DEPARTMENT OF SOILS MIMEOGR H SERIES NO. _5-2 J 1IN 14 1955
NOTES ON THE PROBLEM ~ MIXING MINOR EIEMENT MATERIALS IN FERTILIZER.1
J. G. A. Fiskel and W. D. Hanson2
There are two problems that arise in the mixing of a declared amount of
minor element, such as copper in the sulfate form, with the main goods in the
mix. First, there is the problem of getting a uniform mix and secondly, of main-
taining a uniform mix in shipment. In the latter case there is the marked tend-
ency of larger particles of the mix to shift to the top. The sampling of this
mix, its analysis, and its final application to the land, involve these two prob-
Let us consider some obvious fundamentals relating to the chances of getting
a true sample from a supposedly uniform mix. This true sample may also be ex-
pressed as the distribution of the fertilizer ingredients per unit length in the
In any sample volume, the number of particles is limited by the size of the
particles. Each particle occupies a certain volume in the mix and a certain
volume of the mix can be occupied by each ingredient of the mix. The number of
particles per unit volume depends on the particle size and the particle shape.
Case I. Particles the Same Size
Certain fertilizer materials have about the same distribution of particle
sizes, some materials are very fine and others rather coarse. For the sake of a
discussion of probabilities, let us assume that the particles are spherical or
balls and the same size.
Suppose, for example, we represent the ingredient containing the minor ele-
ment by a blue balls, a carrier or conditioner by b green balls and the main
ingredient, for example superphosphate, by c white balls, the balls being
1 Contribution from the Department of Soils and Department of Agronomy of the
University of Florida.
2 Assistant Biochemist, and Assistant Professor of Agronomy, respectively.
indistinguishable apart from their color. Suppose further that a is very much
less than b or c and b equals c and an b and c are large numbers as would be
found in a very small particle size (say 1010, 1020, and 1020, respectively).
The a blue balls and b green balls are put in Bin I (Fertilizer) and mixed well.
The c white balls are placed in Bin II. In any random sample withdrawn from Bin I,
the blue and green balls are in the ratio a/b and in fact, if we find x blue
balls in a grab sample there will probably be x X b/a green balls accompanying it.
With the large numbers suggested above, statistical fluctuations till be very
small, e.g., in a grab sample containing one hundredth of the balls in the bag,
the standard variation is 0.01% (1).
Suppose now that we add the contents of Bin I to Bin II and mix well. This
corresponds to adding the premix of minor element and carrier or conditioner to
the fertilizer mix. There are now a + b + c balls that are indistinguishable
while in the bin. If we withdraw one ball, the probability of it being blue,
green or white is in the ratio a r b s c and if we withdraw y balls in the sample,
there will probably be a.y/(a+b*c) blue, b.y/(a+btc) green and c.y/(a+b+c) white
balls. The ratio blue/green = a.y b.y = a/b as before and correspondingly
if we find z blue balls in a grab sample, there will be z.b/a green balls accom-
panying it, which came originally out of Bin I, plus some white balls. Thus the
number of green balls in a sample may be obtained by multiplying the number of
blue balls found to be present by the factor b/a. The ratio a/b and hence b/a
is known from a knowledge of how many blue and green balls were put in Bin I or
how much minor element ingredient was added to a given proportion of another
fertilizer ingredient in the premix.
The case of caking of the mix or of shifting of the mix in the bag or bin can
also be considered. Suppose that in certain time Q balls out of the added
(a + b) fertilizer from Bin I cake or shift in the mix. Then of these Q balls
(1) Yule, V. and M. G. Kendall, Theory of Statistics. Griffin, London. 1965.
(a/a+b) Q will be blue balls and (b/a+b)Q will be green balls and the ratio of
blue/green balls left is a (a/a+b)Q
.- - a/b as before. If R of these balls sub-
_b _b ab Q -
sequently break up in the cake or reshift, the ratio blue/green balls will still
be unchanged. Likewise the ratio blue/white balls will be unchanged in the mix
after the primary uniform mix.
Sampling of n balls in succession from w in a fertilizer bag containing p w
blue and g n white balls will be such that the square of the deviation, o- in
the number of blue balls is n. p. g X(w n). If the sample contains n 1018
balls and w 1020 balls or 100 times the sample taken and p = 101, g = 1 from
which it can be shown that the standard deviation, o" is 0.01%. Probability of
a sample showing a variation greater than 0.05% is 0.00001. (2)
Thus by a thorough incorporation of finely ground ingredient with a portion
of the mix also finely ground, the chances of drawing an accurate sample of the
minor element are very probable. Note that such a fertilizer when passing the
fertilizer drill should spread a uniform distribution of a, b and c particles per
unit distance in the row. Data are not available on such sampling studies.
Case II. Particles Varying Greatly in Size Distribution
In many fertilizer materials such as superphosphate the particle size is far
from uniform, the weight of particles passing 100 mesh screens being about the
same as the weight of particles not passing the screen. In the cases of sulfates
of copper, iron, zinc, or manganese, the average particle sizes are likely to be
above 20 mesh for reasons of hydroscopic qualities or visual identification by
the buyer. The large particle size makes the task of achieving a uniform mix and
adequate sampling a difficult one. In any sample volume, the large particle of
one ingredient a may occupy more space in the sample than the declared ratio for
the whole mix. This case will then result in the ingredient a having a higher
(2) Spinks, J. W. T. and S. A. Barber. Sci, Agric. 27: 145-156. (1947)
analysis in the sample than the declared value. On the other hand, there are
quite likely to be samples where the large particle or particles may be below the
declared ratio of ingredient to whole mix because they were missed in sampling.
This results in a value lower than the declared value for ingredient a. What
tolerance or confidence limits can we expect with respect to the size of the
For illustration of this point let us consider the following data on parti-
cle size of a sample of copper sulfate and apply these data to the sampling prob-
lem. The data are shown in Table I.
TABLE I. General Information on the Particle Size Distribution of a Sample of
Copper Sulfate and on Fertilizer Volumes.
(a) Copper Sulfate Size Distribution:
Particle size Weight Volume Number of particles Average volume
mm. % of total per 100 gm. per pound per particle, c.c.
4.8 or greater 9.0 85 3.90 x 103 0.12
4.8 to 2.0 60.8 90 2.04 x 104 0.002
2.0 or less 30.2 90 3.54 x 105 0.001
(b) A sample of 4-7-5 fertilizer:
100 pounds of 4-7-5 fertilizer occupying 1.43 cubic feet, or 4.05 x 10t c.c.
and containing 1% or 1 pound of copper sulfate in each 100 pounds of fertilizer
(c) Sampling volumes:
In a sampling procedure drawing 1 pint from each of ten one-hundred pound
bags of the fertilizer, the sample then is cut to one quart. This is split, one-
half for official analysis and one-half kept in reserve.
(d) General assumptions:
General assumptions are that we have a volume V (large) of fertilizer con-
taining N (large) particles of copper sulfate. Secondly, the particles of
copper sulfate and fertilizer particles have approximately the same specific
gravity. Thirdly, assume that the particles of copper sulfate are of a constant
size s and randomly distributed in the fertilizer mix. This situation may be
likened to balls of a particular size suspended in a liquid. Let p be the prob-
ability of a particle of copper sulfate being selected in a sample volume v,
this sample being small compared to the whole bulk. Then Np s n number of
particles expected in the volume v. Since a Poisson distribution is encountered
then the variance in numbers becomes equal to the expected number, so that
variance o-2 N = n.
From the information in Table I we have calculated the pure or random sam-
pling error resolving from particular particle size and volume. These data are
given in Table II.
The percentage errors, %, given in Table II arise from random fluctuations
of the balls of the given size and density with respect to a selected volume of
sample. Let the variability arising from random fluctuation of particle number
in the sample be ~. Further errors in determination arise from non-random dis-
atibution of copper sulfate particles in the fertilizer due to shifts or poor
mixing. Let this source of variability be ~2. After the original sample is
ground for the homogeneity required for accurate analysis, errors arise from
sub-sampling of this ground material and in analysis. Let this source of varia-
bility be f2. Then the total variance for a determination of the copper sulfate
content in the mix would be
2 2 2 2
S t + 2M + 2 when errors are independent.
Although OA2 would not be expected to be too large, d2 could be large as compared
to 2, thus appreciably increasing the errors expected for any one determination.
The variability of actual analysis would, therefore, be expected to be greater
than shown in Table II. A similar situation and statistics could be applied to
Variability Expected in Sample Analysis as Particle Size of Copper
Sulfate and Sample Size Varies from a Uniform Mix Oontaining 1%
Copper Sulfate by Weight.
Analysis of Sample for Copper Sulfate :
Expected Range as % CuSO),J. 10 H _
0.05 probability : 0.01 probability I s $ sI
: 0.12 c.c. : qt. 1 + .29% 1 + .38% : .18 :
S: qt. : 17 .41% 1 7 % : .209
: 1/8 qt. : 1 .58% : 1T .76% : .296
0.002 c, qt. + .13% 1 + 17% : 06 :
46 qt. 1: 7 .18% : 17 .24% .092 =
S: 1/8 qt. 1 .25% : 1 .34% : 130:
: 0.001 c.c. : qt. : 1+ .03% 1 + .04% .016
: : qt. 1 7.0% : 17 .06% .022
S1/8 qt. : 1 .06% 1 .08% .031
Particle Number Range
: 0.01 probability >-
.002 qt : 238 + 30 : 238 + 40
Sqt. 119 7 21 : 119 + 28
1/8 qt. : 59 15 59 20 :
001 : qt. : 130 + 126 : 4130 + 166
qt. : 2070 7 89 t 2070 7 117
1/8 qt. :1035 l 63 1035 7 83 :
any of the fertilizer ingredients, copper sulfate being mentioned only for pur-
poses of illustration.
An estimate of q2 for the general case of unequal particle size is not
available. A study of sampling procedure is necessary to establish the confidence
limits on a statistical basis. Obviously the mix with large particle size of the
ingredient or ingredients containing the minor elements will be more difficult
to sample accurately than the sample whose content of minor elements was homog-
enized by pregrinding and/or pre-mixing with a portion of the mix.
The distribution of the fertilizer mix to the land is again a problem of
sampling, because each unit length that the fertilizer is spread is equivalent
to drawing a sample from the fertilizer mix. For each rate of application, the
volume of fertilizer applied per unit distance is nearly constant. But the dis-
tribution of the ingredients of the fertilizer in this volume presents some in-
teresting possibilities. For instance, the frequency with which the ingredients
containing the minor elements occurs is proportional to two variables: (1) the
uniformity of the mix and (2) the particle size of these ingredients. If large
particles of the minor element are involved, this intermittency results in con-
ditions conducive to alternate toxicity and deficiency with respect to this in-
gredient. How big a factor is this in our crops? Is germination affected? These
answers we do not know and, indeed, it does not seem that such studies are re-
ported in the literature. Nevertheless, it may well be a factor. In Table III
is presented the particle size and the diameter of soil into which this particle
must dissolve and diffuse that a toxic situation become unlikely, e.g., one-half
TABLE III. Effective Soil Volume and Diameter Necessary to Reduce to One Half
the Toxicity Level from a Particle of a inor Element of Different
I IJ I Ii----II1
Minor Element Ingredient :
Mean : Toxicity level : Effective:
pa&tidl : as parts per : soil :
volume : million weight :
c.c. : : gn.
0.12 : 500 : 1,220
0.12 : 200 : 3,060
0.12 50 12,200
0.002 : 500 2 20
0.002 200 : 50 :
0.002 : 50 200
0.001 500 10
0.001 : 200 : 25
0.001 : 50 100
Soil taken to occupy 65 c.c. per 100
particle of minor element as 2.54.
soil volume to
grams and specific
4 3 6/8
: 0.073 o:
gravity of the
The toxicity levels given in this table are hypothetical since soil varia-
tion and availability of the minor element in the soil will not be constant.
However, where toxicity levels of one minor element are known to be some value
within the range given in the table, the role of particle size in the fertilizer
can be appreciated. These roles can be solved by suitable research on the
SOILS F.B.S.- 10/7/53 100