Residence time moments

Matis, J. and Wehrly, T., On the use of residence time moments in the statistical analysis of age-department stochastic compartmental systems, Mathematics in Biology and Medicine, edited by V. Capasso, E. Grosso, and S. Paveri-Fontana, Springer-Verlag, New York, 1985, pp. 386-398. 

Residence time, mean The average time spent by the molecules in a vessel. Mathematically, it is the first moment of the effluent concentration from a vessel with impulse input, or ... 

Skewness The third moment of a residence time distribution ... 

Residence time distributions can be described by any of the functions W(t), F(t), or f t). They can also be described using an infinite set of parameters known as moments. ... 

The first moment is the mean of the distribution or the mean residence time. 

Roughly speaking, the first moment, t, measures the size of a residence time distribution, while higher moments measure its shape. The ability to characterize shape is enhanced by using moments about the mean ... 

The entire residence time distribution can be made dimensionless. A normalized distribution has the residence time replaced by the dimensionless residence time, X = t/t. The first moment of a normalized distribution is 1, and all the moments are dimensionless. Normalized distributions allow flow systems to be compared in a manner that is independent of their volume and throughput. For example, all CSTRs have the same normalized residence time distribution, W(x) = exp(—t). Similarly, all PFRs have f(r) = S(x — 1). 

Example 15.3 Determine the first three moments about the origin and about the mean for the residence time distribution of a CSTR. 

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. 

The variance of the residence times VRT is derived from the area under the second moment of the plasma concentration curve AUSC ... 

You have been asked to carry out a residence time distribution study on a reactor network that has evolved over the years by adding whatever size and type of reactor was available at the moment. The feed stream presently contains 1... 

This involves obtaining the mean-residence time, 0, and the variance, (t, of the distribution represented by equation 19.4-14. Since, in general, these are related to the first and second moments, respectively, of the distribution, it is convenient to connect the determination of moments in the time domain to that in the Laplace domain. By definition of a Laplace transform,... 

Experimentally, VDSS is determined by calculating the area under the first moment of the plasma versus time curve (AUMC), which when combined with AUC will yield the mean residence time. 

Chanter DO. The determination of mean residence time using statistical moments is it correct J Pharmacokinet Biopharm 1985 13 93-100. 

Level B utilizes the principles of statistical moment analysis. The mean in vitro dissolution time is compared to either the mean residence time or the mean in vivo dissolution time. Like correlation Level A, Level B utilizes all of the in vitro and in vivo data, but unlike Level A it is not a point-to-point correlation because it does not reflect the actual in vivo plasma level curve. It should also be kept in mind that there are a number of different in vivo curves that will produce similar mean residence time values, so a unique correlation is not guaranteed. 

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. 

These relationships are of profound importance for, once a reactor has been described by means of a transfer function, they enable the residence time distribution for that reactor to be chsiracterised in terms of its mean, variance, skewness, etc. Such a characterisation in terms of a few low-order moments is often entirely adequate for the requirements of chemical reaction engineering. 

In these equations kei is the elimination rate constant and AUMC is the area under the first moment curve. A treatment of the statistical moment analysis is of course beyond the scope of this chapter and those concepts may not be very intuitive, but AUMC could be thought of, in a simplified way, as a measure of the concentration-time average of the time-concentration profile and AUC as a measure of the concentration average of the profile. Their ratio would yield MRT, a measure of the time average of the profile termed in fact mean residence time. Or, in other words, the time-concentration profile can be considered a statistical distribution curve and the AUC and MRT represent the zero and first moment with the latter being calculated from the ratio of AUMC and AUC. 

Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

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 second method is an indirect method, based on the liquid s average residence time evaluated with the tracer injection technique. From the first moment of the RTD curve the total external liquid hold-up can be calculated. 

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