MATHEMATIC-PHYSICAL MODEL OF DIMENSIONING SYSTEM IN THE PROPAGATION OF MICROWAVE "WAVEGUIDE-SLUDGE FROM WASTEWATER TREATMENT PLANTS

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1 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- MATHEMATIC-PHYSICAL MODEL OF DIMENSIONING SYSTEM IN THE PROPAGATION OF MICROWAVE "WAVEGUIDE-SLUDGE FROM WASTEWATER TREATMENT PLANTS Emiin-Nri Riți-Miho, Phd. En. Emi Riți-Miho, Phd. En. Dn Porr, Phd. En. Tehni Univerity of Cuj-No, Roumnie Abtrt: Thi er reent hyio-mthemti mode izin ytem mirowve fied rotion throuh wveuide of the wveuide omoed of retnur etion nd yrmid funne etion retnur vribe ditribution. Thi mode w dted to the ondition of "wveuideond ude bed" trety. The ytem i to define rmeter nd rotion eqution for dttion. Key Word: Phyi nd mthemti mode, mirowve, ude ettin, wveuide Introdution Mkin the hyi nd mthemti mode w deveoed in order to rete mthemti too for utin the ize of the wve uide tnd for bortory reerh on the fied of mirowve therm roein of ude from wtewter tretment ond. The mode i bed on the ener theory of mirowve rotion reented in the iterture [], [], [3], [4], [5], dtin to the eifi mthemti retion of "wveuide-bed mud ond. It tke into ount henomen ourrin in eetromneti wve rotion throuh retnur uide nd mthemti retionhi tht define thee henomen. Determintion of mode rmeter For the eifi e of the nt deined to etbih hyi nd mthemti mode of mirowve rotion ytem we took trutur nyi with the foowin omoition: the mnetron, wve uide, funne ditribution nd reonnt vity, hown hemtiy in Fiure. Mirowve trnmiion ytem, retnur etion; work on the fundment rotion modete. Determintion of hyi-mthemti mode of mirowve rotion in the ytem invove definin the rmeter of the retnur uide rotion. Criti frequeny i defined the frequeny beow whih no eetromneti wve rotion our. Thi frequeny i uted with: f m b n + Criti wveenth i the wveenth orreondin riti frequeny i determined with: () f () 8

2 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- Fiure. - The rotion of mirowve - mnetron emby, - wveuide, 3 - Funne ditribution, 4 - vity reonnt Guide wveenth i uted with: ( ) Vuum wveenth, rorite workin frequeny i uted with: f Dehin ontnt i uted with: f β π ( f f ) (5), Dehin ontnt hne retive to π (i.e. β / π ), the frequeny retive to the eed of rotion in free e (i.e. f / ), i rereented in Fiure. In thi e the dehin ontnt doe not vry in roortion to frequeny, o the uer i dierive wve. (3) (4) Fiure. - Vrition of ontnt in the reorted frequeny [] Grou veoity i the tu eed of rotion of mirowve enery i uted with: v ( f f ) (6) Grou veoity i e thn the eed of iht nd by the ymtoti (fiure 3.). Wve imedne If the wve mneti fied H (for mode rotion TE ), the wve imedne i uted with: (7) Z Z ( f f ) The: Z µ ε π 377 i the vuum wve imedne. Fiure 3.- Grou-veoity vrition with frequeny [] 83

3 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- In fiure 4 i reented the wve imedne vrition with frequeny. The exreion for utin the wve imedne (7) i vid for uniform wveuide (oe) nd ir inide the rotion environment. Thu the frequeny bnd f > f wve imedne i urey reitive. Ditribution of eetromneti fied omonent in the uide i iven by retion (8)...(): H o π x ( ωt β z) (8) H z H E o π H in x in( ωt β z) π Z H in x in ωt β z (9) x ( ) () y Fiure 4. - Vrition of imedne with frequeny [] Curve of vrition of the mitude omonent x re reented ordin to fiure 5, where it i hown ine of eetri nd mneti fied in the wveuide where the wve H Fiure 5. - Vrition of intenity omonent H, z H, nd x E deendin on x [] y From thi fiure two onuion n be drwn: the eetri fied tnenti to the w i zero nd how mny in m nd n of H m, the emi-ine wve i reent on the be nd the heiht b. n If H for x i emi-inuoid; nd b the fied i ontnt, i.e. not deendin on y. Attenution ontnt determine the ower oe in the wveuide. Thi ontnt i uted from the dimenion of the wveuide etion. For the wve H, the exreion for utin the ttenution ontnt i: α H + + b δ ψ Z / Penetrtion deth δ i determined by the: d f ψ From thi retion reut the enetrtion deth [ m ], if the frequeny i meured in [GHz ] nd ontnt w mteri formin the uide, in [ m ]; The diited ower in the w of the uidne ytem i determined by the tot ttenution of the mirowve route uidne ytem, nd i iven by: α α (3) H () () 84

4 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- - i the enth of the tri nd mirowve to the mteri i omoed of: mnetron wveuide enth, nd the enth of the ditribution funne. For the ditribution funne w tken overe ftor of, o tht the tot equivent enth i: + (4) Sine, α o X nd Pi X Pe X - i the rtio of ower of entry into the route P nd exit i P. e reut: α X (5) The diited ower (whih i onverted into het in the met w of the uidne ytem by the Joue-Lentz effet) i iven by: X Pd P (6) i X The mximum owbe ower i imited, deendin on the uide ize nd eted defut oe. The mximum ermiibe ower, trnmitted throuh the wveuide for the wve H, in, ondition of dttion: P 5 η b (7) In e of indttion, ower in od i the differene between the orreondin ower of the inident wve nd ower of the refeted wve, i.e.: b P ( Ei + Er ) ( Ei Er ) (8) 4 Z Beue Ei + Er E mx ; Ei Er E nd E mx min σ (Defined tndin wve ftor), we obtin: Emin b Emx P (9) 4 Z σ Tht trnmitted ower i σ time e, thn, for the me mount of eetri fied trenth E. mx The ower vrition (or the qured eetri fied trenth, E ) in the e of dttion i hown in fiure 6. Biy, for the modein of mirowve izin uide mut wy onider the ondition of dttion, the ondition i tified if the tndin wve ftor i σ,,. Fiure 6. - Vrition of the qure intenity of the eetri fied, in e of dttion [] For the mximum vue of the tndin wve ftor σ,, refetion oeffiient beome σ, Γ,9 nd o the refeted ower (reorted to the inident ower) rereent %, whih σ +, i rtiy etbe. Determintion of the eifi eqution for the rimry hyi mode of dttion In the e of yrmid heme ditribution funne (fiure ), doted when deinin mobie itor bortory tnd, we n define the hyi mode of rimry dttion, whoe rinie heme i hown in fiure 7. Thi mode hve of three re of rotion: rotion in e uided, whih i omoed of the ontnt etion uide nd the vribe ro-etion uide, the yrmid funne; the rotion in free e 85

5 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- from the outut of funne nd red to the urfe of ude AA ' nd in the interior of the ude bed, whih i hrterized mixture of two omonent, wter nd oid rt. Fiure 7. - Pyrmid funne rotion ytem [] Thi rotion ytem n be equivent with n eetri iruit imir to tht in fiure 8. Sude bed i o medium hrterized in term of dieetri ontnt nd ondutivity eetromneti time exty tnent of o ne: σ () tδ ω ε A the ne i m, it i onidered to be equ to the vue of ne tnent of o ne in rdin: tδ δ. Fiure 8. - Equivent iruit of the dttion ytem Adttion i the eimintion of refeted wve (refeted ower tht i roortion to the qure refeted wve hown). Eimintion of refeted wve our when the trnition from one environment to nother i eiminted diontinuitie, imedne i.e. t the eft nd riht of the urfe AA ', ' nd bb ' i onvenienty turn on the me hown. It i neery therefore to ute the inut imedne t the urfe AA '. Etbihin the imedne ution retionhi of the t the entrne in the ude bed: Fied ditribution in the ude bed, t the riht of the urfe AA ' i exonenti nd there i no refeted wve, due to the hih ttenution in the treted ude. In thi e the urfe imedne ε δ i the imedne of emi-infinite e with the dieetri ontnt nd the o ne. Imedne of free e i Z µ ε [Ω] nd in the e of dieetri environment with o, the imedne beome: µ () Z ε ( ε jδ ) where the ne of o i iven by: σ δ () j ω ε In thi e t, 45 GHz frequeny for the wter ε 77 nd δ, 5, nd the oid rt ε 4 şi δ,. Eqution for utin the vere eetromneti ontnt nd inut imedne of ude bed: Sude n be treted mixture of two omonent: wter nd the oid, o the vere dieetri ontnt ( ε m ) i uted with: ε m ε + ε (3) Imedne t the entrne in the ude bed i determined by the retionhi: 86

6 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- Z Z AA' (4) ε m The eqution of vrition of mirowve refetion: Inut imedne in the ude bed h different vue deendin on the wter ontent, durin dryin roe the wter ontent i vribe nd the vere dieetri ontnt i vribe o tht refetion of mirowve wi our. Genery the refetion oeffiient i iven by: Z Z Γ (5) Z + Z Z - i the od imedne (for thi e, the inut imedne in the ude bed); Z - i the referene imedne, where the refetion oeffiient i zero (in thi e, the imedne of the wter ontent for whih i the dttion). The djutment i mde for the referene wter ontent, utin the refetivity for the minimum nd mximum wter ontent in ude nd with (6) i determined the refeted ower: Pr Γ P i (6) With retion (7) i determined the retionhi between the mximum eetri fied nd minimum eetri fied in the wve uide for the two wter ontent. Γ (7) + k Γ Permiibe vue of the wter ontent in the ude re iven by the owbe mount of refeted ower ( P r ). In the ener e of dryin mteri with hih humidity, it n et ower refeted ermiibe vue of % of rted outut ( P i ) of the mnetron, in thi e, refetivity oeffiient i: P r Γ, i.e. Γ, 36 nd,36 k,93 Pi,684 Etbihin the ine equivent retion ude-funne: The e between the ude nd funne ditribution mouth (fiure 7) the ditne, between the urfe AA ' nd ', h the equivent iruit two-wire ine with th enth, between the od Z (inut imedne of the ude bed) nd termin ' (fiure. 8). Puttin on tht, the ortion rovided tht rotion i oe, the retionhi for utin the inut imedne of oe ine i: Z + j Z tβ Z Z (8) Z + j Z tβ Z - i the hrteriti imedne of the ine; Z - i the od imedne; β - i the dehin ontnt, with: π β ; - i the enth of the ine. Shoud be rovided tβ nd when Z Z, i.e. the od imedne i equ to the ditribution i t the mouth of the funne. The ditne between the he nd urfe re t the mouth of the ditribution funne i derived from the retion: β N π, N,,,3... (9) And reut: N (3) So the ditne between the ditribution funne mouth re nd urfe of the ude i n inteer vue equ to hf the ize of the wveenth. 87

7 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- Funne ize otimiztion eqution: To etbih the eqution for otimiztion of the funne wveuide dimenion mut onider the vrition of the modue of the refetne oeffiient (Γ), ordin to the rtio /, hown in fiure 9. Thu, the nyi of fiure 9, reut tht (Γ) i m when the rtio / i round N. Fiure 9. - The vrition of the modue (Γ) ordin to the rtio / [] If the funne etion i quoted ( x) in ny oint of it xi x (Fiure ), tht n exre thi vue by the retion: ( x) x + (3) Genery: π dx ( N ) π ( x) (3) Fiure. - The ditribution funne dimenion And the uide wveenth t ditne x, we n write the retionhi: So reut: ( x) / ( x) π π π (33) dx ( ) dx dx x ( x) ( x) Notin: (34) T dx Reut: ( x) + T ro ro ( ) (35) but: π T ( N ) π (36) It foow the retionhi for utin the enth of the ditribution funne: ( N ) π [ m ] (37) T π Cution of tit ne of the ditribution funne: (3) 88

8 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- - the on ide of the etion: t b b α (38) - the hort ide of the etion: t β (39) The ution retionhi for the refetne oeffiient: The refetion oeffiient in the ditribution funne i: K + K Γ 64π K b b b b b K b K K 3π ( ) ( ) [ ] ( ) [ ( ) ] Nottion: TE - fundment trnvere mode of rotion of mirowve ower; f - riti frequeny, in [ Hz ]; f - oertin frequeny, in [ Hz ]. - eed of iht, 3 in [ m / ];, b - the be dimenion of the retnur uide, in [ m ]; - riti wveenth, in [ m ]; - uide wveenth, in [ m ]; - orreondin vuum wveenth for the work frequeny, in [ m ]; β - dehin ontnt, in [ rd / m ];, v - rou veoity, in [ m / ]; Z - imedne mirowve rotion in vriou mteri, in [ Ω ]; Z - imedne to the rotion of mirowve in vuum e, in [ Ω ]; Z - vere wve imedne, in [ Ω ]; m E m,, n m n H, - eetri nd mneti intenity of the rotion mode indie m nd n of eetromneti wve; H - mneti intenity for the fundment wve, m nd n ; α - ttenution ontnt, in [ db / m ]; H ψ - ontnt uide w formin mteri, in [ m ]; d - deth of enetrtion of wve into the mteri, in [ µ m ]; - enth of the route uidne ytem of the mirowve, in [ m ]; α - tot ttenution of the mirowve, in [ db ]; X - rtio between the ower t the entrne nd the ower t the exit of the uide route; P - ower diited in the w of the uidne ytem, in [W ]; d P - imit ower (mximum owbe by the uide), in, [W ]; P - refeted ower, in r7% [W ]; m, n - vue of eetromneti wve for exme in the e of the trnvere eetri rotion mode TE, we hve m nd n ; η - Fied effiieny of trnmiion defined : η ; f f (4) (4) (4) 89

9 t Annu Interntion Interdiiinry Conferene, AIIC 3, 4-6 Ari, Azore, Portu - Proeedin- σ - ttionry wve ftor; Z - od imedne, in [ Ω ]; Z - referene imedne, in [ Ω ];, Γ - refetion oeffiient modue; ε - vere dieetri ontnt of mixture; m ε, ε - dieetri ontnt of mixture omonent; δ - ne of o, in [ rd ];, - roortion of omonent in mixture, in [ % ]; N - wve number of the rnk;, b - funne entrne dimenion, in [ m ];, b - funne exit dimenion, in [ m ]; - funne enth, in [ m ]; K ; K - refetion rmeter uted ordin to the funne etion ize t the entrne nd exit; Conuion Thi hyi-mthemti mode n be ued to uide ytem izin mirowve itor for mobie inttion in the fied of mirowve roein of ude from the wtewter tretment nt. Referene: []. Riti-Miho. E.N., - Studie nd reerhe on the therm roein of mirowve fied to ewe ude of tretment nt, Phd Thei, Dertment of Enineerin nd Sutinbe Deveoment Entrereneurhi, Futy of Mteri nd Environment Enineerin, Tehni Univerity of Cuj-No, Cuj-No,. []. Dn, V., Riţi-Miho, E., Riţi-Miho, E. N. Therm tretment of erin ude in mirowve fied, 4 th Reerh/Exert Conferene with Interntion Prtiition QUALITY 5, Fojni, B&H, 5. [3]. Hi, J. M., Mrhnt, T. R. Modein mirowve hetin. Aied Mthemti Modein; ():3 5, 996. [4]. Rue, G. Tehni Miroundeor, E.D.P., Buureşti, 98. [5]. Metx, A. C., Meredith R. J. - Indutri Mirowve Hetin. Power Enineerin Serie 4. Peter Pererinu Ltd. (on behf of the IEE), 993. [6]. Prei, A. Etude de méiortion du rendement de tehnique de éhe de mtériux éi r ort d énerie éetromnétique. Conetion d iteur, Thèe de Dotort, L Intitut Ntion Poytehnique de Tououe,