Drag force equilibrium model

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

Consider a two-phase nonisothermal turbulent flow in which droplets move under the influence of fluid drag force and their temperature, Tj, changes due to evaporation and the thermal interaction (driven by the temperature difference, T — Td) with the carrier fluid. Here, T is the temperature of the fluid in the vicinity of the droplet. The rate of evaporation governs the size (diameter) of the droplets. A variety of equilibrium and nonequilibrium evaporation models available in the literature were recently evaluated by. Miller et al. [16]. Here, the model which was used in the previous DNS work is selected [17]. The Lagrangian equations governing the time variation of the position X. velocity V, temperature Td, and diameter dd of the droplet at time t can be written as... 

Many researches adopted one of the aforanentioned approaches and modified it to include various aspects of the pneumatic drying process. Andrieu and Bressat [16] presented a simple model for pneumatic drying of polyvinyl chloride (PVC), particles. Their model was based on elementary momentum, heat and mass transfer between the fluid and the particles. In order to simplify their model, they assumed that the flow is unidirectional, the relative velocity is a function of the buoyancy and drag forces, solid temperature is uniform and equal to the evaporation temperature, and that evaporation of free water occurs in a constant rate period. Based on their simplifying assumptions, six balance equations were written for six unknowns, namely, relative velocity, air humidity, solid moisture content, equilibrium humidity, and both solid and fluid temperatures. The model was then solved numerically, and satisfactory agreanent with their experimental results was obtained. A similar model was presented by Tanthapanichakoon and Srivotanai [24]. Their model was solved numerically and compared with their experimental data. Their comparison between the experimental data and their model predictions showed large scattering for the gas temperature and absolute humidity. However, their comparisons for the solid temperature and the water content were failed. 

The drop deformation is modeled by Taylor s drop oscillator [56], as introduced by O Rourke and Amsden [37] into the context of sprays. In this approach, the drop distortion is described by a forced, damped, harmonic oscillator in which the forcing term is given by the aerodynamic drag, the damping is due to the liquid viscosity, and the restoring force is supplied by the surface tension. More specifically, the drop distortion is described by the deformation parameter, y = 2x/r, where x denotes the maximum radial distortion from the spherical equilibrium surface, and r is the drop radius. The deformation equation in terms of the normalized distortion parameter, y, is... 

The one-dimensional particle bed model has been formulated in terms of the primary fluid-particle interaction forces, which alone may be considered to support a fluidized particle under steady-state equilibrium conditions, together with particle-phase elasticity, which provides a force proportional to void fraction gradient (or particle concentration gradient) and so comes into play under non-equilibrium conditions. Only axial components of these interactions have been considered so far. Generalizing these considerations to encompass lateral force components is a straightforward matter, but, as we now see, calls for some modification in the constitutive relation for drag in order to unify the axial and lateral constitutive expressions. The following derivations are expressed in terms of volumetric particle concentration a rather than void fraction e a= - . 

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