First appearance time

This equation can be fit to experimental data in several ways. The model exhibits a sharp first appearance time, tf st = rpt, which corresponds to the fastest material moving through the system. The mean residence time is found using Equation (15.13), and Xp = tf,rsi/1 is found by observing the time when the experimental washout function first drops below 1.0. It can also be fit from the slope of a plot of In W versus t. This should give a straight line (for t > tfirst) with slope = 1/(F— tfirst)- Another approach is to calculate the dimen-... 

The fractional tubularity model has been used to fit residence time data in flui-dized-bed reactors. It is also appropriate for modeling real stirred tank reactors that have small amounts of dead time, as would perhaps be caused by the inlet and outlet piping. It is not well suited to modeling systems that are nearly in piston flow since such systems rarely have sharp first appearance times. 

The Tanks-in-Series Model. A simple model having fuzzy first appearance times is the tanks-in-series model illustrated in Figure 15.2. The washout function is... 

This function is shown in Figure 15.9. It has a sharp first appearance time at tflrst = tj2. and a slowly decreasing tail. When t > 4.3f, the washout function for parabohc flow decreases more slowly than that for an exponential distribution. Long residence times are associated with material near the tube wall rjR = 0.94 for t = 4.3t. This material is relatively stagnant and causes a very broad distribution of residence times. In fact, the second moment and thus the variance of the residence time distribution would be infinite in the complete absence of diffusion. 

In Figure 4.28, the model predictions are plotted for different breakpoint concentrations. Note that while the model works quite well for low Cbr, 0.01% in our case, it fails to represent the data for higher values. For example, for Cbl. = 1.7 mg/L (10%), it predicts a breakpoint time of only 47.2 min instead of 100 min, which is the approximate experimental value. This is an expected result as normally, this kind of breakpoint models are designed to work at relatively low breakpoint concentrations. On the other hand, by setting the first appearance at lower values of exit concentration, the model gradually predicts a much lower first appearance time than the experimental one. Thus, it seems that a breakpoint or first appearance concentration in the vicinity of 0.01-1% is adequate in order to have representative results (filled squares). 

Equation (1-4) is a theoretical result calculated from a hydrodynamic model, albeit a very simple one. It has a sharp first appearance time, tfl t, where the washout function first falls below 1.0. Real systems, such as that for the static mixer illustrated in Figure 1-1, may have a fuzzy first appearance time. For the fuzzy case, a 5% response time [i.e., W(t) = 0.95] is used instead. Table 1-3 shows first appearance times for some laminar flow systems. 

Table 1-3 First Appearance Times in Laminar Flow Systems...

A system with a sharp first appearance time and 0 < 1 can be approximated as a PFR in series with a CSTR. This model is used for residence times in a fluidized bed reactor. If the system has a fuzzy first appearance time and 0 1,... 

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