Momentum transport molecular

The present approach to the prediction of thermal transport in turbulent flow neglects the effect of thermal flux and temperature distribution upon the relationship of thermal to momentum transport. The influence of the temperature variation upon the important molecular properties of the fluid in both momentum and thermal transport may be taken into account without difficulty if such refinement is necessary. 

When the gas density between two plates at different temperatures is such that the mean free path of the gas is much greater than the distance between the plates, the transport of thermal energy is directly by molecular impacts upon the plates. This process can be analyzed by following the procedure used for momentum transport at low densities (Sec. VIII.5). 

The equation for the momentum transport in vectorial form, gives (by particularization) the famous Navier-Stokes equation. This equation is obtained considering the conservation law of the property of movement quantity in the differential form P = mw. At the same time, if we consider the expression of the transport vector Jt = f + w(pw) and that the molecular momentum generation rate is given with the help of one external force F, which is active in the balance point, the par-d(pw)... 

The ratio between both fluxes shows that the Prandtl number is an index giving the relative quantity of the momentum transported by the molecular mechanism and of the heat transported by the same mechanism at the interface ... 

The Schmidt number is the mass transfer analogue of the Prandtl number. Indeed, by analogy, we can note that the Schmidt number is a measure characterizing the ratio between the quantity of the momentum transported to the interface by the molecular mechanism and the quantity of species A transported to the interface by the same mechanism. Equation (6.171) shows this statement ... 

The work of Nee and Kovasznay (73) and, more recently, of Nee 74) proposes a much simpler approach for two-dimensional flows, one in which a single transport equation is written for the full shear viscosity (molecular plus turbulent). Much of the simplicity of this approach is gained by introducing postulated relationships for the production and decay of viscosity. Although certain limitations are clearly inherent in this approach, the results of various tests suggest that the model may be sufficiently detailed to account for the most relevant mechanisms of the turbulent momentum transport. 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

One approach to deriving correlations for mass transfer coefficients in process systems is to generate experimental data in momentum transport studies. In this approach, it is assumed that both molecular and eddy diffusions play a role in the intermediate region. Then at any distance y from the wall, the rate of mass transfer can be expressed as a function of both the molecular and eddy diffusivities. However, applications of these models rely on a knowledge of the eddy diffusivity, Ed, as a function of y, a relationship that is usually inferred from the experimental data [7-10]. There the eddy diffusivity can be inferred from the eddy viscosity by similarity arguments. A substantial amount of published works is along this line [11-13]. 

Now let s consider the molecular transport of momentum. The molecular mechanism is given by the stress tensor or molecular momentum flux tensor, r. Each element Ty can be interpreted as the component of momentum flux transfer in the direction. We are therefore interested in the terms tix- The rate at which the x component of momentum enters the volume element at face x is XxxAyAx Ij, the rate at which it leaves at face x + Ax is XxxAyAx i+ax, and the rate at which it enters at face y is TyxAxAz y. The net molecular contribution is therefore... 

Newton s viscosity law 5. Alternative formulation Molecular momentum transport. dv, T/A = v=-p— f/Ax -yd(P A) Velocity Momentum concentration... 

The field of transport phenomena traditionally encompasses the subjects of momentum transport (viscous flow), energy transport (heat conduction, convection, and radiation), and mass transport (diffusion). In this section the media in which the transport occurs is regarded as continua however, some molecular explanations are discussed. The continuum approach is of more immediate interest to engineers, but both approaches are required to thoroughly master the subject. The current emphasis in engineering education is on understanding basic physical principles versus blind use of empiricism. Consequently, it is imperative that the reader seek further edification in classical transport phenomena... 

---