F, cumulative residence-time

F Cumulative residence time distribution Dimensionless Dimensionless... 

Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(0) is given by... 

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],...

The cumulative residence-time distribution function F(t) is defined as the fraction of exit stream that is of age 0 to t (i.e., of age t) it is also the probability that a fluid element that entered at t = 0 has left at or by time t. Since it is defined as a fraction, it is dimensionless. Furthermore, since F(O) = 0, that is, no fluid (of age 0) leaves the vessel before time 0 and F( ) = 1, that is, all fluid leaving the vessel is of age 0 to or all fluid entering at time 0 has left by time then... 

Derive the equation for the cumulative residence time distribution, F(t), for the fluid driven by pressure flow inside a slit. Assume a volumetric flow rate of Q and a Newtonian viscosity of /t. Use the notation used in the schematic of Fig. 6.78. 

The derivation of the residence time behavior of the single streamlines now allows the formulation of the cumulative residence-time distribution function F(t) according to the following formula ... 

Since the foundations of residence time theory are rigorously given elsewhere (5,6, 7), only those features which are essential to the present treatment will be given here. The residence time distribution (residence time frequency function exit age distribution), f(t), is defined such that f(t)dt is the fraction of fluid at any instant leaving the system, having spent time between t and t + dt within the system. The cumulative residence time distribution is... 

After studying this chapter the reader will be able to describe the cumulative F t) and external age (f) and residence-time distribution functions, and to recognize these functions for PER, CSTR, and laminar flow reactors. The reader will also be able to apply these functions to calculate the conversion and concentrations exiting a reactor using the segregation model and the maximum mixedness model for both single and multiple reactions. 

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

Fig. 2.2-4 Cumulative residence-time function F t/z) for an ideal continuous stirred tank (1), an ideal tubular reactor with plug flow (2), and laminar flow in a tubular reactor (3).

Differentiation of the cumulative residence-time function F t/r) (Equations 2.2-23 and 2.2-37) gives the corresponding residence-time distribution function (Equation 2.2-39) ... 

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... 

The cumulative residence time distribution function [F(f)] for a cascade consisting of two nonidentical stirred-tank reactors can be represented by a mathematical expression of the form... 

To determine the function F(t) from experimental data, we use the property G, as shown in Chapter 1. If G is any property (conductivity, ionization, wavelength, etc.) proportional to the concentration, in which Gi is the magnitude at the inlet and Gi at the outlet, then the cumulative residence time distribution function, which remained in the reactor at an instant shorter than t, will be ... 

The reactor volume is 1 and the feed volumetric flow is 0.2 m /min. Determine the cumulative residence time distribution function (F) and the mean residence time. 

Positive Step Changes and the Cumulative Distribution. Residence time distributions can also be measured by applying a positive step change to the inlet of the reactor Cm = Cout = 0 for r<0 and C = Co for r>0. Then the outlet response, F i) = CouMICq, gives the cumulative distribution function. ... 

Material flowing at a position less than r has a residence time less than t because the velocity will be higher closer to the centerline. Thus, F(r) = F t) gives the fraction of material leaving the reactor with a residence time less that t where Equation (15.31) relates to r to t. F i) satisfies the definition. Equation (15.3), of a cumulative distribution function. Integrate Equation (15.30) to get F r). Then solve Equation (15.31) for r and substitute the result to replace r with t. When the velocity profile is parabolic, the equations become... 

Note that, in a laminar-flow tubular reactor, the material on the reactor centre line has the highest velocity, this being exactly twice the average velocity, Q/A, for the whole reactor. This means that, following any tracer test, no response will be observed until the elapsed time exceeds one half of the reactor space time or mean residence time. The following values for 0 and F(0) emphasise the form of the cumulative RTD and the fact that, even up to 10 residence times after a tracer impulse test, 0.25% of the tracer will not have been eluted from the system. 

The definition of RTD functions is credited to Danckwerts (55). We first differentiate between the internal RTD function g(t) dt and the external RTD function /(f) dt. The former is defined as the fraction of fluid volume in the system with a residence time between t and t + dt, and the latter is defined as the fraction of exiting flow rate with a residence time between t and t + dt. From these definitions we can define the cumulative functions G(t) and F(t), respectively, as follows ... 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

The cumulative RTD function F(t) is obtained by integrating Eq. 9.2-24 while recalbng that the fraction of flow rate with a residence time less than t is located in the region is located between = 2/3 and... 

F(t) Cumulative exit residence time distribution function (7.3-12)... 

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