Residence-time distribution variance

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. 

Figure 8-38. Residence time distributions of some commerciai and fixed bed reactors. The variance, equivaient number of CSTR stages, and Peciet number are given for each reactor. (Source Wales, S. M., Chemicai Process Equipment—Seiection and Design, Butterworths, 1990.)...

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. 

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 ... 

In addition to the aforementioned slope and variance methods for estimating the dispersion parameter, it is possible to use transfer functions in the analysis of residence time distribution curves. This approach reduces the error in the variance approach that arises from the tails of the concentration versus time curves. These tails contribute significantly to the variance and can be responsible for significant errors in the determination of Q)L. 

The variance approach may also be used to determine n. From Illustration 11.2 the variance of the response data based on dimensionless time is 30609/(374.4)2, or 0.218. From equation 11.1.76 it is evident that n is 4.59. Thus the results of the two approaches are consistent. However, a comparison of the F(t) curves for n = 4 and n = 5 with the experimental data indicates that these approaches do not provide very good representations of the data. For the reactor network in question it is difficult to model the residence time distribution function in terms of a single parameter. This is one of the potential difficulties inherent in using such simple models of reactor behavior. For more advanced methods of modeling residence time effects, consult the review article by Levenspiel and Bischoff (3) and textbooks written by these authors (2, 4). 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

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. 

The mixing pattern in an n-stage CSTR battery is intermediate between segregated and maximum mixed flow and is characterized by residence time distribution with variance o2 = 1/n. Conversion in the CSTR battery is found by solving n successive equations... 

Figure 17.3. Ratio of volumes of an n-stage CSTR battery and a segregated flow reactor characterized by a residence time distribution with variance a2 = 1/n. Second-order reaction.

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. 

Using the experimental residence time distribution data of Levenspiel and Smith in Example 8-2, determine the number of ideal tanks N, the variance, dispersion number, and Peclet number. 

The liquid velocity should be examined also with respect to the residence-time distribution of the liquid phase. Too slow or too fast liquid velocity would increase the variance, as proved experimentally [11], Values of liquid load of 5 to 20m3/m2 h seem the most practical, with the optimum around 10m3/m2 h. 

Knowledge of the hydrodynamics of liquid flow and particle movement are required for scale-up and optimization of expanded-bed processes. Residence time distribution (RTD) analysis i.e., a plot of the dimensionless tracer concentration in the effluent stream versus the dimensionless time, can determine whether the liquid flow in the expanded bed is plug flow or well mixed. Using the method described by Levenspiel,6 values of mean residence time in the expanded bed (t), the dimensionless variance of the RTD curve,... 

The definition of symbols is in the Table of Nomenclature. Basically SD is a number proportional to the reactor length, made dimensionless by a proper combination of thermal and reaction kinetic paramters. t is proportional to the temperature rise, made dimensionless by a combination of inlet temperature and activation energy, y and a2 are the mean and variance, respectively, of the residence-time distribution in the reactor. 

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. 

Levenspiel [9] gives the variance of residence time distribution, o- as... 

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 ... 

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 ... 

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