Washout function, residence time

The quality of the packed bed may also be determined by frontal analysis where the sample is applied until it reaches a plateau to give the residence time function and then the solution is momentarily switched to wash to give the washout function. The latter is used to calculate the plate height of the column... 

Real reactors can have 0 < cr < 1, and a model that reflects this possibility consists of a stirred tank in series with a piston flow reactor as indicated in Figure 15.1(a). Other than the mean residence time itself, the model contains only one adjustable parameter. This parameter is called the fractional tubularity, Xp, and is the fraction of the system volume that is occupied by the piston flow element. Figure 15.1(b) shows the washout function for the fractional tubularity model. Its equation is... 

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-... 

Now select a few hundred thousand molecules. Twenty-five percent will leave after one pass through the reactor. For each of them, pick a random number, 0 < Rnd < 1, and use the washout function to find a corresponding value for their residence time in the system, t. This requires a numerical solution when W t) is a complicated function, but for the case at hand... 

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. 

FIGURE 15.9 Residence time distribution for laminar flow in a circular tube (a) physical representation b) washout function. 

The completely segregated stirred tank can be modeled as a set of piston flow reactors in parallel, with the lengths of the individual piston flow elements being distributed exponentially. Any residence time distribution can be modeled as piston flow elements in parallel. Simply divide the flow evenly between the elements and then cut the tubes so that they match the shape of the washout function. See Figure 15.12. A reactor modeled in this way is said to be completely segregated. Its outlet concentration is found by averaging the concentrations of the individual PFRs ... 

A washout experiment is performed on a CSTR to measure its mean residence time. What is the effect of starting the experiment before the outlet concentration has fully reached Co Assume that the normalized output response is based on the outlet concentration measured at I = 0 so that the experimental washout function starts at 1.0. 

The washout residence-time distribution function W(t) is defined as the fraction of the exit stream of age s t (and similarly for W(0)). It is also the probability that an element of fluid that entered a vessel at t = 0 has not left at time t. By comparison, F(t) (or F(6)) is the probability that a fluid element has left by time t (or 13) (Section 13.3.2.)... 

Figure 6.14 shows these stationary-state solutions as a function of residence time for various small values of k2. The non-zero states exist over a limited range of ires they lie on the upper and lower shores of a closed curve, known as an isola . The size of the isola decreases as k2 increases. At each end of the isola there is a turning point in the locus, corresponding to extinction or washout. There are no ignition points in these curves. 

FIGURE 15.5 Pathological residence time behavior in a poorly designed stirred tank (a) physical representation (b) washout function. 

Tracer molecules originally in the system at time t = 0 gradually wash out. The exponential form of Equation 15.1 is specific to a CSTR, but the concept of washout applies to any flow system. Consider some time t > 0 when the fraction of molecules remaining in the system is W t) = C(t)/Co. These molecules must necessarily have entered the reactor before time t = 0 since no tracer was fed after that time. Thus these molecules have residence times of t or longer. The residence time washout function is defined as... 

The RTD is normally considered a steady-state property of a flow system, but material leaving a reactor at some time 0 will have a distribution of residence times regardless of whether the reactor is at steady state. The washout function for an unsteady reactor is defined as... 

We specify the washout function as the ratio [/]/[/]o = W(T), which represents the fraction of molecules that had a residence time of t or longer. As can be inferred from Figure 3.5a, the washout function for a PFR is... 

Call that t = 0 and monitor the outflow concentration C (t). Now, if the system is at steady state and the tracer is perfect, C (t)/CQ will yield the washout function, W(t), which is the fraction of the fluid outflow with residence times larger than t. 

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