Mean residence times and variance

The aforementioned investigators (10-12) have derived equations relating the measured mean residence times and variances to the Peclet number or dispersion parameter for the test section. For the case where the conditions at both monitoring probes correspond to a doubly infinite pipe, it can be shown that... 

Asif et al. (1991) studied distributor effects in liquid-fluidized beds of low-density particles by measuring RTDs of the system by pulse injection of methylene blue. If PF leads into and follows the fluidized bed with a total time delay of 10 s, use the following data to calculate the mean-residence time and variance of a fluid element, and find N for the US model. 

Statistical Moments Parameters that describe the characteristics of the time courses of plasma concentration (area, mean residence time, and variance of mean residence time) and of urinary excretion rate. 

The Microsoft Excel spreadsheet (Example8-l.xls) was developed to calculate the moments (mean residence time and variance) and the... 

Example 13-2 Mean Residence Time and Variance Calculations... 

Then the mean residence time and variance of residence times (the same as would be given by the ideal tracer experiment) are... 

DOWNSTREAM PROCESSING AND BIOSEPARATION Table 2. Mean Residence Times and Variance... 

These expressions demonstrate that the normalized mean residence time and variance of the normalized residence time distribution increase with increased values of the axial dispersion number Dj. In the limit of = 0, the signal is convected and behavior corresponding to the parallel tube model is approximated the normalized residence time fi= 1 and Act = 0. For very large values of D, the behavior corresponds to a single well-mixed compartment. 

The dispersion coefficient can be detennined from a pulse tracer experiment. Here, we will use / and a to solve for the dispersion coefficient D, and then the Peclet number, Pe Here the effluent concentration of the reactor is measured as a function of time. From the effluent concentration data, the mean residence time. and variance, o, are calculated, and these values are then used to determine Dg. To show how this is accomplished, we will write... 

The estimation of mean residence time and variance are illustrated in Example 3.1. The situation is different for systems open for dispersion as discussed in Section 3.4.2. Under these conditions the experimentally determined C-curve is... 

Figure 2.8 Calculated mean residence time and variance for 3.66 m diameter kiln with L/D = 10, slope = 3°, and material angle of repose = 40°.

The normalized mean residence time and variances are given here, since they are important firom the calculation point of view ... 

A table was constructed to determine the mean residence time t, variance E(6), F(6), and 1(6) parameters from the effluent tracer versus time data. 

Alternative methods of estimating Q)L are based on the response of the reactor to an ideal pulse input. For example, equation 11.1.39 may be used to calculate the mean residence time and its variance. Levenspiel and Bischoff (9) indicate that for the boundary conditions cited,... 

Measurement of axial mixing in the liquid phase of a fluidized bed is performed by analysis of the residence time distribution of step or pulse signals [55], By plotting the dimensionless E-function of the output signal versus the dimensionless time, the moments of the residence time distribution may be calculated according to Eqs. (7) and (8), the first dimensionless moment /q describing the mean residence time and the second dimensionless moment U2 standing for the variance of the distribution. 

Calculate the mean residence time and the variance for the reactor characterized in Example 13 -1 by the RTD obtained from a pulse input at 320 K. 

Equation (14-34) has been solved numerically for a pulse injection, and the resulting dimensionless effluent tracer concentration, is shown as a function of the dimensionless time 0 in Figure 14-13 for various Peciet numbers. Although analytical solutions for ih can be found, the result is an infinite series. The corresponding equations for the mean residence time. and the variance. [Pg.967]

For practical purposes, it is beneficial to characterize distribution functions by a few characteristic terms as the mean residence time and the variance around the mean. The mean residence time t) corresponds to the first moment of the distribution function, E t). 

The additive property of variances also allows to treat any measured tracer pulse input and to extract from it the mean residence time and the variance of the measured outlet curve as indicated in Figure 3.13 and Equation 3.56. 

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. 

The mean residence time, and the variance of the tracer entering vessel 1 can be calculated from the curve of Co(0 versus time that is measured at detector 1. Thus,... 

In these equations, the subscript overall denotes the combination of all of the vessels plus the injection system. In other words, toveraii and o yeran are the mean residence time and the variance that would be computed from the signal of the detector on the stream leaving the last (A ) vessel. 

The exit concentration data shown in Table P4.8 were obtained from a tracer experiment studying the mixing characteristics of a continuous flow reactor. Calculate the RTT> function, cumulative distribution function, mean residence time, and the variance of the RTD function of this reactor. 

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. 

Levenspiel and Smith Chem. Eng. Sci., 6 (227), 1957] have reported the data below for a residence time experiment involving a length of 2.85 cm diameter pyrex tubing. A volume of KMn04 solution that would fill 2.54 cm of the tube was rapidly injected into a water stream with a linear velocity of 35.7 cm/sec. A photoelectric cell 2.74 m downstream from the injection point is used to monitor the local KMn04 concentration. Use slope, variance, and maximum concentration approaches to determine the dispersion parameter. What is the mean residence time of the fluid ... 

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