ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT AZOJETE - CIGR Section VI Special Issue: Innovation & Technologies for Sustainable Agricultural Production & Food Sufficiency AZOJETE, December, 2018. Vol. 14(SP.i4):20-36 Published by the Faculty of Engineering, University of Maidiguri, Maidiguri, Nigeria. Print ISSN: 1596-2490, Electronic ISSN: 2545-5818 www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 20 ORIGINAL RESEARCH ARTICLE NUMERICAL AND EXPERIMENTAL STUDIES ON THE CUTTING ENERGY REQUIREMENTS OF OKRA (ABELMOSCHUS ESCULENTUS L.) U. G. Asonye1*, N. R. Nwakuba2, S. N. Asoegwu1 1*Agricultural and Bioresources Engineering Department, School of Engineering and Engineering Technology, Federal University of Technology, Owerri, Nigeria. 2Agricultural and Bioresources Engineering Department, College of Engineering and Engineering Technology, Michael Okpara University of Agriculture, Umudike, Nigeria. *Corresponding author’s e-mail address: u.asonye@yahoo.com ARTICLE INFORMATION Received: October, 2018 Accepted: December, 2018 Keywords: Modeling Okra Cutting energy Dimensional analysis Crop size Nigeria ABSTRACT Cutting energy requirements for vegetable crops is a prerequisite in the engineering design of appropriate cost effective cutting systems consuming minimum amount of energy while still providing high quality products. This study attempts the development of predictive equations describing the cutting energy for okra (Abelmoschus esculentus L.). Dimensional analysis based on the Buckingham pi theorem was used to obtain the functional relationship between the cutting energy of the selected vegetable and the independent variables such as tool weight (w), height of tool drop (Hd), tool edge thickness (t), cutting speed (v), crop size (s), crop moisture content (φ), crop contact area (A) and crop density (σ). The developed model was validated with experimental data. A high coefficient of determination of R2 value of 0.973 between the predicted and measured energy values showed that the method is good. Hence the obtained predictive model is appropriate for determining the cutting energy requirements of okra up to 97%. © 2018 Faculty of Engineering, University of Maiduguri, Nigeria, All rights reserved. 1.0 Introduction Agricultural products often occur in sizes too large to be used, and therefore they need to be reduced and put into different sizes and shapes like cubes, thin slices or rings to facilitate further processing. Cutting, which is a size reduction activity, is an energy consuming operation. However, the cutting energy requirements of vegetables have been estimated using different cutting systems and approaches (McGorry et al., 2003, Blahovec, 2007; McCarthy et al., 2007 and Atkins, 2009) with the following objectives; to know the most efficient cutting conditions necessary for optimization of the cutting process; to determine the optimal energy required to cut and to achieve an overall efficient cutting operation. Ciulica and Rus 2011, Elżbieta and Agnieszka, 2012; and Singh et al., 2016 have identified blade sharpness, slicing angle, contact area, depth of cut, cutting speed, and the engineering properties of vegetables such as crop variety, maturity stage, moisture content, mailto:u.asonye@yahoo.com http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 21 orthogonal dimensions and fiber orientation as some of the parameters affecting the energy requirements for cutting of vegetables. Okra (Abelmoschus esculentus L.) is a fruit vegetable of interest in this work. Okra or lady’s finger, deriving its name from the ‘Igbo’ word ‘툘 ko ro ’ (McWhorter, 2000) is a widely cultivated vegetable crop in most parts of the world and one of the most popular vegetables in many West African markets and some other regions of the world (Oyelade et al., 2003; Andras et al., 2005; Saifullah and Rabbani, 2009). About 10% of the world’s annual okra production comes from West Africa with Nigeria as the leading producer (Burkil, 1997). In South-Western Nigeria, the three known cultivars of okra are yaaya or kògbóyè, which has long slender pods; kúdìkán or ilá-òjò, which has short, sturdy pods and ilá-ìròkò which has long sturdy pods (Farinde et al., 2007). Okra is a multipurpose crop due to the various uses of the fresh leaves, buds, flowers, pods, stems and seeds. The benefits derivable from the okra fruit include nutritional benefits; a good source of minerals, vitamins, carbohydrate, protein and essential amino acids; (Adeboye and Oputa, 1996; Adom et al.,1996; Okra Food, 2003; Sobukola, 2009), medicinal benefits; detoxifies and aids digestion in human nutrition (Makose and Peter, 1990; Owolarafe and Shotonde, 2004) industrial benefits; valuable material for gum, paper glazing and quality soap making, contains glycans used in aqueous suspensions, an excellent source of iodine and raw material for flocculant used for removal of solid wastes from tannery effluent (Owoeye et al., 1990; Agarwal et al., 2003; Falade and Omojola, 2010) and economically it is a high income earner for vegetable farmers in producing countries (Schippers, 2000). Okra, like many other vegetables is highly perishable due to its relatively high moisture content, soft texture and high respiration rate. (Atanda et al., 2011) and the traditional method of preserving fresh okra pods which involves spreading it in the air for few hours during cold weather only makes it preserve for just 3-4 days (Schippers, 2000 and Sobukola, 2009) after which, it becomes tough and unsuitable for use as a fresh vegetable. The unsold lots are processed by manually cutting into slices and sun-drying to preserve it, thereby extending its availability from one season to the other, improves the shelf life and also forestalling seasonal wastage. This manual cutting operation is a most laborious, time consuming one with the challenges of contamination of the product and injury to the hand. There have been efforts made by some researchers at developing equipment for handling the cutting operation (Owolarafe et al., 2007, Ogbobe et al., 2007). However, in spite of its many industrial, medicinal, economic and nutritional benefits, studies reveal that it is among the least studied vegetables, which portends a danger of gradual disappearance from our meal menu and eventual extinction unless urgent attention is given to it. Also, it suffers a lot of postharvest losses due to inadequate post-harvest handling technologies which makes the crop scarce and expensive when it is not in season. Hence cutting of okra as a postharvest handling operation will enhance further processing such as drying, packaging and preparation into several food forms. Furthermore, extensive documentation on the properties of foods and food products exist, however data related to the cutting energy of different vegetables such as okra is scarce, even though such data is important in the design of cutters. This observation is affirmed by researchers like Saravacos https://en.wikipedia.org/wiki/Igbo_language http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3614041/ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3614041/ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3614041/ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3614041/ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3614041/ file:///C:/Users/user/Downloads/azojete143/www.azojete.com.ng Asonye, et al: A mathematical model for predicting the cutting energy requirements of okra (Abelmoschus esculentus l.), AZOJETE, 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 22 and Kostaropoulos (2002) in their assessment that less work has been performed on energy involved for cutting of different food materials, and Brown et al. (2005) who opined that limited published literatures on specific energy requirement in cutting of fruits and vegetables are available. Also, Mitcham et al. (1996) agreed that literature related to cutting of fruits and vegetables are limited, Hence the need to intensify research activities in this post harvest area of cutting and determining the energy requirements for cutting vegetables such as the (ilá-ìròkò) variety of okra. 1.1 Modeling Agricultural Processes Modeling, simply defined is a representation in mathematical terms of the behavior of real devices and objects, or a system of postulates, data and inferences presented as a mathematical description of an entity or state of affairs. It could be descriptive, explanatory or predictive. In the prediction part, which can be envisaged as organized thinking of the possible, models are exercised to give information on a yet-to-be-conducted experiment. These predictions are then followed by observations that serve either to validate the model or to suggest reasons that the model is inadequate (Dym and Ivey, 1980; Dym, 1994 and Cha et al., 2000). Various mathematical modeling techniques that are in use include: dimensional homogeneity and analysis, abstraction and scaling, conservation and balance principles and consequences of linearity. However, Dimensional analysis has become a widely applicable and very powerful technique that is adopted in mathematical modeling. This is due to its simplicity in planning, presentation, and interpretation of experimental data, providing an organised way to plan and carry out experiments, and enables one to scale up results from model to prototype (Bahrami et al., 2006). The dimensional analysis is a mathematical technique used in identifying the factors involved in a physical situation or phenomenon and forming a relationship between them. It offers a method for reducing complex physical problems to the simplest form prior to obtaining a quantitative answer (Andrzej, 2015). Although there are other methods of performing dimensional analysis, notably the indicial method, the method based on the Buckingham pi theorems gives a well organized procedure for obtaining a solution. The Buckingham pi theorem states that the number of dimensionless and independent quantities required to express a relationship among variables in any phenomenon is equal to the number of quantities involved minus the number of dimensions in which those quantities may be measured (Fox and McDonald, 1992). Mathematically speaking, if there is a physically meaningful equation involving a certain number (n) of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters (π1, π2, ..., πp) constructed from the original variables (Hart, 1995), (Here k is the number of physical dimensions involved; which in this situation, (K = 3 i.e. Mass, Length and Time). The application of mathematical models in different aspects of agricultural engineering including tillage operations (Fielke, 1999), spraying machines (Teske et al., 1991), crop handling machines (Gorial and O’Callaghan, 1991), harvesting (Baruah and Panesar, 2005a and b) and many versatile topics on post harvesting aspects have been successfully attempted such as; modeling flow rate of egusi-melon (Colocynthis citrullus) through circular horizontal hopper orifice (Asoegwu et al., 2010); modeling the grain cleaning process of a stationary sorghum thresher (Simonyan et al., 2006) , modeling the drying process of a hybrid convective vegetable crop dryer (Nwakuba, 2018) and http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 23 development of a model to describe infrared radiative and convective drying characteristics of onion slices for optimum management of operation parameters (Jain and Pathare, 2004). Although a few research attempts have been made to model cutting processes like mathematical models and laboratory tests of Impact cutting behaviour of forage crops, (McRandal and McNulty, 1978); performance modeling of the cutting process in sorghum harvesting (Mohammed, 2002), mathematical modeling of laser based potato cutting and modeling yield efficiency of peeling (Ferraz et al., 2007, Somsen et al., 2004), much is left undone on modeling the energy requirement for cutting vegetables. Some researchers used the dimensional analysis based on the Buckingham’s pi theorem as veritable instrument in establishing a prediction equation of various systems which include the development of screw-conveyor performance models using dimensional analysis, (Degrimencioglu and Srivastava, 1996); a mathematical model for predicting output capacity of selected stationary grain threshers (Ndirika, 2006); a mathematical model for predicting the cracking efficiency of vertical-shaft centrifugal palm nut cracker (Ndukwu and Asoegwu, 2011); and modeling flow rate of egusi-melon (Colocynthis citrullus) through circular horizontal hopper orifice (Asoegwu et al., 2010). The size reduction operation by cutting in most postharvest processes is a most laborious, time wasting and energy consuming task which deserves research attention. These studies are necessary in order to discover methods of efficiently improving the cutting process by maximizing the scarce energy, time and resources available and optimally enhancing the post harvest operations concerned. Hence the need to undertake the present study which will establish a mathematical model using dimensional analysis based on the Buckingham’s pi theorem to predict the cutting energy requirements for okra (Abelmoschus esculentus L.). The mathematical model will become a design tool for machine designers in the development of energy saving and cost effective cutting systems that will provide good quality cut products. 2. Materials and Methods 2.1 Theoretical Development Factors affecting the cutting energy requirements of vegetables obtained from literature include tool parameters like (materials of construction, sharpness, rigidity of cutting tools, knife speeds, etc.) and physical properties of the plant material like (crop variety, size, maturity stage, crop moisture content , crop density, fibre orientation, etc.) (Nadulski, 2001; Szot et al., 1987; Atkins, 2009; Blahovec, 2007; McCarthy et al., 2007; and McGorry et al., 2003). In the development of the model, dimensional analysis based on the Buckingham pi theorem was employed which is a technique used in identifying factors involved in a physical situation and forming a relationship among them. The Buckingham pi theorem states that the number of dimensionless and independent quantities required to express a relationship among variables in any phenomenon is equal to the number of quantities involved minus the number of dimensions in which those quantities may be measured’(Fox and McDonald, 1992). i.e. if N = number of variables involved in a physical situation x = number of dimensions in which quantities may be expressed = (3), M, L, T M = Mass; L = Length; T = Time file:///C:/Users/user/Downloads/azojete143/www.azojete.com.ng Asonye, et al: A mathematical model for predicting the cutting energy requirements of okra (Abelmoschus esculentus l.), AZOJETE, 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 24 n =number of dimensionless groups as given in Equation 1. Hence, n = N - x (1) However, due to the large number of variables influencing the energy of cut of the selected fruit vegetable, assumptions will be made in order to bring these large numbers to a reasonable and manageable number. (Simonyan, et al., 2006). 2.2 Assumptions made in Model Development i. Variables that are design parameters which are measurable were considered. ii. Variables that are functions of other variables were not considered e.g. volume which is a function of crop size. However, cutting speed which is a function of time was considered instead of time. iii. The fiber orientation, textural properties (inner hollow texture) and crop variety are considered negligible in the model development, since only the horizontal orientation of the okra fruit is considered. 2.3 Development of the Functional Energy Equation. These assumptions helped in reducing the number of variables involved to the underlisted as these were considered to have greater influence on the energy of cut of the selected vegetable (okra) and are measurable. The chosen variables are: tool weight (w), height of tool drop (Hd), tool edge thickness (t), cutting speed (v), crop size(s), crop moisture content (φ), crop contact area (A) and crop density (σ). Having identified the core variables influencing the energy required for cutting the selected vegetable, Equation 2 represents the functional equation of the predictive model. i.e � � ������� ������φ�A�σ) (2) Where E is the cutting energy of the okra crop (J) w = Tool weight; Hd = Height of tool drop; t = Tool edge thickness; v = Cutting speed; s = Crop size; φ = Crop moisture content; A = Crop contact area and σ = Crop density. Three primary dimensions M= Mass, L= Length, T= Time were chosen in the description of the variables. From the Buckingham pi theorem (Fox and McDonald, 1992) the total number of dimensionless groups to be formed is as given in Equation 1 above. i.e � � � − � N = number of variables involve in the situation studied = 9 x = number of dimensions for describing these variables = 3 Hence � � � � � � h П1, П2, П3, П4, П5, П6 Hence six dimensionless groups were formed namely; П1, П2, П3, П4, П5 and П6. In determining the dimensionless groups, the following procedure was adopted (Fox and McDonald, 1992). The variables utilized in the establishment of the model equation for the cutting energy of the selected vegetable were expressed in terms of their dimensions as shown in Table 1. http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 25 Table 1. List of variables and their dimensions S/N Variable Symbol Unit Dimension 1 Energy E kgm2s-2 ML2T-2 2 Tool weight w kgms-2 MLT-2 3 Height of tool drop Hd m L 4 Tool edge thickness t m L 5 Cutting speed v ms-1 LT-1 6 Crop size s m L 7 Crop moisture content φ φ M0L0T0 8 Crop contact area A m2 L2 9 Crop density σ kgm-3 ML-3 The dimensional matrix of the variables is shown in Table 2. This is needed to develop the indices of the involved variables. Table 2. Dimensional matrix of variables S/N Variable Symbol M L T 1 Energy E 1 2 -2 2 Tool Weight w 1 1 -2 3 Height of tool drop Hd 0 1 0 4 Tool edge thickness t 0 1 0 5 Cutting speed v 0 1 -1 6 Crop size s 0 1 0 7 Crop moisture content φ 0 0 0 8 Crop contact area A 0 2 0 9 Crop density σ 1 -3 0 From the above matrix, φ is dimensionless and therefore excluded from the dimensionless terms determination and is to be added later (Simonyan et al, 2006) while the other variables were combined to form the П groups. Cutting speed (v), tool weight (w) and crop size (s) were selected as the major parameters (i.e. recurring set) because they contain all the primary dimensions involved in this problem and their combination does not form a dimensionless group. Having selected w, v and s as the recurring set, the exponents a, b and c are attached to them respectively so that when their product ������ divide the remaining variables E, Hd, t, σ and A, the dimensionless groups П1, П2, П3, П4 and П5 are obtained as given in Equations 3 to 7 (Ndirika, 2006; Simoyan et al., 2006; Asoegwu et al., 2010; Ndukwu and Asoegwu, 2011) .This is the basis of the Buckingham pi theorem of dimensionless groups. П� = � ��vbsc (3) П� � �� ��vbsc (4) П� � � ��vbsc (5) file:///C:/Users/user/Downloads/azojete143/www.azojete.com.ng Asonye, et al: A mathematical model for predicting the cutting energy requirements of okra (Abelmoschus esculentus l.), AZOJETE, 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 26 П� � A ��vbsc (6) П� � σ ��vbsc (7) Where a, b, and c are exponents needed to make the groups non-dimensional. The variables are substituted with their dimensions and the non-dimensional Пs are replaced with M0L0T0 which is a dimensionless group. In order to obtain values for the exponents, the principle of dimensional homogeneity is used to equate the dimensions on each side of the equations of the П groups. Equation 3 being expressed in terms of the dimensions on both sides becomes Equations 8 ������ � ����−� ����−�)����−�)��� (8) Cross- multiplying Equation 8 gives Equation 9. ������� ���−�)� ��−� ��� � ����−� (9) Using dimensional homogeneity for M, L and T, the exponents a, b and c are got in equations 10, 11, 12 and 13. For M; ��U� � �� 0 + a = 1 a = 1 (10) For L; ��U�U�U� � �� 0+a+b+c = 2; from 3.10, a = 1 1+b+c =2 b+c = 2-1 =1 b+c = 1 (11) for T; ��−��−� � �−� -2a – b =-2 Since a =1 -2(1) – b = -2 -b = -2+2 =0, b = 0 (12) From Equation 11, b + c = 1, put b=0 0+c=1; c=1 (13) Hence, a = 1, b = 0, c = 1; replacing the exponents with their values, П� becomes Equation 14 П� = � ��v�s� = � �s (14) Similarly, solving for the remaining П groups, we obtain Equations 15 to 19. http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 27 П� = �� s (15) П� � � s (16) П� � A s� (17) П� � σv�s� � (18) Пh = φ (19) Combining these equations gives Equation 20, whose components are dimensionless. i.e. � �s = f ( �� s � � s � A s� � σv �s� � � φ ) (20) Combining the dimensionless terms to reduce it to a manageable level (Shefii et al. 1996) by multiplication and/or division, we obtain Equations 21 to 23 which are dimensionless. П�� = П� П� = � �s × s �� = � ��� (21) П�� = П3 П4 = t s × s2 A = ts A (22) П56 = П5 П6 = σv2s2 w × 1 φ = σv2s2 wφ (23) The new dimensionless functional relationship becomes Equations 24 and 25. П�� = f (П�� , П�h) (24) E wHd = f (ts A , σv�s� wφ ) (25) From Equation 25, the Equations for E are obtained as Equations 26 and 27. E= wHd ( ts A , σv�s� wφ ) (26) E = f (tswHd A � σv �s�Hd φ ) (27) Equation 27 gives the cutting energy E, with all the parameters in Equation 2, as a function of two energy components tswHdA and σv2s2Hd φ which are represented as � and � in Equation 28. E = f (���) (28) 2.4 Experimental Procedure Green, mature and freshly harvested okra fruits (ilá-ìròkò variety) that were healthy and free from mechanical injuries were purchased from a local farmer in Owerri west Local government area of Imo state, Nigeria. They were thoroughly cleaned of all impurities and unwholesome fruits and thereafter sorted into four size ranges of 10 fruits each and some physical properties were obtained at an initial moisture content of 89.41± 0.4% (wb) (AOAC, 2000). With an automated vegetable cutter, at a preset speed of 30mm/min, the cutter drops from a height of 20cm, cutting the sampleElectric motor file:///C:/Users/user/Downloads/azojete143/www.azojete.com.ng Asonye, et al: A mathematical model for predicting the cutting energy requirements of okra (Abelmoschus esculentus l.), AZOJETE, 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 28 as it travels through it. The force-time relationship displayed on the monitor was used to calculate the cutting energy. Plate 1.0 shows the experimental set-up of the cutting mechanism. Plate 1.0: Experimental set-up of the cutting mechanism The automated vegetable cutter consists of the hardware and software components. Arduino controlled processors automatically and effectively measures, records and stores cutting variables and other basic parameters with minimum human supervision, thus making the entire cutting process automated. Connecting to an electric power source switches on the cutter and a predetermined speed value is selected on the keypad. As the knife presses against the sample, the reactive force exerted on the load cell is amplified and measured. Connecting to a computer using USB port the measured values are relayed and MATLAB intercepts the values and plots the resulting graph of force of cut against time travelled which is used in calculating the energy of cut. 2.5 Determination of Validation Parameters In the determination of validation parameters, every other parameter was held constant while the crop size varied. The selected tool edge thickness (t) measured with a digital venier caliper (Model 500-196, Mitutoyo Products, America) was 0.1cm. The weight (w) of the cutting tool determined with an electronic weighing balance (Model GF-200, A & D company Ltd, Japan of an accuracy of ±0.01) was 23.1g. The cutting tool was dropped from a pre-determined height (h) of 20cm measured with a meter rule. The crop density (σ) determined by dividing the weight of the crop by the volume of water displaced when sample is placed inside a measuring cylinder was taken as 1.107g/cm3. The orthogonal dimensions of the sample were determined with the digital venier caliper (Model 500- 196, Mitutoyo Products, America). The crop contact area (A) taken as 28.75cm2 was determined by multiplying the circumference at the point of cut (obtained by winding round with a tape and measuring out on the meter rule) with the major diameter of the crop sample. The moisture content Cutter frame Arduino processor Cutting knife Okra sample Monitor display http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 29 of the crop sample determined with the method described in AOAC (2000) was taken as 89.41± 0.4% (wb). 2.6 Prediction Energy Equation The prediction energy equation was established by allowing �th � to vary at a time while keeping the other constant and observing the resulting changes in the function (Shefii et al., 1996). This was achieved by plotting the experimental values of E against P = (����� A ) while keeping � constant. P = ( ����� A ) was evaluated by substituting the measured values for tool edge thickness (t), crop size(s), tool weight (w), height of tool drop (Hd) and crop contact area (A) into �� Also, E against � � � σv �s�Hd φ ) was plotted while keeping P constant. Values for � � � σv �s�Hd φ ) were obtained by substituting the measured values for crop density (σ), cutting speed (v), crop size(s), height of tool drop (Hd) and crop moisture content (φ). 3. Results and Discussion Average values and standard deviation of the measured physical parameters of the okra fruits are shown in Table 3. Table 3. Average values for physical properties of okra fruits at 89.41± 0.4% (w.b) S/N Okra Mass (g) Diameter(mm) Circumference (mm) Area (mm2) 1 Ok1 18.16±2.17 24.90±0.66 89.75±4.76 487.3±26.15 2 Ok2 8.70±0.94 18.26±0.86 67.5±3.64 262.39±24.79 3 Ok3 5.15±0.07 14.99±1.02 59.5±2.29 177.35±24.90 4 Ok4 1.72±0.14 12.44±0.84 47.75±1.92 121.99±15.71 Table 4 shows the experimental values of the cutting energy obtained with the automated vegetable cutter and the calculated values of the cutting energy obtained by substituting values of the cutting variables into the energy equations � � (����� A ) and Q � � σv �s�Hd φ ). Note : Crop size was varied while the other factors remained constant. Table 4. Experimental values (Emeasured) and calculated values of cutting energy for okra fruits. S/N Okra Crop size (mm) (Emeas)       A tswH P d          dHsvQ 22 1 Ok1 24.90±0.66 4.16 23.63 13.82 2 Ok2 18.26±0.86 2.75 17.33 7.43 3 Ok3 14.99±1.02 1.98 14.23 5.01 4 Ok4 12.44±0.84 0.70 11.81 3.45 The plots of the cutting energy (Emeasured) against ����� are shown in Figures 1 and 2 with their linear equations and R2 values expressed in Equations 29 and 30. file:///C:/Users/user/Downloads/azojete143/www.azojete.com.ng Asonye, et al: A mathematical model for predicting the cutting energy requirements of okra (Abelmoschus esculentus l.), AZOJETE, 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 30 Figure 1. Variation of cutting energy against � � tswHd A � keeping � � � σv �s�Hd φ ) constant. Figure 2. Variation of cutting energy withQ � (σv �s�Hd φ )� keeping � � � tswHd A ) constant. E = 3.462P + 8.448 R� � ���h� (29) E = 3.026Q + 0.170; R� � ����� (30) The plot of the P and Q terms in Figures 1 and 2 forms a plane surface in linear space and according to Mohammed (2002), it implies that their combination favors summation or subtraction. Therefore, the component equations formed by the subtraction and summation of Equations 29 and 30 give Equations 31 and 32 respectively. E = �� (��� ) - �� (��� ) + K (31) E = �� (��� ) + �� (��� ) + K (32) It must be noted that; at f�, Q was kept constant while P varied at f�, P was kept constant while � varied Substituting Equations 29 and 30 into Equation 31 and performing some algebraic manipulations yields Equation 33. http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 31 E = 3.462P – 3.026Q + 8.278 (33) Also, Substituting Equations 29 and 30 into Equation 32 and performing some algebraic manipulations yields Equation 34. E = 3.462P + 3.026Q + 8.618 (34) A further manipulation as permitted under the rules of the Burkingham pi theorem (Shefii et al., 1996) is manipulating with a constant factor. Hence, Equations 33 and 34 were divided with a constant factor of 10 which yields the predicted model equations expressed in Equations 35 and 36 respectively. Dividing equations 33 and 34 with the constant factor of 10 yielded predicted values close to the actual ones. E = 0.3462P – 0.3026Q + 0.8278 (35) E = 0.3462P + 0.3026Q + 0.8618 (36) Substituting the variables for P and � into Equations 35 and 36 yield Equations 37 and 38 respectively. E = 0.3462����� A – 0.3026σv �s��� φ + 0.8278 (37) E = 0.3462����� A + 0.3026σv �s��� φ + 0.8618 (38) However, the final predicted model equation will be either of the above two equations that gives the better statistical inference 3.1 Model Validation The mathematical model was validated using the data generated from the vegetable cutter. The model validation was done at four ranges of crop sizes (23-26mm, 17-20mm, 14-17mm and 10- 13mm) and a constant cutting speed of 30mm/min. The method of regression analysis as computed using Microsoft Excel environment was used to describe the relationships, plot the graphs and compute the coefficients of determination (R2). Measured values of parameters were substituted into Equations 37 and 38 to yield the predicted cutting energy values which were plotted against the experimental energy values on a regression curve in order to obtain the coefficients of determination as shown in Figures 3 and 4 respectively. Equations 39 and 40 express the relationship between the predicted cutting and experimental cutting energy with R2 values of 0.973 and 0.945 respectively. The high R2 values of 0.961, 0.921 and 0.973 obtained for the individual predictions equations and the Equation for subtraction of component equations is an indication that the method adopted in the development of the mathematical models is acceptable and can be translated in the development of the other varieties of the okra fruit in particular and other vegetables in general. Table 4 presents values of the predicted and experimental cutting energy and Figure 3 presents the graphical relationship between the experimental and predicted cutting energy. Equation 35 expresses the relationship between the predicted cutting and the experimental cutting energy with a very high correlation with R2 value of 0.973. file:///C:/Users/user/Downloads/azojete143/www.azojete.com.ng Asonye, et al: A mathematical model for predicting the cutting energy requirements of okra (Abelmoschus esculentus l.), AZOJETE, 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: u.asonye@yahoo.com 32 Figure 3. The graph of the relationship between experimental and predicted cutting energy (for subtraction of component energy equations) Epred = 3.423 Eexp – 12.59; R2 = 0.973 (39) Where Epred = predicted cutting energy Eexp = experimental cutting energy Figure 4. The graph of the relationship between experimental and predicted cutting energy. (for summation of component energy equations) Epred = 0.447Eexp – 1.584; R2 = 0.945 (40) From the statistical inference carried out, the predictive model equation derived from the subtraction of component energy equations gave a higher coefficient of determination (R2) value of 0.973, a lower mean difference of 1.983 between the predicted and experimental energy values and a lower standard error of 0.127 as compared to an R2 value of 0.945 obtained from the summation of component energy equations, a mean difference of 6.508 between the predicted and experimental energy values and a standard error of 0.199. Hence, the predicted model equation which gives the better statistical inference of a higher R2 value of 0.973 and lower values of 1.983 and 0.127 for mean difference and standard error respectively is chosen as the predicted model equation for the cutting energy requirement for okra (Abelmoschus esculentus L.) ‘Ila-iroko’ variety and is given as Equation 41. E = 0.3462����� A – 0.3026σv �s��� φ + 0.8278 (41) http://www.azojete.com.ng mailto:u.asonye@yahoo.com Arid Zone Journal of Engineering, Technology and Environment, December, 2018; Vol. 14(sp.i4):20-36. ISSN 1596-2490; e-ISSN 2545-5818; www.azojete.com.ng 33 4.0 Conclusion A mathematical model for predicting the cutting energy of okra (Abelmoschus esculentus L) was presented using dimensional analysis based on the Buckingham’s П theorem. The model equation expressed as E = 0.3462����� A – 0.3026σv �s�Hd φ + 0.8278 was validated with data from an automated vegetable cutter. Results obtained showed a high coefficient of determination (R2 = 0.973), a low mean difference of 1.983 between the predicted and experimental energy values and a low standard error of 0.127. This is an indication that the method adopted in the development of the mathematical model is acceptable and can be translated in the development of predictive models for the other varieties of the okra fruit in particular and other vegetables in general. Also, this expression will also help designers of the cutting equipment for okra (ilá-ìròkò) variety to avoid the rigors of experimentations and at the same time obtain efficient cutters. Hence, the developed model could be used to predict the cutting energy for okra (ilá-ìròkò) variety up to 97%. References Adeboye, OC. and Oputa, CO. 1996. Effect of galex on growth and fruit nutrient composition of okra. Ife Journal of Agriculture. 18 (12): 1-9. Adom, KK., Dzogbefia, VP., Ellis, WO. and Simpson, BK. 1996. Solar drying of okra- effects of selected package materials on storage stability. Food Research International.29(7):589–593. Agarwal, M., Rajani, S., Mishra, A. and Rai, JSP. 2003. 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St. Josef, Michigan. http://www.azojete.com.ng mailto:u.asonye@yahoo.com