Momentum exchange coefficient

From Tolmin s theory and experimental data (e.g., Reichardtthe relationship between velocity profile and temperature profile in the jet cross-section can be expressed using an overall turbulent Prandtl number Pr = v /a, where Vf is a turbulent momentum exchange coefficient and a, is a turbulent heat exchange coefficient ... 

The stress tensor depends on the type of flow. In laminar flow, it is proportional to the viscosity, which depends on the temperature, while in turbulent flow, it depends on the momentum exchange coefficient e. That is ... 

With the help of the dispersion model one can readily calculate energy and momentum exchange coefficients and aerodynamical characteristics of different objects. 

The transport of a sub-critical Lennard-Jones fluid in a cylindrical mesopore is investigated here, using a combination of equilibrium and non-equilibrium as well as dual control volume grand canonical molecular dynamics methods. It is shown that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. It is also demonstrated that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities. 

The first term on the right-hand side represents momentum exchange between solid phases I and s and Kis is the solid-solid exchange coefficient. The last term represent additional shear stresses, which appear in granular flows (due to particle translation and collisions). Expressions for solids pressure, solids viscosity (shear and bulk) and solid-solid exchange coefficients are derived from the kinetic theory of granular flows. 

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

The above theoretical analysis of penetrable roughnesses and their interaction with the flow was based on the introduction of a distributed momentum sink (i.e. the force) and heat and mass sources, and was sufficient for discovering some important features of the phenomenon under consideration. It was a simplified consideration with mainly constant coefficients. However, in order to be applied to real environmental and engineering problems, realistic exchange coefficients are to be known. 

Experiments using porous materials with characteristic average density p o 20 to 200 kg/m and porosity coefficients tto=Vg/v =0.95-0.98 (the diameter of pores being 10 to 10 m ) show that momentum exchange between phases increases the width of the shock wave front to 30-50 10 m. Accordingly, the dynamic relaxation time was 1.5 10 s (the shock wave speed in the material being 2-lO m/s). 

The moleculai beam experiments on rainbow scattering are rather expensive. Therefore the efficiency of the gas-surface interaction is described frequently in terms of the exchange coefficients of momentum... 

Let us look into some average parameter for defining gas-solid interaction from macroscopic point of view. Tangential momentum accommodation coefficient (momentum exchange of gas molecules with the solid surface defined as... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

While electrical conductivity, diffusion coefficients, and shear viscosity are determined by weak perturbations of the fundamental diffu-sional motions, thermal conductivity is dominated by the vibrational motions of ions. Heat can be transmitted through material substances without any bulk flow or long-range diffusion occurring, simply by the exchange of momentum via collisions of particles. It is for this reason that in liquids in which the rate constants for viscous flow and electrical conductivity are highly temperature dependent, the thermal conductivity remains essentially the same at lower as at much higher temperatures and more fluid conditions. 

The axial Poiseuille flow occurs when 6 = n/2, with x taking the role of z, and y taking the role of De/2 — r. The metric coefficient reduces to h = De/2 — y = r. The expected axisymmetric momentum equations for axial Poiseuille flow can be recovered by substituting f — ur and carrying out the independent variable transformation to exchange r for v. The chain rule for the independent variable transformation provides that... 

In process operations, simultaneous transfer of momentum, heat, and mass occur within the walls of the equipment vessels and exchangers. Transfer processes usually take place with turbulent flow, under forced convection, with or without radiation heat transfer. One of the purposes of engineering science is to provide measurements, interpretations and theories which are useful in the design of equipment and processes, in terms of the residence time required in a given process apparatus. This is why we are concerned here with the coefficients of the governing rate laws that permit such design calculations. 

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