Mathematical model of the shape of broad bean seed

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 Nadwilaski 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, uorine. According to wicicki, after Jerzak et al. [2012], the annual demand for plant protein in Poland amounts to about 1 million tons; the imported ground soya bean satises the demand for about 0.8 million tons of protein. Podleny [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 Podleny and Sowiski [2004], the traditional Nadwilaski variety yields better than some self-ending varieties, while arrangement of plants on area unit signicantly 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 [Podleny 2006]. The seed geometric features of broad bean are very important in many processing operations: precision drilling [Podleny 2006], hulling [Mieszkalski 1993, 1999] and seed grinding [ysiak 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 Nadwilaski 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 Nadwilaski variety amounts to 463 g [Kulig et al. 2007] or 475.3 g [Podleny 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 Nadwilaski 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 verication 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 Frczek 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 Nadwilaski variety are given in Table 1. Exemplary photograph of broad bean seed of Nadwilaski 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 Nadwilaski variety (moisture content 13.6%) Designation of broad bean seed Dimension 1 2 3 4 5 6 mm mm mm mm mm mm Length (a) 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 FIGURE 1. Exemplary photograph of broad bean seed of Nadwilaski variety (own elaboration, dimensions in Table 1) 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 ij, a Asin( i) d (cos( )) 2 i ecos( j) Y ij, b sin( i) sin( j) f (cos( )) 2 i sin( i) g sin( i) (1) (2) Z ij, c cos( i) (3) where: i i (4) N i 2 j (5) N The measured basic dimensions of broad bean seeds (a, b, c) included in equations 1, 2, 3 are written down in matrix 6: a1 b1 c1 12.8 10 8.2 a2 b2 c2 13.5 12.1 9.6 a3 b3 c3 12.1 10.1 8.3 a4 b4 c4 10.3 9.5 7.5 a5 b5 c5 11.9 10.3 8.2 a6 b6 c6 11.1 9.1 7.6 (6)

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: N 25 A 1.5 d 1 e 1.5 f 2 g 2 i 0... N j 0... N (7) (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 dimensions of broad bean seed model for a given shape (equations 1, 2, 3) have the following form: XN YN ZN a X max( X) min( X) b Y max( Y) min( Y) c max( Z) min( Z) Z (9) (10) (11) 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 Nadwilaski 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 Nadwilaski 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 verication. 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 veried, 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 verication 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 veried; therefore, it can be used for description of broad bean seed shape. TABLE 2. Verication results for models of broad bean seeds of Nadwilaski variety represented with the discrete spatial surfaces Dimension under verication Representation with discrete spatial surface (equations 9, 10, 11) formula for basic result [mm] dimension of seed 1 2 3 4 5 6 Length (a) max(xn) min(xn) 12.8 13.5 12.1 10.3 11.9 11.1 Width (b) max(yn) min(yn) 10 12.1 10.1 9.5 10.9 9.1 Thickness (h) max(zn) min(zn) 8.2 9.6 8.3 7.5 8.2 7.6

46 L. Mieszkalski TABLE 3. Results of calculations on selected shape factors for broad bean seeds of Nadwilaski variety together with mathematical models Reference Grochowicz (1994) Mohsenin (1986) Donev et al. (2004) Grochowicz (1994) Mohsenin (1986) Donev et al. (2004) Designations of seeds and their models Shape factor 1 2 3 4 5 6 Real seeds K m 0.78 0.9 0.83 0.9 0.87 0.82 K w 0.64 0.71 0.69 0.73 0.69 0.69 S n 0.79 0.86 0.83 0.88 0.84 0.83 1.56 1.41 1.46 1.37 1.45 1.46 1.22 1.26 1.22 1.27 1.26 1.2 Models of seeds K m 0.78 0.9 0.83 0.9 0.87 0.82 K w 0.64 0.71 0.69 0.73 0.69 0.69 S n 0.79 0.86 0.83 0.88 0.84 0.83 1.56 1.41 1.46 1.37 1.45 1.46 1.22 1.26 1.22 1.27 1.26 1.2 CONCLUSIONS 1. 2. The proposed mathematical model described with parametric spatial surface can serve representing the 3D solids that are similar to broad bean seeds of Nadwilaski variety in respect to the shape and basic dimensions. 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. REFERENCES ANDERS A., KALINIEWICZ Z., MAR- KOWSKI P. 2012: Zastosowanie skanera 3D do monitorowania ksztatu produktów spoywczych na przykadzie pieczywa. Inynieria Rolnicza 3(138): 7 14. ANDERS A., KALINIEWICZ Z., MAR- KOWSKI P. 2013: Porównanie cech geometrycznych nasion pieprzycy siewnej (Lepidium sativum L.) z okryw oraz poddanych obuskiwaniu. Acta Agrophysica 20(1): 17 28. DONEV A., CISSE I., SACHS D., VARIA- NO E.A., STILLINGER F.H., CONNEL- LY R., TORQUATO S., CHAIKIN P.M. 2004: Improving the density of Jammed Disordered Packings using Elipsoids. Science 303: 990 993. FLIS M., ZDUCZYK Z., SOBOTKA W. 1996: Moliwoci zwikszenia przydatnoci paszowej bobiku i ubinu poprzez obuskanie nasion. Post. Nauk Rol. 5: 104 114. FOLEY J.D., VAN DAM A., FEINER S.K., HUGHES, J.F., PHILLIPS R.L. 2001: Wprowadzenie do graki komputerowej. WNT, Warszawa. FRCZEK J., WRÓBEL M. 2006: Metodyczne aspekty oceny ksztatu nasion. Inynieria Rolnicza 12(87): 155 163.

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48 L. Mieszkalski wprowadzono trzy parametry (a, b, c), za których pomoc ustalano podstawowe wymiary zewntrzne modelu bryy nasiona bobiku (dugo, szeroko, grubo). Ksztat nasiona mona zmienia picioma parametrami (A, d, e, f, g), a liczb poudników i równoleników na dyskretnej powierzchni przestrzennej zmienia si parametrem N. Wizualizacji modeli 3D bry nasion bobiku dokonano za pomoc programu komputerowego Mathcad. MS. received December 2013 Author s address: Leszek Mieszkalski Wydzia Inynierii Produkcji SGGW Katedra Organizacji i Inynierii Produkcji 02-787 Warszawa, ul. Nowoursynowska 164 Poland e-mail: leszek_mieszkalski@sggw.pl