Particle size density function

The particle size density function U r) is obtained by combining the growth equation with the age distribution. 

The composition of the particle population is usually indicated by the particle size density function f(rp), where tp is the characteristic particle dimension of importance. We denote the values of this function for the feed stream and the product streams by fj rp), fi rp) andfftp), respectively. Since the fraction of particles in the size range rptotp-y dtp is given byV(t>) dtp when the particle size density function is/(rp), such that... 

Often, moments of distribution/density functions are used in particulate systems. The th moment of the particle size density function is defined by... 

Figure 2.4.3. (a) Particle size density function and distribution function, (b) Typical plots of population density function and cumulative particle numbers against the particle size. 

It is known that the feed particle size density function is Gaussian with an average value Tp and a standeird deviation of Or- (For the liinits of integration, a value of infinity may be substituted for r ax- However, in the integrand, retain rn ax-) Obtain an expression for a total efficiency 7- as a function of 7p, r ax and Cr. 

The notion of a size-distributed particle population was introduced in Section 2.4 via a particle size density function/(rp) the quantity/(rp)drp represents the fraction of particles in the size range of Vp to Vp + dVp in a unit fluid volume. It is also the probability of finding a particle having a size in the size range Vp to Vp + dVp in a unit fluid volume. 

We consider now briefly one of the effects not considered in the derivation of the Deutsch equation (7.3.42), namely the presence of a particle size distribution. Let J rp) be the particle size density function and let f tp) be the corresponding distribution function of the dust cloud entering the precipitator. Then... 

A particle trajectory, whose r value is less than Tq but z = L, will not hit the centrifuge wall at r = ro this particle will leave the centrifuge with the liquid overflow and is not captured by the device. A separation analysis should be able to predict the particle size density function, ffr, in the liquid overflow for a given feed liquid particle size density function f/(rp). Alternatively, the separation analysis should provide an expression for the grade efficiency function, Gfrp) of the device. 

When a suspension is introduced into the inclined lamella settler, the feed suspension may be characterized by means of its solids volume fraction and its particle size density function fjO p)- The corresponding quantities for the overflow and underflow streams are ji,/i(rp) (rp). Often such problems are analyzed instead using the solids volume fraction and the particle settling (terminal) velocity density function /[t/pzt), where the particle settling velocity Upzt in the Stokes law range is related to the particle radius tp by relation (6.3.1) ... 

Note We have generally employed particle size density functions ff rp), fi rp) and fzirp), where the probability density function depends on the random variable tp, the particle radius. In the analysis considered here for inclined settlers, we are dealing with density functions ffJJpa), ( Upzt) and f-JJJp. Since, by relation (6.3.1), the relation between Up and tp is (if Stokes law is valid)... 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

In summary the advantage of using neutrons for catalyst particle-size-distribution function measurements, is that, unlike X-rays, they can be applied to catalysts dispersed on high-electron-density supports such as a-Al203. This is because the technique of contrast matching to mask-out one component of the scattering is much more versatile with neutrons than with X-rays. In part this is due to the ready availability of suitable deuteriated solvents. 

Particle behavior is a function of particle size, density, surface area, and shape. These interact in a complex manner to give the total particle behavior pattern [28], The shape of a particle is probably the most difficult characteristic to be determined because there is such diversity in relation to particle shape. However, particle shape is a fundamental factor in powder characterization that will influence important properties such as bulk density, permeability, flowability, coatablility, particle packing arrangements, attrition, and cohesion [33-36], Consequently it is pertinent to the successful manipulation of pharmaceutical powders that an accurate definition of particle shape is obtained prior to powder processing. 

The seas may also act as a receptor for depositing aerosol. Deposition velocities of particles to the sea are a function of particle size, density, and shape, as well as the state of the sea. Experimental determination of aerosol deposition velocities to the sea is almost impossible and has to rely upon data derived from wind tunnel studies and theoretical models. The results from two such models appear in Figure 4, in which particle size is expressed as aerodynamic diameter, or the diameter of an aero-dynamically equivalent sphere of unit specific gravity.If the airborne concentration in size fraction of diameter d is c then... 

The parameters may be associated with the properties of the explosive as incorporated in the detonator, such as its composition, particle size, density (both in real terms and as a function of filling pressure), and with the volume. 

To describe the polymerization process we introduce a particle size distribution function / = /(t, v) depending on time t > 0 and a volume variable t > 0. The physical interpretation of / may be given in a heuristic way as follows The differential /(t, v)dv is the average number of particles whose volumes at time t belong to the infinitesimal volume interval (v, v + dv). That is, we make the usual assumption of statistical physics that the particle number has a density - which is justified by the large number of particles. 

The particle size distribution (PSD) significantly impacts the reactant conversion in a fluidized bed reactor. Sun and Grace (1990) examined the three different particle size distributions, wide, narrow, and bimodal, on the performance of a catalytic fluidized bed reactor using the ozone decomposition reaction. They found that a fluidized bed with particles of wide size distribution yields the highest reactant conversion. Of further interest, the property of particle entrainment and elu-triation is a function of particle size, density, and shape. Both entrainment and elutriation rates increase... 

Here, r,nax min refer to the maximum and minimum particle sizes in the feed stream. The nature of such a density function is illustrated in Figure 2.4.3(a). The particle size distribution function F(r is defined by... 

If there is a particle size distribution indicated by n(rp), the particle number density function (equation (2.4.2a)), such that Up,(rp) is the terminal velocity of particles in the size range of tp to tp + Atp, the particle flux Up across a surface area perpendicular to Up,(rp) is... 

We are interested here in the actual number of particles dA/(rp) in the size range of Vp to Vp + dVp per unit volume of the fluid. This quantity is related to a particle number density function n(rp) (the population density junction, see Figure 2.4.3(b) and definition (2.4.2a)) by... 

The analysis based on Clift et ed. (1991) and equantions (7.3.140) (7.3.144) led to result (7.3.144), based on a particular size. If one is interested in the total particle collection efficiency j> we have to integrate over all sizes using the incoming particle numher density function nj(rp) as follows ... 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

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