APPLICATION OF GRAPH THEORY MATRIX APPROACH TO SELECT OPTIMAL COMBINATION OF OPERATING PARAMETERS ON DIESEL ENGINE TO REDUCE EMISSIONS

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1 Int. J. Che. Sci.: 14(2), 2016, ISSN X APPLICATION OF GAPH THEOY MATIX APPOACH TO SELECT OPTIMAL COMBINATION OF OPEATING PAAMETES ON DIESEL ENGINE TO EDUCE EMISSIONS N. K. GEETHA,* nd P. SEKA b Deprtent of Mthetics, Sveeth School of Engineering, Sveeth University, CHENNAI (T.N.) INDIA b Deprtent of Mthetics, C. Kndswi Nidu College for Men, Ann Ngr, CHENNAI (T.N.) INDIA ABSTACT The objective of this pper is to dopt grph theory trix pproch to find the optil cobintion of operting preters on diesel engine. Bsed on the engine perfornce preters like brke power, brke specific fuel consuption nd brke therl efficiency, the cobintion of 18A lod, 27 0 btdc injection tiing nd 200 br Injection pressure gives the best result. Bsed on the eission preters like nitric oxide, hydro crbon, crbon onoxide, crbon dioxide nd oxygen, the cobintion of 9A lod, 19 0 btdc injection tiing nd 240 br injection pressure gives the best result out of the engine. Key words: Digrph, Mtrix, Pernent function, Pernent index, MADM, GTMA. INTODUCTION Grph theory is the study of grphs. A grph is collection of nodes or vertices nd collection of edges tht connect pirs of vertices nd is brnch of grph theory tht pplies topology nd geoetry to derive two diensionl representtions of grphs. A drwing of grph is bsiclly pictoril representtion usully ied t visuliztion of certin properties of process or syste odeled by the grph. A stright-line ebedding of grph G = (V, E) suggested by Geeth 1 is n injective function π: V 2 such tht for ny two distinct edges b nd cd the stright line segents π() π(b) nd π(c) π(d) re internlly disjoint (i.e., they y only intersect t their endpoints). * Author for correspondence; E-il: nkgeeth@gil.co, cicesekr@yhoo.co.in

2 596 N. K. Geeth nd P. Sekr: Appliction of Grph Theory. Grph theory trix pproch (GTMA) is used in odeling nd solving decisionking proble with ultiple nd interrelted ttributes. Gndhi nd Agrwl 2 proposed grph theory trix pproch to find the relibility of echnicl nd hydrulic syste. o 3 eployed GTMA for developing perfornce evlution syste for technicl eduction institutions, which is used for rnking of the technicl institutions. Grover nd Agrwl 4 developed TQM index to quntify degree of TQM concepts ipleenttion in n industry. Kur et l. 5 proposed supply chin coordintion index to evlute severl coordintion echniss. Updhyy 6 proposed systetic procedure for nlyses of object oriented softwre systes, which is useful to void pitflls in the qulity of softwre developent life cycle. Grph theory hs served n iportnt purpose in the odeling of systes, network nlysis, functionl representtion, conceptul odeling, nd dignosis etc. Grph theory hs proved its ettle in vrious fields of science nd technology s suggested by o 7. In the present study, Grph theory trix pproch is dopted to find the optil cobintion of operting preters i.e., lod, injection pressure nd Injection tiing on diesel engine. This is ttined by considering the perfornce preters like Brke power (BP), Brke specific fuel consuption (BSFC), Brke therl efficiency (BTE) nd eission preters like oxides of nitrogen (NO x ), crbon onoxide (CO), hydrocrbon (HC), crbon dioxide (CO 2 ), nd oxygen (O 2 ) s worked by Bridjesh To eticulously nlyze the bove preters nd their effect, theticl odel is needed to correlte different fctors, which give ccurte results. Generl Methodology of Grph Theory Mtrix Approch Grph theory helps to nlyze nd understnd the syste s whole by identifying the syste nd subsyste up to the coponent level. So, it is verstile tool tht hs been used in vrious pplictions copiled by Singh et l. 11 The theticl odel developed by grph theoretic pproch considers both the contribution of ttributes itself nd the extent of dependence ong the ttributes. Digrph representtion is useful for odeling nd visul nlysis. Mtrix representtion is useful in nlyzing the digrph odel. Pernent function chrcterizes the syste. Pernent function index represents the unique nuber useful for coprison, rnking nd optiu cobintion selection. The grph theory trix pproch is divided into three prts: (i) (ii) Digrph representtion Mtrix representtion (iii) Pernent function representtion

3 Int. J. Che. Sci.: 14(2), Digrph representtion Digrph is finite set of objects clled vertices together with finite set of directed edges or rcs, which re ordered pirs of vertices 12. In the present work, digrph represents the ttributes nd their inter dependencies in ters of nodes nd edges. This digrph consists of set of nodes V = {v i }, with I = 1,2,3,..M nd set of edges D = {d ij }. If node i hs reltive iportnce over nother node j, directed edge or rrow is drwn fro node i to j (d ij ). If node j hs reltive iportnce over i, then directed edge is drwn fro node j to i (d ji ) proposed by o nd Gndhi 13. Fig. 1 shows the digrph of perfornce nd eission preters. It consists of 8 nodes, which represent BP, BSFC, BTE, NO x, CO, HC CO 2 nd O 2 nd their interdependence. As the nuber of nodes nd their reltive iportnce increses, the digrph becoes coplex. Mtrix representtion Mtrix representtion of digrph is developed to reduce its coplexity if the nuber of nodes is ore nd lso trix representtion presents one-to-one reltionship between ttributes nd their reltive iportnce. If the digrph contins M nodes, the ttribute s trix is of size NxN. The ttributes re represented s digonl eleents i nd the reltive iportnce between ttributes is represented s off digonl eleents ij. The ttributes trix for the digrph is given in Eq. (1). A = (1) Pernent function representtion nd pernent index The pernent of trix is polynoil in the entries of trix. It is interpreted s the su of weights of cycle covers of directed grph nd s the su of weights of perfect tching in biprtite grph suggested by Muir nd Thos 14. Pernents re used in cobintoril thetics s no infortion is lost due to the positive signture of the peruttions. The pernent function for trix shown in Eq. (1) is written s:

4 598 N. K. Geeth nd P. Sekr: Appliction of Grph Theory. Per(A) = M + + Si+ i= 1 i j k i j k ( ) + ( )( ) + ij ji ( )( + ) + ij ji ij ji kl lk k l kl l k n k l lk i j k i j k n o ( + ) ij ji ki ik kj ji ( ij jkklli + illkkj ji)n i j k l ( ij jkkll i + illkkj ji)no + (2) The vlues of i re obtined fro experientl results, which re objective. These objective vlues will hve different units. So, they re to be norlized on the se scle s the subjective vlues i.e., 0 to 1. The reltive iportnce ( ij ) between ttributes cn lso be ssigned vlue between 0 nd 1 on scle shown in Tble 1 proposed by Adil Byksoglu 15. The vlues of ji re clculted using the following Eq. ji = 1- ij or ji = 1/ ij (3) Tble 1: eltive iportnce of ttributes Description eltive iportnce ij ji = 1/ ij ji = 1- ij Two ttributes re eqully iportnt One ttribute is slightly ore iportnt over the other One ttribute is strongly ore iportnt over the other One ttribute is very strongly iportnt over the other One ttribute is extreely iportnt over the other One ttribute is exceptionlly ore iportnt over the other The preter index for ech experient is evluted using Eq. (2) nd tbulted in the descending or scending order to rnk the. The experient, for which the pernent index is highest, is the best or optil cobintion of operting preters.

5 Int. J. Che. Sci.: 14(2), Appliction of grph theory trix pproch Considering perfornce preters Grph theory trix pproch is dopted to find the optil cobintion of operting preters i.e., lod, injection tiing nd Injection pressure bsed on the perfornce preters i.e., brke power, brke specific fuel consuption nd brke therl efficiency. Digrph representtion The ttributes for the representtion of digrph re BP, BSFC nd BTE, which re represented s nodes 1, 2 nd 3, respectively. The ttribute BP is influenced by other ttributes BSFC nd BTE. Hence, directed edges re drwn fro node 1 to the other nodes 2 nd 3. The ttribute BSFC is lso dependent on BP nd BTE. Hence, directed edges re drwn fro node 2 to other nodes 3 nd 1. The ttribute BTE is lso influenced by BP nd BSFC. Hence, directed edges re drwn fro node 3 to nodes 1 nd 2. The perfornce ttributes digrph is shown in Fig Mtrix representtion 3 2 Fig. 1: Perfornce preter digrph The perfornce preter ttributes trix, B, for the digrph is shown in Eq. (4). It is 3 x 3 trix with digonl eleents i, which represents the ttributes nd off digonl eleents ij represents the reltive iportnce. B = (4) The vlues of 1, 2 nd 3 re tken fro Tble 2, which re norlized between 0 nd 1. The vlues of 12, 13, 21, 23, 31 nd 32 re tken fro Tble 1.

6 600 N. K. Geeth nd P. Sekr: Appliction of Grph Theory. Tble 2: Norlized vlues of ttributes Exp. No. Operting preter Lod IT IP BP (kw) Engine perfornce vlues BSFC (kg/h kw) BTE Norlized perfornce vlues BP (kw) BSFC (kg/h kw) BTE

7 Int. J. Che. Sci.: 14(2), Pernent function representtion nd pernent index The pernent function for the trix shown in Eq. (4) is given in Eq. (5). 3 Per (B) = i + ( ij * ji ) k + ( * * + * * ) (5) i= 1 ijk ijk ij = 1 * 2 * * 23 * * 21 * * 2 * * 21 * * 23 * 32 Substituting the vlues of ttributes i nd their reltive iportnce ij in Eq. (5), the pernent index is evluted. Exple: For Experient No. 1, jk ki ik ij ji B = Per(B) = *0.9068* *0.8* *2.0* *0.9068*2.0 = *2.0* *0.8*1.25 The pernent index vlues for 27 experients re given in descending order in Tble 3. Tble 3: Pernent index vlues nd nk Exp. No. Operting preter Lod IT IP Pernent index nk Cont

8 602 N. K. Geeth nd P. Sekr: Appliction of Grph Theory. Exp. No. Operting preter Lod IT IP Pernent index nk Fro Tble 3, it cn be seen tht the Exp. No. 22 hs the highest vlue (4.614) of Pernent index. Hence, the engine operting preter cobintion of 18A Lod, 27 0 btdc Injection tiing nd 200 br Injection pressure gives the best result. Considering eission preters The eission preters like NO x, CO, HC, CO 2 nd O 2 re considered to find the optil cobintion of operting preters i.e., Lod, Injection tiing nd Injection pressure.

9 Int. J. Che. Sci.: 14(2), Digrph representtion The ttributes chosen for representtion of eission preters digrph re NO x, CO, HC, CO 2 nd O 2. These re represented s nodes 1, 2, 3, 4 nd 5 respectively. Bsed on the interdependence ong the ttributes, directed edges re drwn fro vrious nodes. The ttribute s digrph is shown in Fig Mtrix representtion 4 3 Fig. 2: Eission preters digrph As stted erlier, trix representtion of digrph gives one-to-one representtion. The trix clled eission preter trix, C, is shown in Eq. (6). C = = (6) The vlues of 1, 2, 3, 4 nd 5 re tken fro Tble 4, which re norlized between 0 nd 1. The vlues of 12, 13, 21, 23, 24, 25, 31, 32, 42, 43 nd 54 re tken fro Tble 1. Pernent function nd pernent index The pernent function for the ttribute s trix shown in Eq. 6 is evluted using the Eq. (2). A coputer progr is developed in C++ to copute the vlues of Pernent index for ll experients nd tbulted in descending order in Tble 5. Fro Tble 5, it cn be seen tht the experient nuber 3 hs the highest vlue (6.3750) of pernent index. Hence, bsed on the engine eission preters, cobintion of 9A Lod, 19 0 btdc Injection tiing nd 240 br Injection pressure gives the best result.

10 604 N. K. Geeth nd P. Sekr: Appliction of Grph Theory. Tble 4: Norlized vlues of ttributes Exp. No. Operting preters Lod IT IP Nox (pp) Eission chrcteristics vlues HC (pp) CO CO 2 O 2 Nox (pp) Norlized chrcteristics vlues HC (pp) CO CO 2 O

11 Int. J. Che. Sci.: 14(2), Tble 5: Pernent index nd rnk Exp. No. Operting preters Lod IT IP Pernent index nk

12 606 N. K. Geeth nd P. Sekr: Appliction of Grph Theory. CONCLUSION In this reserch, Grph theory trix pproch ws used to find the optil cobintion of operting preters on diesel engine. Bsed on the engine perfornce preters like Brke power, Brke specific fuel consuption nd Brke therl efficiency, the cobintion of 18A Lod, 27 0 btdc Injection tiing nd 200 br Injection pressure fors the optil cobintion of operting preters. Bsed on the eission preters like nitric oxide, hydrocrbon, crbon onoxide, crbon dioxide nd oxygen the cobintion of 9A Lod, 19 0 btdc Injection tiing nd 240 br Injection pressure fors the optil cobintion of operting preters. EFEENCES 1. N. K. Geeth, Verifiction on Given Point Set for Cubic Plne Grph, Indin J. Sci. Technol., 8(13), (2015). 2. O. P. Gndhi nd V. P. Agrwl, FMEA A Digrph nd Mtrix Approch, elib. Eng. Syst. Sf., 35, (1992). 3.. V. o, Grph Theory Mtrix Approch for the Perfornce Evlution of Technicl Institutions, Indin J. Tech. Edu., 23(2), (2000). 4. S. Grover nd V. P. Agrwl, A Digrph Approch to TQM Evlution of n Industry, Int. J. Prod. es., 42(19), (2004). 5. A. Kur, A. Knd nd S. G. Deshukh, A Grph Theoretic Approch for Supply Chin Coordintion, Int. J. Logist. Syst. Mng., 2(4), (2006). 6. N. Updhyy, Structurl Modeling nd Anlysis of Object Oriented Systes, Int. J. Control Coun., 1(2), (2008). 7.. V. o, Decision Mking in the Mnufcturing Environent: using Grph Theory nd Fuzzy Multiple Attribute Decision Mking Methods, Fourth Ed., Springer, London (2007). 8. Bridjesh, A Survey on Low Copression tio Diesel Engine, APN J. Engg. Appl. Sci., 10(22), (2015). 9. Bridjesh, Study on the Effects of Vrition of Fuel Injection Pressure on Single Cylinder Diesel Engine, APN J. Engg. Appl. Sci., 10(1), (2015). 10. Bridjesh, Tuning of Conventionl Diesel Engine into Low Copression tio Diesel Engine, Int. J. Appl. Engg. es., 10(5), (2015).

13 Int. J. Che. Sci.: 14(2), M. Singh, I. A. Khn nd S. Grover, Selection of Mnufcturing Process using Grph Theoretic Approch, Int. J. Syst. Assur. Eng. Mng., 2(4), (2011). 12. W. D. Wllis, A Beginner s Guide to Grph Theory, Springer Science Business Medi LLC (2000) Venkt o nd O. P. Gndhi, Digrph nd Mtrix Methods for the Mchinbility Evlution of Work Mterils, Int. J. Mchine Tools nd Mnufcturing, 42, (2002). 14. Thos Muir, A Tretise on the Theory of Deterinnts, Dover, New York (1882). 15. Adil Byksoglu, A eview nd Anlysis of Grph Theoreticl Mtrix Pernent Approch to Decision Mking with Exple Applictions, Artif Intell. ev., 42, (2014). Accepted :

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