Distribution of the residence time

Figure 4.56. Pattern of time-space distribution of the residence time in filling of a mold by a Newtonian liquid. Figures on the curves are t values.

Most of the indices of the mixing capacity in the left-hand side column in Table 2.1 are related to the mixing rate—residence time for the flow system (e.g., ratio of the standard deviation of the probability density distribution of the residence time to the average residence time residence time is the stay time of the inner substance in an equipment), circulation time for a batch system (e.g., ratio of the standard deviation of the probability density distribution of the circulation time to the average circulation time circulation time is the time required for one circulation of the inner substance in an equipment), mixing time (e.g., the time required for the concentration of the inner substances at a specific position in the equipment to reach a final constant value within some permissible deviation), and so on. 

The function of the distribution of the residence time from 0 up to H can be obtained by the sum of the probabilities of the exit from the way. This is possible at z = H with an elementary action of type I and at z = 0 with a standard elementary action II. Thus, for the function of residence time distribution, the following equation can be written ... 

The distribution of the residence time for the gas evolution inside the bed takes into account the statement above, which concerns the possibility for the gas to exit the bed through any of the elementary processes ... 

The multiplication P (n)AT gives the existence probability or the probability to have the fluid element in compartment k in the interval of time defined by nAt and (n + 1)At. In other words, it is the response of compartment k to an impulse signal. For k = N, we can observe that the probability Pjj(n)AT makes it possible for the fluid element to leave the cells assembly in the same interval of time T = nAr and t + At = (n + 1)At. Furthermore, because PN-i(n) gives the distribution of the residence time for our assembly of compartments, then we can conclude that the response to one step impulse can be written as ... 

An important number of factors having a key influence on this unit operation [6.19, 6.20], together vith the examples described above, show how difficult it is to formulate a complete and unitary mixing theory responding to the various technical questions such as mixing time, distribution of the residence time, power consumption, heat and mass transfer kinetics in mixed media, scaling-up of a laboratory mixing plant etc. 

The curves in the bottom series in Fig. 79 are obtained by differentiating the integral concentration changes, and the area below them is equal to unity. They represent the distribution of the residence time ( age ) of volume elements in a given system. The same information is obtained by direct monitoring of the pulse response curve. 

The most important problem of rheokinetics remains the analysis of distribution of the residence times, especially for the stirred tank reactors. To develop a reasonable hydrodynamic model, which is physically proved and mathematically simple and takes into account the effect of a sharp viscosity growth is the main problem. 

FIGURE 15.4 A typical distribution of the residence time of particles. Tc = 2ti + T2. 

FIGURE 14.10 (a) The effect of rotational speed on coating variability (CV) at spray rate of 2.316ml/min and (b) the frequency distribution of the residence time of the coated particles for the effect of speed at 32° tilt. Adapted from Ref. [13] with permission. Copyright 2011, Elsevier. 

In reactors with some degree of backmixing, however, this cannot be achieved, and there is always a distribution of the residence times of the volume elements. The largest possible distribution of residence times is found in CSTRs, in which the residence times of individual volume elements are spread throughout the time frame, from zero to infinity. 

Oosterhuis et al. (1985) reported a mean circulation time of approximately 12 s in an unaerated 19 m fermenter using a radio pill flow follower. The experimental cumulative CTD associated with this mean is shown in Figure 18-7. This is the distribution of the residence time of the liqnid outside the impeller region. If it is... 

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