Residence time probability density distribution

However, it is clear that the probability density distributions of the classical and quantum oscillations are quite different, particularly for low values of v the quantum oscillation has the largest probability density near r = re the classical probability density (the residence time ) is greatest at the turning points. 

Most of the indices of the mixing capacity in the left-hand side column in Table 2.1 are related to the mixing rate—residence time for the flow system (e.g., ratio of the standard deviation of the probability density distribution of the residence time to the average residence time residence time is the stay time of the inner substance in an equipment), circulation time for a batch system (e.g., ratio of the standard deviation of the probability density distribution of the circulation time to the average circulation time circulation time is the time required for one circulation of the inner substance in an equipment), mixing time (e.g., the time required for the concentration of the inner substances at a specific position in the equipment to reach a final constant value within some permissible deviation), and so on. 

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. 

Mixedness based on the residence time probability density distribution. 

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). 

A minimum residence time of 10 to 30 minutes should be provided to assure that surges do not upset the system and to provide for some coalescence. As discussed previously, potential benefits of providing more residence time probably will not be cost efficient beyond this point. Skimmers with large residence times require baffles to attempt to distribute flow and eliminate short-circuiting. Tracer studies have shown dial skniimei tanks, even those with carefully designed spreaders and baffles, exhibit poor flow behavior and short-circuiting This is probably due to density and temper-atuie differences, deposition of solids, corrosion of spreaders. etc... 

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... 

Hydrocarbon distributions in the Fischer-Tropsch (FT) synthesis on Ru, Co, and Fe catalysts often do not obey simple Flory kinetics. Flory plots are curved and the chain growth parameter a increases with increasing carbon number until it reaches an asymptotic value. a-Olefin/n-paraffin ratios on all three types of catalysts decrease asymptotically to zero as carbon number increases. These data are consistent with diffusion-enhanced readsorption of a-olefins within catalyst particles. Diffusion limitations within liquid-filled catalyst particles slow down the removal of a-olefins. This increases the residence time and the fugacity of a-olefins within catalyst pores, enhances their probability of readsorption and chain initiation, and leads to the formation of heavier and more paraffinic products. Structural catalyst properties, such as pellet size, porosity, and site density, and the kinetics of readsorption, chain termination and growth, determine the extent of a-olefin readsorption within catalyst particles and control FT selectivity. 

Non-Flory molecular weight distributions have also been attributed to the presence of several types of active sites with different probabilities for chain growth and for chain termination to olefins and paraffins (45). Two-site models have been used to explain the sharp changes in chain growth probability that occur for intermediate-size hydrocarbons on Fe-based catalysts (46,47). Many of these reports of non-Flory distributions may instead reflect ineffective dispersal of alkali promoters on Fe catalysts or inadequate mass balances and product collection protocols. Recently, we have shown that multisite models alone cannot explain the selectivity changes that occur with increasing chain size, bed residence time, and site density on Ru and Co catalysts (4,5,40,44). 

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. 

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 ... 

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. 

For an isothermal, homogeneous system of constant density with well-defined boundaries and a single inlet and outlet stream, a residence time distribution, F(t), is defined as the probability that a fluid element has a residence time less than t. At steady state F(t) is the fraction of the fluid in the outflow that has resided in the vessel less than time t. Here we assume that each fluid element has a definite point of entry into the vessel and that age (residence time) is acquired by the fluid element only while it resides within the system s boundaries. 

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