Particle mass material density

In Eulerian coordinates x and t, the mass and momentum conservation laws and material constitutive equation are given by (u = = particle velocity,, = longitudinal stress, and p = material density)... 

Equation (14.91) contains only the mass flow ratio /u as a characteristic number of the mechanics of similitude of the mixture. All the other irnpor rant factors, such as particle size, solid density, etc., are contained in the additional pressure-loss coefficient of the solid particles, A, which is determined separately for each material. 

Bulk density, or the apparent density, is the total mass per unit of total volume. It is not an intrinsic property of a material since it varies with the size distribution of the particles and their environment. The porosity of the solid and the material with which the pores, or voids, are filled also affect the bulk density. For a single nonporous particle, the true density equals the bulk density. 

X = himv, where h is Planck s constant, m the particle mass and v the particle speed. As the speed is proportional to the square root of the temperature mv 12 = kT),vit see that the quantum effect is much more pronounced at high densities and low temperatures, and when the particle in question is very light. The pressure then becomes independent of the temperature. Conversely, for a given density, the quantum effects disappear above a certain critical temperature and the stellar material reassumes its initial flexibility. 

The solids density Ps is the density of fhe solid material from which fhe parficle is made and excludes any pore spaces within the particle. It can be measured using a specific gravity bottle and a liquid in which the particle does not dissolve. The envelope density of a particle is that which would be measured if an envelope covered the external particle surface, i.e. it is equal to the particle mass divided by the external volume. In most analyses the envelope and solids densities are assumed to be equivalent. The bulk density of a powder ps is the effective density of the particle bed defined by... 

Human hemoglobin, for example, has a sedimentation coefficient of 4.48 S and a diffusion coefficient of 6.9 10 m2 s l in aqueous solution at 20°C. The density of this material is 1.34 g cm -3. Substituting these values into Equation (34) shows the particle mass to be... 

Catalyst-supporting materials are used to immobilize catalysts and to eliminate separation processes. The reasons to use a catalyst support include (1) to increase the surface area of the catalyst so the reactant can contact the active species easily due to a higher per unit mass of active ingredients (2) to stabilize the catalyst against agglomeration and coalescence (fuse or unite), usually referred to as a thermal stabilization (3) to decrease the density of the catalyst and (4) to eliminate the separation of catalysts from products. Catalyst-supporting materials are frequently porous, which means that most of the active catalysts are located inside the physical boundary of the catalyst particles. These materials include granular, powder, colloidal, coprecipitated, extruded, pelleted, and spherical materials. Three solids widely used as catalyst supports are activated carbon, silica gel, and alumina ... 

One of the most important properties of a material is its density, for which there are several expressions, namely, bulk, particle, and skeletal densities. The bulk density of solids is the overall density of the material including the interparticle distance of separation. It is defined as the overall mass of the material per unit volume, which can be determined by simply pouring a preweighed sample of particles into a graduated cylinder and measuring the volume occupied. The material can become denser with time and settling, and its bulk density reaches a certain limiting value, known as the tapped or packed bulk density. 

It is convenient to express the particle loading by the particle mass fraction, which depends on the material densities of the phases as well as the volume fractions of the phases and is defined by... 

Solution From Eq. (6.29), the particle volume fraction can be expressed in terms of material density ratio of particle to gas and particle mass fraction as... 

The material density ratio of particle to gas from the given conditions is 2,000. Thus, for a particle mass fraction of 99 percent, we have... 

Another approach for estimating am is based on the pseudothermodynamic properties of the mixture, as suggested by Rudinger (1980). The equation for the isentropic changes of state of a gas-solid mixture is given by Eq. (6.53). Note that for a closed system the material density of particles and the mass fraction of particles can be treated as constant. Hence, in terms of the case for a single-phase fluid, the speed of sound in a gas-solid mixture can be expressed as... 

The interstellar medium is the medium that fills the space between the stars. This space is far from empty. It includes magnetic fields, gas composed of atoms and ions at several different temperatures and densities, cosmic rays, and dust particles. The material content of the ISM changes with time owing to the formation of new stars from it and the ejection of matter from stars into it. The latter include the new nuclei thathave just been assembled by nucleosynthesis in the stars. The state of this medium is turbulent, driven by the shock waves from exploding supernovae. Dust comprises about one percent of the mass of the interstellar matter. It is measured by its infrared radiation and by its obscuration and reddening of starlight. 

Given that the bulk volume associated with the particle mass is a mixture of air and solid material, the bulk density value is highly dependent on sample history prior to measurement. Calculation of the tapped density can then be achieved by tapping the bulk powder a specified number of times (to overcome cohesive forces and remove entrapped air) to determine the tapped volume of the powder. The tapped and bulk density values can be used to define the flowability and compressibility of a powder using Carr s index and the Hausner ratio. 

Apart from the surface composition the bulk properties of a particle material will affect composite deposition. Particle mass transfer and the particle-electrode interaction depend on the particle density, because of gravity acting on the particles. Since the particle density can not be varied without changing the particle material, experimental investigations on the effect of particle density have not been performed. However, it has been found that the orientation of the plated surface to the direction of gravity combined with the difference in particle and electrolyte density influences the composite composition. In practice it can be difficult to deposit composites of homogeneous composition on products where differently oriented surfaces have to be plated. 

Materials analyzed by FFF range from high-density metals and low-density latex microspheres to deformable particles such as emulsions and biological cells. The particles need not be spherical since separation is based on effective particle mass. 

The bulk density of a powder is obtained by dividing its mass by the bulk volume it occupies. The volume includes the spaces between particles as well as the envelope volumes of the particles themselves. The true density of a material (i.e., the density of the actual solid material) can be obtained with a gas pycnometer. The bulk density of a powder is not a definite number like true density or specific gravity but an indirect measurement of a number of factors, including particle size and size distribution, particle shape, true density, and especially the method of measurement. Although there is no direct linear relationship between the flowability of a powder and its bulk density, the latter is extremely important in determining the capacity of mixers and hoppers and providing an easily obtained valuable characterization of powders. 

Pp is the density of the particles of packing material, kfp is the fluid to particle mass transfer coefficient, is the adsorption equilibrium constant of compoimd i. 

If the disperse phase has particles with different volumes and different masses (or material densities), at least two internal coordinates are necessary to describe a particle state. In order to derive a transport equation of the disperse-phase mass density pp, we will let... 

For the case in which all particles have the same mass, Um = Un for the case in which all particles have the same material density, Um = Uy. Note that, like Un and Uy, the mass-average disperse-phase velocity will not usually be in closed form. In most applications, the mean particle velocity Up will be set to the mass-average disperse-phase velocity. Thus, unless noted otherwise, we will set Up = Um throughout the rest of this book. The particle-mass source terms are defined by... 

The particle mass can be written as Mp = PpVp, where pp is the material density of the particle and Vp is its volume. Note that, in addition to mass, either material density or volume could be included in the internal-coordinate vector. (In general, we will use mass and volume as the internal coordinates.) Thus, for example, fixing the particle masses to be equal does not imply that all particles have the same volume. 

Note that, in order to be consistent with the assumption that all particles have the same volume and mass, the right-hand side of this expression is null. If the material density of the fluid is also constant (i.e. the fluid is incompressible), then... 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

When the elements of the disperse phase can be classified as equidimensional, namely they have nearly the same size or spread in multiple directions, and have constant material density, typically a single internal coordinate is used to identify the size of the elements. This could be particle mass (or volume), particle surface area or particle length. In fact, in the case of equidimensional particles these quantities are all related to each other. For example, in the trivial cases of spherical or cubic particles, particle volume and particle surface area can be easily written as Vp = k d and Ap = k d, or, in other words, as functions of a characteristic length, d (i.e. the diameter for the sphere and the edge for the cube), a volume shape factor, k, and a surface-area shape factor, k. For equidimensional objects the choice of the characteristic length is straightforward and the ratio between kp, and k is always equal to six. The approach can, however, be extended also to non-equidimensional objects. In this context, the extension turns out to be very useful only if... 

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