Annals of Warsaw University of Life Sciences – SGGW
Agriculture No 63 (Agricultural and Forest Engineering) 2014: 41–48
(Ann. Warsaw Univ. Life Sci. – SGGW, Agricult. 63, 2014)
Mathematical model of the shape of broad bean seed
LESZEK MIESZKALSKI
Department of Production Management and Engineering, Warsaw University of Life Sciences – SGGW
Abstract: Mathematical model of the shape of
broad bean seed. A method for mathematical
modeling of the shape of broad bean seed solid
(Vicia faba minor) of NadwiĞlaĔski variety has
been proposed. To describe the seed solid, in
mathematical model the spatial surface parametric equations were used. The discrete spatial surface subjected to modeling surrounds the volume
situated near the external surface of broad bean
seed. In equations there were introduced three parameters (a, b, c), used for determination the basic
external dimensions of broad bean seed (length,
width, thickness). The seed shape can be changed
with 5 parameters (A, d, e, f, g), while the number
of meridians and parallels on discrete spatial surface can be changed with parameter N. The visualization of 3D models for broad bean seed solids
was performed with the use of computer software
Mathcad.
Key words: broad bean seeds, shape, shape factors, mathematical model, discrete spatial surface,
model 3D
INTRODUCTION
The seeds of broad bean (Vicia faba minor) contain a high quantity of valuable
protein (about 33%) and are important
raw material for food industry. Apart from
protein, they contain also from 1 to 1.6%
of fat, from 7.7 to 9.6% of ¿ber, about
55% of carbohydrates, including about
41% of starch. They also contain many
mineral components: calcium, magnesium, potassium, phosphorus, sodium,
iron, zinc, copper, Àuorine. According
to ĝwiĊcicki, after Jerzak et al. [2012],
the annual demand for plant protein in
Poland amounts to about 1 million tons;
the imported ground soya bean satis¿es
the demand for about 0.8 million tons of
protein. PodleĞny [2005] presented the
perspectives of cultivation and utilization of leguminous plants seeds in Poland; the cultivated area of these plants
in Poland drastically decreased.
The scientists pointed out at connections of Bovine Spongiform Encephalopathy in cattle and Creutzfeldt-Jakob
disease in human with application of
meat-and-bone meal in feeding animals;
therefore, the import and application
of this meal have been prohibited. According to Majchrzycki et al. [2002], it
caused an increased demand for the high-protein feeds of vegetable origin. Broad
bean can be an alternative for soybean in
providing the feed protein. The involucre of broad bean varieties of dark colour
contains the antinutritional substances,
therefore, the broad bean seeds are usually subjected to the process of hulling
[Mieszkalski 1993]. Flis et al. [1996]
maintain that this process can increase
the seeds’ feeding usability. According
to PodleĞny and SowiĔski [2004], the
traditional NadwiĞlaĔski variety yields
better than some self-ending varieties,
while arrangement of plants on area unit
signi¿cantly affects their morphological
features. Sowing of broad bean seeds
42
L. Mieszkalski
with a precision drill enables to obtain
the higher seed yield by about 22%, than
in application of general purpose sawing
machines [PodleĞny 2006].
The seed geometric features of
broad bean are very important in many
processing operations: precision drilling
[PodleĞny 2006], hulling [Mieszkalski
1993, 1999] and seed grinding [àysiak
and Laskowski 2004].
The seeds make a set of unrepeated
elements in respect of the shape and dimensions. Grzesiuk and Kulka [1981]
and Szot [1987] proposed determination
of the three basic dimensions of seeds
(length, width and thickness) in their
geometric characteristics. According to
Mieszkalski [1993], the ratio between
length, width and thickness of broad bean
seeds of NadwiĞlaĔski variety amounts
to 1 : 0.83 : 0.65. The seed length range
amounts to 10.1–13.6 mm, width 8.3–
–11.9 mm, and thickness 7.4–9.7 mm
[Mieszkalski and Lewandowski 1996].
The mass of 1,000 seeds of NadwiĞlaĔski
variety amounts to 463 g [Kulig et al.
2007] or 475.3 g [PodleĞny 2009].
Modelling of broad bean seed shape
consisted in determination of the model of
a solid that represented the road bean seeds.
In 1993 Mieszkalski proposed a sphere as
the model of broad bean seed in the process of its hulling. An ellipsoid model was
considered also [Mieszkalski and Lewandowski 1996, Mieszkalski 1999]. Neither
the sphere nor the ellipsoid represented
precisely the shape of broad bean seed.
Dynamic development of computer
graphics [Kiciak 2000, Foley et al. 2001]
and the methods for mathematical shape
modelling [Gielis 2000, Gielis and Gerats
2004] allow for more precise working out
a shape of broad bean seed solids.
This work aims at formulating a mathematical model for the broad bean seed
shape with consideration to a change in
basic dimensions of seeds.
MATERIAL AND METHODS
The research material was broad bean
seeds of NadwiĞlaĔski variety. There
were selected six seeds of broad bean of
different dimensions. The seed moisture
content was determined by drying and
weighing method, while length, width
and thickness were measured with a slide
caliper with accuracy 0.1 mm.
A mathematical model was formulated with the use of parametric equations,
that enabled to create the solid surfaces
of the shape similar to broad bean seeds
of given dimensions (length, width,
thickness). The basic dimensions of solids obtained from mathematical model
were compared with dimensions of the
real broad bean seeds. In veri¿cation
of mathematical model for broad bean
shape there were used the shape factors
according to Mohsenin [1986], Grochowicz [1994], Donev et al. [2004]. Description of these factors can be found in
works of Anders et al. [2012, 2013] and
Frączek et al. [2006]. The visualization
of solid models was performed with the
use of computer software Mathcad.
RESULTS OF MEASUREMENTS
The basic dimensions of broad bean
seeds of NadwiĞlaĔski variety are given
in Table 1.
Exemplary photograph of broad bean
seed of NadwiĞlaĔski variety is presented in Figure 1.
Mathematical model of the shape of broad bean seed
43
TABLE 1. Basic dimensions of broad bean seeds of NadwiĞlaĔski variety (moisture content 13.6%)
Designation of broad bean seed
Dimension
1
2
3
4
5
6
mm
mm
mm
mm
mm
mm
12.8
13.5
12.1
10.3
11.9
11.1
Width (b)
10
12.1
10.1
9.5
10.3
9.1
Thickness (c)
8.2
9.6
8.3
7.5
8.2
7.6
Length (a)
FIGURE 1. Exemplary photograph of broad bean seed of NadwiĞlaĔski variety (own elaboration, dimensions in Table 1)
where:
MATHEMATICAL MODEL FOR
BROAD BEAN SEED SOLID
The matrix equations of coordinates X,
Y, Z of points on the surface describing
broad bean seed solid are of the form:
X i, j
a ˜ A ˜ sin(M i ) ˜
˜ ª«d ˜ (cos(M i )) 2 e ˜ cos(- j )º»
¬
¼
Y i, j
˜ ªf
«¬
(1)
b ˜ sin(M i ) ˜ sin(- j ) ˜
˜ (cos(M i )) 2 ˜ sin(M i ) g ˜ sin(M i )º
»¼
(2)
Z i, j
c ˜ cos(M i )
(3)
Mi
i ˜S
N
(4)
-j
i ˜ 2˜S
N
(5)
The measured basic dimensions of
broad bean seeds (a, b, c) included in
equations 1, 2, 3 are written down in matrix 6:
ª a1
«a 2
«
«a 3
«a 4
«
«a5
«¬a 6
b1
b2
b3
b4
b5
b6
c1º
c 2»
»
c3»
c 4»
»
c5»
c6»¼
ª12.8 10 8.2º
«13.5 12.1 9.6»
«
»
«12.1 10.1 8.3» (6)
«10.3 9.5 7.5»
«
»
«11.9 10.3 8.2»
«¬11.1 9.1 7.6»¼
44
L. Mieszkalski
In vector 7 there is given the number of
meridians and parallels on the surface of
broad bean seed and the shape parameters,
while in vector 8 the range variables:
ªN º
« A»
« »
«d »
«e»
« »
«f»
«¬ g »¼
ª 25º
«1.5»
« »
«1»
«1.5»
« »
«2»
«¬ 2 »¼
(7)
ªi º
«¬ j»¼
ª0... N º
«¬0... N »¼
(8)
To obtain the required dimensions included in matrix 6 for broad bean seed,
the equations 1, 2, 3 were scaled. The
matrix equations describing basic di-
mensions of broad bean seed model for
a given shape (equations 1, 2, 3) have the
following form:
XN
a
˜X
max( X ) min( X )
YN
b
˜Y
max(Y ) min(Y )
(10)
ZN
c
˜Z
max( Z ) min( Z )
(11)
(9)
The 3D models for broad bean seeds
are presented in Figure 2.
Figure 3 presents the main and right
side projections of the model for broad
bean seed of NadwiĞlaĔski variety. Comparing the solids on Figures 1 and 3 one
can ¿nd that the shape of broad bean seed
FIGURE 2. 3D models for broad bean seeds of subsequent numbers from 1 to 6 and dimensions included in matrix 6
Mathematical model of the shape of broad bean seed
45
FIGURE 3. Main and right side projections of model for broad bean seed of NadwiĞlaĔski variety
solid obtained from mathematical model
is similar to the shape of real broad bean
seed.
VERIFICATION OF MODELS FOR
BROAD BEAN SEED SOLIDS
The mathematical model that describes
the shape of broad bean seeds with the
use of spatial parametric surface was
subjected to veri¿cation. The characteristic verifying dimensions were: seed
length (a1, …, a6), seed width (b1, …,
b6) and seed thickness (c1, …, c6). This
mathematical model could be regarded
as veri¿ed, if with the use of discrete
spatial surface it is possible to determine
the three basic dimensions of broad bean
seed similar to the measurement results.
The veri¿cation results for the models of broad bean seeds represented with
the discrete spatial surfaces are listed in
Table 2; it is evident that these surfaces
precisely go through the points that determine the basic dimensions of broad
bean seeds and are the same as measured
seed dimensions. The values of shape
factors (Table 3) of real seeds are such as
these of their models.
Since the spatial surface determined
with the presented mathematical model
always go through points that determine
the real basic dimensions of broad bean
seeds, the proposed mathematical model
can be recognized as veri¿ed; therefore,
it can be used for description of broad
bean seed shape.
TABLE 2. Veri¿cation results for models of broad bean seeds of NadwiĞlaĔski variety represented with
the discrete spatial surfaces
Dimension
under
veri¿cation
Representation with discrete spatial surface
(equations 9, 10, 11)
result [mm]
formula for basic
dimension of seed
1
2
3
4
5
6
12.8
13.5
12.1
10.3
11.9
11.1
max(YN) ņ min(YN)
10
12.1
10.1
9.5
10.9
9.1
max(ZN) ņ min(ZN)
8.2
9.6
8.3
7.5
8.2
7.6
Length (a)
max(XN) ņ min(XN)
Width (b)
Thickness (h)
46
L. Mieszkalski
TABLE 3. Results of calculations on selected shape factors for broad bean seeds of NadwiĞlaĔski variety together with mathematical models
Designations of seeds and their models
Reference
Shape
factor
1
2
3
Km
0.78
0.9
Kw
0.64
Sn
4
5
6
0.83
0.9
0.87
0.82
0.71
0.69
0.73
0.69
0.69
0.79
0.86
0.83
0.88
0.84
0.83
Į
1.56
1.41
1.46
1.37
1.45
1.46
ȕ
1.22
1.26
1.22
1.27
1.26
1.2
Real seeds
Grochowicz
(1994)
Mohsenin
(1986)
Donev et al.
(2004)
×
Grochowicz
(1994)
Models of seeds
Km
0.78
0.9
0.83
0.9
0.87
0.82
Kw
0.64
0.71
0.69
0.73
0.69
0.69
Sn
0.79
0.86
0.83
0.88
0.84
0.83
Į
1.56
1.41
1.46
1.37
1.45
1.46
ȕ
1.22
1.26
1.22
1.27
1.26
1.2
Mohsenin
(1986)
Donev et al.
(2004)
CONCLUSIONS
1. The proposed mathematical model
described with parametric spatial surface can serve representing the 3D
solids that are similar to broad bean
seeds of NadwiĞlaĔski variety in respect to the shape and basic dimensions.
2. Changing the values of control parameters in the proposed model one
can generate any (within Vicia faba
minor) solids that are similar to broad
bean seeds in respect of shape and basic dimensions.
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Streszczenie: Matematyczny model ksztaátu nasion bobiku. Zaproponowano metodĊ matematycznego modelowania ksztaátu bryáy nasion bobiku (Vicia faba minor) odmiany NadwiĞlaĔski.
W modelu matematycznym do opisu ksztaátu bryáy nasion bobiku zastosowano równania parametryczne powierzchni przestrzennej. Modelowana
dyskretna powierzchnia przestrzenna otacza objĊtoĞü leĪącą w bliskiej odlegáoĞci od powierzchni zewnĊtrznej nasiona bobiku. W równaniach
48
L. Mieszkalski
wprowadzono trzy parametry (a, b, c), za których pomocą ustalano podstawowe wymiary zewnĊtrzne modelu bryáy nasiona bobiku (dáugoĞü,
szerokoĞü, gruboĞü). Ksztaát nasiona moĪna zmieniaü piĊcioma parametrami (A, d, e, f, g), a liczbĊ
poáudników i równoleĪników na dyskretnej powierzchni przestrzennej zmienia siĊ parametrem
N. Wizualizacji modeli 3D bryá nasion bobiku
dokonano za pomocą programu komputerowego
Mathcad.
MS. received December 2013
Author’s address:
Leszek Mieszkalski
Wydziaá InĪynierii Produkcji SGGW
Katedra Organizacji i InĪynierii Produkcji
02-787 Warszawa, ul. Nowoursynowska 164
Poland
e-mail: [email protected]