Streaming term

It is less forbidding than it looks. The first three lines are the linearized Boltzmann operator acting on the factor u(rin the factorial cumulant the next three lines are the same operator acting on the factor u(r2,p2) lines seven and eight represent the sources of the fluctuations and on the last line the flow terms have been added for both factors. The equation has therefore the general form (VIII.6.8), when A is identified with the linearized Boltzmann operator including the streaming term. 

Reference [51] proposes to split the flux term from the Fokker-Planck equation into two terms one accounting for the dynamics of a single neuron independently of all other neurons in the population (the streaming term) and another including the interactions of a neuron with the other neurons in the network (the interaction term). Before going further, let us denote by w the state of a neuron, a vector in the general case. The proposed general expression for these two fluxes is [51] ... 

QUICKEST QUICK with Estimated Streaming Terms... 

For clarity the streaming term for subsystem 1 has been rewritten with respect to the total V (obviously V2l x =0). If the equilibrium is required to be independent of the interaction energy, that is. 

In the kinetic equation, one might have expected the streaming term to contain... 

The free streaming term can be written as the difference between the number of particles entering and leaving the small region in time 5t. Consider, for example, a cubic cell and look at the faces perpendicular to the x-axis. The flow of particles across the faces at x - and at x -i- is... 

O,. The streaming terms (la) are the same in all of them. However, the transference fimction S( c, E E, SI —> 2) becomes a four by four matrix, with rows and columns labeled by 0, x, y, z. The 0-0 component of this matrix has the same structure as the S of (lb) in particular it depends in an isotropic medium only via the scalar product 2 S2 on the velocity-directions before and after the collision process. The O-o component of the matrix S is also a function of 2 2, multiplied, however, with the x component of the vector product 2x 2 and the x-0 component has the same form. Similar statements apply to the 0-y, 0-2,2/-0, 2-0 components of the matrix S. Finally, the ij components (where i and j may be x, y, or z) contain five terms ... 

The principal difficulty in solving the Boltzmann equation lies in the analytically intractable collision term. For small disturbances from equilibrium, the collision term may be linearized. Another approach is the calculation of transfer processes about a particle using a relaxation model for the collision term. It would be expected that such models would be most successful in near-free-molecular conditions where the "free-streaming" terms are much more important than collisions between host-gas molecules. The so-called BGK model is perhaps the most widely applied of these models [2.5,6]. 

Of = There are four Boltzmann equations for the four quantities Oq, O, O, Oj,. The streaming terms (la) are the same in all of them. However, the transference function E - E, 2) becomes a four by four matrix, with rows and columns... 

Gas-liquid chromatography is a form of column chromatography where the stationary phase is a non-volatile liquid. The stationary phase here is known as the liquid phase. This phase is dispersed over a surface of an inert solid support. The solid support which is coated on to the inside surface of a long column is inert to the sample components and does not react with them in any way it merely acts to hold the liquid phase in a stable dispersed form so as to present a large contact area. It is the liquid phase that Interacts with sample components. A gas stream termed as the carrier gas flows continuously through this column at a flow rate which is controlled (Figure 11.7). When a small quantity of the volatile sample is introduced into the gas, the gas promptly carries it on to the column (hence the name carrier gas"). In the... 

The PFD for the toluene hydrodealkylation process (Figure 1.51 represents the battery-limits plant. The equipment necessary to produce the various service or utility streams, which are used in the process and are necessary for the plant to operate, are not shown on the PFD. However, the utility streams such as cooling water and steam for heating are shown on the PFD. These streams, termed utilities, are necessary for the control of stream temperatures as required by the process. These utilities can be supplied in a number of ways. 

Whether heat integration is restricted to the separation system or allowed with the rest of the process, integration always benefits from colder reboiler streams and hotter condenser streams. This point is dealt with in more general terms in Chap. 12. In addition, when column pressures are allowed to vary, columns with smaller temperature differences are easier to integrate, since smaller changes in pressure are required to achieve suitable integration. This second point is explained in more detail in Chap. 14. 

The analysis of the heat exchanger network first identifies sources of heat (termed hot streams) and sinks (termed cold streams) from the material and energy balance. Consider first a very simple problem with just one hot stream (heat source) and one cold stream (heat sink). The initial temperature (termed supply temperature), final temperature (termed target temperature), and enthalpy change of both streams are given in Table 6.1. 

In this section we describe hydrocarbon processing in preparation for evacuation, either from a production platform or land based facilities. In simple terms this means splitting the hydrocarbon well stream into liquid and vapour phases and treating each phase so... 

A gas/gas heat exchanger at a refinery had a known leakage, which it for technical reasons had been impossible to repair completely. The objective of the survey was to determine the leakage size in terms of the percentage of process stream crossing it. 

The size of the droplets formed in an aerosol has been examined for a range of conditions important in ICP/MS and can be predicted from an experimentally determined empirical formula (Figure 19.6). Of the two terms in the formula, the first is most important, except at very low relative flow rates. At low relative velocity of liquid and gas, simple droplet formation is observed, but as the relative velocity increases, the stream of liquid begins to flutter and to break apart into long thinner streamlets, which then break into droplets. At even higher relative velocity, the liquid surface is stripped off, and the thin films so-formed are broken down into... 

A stream of a liquid solution can be broken up into a spray of fine drops from which, under the action of aligned nozzles (skimmers) and vacuum regions, the solvent is removed to leave a beam of solute molecules, ready for ionization. The collimation of the initial spray into a linearly directed assembly of droplets, which become clusters and then single molecules, gives rise to the term particle beam interface. 

Classically, aerosols are particles or droplets that range from about 0.15 to 5 p.m ia size and are suspended or dispersed ia a gaseous medium such as air. However, the term aerosol, as used ia this discussion, identifies a large number of products which are pressure-dispensed as a Hquid or semisohd stream, a mist, a fairly dry to wet spray, a powder, or even a foam. This definition of aerosol focuses on the container and the method of dispensiag, rather than on the form of the product. 

Other Centrifugal Collectors. Cyclones and modified centrifugal collectors are often used to remove entrained Hquids from a gas stream. Cyclones for this purpose have been described (167—169). The rotary stream dust separator (170,171), a newer dry centrifugal collector with improved collection efficiency on particles down to 1—2 pm, is considered more expensive and hence has been found less attractive than cyclones unless improved collection in the 2—10-pm particle range is a necessity. A number of inertial centrifugal force devices as well as some others termed dynamic collectors have been described in the Hterature (170). 

Referring to Figure 2, by considering solute mass balances over n, (n — 1),. .. 2, 1 units in turn and eliminating intermediate solute mass fractions and flow rates, the amount of solute associated with the leached sohd may be calculated in terms of the composition of the sohd and solvent streams fed to the system. The resulting equation is (2)... 

The term pressure drop usually refers to the pressure loss that is not recoverable in the circuit, and it is lost energy that is dissipated into the fluid stream in the form of heat energy. The pressure drop in a flow circuit is associated with various forms of energy dissipation owing to friction, change in flow area, flow turning, and others ... 

BulkSepa.ra.tlon, The adsorptive separation of process streams into two or more main components is termed bulk separation (see Fig. 12). The development of processes and products is complex. Consequently, these processes are proprietary and are purchased as a complete package under licensing agreements. High purities and yields can be achieved. 

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