Phase space velocity

Figure 2 Orbits in phase space (velocity, radius) for cases (c).

In order to account for variable particle numbers, we generalize the collision term iSi to include changes in IVp due to nucleation, aggregation, and breakage. These processes will also require models in order to close Eq. (4.39). This equation can be compared with Eq. (2.16) on page 37, and it can be observed that they have the same general form. However, it is now clear that the GPBE cannot be solved until mesoscale closures are provided for the conditional phase-space velocities Afp)i, (Ap)i, (Gp)i, source term 5i. Note that we have dropped the superscript on the conditional phase-space velocities in Eq. (4.39). Formally, this implies that the definition of (for example) [Pg.113]

This relation is denoting that the rate of change of / for a system of a number of particles moving with the phase space velocity vector (, equals the rate at which / is altered by collisions. 

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

B(A) is the probability of observing the system in state A, and B(B) is the probability of observing state B. In this model, the space is divided exactly into A and B. The dividing hyper-surface between the two is employed in Transition State Theory for rate calculations [19]. The identification of the dividing surface, which is usually assumed to depend on coordinates only, is a non-trivial task. Moreover, in principle, the dividing surface is a function of the whole phase space - coordinates and velocities, and therefore the exact calculation of it can be even more complex. Nevertheless, it is a crucial ingredient of the IVansition State Theory and variants of it. 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

Consider a quantity of some liquid, say, a drop of water, that is composed of N individual molecules. To describe the geometry of this system if we assume the molecules are rigid, each molecule must be described by six numbers three to give its position and three to describe its rotational orientation. This 6N-dimensional space is called phase space. Dynamical calculations must additionally maintain a list of velocities. 

In recent years alkylations have been accompHshed with acidic zeoHte catalysts, most nobably ZSM-5. A ZSM-5 ethylbenzene process was commercialized joiatiy by Mobil Co. and Badger America ia 1976 (24). The vapor-phase reaction occurs at temperatures above 370°C over a fixed bed of catalyst at 1.4—2.8 MPa (200—400 psi) with high ethylene space velocities. A typical molar ethylene to benzene ratio is about 1—1.2. The conversion to ethylbenzene is quantitative. The principal advantages of zeoHte-based routes are easy recovery of products, elimination of corrosive or environmentally unacceptable by-products, high product yields and selectivities, and high process heat recovery (25,26). 

Gas Phase. The gas-phase methanol hydrochlorination process is used more in Europe and Japan than in the United States, though there is a considerable body of Hterature available. The process is typicaHy carried out as foHows vaporized methanol and hydrogen chloride, mixed in equimolar proportions, are preheated to 180—200°C. Reaction occurs on passage through a converter packed with 1.68—2.38 mm (8—12 mesh) alumina gel at ca 350°C. The product gas is cooled, water-scmbbed, and Hquefied. Conversions of over 95% of the methanol are commonly obtained. Garnma-alurnina has been used as a catalyst at 295—340°C to obtain 97.8% yields of methyl chloride (25). Other catalysts may be used, eg, cuprous or zinc chloride on active alumina, carbon, sHica, or pumice (26—30) sHica—aluminas (31,32) zeoHtes (33) attapulgus clay (34) or carbon (35,36). Space velocities of up to 300 h , with volumes of gas at STP per hour per volume catalyst space, are employed. 

Product (raw materials) Type Reactor phase Catalyst T c(2 P, atm Residence time or space velocity Source and page ... 

ALLreviations reactors Latch (B), continuous stirred tank (CST), fixed Led of catalyst (FB), fluidized Led of catalyst (FL), furnace (Furn.), multituLular (MT), semicontinuous stirred tank (SCST), tower (TO), tuLular (TU). Phases liquid (L), gas (G), Loth (LG). Space velocities (hourly) gas (GHSV), liquid (LHSV), weight ( VHSV). Not available, NA. To convert atm to kPa, multiply Ly 101.3. 

Both phases are siibstantiaUy in plug flow. Dispersion measurements of the hquid phase usuaUy report Peclet numbers, Uid /D, less than 0.2. With the usual smaU particles, the waU effect is negligible in commercial vessels of a meter or so in diameter, but may be appreciable in lab units of 50 mm (1.97 in) diameter. Laboratory and commercial units usuaUy are operated at the same space velocity, LHSy but for practical reasons the lengths of lab units may be only 0.1 those of commercial units. 

A completely discrete phase space (i.e. discrete values of lattice-site positions, particle velocities and time, so that particles move from site-to-site and collisions taken within discrete time steps). 

The objective of this phase of the methanation system s operation is to determine space velocity requirements, recycle ratio, and analytical and control systems. 

Ordinarily, the term phase space refers to the conjunction of configuration and momentum space. We use it here for configuration-velocity space. 

If only one type of particle is present, mx = m2 however, the expressions relating the velocities before and after collision do not simplify to any great extent. If several types of particles are present, then there results one Boltzmann equation for the distribution function for each type of particle in each equation, integrals will appear for collisions with each type of particle. That is, if there are P types of particles, numbered i = 1,2,- , P, there are P distribution functions, ft /(r,vt, ), describing the system ftdrdvt is the number of particles of type i in the differential phase space volume around (r,v(). The set of Boltzmann equations for the system would then be ... 

This is the probability of finding particle 1 with coordinate rx and velocity vx (within drx and dVj), particle 2 with coordinate r2 and velocity v2 (within phase space with velocity rather than momentum for convenience since only one type of particle is being considered, this causes no difficulties in Liouville s equation.) The -particle probability distribution function ( < N) is... 

Peculiar particle velocity, 19 Pendulum problem, 382 Periodicity conditions, 377 Perturbed solution, 344 Pessimism-optimism rule, 316 Petermann, A., 723 Peterson, W., 212 Phase plane, 323 "Phase portrait, 336 Phase space, 13 Photons, 547... 

Zabor et al. (Zl) have described studies of the catalytic hydration of propylene under such conditions (temperature 279°C, pressure 3675 psig) that both liquid and vapor phases are present in the packed catalyst bed. Conversions are reported for cocurrent upflow and cocurrent downflow, it being assumed in that paper that the former mode corresponds to bubble flow and the latter to trickle-flow conditions. Trickle flow resulted in the higher conversions, and conversion was influenced by changes in bed height (for unchanged space velocity), in contrast to the case for bubble-flow operation. The differences are assumed to be effects of mass transfer or liquid distribution. 

A vapor phase process for deparaffmization of light gas oils performed by the BP works in this way The gas oil, boiling range 230-320°C, is passed over a 5-A molecular sieve at 320°C and a pressure of 3.6 bar. The space velocity is 0.63 vol liquid gas oil per vol molecular sieve and per hour, [liquid hourly space velocity (lhsv) = 0.63] the adsorption period takes 6 min. Together with the gas oil vapor 120 vol N2 per vol liquid gas oil is led over the sieve as carrier and purge gas. After the adsorption period the loaded molecular sieve is purged at the same temperature with pure N2 for 6 min. Subsequently, the adsorbed /z-alkanes are desorbed by 1 vol liquid /z-pentane per vol molecular sieve and per hour. The /z-pentane is recovered from the /z-alkane//z-pentane mixture by simple distillation [15]. The IsoSiv process of the Union Carbide Corporation works in a similar way [16]. The purity of the isolated /z-alkanes is >98%. 

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