Determination of Energy Involved In a Stepwise Size Reduction of Maize, Using a Numerical Approach.

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1 IOSR Joural of Matematics (IOSR-JM) e-issn: , p-issn: 9-765X. Volume, Issue Ver. V (Ja - Feb. 5), PP Determiatio of ergy Ivolved I a Stepwise Size Reductio of Maize, Usig a Numerical Approac. Famurewa O. K.. ad Famurewa J. A. V. Departmet of Matematical Scieces, Wesley Uiversity of Sciece ad Tecology, Odo, Nigeria. Departmet of Food Sciece ad Tecology, Federal Uiversity of Tecology, Akure, Nigeria Abstract: Tis work described te determiatio of eergy ivolved i a stepwise size reductio of maize, usig a umerical approac. It adopts a implicit oe-step tird-derivative metod of te form; () () () () () (). Te size to wic maize grai is to be reduced relative to its iitial size affects te eergy required of te millig macie. Te smaller te size of te milled product te iger te eergy required of te macie. Te eergy required is evaluated by te size reductio formula: d = dl GL. ompariso of te umerical result wit te exact solutio sowed tat te tird- derivative metod is accurate ad tus recommeded for solvig egieerig problems o rate of cage. Keywords: ergy, Maize, Size reductio, Oe-step metod, Millig. I. Itroductio Maize is a ecoomic crop. Primary processig metods of tis food i dry state requires size reductio, a eergy drive uit operatio. Maize is a importat agricultural crop tat plays sigificat roles i te diet of te people all over te world particularly i te developig atios. FAO () respectively raked maize as tird, importat cereal i te world. Accordig to Ket ad vers (994), maize grai cotais about.8% moisture,.% protei, 4.% fat,.7% fiber,.5% as, ad 7.7% starc Size reductio is te uit operatio i wic te average size of solid pieces of food is reduced by te applicatio of gridig, compressio or impact forces (Famurewa, 7). Size reductio ito flour is a importat food processig operatio because a large proportio of food materials are reduced ito flour before coversio ito fiised products. Oe measure of te efficiecy of te millig operatio is based o eergy required to create ew surfaces. Size reductio is oe of te least eergy-efficiet of all te uit operatio ad te cost of power is a maor expese i crusig ad gridig, so te factors tat cotrol tis cost are importat (Mcabe et al., 5). Durig millig, Kietic ergy is dissipated o te material i excess of its iteral stregt wic causes rupture alog te lie of cleavages resultig ito smaller particles (Famurewa, 7). Some beefits of size reductio i food processig iclude; icrease i te surface-area-to-volume ratio of te food wic icreases te rate of dryig, eatig or coolig, improvemet i te eatig quality or suitability of foods for furter processig ad icrease i te rage of products available. Te relatiosip betwee te commiutio eergy ad te product size obtaied for a give feed size as bee researced extesively over te last cetury. Teoretical ad empirical eergy-size reductio equatios were proposed by Rittiger (867), Kick (885) ad Bod (95), kow as te tree teories of commiutio; ad teir geeral formulatio by Walker et al. (97). Fially, Hukki (96) proposed te revised form of te geeral form of commiutio ad suggested tat te eergy-size relatio is a combied form of tese tree laws. Walker et al. (97) proposed te followig equatio, for a geeral form of commiutio. [] Were is te et specific eergy; x is te caracteristic dimesio of te product; is te expoet; ad is a costat related to te material. quatio [] states tat te required eergy for a differetial decrease i size is proportioal to te size cage (dx) ad iversely proportioal to te size to some power. If te expoet i quatio [] is replaced by te values of, ad.5 ad te itegrated, te well-kow equatios of Rittiger, Kick ad Bod, are obtaied respectively. Rittiger (867) stated tat te eergy required for size reductio is proportioal to te ew surface area geerated. Sice te specific surface area is iversely proportioal to te particle size, Rittiger s ypotesis ca be writte i te followig form: DOI:.979/ Page

2 Determiatio Of ergy Ivolved I A Stepwise Size Reductio Of Maize, Usig [] were is te et specific eergy; x f ad x p are te feed ad product size idices, respectively; ad K is a costat. Kick (885) proposed te teory tat te equivalet relative reductios i sizes require equal eergy. Kick s equatio is as follows: [] were is te et specific eergy; x f ad x p are te feed ad product size idices, respectively, ad K is a costat. Bod (95) proposed te Tird Law of gridig. Te Tird Law states tat te et eergy required i commiutio is proportioal to te total legt of te ew cracksformed. Te resultig equatio is: Accordig to Fellows (), combied impact ad searig forces are ecessary for fibrous foods like maize, terefore Rittiger s equatio was adopted for tis study. A differetial equatio of te form: y = f (x, y), y( x ) = y, were f is assumed to be Lipsciz cotiuous, ca be solved usig various approximatio metods, amog wic is te Implicit oe- step tird derivative umerical metod of order 6, wic is a improvemet o te covetioal implicit family of liear multistep metods. Tis metod icludes more derivative properties of te differetial equatio ito te existig metod wic makes it more accurate ad of more practical values. II. Metodology Maize grais were milled from 6 mm to. mm usig a p (7.5 KW) motorised plate millig macie. Te eergy required i a stepwise size reductio is evaluated by te size reductio formula: d dl = GL, were d dl is te rate of cage of eergy wit legt (Jm- ). G is te Rittiger s costat (Jm - ). L is te legt of te maize grai (m). Implicit oe- step tird derivative umerical metod of order 6, of te form; () () () () () () was adopted for te measuremet of te eergy required for te size reductio. III. Derivatio of te Numerical Metod. Teorem Settig k=, l = i te geeralized implicit multi-derivative liear multistep metod of te form: k k k i i βi i k (Famurewa ad Olorusola, ) gives a oe-step tird derivative metod of te form; () () () () () () [5] [4]. [6] Proof Settig k=, l= i (5) gives [7] wit local trucatio error () () () () () () DOI:.979/ Page

3 T () Determiatio Of ergy Ivolved I A Stepwise Size Reductio Of Maize, Usig ) ( ( () () () () ) ) ( () Adoptig Taylor s series expasio of +, () +, () + ad () + i (8) ad combiig terms i equal powers of gives T were () () ()... ( [9] 8 ) [8]! 4 4!! (8) [] 5 5! 4!! 6 6! 5! 4!! 7 7! 6! 5! 4! Imposig accuracy of order 7 o T + to ave = = = = 4 = 5 = 6 = ad T + = ( 7 ) wic gives!! 4! 4!!! 5! 4!! 6! Solvig gives; α =, β =.5, β =.5, β =., β =., β =.8 ad β =.8 puttig tese values ito equatio (7) gives te oe step tird derivative metod of equatio (6). Implemetatio: To access te accuracy of te sceme, te metod was re-writte i FORTRAN programmig laguage ad implemeted o a digital computer. Te program was used to solve te size reductio equatio. Te results ad errors are sow i Table. Problem; d = dl GL,.6 mm = 5 J (we te macie is operatig empty), xact solutio; = 8 L + 5 Numerical metod; () () () () () () IV. DOI:.979/ Page Results Ad Discussio

4 Determiatio Of ergy Ivolved I A Stepwise Size Reductio Of Maize, Usig Table : Te Result Obtaied for te Solutios. x-value exact-solutio computed-solutio error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-4 It was observed tat te computed result is very close to te exact solutio as sow i Table. Hece, te metod is recommeded for solvig practical problems of Sciece ad gieerig. V. oclusio I tis study, oe step tird derivative metod was used to determie te eergy required for te stepwise size reductio of maize grais. Te result obtaied sowed tat te umerical metod is accurate, effective ad efficiet, ece, ca be adopted i solvig practical problems o rate of cage. Refereces DOI:.979/ Page

5 Determiatio Of ergy Ivolved I A Stepwise Size Reductio Of Maize, Usig []. Bod, F Te tird teory of commiutio. Trasactios of te AIM 9(): 484 []. Famurewa J.A.V(7). Te Developmet ad valuatio of Four Roller Flour Mill wit Parallelogram ofiguratio, Iteatioal Joural of Food gieerig. (6): -7. []. Famurewa O K ad Olorusola S. A (). Te effect of ivolvig more derivative properties o te accuracy of a implicit oe step metod. Asia Joural of Natural ad Applied Scieces. (): [4]. FAO.(). Food ad Agriculture Orgaizatio of te Uited Natios coomic ad Social Departmet. ttp:// Accessed o April,. [5]. Fellows P.J (). Food Processig Tecology. Priciple ad Practice Secod ditio, Oxford Brookes Uiversity. [6]. Hukki, R.T.(96). Proposal for a solomoic settlemet betwee te teories of vo Rittiger, Kick, ad Bod. Tras. AIM, : [7]. Ket, N. L. ad vers, A. D Tecology of cereals, a itroductio for studets of Food Sciece ad Agriculture, 4t d. Pergamo, Oxford, UK pp [8]. Kick, F. (885). Das Gesetz der Proportioalem Widerstad ud Seie Awedug (Priciple of Proportioal Resistace ad Its Applicatio). Leipzig, Germay: Felix. [9]. Mcabe, W. L, Smit, J.. ad Harriott, P. (5). Uit operatios of cemical egieerig. 7t d. McGraw-Hill Iteratioal New York pp []. Rittiger, P. R. (867). Lerbuc der Aufbreitugskude(Textbook of Uit Operatios). Berli, Germay: rest & Kok. []. Walker, W.H., Lewis, W.K., Mcadams, W.H., ad Gillilad,.R. (97). Priciples of emical gieerig. McGraw-Hill, NY, USA.. DOI:.979/ Page