CIGR Ejournal Style and Format Guidelines 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): 247-256 Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria. Print ISSN: 1596-2490, Electronic ISSN: 2545-5818 www.azojete.com.ng Corresponding Author’s Email: ajaotaiwooluwatoyin@gmail.com 247 ORIGINAL RESEARCH ARTICLE NUMERICAL ANALYSIS OF TWO TERM MODEL USING NEWTON’S METHOD ON DRYING KINETICS OF AN IMPROVED COWPEA (IT 97K-56S-IS) Ajao, T. O.* and Raji, A. O. Department of Agricultural and Environmental Engineering, Faculty of Technology, University of Ibadan ARTICLE INFORMATION Received: October, 2018 Accepted: December, 2018 Keywords: Two term model Newton Raphson Numerical solution Drying kinetics IT 97K-56S-IS Maturity date Modelling Nigeria ABSTRACT Invention of computers has made numerical methods prevalent and widely accepted. Different numerical methods exist but all aid in solving large numbers of tedious arithmetic calculations involving complex differential equations and so an analytical solution cannot be applicable, hence, computer simulations and numerical solution become functional. IT 97K-56S-IS, a high yielding, multiple disease resistant cowpea variety with maturity date of 60Days was gotten from the International Institute of Tropical Agriculture (IITA). Experiments were carried out on samples after harvesting at maturity in a convective dryer at drying temperature of 55, 65, 75 and 85ºC and drying kinetics was performed utilising the Two term model. The model was numerically solved using Newton Raphson’s iterative method on account that it is a non linear equation, its fast convergence to the roots and requirement of a single guess. A well structured algorithm was written. The performance of the models was evaluated using the coefficient of determination (R^2) and root mean square error (RMSE) showing the relationship between the experimental and predicted moisture ratios. The result shows that Newton Raphson’s method would be suitable for Two-Term model if used with a multiplication factor or constant. © 2018 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved 1.0 INTRODUCTION Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations. Although there are many kinds of numerical methods, they have one common characteristic which is that they invariably have large numbers of tedious arithmetic calculations. Hence, with the development of fast, efficient digital computers, the role of numerical methods in engineering problem solving has increased dramatically in recent years (Steven and Raymond, 2010). The purpose of modelling is to allow engineers to select the most appropriate method of drying for a given product as well as to determine the most suitable conditions (Aremu et al., 2013). The principle of modelling is based on having a set of mathematical http://www.azojete.com.ng Ajao and Raji: NUMERICAL ANALYSIS OF TWO TERM MODEL USING NEWTON’S METHOD ON DRYING KINETICS OF AN IMPROVED COWPEA (IT 97K-56S-IS). AZOJETE, 14(sp.i4): 247-256. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding Author ‘s: Email: ajaotaiwooluwatoyin@gmail.com 248 equations which can satisfactorily explain the system. Models are often used to study the variables involved in the process, predict drying kinetics of the product and to optimize the operating parameters and conditions (Karathanos and Belessiotis, 1999; Raji and Olanrewaju, 2015). They predict the drying times of several products and also generalize drying curves (Meisami-asl et al., 200; Raji and Olanrewaju, 2015). A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. Differential equations can be used in describing nearly all systems undergoing change (Steven and Raymond, 2010). Thin layer drying process of food products has been described by many mathematical models according to Ojediran and Raji (2010) and these include: cowpea (Jianfang et al., 2013; Raji and Olanrewaju, 2015), green bean and onion (Yaldiz and Ertekin, 2001), millet (Ojediran and Raji), 2010), soybean (Gely and Santalla, 2000), Grains (Tagawa et al., 1996). High yielding, multiple disease resistant cowpea varieties with varying maturity periods as well as different seed colours, adapted to various Nigerian agro-ecological zones have been developed in IITA, in collaboration with several National Research Institutes and Universities. Several of these varieties have been released in Nigeria and are promoted by the State Agricultural Development Projects (ADPs), farmers’ groups and seed companies (Boukar and Ajeigbe, 2010). The most important factor in crop production is the choice of a good variety. Varieties that have resistance to the prevailing biotic and abiotic stresses in the areas should be planted. Maturity, growth pattern, market value, seed size, and seed colour should also be considered in variety selection (DPP, 2011). The demand for quality farm products increases, therefore, mechanical drying becomes more popular and acceptable as the advantages outweigh the disadvantages. Matured cowpea can thus be harvested promptly and subjected to mechanical drying so as to reduce harvest moisture content to safe level for proper storage (Yakubu et al., 2012). Also, the error present in the solution of models is dependent on how the problem is solved. Numerical methods are used when evaluating empirical information such as experimental data. Such data, no matter how exact and regulated the experiment was conducted, will involve some degree of error. It would be a waste of time to employ exact analytical techniques in such a situation because the answer can never be more “correct” than the input data (Swick, 2013). Therefore, this work investigated the drying kinetics of an improved variety cowpea (IT 97K-56S-IS) at varying drying temperature using the application of numerical method for a complex uncommonly used model (Two term model) thereby developing an efficient algorithm for finding its successive better approximations to the solutions of the complex model. http://www.azojete.com.ng 2.0 MATERIALS 2.1 Sample preparation A high yielding cowpea, disease resistant, IT 97K-56S-IS with a maturity date of 60days was gotten from the International Institute of Tropical Agriculture (IITA). It was propagated by seeds during the period of a partially wet season on 5th of May, 2014 at the Demonstration Area, Research Farms Unit, IITA, Ibadan, also as stated by Raji and Olanrewaju (2015) and Raji and Olanrewaju (2016). This was to ensure that the seeds tested were harvested at the period needed. For each drying experiment, two hundred (200) grams of the freshly harvested sample was used according to Aremu and Akintola (2014); Tunde-Akintunde and Afon (2009); Raji and Olanrewaju (2015); Raji and Olanrewaju (2016). Each experiment was replicated three times (Aremu et al., 2013) and triplicate samples were spread out in thin layer and placed in the dryer. The drying temperature used for the experiment were 55, 65, 75, 85°C which are within the range of temperatures used by Mario et al. (2003), Mc Watters et al. (1988) and Wilton et al. (2008) for drying of cowpea. The drying process was monitored by weighing the samples every 10mins for the first one hour; then every 30mins for the next three hours and every 1hr for the next three hours till the end of drying according to Ojediran and Raji (2010) Raji and Olanrewaju (2015); Raji and Olanrewaju (2016). Weighing continued until constant weights were obtained being the period that equilibrium with environment was assumed to have been reached and the test was terminated. Moisture content determined at this point is the dynamic equilibrium moisture content. With the initial moisture already known, weight loss was used to calculate the moisture content using the equation used by Ojediran and Raji (2010) given as: �� � ������� ����� (1) where, Mt is the moisture content (m.c.) at time t, (% w.b.), Mi ,the initial m.c. (%w.b.), mi, the initial weight, (g) and wi is the weight loss at time, t (g). The moisture ratio (MR) of the samples during the thin layer drying experiments was calculated using Equation 2.2 according to Ojediran and Raji (2010): �瀀 � ���� ����� (2) Matured pods were harvested by hand at period of harvest of 60 Days After Planting (DAP) in line with the recommendation of Directorate Plant Production (DPP, 2011) and to avoid shattering of the pods. Freshly harvested cowpea pods were then cleaned and sorted to remove foreign materials. The initial moisture content of the samples was determined using the drying method. Samples of known weight (200g) were measured with the use of a top loading Scout Pro sensitive weighing scale and were placed in the cabinet tray dryer at 130°C for about 16 – 18 hours as required by ASABE standards (ASABE, 2003) as stated also by Raji and Olanrewaju (2015) and Raji and Olanrewaju (2016). Ajao and Raji: NUMERICAL ANALYSIS OF TWO TERM MODEL USING NEWTON’S METHOD ON DRYING KINETICS OF AN IMPROVED COWPEA (IT 97K-56S-IS). AZOJETE, 14(sp.i4): 247-256. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding Author ‘s: Email: ajaotaiwooluwatoyin@gmail.com 250 3.0 METHOD 3.1 Iterative methods Iterative methods can be used in solving equations by methods of successive approximations to the roots (variable or parameter that satisfy a single nonlinear equation). Three methods of successive approximations include; Newton-Raphson’s formula, Bisection method and an algebraic method. Each successive approximation method relies on a reasonably good first estimate of the value of a root being made. One way of determining this is to sketch a graph of the function, say y=f(x), then determine the approximate values of roots from the points where the graph cuts the x- axis. Another way is by using a functional notation method. This method uses the property that the value of the graph of f(x)=0. It changes sign for values of x just before and just after the value of a root (John, 2003). 3.2 Reasons for the choice of Newton Raphson’s method Newton Raphson method was chosen for numerical solution of the Two Term Model due to the following reasons: 1. It can be used for solving nonlinear equations. 2. It converges fast to the roots, if it converges. 3. It requires only one guess. 3.3 Newton’s method The Newton–Raphson’s formula, often referred to as Newton’s method, may be stated as follows: If �� is the approximate value of a real root of the equation � � � t , then a closer approximation to the root �� is given by: ��= ��+ �晦��� ��晦��� (3) The advantages of Newton’s method over other methods of successive approximations is that it can be used for any type of mathematical equation (i.e. ones containing trigonometric, exponential, logarithmic, hyperbolic and algebraic functions), and it is usually easier to apply than other methods (John, 2003). 3.4 Newton Raphson’s Iteration Let �t be a good estimate of r and let r = �t + h. Since the true root is r and h = r - �t , the number, h, measures how far the estimate �t is from the truth. Since h is ‘small,’ we can use the linear (tangent line) approximation to conclude that t � � � � �晦�� � ��t ≈ f (�t) + �f�晦xt� (4) http://www.azojete.com.ng Therefore, unless fˈ (�t) is close to 0, h ≈ − �晦�t� ��晦�t� (5) It follows that, � = �t � � ≈ �t − �晦�t� ��晦�t� (6) Then the new improved estimate of r is therefore given by; �� = �t+ �晦�t� ��晦�t� (7) The next estimate �� is obtained from �� in exactly the same way as �� was obtained from �t: �� = ��+ �晦��� ��晦��� (8) Continuing in this way. If �� is the current estimate, then the next estimate ���� is given by: n+1is given by; ���� � ��+ �晦��� ��晦��� (9) 3.5 Computer Programming and Algorithm Computer programs are merely a set of instructions that direct the computer to perform a certain task. A well structured algorithm is easier to debug and test, resulting in programs that take a shorter time to develop, test, and update. Structured programming is a set of rules that prescribe good style habits for the programmer. It is flexible enough to allow considerable creativity and personal expression. A key idea behind structured programming is that any numerical algorithm can be composed using the three fundamental control structures: sequence, selection, and repetition (Steven and Raymond, 2010). To keep this description generic, pseudocode was employed. 4 Results and Discussions 4.1 Results The non linear model of the form (Two Term model): 0 1k t k tY e e    (10) Solving the non linear regression model using Newton Raphson’s Method, the parameters can be estimated using the non linear least square method, the sum of squares is given as: 0 1 2 ( ) ( ( ))i ik t k t iF x y e e     (11) The partial derivatives with respect to the parameters `were found and also their second derivatives. Recall from equation 8, the Newton Raphson formula is given as: 1 ( ) ( ) n n n n f xx x f x    This can be rewritten as: Ajao and Raji: NUMERICAL ANALYSIS OF TWO TERM MODEL USING NEWTON’S METHOD ON DRYING KINETICS OF AN IMPROVED COWPEA (IT 97K-56S-IS). AZOJETE, 14(sp.i4): 247-256. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding Author ‘s: Email: ajaotaiwooluwatoyin@gmail.com 252 1 ( ) ( ) n n n n g yy y H y    (12) Therefore, 0 11 ( ) ( ) n n n k t k tn n g yy e e H y       (13) 4.2 Discussions 4.2.1 Drying kinetics of an improved variety cowpea (IT 97K-56S-IS) The initial moisture content and equilibrium moisture content of (IT 97K-56S-IS) at period of harvest of 60DAP and at drying temperatures of 55, 65, 75 and 85°C is presented in Table 4.1 according to (Raji and Olanrewaju, 2015). The moisture ratio decreased exponentially with time and the time required to reaching equilibrium moisture content decreases with increasing temperature as illustrated in Figure 4.1. This is a general trend reported for other food products e.g. mulberry, tomatoes, sweet pepper and peach slices. (Doymaz, 2004; Doymaz, 2007: Vengaiah and Pandey, 2007; Kingsly et al., 2007, Raji and Olanrewaju, 2015). 4.2.2 Model Fitting The model constants and the coefficients for the Two term model at period of harvest of 60DAP and at 55, 65, 75 and 85°C is presented in Table 4.2. The model fittings are illustrated as presented in Figure 4.1a-d. 4.2.3 Limitation of Newton Raphson’s Method Often times, the Newton Raphson’s method is very efficient but sometimes there are situations when it performs poorly especially in special case of multiple roots. There is tendency of the Newton Raphson technique to oscillate around a local maximum or minimum. Such oscillations may persist, or as a near-zero slope is reached, whereupon the solution is sent far from the area of interest. Obviously, a zero slope [f(x) = 0] is not suitable because it causes division by zero in the Newton-Raphson formula, it implies that the solution shoots off horizontally and never hits the x axis. Therefore, there is no general convergence criterion for Newton-Raphson. Its convergence depends on the nature of the function and on the accuracy of the initial guess. An initial guess sufficiently close to the root may be required (Steven and Raymond, 2010). 5 Conclusions To select and apply the most appropriate method for a particular problem, it requires a good understanding of the characteristics of the method and of the problem being solved (Craft, 2010). However, from the results gotten from this research it can be concluded that Newton Raphson’s method will be suitable for numerically solving Two- Term model but due to its limitation a multiplying factor (constant) was applied. Other numerical methods may also be applied. http://www.azojete.com.ng Table 4.1 Initial moisture content and equilibrium moisture content of (IT 97K-56S-IS) at period of harvest of 60DAP and at 55, 65, 75, 85°C Period of Harvest Initial M.C. (X100%d.b.) Equilibrium M.C. (X100%d.b) 55°C 65°C 75°C 85°C 60 2.26 0.039 0.028 0.030 0.021 Table 4.2 Drying constants and coefficients of the models for IT 97K-56S-IS at period of harvest of 60DAP Multiplication factor Constants and coefficients Two term model 55°C 4.1354 a k b r 0.3050 0.6377 -0.0632 0.4128 R2 RMSE 0.9423 0.0015 65°C 3.3847 a k b r -0.0716 0.7419 0.3670 0.9754 R2 RMSE 0.9053 0.0025 75°C 3.5064 a k b r -0.0684 0.6513 0.3536 0.8861 R2 RMSE 0.9839 0.0008 85°C 3.4130 a k b r 0.3617 0.7728 -0.0687 0.5356 R2 RMSE 0.9946 0.0002 Ajao and Raji: NUMERICAL ANALYSIS OF TWO TERM MODEL USING NEWTON’S METHOD ON DRYING KINETICS OF AN IMPROVED COWPEA (IT 97K-56S-IS). AZOJETE, 14(sp.i4): 247-256. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding Author ‘s: Email: ajaotaiwooluwatoyin@gmail.com 254 a b c d Figure 4.1a: Drying model fittings at period of harvest of 60DAP at 55°C b: Drying model fittings at period of harvest of 60DAP at 65°C c: Drying model fittings at period of harvest of 60DAP at 75°C d: Drying model fittings at period of harvest of 60DAP at 55°C 6 Acknowledgement Special thanks to the International Institute of Tropical Agriculture for the priviledge given to the authors to carry out this research in their institute. References Aremu, A. K., Adedokun A. J. and Raji, A. O. 2013. ″Effect of slice thickness and temperature on the drying kinetics of mango (Mangifera indica), IJRRAS, 15(1): 41-50. Aremu, A. K. and A. Akintola. 2014. Effects of some drying methods on nutritional characteristics of moringa (Moringa oleifera) seeds. 2014 5th International Conference on Agriculture and Animal Science (ICAAS 2014) 2014. http://www.azojete.com.ng 255 Boukar, Ousmane and Hakeem A. Ajeigbe. 2010. Improved agronomic practices for cowpea production and prospects for mechanized cultivation pp.1-10 in Farmers’ guide to increased productivity of improved legume–cereal cropping systems in the savannas of Nigeria IITA, Nigeria. Directorate Plant Production, DPP. 2011. Production guidelines for Cowpeas, Directorate Agricultural Information Services, Department of Agriculture, Forestry and Fisheries, South Africa. Gely, M.C. and Santalla M. 2000. Effect of some physical and chemical properties of oilseeds on drying kinetics parameters. Drying Technology, 18(9), 2155-2166. Jianfang, Shi, Wu Huihuang, Lou Zheng, Wu Zhonghua, Liu Qing1. 2013. Drying characteristics and model of cowpea in tunnel hot air dryer. Vol. 29, No.11, 232, 2013, 6, Transactions of the Chinese Society of Agricultural Engineering, Jun. 2013. John Bird, (2003). Engineering Mathematics, Fourth Edition, An imprint of Elsevier Science, ISBN 0 7506 5776 6, Copyright 2001, 2003. Karathanos, V. T. and V. G. Belessiotis. 1999. Application of a thin layer equation to drying data fresh and semi-dried fruits. J. Agric. Eng. Res., 74: 355-361. Mario, Eduardo Range, Moreira Cavalcanti Mata, Maria Elita Martins Duarte. 2003. Drying Simulation Theory of the Cowpea Considering the Grains Shrinkage, Revista. Brasileira de Produtos Agroindustriais, Campina Grande, 5(2).179 - 185. McWatters, K. H. M., Chinnan, M. S., Hung, Y. C. and A. I. Branch. 1988. The effect of pre- decortication drying temperature on cowpea paste characteristics and functionality in preparation of akara. Cereal Chemistry, 65(1), 1988. Meisami-asl, E., Shahin R., Alireza K. and Ahmad T. 2009. Mathematical Modeling of Moisture Content of Apple Slices (Var. Golab) during drying. Pakistan Journal of Nutrition 8 (6): 804-809, 2009. ISSN 1680-5194. Ojediran, J. O. and Raji, A. O. 2010, ″Thin layer drying of millet and effect of temperature on drying characteristics″ International Food Research Journal, 17: 1095-1106. Raji, A. O., and *T. O. Olanrewaju. 2015. Effect of period of harvest on drying characteristics of an improved variety cowpea (IT 97K-56S-IS). Agric Eng Int: CIGR Journal, 17(4):291 305. Raji, A. O. and *Olanrewaju, T. O., (2016). Moisture Diffusivity and Activation Energy of an Improved Variety of Cowpea as affected by period of harvest and drying conditions. Journal of Applied Science, Engineering and Technology (2016) vol. 16(1), pp10-18. Steven, C. Chapra and Raymond, P. Canale. 2010. Numerical Methods for Engineers, Sixth Edition Ajao and Raji: NUMERICAL ANALYSIS OF TWO TERM MODEL USING NEWTON’S METHOD ON DRYING KINETICS OF AN IMPROVED COWPEA (IT 97K-56S-IS). AZOJETE, 14(sp.i4): 247-256. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng Corresponding Author ‘s: Email: ajaotaiwooluwatoyin@gmail.com 256 Swick, Justin. 2013. An Introduction to Numerical Methods. STEM Articles. http://www.crosscuttingconcepts.com/stem-articles/31-an-introduction-to-numerical- methods. Created on Wednesday, 10 April 2013 14:57. 2016 Crosscutting Concepts, LLC. Tagawa, A., Kitamura, Y. and Murata, S. 1996. Thin layer drying characteristics of Adzuki beans. Transactions of the ASAE 39(12): 605 - 609. Tunde-Akintunde, T. Y. and A. A. Afon (2009). “Modelling of Hot-Air Drying of Pretreated Cassava Chips”. Agricultural Engineering International: The CIGR Ejournal. Manuscript 1493 Vol. August, 2009. Wilton, P. da Silva, Mário, E. R. Mata, M. C, Cleiton, D. P. S. e Silva, Manoe A. Guedes, Antonio G. B. Lima (2008); Journal of Biology, Agriculture and Healthcare, www.iiste.org, 2(2), 2012 . http://www.azojete.com.ng 1.0INTRODUCTION 3.0 METHOD 3.1 Iterative methods