Probability residence time

In Figures 3.4 and 3.5, the RTDs of ideal reactors are presented together with the RTD of a real reactor. The ideal, continuously operated stirred tank reactor (CSTR) has the broadest RTD between all reactor types. The most probable residence time for an entering volume element is t = 0. After a mean residence time t = t), 37% of the tracer injected at time t = 0 is still present in the reactor. After five mean residence times, a residue of about 1% still remains in the reactor. This means that at least five mean residence times must pass after a change in the inlet conditions before the CSTR effectively reaches its new stationary state. 

In the higher pressure sub-region, which may be extended to relative pressure up to 01 to 0-2, the enhancement of the interaction energy and of the enthalpy of adsorption is relatively small, and the increased adsorption is now the result of a cooperative effect. The nature of this secondary process may be appreciated from the simplified model of a slit in Fig. 4.33. Once a monolayer has been formed on the walls, then if molecules (1) and (2) happen to condense opposite one another, the probability that (3) will condense is increased. The increased residence time of (1), (2) and (3) will promote the condensation of (4) and of still further molecules. Because of the cooperative nature of the mechanism, the separate stages occur in such rapid succession that in effect they constitute a single process. The model is necessarily very crude and the details for any particular pore will depend on the pore geometry. 

The classical experiment tracks the off-gas composition as a function of temperature at fixed residence time and oxidant level. Treating feed disappearance as first order, the pre-exponential factor and activation energy, E, in the Arrhenius expression (eq. 35) can be obtained. These studies tend to confirm large activation energies typical of the bond mpture mechanism assumed earlier. However, an accelerating effect of the oxidant is also evident in some results, so that the thermal mpture mechanism probably overestimates the time requirement by as much as several orders of magnitude (39). Measurements at several levels of oxidant concentration are useful for determining how important it is to maintain spatial uniformity of oxidant concentration in the incinerator. 

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. 

To specify these transition probabilities we make the further assumption that the residence time of a particle in a given adsorption site is much longer than the time of an individual transition to or from that state, either in exchange with the gas phase in adsorption and desorption or for hopping across the surface in diffusion. In such situtations there will be only one individual transition at any instant of time and the transition probabilities can be summed, one at a time, over all possible processes (adsorption, desorption, diffusion) and over all adsorption sites on the surface. To implement this we first write... 

The results of Massimilla et al., 0stergaard, and Adlington and Thompson are in substantial agreement on the fact that gas-liquid fluidized beds are characterized by higher rates of bubble coalescence and, as a consequence, lower gas-liquid interfacial areas than those observed in equivalent gas-liquid systems with no solid particles present. This supports the observations of gas absorption rate by Massimilla et al. It may be assumed that the absorption rate depends upon the interfacial area, the gas residence-time, and a mass-transfer coefficient. The last of these factors is probably higher in a gas-liquid fluidized bed because the bubble Reynolds number is higher, but the interfacial area is lower and the gas residence-time is also lower, as will be further discussed in Section V,E,3. 

The dynamics of these models depend strictly on carbon fluxes, but the fluxes are poorly measured or are calculated from carbon reservoir size and assumptions about the residence time of the carbon in the reservoir. In addition, model fluxes are linear functions while in reality few, if any, probably are linear. 

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... 

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. 

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). 

To find the conversion for the reactor, we need the average reaction probability for a great many molecules that have flowed through the system. The averaging is done with respect to residence time since residence time is what determines the individual reaction probabilities ... 

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. 

An increase from 2 to 5 bar total pressure increases the space-time yield by about 20% (15 vol.-% ethylene, 85 vol.-% oxygen, 2-20 bar 0.235-3.350 s 11 h ) [4], At higher pressures, 10 and 20 bar, a decrease activity is observed. Since industrial processes occur at up to 30 bar, at first sight this result is surprising. The decreasing activity with pressure was partially explained by catalyst deactivation, probably as a consequence of the longer residence times applied. 

P 69] No details on the solvent used and concentrations are given in [127], as the process most likely is proprietary (Figure 4.96). Probably the process is solvent-free as obviously one of the reactants has also the function of dissolving the other. The temperature for micro-channel processing was set to 0 °C. The residence time between the pre-reactor and micro mixer was 1 s and between the micro mixer and quench 5 s, totalling 6 s. 

Release of superoxide during ORR catalysis indicates that the ferric-superoxo intermediate (Fig. 18.20) has a substantial residence time at 0.2 V (the potential of the maximum production of superoxide), suggesting that the potential of the ferric-superoxo/ferric-peroxo couple, (Fig. 18.20), is more reducing than 0.2 V. The fraction of superoxide detected at potentials >0.2 V probably reflects the fact that 02, which is a strong outer-sphere reductant [Huie and Neta, 1999], was oxidized by the mostly ferric catalytic film before it could escape the film. There are two plausible explanations for the decrease in the fraction of superoxide byproduct released at... 

The gas flow rate is usually presented as a deposition parameter however, it is much more instructive to report the gas residence time [6], which is determined from the flow rate and the geometry of the system. The residence time is a measure of the probability of a molecule to be incorporated into the film. The gas depletion, which is determined by the residence time, is a critical parameter for deposition. At high flow rates, and thus low residence times and low depletion [303], the deposition rate is increased [357, 365] (see Figure 39) and better film quality is obtained, as is deduced from low microstructure parameter values [366],... 

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