Double-Relaxation Solute Transport in Porous Media
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1 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 ouble-relaaon Solue Transpor n Porous Meda Bakyor Kuzayorov Tursunpulo zyanov Odl Kaydarov Professor eparmen of Maemaal modellng Samarkand Sae Unversy Samarkand Uzbeksan Asssan Professor eparmen of Maemaal modellng Samarkand Sae Unversy Samarkand Uzbeksan Asssan Professor eparmen of Maemaal modellng Samarkand Sae Unversy Samarkand Uzbeksan ABSTRACT: In e paper a problem of double relang solue ranspor n a porous medum s onsdered. On e bass of mass balane equaon and e double relang Fk s law a solue ranspor equaon s derved. Te equaon s numerally solved and e nfluene of e relaaon parameers on solue ranspor araerss s esablsed. KEYWORS: Flraon; maemaal model; porous meda; relaaon; suspenson; solue ranspor; I. INTROUCTION In reen mes e nensy of sudes of anomalous penomena a dffuson and dsperson of parles n porous meda as sgnfanly nreased. One of e orgnal works were volaon of lassal Fkan dffuson law was ndaed s []. In [] e mass solue flow s gven by a sum wo members - e lass Fk s dffuson and relaaon erm. A e ranspor of solue or oer suspended fne parles n porous meda breakroug urves an be non- Gaussan w onsderable alng effe [ 3 4]. A Non-Fkan generalzed eory dsperson s proposed n [5]. A general equaon from w equaons onanng dfferen dervaves of e mass dsperson flow w respe o me an be obaned n parular ases was derved and soas models desrbed by negrodfferenal equaon n w e dsperson ensor nreases asympoally w me and as a funon of e spae oordnae were proposed. In [6] s found a low-onenraon-graden epermens an be smulaed sasfaorly usng e Fkan-ype dsperson equaon. However alulaed breakroug urves for g-onenraon-graden epermens devae subsanally from e measured urves. I appears a a sasfaory f o g-onenraon-graden daa an be obaned only f e value of longudnal dspersvy s redued by a faor of ree. Usng e non-lnear eory owever s possble o smulae bo low- and g-onenraon-graden epermens w a unque se of parameer values. Fk s law epresses e proporonaly of solue flu w respe o onenraon graden. Smlar relaons are gven by ary s law for e flud flow n porous meda[7]. In [8] w usng an neral Fk`s law yperbol solue ranspor equaon s derved. I was sown a su a yperbol desrpon s vald bu for ransen solue flows w very sor araers mes. In [9] e problem of e marosop smulaon of e moon of a vsous flud and mass ranspor n a porous medum s onsdered under e assumpon a e mass ranspor an loally be desrbed by e Fk relaaon law. Several ases deermned by e loal nera number of e mass flow and e Pēle number are nvesgaed. Te marosop ranspor models are analyzed and ompared w well-known penomenologal models. I s well known a e flow of aqueous polymer fluds roug porous meda plays mporan role n enaned ol reovery proesses []. We an noe [] a e n su reology of e flud does no eplly depend upon e bulk reology of e aqueous polymer soluon. For polymer soluons e apparen vsosy s a funon of flow rae roug e porous medum and flow rae may be nerrelaed w e flud memory n e pore nework []. Formaon of alng n breakroug urves.e. non-gaussan onenraon profles nvesgaed also n [ 3] a eplly demonsraes e volaon of lassal Fk`s law. Sasal approa o model anomalous dsperson n porous meda s used n [3 4]. Models are presened by negrodfferenal equaons n w e dsperson ensor s asympoally nreases w me and s a funon of e spae oordnae. Copyrg o IJARSET 594
2 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 Some ypoe flraon laws for omogeneous lquds n porous meda a ake no aoun e relaaon of e pressure graden and e veloy of flraon are gven n [56]. On e bass of ese models non-seady flraon equaons are derved a makes enable o deermne pressure dsrbuon and oer flraon araerss. In s paper prnpal model approaes and meodology used n ese works for flraon proesses we aemp o adap for solue ranspor problems. Frs for double-relaaon Fk`s law a nludes bo e relaaon of solue mass flow and onenraon graden we derve a solue ranspor equaon and pose an nal-boundary problem. Ten we numerally solve e problem. Ne we presen some resuls and er sor analyss. II. ERIVATION OF TRANSPORT EQUATION AN FORMULATION OF THE PROBLEM Te mass balane of e solue n e one-dmensonal ase s epressed by e equaon F m ( were m e porosy of e medum e onenraon of solue dssolved n e flraed lqud F e oal solue mass flow onssng of e onveon (F and dffuson (F d flows F F F d me spae oordnae. Conveve flow as e followng form [7] were v f flraon veloy v e pysal flud veloy. F v vm ( Te dffuson law w double-relaaon we wre as Fd Fd m (3 were e relaaon mes dffuson oeffen w we ake as a onsan. In more general form nsead of dffuson oeffen one an use dsperson oeffen a depends on flraon veloy dsrbuon [7]. Equaon ( akng no aoun ( (3 beomes Fd m vm. (4 fferenang (3 w respe we ave Fd m. (5 f From (4 and (5 we oban or v 3 v v. (6 In order o assess e relaaon effes on e solue ranspor araerss we pose e followng smple problem. Le n e sem nfne porous medum nally flled w pure (wou solu flud sne nflows lqud w onsan onenraon of solue. Ten e nal and boundary ondons ake e from Copyrg o IJARSET 595
3 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 Copyrg o IJARSET ( ( (7. ( ( (8 III. NUMERICAL SOLUTION OF THE PROBLEM To solve e problem (6 - (8 we apply e meod of fne dfferenes [7]. In e area ( T we nrodue e followng grd... ( J J T were - seps w respe and respevely. Numeral soluon for e onenraon on e grd we denoe as and solue dffuson flow rae d F as. F We appromae equaon (6 usng e mpl fne dfferene seme on e grd n e form. v v (9 Te nal and boundary ondons (7 (8 are appromaed as... ( J I.... ( Fne dfferene seme (8 leads o e sysem of lnear algebra equaons G E B A I J ( were v v A v v B E ( ( v v G I enoug large neger. Te sysem of equaons ( we solve by e Tomas algorm [7]. In aordane w (3 we defne e solue flow rae. W F d we ave d d e me F. (3 Te F d an be deermned by alulang e negral (3 a e known onenraon dsrbuon. We an also drely dsrezaze equaon (5 on e grd.
4 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 So equaon (3 s appromaed as F F F m m from w F s deermned F F m m. (4 Sne e fne dfferene seme (4 s sable. A known... J. A e lowes grd layer we ave.... F F с from (4 we fnd IV. RESULTS In e alulaons e followng nal values of parameers are used: m 3 and e varous v. Some resuls of numeral alulaons are sown n Fg W e nreasng of a gven e developmen of onenraon profles s delayed. I an be seen from e fgures a n e ranson perod e duraon of w s deermned by e relaaon mes e onenraon profles lag bend e orrespondng profles wou akng no aoun e relaaon. W nreasng of a e gven e developmen of onenraon profles nensfes (Fg.. In s ase also a large mes we an observe e weakenng of relaaon penomena. Comparson of wo ases sows a nfluenes of and on onenraon profles are que oppose. Jon nfluenes of wo relaaon parameers on solue ranspor araerss are deermned by domnan values of and. Influene of relaaon parameers on dffuson mass flu rae F d also s suded. ynams of F d a dfferen pons and for dfferen values of relaaon mes s sown n Fg Resuls ndae non-monoonous dependenes of F d a gven pons. Te araer of relaaon parameers nfluenes on F d s same as for onenraons. We an see dereasng dynams of F d for large mes. I ours due o dereasng of onenraon gradens for large mes a a gven pons of e area. On e bass of presened resuls we an onlude a e relaaon araer of dffuson law onsderably alers all solue ranspor araerss. а b m m Copyrg o IJARSET 597
5 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 m Fgure. Profles of с a 5 6 v m s m s s а 36 s; b s; s; s. a b m m m Fgure. Profles of с a 5 6 v m s m s s а 36 s; b s; s; s. Copyrg o IJARSET 598
6 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 8 m s F d F 8 m s a d b 4 s 4 s 8 m s F d 4 s Fgure 3. ynams of F d a s 5 6 v m s m s (а; m (b; m (; (( 3(3s. 8 m s F d а 8 m s F d b 4 s 4 s Copyrg o IJARSET 599
7 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 8 m s F d 4 s Fgure 4. ynams of F d a s 5 6 v m s m s (а; m (b; m (; (( 3(3s. V. CONCLUSION In s paper a solue ranspor equaon for double-relaaon solue dffuson law s derved. Solue dffuson law akes no aoun bo e relaaon of e dffuson solue flu and onenraon graden. An nal-boundary problem for s equaon s posed and numerally solved. On e bass of numeral soluons e nfluene of relaaon mes on solue ranspor araerss s suded. I was sown a e relaaon of e solue dffuson mass leads o delayng of ranspor araerss wle e relaaon of solue onenraon gradens leads o e advane dsrbuon of ranspor araerss. So nfluenes of and on solue ranspor araerss are que oppose. All obaned resuls sow a relaaon beavour of e dffuson law onsderably alers all solue ranspor araerss. REFERENCES []. Sra O..L. A maemaal model for dsperson w a movng fron n groundwaer // Waer Resours. Res. 8( []. Bar. J.-C. Bouaud J.P. Georges A. Guyon E. Hulen J.P. Rakoomaala N. and Saln. Transen non-gaussan rae dsperson n porous meda n: Hulen J.P. e al. (Eds. Hydrodynams of spersed Meda. Elsever. Amserdam p. [3]. Brady J.F. sperson n eerogeneous meda n: Hulen J.P. e al. (Eds.. Hydrodynams of spersed Meda. Elsever. Amserdam p. [4]. Maeron G. and de Marsly G. Is ranspor n porous meda always dffusve? A ounereample. Waer Resours. Res p. [5]. Hassanzade S.M. On e ransen non-fkan dsperson eory // Transpor n Porous Meda. 3( [6]. Hassanzade S.M. A. Lense. A non-lnear eory g-onenraon-graden dsperson n porous meda // Advanes n Waer Resoures. 8( [7]. Bear J. ynam of fluds n porous meda. over New York. 97. [8]. Auraul J.-L. Lewandowska J. Royer P. On non-fkan yperbol dffuson. Suda Geoena e Meana Vol. XXX No [9]. Kuzayorov B.K. Marasop Smulaon of Relaaon Mass Transpor n a Porous Medum. Flud ynams Vol. 39. No pp. []. Cen Z. Huan G. Ma Y. Compuaon Meods for Mulpase Flows n Porous Meda. Soey for Indusral and Appled Maemas SIAM. Pladelpa. 6. []. Kuzayorov B. Flraon of non-omogeneous lquds n porous meda. Tasken FAN Publser p. []. Hossan M.E. L. Lu. M. Rafqul Islam. Inluson of e Memory Funon n esrbng e Flow of Sear-Tnnng Fluds n Porous Meda. Inernaonal Journal of Engneerng (IJE Vol.3. Issue pp. Copyrg o IJARSET 5
8 ISSN: Inernaonal Journal of Advaned Resear n Sene Engneerng and Tenology Vol. 5 Issue January 8 [3]. Cusman J.H. and Gnn T.R. Non-loal dsperson n meda w onnuously evolvng sales of eerogeney Transp. Porous Meda p. [4]. Cusman J.H. Hu B.H. and Gnn T. R. Nonequlbrum sasal means of preasympo dsperson. J. Sa. Pys p. [5]. Moloov Ju.M. Relaaon flraon. Kazan. KSU Publser p. [6]. Aklov J.A. Non-seady flow of vsoelas lquds. Tasken: FAN Publser p. [7]. Azz K. Seary A. Peroleum Reservor Smulaon. Appled Sene Publser LT p. Copyrg o IJARSET 5
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