Residence-time distribution tailing

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 the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

In a real stirred tank with bypassing or short-cut flow (Fig. 3.22), highly concentrated tracer comes out early, and the residence time distribution depends on the fraction a of the flow in the bypass (Fig. 3.23). The tailing of the response curve is caused by the perfect mixing in the main part of the tank. 

Figure 5.1.7 shows the propagator of the motion measured for a clean and a biofilm impacted capillary [14,15] and the residence time distributions calculated for each from these velocity distributions. The clean capillary gives an experimental propagator equal to the theoretical velocity distribution convolved with a Gaussian diffusion curve [14], as shown in Figure 5.1.2. For the flow around the biofilm structure note the appearance of a high velocity tail indicating higher probability of large displacements relative to the clean capillary. The slow flow peak near zero displacement results from the protons trapped within the EPS gel matrix where the...

In addition to the aforementioned slope and variance methods for estimating the dispersion parameter, it is possible to use transfer functions in the analysis of residence time distribution curves. This approach reduces the error in the variance approach that arises from the tails of the concentration versus time curves. These tails contribute significantly to the variance and can be responsible for significant errors in the determination of Q)L. 

In a final RTD experiment, a sheet of dye was frozen as before and positioned in the feed channel perpendicular to the flight tip. The sheet positioned the dye evenly across the entire cross section. After the dye thawed, the extruder was operated at five rpm in extrusion mode. The experimental and numerical RTDs for this experiment are shown in Fig. 8.12, and they show the characteristic residence-time distribution for a single-screw extruder. The long peak indicates that most of the dye exits at one time. The shallow decay function indicates wall effects pulling the fluid back up the channel of the extruder, while the extended tail describes dye trapped in the Moffat eddies that greatly impede the down-channel movement of the dye at the flight corners. Moffat eddies will be discussed more next. Due to the physical limitations of the process, sampling was stopped before the tail had completely decreased to zero concentration. 

Now that a combination of the tabulated data and exponential tail allows a complete description of the residence time distribution, we are in a position to evaluate the moments of this RTD, i.e. the moments of the system being tested [see Appendix 1, eqn. (A.5)] The RTD data are used directly in Example 4 (p. 244) to predict the conversion which this reactor would achieve under specific conditions when a first-order reaction is occurring. Alternatively, in Sect. 5.5, the system moments are used to evaluate parameters in a flexible flow-mixing transfer function which is then used to describe the system under test. This model is shown to give the same prediction of reactor conversion for the specified conditions chosen. 

This is usually indicated by a residence time distribution which rises to a maximum very much before o = 1 and is followed by a long tail due to the exchange of tracer with this stagnant region. This particular fault is especially serious in wash tanks and skim tanks where the ratio of height to diameter is small. 

Regions of stagnancy are often caused by baffles and by interference due to pipes and fittings in corners and other places where abrupt changes in flow paths can occur. It is evidenced principally by long tails in the residence time distribution curve and in extreme cases, by a mean residence time which is very much shorter than that calculated by the volume divided by the flow rate. 

If maximum product size is desired, the process stream flow is parallel, with a fresh feed and slurry discharge to product recovery equipment from each crystallizer. Cascaded product flow is avoided because flow through a series of tanks narrows the residence time distribution and the CSD and reduces the mass mean size by reducing the number of larger crystals in the distribution tail. This tail represents the major mass-weighted fraction. Population models have been solved to verify this effect, assuming an equal nucleation rate in each stage (Randolph and Larson 1988). The actual mean in industrial practice is lower because transfer of... 

In moment analysis technique the second moment,y, is usually not considered due to its complexity and high uncertainity coming from the long tails of the residence time distribution (RTD) curves. Nevertheless the functional relationships show that the transport rates such as liquid-solid mass transfer coefficient, kg, effective diffusivity D, and adsorption rate, k, can be estimated from y2 ... 

The conductivity probe technique can also be used to measure the residence time distribution of continuous flow systems by installing probes at the inlet and outlet of the mixing vessel. The probe response can be normalized and interpreted as ouUined by Levenspiel (1972) or as discussed in Chapter 1 of this book. Care should be taken to ensure that the data are collected over a sufficiently long time, because the tail can have a large effect on the measured mean residence time and the derived variance. 

If the tail is truncated, then the tracer impulse response should be normalized based on the area under the curve to obtain a proper density function that approximately describes the distribution of residence times in regions through which there is active flow. 

---