Mixture momentum

One-dimensional flow models are adopted in the early stages of model development for predicting the solids holdup and pressure drop in the riser. These models consider the steady flow of a uniform suspension. Four differential equations, including the gas continuity equation, solids phase continuity equation, gas-solid mixture momentum equation, and solids phase momentum equation, are used to describe the flow dynamics. The formulation of the solids phase momentum equation varies with the models employed [e.g., Arastoopour and Gidaspow, 1979 Gidaspow, 1994], The one-dimensional model does not simulate the prevailing characteristics of radial nonhomogeneity in the riser. Thus, two- or three-dimensional models are required. 

Equation(25) involves the continuity equations for the gas phase and the mixture, two mixture momentum balance equations for the coordinate directions ( ), and the energy equation. 

The mixture momentum associated with an Eulerian control volume E, at... 

The mixture momentum equation can be deduced in a similar way from the sum of the momentum balances of the individual phases ... 

Using the given definition of the diffusion velocity (3.438), the Favre form of the mixture density (3.418) and the Favre form of the mixture velocity (3.421), the second term in the mixture momentum balance (3.437) can be reformulated ... 

The last term on the right hand side of the mixture momentum equation (3.440) accounts for the influence of the surface tension force F/ on the mixture... 

The mixture continuity equation given in Eq. (4.69) can be combined with a mixture momentum equation found by adding together Eqs. (4.85) and (4.92) ... 

At the interface the total mass is balanced in accordance with the jump mass condition (3.441), thus the RHS of (3.443) must vanish. Hence, by use of the Favre averaged form of (3.425) and (3.428) we obtain the continuity equation for the mixture (3 431) The mixture momentum equation can be deduced in a similar way from the sum of the momentum balances of the individual phases ... 

The mixture model is based on the solution of a single mixture momentum equation for all phases, which significantly reduces computational effort. Saalbach and Hunze (2008) used the Algebraic-Slip-Mixture model to simulate the interactions of three phases water, air and sludge in pilot and full-scale plants. The mixture model could account for the slip velocities of the dispersed phase and the continuous phase relative to the mixture. 

ATHLET offers the possibility of choosing between different models for the simulation of fluid dynamics.In the current released code version, the basic fluid-dynamic option is a five-equation model, with separate conservation equations for liquid and vapour mass and energy, and a mixture momentum equation, accounting for thermal and mechanical non-equilibrium, and including a mixture level tracking capability. 

Another fluid-dynamic option in ATHLET consists of a four-equation model, with balance equations for liquid mass, vapour mass, mixture energy and mixture momentum. It is based on a lumped-parameter approach. The solution variables are the pressure, mass quality and enthalpy of the dominant phase within a control volume, and the mass flow rates at the junctions. The entire range of fluid conditions, from sub-cooled liquid to superheated vapour, including thermodynamic non-equilibrium is taken into account, assuming the non-dominant phase to be at saturation. The option has also a mixture level tracking capability. 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

A good exposition of the momentum transfer arguments is given by Present [9], but for our purpose it is necessary only to quote the result. For a binary mixture this takes the form... 

There are n Stefan-Maxwell relations in an n-component mixture, but they are not independent since each side of (2.16) yields zero on summing over r from 1 to n. Physically this is not surprising, since they describe only momentum exchange between pairs of species, and say nothing about the total momentum of the mixture. In order to complete the determination of the fluxes N.... N the Stefan-Maxwell relations must be supple-I n... 

Ocily n. - 1 of the n equations (4.1) are independent, since both sides vanish on suinming over r, so a further relation between the velocity vectors V is required. It is provided by the overall momentum balance for the mixture, and a well known result of dilute gas kinetic theory shows that this takes the form of the Navier-Stokes equation... 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

When M is an atom the total change in angular momentum for the process M + /zv M+ + e must obey the electric dipole selection mle Af = 1 (see Equation 7.21), but the photoelectron can take away any amount of momentum. If, for example, the electron removed is from a d orbital ( = 2) of M it carries away one or three quanta of angular momentum depending on whether Af = — 1 or +1, respectively. The wave function of a free electron can be described, in general, as a mixture of x, p, d,f,... wave functions but, in this case, the ejected electron has just p and/ character. 

Most of the commercial gas—air premixed burners are basically laminar-dow Bunsen burners and operate at atmospheric pressure. This means that the primary air is induced from the atmosphere by the fuel dow with which it mixes in the burner passage leading to the burner ports, where the mixture is ignited and the dame stabilized. The induced air dow is determined by the fuel dow through momentum exchange and by the position of a shutter or throtde at the air inlet. Hence, the air dow is a function of the fuel velocity as it issues from the orifice or nozzle, or of the fuel supply pressure at the orifice. With a fixed fuel dow rate, the equivalence ratio is adjusted by the shutter, and the resulting induced air dow also determines the total mixture dow rate. 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

Fuel from a fiilly unobstructed jet would be diluted to a level below its lower flammability limit, and the flammable portion of the cloud would be limited to the jet itself. In practice, however, jets are usually somehow obstructed by objects such as the earth s surface, surrounding structures, or equipment. In such cases, a large cloud of flammable mixture will probably develop. Generally, such a cloud will be far from stagnant but rather in recirculating (turbulent) motion driven by the momentum of the jet. 

A term label like for example, is thus no longer strictly meaningful for it implies constant spin- and orbital angular momentum properties (5 = 1, L = 3). One consequence of spin-orbit coupling is a scrambling of the two kinds of angular momentum. So a nominal term may really more properly be described as a mixture of terms of different spin-multiplicity as, for example, in Eq. (4.10). 

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