The phenomenon of axial dispersion has been investigated analytically, experimentally and numerically over the last six decades. Taylor (1953) was perhaps the rst person to address the issue. Taylors's pioneering work to establish the degree of axial dispersion in laminar ows through circular pipes was essentially an analytical approach. The elegance of such an approach leads to a closed form expression for the axial dispersion coecient. Taylor's work required deep insights into the factors that aect axial dispersion in laminar ows. Although it could be argued that his insights could have been stated somewhat more explicitly, they led him to develop a profoundly important result. It was reported that the spreading of solute in a uid owing slowly through a circular cross sectioned capillary tube can be characterized
be approximated by means of an eective longitudinal dispersion coecient, DL , given by DL = R2U2 48Dm = P e 2D m 48 (2.1)
Here R is the pipe radius, Uavg is mean velocity of the ow, Dm is the molecular
diusivity of the solute andP e(=RUavg/Dm ) is the Peclet number, a dimensionless
number dening the relative eect of convection to diusion in the transport process. Inspired by the consistency of the solution with the assumptions made during the solution, Taylor performed an experimental investigation for a dye-water system in a capillary tube. It was concluded that the analytical expression for dispersion coecient concur with the experimental observation under the limiting conditions, (Taylor, 1954a) 4L R RUavg Dm 6.9 (2.2)
Although this analysis of scalar transport provides useful insight and guidance, it is based on several simplifying assumptions. For example, only radial diusion is taken into account and axial diusion is neglected. By an order-of-magnitude analysis (or scale analysis), it was claimed that the solute transport by axial diusion is very small compared to transport by the convection and radial diusion. However, in certain cases, such as at low Reynolds number, this may be an erroneous assumption. Nevertheless, Taylor's work established a niche for further investigation in the eld, and inspired a series of researchers to apply the techniques of applied mathematics to relax some of the constraints on the original solution and to improve its accuracy and range of applicability. For example, to seek a solution which includes axial diusion,
Aris (1956) incorporated the method of moment analysis as a modication to Taylor's work. He suggested that the eective dispersion coecient can be expressed as
DL=Dm+
R2U2
avg 48Dm
(2.3) Evans and Kenney (1965)have veried this result with their experimental obser- vations of several gas-liquid systems. Although this solution relaxed the restriction of high Peclet number (P e ) in Taylor's work, it failed to estimate the dispersion coecient for highly diusion dominated ows with Peclet number of the order of less than 10. To increase the accuracy of the solution developed by Aris, Andersson and Berglin (1981) extended his work by calculating the rst four moments instead of calculating only the rst moment, and approximating the second moment as in the original work. These methods have been widely used to determine the diusion coecient of substances because of its simple and straightforward technique.
Reejhsinghani et al. (1966)reported the eect of wider range of Peclet number,P e (12-50000) and dimensionless time, τ (Dmt/R2) (0.01-60) by conducting an experi-
mental study of a dye and water system. Their study suggests that at small values ofτ(7.45×10−3)and highP e (36.4×103), pure convection dominates the transport
process, however as τ increases, diusion also contributes to the overall dispersion process.
Other techniques from applied mathematics have been used for the solution of laminar dispersion for the ow through circular pipes. For instance, a series solution method for transient shear dispersion phenomena reported by Gill and Ananthakr- ishnan (1967) and this has been later extended to develop a representation of the dispersion coecient as a function of time (Gill and Sankarasubramanian, 1970).
on the solute that is being dispersed and concluded that the value of the peak mean concentration is proportional to the slug length Gill and Sankarasubramanian (1971). Likewise, Tseng and Besant (1970) developed a solution to the convection diusion equation from the linear combination of the eigenvalues and eigenvectors of the matrix form of the equation for fully developed laminar ow. This solution was found to be in good agreement with the experimental results of Reejhsinghani et al. (1966) at high Peclet number and high dimensionless time. However this solution failed to be applicable for low Peclet numbers and short dimensionless times. Similarly De Gance and Johns (1978) calculated solute concentration proles using a Hermite polynomial and they developed an expression for the time dependent dispersion coecient for chemically active solute in rectilinear domain.
The criteria for the validity of Taylor's solution have been examined by Yu (1976) who developed a series solution in terms of zero-order Bessel functions and claimed that Taylors's analysis is valid when τ > 0.7. Furthermore, the solution has been
extended by Yu (1979) to include the non-uniform injection of solute who concluded that the position of local maxima of the mean concentration depends on dimensionless timeτ for dierent Peclet numbers. Although the solution proposed by Yu (1979) has a wider range of applicability, its computational complexity prevents the solution from converging towards quick and accurate solutions. Considering the fact that most of the previous solutions are applicable only at large times, Vrentas and Vrentas (1988) developed a perturbation solution for small values of τ and low P e which exhibits superior convergence. However, the narrow range of applicability limits the approach from being a complete complementary version of the exact solution. Nevertheless, this study highlighted the limitation of Taylor's theory at small times and eliminated the limitation by providing solutions applicable to short times. In addition, they have
developed an alternative approach to study dispersion during its initial development using an exponential Fourier transform (Vrentas and Vrentas, 2000) which provides an asymptotic solution at short times. Other studies on dispersion for short periods of time and low Peclet numbers have been proposed by Shankar and Lenho (1989), Lighthill (1966) and Yu (1981). Additionally, a few analytical studies have been reported on dispersion in pulsed systems Paine et al. (1983)and dispersion in short tubes by Shankar and Lenho (1989).
Most of the studies mentioned above are concerned with the dispersion in straight circular pipes. However, in practice, pipes may not always be straight and they often contain bends and curved sections. Flow through pipes with bends sometimes gives rise to secondary ows in the radial direction which enhances lateral transport of solute. As a consequence dispersion is usually smaller in curved tubes than straight tubes Daskopoulos and Lenho (1988). This eect of secondary ow in reducing dispersion has been utilized in chromatographic separation and purication process where it is desired to minimize dispersion to increase instrumental eectiveness Zhao and Bau (2007). Apart from in curved tubes, secondary ows can be generated by turbulence, particularly in non-circular shaped ducts. Hence, dispersion is usually expected to be reduced in turbulent ow in non-circular ducts. This topic is discussed in detail in forthcoming chapters of this thesis.