Residence time distribution probability function

From the assumption of perfect mixing, the corresponding residence time distribution probability density function is well known as... 

The movement of the particles in this stage is very complex and extremely random, so that to determine accurately the residence time distribution and the mean residence time is difficult, whether by theoretical analysis or experimental measurement. On the other hand, the residence time distribution in this stage is unimportant because this subspace is essentially inert for heat and mass transfer. Considering the presence of significant back-mixing, the flow of the particles in this stage is assumed also to be in perfect mixing, as a first-order approximation, and thus the residence time distribution probability density function is of the form below ... 

It is noted that the right-hand side of Eq. (10.20) is just the series expansion of an exponential function. Therefore the overall residence time distribution probability density function in the SCISR is obtained to be... 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

Since F(t + dt) represents the volume fraction of the fluid having a residence time less than t + dt, and F(t) represents that having a residence time less than r, the differential of F(t dF(t will be the volume fraction of the effluent stream having a residence time between t and t + dt. Hence dF(t) is known as the residence time distribution function. From the principles of probability the average residence time (t) of a fluid element is given by... 

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... 

The washout residence-time distribution function W(t) is defined as the fraction of the exit stream of age s t (and similarly for W(0)). It is also the probability that an element of fluid that entered a vessel at t = 0 has not left at time t. By comparison, F(t) (or F(6)) is the probability that a fluid element has left by time t (or 13) (Section 13.3.2.)... 

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 three ideal reactors form the building blocks for analysis of laboratory and commercial catalytic reactors. In practice, an actual flow reactor may be more complex than a CSTR or PFR. Such a reactor may be described by a residence time distribution function F(t) that gives the probability that a given fluid element has resided in the reactor for a time longer than t. The reactor is then defined further by specifying the origin of the observed residence time distribution function (e.g., axial dispersion in a tubular reactor or incomplete mixing in a tank reactor). 

The last equations prove that the Markov chains [4.6] are able to predict the evolution of a system with only the data of the current state (without taking into account the system history). In this case, where the system presents perfect mixing cells, probabilities p and p j are described with the same equations as those applied to describe a unique perfectly stirred cell. Here, the exponential function of the residence time distribution (p in this case, see Section 3.3) defines the probability of exit from this cell. In addition, the computation of this probability is coupled with the knowledge of the flows conveyed between the cells. For the time interval At and for i= 1,2,3,. ..N and j = 1,2,3,..N - 1 we can write ... 

The function of the distribution of the residence time from 0 up to H can be obtained by the sum of the probabilities of the exit from the way. This is possible at z = H with an elementary action of type I and at z = 0 with a standard elementary action II. Thus, for the function of residence time distribution, the following equation can be written ... 

The residence time distribution function is found as a result of the addition of the probabilities showing the possibility for a liquid element to leave the MWPB (see also Section 4.3.1) ... 

The age of an atom or molecule in a reservoir is the time since it entered the reservoir. Age is defined for all molecules, whether they are leaving the reservoir or not. As with residence times, the probability density function of ages [ (r)] can have different shapes. In a steady-state reservoir, however, y>(r) is always a non-increasing function. The shapes of V(t) corresponding to the three residence time distributions discussed above are induded in Fig. 

Individual movement by walking was modeled as a jump from 1 cell to a randomly selected neighboring cell at a time set by the (probabilistic) residence time. The probability density function was obtained from a simulation of a random walk process with parameters derived from experimental work (Englund and Hamback 2004). The model incorporated passive movement downstream by implying that 1% of the movement to other cells was long-distance movement (drift) in a downstream direction. Drift distance was incorporated as an exponential distribution, with an assumed average of 10 m. 

We will use the NormalDistribution to make the representations of the residence time distribution. The Probability Density Function (PDF) is made up of the Normal Distribution and the variable 9. This can be integrated in closed form ... 

Figure 9 displays the probability distribution function (p r) and the effectiveness factor r] k), which have been calculated via Eqs. 36 and 34 from the tracer exchange curves in the limiting cases of single-hle diffusion, normal diffusion and barrier confinement. The fact that in all cases the residence time distribution function is found to decrease monotonically may be easily rationalized as a quite general property. Due to the assumed stationarity of the residence time distribution function, the number of molecules with a residence time r is clearly the same at any instant of time. The number of molecules with a residence time r + At may therefore be considered as the number of molecules with a residence time r minus the number of molecules which will leave the system in the subsequent time interval At. Therefore, (p x) must quite generally be a monotonically decaying function. 

The residence time distribution (RTD) is a probability distribution function used to characterize the time of contact and contacting pattern (such as for plug-flow or complete backmixing) within the reactors. Excessive retention of some elements and shortdrcuiting of others due to backmixing and other dispersive phenomena lead to a broad distribution in the residence times of individual molecules in the reactor. This tends to decrease conversion and exerts a negative influence on product selectivity/yield. The RTD depends on the flow regime and is characterized by Reynolds (Re) and Schmidt (Sc) numbers. 

Choose between different theories for modeling nonidealities in chemical reactors snch as residence time distribution (RTD), interaction by exchange with the mean (lEM), engulfment deformation (E), and probability density function (PDF). 

Major methods used to account for mixing in reactors. Illustrations on statistically stationary field of a velocity component. DNS Direct Numerical Simulation PDF Probability Density Function I Internal distribution function RTD Residence Time Distribution <> macro-scale averaged reactor-scale averaged. 

It is normally called the differential distribution function (of residence times). It is also known as the density function or frequency function. It is the analog for a continuous variable (e.g., residence time i) of the probabiUty distribution for a discrete variable (e.g., chain length /). The fraction that appears in Equations (15.2), (15.3), and (15.6) can be interpreted as a probability, but now it is the probability that t will fall within a specified range rather than the probability that t will have some specific value. Compare Equations (13.8) and (15.5). 

For any quantity that is a function of time we can describe its properties in terms of its distribution function and the moments of this function. We first define the probability distribution function p(t) as the probability that a molecule entering the reactor wih reside there for a time t. This function must be normalized... 

Impulse (delta) response method The input signal is changed in the form of a delta function. This method is widely used in chemical engineering to investigate the residence time probability density distribution function. 

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