Residence time fractional

Another useful measure of biogeochemical processing is the fractional residence time or turnover time of a material in a reservoir. Computation of this time is similar to that of a residence time except that some subset of the input or output processes is substituted into the denominator of Eq. 1.2. The resulting turnover time represents how long it would take for that subset of processes by itself to either supply or remove all of the material from the reservoir. Turnover times can be calculated for reservoirs that are not in steady state. As will be shown in Chapter 21, the residence time can be computed by summing the reciprocals of the turnover times. 

Using the rock cycle as an example, we can compute the turnover time of marine sediments with respect to river input of solid particles from (1) the mass of solids in the marine sediment reservoir (1.0 x 10 g) and (2) the annual rate of river input of particles (1.4 X lO g/y). This yields a turnover time of (1.0 x 10 " g)/(14 x lO g/y) = 71 X lo y. On a global basis, riverine input is the major source of solids buried in marine sediments lesser inputs are contributed by atmospheric feUout, glacial ice debris, hydrothermal processes, and in situ production, primarily by marine plankton. As shown in Figure 1.2, sediments are removed from the ocean by deep burial into the seafloor. The resulting sedimentary rock is either uplifted onto land or subducted into the mantle so the ocean basins never fill up with sediment. As discussed in Chapter 21, if all of the fractional residence times of a substance are known, the sum of their reciprocals provides an estimate of the residence time (Equation 21.17). 

For elements that have multiple sources or sinks, a fractional residence time (t, ), or turnover time, can be calculated for each supply or removal process. The residence time of the element (t) is then given by... 

We construct the attainable region by noting that the concentration space is a vector field with a rate vector (e.g., in Fig. 1, dC /dC/ = RB/R ) defined at each point. Moreover, we are not restricted to concentration space, but can consider any other variable that satisfies a linear conservation law (e.g., mass fractions, residence time, energy, and temperature—for constant heat capacity and density). The attainable region is an especially powerful concept once it is known, performance of the network can often be determined without the network itself. 

Equation 7 shows that the stronger fluxes (smaller fractional residence times) make the contributions of the weaker fluxes insignificant. 

Sieye no. Particle diameter, microns Weight fraction, Residence time, e/e... 

To show that residence time in mass fraction space obeys a linear mixing law, the traditional definition for t written in terms of volumetric flow rate is no longer suitable. Instead, we define, in an analogous fashion, the equivalent mass fraction residence time for a reactor i as follows ... 

Next, the average mass fraction residence time a is defined as the total reactor volume divided by the total mass flow,... 

Develop an expression for the mass fraction residence time a in terms of standard volumetric residence time t. 

Similar to that given in the development for the PFR expression, r,(z) = r,(C(z)) which is still the molar reaction rate now expressed in terms of the mass fi action vector z instead of the concentration vector C. Also, the mass fraction residence time a may be substituted in to give... 

In order to compute ARs in mass fraction residence time space, we must also be able to express o as a component in the rate vector. From the definition of reaction rate, we know... 

From this expression, we observe that when constructions involving o are carried out, the component corresponding to mass fraction residence time is equal to one. That is, dtr/do- = 1. This is analogous to residence time constructions involving concentration, where the analogous component in the molar rate vector is also one (dv/dr = 1)... 

CO mass fraction predicted by the AR is approximately 0.21. This value is achievable for a mass fraction residence time of approximately 0.5m. h/g as predicted by the AR constmction. Figure 9.9(b) compares the size of the AR in relation to the stoichiometric subspace. 

It is generally true that the length of the simulation needs to be greater to obtain statistically converged results for a potential of mean force than for a free energy difference, since fractional residence times are collected for each small interval. Regions of space with small relative probabilities are prone to have poorly determined potentials of mean force. 

Olefins are produced primarily by thermal cracking of a hydrocarbon feedstock which takes place at low residence time in the presence of steam in the tubes of a furnace. In the United States, natural gas Hquids derived from natural gas processing, primarily ethane [74-84-0] and propane [74-98-6] have been the dominant feedstock for olefins plants, accounting for about 50 to 70% of ethylene production. Most of the remainder has been based on cracking naphtha or gas oil hydrocarbon streams which are derived from cmde oil. Naphtha is a hydrocarbon fraction boiling between 40 and 170°C, whereas the gas oil fraction bods between about 310 and 490°C. These feedstocks, which have been used primarily by producers with refinery affiliations, account for most of the remainder of olefins production. In addition a substantial amount of propylene and a small amount of ethylene ate recovered from waste gases produced in petroleum refineries. 

Oxidation of cumene to cumene hydroperoxide is usually achieved in three to four oxidizers in series, where the fractional conversion is about the same for each reactor. Fresh cumene and recycled cumene are fed to the first reactor. Air is bubbled in at the bottom of the reactor and leaves at the top of each reactor. The oxidizers are operated at low to moderate pressure. Due to the exothermic nature of the oxidation reaction, heat is generated and must be removed by external cooling. A portion of cumene reacts to form dimethylbenzyl alcohol and acetophenone. Methanol is formed in the acetophenone reaction and is further oxidized to formaldehyde and formic acid. A small amount of water is also formed by the various reactions. The selectivity of the oxidation reaction is a function of oxidation conditions temperature, conversion level, residence time, and oxygen partial pressure. Typical commercial yield of cumene hydroperoxide is about 95 mol % in the oxidizers. The reaction effluent is stripped off unreacted cumene which is then recycled as feedstock. Spent air from the oxidizers is treated to recover 99.99% of the cumene and other volatile organic compounds. 

These design fundamentals result in the requirement that space velocity, effective space—time, fraction of bubble gas exchanged with the emulsion gas, bubble residence time, bed expansion relative to settled bed height, and length-to-diameter ratio be held constant. Effective space—time, the product of bubble residence time and fraction of bubble gas exchanged, accounts for the reduction in gas residence time because of the rapid ascent of bubbles, and thereby for the lower conversions compared with a fixed bed with equal gas flow rates and catalyst weights. 

Equation 6 relates the catalytic coke yield (as a fraction of the feed) to the delta coke, to the conversion, and to the catalyst residence time. 

FIG. 7-3 Concentration profiles in fiatch and continuous flow a) fiatch time profile, (h) semifiatcli time profile, (c) five-stage distance profile, (d) tubular flow distance profile, (e) residence time distributions in single, five-stage, and PFR the shaded area represents the fraction of the feed that has a residence time between the indicated abscissas. 

Both types of coalescence can be important in the foam separations characterized by low gas flow rate, such as batchwise ion flotation producing a scum-bearing froth of comparatively long residence time. On the other hand, with the relatively higher gas flow rate of foam fractionation, the residence time may be too short for the first type to be important, and if the foam is sufficiently stable, even the second type of coalescence may be unimportant. 

Residence time distiabution (RTD) In the case of elutriation of tracer from a vessel that contained an initial average concentration the area under a plot of E t) = between the ordinates at ti and to is the fraction of the... 

FIG. 23-7 Imp ulse and step inputs and responses. Typical, PFR and CSTR. (a) Experiment with impulse input of tracer, (h) Typical behavior area between ordinates at tg and ty equals the fraction of the tracer with residence time in that range, (c) Plug flow behavior all molecules have the same residence time, (d) Completely mixed vessel residence times range between zero and infinity, e) Experiment with step input of tracer initial concentration zero. (/) Typical behavior fraction with ages between and ty equals the difference between the ordinates, h — a. (g) Plug flow behavior zero response until t =t has elapsed, then constant concentration Cy. (h) Completely mixed behavior response begins at once, and ultimately reaches feed concentration. 

Limiting flow rates are hsted in Table 23-16. The residence times of the combined fluids are figured for 50 atm (735 psi), 400°C (752°F), and a fraction free volume between particles of 0.4. In a 20-m (66-ft) depth, accordingly, the contact times range from 6.9 to 960 s in commercial units. In pilot units the packing depth is reduced to make the contact times about the same. 

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