Washout function

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... 

Negative Step Changes and the Washout Function. Suppose that an inert tracer has been fed to a CSTR for an extended period of time, giving C, = Cout = Co for r < 0. At time r = 0, the tracer supply is suddenly stopped so that = 0 for r > 0. Equation (14.2) governs the transient response of the system. For t > 0,... 

The differential distribution is related to the cumulative distribution and to the washout function by... 

The Single CSTR. The washout function for a CSTR is found from its response to a negative step change in tracer concentration from Equation (15.1) ... 

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-... 

FIGURE 15.1 The fractional tubularity model (a) physical representation (b) washout function. 

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... 

Example 15.6 Determine the washout function if a diffusion-free, laminar flow reactor is put in a recycle loop. Assume that 75% of the reactor effluent is recycled per pass. 

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... 

Example 15.7 Determine the washout function for the side capacity model... 

FIGURE 15.8 Semilog plot of washout function showing two slopes that correspond to the two time constants in the side capacity model. 

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. 

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 ... 

State. The washout function for an unsteady reactor is dehned as... 

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 function also provides the basis for a method of experimentally measuring RTD (Chapter 19 see also problem 13-1). Consequences of the definition of W and its relation to F and E are explored in problem 13-5. 

In practice, it may be more simple to evaluate E(t) either through the washout function or the cumulative RTD F(t) by making a step change in inlet tracer concentration and measuring outlet concentration by the mixing cup average. Such a procedure is advocated by Nauman [4]. 

Solution Refer to Figure 4.2 and set Q = Q = Qout = 0.25m3/s, q = 0.75 m3/s, and V = I m3. Then t = 4 s for the overall system and 1 s for the once-through distribution. The differential distribution corresponding to laminar flow in a tube was found in Section 8.1.3. The corresponding washout function can be found using Equation (15.7). See also Section 15.2.2. The once-through washout function is... 

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