Mass transport processes eddy diffusion

During food engineering operations, many fluids deviate from laminar flow when subjected to high shear rates. The resulting turbulent flow gives rise to an apparent increase in viscosity as the shear rate increases in laminar flow, i.e., shear stress = viscosity x shear rate. In turbulent flow, it would appear that total shear stress = (laminar stress + turbulent stress) x shear rate. The most important part of turbulent stress is related to the eddies diffusivity of momentum. This can be recognized as the atomic-scale mechanism of energy conversion and its redistribution to the dynamics of mass transport processes, responsible for the spatial and temporal evolution of the food system. 

In seawater, physical processes that transport water can also cause mass fluxes and, hence, are another means by which the salinity of seawater can be conservatively altered. The physical processes responsible for water movement within the ocean are turbulent mixing and water-mass advection. Turbulent mixing has been observed to follow Pick s first law and, hence, is also known as eddy diffusion. The rate at which solutes are transported by turbulent mixing and advection is usually much faster than that of molecular diffusion. Exceptions to this occur in locations where water motion is relatively slow, such as the pore waters of marine sediments. The effects of advection and turbulent mixing on the transport of chemicals are discussed further in Chapter 4. 

The physical transport of mass is essential to many kinetic and d3mamic processes. For example, bubble growth in magma or beer requires mass transfer to bring the gas components to the bubbles radiogenic Ar in a mineral can be lost due to diffusion pollutants in rivers are transported by river flow and diluted by eddy diffusion. Although fluid flow is also important or more important in mass transfer, in this book, we will not deal with fluid flow much because it is the realm of fluid dynamics, not of kinetics. We will focus on diffusive mass transfer, and discuss fluid flow only in relation to diffusion. 

There is a sound experimental basis (B6, LI) for the conclusion that in packed tubes when the Reynolds number is above 100, the transport of both matter and heat transverse to the flow is dominated by eddy diffusive processes. To the extent that these processes dominate, all conserved entities in the fluid diffuse alike, provided the gradients of the concentrations of these entities in amount per unit mass of fluid are used as driving forces for the diffusion. In all that follows, it will be assumed... 

The same physical principles are utilized to develop isotopic models which better account for the transport of air masses at a regional scale, such as done by Fisher (1992) using a regional stable isotope model coupled to a zonally averaged global model. Other authors such as Eriksson (1965) and more recently Hendricks et al. (2000) considered the transport of water both by advective and eddy diffusive processes, the latter inducing less fractionation. 

ANALYSIS NIq are asked to determine the rate of production of methane. We shall use the turbulent eddy diffusivity model to represent the transport processes in the gas phase. The mass fluxes are given by Eq. 10.4.24... 

In this article a simplified mass balance has been used to describe the net transport of sand over an accreting mud bottom. The combination of these two sedimentary processes controls the transition from sand to mud on the floor of the Sound. The distribution of sand may be described with three parameters an advection velocity of sand grains, an eddy-diffusion coefficient for mobile sand, and a rate of accumulation of marine mud. (Only the ratios of these quantities are needed if the distribution is in a steady state.) The motion of sand is thereby represented with both a deterministic part and a statistical part. The net, one-way advection of sand is the result of the superposition of an estuarine circulation on the tidal stream, and unpredictable variations in the rate of sand transport are represented as an eddy-diffusion process. Sand is immobilized when it is incorporated into the permanent deposit of marine mud. 

Tha transport mechanisms of molecular diffusion and mass carried by eddy motion are again assumed edditive although the contribution of the molecular diffusivity term is quite small except in the region nenr a wall where eddy motion is limited. The eddy diffusivity is directly applicable to problems snch as the dispersion of particles or species (pollutants) from a souree into a homogeneously turbulent air stream in which there is little shear stress. The theories developed by Taylor.36 which have been confirmed by a number of experimental investigations, can describe these phenomena. Of more interest in chemical engineering applications is mass transfer from a turbolent fluid to a surface or an interface. In this instance, turbulent motion may he damped oni as the interface is approached aed the contributions of both molecolar and eddy diffusion processes must he considered. To accomplish this. 9ome description of the velocity profile as the interface is approached must be available. 

Movement of a soluble chemical throughout a water body such as a lake or river is governed by thermal, gravitational, or wind-induced convection currents that set up laminar, or nearly frictionless, flows, and also by turbulent effects caused by inhomogeneities at the boundaries of the aqueous phase. In a river, for example, convective flows transport solutes in a nearly uniform, constant-velocity manner near the center of the stream due to the mass motion of the current, but the friction between the water and the bottom also sets up eddies that move parcels of water about in more randomized and less precisely describable patterns where the instantaneous velocity of the fluid fluctuates rapidly over a relatively short spatial distance. The dissolved constituents of the water parcel move with them in a process called eddy diffusion, or eddy dispersion. Horizontal eddy diffusion is often many times faster than vertical diffusion, so that chemicals spread sideways from a point of discharge much faster than perpendicular to it (Thomas, 1990). In a temperature- and density-stratified water body such as a lake or the ocean, movement of water parcels and their associated solutes will be restricted by currents confined to the stratified layers, and rates of exchange of materials between the layers will be slow. 

Introduction. In molecular transport of momentum, heat, or mass there are many similarities, which were pointed out in Chapters 2 to 6. The molecular diffusion equations of Newton for momentum, Fourier for heat, and Fick for mass are very similar and we can say that we have analogies among these three molecular transport processes. There are also similarities in turbulent transport, as discussed in Sections 5.7C and 6.1A, where the flux equations were written using the turbulent eddy momentum diffusivity e, the turbulent eddy thermal diffusivity a, and the turbulent eddy mass diffusivity. However, these similarities are not as well defined mathematically or physically and are more difficult to relate to each other. 

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]. 

In the turbulent mass transfer process, the velocity gradient and concentration gradient are established as well as the mJc gradient. The transport of u d is implemented by the turbulent fluid flow and the fluctuated mass flux diffusion. As the velocity eddy, which is the elements of turbulent flow, is the carrier of m-c, the transport of mJc also follows the pattern of the velocity eddy flow and the fluctuated diffusion. 

Air-water transfer rate of chemicals is dependent upon the rate coefficient and the equilibrium that the concentrations in each phase are moving towards. In environmental air-water mass transfer, the flow is generally turbulent in both phases. However, there is no turbulence across the interface in the diffusive sublayer, and the problem becomes one of rate of diffusion. Temporal mean turbulence quantities, such as eddy diffusion coefficient, provide a semiquantitative description of the flux across the air-water interface, however the unsteady character of turbulence near the diffusive sublayer is cmcial to understanding and characterizing interfacial transport processes. 

Mass transfer can be definnd simply as the movement of any identifiable species from one spatial location to another. Tha mechanism of movement can be macroscopic as in the flow of a fluid in a pipe (convection) or in the mechanical transport of solids by a conveyor belt. In addition, the transport of a panicolar species may be the result of madom molecular motion (molecular diffusion) or randum microscopic fluid motion (eddy or turbulent diffusion) in the presence of a composition gradient within a phase. This chapter is concerned primarily with mass transfer owing to molecular or microscopic processes. 

Reynolds argued that mass or heat transport into a flowing fluid must involve two simultaneous processes 1. the natural diffusion of the fluid when at rest [and] 2. the eddies caused by visible motion which mixes the fluid up and brings fresh particles into contact with the surface. He went on The first of these causes is independent of the velocity of the fluid [but] the second cause, the effect of eddies, arises entirely from the motion of the fluid. Note that Reynolds implies that any flowing fluid contains eddies. Nine years later, Reynolds discovered the distinction between laminar flow and turbulent flow, and that eddies occur only in the latter. 

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