Transport of momentum

If we consider the transport of momentum between a gas and a wall which are in relative motion, there may be two types of effects. One of these is the transport of momentum normal to the surface, which is manifested as a force perpendicular to the surface or a pressure. The other is a tangential force which may be exerted on the wall and is referred to as a viscous drag. 

This flux vector denotes the transport of momentum relative to v in the /-direction. In a 3D system the direction index / takes three different values, so there are a total of three flux vectors associated with momentum transfer. 

Together the three flux vectors constitute a symmetric second-order tensor with nine components. This tensor is usually referred to as the pressure tensor, p. The pressure tensor is expressed by  

The diagonal elements Cf)M represent the normal stresses, in general denoting the sum of the static and the viscous forces per unit area acting on a surface at an instant in time. The non-diagonal elements, denoted by CiCj)M for i j, represent the shear stresses or the viscous shear forces per unit area. 

We define the deviatoric—or viscous stresses as the negative thermal flux, determined by the difference between the thermodynamic pressure and the pressure tensor elements, as follows  

The mean pressure (or pressure) is defined as the mean value of the normal stresses across any three orthogonal planes. The mean pressure, p, is thus one third of the trace of the pressure tensor  

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These operations are characterized by different reaction engineering properties. The transport of momentum, heat, and mass take place by different rates in the different operations, and the yield and selectivity obtained for a given chemical reaction will depend upon the type of operation employed. The operations also differ with respect to more loosely defined characteristics, such as ease of operation, and it can be noted in particular that some operations have been studied with considerably more thoroughness than others, and may consequently be designed with greater accuracy and reliability. 

It is the large scale eddies that are responsible for the very rapid transport of momentum, energy and mass across the whole flow field in turbulent flow, while the smallest eddies and their destruction by viscosity are responsible for the uniformity of properties on the fine scale. Although it is the fluctuations in the flow that promote these high transfer rates, it is... 

The diffusion process in general may be viewed as the model for specific well-defined transport problems. In particle diffusion, one is concerned with the transport of particles through systems of particles in a direction perpendicular to surfaces of constant concentration in a viscous fluid flow, with the transport of momentum by particles in a direction perpendicular to the flow and in electrical conductivity, with the transport of charges by particles in a direction perpendicular to equal-potential surfaces. 

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and diffu-sivity of the medium for both air and water. In air, these numbers are all of the same order of magnitude, meaning that air provides a similar resistance to the transport of momentum, heat, and mass. In fact, there are two dimensionless numbers that will tell us these ratios the Prandtl number (Pr = pCpv/kj = v/a) and the Schmidt number (Sc = v/D). The Prandtl number for air at 20°C is 0.7. The Schmidt number for air is between 0.2 and 2 for helium and hexane, respectively. The magnitude of both of these numbers are on the order of 1, meaning that whether it is momentum transport, heat transport, or mass transport that we are concerned with, the results will be on the same order once the boundary conditions have been made dimensionless. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

Let s consider the fully developed velocity profile in the middle of a wide open channel, with X-, y-, and z-components in the longitudinal, lateral, and vertical directions, respectively. It is fully developed because du/dx is close to zero. The fact that it is a wide channel means that du/dy is also very small in the middle. From equations (5.22) and (5.23), we can see that the turbulent transport of momentum in the x- and... 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

To an engineer working in a chemical or explosives(or ammunition) plant, fluid-mechanics is useful not only in predicting friction losses and interconversions of pressure and velocity, but also in producing analogies among the transport of momentum. 

The foregoing discussion relates to the transport of momentum under steady, uniform conditions in turbulently flowing streams. Such matters are of direct interest to the chemical engineer but usually only as they influence the power requirements for the movement of fluids. A deeper interest exists in prediction of the thermal transport and temperature... 

There are shown in Fig. 19 values of the eddy diffusivity calculated from the measurements by Sherwood (SI6). These data show the same trends as were found in thermal transport, indicating that the values of eddy diffusivity are determined primarily from the transport of momentum for situations where the molecular Schmidt numbers of the components do not differ markedly from each other. 

Most nonequilibrium systems are characterized by variation of velocity, temperature, composition, or electrical potential with position and the consequent transport of momentum, energy, mass, or electric charge. Naturally, transport of two or more of these may occur simultaneously. Attention is focused here, however, on situations where only one transport process occurs and a transport coefficient can be calculated from its measured rate. For example, thermal conductivity can be calculated if the rate of energy transport and the temperature variation in the system are measured. 

The magnetic vector potential AB is identified with the convective transport of momentum by individual preons ... 

In addition to mass and energy, other quantities can also experience transfer. Flowing layers with different flow rates in a convection stream can influence one another. The slower flowing layer acts as a brake on the faster layer, while at the same time the faster layer acts to accelerate the slower one. The cause of this behavior is the inner friction of the liquid appearing as a viscosity difference, which is a consequence of the attractive forces between the molecules. Viscosity can be explained as the transport of momentum. The viscosity of different media can be very different and thus plays an important role in transport processes. 

In this type of three-phase catalytic reactor, centrifugal force is employed to vary the hydrodynamics and transport characteristics of the conventional gas-liquid-solid reactor. Interphase transport of momentum and mass in such a reactor is governed by the centrifugal forces. Dudukovic and co-... 

The heat-conduction equation describes the transport of energy, the viscous-shear equation describes the transport of momentum across fluid layers, and the diffusion law describes the transport of mass. 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

If we now consider an ideal gas made up of point particles, the particles cannot suffer collisions with each other and their velocities in the gas phase cannot change except after collisions with the walls of the container. The total transport of momentum in such a gas is by the individual molecules themselves. Let us calculate for such a gas the average pressure exerted on an element of wall surface dS. 

Fig. VIII.2. Transport of momentum between parallel moving plates.

If now we consider an clement of area AaS in this plane, there will be a flow of faster-moving molecules going through it from below and, under steady conditions, an equal flow of slower-moving molecules going through it from above. The difference between the two flows gives the net transport of momentum across AaS. 

The computer model is based upon a continuum description of fluidization in coal gasification reactors. In general, fluidized flows are dominated by specific physicochemical processes and, hence, require particular theoretical representations. For example, in the heavily loaded gas-particle regime appropriate to fluidization, the solid particles dominate the transport of momentum and energy. This aspect of fluidization is reflected in the mathematical descriptions which have been used in the fluidized bed model. 

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