Cumulative residence-time curve

The result of the measurement is the cumulative residence-time curve F(t) (Equation 2.2-22) ... 

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

Figure 6.52 compares the cumulative residence time distributions of a perfect mixer to Poiseuille flow. Disregarding the fact that the stirring tank s output starts at t = 0, we can see that the overall shape of the curve with the prefect mixer is much broader, pointing to a more homogeneous output, or simply, a better mixer. 

The mathematical relations expressing the different amounts of time that fluid elements spend in a given reactor may be expressed in a variety of forms [see, e.g., Leven-spiel (1-3) and Himmelblau and Bischoff (4)]. In this book we utilize the cumulative residence-time distribution curve [F(0], as defined by Danckwerts (5) for this purpose. For a continuous flow system. Fit) is the volume fraction of... 

Figure 11.9 Cumulative residence time distribution curves for the /i-CSTR model.

The variation of the cumulative residence time distribution function is represented by a Gauss curve, indicating the concentration variation C t) as a function of time according to Figure 14.5 ... 

Figure 3.7 Cumulative residence time distribution curves of a cascade of stirred tanks, parameter number of tanks. (Adapted from Ref. [4], Figure 6.30b Copyright 2013, Wiley-...

Overview In this chapter we learn about nonideal reactors, that is, reactors that do not follow the models we have developed for ideal CSTRs, PFRs, and PBRs. In Pan I we describe how to characterize these nonideal reactors using the residence time distribution function (/), the mean residence time the cumulative distribution function Fit), and the variance a. Next we evaluate E t), F(t), and for idea) reactors, so that we have a reference proint as to how far our real (i.e., nonideal) reactor is off the norm from an ideal reactor. The functions (f) and F(r) will be developed for ideal PPRs. CSTRs and laminar flow reactors, Examples are given for diagnosing problems with real reactors by comparing and E(i) with ideal reactors. We will then use these ideal curves to help diagnose and troubleshoot bypassing and dead volume in real reactors. 

Figure I. Cumulative distribution Junction (CDF) and complementary cumulative distribution function (CCDF) for P on (a) 100 m scale, and (b) 1000 m scale. The solid curves have been obtained from DFN simulations based on site-specific data, whereas the dashed curves were obtained from simulated distributions of water residence time T, and assuming P = kz applicable. For the 100 m, k was estimated from field arui simulated data as described in the text as 11628 1/m. whereas on the 1000 m scale, k was obtained assuming linear correlation between Zand P as 3241 l/m.

It should be pointed out that the determination of vapor fractions can begin at any convenient time other than time zero. The volume fraction liquid is merely the slope of the cumulative liquid residence time time curve. Hence, summing the liquid residence times after slug flow becomes well developed will avoid the experimental errors inherent in this technique for the mist flow regime. 

The dimensionless Ctracer curve of a step experiment is the cumulative F(f) function, and represents the fraction of elements in the exit stream with a residence time shorter than t ... 

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