Variance of the residence time

FIG. 23-10 Residence time distributions of pilot and commercial reactors. <3 = variance of the residence time distribution, n = number of stirred tanks with the same variance, Pe = Peclet number. 

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. 

Determine the dimensionless variance of the residence time distribution in Problem 15.1. Then use Equation (15.40) to fit the axial dispersion model to this system. Is axial dispersion a reasonable model for this situation ... 

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

This common measure is the variance of the residence-time distribution. In the absence of reaction, a sudden change in inlet conditions will be followed by a spread-out change in outlet conditions. The spreading can be described by common statistical parameters, the mean, variance, skewness, and so on. 

Ways of calculating these parameters are well-known. For example, they are simply related to the coefficients in the Taylor1s Series expansion of the Laplace Transform of the equation which describes the temperature transient without reaction. With each of the six reactor models, an expression for the ratio of the variance of the residence-time distribution to the square of the mean can be derived analytically by finding the Laplace Transform. The results of such an analysis are listed in Table X. 

At small values of the dispersion number the variance of the residence time distribution decreases and approaches plug flow, where the following approximation can be applied ... 

The liquid flowing inside a MWPB can be described with a one-parameter dispersion flow model. As we show in Section 3.3, the axial mixing coefficient or, more correctly, the axial dispersion coefficient is the specific parameter for this model. Relation (3.112) contains the link between the variance of the residence time of liquid elements and the Peclet number. We can rewrite this relation so as to particularize it to the case of a MWPB. Here, we have the possibility to compute the variance of the residence time of the liquid through the stochastic model for the liquid flow developed previously in order to obtain the value of the axial dispersion coefficient ... 

The variance of the residence times of the tracer gas with respect to mean residence time is obtained from Eqs. (6-26) and (6-27), according to van der Laan (V3), as follows ... 

Once the variance of the residence time distribution has been obtained, the axial dispersion coefficient of the tracer g sE is calculated by combining the following equation for the one-dimensional diffusion model (V3) with Eq. (6-28) or Eq. (6-31) ... 

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 ... 

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. 

C dispersion m laminar flow repme In laminar flow regime = axial dispersion in a packed column is expected due to the local velocity distribution of fluid in void spaces. In this case, variance of the residence time of the fluid passing one layer of particles, o, is considered to be proportional to (dfl uY with a proportionality constant of 2. ... 

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