Moments of RTDs

This type of comparison of the moments of RTD curves of two models has wide applicability. For large the RTD curve becomes increasingly symmetrical and approaches the normal curve of the dispersion model and a comparison of these two curves allows one to relate the two models. The range where these conversion equations are valid is determined by the range of the validity of each of the models. 

The Erlang number /leriang. nd the variances O (t ) and 0 (t) are single parameter characterizations of RTD curves. The skewness y (t), and higher moments can be used to represent RTD curves more closely if the data are accurate enough. 

Although a transfer function relation may not be always invertible analytically, it has value in that the moments of the RTD may be derived from it, and it is thus able to represent an RTD curve. For instance, if Gq and Gq are the limits of the first and second derivatives of the transfer function G(.s) as. s 0, the variance is... 

A transfer function may not be always analytically invertable, but it has nevertheless value in that the moments of an RTD may be derived from it, notably the variance.. One or two of the moments often are adequate characterizations of an RTD curve and enable useful deductions about the behavior of a vessel as a chemical reactor. Problem P5.02.01 covers the basic theory and P5.02.07 is another application. Figure 5.3 is of a simple process flow diagram, individual transfer functions, and the overall transfer function. 

The theory necessary for understanding two-station tracer measuring techniques is outlined in Appendix 1. An arbitrary, but unimodal, impulse of tracer is created in a system inlet and the outlet response recorded, see Fig. 21 (Appendix 1). Then, the mean, Mj, of that which resides between the points at which inlet and outlet pulses are observed and recorded is equal to the difference in means of these two signals. Similarly, the variance, T2, and the skewness, T3 are equal to the differences in these respective moments between inlet and outlet. This enables the system transfer function to be defined in terms of a few low-order moments via eqns. (A.5) or (A.9) of Appendix 1, this in turn defining the system RTD. Recall that system moments and moments of the system RTD are one and the same. 

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. 

Therefore the moments M, T2 and T3 of the RTD of the whole reactor system are simply the sums of the moments of the RTDs of its constituent parts. This, of course, can be generalised to any series combination of reactors or other flow vessels provided that they are linked together in a statistically independent and non-interactive manner this latter phrase implies that changes in operating conditions of a downstream unit are unable to influence the way in which an upstream unit behaves. 

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. 

Next, we define a parallel set of NPD function in continuous flow recirculating systems. We restrict our discussion to steady flow systems. Here, as in the case of RTD, we distinguish between external and internal NPD functions. We define fk and 4 as the fraction of exiting volumetric flow rate and the fraction of material volume, respectively, that have experienced exactly k passages in the specified region of the system. The respective cumulative distribution functions, and /, the means of the distributions, the variances, and the moments of distributions, parallel the definitions given for the batch system. 

These equations allow various special cases to be treated. This links the n order moment of the RTD to the n-l order moment of the IAD. 

The Laplace transform may be inverted to provide a tracer response in the time domain. In many cases, the overall transfer function cannot be analytically inverted. Even in this case, moments of the RTD may be derived from the overall transfer function. For instance, if Go and GJare the limits of the first and... 

Here, C (t) is the concentration of the tracer at time t on the RTD curve at position i, U is the real mean axial velocity of the phase being considered, L is the distance between two measuring points, fit is the first moment and zrfjs the second moment of the RTD curve. Since the first moment of the response curve is essentially the mean of that curve, the average residence time of the tracer can be calculated by taking the difference of the first moment of the response curves... 

The second moment of the response curves indicates the spread of the RTD curve and the difference of the second moments is a measure of the amount of backmixing occurring between the two measuring points. For the.opea system, the second moment has been derived analytically by Levenspiel and Smith58 as... 

Variance The second moment of the RTD. There are two forms one in terms of the absolute time, designated G (f) and the other in terms of reduced time, tr = tit, designated G (fr) ... 

As is the case with other variables described by distribution frmctions, the mean value of the variable is equal to the first moment of the RTD fimc-tion, E(,t), Thus the mean residence time is... 

The entire RTD can be made dimensionless. A normalized distribution has the residence time replaced by the dimensionless residence time, t = tjt. The first moment of anormalized 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 RTD, W(r) = exp(—t). Similarly, allPFRs have /(t) = S(r — 1). 

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