数学 外文翻译 外文文献 真空压力播种机的数学建模

目录1.Introduction (3)引言 (17)2.Materials and methods (4)材料和方法 (18)3.Results and discussion (7)结果与讨论 (21)4. Conclusions (13)结论 (26)5.Acknowledgements (13)鸣谢 (26)Mathematical Modelling of Vacuum Pressure on a Precision SeederD. Karayel 1, Z. B. Barut 2 and A. Özmerzi 11 Akdeniz University, Faculty of Agriculture, Department of Agricultural Machinery, 07070, Antalya, Turkey2 Çukurova University, Faculty of Agriculture, Department of Agricultural Machinery, Adana, Turkey; e-mail of corresponding author: dkarayel@.trReceived 9 July 2003; accepted 19 January 2004. Available online 13 April 2004. AbstractThe purpose of this research was to determine the optimum vacuum pressure of a precision vacuum seeder and to develop mathematical models by using some physical properties of seeds such as one thousand kernel mass, projected area, sphericity and kernel density. Maize, cotton, soya bean, watermelon, melon, cucumber, sugarbeet and onion seeds were used in laboratory tests. One thousand kernel mass, projected area, sphericity and kernel density of seeds varied from 4.3 to 372.5 g, 5–77 mm2, 38.4–85.8% and 440–1310 kg m−3, respectively. The optimum vacuum pressure was determined as 4.0 kPa for maize I and II; 3.0 kPa for cotton, soya bean and watermelon I; 2.5 kPa for watermelon II, melon and cucumber; 2.0 kPa for sugarbeet; and 1.5 kPa for onion seeds.The vacuum pressure was predicted by mathematical models. According to the results, the final model could satisfactorily describe the vacuum pressure of the precision vacuum seeder with a chi-square of 2.51×10−3, root mean square error of 2.74×10−2 and modelling efficiency of 0.99.NomenclatureNomenclaturea, b, c, d, e regression coefficientsE m modelling efficiencyE rms root mean square errork exp experimental vacuum pressure, kPak exp, mean mean value of experimental vacuum pressure, kPak pre predicted vacuum pressure, kPaL length, mmm1000one thousand kernel mass, gN number of observationn number of constants in the modelP projected area, mm2P v vacuum pressure, kPap probabilityR2coefficient of determinationT thickness, mmW width, mmφsphericity, %χ2chi-squareρk kernel density, kg m−3Article OutlineNomenclature1. Introduction2. Literature review3. Materials and methods4. Results and discussion5. ConclusionsAcknowledgementsReferences1. IntroductionPrecision sowing has been a major thrust of agricultural engineering research for many years; however, most of the research and development work has dealt with seeders for agronomic crops.The main purpose of sowing is to place the seed to a certain space and a depth in the seedbed. Precision seeders place seeds at the required spacing and provide a better growing area per seed. There are two common types of precision seeders: belt and vacuum. Precision vacuum seeders have a metering plate with metering holes on a predetermined radius. A vacuum is applied to these metering holes by means of a race machined in a backing plate. As the plate rotates, the vacuum applied to the metering holes enables them to pick up seeds from the seed hopper. Precision vacuum seeders have the following advantages over the mechanical seeders: better working quality,more precise seed rates with lower rate of seed damage, better control and adjustment of upkeep and drift of seeds, and broader spectrum of applicability (Soos et al., 1989).A seeder should place a seed in an environment in which the seed will reliably germinate and emerge. A number of factors affect the spacing of plants. The seed selection mechanism may fail to select or drop a seed resulting in large spacing between seeds. The mechanism may select and drop multiple seeds resulting in small spacings between seeds. Seed quality, soil conditions, seeder design and the skill of the operator all play a part in determining the final plant stand.The physical properties of seeds are essential for the design of equipment for handling, processing, storing and sowing the kernels. Various types of cleaning, grading, separation and sowing equipment are designed on the basis of the physical properties of seeds. However, no model has been found to describe seeder parameters such as vacuum pressure related with physical properties of seeds.The physical properties of the seeds are the most important factors in determining the optimum vacuum pressure of the precision vacuum seeder. In this study, using some of these, e.g. one thousand kernel mass, projected area, sphericity and kernel density, mathematical models were developed to predict optimum vacuum pressure. The experimental values of vacuum pressure were determined from laboratory test procedure.2. Materials and methodsThe laboratory test procedure involved testing the metering uniformity of the seeder at the different vacuum pressure with the different seeds: two different maize varieties (maize I and maize II), cotton, soya bean, two different watermelon varieties (watermelon I and watermelon II), melon, cucumber, sugarbeet and onion. These seeds represent several seed shapes varying from spherical (soya bean, maize II) to flat and elongated (maize I, melon, watermelon, cucumber). Two different varieties of maize and watermelon seeds were selected, because of the more diverse range of one thousand kernel mass, projected area, sphericity or kernel density than other seeds. All seeds used in this research were uncoated seed. The main dimensions of the seeds are given in Table 1. The seeder was set to space the seeds as closely to the recommended spacing as possible.Table 1. Means and standard errors of the seed dimensionsA grease belt test stand was used to determine sowing uniformity of each seed at the different vacuum pressures. This particular test stand had a 150 mm wide belt with a 7·5 m long horizontal viewing surface. A seeder row unit was mounted on a greased belt test stand which utilised an adjustable speed drive mechanism to operate the seed metering devices at a known constant speed. Sufficient oil was added to the top surface of belt to capture the seed as it was released from seeder unit without rolling or bouncing of seed on the belt surface. A wide variety of measures were used to qualify seeder performance with regard to plant spacing (Brooks & Church, 1987; Karayel & Özmerzi, 2001; Jasa & Dickey, 1982). Some tests used performance measures involving distance between plants in the field. Other tests used performance measures involving distance between seeds on grease belt test stand or by opto-electronic sensor system ( Bracy et al., 1998; Smith et al., 1991; Lan et al., 1999). A few tests used performance measures involving distance between seeds sown into soil ( Panning, 1997).A precision vacuum seeder unit was operated in all treatments (Fig. 1). The seeder unit was a general purpose seeder designed for row crops such as maize and soya beans. Three different vacuum plates with different hole diameters were used in the metering mechanism. The diameter of vacuum plates were 230 mm. The holes were drilled along a 200 mm diameter pitch circle. The holes of the vacuum plate were 3·5 mm in diameter for maize I, II, soya bean and cotton; 2·5 mm in diameter for watermelon I, II, melon, and cucumber and 1·5 mm in diameter for sugarbeet and onion. The seed plate operated in a vertical plane. Air suction from the holes of the seed plate caused the seed to stick to the holes. The stuck seed was released from the rotating plate by temporarily preventing airflow. The absence of suction allowed the seed to be dropped into soil. It had no seed tube and the seed fall height (12 mm) of the seeder was kept low in order to reduce the chance of non-uniform spacing which can occur due to the bouncing of seed, if dropped from high plane. The vacuum level was regulated by adjusting the size of an opening in the vacuum line of seeder and measured with a manometer.Fig. 1. The metering mechanism of the precision vacuum seeder: 1, vacuum plate; 2, seed; 3, seed box; 4, air suction canal; 5, air cut; 6, furrow openerThe seeder was operated over the greased belt at a ground speed of 1 m s−1 and adjusted to four vacuum pressures 2.0, 3.0, 4.0 and 5.0 kPa for maize I, II, soya bean and cotton; 2.0, 2.5，3.0 and 3.5 kPa for melon, watermelon I, II and cucumber; 1.0, 1.5, 2.0 and 2.5 kPa for sugarbeet and onion seeds. Seed spacings were measured over a distance of 7 m. The seeder was adjusted to deliver a nominal seed spacing of 230 mm for maize I and II, 170 mm for cotton, 105 mm for soya bean, 550 mm for watermelon I, II, melon and cucumber, 150 mm for sugarbeet and 85 mm for onion.The sowing uniformity was analysed using the methods as described by Kachman and Smith (1995). The multiple index is the percentage of spacings that are less than or equal to half of the theoretical spacing and indicates the percentage of multiple seed drops. The miss index is the percentage of spacings greater than 1.5 times the theoretical spacing and indicates the percentage of missed seed locations or ‘skips’. Quality of feed index is the percentage of spacings that are more than half but no more than 1.5 times the theoretical spacing. Quality of feed index is 100% minus miss and multiple index and indicates the percentages of single seed drops. Preciseness is the coefficient of variation of the spacings that are classified as singles after omitting the outliers consisting of misses and multiples.Kachman and Smith (1995)recommended using miss index, multiple index, quality of feed index and preciseness for summarising the uniformity of seeder metering rather than mean or sample coefficient of variation. They concluded that several measures were needed to give a true picture of seeder uniformity. For this study, miss index, multiple index, quality of feed index and preciseness are reported.Various physical properties of seeds including kernel density, projected area, sphericity and one thousand kernel mass are the most important factors in determining the optimum vacuum pressure of the precision vacuum seeder (Barut, 1996). The physical properties of the seeds were determined by the following methods: Linear dimensions, i.e.length, thickness and width were measured by using a vernier caliper with a sensitivity of 0.01 mm. Sphericity φ were calculated by using the following equation (Mohsenin, 1970):(1) where: L is the length; W is the width; and T is the thickness in mm.One thousand kernel mass was measured by an electronic balance with a sensitivity of 0.001 g.Kernel density was measured by the liquid displacement method. Toluene (C7H8) was used rather than water because it was not absorbed by fruits (Mohsenin, 1970; Ögöt, 1998).Projected area was determined by using a digital camera (Kodak DC 5000) and Sigma Scan Pro 5 program.For the estimation of the vacuum pressure, in relation to kernel density, projected area, sphericity and one thousand kernel mass, mathematical models were developed. The suitability of the final model was compared and evaluated using chi-square, root mean square error and modelling efficiency. Chi-square χ2, root mean square error E rms and modelling efficiency E m were calculated as follows:(2)(3)(4) where: k exp is the experimental vacuum pressure in kPa; k exp,mean is the mean value of experimental vacuum pressure in kPa; k pre is the predicted vacuum pressure in kPa; N is the number of observations; and n is the number of constants in the model.Reduced chi-square is the mean square of the deviations between the experimental and calculated values for the models and, is used to determine the goodness of the fit. The lower values of the reduced chi-square, the better the goodness of the fit. The root mean square error shows the deviations between the calculated and experimental values and it requires to reach zero. The modelling efficiency also shows the ability of the model and its highest value is 1 (Yaldiz et al., 2001; Ertekin & Yaldiz, 2004).Each experiment was arranged as a randomised complete block (Neter et al., 1990) and replicated five times. An analysis of variance method was applied to analyse data sets using a statistical software package SAS. Duncan's multiple-range tests were used to identify significantly different means within dependent variables.3. Results and discussionThe effect of vacuum pressure on sowing uniformity of the vacuum seeder was analysed relating to the multiple index, miss index, quality of feed index and preciseness. Multiple index, miss index and quality of feed index were combined for analysis of variance to determine the significant difference in the variability among the parameters. The results of this analysis are given in Table 2, Table 3 and Table 4. All measurement of sowing uniformity of the vacuum seeder were affected by vacuum pressure.Table 2. The sowing uniformity of the vacuum seeder with maize I and II, cotton and soya bean seeds for different vacuum pressureNote: Means within a group followed by same letter are not significantly different at probability p=0·05, by Duncan's multiple range test.Table 3. The sowing uniformity of the vacuum seeder with watermelon I and II, melon and cucumber seeds for different vacuum pressureNote: Means within a group followed by same letter are not significantly different at probability p=0·05, by Duncan's multiple range test.Table 4. The sowing uniformity of the vacuum seeder with sugarbeet and onion seeds for different vacuum pressureNote: Means within a group followed by same letter are not significantly different at probability p=0.05, by Duncan's multiple range test.The optimum vacuum pressure was determined for each seed according to quality of feed index and preciseness. As can be seen from laboratory study results in Table 2, Table 3 and Table 4, the highest seed spacing uniformities (quality of feed index) and the lowest preciseness values were obtained at the vacuum pressure of 4.0 kPa for maize I and II; 3.0 kPa for cotton, soya bean and watermelon I; 2.5 kPa for watermelon II, melon and cucumber; 2.0 kPa for sugarbeet and 1.5 kPa for onion seeds. The most uniform sowing uniformity was obtained with soya bean seeds at any vacuum pressures. Uniform, spherical seeds such as soya bean and maize II were easy to meter with the vacuum metering system.The miss index decreased and the multiple index increased with increasing vacuum pressure for all seeds. Multiple seed drops were more common than misses for watermelon I and II, melon, cucumber, onion and sugarbeet seeds. Few ‘skips’ or multiple drops occur at any vacuum pressure for maize I and II, cotton and soya bean seeds.Loss of uniformity of the vacuum seeder was probably a combination of several factors. The results support reports from Barut (1996)who found that the pattern efficiency of the vacuum plate differed most at lower or higher vacuum pressures and faster wheel speeds. In this research, preciseness and quality of feed index of the vacuum seeder were poorer at the lower and higher vacuum pressures than optimum vacuum pressure.One thousand kernel mass, projected area, sphericity and kernel density of seeds are given in Table 5. One thousand kernel mass, projected area, sphericity and kernel density of seeds varied from 4.3 to 372.5 g, 5–77 mm2, 38.4–85.8% and 440–1310 kg m−3, respectively.Table 5. Means and standard errors of the seed dimensionsThe relationship between one thousand kernel mass, projected area, sphericity and kernel density with vacuum pressure presented in Fig. 2, Fig. 3, Fig. 4 and Fig. 5. For the determination of the relationship between the one thousand kernel mass and the projected area with vacuum pressure, the power model was used. For the determination of relationship between the sphericity and the kernel density with the vacuum pressure, the linear model was used. The diagrammatic representation of the models results in a curve that fits well for the description of the vacuum pressure. The relationship between one thousand kernel mass with vacuum pressure is better than the others with the highest coefficient of determination of 0.92.Fig. 2. Vacuum pressure of precision vacuum seeder as a function of one thousand kernel mass; R2, coefficient of determinationFig. 3. Vacuum pressure of vacuum seeder as a function of projected area; R2, coefficient of determinationFig. 4. Vacuum pressure of vacuum seeder as a function of sphericity; R2, coefficient of determinationFig. 5. Vacuum pressure of vacuum seeder as a function of kernel density; R2, coefficient of determinationAll possible combinations of the different variables were tested and included in the regression analysis. The multiple combinations of one thousand kernel mass, projected area, sphericity and kernel density that gave the lowest root mean square error and chi-square and the highest modelling efficiency were finally included in the final model. Based on the multiple regression analysis the accepted model constants, coefficients, chi-square χ2, root mean square error E rms and modelling efficiency E m were as follows: P v=a+b m10000·27+cP−0·02−dφ+eρkwhere: P v is the vacuum pressure in kPa; m1000 is one thousand kernel mass in g; P is the projected area in mm2; φ is the sphericity in %; ρk is the kernel density in kg m−3. The optimum values of the coefficient a, b, c, d, and e, namely 1.00, 0.72, 2.09×10−3, 0.01 and 0.37×10−3, respectively, gave values for χ2 of 2.51×10−3, for E rms of 2.74×10−2, and for E m of 0.99.Validation of the established final model was evaluated by comparing the computed vacuum pressures with the observed vacuum pressures. The performance ofthe model was illustrated in Fig. 6. The predicted data generally banded around the straight line which showed the suitability of the final model in describing vacuum pressure of the seeder.Fig. 6. Experimental versus predicted vacuum pressure values by final model; R2, coefficient of determination4. ConclusionsIn laboratory tests, the optimum vacuum pressure of a precision vacuum seeder was determined as 4.0 kPa for maize I and II; 3.0 kPa for cotton, soya bean and watermelon I; 2.5 kPa for watermelon II, melon and cucumber; 2.0 kPa for sugarbeet and 1.5 kPa for onion seeds.In order to predict vacuum pressure in relation to one thousand kernel mass, projected area, sphericity and kernel density of seeds, mathematical models were developed. The relationship between one thousand kernel mass with vacuum pressure was better than the others with the highest coefficient of determination. 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Bulletin of the University of Agricultural Science, Gödöllo, 1, 166–170Wanjura D F; Hudspeth E B(2005). Performance of vacuum wheels metering individual cottonseed. Transactions of the ASAE, 12(6), 775–777Yaldiz O; Ertekin C; Uzun H I (2006). Mathematical modelling of thin layer solar drying of sultana grapes. Energy, 26, 457–465真空压力播种机的数学建模D. Karayel 1, Z. B. Barut 2 and A. Özmerzi 11 Akdeniz University, Faculty of Agriculture, Department of Agricultural Machinery, 07070, Antalya, Turkey2 Çukurova University, Faculty of Agriculture, Department of Agricultural Machinery, Adana, Turkey; e-mail of corresponding author: dkarayel@.tr引言这项研究的目的是确定最佳的精密真空压力播种机。