Dispersion tracer functions

The figure shows the ratio of the widths of initially delta-like concentration tracers at the reactor exits as a function of the micro-channel Peclet number for different values of the porosity. Taking a value of = 0.4 as standard, it becomes apparent that the dispersion in the micro-channel reactor is smaller than that in the fixed-bed reactor in a Peclet number range from 3 to 100. Minimum dispersion is achieved at a Peclet number of about 14, where the tracer width in the micro-channel reactor is reduced by about 40% compared with its fixed-bed counterpart. Hence the conclusion may be drawn that micro-channel reactors bear the potential of a narrower residence time than fixed-bed reactors, where again it should be stressed that reactors with equivalent chemical conversion were chosen for the comparison. 

Figure 5.6 The quasi-hard sphere diffusion data of Ottewill and Williams14 as a function of volume fraction. The long (DL) and short (Ds) time tracer diffusion coefficients are shown (symbols). The dotted line is representative of the relative fluidity of a hard sphere dispersion...

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

For a particular reactor, the dispersion number can be determined by analysis of the response at the reactor outlet to the injection of a tracer at the inlet. The procedure is fully described in Chap. 6. Alternatively, use may be made of published correlations [2], which give the reactor dispersion number as a function of either the Reynolds number or the product of the Reynolds and Schmidt numbers. Once a value of the reactor dispersion number is available, it can be used in one of the following ways to determine reactor performance for particular cases. 

Fortunately, it is not always necessary to recover the system RTD curve from the impulse response, so the complications alluded to above are often of theoretical rather than practical concern. In addition, the dispersion model is most appropriately used to describe small extents of dispersion, i.e. minor deviations from plug flow. In this case, particularly if the inlet pipe is of small diameter compared with the reactor itself, the vessel can be satisfactorily assumed to possess closed boundaries [62]. An impulse of tracer will enter the system and broaden as it passes along the reactor so that the observed response at the outlet will be an RTD and will be a symmetrical pulse, the width of which is a function of DjuL alone. 

The next problem is to find the functional relationship between the variance of the tracer curve and the dispersion coefficient. This is done by solving the partial differential equation for the concentration, with the dispersion coefficient as a parameter, and finding the variance of this theoretical expression for the boundary conditions corresponding to any given experimental setup. The dispersion coefficient for the system can then be calculated from the above function and the experimentally found variance. 

Aris (A8), Bischoff (Bll), and Bischoff and Levenspiel (B14) have utilized a method that does not require a perfect delta-function input. The method involves taking concentration measurements at two points, both within the test section, rather than at only one as was previously done. The remaining sketches in Table II show the systems considered. The variances of the experimental concentration curves at the two points are calculated, and the difference between them found. This difference can be related to the parameter and thus to the dispersion coeflScient. It does not matter where the tracer is injected into the system as long as it is upstream of the two measurement points. The injection may be any type of pulse input, not necessarily a delta function, although this special case is also covered by the method. 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by... 

Consider a fluid flowing steadily along a uniform pipe as depicted in Fig. 2.13 the fluid will be assumed to have a constant density so that the mean velocity u is constant. Let the fluid be carrying along the pipe a small amount of a tracer which has been injected at some point upstream as a pulse distributed uniformly over the cross-section the concentration C of the tracer is sufficiently small not to affect the density. Because the system is not in a steady state with respect to the tracer distribution, the concentration will vary with both z the position in the pipe and, at any fixed position, with time i.e. C is a function of both z and t but, at any given value of z and t, C is assumed to be uniform across that section of pipe. Consider a material balance on the tracer over an element of the pipe between z and (z + Sz), as shown in Fig. 2.13, in a time interval St. For convenience the pipe will be considered to have unit area of cross-section. The flux of tracer into and out of the element will be written in terms of the dispersion coefficient DL in accordance with equation 2.12. For completeness and for later application to reactors (see Section 2.3.7) the possibility of disappearance of the tracer by chemical reaction is also taken into account through a rate of reaction term 9L... 

We conclude therefore that, for small values of the dispersion number (D/uL < 0.01), the C-curve for a pulse input of tracer into a pipe is symmetrical and corresponds exactly to equation 2.20 for a Gaussian distribution function ... 

In several tests series the point of tracer feeding and the column installations are changed and the axial dispersion coefficient is measured as a function of superficial gas velocity. The radial rate profils are neglected. In a first approximation propane is taken incrompres-sible as the pressure loss is neglectable compared to the operating pressure of 12 MPa. 

The extent of gas dispersion can usually be computed from experimentally measured gas residence time distribution. The dual probe detection method followed by least square regression of data in the time domain is effective in eliminating error introduced from the usual pulse technique which could not produce an ideal Delta function input (Wu, 1988). By this method, tracer is injected at a point in the fast bed, and tracer concentration is monitored downstream of the injection point by two sampling probes spaced a given distance apart, which are connected to two individual thermal conductivity cells. The response signal produced by the first probe is taken as the input to the second probe. The difference between the concentration-versus-time curves is used to describe gas mixing. 

In a third paper by the Bernard and Holm group, visual studies (in a sand-packed capillary tube, 0.25 mm in diameter) and gas tracer measurements were also used to elucidate flow mechanisms ( ). Bubbles were observed to break into smaller bubbles at the exits of constrictions between sand grains (see Capillary Snap-Off, below), and bubbles tended to coalesce in pore spaces as they entered constrictions (see Coalescence, below). It was concluded that liquid moved through the film network between bubbles, that gas moved by a dynamic process of the breakage and formation of films (lamellae) between bubbles, that there were no continuous gas path, and that flow rates were a function of the number and strength of the aqueous films between the bubbles. As in the previous studies (it is important to note), flow measurements were made at low pressures with a steady-state method. Thus, the dispersions studied were true foams (dispersions of a gaseous phase in a liquid phase), and the experimental technique avoided long-lived transient effects, which are produced by nonsteady-state flow and are extremely difficult to interpret. 

In a pulse input, an amount of tracer Nq is suddenly injected in one shot into the feedstreain entering the reactor in as short a time as possible. The outlet concentration is then measured as a function of time. Typical concentration-time curves at the inlet and outlet of an arbitrary reactor are shown in The c curve Figure 13-4. The effluent concentration-time curve is referred to as the C curve in RTD analysis. We shall analyze the injection of a tracer pulse for a single-input and single-output system in which only flow (i.e., no dispersion) carries the tracer imaterial across system boundaries. First, we choose an increment of time At sufficiently small that the concentration of tracer, C(t), exiting between time t and t + At is essentially constant. The mount of tracer material, Ah leaving Ihe reactor between time t and t At vs then... 

FIGURE 3-18 Dispersion of a continuous tracer injection in a sand column experiment. The behavior of a front of the tracer is shown in the next to last panel tracer concentration is presented as a function of distance at fixed times t0 and t1. A breakthrough curve, a plot of concentration as a function of time at a fixed point, is shown in the bottom panel. (Compare with Fig. 3-28, which shows breakthrough curves for pulse inputs.)... 

Rhodamine B was added to the tank inflow, and the resulting effluent concentration was recorded as a function of time in a double-beam (Zeiss) spectral photometer. These measurements were evaluated statistically in order to describe the hydraulic characteristics of the tanks. Effluent concentrations of the tracer, dosed as a delta impulse, observed for different (hydraulic) surface loading (q = 0.28, 0.58, and 1.0 m/h) were evaluated in terms of a characteristic (dimensionless) number, the dispersion number (4), and shown in Figure 3. 

A logical next step is the formulation of correlation functions between suspension characteristics (as presently described by a calculated collision efficiency value), specific tank parameters (as presently expressed in terms of the dispersion number, derived from tracer analysis), and solids separation... 

Developing concentration profiles in a soil column for cyclical boundary loading functions is important for several reasons. One reason is that the solution increases the repertoire of mathematical models that are available for which someone may find a use. A second reason is that the solution can be used by those who set up experiments to estimate parameters such as the dispersion coefficient. Another reason is that analytical solutions to tracer transport equations are desired because they can be used easily lot some simple flow cases to quickly estimate what... 

The dispersion coefficient can be detennined from a pulse tracer experiment. Here, we will use / and a to solve for the dispersion coefficient D, and then the Peclet number, Pe Here the effluent concentration of the reactor is measured as a function of time. From the effluent concentration data, the mean residence time. and variance, o, are calculated, and these values are then used to determine Dg. To show how this is accomplished, we will write... 

Figure 7. Longitudinal dispersion (Dl) divided by the diffusion coefficient (Df) for tracers measured in column experiments as a function of the particle scale Peclet number (Npe). It is defined as the product of the average pore fluid velocity, u, and the grain diameter, d, divided by the free fluid diffusion coefficient, D/. The magnitude of the dispersion is independent of the pore fluid velocity (Vp) for very small Peclet numbers (or fluid velocities). Note that the effective diffusion coefficient in a porous media is smaller than the diffusion coefficient in a free fluid phase due to the tortuosity. The dispersion increases linearly with increasing flow velocity (increasing Peclet number). Modified from Appelo and Postma (1999).

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