142 Int. J Bio-Inspired Computation, Vol. 2, No.2. 2010
Fractal dimension and time factors of sawdust
pattern formation in sawmills
Tajudeen A.O. Salau
Department of Mechanical Engineering,
Faculty of Technology, (1st Floor, Annex),
University ofIbadan, Ibadan, Nigeria
E-mail: tajudeen_salau@yahoo.com
S.A. Oke*
Department of Mechanical Engineering,
Faculty of Engineering, University of Lagos,
Room 10, Mezzanine Complex,
Akoka-Yaba, Lagos, Nigeria
E-mail: sa_oke@yahoo.com
*Corresponding author
Abstract: This paper presents the application offractal theory, especially fractal dimension to the
formation of sawdust particles during operation with four detailed cases, which helps us to
understand the distribution of the sawdust particles inhaled by sawmill workers and remove the
effect of toxicity on their body quickly. Pattern of formation of sawdust in human lungs and other
parts of the body (in fast branching rate) is described with a practical case study in a developing
country. As these sawdust particles settle down in the human body, cells are destroyed on a very
fast rate by the toxic nature of sawdust particles. Thus, removing the effect of toxicity on the
body will require special skill and is cost intensive. The pattern formation of sawdust particles
follows random walking in 2-D Euclidean space using fractal dimension and time steps.
Percentage total of average time steps required for aggregation of specified n-sawdust particles
varies according to power law of percentage successive aggregation. Case 3 and its rules are the
most reasonable if used in a real project since its percentage absolute error compared with the
standard literature value of 1.71 is zero. The paper may be of great importance to occupational
health scientist and those who control and monitor occupation health problems in small scale
industries particularly where occupational hazards are well pronounced.
Keywords: fractal dimension; chaotic dynamics; woodworking problem; random walk,
aggregation.
Reference to this paper should be made as follows: Salau, TAO. and Oke, S.A. (20 I0) 'Fractal
dimension and time factors of sawdust pattern formation in sawmills', Int. J. Bio-Inspired
Computation, Vol. 2, No.2, pp.142-l50.
Biographical notes: T.A.O. Salau obtained his BSc, MSc, and PhD from the University of
Ibadan. He was a Coordinator of the Department of Mechanical Engineering, University of
Ibadan and also lectures in the same department. His research interests include fractal analysis
and chaos.
S.A. Oke has obtained his BSc, MSc and PhD in Industrial Engineering from the University of
Ibadan, Nigeria. He lectures in the Department of Mechanical Engineering, University of Lagos.
He has publ ished and reviewed papers for several international journals.
1 Introduction et a!., 2006; Hamadi et a!., 2001; Jadhav and Vanjara, 2004;
Taty-costodes et aI., 2003) and has also been used as new
Understanding the chaotic dynamic behaviour of sawdust base material for boilers (Akira et a!., 2002). Other uses
particles emitted through wood processing activities in include as catalyst in the removal of mercuric ion from
sawmill using modelling and experimentation plays an aqueous solutions (Ansari and Raofie, 2006). Studies on
important role in its control and in the reduction of sawdust is increasingly becoming of great interest to
employee health hazards at sawmills. Sawdust is of researchers (Arif et al., 2003; Demers et aI., 1997; Udoeyo
economic and experimental benefits in serving as and Dashibil, 2002; Ajayi and Owolarafe, 2007). For
absorbents and sorption materials (Hamdaoui, 2006; S6iban example, Hamid and Saffle (1965) identified the volatile
Copyright © 2010 Inderscience Enterprises Ltd.
I
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Fractal dimension and time factors of sawdust pattern formation in sawmills 143
fatty (i.e., acetic, propionic, butyric, isovaleric, n-vaJeric, space. The article is structured as follows: the introduction
isocaproie, and n-caproic) acids present in hickory sawdust presents the problem and the literature review. Section 2
smoke as a preliminary step to solving environmental discusses the methodology utili sed for solving the problem.
pollution problem. Yet, relatively little progress has been In Section 3, a practical case study is given to strengthen the
made in the modelling and experimentation of the chaotic quality of the paper and the results and discussions with
dynamic behaviour of sawdust particles as they are released explanations for the pattern of results obtained discussed.
into the work environment through wood processing Section 4 concludes and states the future trends.
activities at the sawmill (see Moon, 1987). Understanding
this dynamics would help in planning and redesigning the
work environment and in reestablishing the maximum 2 Methodology
inhale-able sawdust particles in the sawmill environment.
In the instance that a scientific approach is known with The idea of the Current study was motivated by the classic
which government could control excessive exposure of studies of Feder (1988) and Zmeskal et al. (200 I) -that
sawmill workers to sawdust, a guideline for controlling attempted to simulate the electrolysis process whereby
activities could be issued by goverrunents for easy control variation of voltage or current intensity drives metal atoms
of occupational hazards at sawmills. Thus, "v'hile the in a random manner towards disposition as attachment to
complying sawmills may be commended, those not growing pattern. However, these authors have omitted the
complying may have to be reprimanded. Therefore, it is time factor and the minimum distance (0.3-unit) for
only through a scientific tool that such a proper guidance successive attachment of next random walk atoms
could be obtained. This also suggests that governments may (particles). It is this serious omission and gap that the
enforce the use of protective devices for nose protection and current study attempts to bridge, with application to a
hearing loss avoidance. This indirectly reduces practical problem of wood working problem. Thus,
governments' health costs as those with health problems numerical experiment was carried out for the simulation of
still seek assistance in government hospitals. Also, frequent the sawdust particle problem. It should be noted that the
reassignment of workers to other job points for those comparison of estimated fractal disk dimension is reference
exposed to much sawdust could be arranged. In addition, to the model results of Feder (1988) and Zmeskal et al.
regulation by governments may result in the redesign of a (2001), which yielded 1.71. Thus, the models developed by
workplace that lacks proper ventilation. these researchers are used and the data generated for
Investigations on sawmill activities have however experimentation was tested with the model.
provided some useful insights to the contention that sawmill
workers are at risk. These primarily concern the effects of 2.1 Model development rules
emission of volatile compounds from stored woods on The methodology utilised in the development of the sawdust
sawmill worker (Svedberg et at, 2004), the risk of particle dynamic chaotic model is hinged on a number of
childhood cancer by children of sawmill workers through rules. Four important rules, labelled Cases I, 2, 3 and 4,
their paternal exposure to harmful substances (Heacock et were developed based on random-walking principle applied
aI., 2000), and prevalence of asthma in sawmill workers to sawdust particles in circulation paths. The rules used for
(Siracusa et aI., 2007). The particular details of the above each of the four studied cases are as follow (Zmeskal et al.,
review are now given. Svedberg et al. (2004) investigated 2001; Feder, 1988):
the emission of volatile compounds, particularly hexanal
and carbon monoxide, from large- and small-scale storage
of wood pallets. Such storage systems are predominantly Case 1
found in sawmills. Heacock et al. (2000) established a A random-walking 'sawdust particle' is released from the
relationship between the risk of childhood cancer and point of cutting the timber where the sawdust particles are
paternal occupational exposure to chlorophenate fungicides produced (at the interaction of the powered machine blade
in British Columbian sawmills. Siracusa et al. (2007) and the timber logs 'already pushed to the band saw). The
evaluated the prevalence of asthma and its predictors in slicing of the timber into planks produces' sawdust particles
studies of several male working in cedar sawmills and from the source position (zero-unit away from source). The
observed the prevalence of asthma after employment in the release of the sawdust now positions it at a random location
industry. on a circle of radius ten-unit away from the 'seed sawdust
This gap between theory and practice of sawdust particle' which was initially located at the circle center. This
dynamics in sawmills is carefully addressed in the current newly produced sawdust staggers with variable step size
paper. The objective of this study is to show how much back and forth, up and down, until it is 0.3-unit closer to the
computational time will be required to form a fractal pattern 'seed sawdust particle'. For instance, it may stagger to 12
for sawdust particles in a sawmill environment using units away from the seed position in a forward movement. It
random-walking sawdust particles in 2-D Euclidean space. may decrease by five units (backward movement), move by
In addition, the study aims at understanding if the time four units down and finally stayed ay O.3-unit closer to the
taken to attach an additional sawdust particle will reduce or seed particle. Note that these movements are only four.
increase as fractal pattern formation progress in time and Similarly the second, third, fourth, fifth, hundredth,
UNIVERSITY OF IBADAN LIBRARY
144 T.A.o. Salau and S.A. Oke
thousandth walker is added to the growing pattern. That is a rules very specific (i.e., generic approach) instead of using a
second particle is released from the seed's initial position, generalised form of N units. For example why ten units
follow the same pattern and the addition' continues until the away and not five or 15? Clearly, the choice of R is
thousandth sawdust particle is generated. arbitrary, but the choice of higher value demand for higher
computational' time and vice versa. Note that successive
Case 2 attachment/aggregation is when two particles by random
motion come close as 0.3-unit, which again is arbitrary, but
This follows the description in Case I except that 'least at least ensures fine pattern formation. In essence, five units
20-random steps must have been taken before next 'sawdust and 0.3-unit combination will be faster, but the branching
particle aggregation' may be allowed'. That is, instead of may not show clearly. However, 15 units and 0.3-unit will
the four steps taken by the sawdust particle as in Case 1, the take longer time to form but the branching will show
number of movements extends to 20 in random clearly. It is a matter of compromise. That is, how do you
combinations of back, forth, up and down movements. If want it without loosing focus (i.e.., time factor in
less than 20-random steps are allowed for aggregation, the non-dimensional form).
final fractal pattern formed and the corresponding Consequently, larger values of R implies that longer
dimension results will not compare satisfactorily with the total time steps will be required to generate the ultimate
literature results. The fact is that more particles will fractal pattern, the dimension of which is the same
aggregate nearer to the circle circumference and as such statistically for any R. The relationship between Nand R is
prevent gradual aggregation with the seed particle. Similarly direct proportionality. If we assume that the sawdust
the second, third, fourth, fifth, hundredth, thousandth walker particle size is incompressible then N would increase
is added to the growing pattern. proportionately with R. However, there is need to introduce
the term D in the relationship. From practical observation, a
Case 3 power relationship seems applicable. It should be noted that
the model developed is based on the estimation of chaotic
This is similar to Case 2 in description but the measurement behaviour through fractional dimension. The particular
of the sawdust particle movements are in radians. It is emphasis is the radius elimination method. The model is
described as follows: a random-walking 'sawdust particle' stated as (Feder, 1988; Zmeskal et al., 2001):
is released at random location on a circle of radius ten units
away from the 'seed sawdust particle' located on the circle (1)
centre and staggers with variable step size in any angular
direction picked between zero-radian and 2:r-radian until it where
is 0.3-unit closer to the 'seed sawdust particle'. If less than
20-random steps are allowed for aggregation there is an N= number of sawdust particles counted in a specified
impact on the final result since it will not compare circle radius
satisfactorily with the literature value. Similarly the second, K = constant of proportionality
third, fourth, fifth, hundredth, thousandth walker is added to
the growing pattern. R = specified circle radius
D = fractal dimension of the studied aggregated pattern
Case 4 formed by any of the four cases described above.
This is similar to Case 3 except that 'a/ leas/ 20-random
steps must have been taken before next 'sawdust particle The problem investigated may be viewed to encompass
aggregation' may be allowed'. biological aspects and factors related to human health suchSimilarly the second, third, as age of a person, respiration rate, etc. These may be
fourth, fifth, hundredth, thousandth walker is added to the considered in further studies in order to improve our
growing pattern. understanding of the woodworking problem. In addition,
consideration may be given to environmental factors such as
2.2 Model for the estimation offractal dimension wind velocity, temperature, type of tool used, type of wood
The development of a model for characterising the chaotic used, etc. The argument here is that these parameters (wind
behaviour of sawdust particles in the sawmill is based on the velocity, temperature, etc.) are lumped together to generate
establishment of a relationship among the number of the random walk of the sawdust particles. For example, if
sawdust particles that could be counted in a specified radius the wind velocity is unchanged, then the sawdust particles
(N), the radius of the circle being described (R), and the will experience zero motion. In addition, higher temperature
fractional dimension of the studied aggregated pattern means faster particle movement and haphazard too.
formed using any of cases as the guideline (D). A number of By taking the natural logarithm of both sides of equation
questions may arise on the relationship of the specified (1) equation (2) can be obtained.
circle radius, R, and the results. It may be interesting to Log(N) = Log(K)+ DLog(R) (2)
know if the value of specified circle radius, R, has impact on
the final results. In other words, why are the assumptions for
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Fractal dimension and time factors of sawdust pattern formation in sQwmiils 145
Equation (2) is linear with Log (R) being the independent respectively, while the intercept obtained were 4.54, 4.54,
variable and Log (N) being the dependent variable. The 4.60, and 4.60, respectively for Cases I to 4 (Table I).
slope of line of best fit to log-log plots as suggested in
equation (2) gives the best estimate of the aggregated Table 1 Estimateddimensions and natural logarithm of
pattern fractal dimension. The various fractional patterns intercepts with seed value of 9876
were formulated with the use of random walking particles
Case 1
principle in 2-D Euclidean space. In particular, fractional Case 2 Case 3 Case 4Z*
dimension and time steps analysis are used. The procedure Dim Int Dim lilt Dim Int Dim lilt
is initiated by arbitrarily picking ten different random 500 0.87 4.54 0.87 4.54 0.83 4.60 0.83 4.60
number generating seed values to drive random-walking
particle in 2-D Euclidean space according to random rules 1000 1.26 4.39 1.26 4.39 1.18 446 1.18 4.47
(cases). The algorithm was coded in FORTRAN language 1500 1.47 4.25 1.47 4.26 1.37 4.34 1 38 4.34
while the emerged fractional pattern was analysed for its 2000 1.60 4.16 1.60 4.16 1.47 4.27 1.50 4.26
dimension using radius dimension method.
The empirical equation of the model used in the current 2500 1.67 4.10 1.69 4.10 1.52 4.22 1.59 4.19
work is based on the articles of Feder (1988) and Zrneskal et 3000 1.72 4.06 1.75 4.04 I 56 4.19 1.65 4.14
al. (2001). Although the equations are not new stars in 3500 1.75 4.03 1.81 4.00 1.59 4.16 \.71 4.10research, however, in their applied forms, they present new
information that has not yet been documented. The idea of 4000 1.77 401 1.85 3.97 1.61 4.14 \.75 4.06
the application sprang up from the observation of the 4500 \.79 3.99 1.90 3.93 1.63 4.13 1.79 4.03
structure and procedure of application of the Feder and
Zmeskal et al.'s expressions. The chaotic experimentations 5000 1.81 3.98 1.93 3.90 1.65 4.11 1.83 4.00
of these references suggest a good resemblance of the Note: Z* meansN-5dwdustparticles,Dim* is dimension,
behaviour of sawdust particles during its deposits in the while Ir;t*means intercept.
lungs through the nostrils of the operator and mill workers Consequently, there are ranges of readings for the various
in general. Certainly, different behavioural patterns of sample sizes. It is interesting to note that the sample size of
motion, collision of particles, speed of motion, sizes of 3000 sawdust particles yielded the dimension of 1.72, which
sawdust particles, concentration of sawdust particles, is closer to 1.71 by 0.6% absolute errors. This literature
density of particles and settlement (settling down) patterns value could also be obtained from Feder (1988). An attempt
of sawdust particles are observed in real life cases, which is made in Figure 1 to draw the pattern of movement of
has motivated the current study in the application of these sawdust particles when considering it in X and Y
established theory of fractal to the sawmill environment. coordinates. The distribution pattern of these particles is
interesting to note in Figure 1.
3 Case study, results and discussion Figure I Case I: 3000-agregatedsawdust particles (see online
Since this paper deals with the formation of sawdust fractal versionfor colours)
pattern in human lungs, a case study dealing with the
practical consideration of size fractal theory of emitted
sawdust particles, which gives consideration to the
mechanism of aggregation in the structure and relevant til ,....
description of why and how the suggested cases would U) - ~,11" !too!'! '::'1 .~.tt;o!l
appear in human lungs is necessary. However, only a brief ut:: .1~· .'~-~li/~' ~ .,~
account which is sufficient for understanding the theoretical li-~~'I.'i' iI.;'~' '::t~$"a'".. ~~..~~~ , ~ .. :~
framework is presented. Numerical experimentation carried - ~'~~ :r.'";' I' 1.~.~r." .••0 k~~t:lf·~'.·
out resulted for the generation of results, which are !! I.t~.... -1"'l"~;.. ~.If'I-. ~ '" ~ _Nj~ •. "\]_'
presented as tables, each contains vital information that led ee-.s \" ~ It 1fr. ~~~
to the conclusion made in this work. Table I emerged from 'E ~•l•i'"~~~~"9 ". -X J ~~~,,_0 'It ~,..,
using the seed value 9,876 to estimate dimensions and 0 '.~r".' , ~~... , ,
natural logarithm of intercepts. Ten different >U- ~~, '
experimentations were carried out in which sawdust
particles of a specified number were constrained according
to rules 1 to 4, corresponding to Cases 1 to 4 stated at the
X-Coordinate of Particles
beginning of Section 2. Thus, the number of particles,
dimension and intercept for all the sample sizes ranging
from 500 to 5,000 sawdust particles, and increasing in steps Figure I shared structural visual resemblance with what is
of 500 particles, were considered. For example, when 500 supported by literature. Figures 2, 3 and 4 also relate to the
sawdust particles were experimented on, dimensions of structural visual resemblance of the aggregated sawdust
0.87, 0.87, 0.83 and 0.83 were obtained for Cases 1 to 4, particles.
UNIVERSITY OF IBADAN LIBRARY
146 T.A.o. Salau and S.A. Oke
Figure 2 Case 2: 3,000-agregated sawdust particles (see online The result presented in Table 2 is an outcome of the
version for colours) simulation based on different random number generator
seed values. Recall that in Table 1, the sample size that
produced the near-literature value of 1.71 was chosen,
which corresponds to a sample size in which the seed value
utilised is 9,876. In Table 2, it is observed that sawdust
~"' -¥:{
u.• ~ "\l- (~*'~-'¥
il1.1~~t ., particles counted inside circle of specified radius increases
t ~.t(:r., ~ as the specified radius increases.
-e, ~~.;. ~,.~' t,.~r~.. ..... . :'. ~"r: ...!:".!4fP;~0 '.~ ''~,~~'~~' .'!* ' ·. Table 2 Number of sawdust particles counted for Case I, seed'."..S '~;~."-.s',t I. '.'"I~.-.;. value 9876 using radius dimension method--rf~. '{',,~~ .'1:! q"~ Sawdust
0 ~";'.' '~. "~" particles counted Natural Natural logarithm0
t.l c· ". ". ,~, ;f Radius inside circle of logarithm of sawdust>- '
specified radius of radius particles counted
I 59 0.0000 4.0775
X·Coordinate of Particles 2 185 0.6931 5.2204
3 379 1.0986 5.9375
4 660 13863 6.4922
Figure 3 Case 3: 3000-agregated sawdust particles (see online
version for colours) 5 883 1.6094. 6.7833
6 1246 1.7918 7.1277
7 1648 1.9459 7.4073
8.- 2077 2.0794 7.6387
'" 9 2559 2.1972 7.8474'" 10' 3000 2.3026 8.0064
Table 3 Estimated fractal dimension for four (4) cases with
o
ten-different random number generator seed values
'"
c"' Estimated fractal dimensions
SIN Seed values
o Case I Case 2 Case 3 Case 4
o
o I 9876 1.72 175 1.56 1.65
>-
2 6789 1.63 1.69 1.72 1.77
3 4567 1.67 1.72 175 1.80
X·Coordinate of Particles
I 4 5678 175 179 1.70 1.785 6784 1.60 1.66 1.67 1.72
6 3456 1.72 1.77 1.75 1.81
Figure 4 Case 4: 3000-agregated sawdust particles (see online
version for colours) 7 7865 1.68 1.77 1.81 1.86
8 3789 175 1.82 1.73 177
9 3467 1.61 1.69 1.77 1.83
10 7896 1.65 1.72 1.64 1.68
''"" MEFD* 1.68 174 171 1.77<.>
~ Standard deviation 0.05 0.05 0.07 0.06
a-'". AE* 175 1.75 0.00 3.51o Notes: AE* means % absolute error comparing mean
'"
'" estimated fractal dimension with the standardc (1.71-literature); MEFD* could be written as
~
o mean estimated fractal dimension
o
o What was done in the experiment that generated Table 3
>- was to use the seed 'values of 6,789, 4,567, 5,678, 6,784,
3,456, 7,865, 3,789, 3,467 and 7,896, respectively and
X-Coordinate of Particles closely monitoring the dimensions that are close to the
literature value of 1.71. The results showed the mean
UNIVERSITY OF IBADAN LIBRARY
Fractal dimension and time factors of sawdust pattern formation in sawmills 147
estimated fractal dimension of 1.68, 1.74, 1.71, and 1.77 as Referring to Figure 6, the estimated fractal dimension for
well as the standard deviation of 0.05, 0.05, 0.07, and 0.06, Case 2 was consistently greater or equal to the estimated
for Cases I, 2, 3 and 4, respectively. The percentage fractal dimension for Case 1. Similarly the estimated fractal
absolute errors when the mean estimated fractal dimension dimension for Case 4 was consistently greater or equal to
was compared with the standard (1.7 I-literature) were 1.75, the estimated. fractal dimension for Case 3. The estimated
1.75, 0.00 and 3.51, respectively. In sum, the mean fractal dimension for the four cases tends toward a limiting
estimated fractal dimension obtained for the four studied value lesser than 2.0. The highest estimated fractal
cases range between 1.68 ± 0.05 and 1.77 ± 0.06. The dimension is 1.93 in Case 2.
percentage absolute error recorded comparing the mean The sample results presented in Table 4 was drawn out
estimated fractal dimension with the standard fractal of 3000 computer simulated results. The average was taken
dimension of 1.71 supported by literature range between over ten different simulators represented by ten different
0.00% and 3.51 % for all studied cases. random number generation seed values as in Table 3
Figure 5 is the log-log plot of specified radius and the column 2. It was observed for all studied cases that' the
sawdust particles counted inside the circle which radius has average time steps taken before the successive attachment of
been.specified. The slope of the line of best fit is taken as Nth sawdust particle statistically decreases as the aggregate
the estimated fractal dimension for the aggregated' sawdust size increases. Average time steps for Case I and Case 2 are
particles (fractal pattern) using the radius dimension same at the early stage of the sawdust particle aggregation,
method. The estimated dimension for the aggregated but differed at later stage. Similarly the average time steps
sawdust particles i~, 1.72 to two-decimal as indicated by the for Case 3 and Case 4 are same at the early stage of the
equation of line of best fit in Figure 5. All other dimensions sawdust particle aggregation, but differed at later stage.
reported in this study were similarly obtained. Very shorter average time steps were recorded for all the
cases at later stage which indicate saturation of the inside of
Figure 5 Estimated fractal dimension diagram fOTCase I with specified circle with aggregated sawdust particles. This
seed value of 9876 means that the probability of sawdust particle attachment
increases drastically at later stage. Case 2 utilised highest
number of 'total average time steps of 538080 before
9.00 r--------------, successful aggregation.
o 8.00
E ~ 7.00 Table 4 Sample of average time steps taken before the
~~ 6.00 successive attachment (aggregation) ofN-th sawdust
:;; :;; 500
e>t1. • particle
~-:;: 4.00 y = 1.7171 x + 4.0603
-;;.,; 3.00 Rl = 0.9996 N-th sawdust Average time steps taken before the sllccess of
~ ~ 2.00 particle sawdust particle attachment
;u 1.00 attached Case J Case 2 Case 3 Case 4
0.00 -l-----,.--...,.-------;r----,---~ 2 3343 3343 1433 1433
0.00 0.50 1.00 1.50 2.00 2.50
3 1699 1699 1855 1855
Natural logarithm of Specified Radius 4 2195 2195 1876 1876
5 1486 1486 1241 1241
Figure 6 Estimated fractal dimension for the four cases 6 2320 2320 2302 2302
7 2205 2205 1708 1708
8 2279 2279 2352 2352
c:
o 9 13il 1371 1451 1451
.~ 200
'E"
10 2638 2638 1385 1385
~1.5)
t"i' 2996 4 22 5 21
~
II.. 1.00 2997 3 23 II 26
'C
'" 2998 7 26 4 25
i:: 2999 3 21 29aInr------,.--.---.-----,- 3000 9 23 2 211!m zm 3&Xl 4&ll TAT* 508926 538080 499170 525570
~Patides inl'UTb::r Notes: TAT* means total of average time steps required
to attach all N-sawdust particles specified.
UNIVERSITY OF IBADAN LIBRARY
148 TA.D. Salau and SiA. Gke
Table 5 shows sample of total aggregation and total average Figure 7 was obtained using the whole data set for which
time steps expressed in percentage. The percentage of Table 5 is a sample. The curves of best fit to the data set are
particles aggregation required ranges from 10.00% to obtained respectively for the four cases as shown in Table 6.
100.00% in steps of 1.00%. For example, for a 10.00% of Observe that Table 6 contains the results obtained from the
sawdust particles, the percentage of total of average time derivation of the power law equation and the coefficient of
steps required for aggregation of N-specified sawdust correlation. Four' cases are considered: Cases I to 4, The
particles for Cases I to 4 were 47.72, 45.13,46.48,44.14, correlation coefficient was least for Case 3 (i.e,
respectively with its average being 45.87. The percentage of R2 = 0,8970), while it was the best for Case 2 (i.e.
total of average time steps required for 50% aggregation of R2 = 0,9213), However, the power exponent was least for
N-specified sawdust particles range between 88.42% and Case I (i.e., 0.4528) and the highest was recorded for Case
92.96% while the average over the studied cases is 90.50%. 4 (i.e. 0.4827), It was noticed that the mean value ofthe
Referring to Table 5 the percentage total of average time power law exponent is 0.4683 ± 0,0125,
steps required for 50% aggregation of N-specified sawdust
particles range between 88.42% and 92.69% while the Table 6 Power law equations for cases
average over the studied cases is 90.50%.
Case Power law equation and coefficient Power law
Table 5 Sample (out of 2,999-data set) of total aggregation of correlation exponent
and total averaJ?etime steps expressed in percentage 1 y = 15 x 04528; R2 = 0.9053 0.4528
2 y = 13.787 x 1l4656; R2 = 0.9213 0.4656
% of sawdust % of total of average time steps requiredfor
particles aggregation of Nsspecified particles. 3 y = 13.89 x 1l4722; R2 = 0.8970 0.4722
aggregation Cases 4 y= 12.868x 1l4827; R2 = 0.9113 0.4827
required Case 1 Case 2 Case 3 Case 4 average
Reference to Table 6, it is shown that the power law
10.00 47.72 45.13 46.48 44.14 45.87 exponent for the cases studied range between 0.4528 and
20.00 68.17 64.49 67.11 63.77 65.88 0.4827 while the mean value is 0.4683 ± 0.0125.
30.00 80.23 76.02 79.73 75.76 77.94
40.00 87.63 83.46 87.70 83.31 85.52 7 Conclusions and future work
50.00 92.41 88.47 92.69 88.42 90.50
This paper describes the formation of sawdust particles in
60.00 95.49 92.10 95.76 91.93 93.82 the lungs of humans and in other parts of the human body
70.00 97.49 94.71 97.65 94.58 96.11 where harmful residue sawdust may accumulate as a fast
branching rate in fractal pattems and also considers the
80.00 98.70 96.75 98.81 96.66 97.73 mechanism of aggregation in structure, These formed
90.00 99.52 98.48 99.56 98.45 99.00 patterns (Cases I to 4) describe the settled state of the
100.00 100.00 \00.00 100.00 100.00 100.00 sawdust particles. The implication is that as these sawdust
particles settle down in the human body these cells are
destroyed on very fast rate by the toxic nature of sawdust
Figure 7 Percentage of successive aggregation and percentage
total of average time steps required for aggregationof, particles, Thus, an attempt to remove the effect of toxicity
N-specified sawdust particles (see online version for on the body will require special skill and it will be cost
colours) intensive,
The estimated fractal dimension of emerged fractal
Porcontago Aggroagatlon/Total of Avo rago Tlmo Stops pattem of 3,000-sawdust particles aggregated in a circle of
t Caso 1) ten-unit radius, sawdust particles closeness not greater than
0.3-unit, random number generating seed value of 9,876 and
. using Case I rule agreed 'by 0.6% absolute error comparing'a" .z 100
"- with literature result of 1,71, The mean of the estimated-:;; 0
" c 80 fractal dimension for all studied cases range between-E:;O::'"..!.
-.• •• .u• 1.68 ± 0.05 and 1.77 ± 0,06 while the percentage absolute~~t:~ "~,0. so error comparing with standard fractal dimension of 1.71
" "'.., y = 15xD,ma~0_~:~; R' =
range between 0,00% and 3,51%. This study further shows
0.9053
- 0'- 40 that average time steps required to successively attach an
~~ ~ additional sawdust particle decreases as the percentage20
- rr successive attachment increases, That is the probability ofo ••
~a:::: an additional sawdust particle attachment increases
0
so drastically at later stage than at earl ier stage of growing0 20 40 80 100
fractal pattern using any of the studied rules. Also
% of Succeslve Aggragatlon out of N-specifiod P3.rtic195
successive aggregation of 3,000 particles took total average
time steps of 538,080, the highest recorded for Case 2. This
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Fractal dimension and time factors of sawdust pattern formation in sawmills 149
study also shows that the percentage of total of average time operators at sawmills as a means to further understand this
steps required for execution of 50% aggregation range problem. Simulation experiments would be performed and
between 88.42% and 92.69% while the average taken over the work will be analytically treated for good results. In
the studied cases is 90.50%. Percentage of total of average addition, since there is an increased level 0 f operational
time steps required for aggregation of specified n-sawdust activities at SL vmills, which translates into increased
particles varies according to power law to percentage sawdust generation. There is a further motivation to study
successive aggregation. The mean value of the power the current problem. A method that studies the dynamics of
exponents was 0.4683 ± 0.0125. Case 3 and its rules are sawdust movements in sawmills which could aid
most reasonable if used in a real project since its percentage understanding its control strategy using defined parameters
absolute error compared with mean fractal dimension with space of forced Duffing's dynamic system could be
the standard literature value is 1.71 was zero. This is statistically investigated for the probability of total
evidenced in Table 3. Thus, growing fractal pattern in 2-D parameter points that would exhibit chaotic behaviour using
Euclidean space using any of the studied rules demand lot of maximum Lyapunov exponent value indicator. This
computational time and nothing better can be projected in approach would involve randomly picking parameter points
3-D Euclidean space. that would be used to solve a second order non-linear
Concerning future investigations, we submit as follows. differential Duffing's equation numerically using
A first future investigation that must be pursued rigorously Runge-Kutta Algorithms, which may be coded in a
concerns detailed practical studies in a number of sawmills computer language. Thus, cycles of solutions based on
where actual real life data could be collected and analysed forcing period would be obtained.
in order to verify the working of the model in reality. Recall
that only numerical experiments wer~ conducted here and
the study was related to practice. but not to the extent of Acknowledgements
collecting practical data from the field. Implementing a
practical study would be exciting as some interesting results The authors are greatly indebted to a number of people that
may be obtained. Viewing from another perspective, it is assisted in bringing up the quality of this work to the current
observed that extensive analysis based on R2 has been done standard. We specially recognise the great efforts of Asim
in this work. This reflects the coefficients of determination, Kumar Pal, Jian Lu and Jun Wu. Other people who
which shows the relationship between two sets of values. co~tributed immensely to the refinement of the work are
However, for model accuracy the use of R2 may not be Petr Konas and Mohammed Al-Nawafleh, We also thank
sufficient for testing the statistical significance of models. the reviewers and those that we forgot to acknowledge but
Several other statistical parameter error indicators which contributed to the manuscript's improvement.
were not considered in the current work may be evaluated
with their results compared to R2. This way, the error
measurements may be monitored and the model adjusted References
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