Harmonic mean size

The harmonic mean size of a number distribution is the number of particles divided by the sum of the reciprocals of the sizes of the individual particles this is related to specific surface and is of importance where surface area of the sample is concerned [4]. 

Intensity weighted harmonic mean size from DLS experiments (m) Hydrodynamic (equivalent) diameter of translation/rotation (m) Stokes diameter or sedimentation equivalent diameter (-)... 

Calculate the arithmetic, geometric, and harmonic mean sizes. 

It may be mentioned here that the mode which represents the most commonly occurring size in a given distribution is not of much use in mineral processing since it does not describe fully the characteristics of a group of particles. The arithmetic mean diameter suffers from the same limitation except when the distribution is a normal one. The harmonic mean diameter is related to the specific surface area. It is, therefore, useful in such mineral processing operations where surface area is an important parameter. 

Polyethylene beads of relatively narrow size distribution with a harmonic mean diameter of 2800 mm and a particle density of 910 kg/m3 were used as the bed material. A static bed height of 1.4 m was employed. 

Results of this type have proved of value in experimental investigations involving surface-volume relations. Of particular interest is the fact that specific surface is inversely proportional to the first moment of the surface-weighted size distribution, and this moment, in turn, is equal to the harmonic mean of the volume-weighted size distribution. 

Table I lists some of the basic mathematical expressions of importance in droplet statistics. The expressions are given in terms of an arbitrary ptb-weighted size distribution. The specific forms are obtained for various integral values of p. For example, the substitution of p = 2 into the equations of Table I yields the cumulative distribution, arithmetic mean, variance, geometric mean, and harmonic mean of the surface-weighted size distribution. Analogous expressions valid for frequencies or mass distributions are obtained by setting p equal to 0 or 3, respectively.

In both foams, from Triton-X-100 and NaDoS solutions, the bubble sizes during 5 min of centrifugation did not exceed avL = 2 to 2.5-1 O 2 cm. The dispersity of a NaDoS foam at the 15 to 20th minute of centrifugation was avL = 310"2 cm. For small angular velocity of rotation co = 52.3 s 1 and A1= 1.5 cm, the highest expansion ratio exceeded the lowest by a factor of 5 times and by a factor of 1.5 its harmonic mean value (Table 6.2). The destruction of a foam layer with A/ = 1.5 cm from NaDoS begins at capillary pressure 8.95 kPa which corresponds as well to a harmonic mean value of 10.9 kPa. 

Park, S. H., et al. (1999). Log-normal size distribution theory of brownian aerosol coagulation for the entire particle size range Part I—Analytical solution using the harmonic mean coagulation kernel. J. Aerosol Science. 30, 1, 3-16. 

Example 4. The number of particles that fall between different size ranges are counted using a microscope and shown in Table 2. The arithmetic, geometric, and harmonic mean diameters along with their arithmetic and geometric standard deviations can be calculated using Equations (24)-<33) and is shown in Table 2. 

In which Atfe 1/2 is the thermal conductivity at temperature f 1//2- This requires a suitable mean value to be chosen, the arithmetic, geometric or harmonic mean of the thermal conductivities at the known temperatures and tA+1 or nd The type of mean value formation does not play a decisive role if A is only weakly dependent on d or if the step size Ax is chosen to be very small. D. Marsal [2.53] recommends the use of the harmonic mean, so... 

If a population of particles is to be represented by a single number, there are many different measures of central tendency or mean sizes. Those include the median, the mode and many different means arithmetic, geometric, quadratic, cubic, bi-quadratic, harmonic (ref. 1) to name just a few. As to which is to be chosen to represent the population, once again this depends on what property is of importance the real system is in effect to be represented by an artificial mono-sized system of particle size equal to the mean. Thus, for example, in precipitation of fine particles due to turbulence or in total recovery predictions in gas cleaning, a simple analysis may be used to show that the most relevant mean size is the arithmetic mean of the mass distribution (this is the same as the bi-quadratic mean of the number distribution). In flow through packed beds (relevant to powder aeration or de-aeration), it is the arithmetic mean of the surface distribution, which is identical to the harmonic mean of the mass distribution. 

Heating value, Btu/dry pound Particle size, Harmonic mean diameter, /x... 

N is the number of particles in the system. According to the arithmetic-geometric-harmonic mean inequalities, (1/r) is always greater than 1 jf and, flius, the right-hand side of the above equation is negative. This shows that the size distribution is always narrowed when the growth rate is proportional to 1 jr and Q and are constants, flinf. use the relation cr = r — (r). )... 

The mean decay rate (F), which can be measured with high accuracy, corresponds to the intensity weighted harmonic mean of the size distribution (xcum)... 

Equation (3.6) is called Young-Laplace equation, in which R is the harmonic mean of the principal radii of curvature. The capillary pressure promotes the release of atoms or molecules from the particle surface. This leads to a decrease of the equilibrium vapour pressure with increasing droplet size Kelvin equation) ... 

There are a great number of different mean sizes and a question arises which of those is to be chosen to represent the population. The selection is of course based on the application, namely what property is of importance and should be represented. In liquid filtration for example, it is the surface volume mean Xsv (surface arithmetic mean Xa surface) because the resistance to flow through packed beds depends on the specific surface of the particles that make up the bed (see equation 9.36). It can be shown that Xsv is equal to the mass harmonic mean Xh (see Appendix 2.2). For distributions that follow closely the log-normal equation (see section 2.5) the geometric mean Xg is equal to the median. 

The remaining task is to determine Xa from the distribution by surface our size distribution is by mass, however, and a conversion has to be made. Rather than converting the whole distribution (from mass to surface) in this case, because the distribution is log-normal (see Figure 2.8), we can use the fact that the arithmetic mean by surface is equal to the harmonic mean by mass... 

Summarizing, then, the surface-volume mean may be calculated as the arithmetic mean of the surface distribution or the harmonic mean of the volume distribution. The practical significance of the equivalence of means is that it permits useful means to be calculated easily from a single size analysis. 

For a give particle size distribution, the mode, the arithmetic mean, the harmonic mean and the quadratic mean all have quite different numberical values. How do we decide which mean is appropriate for describing the powder s behaviour in a given process ... 

The correct size for the equivalent sphere etc. is then got by working out an appropriate mean radius or half-width of the real body. Semenov s route would lead us to consider simply the quotient (volume/ surface area). A more reliable route is to work out the square root of the harmonic mean-square radius. That is, narrow dimensions are to be emphasized. A compromise, probably entirely adequate for work involving factors of safety, is often the sphere of equal volume. This will inevitably be more explosive than, and inferior in stability to, the non-spherical system. The presumed superior stability of the inscribed sphere sets the other bound. 

The permeability of a packed bed with polydisperse spherical or nonspherical particles can also be estimated using Eq. (95) with particle size calculated through the harmonic mean or the Sauter mean if the size distribution is not very broad. For wide size distributions, Li and Park (1998) have proposed equations for calculating the permeability for both spherical and nonspherical particles. 

The volume-mean particle diameter rather than the harmonic mean diameter should be used in Eq. (202). Using a wide variety of particles of different densities and sizes, these authors found that the minimum fluidization velocity depended more strongly on the volumetric fraction rather than on the mass fraction of the particles. The minimum fluidization velocity also is closely related to the mixing state of the mixture, confirming the observation by others in earlier investigations. 

Olaj and Zifferer [229] have performed Monte Carlo simulations in which they placed 300 polymer chains in a lattice, considering excluded volume effects. From the obtained configurations they evaluated the shielding factor which describes how severely the presence of a polymer coil retards the termination of its own radical compared to small (unshielded) radicals. For chains of unequal size Olaj and Zifferer showed that the geometric mean model provided a reasonable mathematical description of their results, although the harmonic mean model. 

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