Solution of Advection-Diffusion Equation for Concentration of Pollution and Dissolved Oxygen in the River Water by Elzaki Transform

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1 Ameican Jonal of Engineeing Reeach (AJER) 016 Ameican Jonal of Engineeing Reeach (AJER) e-issn: p-issn : Volme-5, Ie-9, pp Reeach Pape Open Acce Soltion of Advection-Diffion Eqation fo Concentation of Polltion and Diolved Oxygen in the Rive Wate by Elzaki Tanfom Padip R. Bhadane 1, Kitiwant P. Ghadle, 1 Aitant Pofeo, Depatment of Mathematic, At, Commece and Science College Oza (Mig) Naik (MS) INDIA Aociate Pofeo, Depatment of Mathematic, D. Babaaheb Ambedka Maathwada Univeity, Aangabad (MS) INDIA ABSTRACT : In thi aticle, a new integal tanfom called a Elzaki tanfom i ed to olve the model conit of a cople of advection-diffion eqation. Thee advection-diffion eqation have mch impotance in chemity, engineeing and cience. Thee advection-diffion eqation play a vey impotant ole to tdy the paamete egading ive polltion and ae ed to pedict the level of polltion concentation and level of diolved oxygen concentation in the ive wate. We have obtained the oltion fo advection-diffion eqation by Elzaki tanfom. Thee elt pove that the Elzaki tanfom i qite applicable and efl fo finding oltion of the poblem elated to polltion and diolved oxygen level in ive wate. Keywod: Elzaki tanfom, Advection-diffion eqation, Chemical paamete, Wate polltion. I. INTRODUCTION Thee ae many patial diffeential eqation ch a Laplace eqation, wave eqation etc. which ae qite efl and applicable in Engineeing, Phyic, Chemity and Science. Sch patial diffeential eqation ae vey difficlt and complicated to olve to obtain it exact oltion. In ecent yea many eeache have paid attention to find the oltion of patial diffeential eqation by ing vaio method. One of ch method i Elzaki tanfom, which i a vey efl techniqe. Elzaki tanfom wa popoed oiginally by Taig M. Elzaki in 011. He developed the method fo olving odinay and patial diffeential eqation in the time domain [1] [] [3]. Thee advection-diffion eqation ae the patial diffeential eqation and alo known a Mathematical Model fo ive polltion. In Chemity, advection i the tanpot of polltant in a ive by blk wate flow downteam and diffion i the pead ove of ga thogh wate. Diffeent type of Mathematical Model in Engineeing, Chemity and Eath Science have been olved by B. Pimpnchat in 009 [4]. Recently in 01 Saleem Azaa Hain peented a imple Mathematical model fo the concentation of Polltion and diolved oxygen in ive wate [5]. In thi pape, Elzaki tanfom i ed to olve advection-diffion eqation fo tdying the level of polltant and diolved oxygen concentation in ive wate and the elt i peented. 1.1 Elzaki Tanfom Mainly Foie, Laplace and Smd tanfom ae ed a convenient mathematical tool fo olving diffeential eqation. Recently a new tanfom called a Elzaki tanfom i deived fom the claical Foie integal and ome of it i fndamental popetie ae ed to olve the diffeential eqation. A new tanfom called the Elzaki tanfom defined fo fnction of exponential ode, we conide fnction in the et A defined by t k j j A f ( t ) : M, K, K 0, f ( t ) M e, if, t ( 1) X [0, ) (1) 1 Fo a given fnction in the et A, the contant M mt be finite nmbe, K 1, K may be finite o infinite w w w. a j e. o g Page 116

2 Ameican Jonal of Engineeing Reeach (AJER) 016 The Elzaki tanfom denoted by the opeato E (. ) defined by the integal eqation. [ ( )] ( ) ( ) E f t T f t e d t, t 0, K K., 0 t ( ) 1 0 t The vaiable in thi tanfom i ed to facto the vaiable t in the agment of the fnction; thi tanfom ha deepe connection with the Laplace tanfom. Elzaki Tanfom of the Some Fnction We have Elzaki tanfom of imple fnction. 1) ) 3) t t If, f ( t ) 1, th e n ; E (1) e d t e 0 If, f ( t ) t, th en; E [ t ] te d t a t a t a t If, f ( t ) e, th en ; E ( e ) e e d t 1 a Theoem: - If E [ f ( t )] T ( ) ' 1) ) 0 t 9 then t d f T ( ) E E f ( t ) f (0 ) d t d f " T ( ) E E f ( t ) f (0 ) f '(0 ) d t 3 3) n n 1 d f n T ( ) n k k E E f ( t ) f (0 ) n n d t k 0 1. Mathematical Model In thi model we amed that the advection diffion-eqation may be a good appoximation to tdy model of ive polltion. We amed that the ive i linea and it ha a nifom co-ectional aea. The ive co-ection potion conideed nde tdy i one with an abitay inteio and endpoint at x = 0 and x = L. Thee eqation accont expanion of the polltant and the diolved oxygen concentation epectively ae a R G ( x, t ) ( R G ( x, t )) ( w A R G ( x, t )) M G ( x, t ) d x (3) p t x x with bonday condition G ( 0 ) R F ( x, t ) ( R F ( x, t )) ( F ( x, t )) ( S F ( x, t )) d x (4 ) x t x x with bonday condition F ( 0 ) S The fit eqation inclde the ma tanfeed of olid to the wate (ive) in m /day, and it i the concentation of polltion. The econd eqation i a ma balance fo diolved oxygen, with addition thogh the face at a ate popotional to the degee of atation of diolved oxygen (S-F), R F ( x, t ) ( R F ( x, t )) ( F ( x, t )) ( S F ( x, t )) x t x x (5) with bonday condition G (0 ) with bonday condition F ( 0 ) S Fo thee the only vaiation i with the ditance downteam on the ive and o we wite G(x, t) = G(x) and F S (x, t) = F S (x). w w w. a j e. o g Page 117

3 Ameican Jonal of Engineeing Reeach (AJER) 016 To implify the eqation, we will conide only the teady tate oltion. whee L - Pollted length of ive (15787m) θ x - dipeion coefficient of diolved oxygen in the x diection. (m / day) θ p - dipeion coefficient of polltant in the x diection (m /day) R - co-ection aea of the ive (m ) S - Satated oxygen concentation(le than 5) (kg/m 3 ) γ - Ma tanfe of oxygen fom ai to wate of the ive (m /day) M - Ma tanfe of olid (olte) to the wate of the ive (m /day) k - Degadation ate coefficient (pe Day) - Rate of polltant addition along the ive (kg/m/day) w - Wate velocity in the x- diection. (m ) II. ANALYTIC STEADY-STATE SOLUTIONS FOR SPECIAL CASE (ZERO DISPERSION) Conide the eqation (5), when dipeion θ p = 0, with bonday condition G (0 ) [that i no polltion pteam becae of the abence of dipeion.] d ( G ( x, t )) MG 0 ( x, t ) dx d ( G ( x, t )) M G 0 ( x, t ) dx d ( G ( x, t )) M G 0 ( x, t ) d x Appling Elzaki tanfom on both ide, we get d ( G ( x, t )) M E E [ G ] 0 ( x, t ) d x T ( ) M. G ( o ) T ( ) 0, T ( ). M T ( ) 1 w T ( ) k ( M ) by initial condition G (0 ) Now, by appling invee Elzaki tanfom on both ide, we get w E [ T ( )] E k ( M ) 1 w 1 E [ T ( )] E w k R M 1 wr G ( x ). e M x wr, M x wr G ( x ) C. e, W h ee C Thi how that the polltant concentation downteam of the ive. If x = 0, then G (x) = G (0 ) If x =, then G (x) = G ( ) e and (6) w w w. a j e. o g Page 118

4 Ameican Jonal of Engineeing Reeach (AJER) 016 The oltion of the eqation elt eveal that the amont of polltant ae fond to be inceaed at x = L than x = 0. Now, conide the eqation (6), when dipeion θ x = 0 with bonday condition F (0 ) [fo thi cae thee i no polltion pteam becae of the abence of dipeion.] S d ( F ( x )) ( S F ( x )) 0 dx d F ( x ) S F ( x) d x Appling Elzaki tanfom on both ide, we have d F ( x ) S E E [ F ( x )] E d x T ( ) S. F (0 ). T ( ) E T ( ) S. S. T ( ) E whee F (0 ) S (by initial condition) T ( ). S. S. T ( ) E [1], Since E [1] = S. T ( ). S T ( ) S. [ S ] 1 1 T ( ) S.. 1 wr Now, by appling invee Elzaki tanfom on both ide, we have E [ T ( )] S. E [ ]. E F ( x ) S.(1). e x wr, Since F ( x ) S. e x wr x wr F ( x ). e S. 1 wr E 1 ( ) 1 Thi how that the diolved oxygen concentation downteam of the ive. If x = 0 then ( ) (0 ) F x F S and F x F e S (7) If x = then ( ) ( ). w w w. a j e. o g Page 119

5 Ameican Jonal of Engineeing Reeach (AJER) 016 Thi how that in the downteam of the ive diolved oxygen being contant and it vale i vey cloe to atated vale S, thee mathematical elt ae coincide with the actal fact in mot of the citie in ive wate bt at the pteam of the ive, level of D.O. i vey good. III. FIGURES AND TABLES Table-I: Specification limit fo dinking ive wate Paamete in p.p.m. D.O. B.O.D. T.D.S Slphate Chloide Sodim Nitate COD Limit mg/l Table-II: If we take the coeponding vale fo ome paamete into conideation with help of polltion contol boad banch Naik Mahaahta, then the oltion of model-i and model-ii ae in fll ageement with the oltion. Paamete in p.p.m. Diffeent Sampling tation S1 S S3 S4 S5 S6 S7 D.O B.O.D T.D.S Slphate Chloide Sodim Nitate COD Fige-1 Fige- Fige-3 Fige-4 w w w. a j e. o g Page 10

6 Ameican Jonal of Engineeing Reeach (AJER) 016 Fige-5 Fige-6 Fige-7 Fige-8 IV. CONCLUSION Pevioly advection-diffion model wa ed to get the oltion elated to poblem of the ive polltion. We tied Elzaki tanfom to get the oltion of advection-diffion model. It i fond that the oltion obtained by Elzaki tanfom ae ame a that of the oltion obtained by ing advection-diffion model. Thi eveal that the Elzaki tanfom i alo applicable fo olving ch type of poblem. It may conclde that the new integal Elzaki tanfom i vey convenient tool olving thi type of mathematical model. The elt how that at x = L, the pecentage of diolved oxygen i fond inceaed and it i in bet ageement with the obevation obtained ing advection-diffion model. ACKNOWLEDGEMENTS The atho offe hi gatefl thank to fiend and the efeee fo thei help and active gidance in the pepaation of thi evied pape. REFERENCES [1]. Taig M Elzaki: The new integal tanfom Elzaki Tanfom, Global Jonal of Pe and Applied Mathematic, 7(1), 011, []. Application of new tanfom Elzaki Tanfom to Patial Diffeential Eqation, Global Jonal of Pe & Applied Mathematic, Taig M. Elzaki and Salih M. Elzaki. [3]. Taig M. Elzaki, Salih M. Elzaki and Elayed A. Elno, on the new integal Tanfom Elzaki Tanfom fndamental popetie invetigation and application. [4]. B. Pimpnchat, W. L. Sweatman, G. C. Wake, W. Tiampo and A. Pahotam: Mathematical Model fo polltion in a ive and it emediation by aeation, Applied Mathematical Lette, (009) 304. [5]. Mathematical Model fo the Concentation of Polltion and Diolved Oxygen in the Diwaniya Rive (Iaq) by Saleem Azaa Hain. Ameican Jonal of Scientific Reeach ISSN , Ie-78, Octobe, 01. w w w. a j e. o g Page 11