Residence time distribution probability

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. 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

For reaction other than first order, the reaction probability depends on the time that a molecule has been in the reactor and on the concentration of other molecules encountered during that time. The residence time distribution does not allow a unique estimate of the extent of reaction, but some limits can be found. 

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-... 

Figure 5.1.7 shows the propagator of the motion measured for a clean and a biofilm impacted capillary [14,15] and the residence time distributions calculated for each from these velocity distributions. The clean capillary gives an experimental propagator equal to the theoretical velocity distribution convolved with a Gaussian diffusion curve [14], as shown in Figure 5.1.2. For the flow around the biofilm structure note the appearance of a high velocity tail indicating higher probability of large displacements relative to the clean capillary. The slow flow peak near zero displacement results from the protons trapped within the EPS gel matrix where the...

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.)... 

We will not attempt to solve the preceding equations except in a few simple cases. Instead, we consider nonideal reactors using several simple models that have analytical solutions. For this it is convenient to consider the residence time distribution (RTD), or the probability of a molecule residing in the reactor for a time f. 

There are various ways to classify mathematical models (5). First, according to the nature of the process, they can be classified as deterministic or stochastic. The former refers to a process in which each variable or parameter acquires a certain specific value or sets of values according to the operating conditions. In the latter, an element of uncertainty enters we cannot specify a certain value to a variable, but only a most probable one. Transport-based models are deterministic residence time distribution models in well-stirred tanks are stochastic. 

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). 

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 ... 

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 opposite of the large diameter pipeline with little axial or radial mixing is the perfect backmixed reactor with instantaneous mixing and uniformity. For polystyrene reactors with several hours of residence time, complete mixing in 1-2 min is usually adequate to satisfy a practical definition of perfectly mixed. The probability of exit of any fluid element from this type of reactor is independent of when it entered. The residence time distribution is exponential and the molecular weight distribution in the case of no termination is Mw/Mn = 2.0, which will spread out to 2.3 when chain transfer controls. If product requirements necessitate a narrower residence time distribution, one can utilize several of these reactors in series. This becomes necessary to control the grafting distribution in rubber modified polystyrene. 

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. 

If not is the total probability for reaction, is the probability that a reaction has not occurred during time interval r, which leads directly to Equation 4.94 for choosing the time of the next reaction. We wlU derive this fact in Chapter 8 when we develop the residence-time distribution for a CSTR. Shah, Ramkrishna and Borwanker call this time the interval of quiescence, and us.e it to develop a stochastic. simulation algorithm for particulate. system.dyii.amic.s rather than. c.bem.ical. kinetics.114]. 

Now that we have a model for the residence-time distribution, how shall we use this in the analysis of the unit We need weighting factors for each residence time. These come from the PDF itself. For example, if we integrate the PDF between any two residence times, we obtain the probability density for that range of times ... 

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 ... 

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