Residence time distribution function definition

At the instant r, the residence times of all the particles of the tracer A must be < f while the particles of B consist of two groups in the first group all the particles are fed at instant f0 = 0 or later and so also have residence times < t while the particles in the second group were fed into the device before tQ = 0 and so have residence times > t. If the particles in the first group are denoted by the superscript then, from the definition of the residence time distribution function, F, we have... 

According to the definition of the F-function, the residence time distribution functions of A and B for the case under consideration can be expressed by the corresponding ratios of the amounts of particles coming out from the device to those inputted in the time interval from 0 to t, i.e. 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

It is possible to determine the cumulative residence time distribution function F(t) from either a tracer step-change or a tracer impulse response. From its definition, the properties of F(t) are ... 

The time it takes a molecule to pass through a reactor is called its residence time 6. Two properties of 6 are important the time elapsed since the molecule entered the reactor (its age) and the remaining time it will spend in the reactor (its residual lifetime). We are concerned mainly with the sum of these times, which is 6, but it is important to note that micromixing can occur only between molecules that have the same residual lifetime molecules cannot mix at some point in the reactor and then unmix at a later point in order to have different residual lifetimes. A convenient definition of residence-time distribution function is the fraction J ) of the effluent stream that has a residence time less than 0. None of the fluid can have passed through the reactor in zero time, so / = 0 at 0 = 0. Similarly, none of the fluid can remain in the reactor indefinitely, so that Japproaches 1 as 0 approaches infinity. A plot of J 6) vs 0 has the characteristics shown in Fig. 6-2a. 

Each flow pattern of fluid through a vessel has associated with it a definite clearly defined residence time distribution (RTD), or exit age distribution function E. The converse is not true, however. Each RTD does not define a specific flow pattern hence, a number of flow patterns—some with earlier mixing, others with later mixing of fluids—may be able to give the same RTD. 

This type of curve, then, has an ordinate that gives the fraction of fluid that has a certain residence time, which is plotted on the abscissa. In more formal terms, the curve defines the residence time distribution or exit age distribution. The exact definition uses the common symbol (0) for the exit age-distribution frequency function as defined by Danckwerts [6] (see Himmelblau and BischofT [4] for more details) ... 

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... 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

By definition the exit age distribution function E is such that the fraction of the exit stream with residence times between t and t + St is given by Ed/. 

Water residence times on the protein surface are not directly measurable experimentally, bnt can be defined as the relaxation time of time correlation functions of the popnlation of the hydration shell [5], Due to differences in the definition from one investigation to another, the values reported in the literature exhibit considerable variability. Nonetheless, heterogeneity in water dynamics near the protein surface is clearly manifested in distributions of water residence times. The distribution we have constrncted for the N state of HocLA in solution from the residence time of water next to each residue is plotted up to lOOps in Figure 16.1d. The distribution is very broad, ranging from 2.6 to 241 ps, but highly skewed toward shorter residence times. The mean residence time of 23 ps is about 2.5 times longer than the rotational correlation time for hydration water. 

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