Taylor-Aris equation

Note The Golay equation (2.29) becomes the Taylor-Aris equation when the retention factor becomes 0.]... 

Commenge et al. extended the one-dimensional model of reacting flows to include Taylor-Aris dispersion, i.e. they considered an equation of the form... 

Clearly there is a region between Pe 7 and Pe > 7, with 4L/a > Pe, where both radial and axial diffusion are important. Aris (1956), in a mathematically elegant paper, showed that the governing equation for the mean concentration distribution averaged over the tube cross section can be written in the form of the Taylor dispersion equation, with... 

The functions M (t) were determined from the complete unsteady axially symmetric convective diffusion equation (Eq. 4.6.7), and M (f) were obtained from the Taylor dispersion equation, which was used as the model equation. The phenomenological coefficients U and in the equation were determined by matching the first three moments of the infinite sequence M (t) to M (t) for asymptotically large times [t>a lD). Applying his scheme to the circular capillary problem, Aris showed that D fj, where axial molecular diffusion is not neglected, is given by Eq. (4.6.35). Fried Combarnous (1971) later showed that the satisfaction of the first three moments for t—implies that c x, t), obtained as a solution of the Taylor dispersion equation with = D + Pe /48), is asymptotically the solution of the complete, unsteady, axially symmetric convective diffusion equation averaged over the cross section. 

The Taylor-Aris result can be shown in a somewhat simpler mathematical way by starting with the complete convective diffusion equation (Eq. 4.6.7), including the axial diffusion term. The procedure is essentially the same as Taylor s. Equation (4.6.7) is integrated over the tube cross section, since what is of interest is the average concentration, and the radial concentration distribution is given by Eq. (4.6.21). The replacement of dddx by dctdx is still made. The analysis follows through as before. In addition to requiring we must... 

The contribution of the detector volume is of the same nature. The contribution of the connection tubing is usually estimated through the use of the Taylor-Aris (6,7) equation, which assumes that the flow is laminar and that the tubing is long ... 

This 1-D convection-diffusion equation has features of the Taylor-Aris dispersion equation. However, in contrast to Taylor-Aris, here >eff is a function of temperature and the axial coordinate, as it depends on both D and Up. When/ is a linear function of the axial dimension. Equation 38.39 can be solved in closed form subject to the form of D s. IfT>eff is uniform, the solution is a Gaussian with peak variance = 2Deffrfoc, where Tfoc = l/2 ovo d//dJc. If Detr is also a linear function of X, the solution is... 

On the basis of the Taylor-Aris correlation (Equation 3.75) a Bodenstein number oiBo 24is expected. The experimental results suggest that efficient radial mixing occurs, which may be explained by the used capillary shaped as a coil provoking enhanced radial mixing. 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

This result was first derived by Aris (1956) using the method of moments. While the resulting model now includes both the effects (axial molecular diffusion and dispersion caused by transerverse velocity gradients and molecular diffusion) it has the same deficiency as the Taylor model, i.e. converting a hyperbolic model into a parabolic equation. 

Equation 4.23 is the well-known expression for the dispersion of a solute in flow of fluid, which was first derived by Taylor [10,11] and Aris [12], Its appearance... 

The equations of Golay [71] and Aris [21] can be applied to chromatographic columns which are long after the Taylor [75]-Aris [76] definition, i.e., within which the residence time is long compared to the characteristic time for radial transport. This is the situation found in practically all actual chromatographic systems and certainly in all those used in analytical or preparative practice. 

Analytical solution of this system ofdifferential equations is not possible. Therefore Aris and Amundson [32] linearized it by a Taylor expansion, about the steady-... 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... 

The basic conclusions of Taylor dispersion (and Aris s results) have been supported by solutions of the complete equations by a number of investigators. For a brief introduction to these references and literature, consult Froment and Bischoff (1979, p. 621) and Bird et al. (2002, p. 646). What is more important from our perspective is that such a model of axial dispersion (namely equation (6.2.18), can be effectively used for complex flow situations in separators. The complexity may arise due to a packed bed of particles... 

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