Residence-time distribution cumulative

F Cumulative residence time distribution Dimensionless Dimensionless... 

Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(0) is given by... 

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

To measure a residence-time distribution, a pulse of tagged feed is inserted into a continuous mill and the effluent is sampled on a schedule. If it is a dry miU, a soluble tracer such as salt or dye may be used and the samples analyzed conductimetricaUy or colorimetricaUy. If it is a wet mill, the tracer must be a solid of similar density to the ore. Materials hke copper concentrate, chrome brick, or barites have been used as tracers and analyzed by X-ray fluorescence. To plot results in log-normal coordinates, the concentration data must first be normalized from the form of Fig. 20-15 to the form of cumulative percent discharged, as in Fig. 20-16. For this, one must either know the total amount of pulse fed or determine it by a simple numerical integration... 

Figure 8-3. Cumulative residence time distribution function.

Positive Step Changes and the Cumulative Distribution. Residence time distributions can also be measured by applying a positive step change to the inlet of the reactor Cm = Cout = 0 for r<0 and C = Co for r>0. Then the outlet response, F i) = CouMICq, gives the cumulative distribution function. ... 

Material flowing at a position less than r has a residence time less than t because the velocity will be higher closer to the centerline. Thus, F(r) = F t) gives the fraction of material leaving the reactor with a residence time less that t where Equation (15.31) relates to r to t. F i) satisfies the definition. Equation (15.3), of a cumulative distribution function. Integrate Equation (15.30) to get F r). Then solve Equation (15.31) for r and substitute the result to replace r with t. When the velocity profile is parabolic, the equations become... 

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

The cumulative residence-time distribution function F(t) is defined as the fraction of exit stream that is of age 0 to t (i.e., of age t) it is also the probability that a fluid element that entered at t = 0 has left at or by time t. Since it is defined as a fraction, it is dimensionless. Furthermore, since F(O) = 0, that is, no fluid (of age 0) leaves the vessel before time 0 and F( ) = 1, that is, all fluid leaving the vessel is of age 0 to or all fluid entering at time 0 has left by time then... 

The cumulative residence time distribution function for a perfectly mixed vessel is ... 

The process to be analyzed is represented by Figure 16.4. What will be found are equations for the cumulative and differential size distributions in terms of residence time and growth rate. The principal notation is summarized here. 

The principal quantities related by these equations are tpm, d4>m/dx, L, Lpl, t, n°, and B°. Fixing a certain number of these will fix the remaining one. Size distribution data from a CSTC are analyzed in Example 16.6. In Example 16.7, the values of the predominant length Lpr and the linear growth rate G are fixed. From these values, the residence time and the cumulative and differential mass distributions are found. The effect of some variation in residence time also is found. The values of n° and B° were found, but they are ends in themselves. Another kind of condition is analyzed in Example 16.4. 

The differential distributions are differences between values of crystal length L. The tabulation shows cumulative and differential distributions at the key t = 1.93 hr, and also at 1.5 and 3.0 hr. The differential distributions are plotted and show the shift to larger sizes as residence time is increased, but the heights of the peaks are little affected. 

It is more common to assess mixing using the cumulative residence time distribution, defined by... 

Figure 6.50 presents the cumulative residence time distribution for a tube with a Newtonian model and for a shear thinning fluid with power law indices of 0.5 and 0.1. Plug flow, which represents the worst mixing scenario, is also presented in the graph. A Bingham fluid, with a power law index of 0, would result in plug flow. 

The shape of the residence time or cumulative residence time distributions are used when optimizing the mixing ability of a system. Often, this shape is compared to the residence time in an ideal or perfect mixer. Such a mixer is a well stirred tank, as depicted in Fig. 6.51(a). Here, two components, a primary and secondary component, are fed to the tank at a total flow rate Q. The output can be regarded as a flow rate Q with a concentration (1 — Co) of... 

Figure 6.52 compares the cumulative residence time distributions of a perfect mixer to Poiseuille flow. Disregarding the fact that the stirring tank s output starts at t = 0, we can see that the overall shape of the curve with the prefect mixer is much broader, pointing to a more homogeneous output, or simply, a better mixer. 

Cumulative residence time distribution for a stirring tank with perfect mixing. 

Derive the equation for the cumulative residence time distribution, F(t), for the fluid driven by pressure flow inside a slit. Assume a volumetric flow rate of Q and a Newtonian viscosity of /t. Use the notation used in the schematic of Fig. 6.78. 

Figure 10.35 Cumulative residence time distribution inside three different rhomboidal mixing heads.

The derivation of the residence time behavior of the single streamlines now allows the formulation of the cumulative residence-time distribution function F(t) according to the following formula ... 

A graphical representation of the cumulative residence time distribution function is given in Figure 4.97 for a structured well, a laminar flow reactor and an ideal plug flow reactor assuming the same average residence time and mean velocity in each reactor. 

Figure 4.97 Calculated cumulative residence time distribution function for a multi-channel well, a laminar flow reactor and a plug flow reactor [147] (by courtesy of VDI-Verlag GmbH).

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