Diffusion momentum transfer

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

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. 

The entropy equation can now be used to express the Clausius form of the second law of thermodynamics for open flow systems (e.g., [7] [146], p. 126). The inequality expresses that irreversible phenomena (diffusive momentum transfer, energy transfer, mass transfer, and chemical reactions) lead to entropy production (i.e., energy dissipation) ... 

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 ... 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

These are the flux relations associated with the dusty gas model. As explained above, they would be expected to predict only the diffusive contributions to the flux vectors, so they should be compared with equations (2.25) obtained from simple momentum transfer arguments. Equations (3,16) are then seen to be just the obvious vector generalization of the scalar equations (2.25), so the dusty gas model provides justification for the simple procedure of adding momentum transfer rates. 

In addition to the Burke and Schumann model (34) and the Displacement Distance theory, a comprehensive laminar diffusion flame theory can be written using the equations of conservation of species, energy, and momentum, including diffusion, heat transfer, and chemical reaction. 

An analogy exists between mass transfer (which depends on the diffusion coefficient) and momentum transfer between the sliding hquid layers (which depends on the kinematic viscosity). Calculations show that the ratio of thicknesses of the diffnsion and boundary layer can be written as... 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

Figure 13a shows the contribution of translational diffusion. The translational diffusion only describes the experimental data for the smaller momentum transfer Q = 0.037 A. Figure 13b presents S(Q,t), including the first mode. Obviously, the long-time behavior of the structure factor is now already adequately represented, whereas for shorter times the chain apparently relaxes much faster than calculated. 

It should be highlighted that equation (47) holds for solid particles. In the case of liquid particles, e.g. with emulsions, the convective diffusion process is very different due to interfacial momentum transfer which gives rise to a different velocity profile. Consequently, convective diffusion to/from a liquid particle is more effective than that for a solid particle. Starting again from equation (43),... 

If the fluid in the pipe is in turbulent flow, the effects of molecular diffusion will be supplemented by the action of the turbulent eddies, and a much higher rate of transfer of material will occur within the fluid. Because the turbulent eddies also give rise to momentum transfer, the velocity profile is much flatter and the dispersion due to the effects of the different velocities of the fluid elements will be correspondingly less. 

The data follow the scaling prediction with satisfying precision. The small deviations are related to the translational diffusion of the chains. This becomes evident from Fig. 3.7, where the obtained relaxation rates IXQ) are plotted versus Q in a double logarithmic fashion. The dashed line gives the Rouse prediction JoeWhile at larger momentum transfers the experimental results follow this prediction very well, towards lower Q a systematic relative increase of the relaxation rate is observed. Including translational diffusion, we have ... 

At low momentum transfer A2 describes the translational Rouse diffusion coefficient of the whole diblock, considering N l-f) segments exerting the friction Q and A[fsegments exerting the friction In the high Q hmit, RPA predicts... 

The classical form of the thermodynamic force of diffusion (Xj = — grad /I , /Ij electrochemical potential) can be extended if the influence of diffusion on the momentum transfer is carefully taken into account. This results if the balance equation of momentum for many component systems is formulated as the sum of contributions of all components. The final result is as follows ... 

This value, as expected, relates to the maximum possible momentum transferred from the photon to the microparticle, even if some values of the diffusion angle obviously have a very low or even zero probability. As stated before, this formula for the uncertainty in the momentum of the small particle M after the measurement is precisely the same for both microscopes. In either case, it is necessary to keep in mind that, in this step of the measuring process of the error of the two conjugated observables, the interacting photon behaves like a corpuscle. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

With respect to the physical processes, boundaries can be subdivided into just three classes. The distinction will be made according to the nature of the resistance to mass transfer across the boundary. We must recognize that this transfer is usually mediated by random motions. Thus, the resistance is like the inverse of a generalized diffusivity or transfer velocity, since both these quantities have the function of a conductivity (of mass, heat, momentum, etc.). For simplicity, the following discussion will be focused on the diffusion model (Eq. 18-6), although everything which will be said can also be adapted to the transfer model (Eq. 18-4). 

In the seventies, the growing interest in global geochemical cycles and in the fate of man-made pollutants in the environment triggered numerous studies of air-water exchange in natural systems, especially between the ocean and the atmosphere. In micrometeorology the study of heat and momentum transfer at water surfaces led to the development of detailed models of the structure of turbulence and momentum transfer close to the interface. The best-known outcome of these efforts, Deacon s (1977) boundary layer model, is similar to Whitman s film model. Yet, Deacon replaced the step-like drop in diffusivity (see Fig. 19.8a) by a continuous profile as shown in Fig. 19.8 b. As a result the transfer velocity loses the simple form of Eq. 19-4. Since the turbulence structure close to the interface also depends on the viscosity of the fluid, the model becomes more complex but also more powerful (see below). 

The Debye theory [220] in which a sphere of volume V and radius a rotates in a liquid of coefficient of viscosity t has already been mentioned. There is angular momentum transfer across the sphere—liquid interface that is, the liquid sticks to the sphere so that the velocity of the sphere and liquid are identical at the sphere s surface. Solution of the rotational diffusion equation... 

Liquids with low viscosity or large 3 (high density or efficient momentum transfer across the boundary layer) have a rotational diffusion coefficient close to that of the Debye equation [220], eqn. (110). For viscous liquids, the rotational diffusion coefficient tends to saturate to a viscosity-independent value. Tanabe [235] has found perdeuterobenzene rotational diffusion to be well described by the Hynes et al. theory [221, 222]. 

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