Exit-age distribution function

The exit-age distribution function is a measure of the distribution of the ages of fluid elements leaving a vessel, and hence is an RTD function. As a function of time, f, it is defined as  

The exit-age distribution function may also be expressed in terms of dimensionless time 8 defined by 

Consequences of this definition are analogous to those from equations 13.3-1 to -4. 

Furthermore, for the steady flow of a constant-density fluid, f is constant, and, from equation (13.3-5), dd = dt/t. Thus, on combining equations 13.3-5 and -6, we obtain 

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This is defined as the fraction of material in the outlet stream that has been in the system for the period between t and t + dt, and is equal to E(t)dt, where E(t) is called the exit age distribution function of the fluid elements leaving the system. This is expressed as... 

For all likely operating conditions, (ie., for t < X), the appropriate values of the concentration and the polymerization rate constant are the values calculated at t = t ( 2). To prove this, the exit age distribution function for a backmix reactor was used to weight the functions for Cg and kj and the product was integrated over all exit ages (6). It is enlightening at this point to compare equation 18 with one that describes the yield attainable in a typical laboratory semibatch reactor at comparable conditions. ... 

Figure 13.2 Exit-age distribution function E(t) for arbitrary (nonideal) flow showing significance of area under the E(t) curve...

Problem 1, If the input signal is measured and the exit age distribution functions E, E, and E are known, then Q is the convolution of E with and so on, thus... 

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. 

Scheme Q D, and E, From Fig. 12.1 the exit age distribution function for the two equal-size plug-mixed flow reactor system is...

For a series of equal-sized backmix reactors the exit age distribution function is... 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

The distribution of residence times for a stream of fluid leaving a vessel is called the exit age distribution function E (synonymous with residence time distribution or... 

Fig. 2.3. Exit age distribution function or E-curve also known as the residence time distribution...

Fig. 3.38. Flow of particles through a fluidised bed showing a normalised C-curve which is identical with the exit age distribution function E...

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/. 

Figure 3.78 Exit age distribution function at different measuring points of the screening set-up [111] (by courtesy of AlChE).

Figure 3.79 Dimensionless exit age distribution function using tubular reactors with different inner diameters [111] (by courtesy ofAIChE).

The exit-age distribution function E(t) is approximately obtained by following the experimentally often used method of backward differencing ... 

E E(t) EDM EDTA EDX EHD EKI EO EOF ESI-MS Ez Activation energy Exit-age distribution function Electro-discharge machining Ethylene-diamine-tetraacetic acid Energy dispersive X-ray Electrohydrodynamic Electrokinetic instability Electroosmotic-Electroosmotic flow Electrospray ionization mass spectrometry Electric field... 

The exit age distribution function E(t) is obtained from outside the vessel while the internal age distribution function I(t) is obtained from inside the vessel. I(t) can be represented in terms of the RTD or the F-curve as... 

Sometimes E t) is called the exit-age distribution function. If we regard the age of an atom as the time it has resided in the reaction environment, then E t) concerns the age distribution of the effluent stream. It is tbe most used of the distribution functions connected with reactor analysis because it characterizes ttie lengths of time various atoms spend at reaction conditions. 

I t) and E t) are the internal- and exit-age distribution functions, respectively. Now let us write a material balance on the tracer for a step change in its concentration from zero to 1 (the arbitrary concentration units will not affect the results here) at t = 0. After time t we have... 

Determine the exit-age distribution function for this reactor. What type of ideal reactor model does this most closely resemble ... 

According to this view, then, the conversion in any sort of reactor can be estimated if we know the exit-age distribution function, and indeed this is so. For a further illustration, let us reconsider the laminar-fiow reactor results of Chapter 4. 

As in the situation for tank-type reactors, we need first to define the characteristic time quantities associated with the reactor design. The characteristic diffusion time, tj), is given in equation (8-207), and the extent-of-reaction time, is given in equation (8-208). The third time here is tp, the length of time an element of fluid remains in the reactor. This is reminiscent of the exit-age distribution function developed for homogeneous tubular-flow reactors, but the development of the theory for multiphase reactors has been different. " ... 

Mid-point diameter of particles in size fraction Exit age distribution function, m ... 

E(0) is called exit age distribution function. The age of a fluid element in the reaction vessel at any time instant is defined as the time spent by the fluid element in the vessel at that time instant. Exit age is the total time spent by the fluid element in the vessel from the time of entry to the time of exit. Thus, exit age of the fluid element is also the residence time. E(0)d0 = dF Q) is the fraction of fluid elements (leaving the reactor at any time) whose residence time is between 0 and 0 + A0, which is equal to 0 as d0 0. E(0) = 0 at 0 = 0 as well as at 0 —> oo as fluid elements have finite residence time value greater than zero. Figure 3.45 represents a typical exit age distribution function. 

Using the above equation, the exit age distribution function E(0) is obtained from the C-curve of the impulse tracer test and is shown in Figure 3.50. 

The exit-age distribution function is also known as the external-age distribution function. It is sometimes simply called the residence time distribution function. However, this can cause some confusion. As we shall see shortly, the exit-age distribution function is not the only function that is used to characterize the distribution of residence times. 

Equation (10-3) permits the exit-age distribution function, E t), to be calculated from the tracer response curve that is measured after a pulse injection of tracer. 

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