growth, respectively. Unfortunately, only the steady-state asymptotic values of dispersion coefficients Ki oo) and K2 oo) were determined in Ref. 2, leading to the prediction of the solute peaks much wider than the experimental ones. 

An orthogonal co-location method can be used to convert the above partial differential equahons (PDE) into the ordinary differential equations (ODEs). An ODE solver, EPISODE can be used to solve the (ODEs) (25). In the model, the diffusivity is obtained from a batch kinetic study while the external mass transfer coefficient can be calculated from empirical equations or is available in the literature. The longitudinal dispersion coefficient (D ) is determined by matching the model output with the experimental data. Readers may like to refer to Chen and Wang s work (9) for detailed information. 

The parameters defined in this chapter are divided into model parameters and evaluation parameters. Model parameters are porosity, voidage and axial dispersion coefficient, type and parameters of the isotherm as well as mass transfer and diffusion coefficient. All of them are decisive for the mass transfer and fluid flow within the column. They are needed for process simulation and optimisation. Therefore their values have to be valid over the whole operation range of the chromatographic process. Experimental as well as theoretical methods for determining these parameters are explained and discussed in Chapter 6. 

Based on tracer and solute experiments (Fig. 6.11) the model parameters are determined step-by-step, beginning with the void fraction, total porosity and the axial dispersion coefficient (Section 6.5.6). All experimental data must be corrected for plant effects (Eq. 6.132). 

Experimental extraction curves can be represented by this type of model, by fitting the kinetic coefficients (mass transfer coefficient to the fluid, effective transport coefficient in the solid, effective axial dispersion coefficient representing deviations from plug flow) to the experimental curves obtained fi om laboratory experiments. With optimized parameters, it is possible to model the whole extraction curve with reasonable accuracy. These parameters can be used to model the extraction curve for extractions in larger vessels, such as from a pilot plant. Therefore, the model can be used to determine the kinetic parameters from a laboratory experiment and they can be used for scaling up the extraction. 

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