Scientific Research of the Institute of Mathematics and Computer Science 1(11) 2012, 23-30

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1 Please te ts artle as: Grażyna Kałuża, Te numeral soluton of te transent eat onduton roblem usng te latte Boltzmann metod, Sentf Resear of te Insttute of Matemats and Comuter Sene,, Volume, Issue, ages 3-3. Te webste: tt:// Sentf Resear of te Insttute of Matemats and Comuter Sene (), 3-3 THE NUMERICAL SOLUTION OF THE TRANSIENT HEAT CONDUCTION PROBLEM USING THE LATTICE BOLTZMANN METHOD Grażyna Kałuża Slesan Unversty of Tenology, Poland Abstrat. Te mlementaton of te latte Boltzmann metod (LBM) for te soluton of te transent eat onduton roblem s resented. Te one dmensonal task s onsdered and te dfferent boundary ondtons, sefally te Drlet, Neumann and Robn ones are taken nto aount. Te D latte model s aled. To ek te auray of te LBM algortm, te same roblems ave been solved usng te exlt varant of te fnte dfferene metod. In te fnal art of te aer, te results of omutatons are sown and te onlusons are formulated. Introduton Over te last deade te latte Boltzmann metod (LBM) as been develoed as a romsng omutatonal tool to analyze te large lass of engneerng roblems, among oters, te eat transfer roblems [, ]. In ts aer te LBM as been aled n order to solve te Fourer equaton T ( X, t) X D : λ T ( X, t) + X, t t () were λ s te termal ondutvty, s te volumetr sef eat, (X, t) s te soure funton, T, X, t denote te temerature, satal o-ordnates and tme. Te equaton () s sulemented by boundary ondtons X Γ : T X, t T b (, ) T X t X Γ : q( X, t) λ qb n T( X, t) X Γ 3 : q( X, t) λ α T ( X, t) T n [ ] a ()

2 4 and te ntal one G. Kałuża t : T X, T (3) were T/ n denotes te normal dervatve, n s te normal outward vetor, T b, q b are te known boundary temerature and boundary eat flux, resetvely, α s te eat transfer oeffent, T a s te ambent temerature and T s te ntal temerature.. Knet equaton Te Boltzmann transort equaton an be wrtten as [] f + e f Ω (4) t were f s te dstrbuton funton, e s te veloty and Ω s te ollson oerator. It sould be onted out tat t s dffult to solve equaton (4) beause Ω s a funton of f and n a general ase t s an ntegro-dfferental equaton []. Te startng ont of te latte Boltzmann metod (LBM) s te knet equaton [, 3, 4] f ( X, t) t + e f X, t Ω,,,3,... M were f s te artle dstrbuton funton denotng te number of artles at te latte node x at te tme t movng n dreton wt te veloty e along te latte lnk e onnetng te nearest negbors and M s te number of dretons n a latte troug w te nformaton roagates. Te term Ω reresents te rate of ange of f due to ollsons and s very omlated []. Te smlest model for Ω s te Batnagar-Gross-Krook (BGK) aroxmaton [, 4] Ω f X, t f ( X, t) τ were τ s te relaxaton tme and f (X, t) s te equlbrum dstrbuton funton. It sould be onted out tat n te ase of eat transfer roblems te equlbrum dstrbuton funton s gven by (, ) (, ) (5) (6) f X t w T X t (7) were w are te known wegts, at te same tme w. M

3 Te numeral soluton of te transent eat onduton roblem usng te latte Boltzmann metod 5 Addtonally, te temerature T at te latte node x and for tme t s alulated usng te formula [, 3] M T X, t f X, t (8) From equatons (7), (8) results tat M M M (, ) (, ) (, ) (, ) f X t w T X t T X t f X t (9). Latte Boltzmann metod for D roblem In ts aer, for D roblem, D latte model [, 5] as been used as sown n Fgure. Nodes and n are te boundary ones, wle te nodes,, n are te nternal ones. Fg.. D latte In su ase te latte Boltzmann transort equaton an be wrtten as [4] f x, t f x, t x, t + e f( x, t) f ( x, t) + w,, t x τ () For te D latte, te two velotes e and ter orresondng wegts w are as follows e v, e v w w at te same tme v / t ( s te latte dstane from node to node). Te relaxaton tme τ s omuted from [] () a τ + () v were a λ / s te termal dffusvty and s te tme ste.

4 6 G. Kałuża In oter words, two equatons sould be solved, namely f x, t f x, t x (, ) (, ), t + v f x t f x t + w t x τ f x, t f x, t x, t v f( x, t) f ( x, t) + w t x τ (3) (4) Te aroxmaton of te frst dervatves usng rgt-and sde dfferental quotents s te followng [, 6, 7] f f( x, t + ) f( x, t) t f f( x +, t + ) f( x, t + ) x (5) and usng left-and sde dfferental quotents s of te form f f ( x, t + ) f ( x, t) t f f ( x, t + ) f ( x, t + ) x Introdung (5) nto (3) and (6) nto (4), resetvely, one obtans and (, + ) (, ) ( +, + ) (, + ) f x t t f x t f x t t f x t t + v f ( x, t) f ( x, t) + w τ ( x, t) (, + ) (, ) (, + ) (, + ) f x t t f x t f x t t f x t t v f ( x, t) f ( x, t) + w τ ( x, t) From equatons (7) and (8) results tat ( x, t) f( x +, t + t) f( x, t) + f ( x, t) + w ( x, t) f ( x, t + t) f ( x, t) + f ( x, t) + w (6) (7) (8) (9)

5 ts means Te numeral soluton of te transent eat onduton roblem usng te latte Boltzmann metod 7 f + f + f + w t,,,..., n f f + f + w t, n, n,..., In equatons () (.f. formula (7)):, () f w T f w T () were T denotes te temerature at te node and tme t. Addtonally T f + f () It sould be onted out tat n numeral realzaton t s onvenent to dvde te algortm nto two stes: - ollson ste: for ea node te rgt-and sdes of equatons () are alulated f f + f + w t,,,,..., n f f + f + w t,,,,..., n - streamng ste: obtaned values are assgned to te adequate nodes (Fg. ) + f f, n, n,..., f f,,,..., n (3) (4) It s vsble tat two values are unknown, ts means are determned from te boundary ondtons., f n f and tese values Fg.. Streamng roess

6 8 G. Kałuża For examle, f for x and for x L te Drlet ondtons: T (x, ) T w, T (x, L) T z are assumed ten (.f. formula ()) and w w f + f T f T f (5) n n z n z n f + f T f T f (6) In te ase wen for x L te Neumann ondton: λ T x, t / x qb s aeted, one as and next n n T T λ qb Tn Tn qb (7) λ f f f f q f f f f q λ λ + n + + n n + n b n n + n n b For Robn ondton: x L T( x t) x [ T x t T ] n n : λ, / α (, ) a one as T T λ α λ α( T T ) T T + T λ + α λ + α n a n n a (8) (9) ts means ( ) λ α fn + fn fn + fn + Ta fn λ + α λ + α λ λ + α ( ) α f f f T λ + α n + n n + a (3) 3. Fnte dfferene metod To verfy te LBM te same roblem as been solved usng te exlt seme of te fnte dfferene metod (FDM) [6-8]. For D roblem and nternal node te followng aroxmaton of equaton () s used λ + T T T T T (3)

7 Te numeral soluton of te transent eat onduton roblem usng te latte Boltzmann metod 9 were s te onstant mes ste, T (, ), (, T x t T T x t ) From equaton (3) results tat et. a a T T + ( T + T+ ) + (3) Te rteron of stablty of ts exlt dfferental seme s followng: ( a t ) / [7, 8]. 4. Results of omutatons Te layer of tkness L.5 m made of steel s onsdered. In omutatons te followng nut data are ntrodued: λ 35 W/(mK), MJ/(m 3 K), soure funton. Intal temerature equals to T C, mes ste:.5 m (n 4), tme ste:. s. In te frst examle of omutatons te Drlet ondtons n te form T (, t) C and T (L, t) C are assumed. In Fgure 3 te temerature dstrbutons obtaned by LBM algortm (symbols) and by FDM algortm (sold lnes) for dfferent moments of tme are sown. Fg. 3. LBM (symbols) and FDM (sold lnes) solutons - examle Fgure 4 llustrates te temerature dstrbutons under te assumton tat for x : T (, t) 5 C (Drlet ondton) and for x L: λ T (x, t)/ x α[t(x, t) T a ] (Robn ondton, α W/(m K), T a C).

8 3 G. Kałuża Fg. 4. LBM (symbols) and FDM (sold lnes) solutons - examle Conlusons Te latte Boltzmann metod for te D Fourer equaton sulemented by dfferent boundary ondtons and ntal ondton as been resented. Te exemlary tasks ave been solved bot by te latte Boltzmann metod and by te exlt seme of te fnte dfferent metod. Te good agreement of te solutons obtaned as been observed. Referenes [] Msra S.C., Mondal B., Kus T., Rama Krsna B.S., Solvng transent eat onduton roblems on unform and non-unform lattes usng te latte Boltzmann metod, Internatonal Communatons n Heat and Mass Transfer 9, 36, [] Moamad A.A., Latte Boltzmann Metod Fundamentals and Engneerng Alatons wt Comuter Codes, Srnger, London, Dordret, Hedelberg, New York. [3] Su S., Te Latte Boltzmann Metod for Flud Dynams and Beyond, Oxford Unversty Press, New York. [4] Caabane R., Askr F., Nasralla S.B., Alaton of te latte Boltzmann metod for solvng onduton roblems wt eat flux boundary ondton, Internatonal Renewable Energy Congress, Sousse Tunsa 9. [5] Paseka Belkayat A., Te nterval latte Boltzmann metod for transent eat transort, Sentf Resear of te Insttute of Matemats and Comuter Sene Czestoowa Unversty of Tenology 9, (8), [6] Monak B., Lara-Dzembek S., Wegted resdua metod as a tool of FDM algortm onstruton, Sentf Resear of te Insttute of Matemats and Comuter Sene Czestoowa Unversty of Tenology, (9), [7] Marzak E., Monak B., Metody numeryzne. Podstawy teoretyzne, asekty raktyzne algorytmy, Wyd. Poltenk Śląske, Glwe 4. [8] Monak B., Suy J.S., Numeral Metods n Comutatons of Foundry Proesses, PFTA, Craow 995.