Aust. J. Soil Res., 1993, 31, 67-72
Soil Fertility and Plant Nutrition
Oxidation Rate of Elemental Sulfur
Particles with a Wide Range of Sizes
J. H. Watkinson
New Zealand Pastoral Agriculture Research Institute,
Ruakura Agricultural Centre,
Private Bag 3123, Hamilton, New Zealand.
Abstract
An equation is proposed that describes the oxidation rate of elemental sulfur particles with
a wide range of sizes, such as would be found in fertilizers. The only information needed is
the mean oxidation rate constant over the period of interest and, from a sieve analysis, the
proportions of the total mass in each sieve fraction (ratio of upper to lower size 52).
Keywords: elemental sulfur, oxidation, rate, size.
Introduction
Watkinson (1989) developed equations for the oxidation rate of elemental sulfur
particles of similar blocky shape (similar dimensions along x, y, z axes) but of
different size, based on a constant rate per unit area. These are appropriate to
the oxidation of fertilizer elemental sulfur in soil since processing, transport and
distribution breaks down any particles of a prismatic to a blocky shape. His
general equation is, however, not very useful in practical situations because it
requires the evaluation of two definite integrals involving the mass distribution
with size i.e. the relationship between the mass of particles of a given size, and
their size. Even where the mass distribution is constant so that the integrals can
be readily evaluated, the equation reported by Watkinson (1989) can give low
results if too wide a size range is used. This communication reports on a second
equation that is valid over the range of sizes not covered, so that between the
two equations all sizes are included. In addition, a related equation is developed
that permits estimation of the oxidation rate of a practical mixture of elemental
sulfur. The mixture can comprise any range of sizes, and any mass distribution
measurable by sieve analysis. The only information needed is the sieve analysis,
and the mean oxidation rate constant in the field over the time involved.
Theoretical
The following treatment examines the validity of equations developed for the oxidation rate
of elemental sulfur (Watkinson 1989), and from them develops a simple equation appropriate
to field oxidation of fertilizer mixtures of elemental sulfur. The particles are assumed to be
blocky shaped, which can therefore be treated as 'equivalent spheres' (Watkinson 1989), and
oxidized at a constant rate per unit area. For present purposes it is equivalent, and simpler,
to consider this rate in terms of a constant rate of decrease, a, in the particle diameters per
J. H. Watkinson
unit time (Swartzendruber and Barber 1965; Watkinson 1989). The upper and lower limits of
the initial size range are taken as c and b, respectively. Two cases of mass distribution are
considered. Firstly, where the mass distribution is constant, which facilitates the mathematical
analysis, and secondly where it is variable, as is the case for fertilizer mixtures. The mass
distribution is readily calculated from a sieve analysis of the sulfur particles on the assumption
that the value within each fraction is constant.
Results and Discussion
Constant Mass Distribution
Wide Size Range (c/b >2)
Watkinson (1989) showed that the fraction of unoxidized elemental sulfur,
mlm,, where m,, m are the masses initially and after time t of sulfur particles
having a constant mass distribution at any size x is
m/mo = [l/(c - b)] i c ( l - a t / ~ ) ~ d x .
On integration,
mlm, = 1- 3at[(ln(c/b))/(c - b)]
+ 3a2t2/bc- a3t3[(b+ c)/2b2c2].
(1)
Also, approximately (Watkinson 1989),
1 < c/b
)~,
mlm, = (1 - a t / ( b ~ ) ~ / ~provided
< 2.
(2)
However, equation (1) is strictly applicable only to the time (b/a) when
the smallest particles have just oxidized. For t > b/a the function becomes
increasingly lower than the actual value (as determined by equation (3) below)
and so gives erroneously low values for m/m,. During the time between complete
oxidation of the smallest and largest particles, b/a 5 t 5 c/a, the appropriate
size limits for integration are 'at' and 'c', since the sizes corresponding to this
time are b 5 at 5 c ie,
m/mo = [1/(c - b)] JC(l- a t / ~ ) ~ d x ,
at
Hence the residual elemental sulfur, m/m,, is given by equation (1) for 0 5 t 5 b/a
and by equation (3) for b/a 5 t 5 c/a.
Fig. l ( a ) illustrates the above points for a = 0.14 pm day-', b = 10 pm,
c = 150 pm. In particular, equation (1) is valid up to the time for complete
oxidation of the smallest particles i.e. for times up to b/a = 71 - 4 days. Thereafter
it increasingly deviates negatively from the correct value (equation (3)), becoming
zero after about 230 days. During times between complete oxidation of the
smallest and largest particles ie. between 71.4 days and t = b/a = 1071 days,
equation (3) is valid. For smaller times equation (3) deviates slightly positively
from equation (1) being at most only about 10% greater at zero time (Fig. 1).
Oxidation Rate of Sulfur Particles
Narrow Size Range (1< c/b<2)
Equation (2) is strictly applicable over only the time for complete oxidation
of the smallest particles, i.e. for 0 <_ t <_ b/a. During the time between complete
oxidation of the smallest and largest particles (b/a 5 t c/a), the appropriate
equation, similarly to equation (3), is equation (2) modified by replacing the
term for the lower limit of the size range, b, with the variable size 'at'.
<
Nevertheless, equation (2) is still useful as an approximation for equations (1)
and (3) for particles of the restricted size range 1< c/b < 2. More than 95% of
the mass of the mixture of all particles in this range have oxidized by the time
the smallest particles have completely oxidized after the time b/a (equation (1)).
All particles have oxidized according to the approximation, equation (2), after a
time t = d b c ) / a. At this time using the full equation (3), less than 0 - 5% of
particle mass remains unoxidized. The approximation is, therefore, accurate to
better than 1%and very useful experimentally when a narrow size range can be
used (Watkinson and Lee 1992).
Varying Mass Distribution, Wide Size Range
In measuring the mass distribution with size by sieving, neighbouring sieve
sizes are usually arranged such that the ratio of the greater sieve diameter to the
lesser is 2. Typically most of the mass of elemental sulfur would need to have
a particle diameter <0.25 mm (Fig. I), so sieve sizes could be 1, 0.5, 0.25, 0-15,
0.075, 0.038 mm. This sizing corresponds conveniently to the size restriction
(c/b<2) required for use of equation (2). Within each fraction the assumption of
a constant mass distribution would be a reasonable approximation to the actual
distribution. Hence a mixture of a wide range of particle sizes and varying mass
distribution can be regarded as the sum of particles in several sieve fractions, as
above, each fraction having its own constant mass distribution. In general, for n
successive sieve fractions each of size bi-1 to bi, and weight fraction wi,
<
Validity and use of Equation
(4)
Fig. l(b) illustrates that equation (4) is equivalent to the analytical solution from
a composite of equation (1) (0 t <_ 71.4) plus equation (3) (71.4 5 t 5 1071)
to within 1.5% of m,. The value for a (0.14 pm day-'), is typical for mean
annual oxidation rates in the South Island of New Zealand (Lee and Watkinson
unpubl.). The size range for equation (4) of 10 to 150 pm is divided into the
four size fractions between 10, 20, 38, 75 and 150 pm, which complies with
1 < bi/bi-l
2. To permit comparison with the composite of equations (1) and
(3), the mass distribution, v, must be constant so,
<
<
J. H. Watkinson
Since w1+w2+w3+wq = 1, therefore v = 11140 pm-l and the weight fractions,
wi, are as tabulated below:
Fraction number, i
Size fraction, bi-1 to bi (pm)
Size interval, bi-bi-1 (pm)
Geometric mean, d b i - 1 bi) (pm)
Weight fraction, wi
1
10-20
10
d 1 0 x 20)
10/140
2
20-38
18
d20x38)
181140
3
38-75
37
d38x75)
371140
4
75-150
75
d 7 5 x 150)
751140
Fig. 1 also demonstrates that the particle size of elemental sulfur in the South
Island of New Zealand needs to be <I50 pm for complete oxidation within a
year.
Time (days)
Time (days)
Fig. 1. (a) Comparison of equation (1) (-) with equation (3) (- . -), with a = 0- 14 pm day-',
b = 10 pm and c = 150 pm. Equation (1) is valid at short times (0 5 t 5 b/a (71.4 days)) and
equation (3) at longer times (b/a 5 t c/a (1071 days)). (b) Comparison of equation (4) (---)
with a composite (-) of equation (1) (0 t 5 71.4 days) and equation (3) (71.4 t 5 1071
days), with a, b, c as in Fig. l(a). To facilitate the comparison the mass distribution
is constant at 1/140 pm-I. The four size fractions and corresponding weight fractions of
equation (4) are listed in the text.
<
<
<
Equation (4) is applicable to field or laboratory conditions over time intervals
where a mean rate constant, a, and the initial amount, m,, is valid. In the
Mediterranean climate of New Zealand with warm, dry summers and cool, moist
winters mean annual rate constants in the field are valid since the effects of
temperature and moisture on the rate constant mostly balance out (Watkinson
and Lee, unpubl.). Rate constants increase with temperature and moisture (below
field capacity), and vice versa.
Test of Equation
(4)
using Published Results
Data from a field trial at Pinjarra (rainfall 970 mm) in South West Australia
(Barrow 1971) can be used to test equation (4). Rate constants estimated for
equation (2) using data from three treatments each comprising only one sieve
fraction can be compared with the rate constant estimated using equation (4)
and data from a mixture of six sieve fractions.
From the three treatments using narrow size ranges (20-40, 60-100, <200 mesh),
values for the rate constant, a, and the initial amount, m,, can be estimated
from fitting the data ((m/m0)1/3 against t) to equation (2). The value for the
geometric mean of each fraction is calculated from lower and upper size limits of
Oxidation Rate of Sulfur-Particles
the sieve fraction. The percentage variance accounted for by the three associated
geometric mean sizes 594, 194 and 52 pm (assuming particles 37 to 74 p a ) are 87,
77 and 99%, respectively. This good fit of equation (2) indicates that an annual
mean rate constant is valid for the site. The weighted mean value for the rate
constant for the two sizes used in the mixture of six fractions is 0.159 pm day-l,
while the weighted value for all three size fractions is 0-163 pm day-l.
Using equation (4), the residual elemental sulfur from the mixture of six
fractions, <200 mesh (74 pm) to 40 mesh (420 pm), is given by the fractional
mass in each sieve fraction, the geometric mean of each sieve fraction size, and
the rate constant, a,
Each term corresponds to a sieve fraction, and has a minimum value of zero.
Statistical analysis shows equation (4a) accounts for 92% of variance when fitted
to the experimental data (Fig. 2), with an estimate for the rate constant of
0.154 pm dayF1 (s.e. 0.025). This is in excellent agreement with the mean value
from the other fractions, and is similar to that for New Zealand soils.
The same mixture of elemental sulfur was applied to two other sites. Statistical
analysis, as above, shows equation (4a) accounts for 98% and 89% of variance when
fitted to experimental data from the Darkan and Kwolyin sites, respectively, again
showing that a mean annual rate constant is applicable (Fig. 2). The corresponding
rate constants are 0.146 (s.e. 0.013) and 0.098 (s.e. 0.021) pm day-l.
I
I
I
I
I
100
200
300
400
500
Time (days)
Fig. 2. Fit of equation (4a), with rate constants of 0.154 (-), 0.146
and 0.098
(. - . ) pm day-', to experimental data from oxidation of the elemental sulfur mixture of six
sites,
size fractions (<74 to 420 pm) at the Pinjarra (a), Darkan (A), and Kwolyin (0)
respectively, in South-Western Australia (Barrow 1971).
(--)I
Acknowledgments
I thank M. P. Upsdell for helpful discussion and J. E. Waller for statistical
analyses.
J. H. Watkinson
References
Barrow, N. J. (1971). Slowly available fertilizers in south-western Australia. 1. Elemental
sulphur. Australian Journal of Experimental Agriculture and Animal Husbandry 11, 211-16.
Swartzendruber, D., and Barber, S. A. (1965). Dissolution of limestone particles in soil. Soil
Science 100, 287-91.
Watkinson, J. H. (1989). Measurement of the oxidation rate of elemental sulfur in soil.
Australian Journal of Soil Research 27, 365-75.
Watkinson, J. H., and Lee, A. (1992). A mechanistic model for the oxidation rate of elemental
sulphur in soil tested on results by HPLC. Proceedings Middle East Sulphur Symposium,
163-71.
Manuscript received 3 August 1992, accepted 1 October 1992