Residence-time distributions moments

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. ... 

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. 

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... 

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. 

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. 

The residence-time distribution in the liquid phase of a cocurrent-upflow fixed-bed column was measured at two different flow rates. The column diameter was 5.1 cm and the packing diameter was 0.38 cm. The bed void fraction was 0.354 and the mass flow rate was 50.4 g s l. The RTD data at two axial positions (which were 91 cm apart in Run 1 and 152 cm apart in Run 2) are summarized in Table 3-2. Using the method of moments, estimate the mean residence time and the Peclet number for these two runs. If one assumes that the backmixing characteristics are independent of the distance between two measuring points, what is the effect of gas flow rate on the mean residence time of liquid and the Peclet number Hovv does the measured and the predicted RTD at the downstream positions compare in both cases ... 

If we consider the random variable theory, this solution represents the residence time distribution for a fluid particle flowing in a trajectory, which characterizes the investigated device. When we have the probability distribution of the random variable, then we can complete more characteristics of the random variable such as the non-centred and centred moments. Relations (3.110)-(3.114) give the expressions of the moments obtained using relation (3.108) as a residence time distribution. Relation (3.114) gives the two order centred moment, which is called random variable variance ... 

Note that Mq is the total mass of species i, pj is the mean residence time of the peak (ti), and p 2 is the variance of the residence time distribution (o ). The higher order moments p 3 and p 4 are related to the skewness ... 

For an impulse of a tracer dye, the mean and variance of the experimentally measured residence time distribution are given by the following moment equations... 

Estimates of the dispersion coefficient and time lag can be obtained by integrating the experimental data to obtain the mean and variance of the residence time distribution. These values may be substituted into the moment expressions (Eqs. 30, 31) to obtain Kj and td. In this study, it was convenient to integrate the experimental data using the rectangular rule. The accuracy of the moment analysis can be improved by the use of optimum truncation points (33) or by the use of wei ted moments (32, 34). The disadvantage of these methods is that the expressions relating moments and model parameters become more complicated. 

Experiments were conducted at flowrates of 0.5,1.0,1.5, and 2.0 ml/min and applied voltages of 0, 100, 200, 300, and 400 V across the column. Moment analysis of the residence time distribution of the dyes was used to estimate the electrokinetic dispersion coefficients. The effects of the direction of the electric field and the packing size on Kj were also investigated. 

The residence time distribution curve (RTD) can be inscribed by its statistical moments, of which the centroid of distribution T and spread of distribution a are the most important numerical values. Thus, for a C curve, the zeroth moment is... 

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