Fluid particle diameter

As an illustration, we consider the cluster-size distribution for a monodis-perse Lennard-Jones fluid (particle diameter a, interaction cut-off 2.5(t) at a rather arbitrary density 0.16monotonously decreasing function of cluster size at high temperatures, it becomes bimodal at temperatures around 25% above the critical temperature. The bimodal form is indicative of the formation of large clusters. 

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

Dp = particle diameter, Df = vessel diameter, (note that D /Df has units of foot per foot in the equation), G = superficial mass velocity, k = fluid thermal conductivity, [L = fluid viscosity, and c = fluid specific heat. Other correlations are those of Leva [Jnd. Eng. Chem., 42, 2498 (1950)] ... 

In Eqs. (Il-53r7) and (11-53/ ), is based on particle diameter and superficial fluid velocity. 

H = magnetic-field intensity dB/dz = magnetic-field gradient T = fluid viscosity b = particle diameter... 

This reaction is carried out in tall fluidized beds of high L/dt ratio. Pressures up to 200 kPa are used at temperatures around 300°C. The copper catalyst is deposited onto the surface of the silicon metal particles. The product is a vapor-phase material and the particulate silicon is gradually consumed. As the particle diameter decreases the minimum fluidization velocity decreases also. While the linear velocity decreases, the mass velocity of the fluid increases with conversion. Therefore, the leftover small particles with the copper catalyst and some debris leave the reactor at the top exit. 

Direct interception occurs when the fluid streamline carrying the particle passes within one-half of a particle diameter of the filter element. Regardless of the particle s size, mass, or inertia, it will be collected if the streamline passes sufficiently close. Inertial impaction occurs when the particle would miss the filter element if it followed the streamline, but its inertia resists the change in direction taken by the gas molecules and it continues in a... 

The column length, as well as providing the required efficiency, is also defined by the D Arcy equation. The D Arcy equation describes the flow of a liquid through a packed bed in terms of the particle diameter, the pressure applied across the bed, the viscosity of the fluid and the linear velocity of the fluid. The D Arcy equation for an incompressible fluid is given as follows. 

By integrating Eq. (13.35) step by step in time, the particle trajectory of the particle may be obtained. In the integration, the interaction between the particle and the wall may be approximated as being fully elastic however, when the particle hits the sidewall of the cyclone, the particle may be treated as being collected and the computation for the particle may terminated in order to save the computational time that may be required to track the particle to the bottom of the cyclone. If the particle trajectories for a range of particle diameters at different rates of fluid flow through the cyclone are determined, then the particle efficiency curve and the cut-off particle diameter of the cyclone may be obtained. 

The polydisperse fluid structure is characterized by the total, / (r, a, (j ), and the direct, c(r, a, (jy), correlation function, both being functions of the particle diameters. These functions are related via the OZ equation (17), which is rewritten in the form... 

As we have noted in Sec. II, one of the methods leading to the so-called singlet equations for the density profiles, originally initiated for simple fluids in Ref. 24, starts from considering a mixture of fluid particles and another species of hard spheres at density pq and diameter Dq, taking next the limit Pg - 0,Dg- oo. 

We introduce, for the sake of convenience, species indices 5 and c for the components of the fluid mixture mimicking solvent species and colloids, and species index m for the matrix component. The matrix and both fluid species are at densities p cr, Pccl, and p cr, respectively. The diameter of matrix and fluid species is denoted by cr, cr, and cr, respectively. We choose the diameter of solvent particles as a length unit, = 1. The diameter of matrix species is chosen similar to a simplified model of silica xerogel [39], cr = 7.055. On the other hand, as in previous theoretical works on bulk colloidal dispersions, see e.g.. Ref. 48 and references therein, we choose the diameter of large fluid particles mimicking colloids, cr = 5. As usual for these dispersions, the concentration of large particles, c, must be taken much smaller than that of the solvent. For all the cases in question we assume = 1.25 x 10 . The model for interparticle interactions is... 

We also have studied fluid distribution in the pore H = 6 (Fig. 12(b)) at Ppq = 4.8147 and at two values of Pp, namely at 3.1136 (p cr = 0.4) and at 7.0026 (pqOq = 0.7 Fig. 12(b)). In this pore, we observe layering of the adsorbed fluid at high values of the chemical potential Pp. The maxima of the density profile pi(z) occur at distances that correspond to the diameter of fluid particles. With an increase of the fluid chemical potential, pore filhng takes place primarily at pore walls, but second-order maxima on the density profile pi (z) are also observed. The theory reproduces the computer simulation results quite well. 

There are various methods for the determination of the surface area of solids based on the adsorption of a mono-, or polymolecular layer on the surface of the solid. These methods do not measure the particle diameter or projected area as such, but measure the available surface per gram or milliliter of powder. The surface measured is usually greater than that determined by permeability methods as the latter are effectively concerned with the fluid taking the path of least resistance thru the bed, whereas the adsorbate will penetrate thru the whole of the bed as well as pores in the powder particles. These methods appear to be more accurate than surface areas calculated from weight averages or number averages of particle size because cracks, pores, and capillaries of the particles are included and are independent of particle shape and size... 

Force on particle from Stokes Law = inpdu, where q is the fluid viscosity, d is the particle diameter and u is the velocity of the particle relative so the fluid. 

Mass transfer in a gas-liquid or a liquid-liquid reactor is mainly determined by the size of the fluid particles and the interfacial area. The diffusivity in gas phase is high, and usually no concentration gradients are observed in a bubble, whereas large concentration gradients are observed in drops. An internal circulation enhances the mass transfer in a drop, but it is still the molecular diffusion in the drop that limits the mass transfer. An estimation, from the time constant, of the time it wiU take to empty a 5-mm drop is given by Td = d /4D = (10 ) /4 x 10 = 6000s. The diffusion timescale varies with the square of the diameter of the drop, so... 

The flow pattern of fluids in gas-liquid-solid (catalyst) reactors is often far from ideal. Special care must be taken to avoid by-passing of the catalyst particles near the reactor walls, where the packing density of the catalyst pellets is lower than in the centre of the bed. By-passing becomes negligible if the ratio of reactor to particles diameter is larger than 10 a ratio of 20 is recommended. Flow maldistributions might be serious in the case of shallow beds. Special devices must be used to equalize the velocity over the cross-section of the reactor before reactants are introduced onto the catalyst bed. 

Gr Grashof number D2pg Ap/p)/v2 Dp = particle diameter g = acceleration due to gravity Ap = solid-fluid density difference p = fluid density... 

The Poppe plot is a log-log plot of H/uq = t(JN versus the number of plates with different particle sizes and with lines drawn at constant void time, t(). H is the plate height, Vis the number of plates, and u() is the fluid velocity (assumed equal to the void velocity). The quantity H/u() is called the plate time, which is the time for a theoretical plate to develop and is indicative of the speed of the separation, with units of seconds. In the Poppe plot, a number of parameters including the maximum allowable pressure drop, particle diameter, viscosity, flow resistance, and diffusion coefficient are held constant. 

Given Particle diameter (d) and fluid properties m,n, and p) Particle settling velocity (1/) and fluid properties (m, n, and p)... 

---