Fluid-Solid Forces

The drag coefficient /3 can be expressed in several different limiting forms depending on the flow conditions. At low voidages typical 

In the limit of very high voidage, the drag coefficient can be related to the single particle drag coefficient. For the case of spherical particles, 

In the more general case, CD will also be a function of particle shape, sphericity, surface roughness and turbulence intensity of the fluid. 

The dimensionless drag coefficient fiL/ ps u0) in Eq. (30) can be expressed in terms of other fluid parameters by the use of Eqs. (31-33). For low voidages where the Ergun-like expression holds, 

In this expression dp represents the mean diameter when a distribution of different size particles are in the bed. 

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The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

The study of how fluids interact with porous solids is itself an important area of research [6], The introduction of wall forces and the competition between fluid-fluid and fluid-wall forces, leads to interesting surface-driven phase changes, and the departure of the physical behavior of a fluid from the normal equation of state is often profound [6-9]. Studies of gas-liquid phase equilibria in restricted geometries provide information on finite-size effects and surface forces, as well as the thermodynamic behavior of constrained fluids (i.e., shifts in phase coexistence curves). Furthermore, improved understanding of changes in phase transitions and associated critical points in confined systems allow for material science studies of pore structure variables, such as pore size, surface area/chemistry and connectivity [6, 23-25],... 

Principles and Characteristics The principle of solid-fluid-vortex extraction, a recent development [152], is based on the creation of a relatively high filtration pressure as a result of cooling off a vapour chamber in a boiler vessel in such a way that there is (ideally) complete condensation and the extractive fluid is forced through a filter and/or extraction material at nearly one atmosphere in the case of open extractor systems and at more than one atmosphere in the case of closed extractor systems (cf. hydrostatic pressures up to 0.01 bar in Soxhlet). 

The velocity profile is uniform across the entire width of the channel if the channel is open at the electrodes, as is most often the case. However, if the electric field is applied across a closed channel (or a backpressure exists that just counters that produced by the pump), a recirculation pattern forms in which fluid along the center of the channel moves in a direction opposite to that at the walls further, the velocity along the centerline of the channel is 50% of that at the walls (Fig. 11.32a, see Plate 12 for color version). Figure 11.32b (see Plate 12 for color version) illustrates an electric field generating a net force on the fluid near the interface of the fluid/solid boundary, where a small separation of charge occurs due to the equilibrium between adsorption and desorption of ions. The charge region from excess cations localized near the interface by coulombic... 

For dense gas-solid two-phase flows, a four-way coupling is required however, the coupling between particles is managed in a natural way in DPMs. The task is, therefore, only to find a two-way coupling between the gas and the solid phases, which satisfies Newton s third law. Basically, the gas phase exerts two forces on particle a a drag force Vda due the fluid-solid friction at the surface of the spheres, and a force Vpa = -Va Vp due to the pressure gradient Vp in the gas phase. We will next describe these forces in more detail, along with the procedure to calculate void fraction, which is an essential quantity in the equations for the gas-solid interaction. 

In conduction, heat is conducted by the transfer of energy of motion between adjacent molecules in a liquid, gas, or solid. In a gas, atoms transfer energy to one another through molecular collisions. In metallic solids, the process of energy transfer via free electrons is also important. In convection, heat is transferred by bulk transport and mixing of macroscopic fluid elements. Recall that there can be forced convection, where the fluid is forced to flow via mechanical means, or natural (free) convection, where density differences cause fluid elements to flow. Since convection is found only in fluids, we will deal with it on only a limited basis. Radiation differs from conduction and convection in that no medium is needed for its propagation. As a result, the form of Eq. (4.1) is inappropriate for describing radiative heat transfer. Radiation is... 

Convection involves the transfer of heat by means of a fluid, including gases and liquids. Typically, convection describes heat transfer from a solid surface to an adjacent fluid, but it can also describe the bulk movement of fluid and the associate transport of heat energy, as in the case of a hot, rising gas. Recall that there are two general types of convection forced convection and natural (free) convection. In the former, fluid is forced past an object by mechanical means, such as a pump or a fan, whereas the latter describes the free motion of fluid elements due primarily to density differences. It is common for both types of convection to occur simultaneously in what is termed mixed convection. In such instance, a modified form of Fourier s Law is applied, called Newton s Law of Cooling, where the thermal conductivity is replaced with what is called the heat transfer coefficient, h ... 

Convection in Melt Growth. Convection in the melt is pervasive in all terrestrial melt growth systems. Sources for flows include buoyancy-driven convection caused by the solute and temperature dependence of the density surface tension gradients along melt-fluid menisci forced convection introduced by the motion of solid surfaces, such as crucible and crystal rotation in the CZ and FZ systems and the motion of the melt induced by the solidification of material. These flows are important causes of the convection of heat and species and can have a dominant influence on the temperature field in the system and on solute incorporation into the crystal. Moreover, flow transitions from steady laminar, to time-periodic, chaotic, and turbulent motions cause temporal nonuniformities at the growth interface. These fluctuations in temperature and concentration can cause the melt-crystal interface to melt and resolidify and can lead to solute striations (25) and to the formation of microdefects, which will be described later. 

The primary physical parameters, such as the fluid/fluid and fluid/solid interaction parameters, need apriori evaluation through model calibration using numerical experiments. The fluid/fluid interaction gives rise to the surface tension force and the fluid/ solid interaction manifests in the wall adhesion force. The fluid/fluid and fluid/solid interaction parameters are evaluated by designing two numerical experiments, bubble test in the absence of solid phase... 

The interaction force at the fluid-solid interaction can be expressed by... 

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

Abstract When subjected to a mechanical loading, the solid phase of a saturated porous medium undergoes a dissolution due to strain-stress concentration effects along the fluid-solid interface. Through a micromechanical analysis, the mechanical affinity is shown to be the driving force of the local dissolution. For cracked porous media, the elastic free energy is a dominant component of this driving force. This allows to predict dissolution-induced creep in such materials. 

Even though many of us are familiar with the term fluid, we probably don t know the true scientific definition. You may think that a fluid is a liquid like a normal saline solution. While this is true, a fluid does not have to be a liquid. In the scientific usage of the word, a fluid is any material that has the ability to flow. Thus both liquids and gases are considered fluids. Basic forces, like those that result from gravity or pressure differences, cause fluids to flow. When fluids are placed in a container, they assume the shape of the container, unlike solids that keep their shape. 

Mesomorphic phases have properties between those of solids and liquids. For example, rod-shaped or disk-shaped molecules often melt to form liquid crystals, which are fluid, but contain some order. One form of liquid crystals formed from rod-shaped molecules, called a nematic liquid crystal, is shown in Fig. 11. The molecules in the nematic phase are partially ordered in one dimension, with an orientation angle that can vary from one domain to another in the fluid. The forces... 

Molecular dynamics calculations of Hoover and Ree (25) have indicated that a fluid-solid transition occurs in a system of hard spheres even in the absence of attractive forces. The fluid exists for particle volume fractions up to a value rj = 0.49 and at this point, a solid phase with ij = 0.55 is predicted to coexist in equilibrium with the fluid phase. When the particle volume fraction lies in the range 0.55 < jj < 0.74, the solid phase is stable. The upper limit for ij corresponds to the density at closest packing for a face-centered-cubic (fee) arrangement of the particles. 

The thin film evaporator operates based on action of three forces applied by an electric field to polar molecules of a liquid. The first force pumps the liquid between the two electrodes that are generating the electric field. A second force generated due to the difference between the dielectric constant of the vapor and the liquid phase forces the liquid layer to the surface. A third force pushes down the liquid column raised between the two electrodes. A balance between the first and the third forces, gravity, and viscous forces at the fluid-solid interface determines the height of the liquid between the electrodes. 

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