Momentum transfer conservation

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

Tlie equation of momentum transfer - more commonly called die equation of motion - can be derived from niomentmii consideradons by applying a momentum balance on a rate basis. The total monientmn witliin a system is uiicluinged by an c.xchaiige of momentum between two or more masses of the system. This is known as die principle or law of conservation of monientmn. This differendal equation describes the velocity distribution and pressure drop in a moving fluid. 

Given that a collision takes place, the nature of the momentum transfer between the cells must be specified. This should be done in such a way that the total momentum and kinetic energy on the double cell are conserved. There are many ways to do this. A multiparticle collision event may be carried out on all particles in the pair of cells. Alternatively, a hard sphere collision can be mimicked by exchanging the component of the mean velocities of the two cells along da,... 

Just as diffusive momentum transfer depends on a transport property of the fluid called viscosity, diffusive heat transfer depends on a transport property called thermal conductivity. This section provides a brief discussion on the functional forms of thermal conductivity, with the intent of facilitating the understanding of the heat-transfer discussions in the subsequent sections on the conservation of energy. 

In addition to energy, the semiconductor band gap is characterized by whether or not transfer of an electron from the valence band to the conduction band involves changing the angular momentum of the electron. Since photons do not have angular momentum, they can only carry out transitions in which the electron angular momentum is conserved. These are known as direct transitions. Momentum-changing transitions are quantum-mechanically forbidden and are termed indirect (see Table 28.1). These transitions come about by coupling... 

The quantity K, the momentum transferred to the molecule, is, from the preceding paragraphs, a fundamental quantity in the theory of electron scattering. It may be obtained from purely kinematic considerations. Consider Fig. 2a (i.e., Fig. 1 a of Hamnett et al.23). If k, and ky are the initial and scattered momenta of the incident electron, we have, from conservation of momentum,... 

Of course, predictable differences between He(2 5) and He(23S) occur regarding the polarization of Penning ions and electrons caused by total spin conservation. The fact that a component of spin angular momentum is conserved in Pgl has been demonstrated by the observation of the transfer of spin polarization from optically pumped He(23S) atoms to the Penning ions of cadmium, zinc, and strontium.72 The polarization was detected by measuring the polarization of the light emitted from the excited 2Ds/2 ions in 2Ds/2- 2D2/2 transitions. If a component of spin angular momentum is conserved, we may write the Pgl process as... 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

In direct gap GaAs, an excited electron at the bottom of the conduction band can relax spontaneously back into a hole in the valence band by emitting a photon at the band gap energy. This electron-hole radiative recombination process can only occur in Si if momentum is conserved, i.e., the excited electron wave vector must be reduced to zero. This, in pure Si, occurs via the transfer of momentum to a phonon that is created with equal and opposite wave vector to that of the initial state in the conduction band. Such a three-body process is quite inefficient compared with direct gap recombination.1218 This is why Si is such a poor light emitter. 

The molecular beam deflection method is shown schematically in Figure 3-17 (Buck et al. 1985). It is based on momentum transfer between clusters entrained in a molecular beam and rare gas atoms which are the constituents of a second molecular beam at 90° to the cluster beam. Collisions between the rare gas atoms and the clusters under single-collision conditions deflect a small percentage of the clusters from their original path. The maximum deflection angle depends on the mass of the cluster. For example, binary clusters may be deflected into a broad range of angles with a well defined upper limit set by the momentum conservation... 

Thus, the presence of the fluctuating turbulent velocity components causes the momentum transfer rate to be different from p X (mean velocity)2 x dA. But in applying the momentum conservation principle to the control volume the presence of additional momentum transfer is the equivalent of an additional force on the face of the control volume in the opposite direction to the momentum transfer. Thus, the additional momentum transfer due to the fluctuating velocity leads to an equivalent stress, i.e., force per unit area, of value pu 2 and this is what is termed the turbulent stress on the face. 

The flow of materials is accounted for with two balances conservation of mass and conservation of momentum transfer. The most important is a momentum balance, which is also called the equation of motion. The mass balance (also called the continuity equation) makes sure that mass is conserved. 

The elementary modeling of the diffusion force is outlined in the sequel. For dilute multicomponent mixtures only elastic binary collisions are considered and out of these merely the unlike-molecule collisions result in a net transfer of momentum from one species to another. The overall momentum is conserved since all the collisions are assumed to be elastic (see sect 2.4.2), hence the net force acting on species s from r per unit volume, equals the momentum transferred from r to s per unit time and unit volume [77] (sect 4-2). The diffusion force yields ... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

Analogies Between Mass, Heat and Momentum Transfer Fluxes A comparison of the partial differential equations for the conservation of heat, mass and momentum in a turbulent flow field (5.240), (5.241) and (5.242) shows that the equations are mathematically similar provided that the pressure term in the momentum equation is negligible [135]. If the corresponding boundary contitions are similar too, the normalized solution of these equations will have the same form. 

As mentioned above, we observe a low energy threshold in the electronic energy loss of the projectile. The explanation of this effect is simple and goes as follows. For a binary collision, the minimum momentum transferred during a collision, obtained through energy conservation, is given by [55]... 

If the equilibrium of a semiconductor is disturbed by excitation of an electron from the valence to the conduction band, the system tends to return to its equilibrium state. Various recombination processes are illustrated in Fig. 1.16. For example, the electron may directly recombine with a hole. The excess energy may be transmitted by emission of a photon (radiative process) or the recombination may occur in a radiationless fashion. TTie energy may also be transferred to another free electron or hole (Auger process). Radiative processes associated with direct electron-hole recombination occur mainly in semiconductors with a direct bandgap, because the momentum is conserved (see also Section 1.2). In this case, the corresponding emission occurs at a high quantum yield. The recombination rate is given by... 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

As can be seen from Eq. (5.100), the virtual-mass force reduces the drag and lift forces by a factor of 1 /y. The buoyancy force is not modified because we have chosen to define it in terms of the effective volume Vpy. We remind the reader that the mesoscale acceleration model for the fluid seen by the particle A j must be consistent with the mesoscale model for the particle phase A p in order to ensure that the overall system conserves momentum at the mesoscale. (See Section 4.3.8 for more details.) As discussed near Eq. (5.14) on page 144, this is accomplished in the single-particle model by constraining the model for Apf given the model for Afp (which is derived from the force terms introduced in this section). Thus, as in Eqs. (5.98) and (5.99), it is not necessary to derive separate models for the momentum-transfer terms appearing in Apf. 

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