Moments of RTD curves

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... 

The magnitude of the variance a represents the square of the distribution spread and has the units of (time). The greater the value of this moment, the greater the spread of the RTD. The variance is particularly useful for matching experimental curves to one of a family of theoretical curves. 

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. 

Accurate values of the higher moments are not. easy to obtain. The substantial sensitivity of an RTD curve to small differences in a third moment is brought out in problem P5.02.07A. 

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. 

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... 

Several aspects of the above method of moments are discussed by Butt.17 In this method, also, the tailing in the RTD curves can cause significant errors in the calculation of the Peclet number. An analysis involving the Laplace transform... 

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... 

As described in Sec. 12.3, the moments of the impulse response can be used to characterize the RTD curve. This technique is also useful here for estimating the model parameter, D , although better techniques will be described below for the latter. It is found for a closed system that... 

In moment analysis technique the second moment,y, is usually not considered due to its complexity and high uncertainity coming from the long tails of the residence time distribution (RTD) curves. Nevertheless the functional relationships show that the transport rates such as liquid-solid mass transfer coefficient, kg, effective diffusivity D, and adsorption rate, k, can be estimated from y2 ... 

A fourth model to be discussed is the piston -dispersion-exchange (PDE) model which is a combination of PE model and ADPF model. For this model niQ f Hh IG approach to ADPF model where only the effect of D is observed. For higher moments such as y2 other hydrodynamic model parameters are also effective. In fact this is an expected phenomena, although ADPF model fits the RTD curves at peaks well it fails to fit the tail, where PE model and PDE model are more successfully fitted. 

Besides the dispersion model, the tanks in series model is the other onedimensional model widely used to represent non-ideal flow. Here the fluid is thought to flow through a series of equal-size ideal stirred tanks, and the parameter in this model is the number of tanks in the cascade (iV q). The RTD curves and moments of this model are easy to obtain, since problems of proper boundary conditions and method of tracer injection and measurement do not intrude. need not be an integer for curve-fitting purposes. It is strictly empirical, and no theoretical justification, such as Taylor diffusion, or theoretical estimates of the model parameter, are generally possible. This model starts from the mass balance equation for a series of i stirred vessels with 1 < i < iV, and AT, the number of vessels in the series (or the number of equivalent stages,... 

The dimensionless residence time 0 is used in (12.6.3-1), allowing to compare the widths of different RTD curves at the same location, i.e., 0 = t or = 1. Higher statistical moments of the RTD can also be used in theory, but calculating them from the usually rather scattered tracer data turns out to be quite difficult in... 

A recommended method of characterizing the RTD in flow systems is by using their moments. These are known as the mean, variance, and skewness. The mean value or the centroid of distribution for a concentration versus time curve is... 

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