Rectangular concentration dispersion

Figure 24.7 Two extreme input scenarios for chemical being spilled into a river, (a) Constant input rate J (mass per unit time) during time At leading to a rectangular concentration profile, Cin = J IQ. The dashed line shows how dispersion acts on the edges and leaves the concentration in the middle of the cloud unchanged. (b) Gaussian input scenario. The time integral between t = -2G and t = 2c0 comprises 95% of the total input 3H (see Box 18.2). Dispersion causes the variance to increase according to Eq. 24-60. The maximum concentration is given by Eq. 24-62.

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

Figure 19.14 (a) A pollutant front in a river with an initial rectangular shape. This abrupt change is transformed into a smoother concentration profile while the front is moving downstream (b). The hatched area represents the integrated mass exchange Msx across the front due to longitudinal dispersion. 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

The concentration of methane at room temperature at any channel position is described by the impulse response without reaction [Eq. (3.4)] [the dispersion coefficient for a rectangular channel with channel depth b and aspect ratio s is given in Eq. (3.3)] ... 

Where C, (t, z) is the concentration of the the / th component at position z and L is the column length. Boundary conditions characterize the injection and if the dispersion effects are neglected they can be described by rectangular pulses with duration tp at the column inlet. Assuming that the sample concentration is C0 i... 

When a wide rectangular injection pulse is injected in a column and the width is such that the plateau is not completely eroded when it is eluted, the solution of the system of equations of the ideal model (Eqs. 8.1a and 8.1b) includes a constant state, followed by a simple wave, as shown by the theory of partial differential equations [12,13]. The importance of this result is due to the existence of a relationship between the concentrations of the two components of the binary mixture in the simple wave region. This relationship is independent of the position of the band along the column. We have discussed the properties of the hodograph transform in the case of the ideal model (Oiapter 8, Sections 8.1.2 and 8.8). In the case of the equilibrium-dispersive model (Eqs. 11.1 and 11.2), this result is no longer valid. However, the plots of Ci versus C2 are often close to the simple wave solu-... 

This value is in agreement with the one derived from band profiles calculated with the equilibrium-dispersive model [9]. The time given by Eq. 16.20 provides useful information regarding the specifications for the experimental conditions under which staircase binary frontal analysis must be carried out to give correct results in the determination of competitive isotherms. The concentration of the intermediate plateau is needed to calculate the integral mass balances of the two components, a critical step in the application of the method (Chapter 4). This does not apply to single-pulse frontal analysis in which series of wide rectangular pulses are injected into the column which is washed of solute between successive pulses. 

The dispersion process can be better understood by considering a model single line system into which a solution of dye A is inserted into an inert carrier stream. At the time of insertion (time zero), the plug of this solution is ideally a perfect cylinder and the associated concentration/ time function is hypothetical and rectangular in shape (Fig. 5.9a). 

An extensive but unfortunately, as yet, unpublished study by Yamaoka (28) was concerned with the mode of orientation of several polypeptides in varied solvents under the influence of a rectangular voltage pulse. While measurements could be made in most organic solvents, he was unable to obtain steady-state values for the birefringence of PBLG dissolved in benzene and dioxane, except at low concentrations in dioxane. Extremely long rise times were observed in these solvents, and the 1.4-millisecond limit on his pulse width prevented establishment of equilibrium. Yamaoka showed by means of optical rotatory dispersion that PBLG assumes a helical conformation in benzene. 

Eigure 2.8 shows that the rectangular pulse, which is introduced at the column inlet (x = 0), is symmetrically broadened as it travels along the column. As a consequence of the band broadening, the maximum concentration of the solute is decreased. This causes an unfavorable dilution of the target component fraction. Main factors that influence axial dispersion are discussed below. 

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