MULTIPLE INTEGRAL-BALANCE METHOD Basic Idea and an Example with Mullin s Model of Thermal Grooving

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1 THERMAL SCIENCE: Year 17, Vol. 1, No. 3, Oe forum MULTIPLE INTEGRAL-BALANCE METHOD Basc Idea ad a Examle wth Mull s Model of Thermal Groovg by Jorda HRISTOV * Deartmet of Chemcal Egeerg, Uversty of Chemcal Techology ad Metallurgy, Sofa, Bulgara Orgal scetfc aer htts://do.org/1.98/tsci1711h A multle tegrato techque of the tegral-balace method allowg solvg hgh-order dffuso equatos s coceved ths ote. The ew method termed multle-tegral balace method s based o multle tegrato rocedures wth resect to the sace co-ordate ad s geeralzato of the wdely aled heat-balace tegral method of Goodma ad the double tegrato method of Volkov. The method s demostrated by a soluto of the lear dffuso models of Mulls for thermal groovg. Key words: multle-tegrato method, hgh-order o-lear dffuso, Mulls equato Itroducto The tegral-balace method [1, ] to dffuso models of heat ad mass emloys a cocet of a eetrato deth thus defg a shar fort of roagato. Itegral-balace solutos are extesvely used [3, ] ad stll attractve for solutos of o-lear dffuso roblems [-8]. Ths commucato coceves the dea of the multle-tegral method (MIM) whch ca be cosdered as a geeralzato of the tegral-balace aroach beyod the heat-balace tegral method of Goodma [1] ad the double-tegrato method of Volkov [8]. The tegral-balace method: Exstg backgroud as tegrato techques The classcal aroach: basc cocet Traset dffuso (heat or mass) a homogeeous medum wth a costat trasort coeffcet (dffusvty), a, s modelled by the arabolc equato dmesoless form, wth resect to the temerature (cocetrato) oly: θ a θ =, θ ( xt, ) = for t > (1) t I the case of a sem-fte medum the value of θ(x,t), far away from the boudary x =, s assumed θ(,t) =. The shar frot cocet relaces these codtos by ew oes θ(δ,t) = θ(δ,t)/ =. * Authorʼs e-mal: jorda.hrstov@mal.bg

2 1556 THERMAL SCIENCE: Year 17, Vol. 1, No. 3, The smlest method kow as heat-balace tegral method (HBIM) [1, ] suggests tegrato of eq. (1) wth resect to the sace co-ordate over a fte eetrato deth, δ, amely: δ δ δ d dδ θ d θ ( xt, ) d x (, t) a d x ( xt, ) dx a (, t) dt θ θ δ = dt θ = dt (a,b) Relacg θ by a aroxmate rofle θ a, a ordary dfferetal equato descrbg the tme-evoluto of δ(t) ca be derved. The rcle roblem of HBIM s the aroxmato of the gradet of rght-sde of eq. (b) because t should be defed through the assumed rofle. The double-tegrato method (DIM) [6, 7, 9] ts classcal formulato uses two stes: tegrato wth resect to the sace co-ordate from to x ad a secod ste as tegrato from to δ, eq. (3a). Here wll use a more geeral defto of DIM, eq. (3b), develoed [7] allowg alcato to cases wth teger-order [6] (as the reset case) ad tme-fractoal dervatves [7]. Wth a cosequet alcato of the Lebz rule we have the followg exressos: δ x δ δ d d x ( xt, ) dx dx a (, t) or ( xt, ) dx dx a (, t) dt θ = θ θ θ dt = (3a,b) x Boudary characterstc aroach Recetly the method of boudary characterstcs (MBC) as a mrovemet of the tegral-balace method was roosed by Kot [1] for soluto of the model (1). The rcle stes of MBC suggest alcato of a sequece of tegral oerators (a) the terval [x,δ(t)] ad over a fte terval [,δ(t)] (b), amely: x x x x x x x (...) d... (...) d, (...) d... (...) d L = x x L = x x (a,b) The techology of MBC requres addtoal boudary codtos at x = ad olyomal assumed rofles u to the 8 th order. The rcle ste of MBC s that after alcato of L x the co-ordate x s set to x =. At ths ot we sto to commet MBC because t s beyod the scoe of the reset work but t was commeted oly to show ts dea ad to make clear the exstg backgroud thus allowg demostratg the basc rcle MIM ad ts dffereces wth resect to MBC, as well as avodg ay msuderstadgs terretato of the roosed geeralzato of the tegrato techque. Multle-tegrato method: deftos To reset the basc dea of the method let us cosder a dffuso equato of order : θ θ = b, b>, t Defto of the multle tegral oerators L k The MIM suggests alyg the oerator L k, o a dffuso equato of order defed as ad alyg the Lebtz rule see eqs. () ad (3): (5)

3 THERMAL SCIENCE: Year 17, Vol. 1, No. 3, δ δ δ δ δ δ δ δ d ( x, t) dxdxdx d x b ( x, t) dxdxdx dx dt θ = θ x x x x x x k 1 k 1 k k (6) Therefore, we have to erform k 1 tegratos from x to δ ad the last k th tegrato should be from to δ. Now, alyg L k to the model of eq. (1), by hel of the Lebtz rule, we get eq. (6). The umber of tegratos that should be aled eq. (6) deeds o the order of the dffuso term ad the geeral rule s k =. However, whe k s aled we get: δ δ δ δ k b θ x t x x x x b t k (, ) d d d d = θ(, ) x x x k 1 k I accordace wth eq. (7), f k = 1ad =, we get the tegral relato of the HBIM exressed by eqs. (a,b). Further, for k = = we actually aly DIM exressed by eqs. (3a,b). The MIM to the Mulls equato of thermal groovg Mulls dffuso model I rocesses of groove growths by mechasm of evaorato-codesato the surface rofle satsfy the o-lear dffuso equato (wth boudary ad tal codtos) [11, 1]: (7) D( u ) u u(, t) =, = u = cost., t u 1+ uxt (,) as x, ux (,) = (8a,b,c) Wth the assumto of symmetrc groove shae about x =, the roblem could be cosdered oly alog the half-le x, fg. 1(a). The o-lear model (1) has bee solved exactly by Broadbrdge [1, 13] ad other authors [1, 15] but the solutos are hard to be hyscally aalyzed. I case of a small curvature of the surface ad the assumto that the evaorato s the sole mass trasfer mechasm t s ossble to learze eq. (8a): 3 u u u(, t) u(, t) = B, = m, m>, = (9a,b,c) 3 t I eq. (9b) the dffuso coeffcet s B = D s γω ν/kt, where: D s s the coeffcet of surface dffuso, γ the free surface eergy er ut area, Ω the molecular volume, ν the area where the dffuso takes lace. The codto (9c) at x = corresods to the requremet the flux to equals to zero at the org [1, 1]. Equato (9a) wll be used to demostrate the soluto techology of MIM.

4 1558 THERMAL SCIENCE: Year 17, Vol. 1, No. 3, The MIM soluto To be the soluto more famlar to kow roblems traset dffuso we vert the rofle of the groove from the orgal oe wth a ostve curvature, fg. 1(b), to a mrror rofle wth a egatve curvature, fg. 1(a). The verted groove rofle mmcs the decayg temerature (cocetrato) dstrbuto wth a tycal Neuma roblem, where the boudary codto s u x (,t) = m, the sg eq. (9b) chages sce the curvature of the verted rofle s egatve. Cosequetly the dffuso coeffcet eq. (9a) chages from B to B ad the equato becomes as the oe reseted by eq. (5) wth =. The tegral method, geeral, requres the assumed rofle to satsfy the codtos at both eds of the dffusoal eetrato deth. The assumed rofle wth usecfed exoet [3-7] s defed as u a = u s (1 x/δ) ad used to demostrate the techology of MIM. Its rcle advatage s that there s a freedom to otmze the accuracy of aroxmato by the mea squared error of aroxmato of the goverg equato [, 6, 7, 11]. Next, alyg the boudary codto (a) w/ β w/ d taβ = m Fgure 1. Sketch of the groove real rofle (a) ad the verted (mrror) rofle (b) used the MIM soluto ( )( )( )( ) (9b) we may defe u s = u a (,t) = m(δ/) as the classcal Neuma roblem solved by HBIM ad DIM [3, ]. I ths otato, u s = d + h s the groove deth, whle δ equals the half wdth of the groove (δ = w/) from the symmetry axs to the flecto ot, fg.1(a) ad 1(b), where the followg codtos u x (δ) = u xx (δ) = u xxx (δ) = are vald. Now, deotg z = x/δ that trasforms the movg boudares x δ to fxed Zeer s co-ordates z 1, ad chagg the varables the tegrals the left-had sde of eq. (6) we have for k =, defed by the order of dervatve the rght-had sde of eq. (8a): d N δ z( 1 z) dzdzdz dz = mbu(, t) δ = Bδ dt dt z z z N = =! =Γ ( + k), k = m d 5 (1a,b) x h u s (b) taβ = m d β δ The soluto of eq. (1b) wth the tal codto δ(t = ) = s δ = (Bt) 1/ M 1/, where M = (5/)[Γ( + )/]. Therefore, the groove surface rofle ca be aroxmated by: ua M 1 1 a ( Bt) M m( Bt) ( Bt) M m x x ua( x, t) = ( Bt) M 1 U = = 1 h x (11a, b) The ormalzed rofle U a s reseted a way smlar to that used by Mulls [11], eq. (1), ad mmcs the soluto of the classcal Newma roblem whe =. For x = we get the temoral growth of the groove deth whch the orgal cofgurato, fg., should be u s = (m/)(bt) 1/ M 1/. The mea squared dslacemet x of the dffuso rocess s defed: x xu( x, t)dx =

5 THERMAL SCIENCE: Year 17, Vol. 1, No. 3, Wth the assumed rofle u a ad uer termal of the tegral fxed at δ we have x δ [1, 15]. From the revously develoed soluto t follows that δ t 1/ that mmedately tells us that the surface dffuso rocess modelled by eq. (9) s subdffusve ature as t was aalyzed [16]. The aroxmate rofle of eq. (11) defes a atural way the o-boltzma smlarty varable η M = x/(bt) 1/. Ths smlarly varable was used by Mulls [11] to trasform eq. (1a) a ordary dfferetal equato. The soluto of ths trasformed equato s aalyzed by Robertso [15], but the ma dsadvatage s that the results are hard to be used for egeerg alcatos ad hyscal aalyzes. Mulls [11] develoed a aroxmate soluto as a seres of η M, amely: (, ) ( ) 1 u xt u( xt,) = m( Bt) a( η ) U( xt,) = a( η ), U(,) xt = m Bt M M 1 (1) The soluto was carred out over the rage from m = to m = accetg all cases (Bt) 1/ = 1 (sc!) ad exressg the rofle wth resect to the satal co-ordate x ad 16 terms of the seres (1), tab. 1 [11]. The exoet of the aroxmate soluto (11a,b) ca be easly determed by the least-squared method mmzg the resdual fucto of eq. (9) whe u(x,t) s relaced by u a. That s R = (u a ) t B(u a ) xxxx has to be mmzed wth resect to, whch meas alcato of the Lagford method for tegral-balace solutos []. Detals of ths otmzato rocedure, commoly aled tegral-balace solutos sce 9 [3,, 6] are avalable elsewhere [3-7]. Sce the arameter m s a re-factor of the rofle but t does ot affect δ, the the mmzato of the resdual fucto, R, s deedet of m. I ths cotext, t s oteworthy, that the valdty of the soluto whe x δ moses the requremet >, smlar to the case wth =, eq. (1), where the requremet s > [6, 7]. The reset study establshed a otmal exoet ot.55. The aroxmate solutos by MIM, eq. (11), ad that of Mulls, eq. (1), are reseted at fg.. It clear that the MIM soluto errors of aroxmato fall to the commo rage of -3% [6, 7]. Moreover, MIM as a tegral-balace method wth a fte eetrato deth works oly wth the rage x δ or equvaletly whe ηm [(Bt)M ] 1/. Fgure. Aroxmate solutos; by Mulls ad MIM, ad the otwse errors Cocluso Ths ote coceves a ovel tegral-balace method (soluto techque) amed MIM whch a geeralzato of the exstg heat-balace tegral of Goodma ad the DIM of Volkov. The method allows solvg dffuso equatos of hgh order ad the soluto examle wth the Mulls equato of thermal groove growg s a good demostrato of ts feasblty to solve equato beyod the classcal dffuso models. Refereces [1] Goodma, T. R, The Heat Balace Itegral ad ts Alcato to Problems Ivolvg a Chage of Phase, Trasactos of ASME, 8 (1958), 1-, u a [ ] MIM Mulls η M [ ] Potwse error [ ]

6 156 THERMAL SCIENCE: Year 17, Vol. 1, No. 3, [] Lagford, D., The Heat Balace Itegral Method, It. J. Heat Mass Trasfer, 16 (1973), 1,. -8 [3] Myers, J. G., Otmzg the Exoet the Heat Balace ad Refed Itegral Methods, It Comm Heat Mass Trasfer, 36 (9),, [] Mtchell, S. L., Myers, T. G. Alcato of Stadard ad Refed Heat Balace Itegral Methods to Oe-Dmesoal Stefa Problems, SIAM Revew, 5 (1), 1, [5] Hrstov, J., The Heat-Balace Itegral Method by a Parabolc Profle wth Usecfed Exoet: Aalyss ad Bechmark Exercses, Thermal Scece, 13 (9),,. 7-8 [6] Hrstov J., Itegral Solutos to Traset Nolear Heat (Mass) Dffuso wth a Power-Law Dffusvty: A Sem-Ifte Medum wth Fxed Boudary Codtos, Heat Mass Trasfer, 5 (15), 3, [7] Hrstov, J., Double Itegral-Balace Method to the Fractoal Subdffuso Equato: Aroxmate Solutos, Otmzato Problems to be Resolved ad Numercal Smulatos, J. Vbrato ad Cotrol, Ole frst, htts://do.org/1.1177/ [8] Volkov, V. N., L-Orlov, V. K., A Refemet of the Itegral Method Solvg the Heat Coducto Equato, Heat Trasfer Sov. Res., (197),,. 1-7 [9] Esfaha, J. A., et al., Accuracy Aalyss of Predcted Velocty Profles of Lamar Duct Flow wth Etroy Geerato Method, J.Al. Math. Mech., 3, (13), 8, [1] Kot, V. A., Method of Boudary Characterstcs, J. Eg. Phys. ThermoPhys., 88 (15), 6, [11] Mulls, W. W., Theory of Thermal Groovg, J. Al. Phys, 8 (1957), 3, [1] Broadbrdge, P., Exact Solvablty of the Mulls Nolear Dffuso Model of Groove Develomet, J. Math. Phys, 3 (1989), 7, [13] Broadbrdge, P., Exact Soluto of a Degeerate Fully Nolear Dffuso Equato, Z.agw.Math.Phys, 55 (), 3, [1] Ktada, A., O Proertes of a Classcal Soluto of Nolear Mass Trasort Equato u t = u xx /(1 + u x), J. Math. Phys, 7 (1986), 7, [15] Robertso, W. M., Gra-Boudary Growg by Surface Dffuso for Fte Sloes, J. Al. Phys, (1971), 1, [16] Abu Hamed, M., Neomyashchy, A. A., Physca D: Nolear Pheomea, (15), 1,. -7 Paer submtted: Arl 1, 17 Paer revsed: May, 17 Paer acceted: May 3, Socety of Thermal Egeers of Serba Publshed by the Vča Isttute of Nuclear Sceces, Belgrade, Serba. Ths s a oe access artcle dstrbuted uder the CC BY-NC-ND. terms ad codtos