Density function, residence time

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-... 

Composition The law of mass aclion is expressed as a rate in terms of chemical compositions of the participants, so ultimately the variation of composition with time must be found. The composition is determined in terms of a property that is measured by some instrument and cahbrated in terms of composition. Among the measures that have been used are titration, pressure, refractive index, density, chromatography, spectrometry, polarimetry, conduclimetry, absorbance, and magnetic resonance. In some cases the composition may vary linearly with the observed property, but in every case a calibration is needed. Before kinetic analysis is undertaken, the data are converted to composition as a function of time (C, t), or to composition and temperature as functions of time (C, T, t). In a steady CSTR the rate is observed as a function of residence time. 

In a time period from t = 0 to t = 6t seconds, a quantity m (g) of a tracer is introduced at the system inlet, and the tracer concentration C(t) (g/1) is measured in the exit from the system. Subject to the above conditions, the residence time density function from the measured tracer response is ... 

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... 

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). 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

Vapor-Liquid Gravity Separator Design Fundamentals The critical factors in the performance of a horizontal separator are the vapor residence time and the settling rate of the liquid droplets. However, two other factors enter into the design—the vapor velocity must be limited to avoid liquid entrainment, and there must be sufficient freeboard within the vessel to allow for a feed distributor. For vertical separators, the design is based on a vapor velocity that must be less than the settling velocity of the smallest droplet that is to be collected, with due allowance for turbulence and maldistribution of the feed. The vapor residence time is a function of the vapor flow rate (mass), vapor density, and volume of vapor space in the separator, based on the following ... 

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... 

In a variable-density reactor the residence time depends on the conversion (and on the selectivity in a multiple-reaction system). Also, in ary reactor involving gases, the density is also a function of reactor pressure and temperature, even if there is no change in number of moles in the reaction. Therefore, we frequently base reactor performance on the number of moles or mass of reactants processed per unit time, based on the molar or mass flow rates of the feed into the reactor. These feed variables can be kept constant as reactor parameters such as conversion, T, and P are varied. 

NMR analysis of protein structure suggests that the ordered water molecules seen by X-ray diffraction on protein surfaces have very short residence times in solution. Thus most of these molecules may be of little importance to an understanding of protein function. However, ordered water is of great importance to the crystallographer. As the structure determination progresses, ordered water becomes visible in the electron-density map. For example, in Plate 2, water molecules are implied by small regions of disconnected density. 

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... 

Temperature along the Riser Height Temperature (T) affects the cracking (k.) and deactivation (VO constants (or functions). It also influences the gas density and, as a consequence, the flow rates, the superficial velocities, and the residence time of the gas in the riser. Its effect on ki makes the product distribution (gas, coke, etc.) dependent on T. 

The degree of desalination that can be achieved in passing the feed solution through a stack is a function of the solution concentration, the applied current density, and the residence time of the solution in the stack. If the flow rates of diluate and concentrate through the stack are relatively high the degree of desalination or concentration that can be achieved in a single path is quite low and often not sufficient to meet the required product qualities. 

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. 

If the concentration of tracer in the outlet stream is not measured directly, a quantity Rj(t), which is proportional to C(t), must be measured. For example, Rj(t) may be light absorbance if the tracer is a dye, a conductance if the tracer is an electrolyte, or a counting rate if a radioactive tracer is used. If C(t) = kR[(t), where k is the proportionality constant, it can be substituted in Equation 8-54 to give the residence time density function E(t) as... 

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