Residence time probability density

Sinclair, C. G. and McNaughton, K. J. (1965). The residence time probability density of complex flow systems. Chemical Engineering Science, 20 261-264. 

Sinclair, C. G. and McNaughton, K. J. (1965). The residence time probability density of complex flow systems. Chemical Engineering Science, 20 261-264. Nosseir, N., Pelet, U. and Hildebrand, G. (1968). Pressure field generated by jet on jet impingement. AIChE J, 25 78-84. 

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. 

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... 

We address the probability to find a particle at a certain time at a distinct box as the probability of residence or probability density. We differentiate clearly between the probability density and the transition probability. 

The residence time is the time spent in a reservoir by an individual atom or molecule. It is also the age of a molecule when it leaves the reservoir. If the pathway of a tracer from the source to the sink is characterized by a physical transport, the word transit time can also be used. Even for a single chemical substance, different atoms and molecules will have different residence times in a given reservoir. Let the probability density... 

The shape of the probability density function, depends on the system. Some examples are shown in Fig. 4-4. This figure also contains probability density of age (see Section 4.2.3). Figure 4-4a might correspond to a lake with inlet and outlet on opposite sides of the lake. Most water molecules will then have a residence time in the lake roughly equal to the time it takes for the mean current to carry the water from the... 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

Solution It is easy to begin the solution. In piston flow, molecules that enter together leave together and have the same residence time in the reactor, t. When the kinetics are first order, the probabiUty that a molecule reacts depends only on its residence time. The probability that a particular molecule will leave the system without reacting is exp(— F). For the entire collection of molecules, the probability converts into a deterministic fraction. The fraction unreacted for a variable density flow system is... 

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). 

Considering the tracer entering the vessel at a given instant of time to be the nuclei formed at that time, C, (Do and C(0) can be converted to the number density of nuclei in the whole vessel, n, in the 1st tank, no and that of crystals in the exit stream from the vessel, n(0), respectively. The crystals having the residence time of 0 grow up to the size L, which is given by Equation 1. Therefore, by using Equations a-1 or a-2 and 1, the number basis probability density function of final product crystals, fn(L) is obtained, as follows. 

E(t) is a probability density function or frequency function and E(f)df is the fraction of material which leaves the system with an age of between t and (t + df) units of time. Since all material must have a residence time between zero and infinity... 

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... 

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. 

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. 

The effective diffusivity Dn decreases rapidly as carbon number increases. The readsorption rate constant kr n depends on the intrinsic chemistry of the catalytic site and on experimental conditions but not on chain size. The rest of the equation contains only structural catalyst properties pellet size (L), porosity (e), active site density (0), and pore radius (Rp). High values of the Damkohler number lead to transport-enhanced a-olefin readsorption and chain initiation. The structural parameters in the Damkohler number account for two phenomena that control the extent of an intrapellet secondary reaction the intrapellet residence time of a-olefins and the number of readsorption sites (0) that they encounter as they diffuse through a catalyst particle. For example, high site densities can compensate for low catalyst surface areas, small pellets, and large pores by increasing the probability of readsorption even at short residence times. This is the case, for example, for unsupported Ru, Co, and Fe powders. 

In order to solve the model equation, we must complete it with the univocity conditions. In some cases, relations (3.100)-(3.107) can be used as solutions for the model particularized for the process. The equivalence between both expressions is that c(x,t)/C(j appears here as P(x,t). Extending the equivalence, we can establish that P(x, t) is in fact the density of probability associated with the repartition function of the residence time of the liquid element that evolves inside a uniform porous structure. 

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 probability of readsorption increases as the intrinsic readsorption reactivity of a-olefins (k,) increases and as their effective residence time within catalyst pores and bed interstices increases. The Thiele modulus [Eq. (15)] contains a parameter that contains only structural properties of the support material ( <>, pellet radius Fp, pore radius 4>, porosity) and the density of Ru or Co sites (0m) on the support surface. A similar dimensional analysis of Eqs. (l9)-(24), which describe reactant transport during FT synthesis, shows that a similar structural parameter governs intrapellet concentration gradients of CO and H2 [Eq. (25)]. In this case, the first term in the Thiele modulus (i/>co) reflects the reactive and diffusive properties of CO and H2 and the second term ( ) accounts for the effect of catalyst structure on reactant transport limitations. Not surprisingly, this second term is... 

Diffusive and convective transport processes introduce flexibility in the design of catalyst pellets and in the control of FT synthesis selectivity. Transport restrictions lead to the observed effects of pellet size, site density, bed residence time, and hydrocarbon chain size on chain growth probability and olefin content. The restricted removal of reactive olefins also allows the introduction of other intrapellet catalytic functions that convert olefins to other valuable products by exploiting high intrapellet olefin fugacities. Our proposed model also describes the catalytic behavior of more complex Fe-... 

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