Fluid systems convective dispersion

The theories that have been employed to derive the macroscopic differerrtial equations that describe solute transport through porous media may be grouped into different classes. The most widely used theory for convection-dispersion of chemicals in porous media is that based on fluid mechanics. In this section we introdnce the relevant concepts on fluid mechanics, porous media, and mass transfer in both fluid systems and porous media. These concepts provide the theoretical fundamentals for the modelling of the dyeing process. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

The disposition of a solute in the fluid as it flows through the system is governed by convection and dispersion. The convection takes place with velocity... 

Mass transfer to a particle in a translational flow, considered in Section 4.4, is a good model for many actual processes in disperse systems in which the velocity of the translational motion of particles relative to fluid plays the main role in convective transfer and the gradient of the nonperturbed velocity field can be neglected. 

G.I. Taylor (1953, 1954) first analyzed the dispersion of one fluid injected into a circular capillary tube in which a second fluid was flowing. He showed that the dispersion could be characterized by an unsteady diffusion process with an effective diffusion coefficient, termed a dispersion coefficient, which is not a physical constant but depends on the flow and its properties. The value of the dispersion coefficient is proportional to the ratio of the axial convection to the radial molecular diffusion that is, it is a measure of the rate at which material will spread out axially in the system. Because of Taylor s contribution to the understanding of the process of miscible dispersion, we shall, as is often done, refer to it as Taylor dispersion. 

Efficient heat and mass transfer in dispersed systems and low temperatures required for drying heat-sensitive materials justify the application of a heat-pump system in conjunction with pneumatic, fluid bed, vibrated bed, conveyor, spin-flash, and similar convection dryers. A simplified layout of a heat-pump fluid bed dryer developed at the Norwegian Institute of Technology (NTNU) and thoroughly tested for various products (Jonassen et al., 1994 Strpmmen and Jonasen, 1996 Alves-FiUio and Str0mmen, 1996) is shown in... 

The first two terms and the last term represent accumulation of energy in the fluid, the particle, and the column wall. The third term is convection of energy while the fourth term is the axial dispersion of energy. The fifth term (first term on right hand side) represents the heat transfer from the column walls. Because industrial scale systems have a small ratio of wall area to column volume, the fifth term is often negligible (the column is adiabatic), and the sixth term is often negligible because the mass of the column wall is small compared to the mass of adsorbent. 

An important feature of all mass transfer operations and of a significant number of reaction systems in chemical engineering is the critical role played by interfacial phenomena. Liquid-liquid and gas-liquid systems are characterized by convective-diffusive transfer at interfaces that keep distorting (e g., in distillation, gas absorption, and liquid-liquid extraction). One fluid phase in such systems is often dispersed in another. The dynamic behavior of drops and bubbles, for example their shapes under various flow conditions and their breakage and coalescence, has been smdied for many years, one goal being to predict mass transfer rates in dispersed systems (Azbel, 1981 Clift et al., 1978 Mobius anad Miller, 1998). 

TURBULENCE is chaotic fluid flow characterized by the appearance of three-dimensional, irregular swirls. These swirls are called eddies, and usually turbulence consists of many different sizes of eddies superimposed on each other. In the presence of turbulence, fluid masses with different properties are mixed rapidly. Atmospheric turbulence usually refers to the small-scale chaotic flow of air in the Earth s atmosphere. This type of turbulence results from vertical wind shear and convection and usually occurs in the atmospheric boundary layer and in clouds. On a horizontal scale of order 1000 km, the disturbances by synoptic weather systems are sometimes referred to as two-dimensional turbulence. Deterministic description of turbulence is difficult because of the chaotic character of turbulence and the large range of scales involved. Consequently, turbulence is treated in terms of statistical quantities. Insight in the physics of atmospheric turbulence is important, for instance, for the construction of buildings and structures, the mixing of air properties, and the dispersion of air pollution. Turbulence also plays an... 

The major goal of The direct quadrature method of moments (DQMOM) was to derive transport equations for the weights w and abscissas that can be solved directly and which yield the same moments nk without resorting to the ill-conditioned PD algorithm. Another novel concept imposed is that each phase can be characterized by a weight w and a property vector )i, thus the DQMOM can be employed solving the multi-fluid model describing multi-phase systems. Moreover, since each phase has its own momentum balance in the multi-fluid model, the nodes of the DQMOM quadrature approximation are convected with their own velocities. The DQMOM was proposed by Marchisio and Fox [143] and Fan et al. [53] in order to handle poly-dispersed multi-variate systems. 

The simplest packed bed design arises with a single dilute adsorbate in a carrier fluid when it can be assumed that the process is isothermal, that there is plug flow, and that there are no mass transfer resistances. In such a situation, instantaneous equilibrium exists at all points in the system. Without the axial dispersion term and taking the velocity outside the partial differential term for the convective flow, equation (6.19) is simplifled to ... 

The ratio of reaction and permeation rates is critical in designing an MR. Dimensionless numbers are important in parametric analysis of engineering problems. They allow comparison of two systems that are vastly different by combining the parameters of interest. Dimensionless numbers are used to simplify the meaning of the information in scaUng-up the reactor for real flow conditions and to determine the relative significance of the terms in the equations. The Damkohler number (Da) is the ratio of characteristic fluid motion or residence time to the reaction time, and the Peclet number (Pe) defines the ratio of transport rate by convection to diffusion or dispersion (Basile et al, 2008a Battersby et al., 2006 Moon and Park, 2000 Tosti et al., 2009). In the case of an MR, Da and Pe are defined in Equations [11.1] and [11.2]. 

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