Abstract
The validity of estimation of the three-dimensional size distribution (3-DSD) using the Schwartz-Saltykov (SS) method has been evaluated by applying the discrete two-dimensional size-distribution (2-DSD) model. The coefficient for the SS method has been generalized, and the verification of this method has been studied by focusing on the measured number of sectioned particles on the cross section of the specimen, the step width of the histogram for the SS method, and the oversight of small sectioned particles. The total number of particles per unit volume (total N V (T.N V )) and the number of particles in each size class, estimated by the SS method, are found to be always underestimated compared with the true N V value, without regard to the oversight of small sections. The T.N V value can be reasonably estimated by an extrapolation method, in which the step width of the histogram approaches zero. The relative frequency in a distribution function is correctly determined independent of the step width when the oversight involves a sectioned particle size equal to or less than one-tenth of the true mean diameter of the particles. The quantitative 3-DSD can be obtained from the reasonably estimated value of T.N V and the correctly determined relative frequency.
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Abbreviations
- A −1 ij :
-
reverse coefficient between N V (j) and N A (i)
- A(i, j):
-
shape factor
- d A :
-
diameter of section
- d A (i):
-
class mark of section diameter in the class i
- d A,i :
-
measured diameter of sections
- d A,L :
-
resolution size of sections in microscope
- d A,max :
-
maximum diameter of the measured sections
- d V :
-
diameter of particle
- \(\bar d_V \) :
-
average diameter of particles
- d V (j):
-
class mark of particle diameter in the class j
- d V,max :
-
maximum diameter of particles corresponding to d A,max
- f(x):
-
probability density function of random variable, X
- F(x):
-
cumulative distribution function of random variable, X
- h ij :
-
thickness of the circular slice for particles
- i, j :
-
class number in histogram for sections and particles, respectively
- k :
-
maximum class number
- n :
-
number of sections
- \(\bar n\) :
-
average number of sections
- N A :
-
number of sections per unit area
- N A (i):
-
number of sections in size class d A (i) per unit area
- N A (i, j):
-
a part of N A (i) contributed by particles with diameter of jΔ
- N V :
-
number of particles per unit volume
- N V (j):
-
number of particles in size class d V (j) per unit volume
- N V,A :
-
total N V obtained by sum of the N V (j) values obtained by the Schwartz-Saltykov translation
- N V,D :
-
total N V obtained from the N A (i)-d A (i) histogram
- N V,P :
-
total N V obtained by sum of the only positive values of N V (j) resulted from the Schwartz-Saltykov translation
- p :
-
probability of the plane intersecting a sphere of diameter jΔ
- s :
-
standardized variate
- T(i, j):
-
translation coefficient
- α(i, j):
-
coefficient for the Schwartz-Saltykov translation in Eq. [8a]
- β :
-
parameter for the monodispersed system
- Δ:
-
step width of a histogram
- η :
-
harmonic mean
- λ :
-
parameter for the log-normal distribution
- μ :
-
mean value
- σ :
-
standard deviation
- τ :
-
parameter for the exponential distribution
- ξ :
-
parameter for the log-normal distribution
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Takahashi, J., Suito, H. Evaluation of the accuracy of the three-dimensional size distribution estimated from the schwartz-saltykov method. Metall Mater Trans A 34, 171–181 (2003). https://doi.org/10.1007/s11661-003-0218-6
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DOI: https://doi.org/10.1007/s11661-003-0218-6