T q (heat generation) Figure 0-1: Slab heated with constant source 2 = q k

Transcription

1 IFFERETIL EQUTIOS, PROBLE BOURY VLUE 5. ITROUCTIO s as been noed n e prevous caper, boundary value problems BVP for ordnary dfferenal equaons ave boundary condons specfed a more an one pon of e ndependen varable. Boundary value problems are que common n cemcal engneerng applcaons. Problems were e ndependen varable s e poson raer an me are ofen descrbed n erms of ordnary dfferenal equaons w condons mposed a more an one pon. Eamples of e BVP nclude e seady-sae dsrbuon of emperaure and concenraon n plug flow sysems. Consder e problem of emperaure dsrbuon nsde a slab sold w ermal conducvy eaed by a consan unform ea source q Fgure 5.. ssumng a e ea flows only on e aal drecon,. Usng e mcroscopc energy balance, e seady sae energy equaon s: L L T -L T L T q ea generaon - Fgure -: Slab eaed w consan source d d T q 5. Te -as s cosen so a e dsance s bounded by L and L See Fgure 5.. Varous boundary condons for s problem are possble:

2 Te surface emperaures a L and L can be consan, TL T w 5. TL T w 5.3 Te emperaure a one sde, say L can be consan wle e surface on e oer sde s nsulaed. Te flu φ, gven by Fourrer law, s us zero a s end. In s case e boundary condons are: T L T w 5.4 T L 5.5 Te emperaure a one sde, say L can be consan wle a coolng mecansm ess a e oer end L. Te boundary condons a s end can be epressed by equang e flu, gven by Fourrer law, w e ea absorbed by e coolng mecansm T L T w 5.6 T φ L L T T 5.7 were s e ea ransfer coeffcen. Our sandard form for a boundary value problem s e second order ordnary dfferenal equaon of e form: '' f,,' [a,b] 5.8

3 Te assocaed boundary condons a a and b ave e general form g,' a a 5.9 and g,' a b 5. In mos cemcal engneerng problems usng ranspor equaons mass, energy and momenum, g and g ave muc smpler forms. Le e quany S desgnaes e concenraon C, emperaure T or velocy v and le φ desgnaes e flu of mass, energy or momenum. Some of e mos common boundary condons are e followng:. Te value of S a e boundary s specfed and as a consan value S, S boundary S 5. Tese boundares of ype are consan-value boundares. Eamples of ese ype of boundares nclude consan concenraon C C, consan surface emperaure T T and consan velocy vv o.. Te values of S a bo sdes of e boundary are equal or relaed funconally S lef S rg 5. Or S lef FS rg 5.3 Boundares of ype are consan-value boundares a are connuous across e boundary 3. Te flu a e boundary as a consan value φ : 3

4 φ boundary φ 5.4 Boundares of ype 3 represen cases were flu s specfed a a boundary. Eamples of s boundary nclude e case were e rae of a reacon a a boundary surface s specfed, I ncludes e eamples of ermal nsulang surface were e ea flu φ dt/d s zero. I also ncludes e eample were e momenum flu a a gas-lqud nerface s assumed zero. 4. Te flu a a boundary s connuous,.e., e flu of bo sdes of e boundary are equal. φ lef φ rg 5.5 Boundares of ype 4 represen cases of connuaon of flu ea, mass and momenum across a boundary. 5. Te flu a e boundary s emprcally deermned φ boundary fs 5.6 Boundares of ype 5 represen cases of mass, ea and momenum ranspor across a boundary layer.te mass or ea flu for nsance from a bul flud n moon o a sold surface w emperaure T s and concenraon c * can be descrbed by ea or mass ransfer coeffcen, and c c * 5.7 q T T s 5.8 4

5 5. UERICL ETHOS FOR THE SOLUTIO OF BVP PROBLES To apprecae e dfference beween a boundary value problem BVP and an nal value problem IVP consder e followng BVP n sandard form and w consan-boundary condons '' f,,', [a,b] a α, b β 5.9 and le consder e followng IVP '' f,,' [a,b] a α 'a γ 5. oe a e dfferenal equaon and e frs boundary condon are e same for bo problems. Only e second boundary condon s dfferen. Te mporan dsncon beween e nal value problem Eq 5. and e boundary value problem Eq 5.9 s a n e frs case e negraon sars from e nal value and connues owards e endng pon. In e BVP case a soluon a sasfes e boundary condon a e sarng pon wll no necessarly sasfy e boundary condon a e endng pon. For s reason e meods used for nal value problems can no be used drecly for e soluon of BVP. In e followng wo major meods a are used for e soluon of e BVP are presened. Soong meods Fne dfference meods 5.. Soong eod Soong meod consss n converng e BVP no an erave nal value problem. Consder e BVP w consan-boundary condons Eq Consder also e nal value problem Eq 5., assocaed w e BVP problem were γ s a guessed value of 'a. Ts nal value problem can be negraed usng any meod presened n earler caper o oban a soluon. Ts soluon wll be e soluon of e BVP Eq 5.9 f sasfes e condon b β. Ts s unlely snce e value γ 5

6 was cosen arbrarly. However, f e guessed value of γ 'a s updaed n a raonal way, e procedure can be repeaed unl convergence. Ts process s called soong as can be seen grapcally n Fgure 5.. Te problem of updang e guessed value γ n a suable way s e subjec of e followng secon. Te soluon of e nal-value problem, Eq. 5., depends eplcly on e value of γ and, us, wll be denoed γ. Te objecve of e soong meod s o selec γ so a γ b β 5. γ γ 3 γ γ β γ γ a b Fgure -: Grapcal nerpreaon of e soong meod; γγ : e frs so; γγ : e second so; γγ 3 : e fnal so Ts s equvalen o solvng e followng algebrac equaon for γ φγ : γ b β 5. Te algebrac equaon can be solved by any of e meods suded earler n Caper 3. ewon-rapson eraon, for nsance, appled o Eq 5. yelds e followng eraon sceme: 6

7 γ n γ n φ γ n φ γ n 5.3 Ta s γ n γ n γ n b β φ γ n 5.4 Ts eraon sceme wll updae e value of γ provded a e dervave erm φ'γ n s nown. In e followng secon, a way o deermne s erm s sown. Le us dfferenae parally w respec o γ bo sdes of Eq5.. Ts yelds: d γ df dγ d γ d df dγ d γ dγ df dγ d γ d dγ 5.5 Te same s done for e boundary condons of Eq. 5. o yeld dγ a dγ 5.6 d γ dγa 5.7 Inroducng e aulary varable v d γ /dγ and nong a d/dγ, Eq. 5.5 becomes: df df v v v d d γ γ 5.8 w e followng nal condons va v'a 5.9 7

8 Ts s an nal value problem and s called e frs varaonal equaon. Te soluon of s nal value problem wll yeld e value of vb. Te erm φ'γ can en be compued snce dγ b v b φ γ dγ 5.3 Knowng e value of e dervave φ'γ, e eraons of e ewon-rapson meod Eq. 5.4 can proceed unl e convergence. s can be seen, e soong meod s bes appled o cases of consan-value boundary condons.te soong meod can be cosly n compuaonal effor snce eac value of e funcon φγ s obaned by solvng wo nal-value problems Eq 5. and Eq Organgram 5.: Te Soong eod Te soong algorm for solvng BVP problem of e form of Eq 5.9 s sown n Fgure 5.3. Te algorm ermnaes f one of e followng crera s me: Te soluon a e boundary pon s wn specfc olerance of e boundary value,.e., γ b β δ Te number of eraons eceeded e mamum value. Eample 5.: Temperaure dsrbuon n a ppe wall Consder e problem of fndng e seady sae emperaure dsrbuon n a wall of ppe were a o lqud flowng nsde e ppe s eang a cold lqud flowng ousde of e ppe.te nner r and ouer r o surfaces of e ppe are assumed o be a consan emperaures T and T c. Te ea flu s assumed o occur radally. Performng a seady sae mcroscopc energy balance on an elemen of e ppe as sown by fgure 5.4, e emperaure on e wall of e ppe sasfes e followng dfferenal equaon 8

9 Sar Inpu endpons: a,b; boundary condons: α, β; number of subnerval: ; olerance: δ; amum eraon: b-a/ γ βα/b-a γ γ - b - β/vb solve e followng IVPs: '' f,,' ; a a, 'a γ v'' df/dv df/d'v'; va, va over e nerval [a,b] w sep sze b - β < δ o Yes Prn, > Yes o mamum eraon eceeded Sop Fgure -3: Soong meod for boundary value problem Cold flud a T c Ho flud a T Fgure -4: Temperaure dsrbuon n a ppe wall 9

10 d dr dt r dr 5.3 wc s equvalen o d dr T r dt dr 5.3 w e boundary condons Tr T 5.33 Tr o T c 5.34 Le r.5 cm, r o. cm, T 3 o C and T c o C. Te analycal soluon of Eq. 5.3 can be easly obaned by mang cange of varable w T o yeld: T r C C ln r 5.35 Te consans C and C Eqs o yeld are deermned by mposng e boundary condons of T T r ln r r ln r c T r o T 5.36 For e numercal soluon of e problem, we apply e seps of e algorm. eac eraon e followng IVP s solved: T''r /r T 'r 5.37 W

11 Tr T T 'r γ 5.38 Because of e smplcy of e problem, s IVP can also be solved analycally o yeld r T γ r γ r ln T r 5.39 Te second IVP s consruced as follows: Snce fr,t,t ' /r T 'r, follows a f/ T and f/ T' /r. Tus Eq. 5.8 becomes f f v v v' v T v r 5.4 w vr and v'r 5.4 In s smple eample e second IVP can also be solved analycally agan o yeld v r r ln r r 5.4 Te algorm goes en as follows: Sep : ssume a value for γ, e.g., γ, Te IVP Eq s solved Eq o yeld T. 3.34, a value wc s far from e gven boundary condon. Sep : Te second IVP Eq s solved Eq. 5.4 o yeld v Sep 3: Te value of γ s updaed by Eq. 5.4 o yeld

12 γ γ T. β v and upon subsuon yelds γ Seps roug 3 are repeaed. Te algorm converges n ree eraons as sown n Table 5.. Table -: Soong meod eraons for eample 5. Ieraon γ T. calc. boundaryt. calc Fne-fference eods noer approac o e soluon of e boundary value problem consss n usng fne dfference appromaons of e dervaves o represen e governng dfferenal equaons and e equaons for e boundary condons. Ts resuls n a se of lnear or nonlnear algebrac equaons. Taylor seres epansons are used o develop dfference appromaons for e frs and second dervaves of a funcon. forward fne dfference appromaon of frs order s gven by: ' Δ 5.45 '' Δ 5.46 w

13 5.47 Δ bac fne dfference appromaon of frs order s: 5.48 Δ ' 5.49 '' Δ cenral fne dfference yelds 5.5 Δ ' 5.5 '' Δ beer appromaon s o use a second order appromaon for e frs and e second dervaves. forward fne dfference appromaon of e second order s: 5.5 Δ 4 3 ' '' Δ bac fne dfference of e second order s: 5.54 Δ 4 3 ' 3

14 '' Δ For consderaons of smplcy and sably, e frs order cenral fne dfference sceme s preferred. In e frs sep of e mplemenaon of e fne-dfference meod e nerval [a,b] s dvded no equal subnervals wose endpons are e mes pons a for,,, were b a/. Le e appromae value of be denoed by and defne e boundary condons of Eq. 5.9 by 5.56 α, β pplyng e cenral-dfference formula o equaon 5.9 yelds: 5.57 f,,,,,, L Wrng s equaon for eac,,,, en e followng nonlnear sysem s obaned : 5.58 β β α α,,,,,,,, 3 3 f f f f Te unnowns are,,,. Te ewon's meod can be used for e soluons of s sysem of nonlnear algebrac equaons. However e number of equaons can be large and so s e number of nal guesses. safe bu slow procedure s o sar w 4

15 a sysem. Solve and en use s resuls as nal guesses for e 44 sysem and so on. Eample 5.: ffuson and a second order reacon Consder e problem of dffuson and a omogenous reacon of second order of a componen n a dlue lqud pase of componen B Fgure 5.4. Te lqud pase s sagnan, us, mass s only ranspored by dffuson and no by bul flow. Te seady sae equaon for e dffuson of n e z-drecon can be obaned by drecly applyng e componen balance equaon o yeld: Gas z Lqud B Δz z L Fgure -5: ffuson no a sagnan flud dc c dz 5.56 w boundary condons c c a z c c a z L 5.57 Ts s a non lnear BVP w f dc z, C, dz C

16 pplyng e fne dfference meod o Eq. 5.58, e followng sysem of nonlnear algebrac equaons s obaned: C C C C C C C C C 3 C C α C C C β 5.59 Soluon of s sysem s lsed n Table 5., for,.5-5, 9, L.5, C. and C.. Table -: on-lnear fne dfference meod eraons for eample 5. Z C Fne-fference eods for Lnear Problems For a lnear BVP e funcon f s lnear n and ' and can be represened by e general form 6

17 f,,' w' v u 5.6 pplyng e fne-dfference meod Eq resuls n e followng equaons u v w,,, 5.6 w e boundary condons 5.6 α, β Equaons 5.6 are rearranged o become: /w - v /w u 5.63 Te resulng sysem of equaons s a rdagonal sysem of e form b 5.64 W 5.65 n n n v w w w v w w v L O O O O L 5.66 n n 7

18 and b u w u u n u n w n n 5.67 Te rdagonal sysem can be solved by e meods presened n Caper. I can be proved a f v on e nerval [a,b], en e lnear sysem as a unque soluon provded a L < 5.68 Were L ma w a b 5.69 Eample 5.3: ffuson and frs order reacon Consder e prevous eample of dffuson and omogenous reacon. ssume e reacon s of frs order. Ten e mass balance equaon s dc dz c 5.7 w boundary condons c c a z c c a z L 5.7 Ts lnear BVP can be solved analycally o yeld 8

19 5.7 L z L c z c c sn sn sn Te equaon Eq can be solved usng e algorm descrbed n e prevous secon.te dfferenal equaon s pu under e form of Eq z u c z v dz dc z w dz dc were wz, vz / and uz. Te mar and vecor b ae n s case e smpler form of 5.74 L O O O O L 5.75 n b Ts can be solved usng a lnear sysem roune. Table 5.3 sows e resuls for e same values as n e prevous eample. Te comparson beween e numercal soluon and e acual soluon s also provded. 9

20 Table -3: Lnear fne dfference meod eraons for eample 5.3 C calc. C nal. C calc -C nal z HLIG IFFERET TYPES OF BOURY COITIOS Te fne dfference meod s useful n andlng dfferen ypes of boundary value besdes e consan-value boundary problems dscussed before. Consder, for nsance, e boundary value problem of ype appled o e eample of Fgure 5.. For s ype of BVP we ave: q dt d a L 5.76 Epressng e dervave as a second-order forward fne dfference equaon Eq. 5.5 and denong T-LT, leads o 3T o 4T Δ T 5.77 Ts equaon resuls n e followng epresson for e boundary pon T T 4T T /3 5.78

21 Te boundary condon Type 5 can also be epressed n e same way. For s case, assumng a coolng mecansm, e boundary condon s gven by dt q T T d a L 5.79 Usng agan Eq.5.5 o appromae e dervave yelds e followng equaon for T T Δ T T T Δ Tese epressons for e boundary condon can be, erefore, negraed n e fnedfference sceme. 5.4 OTHER SOLUTIO TECHIQUES Te soong meod dscussed n s caper can be mproved by dvdng e orgnal nerval [a,b] no smaller nervals and solve e BVP no peces. Ts ecnque s called mulple soongs ecnques. Le recas e BVP n s sandard form `` f,,` 5.8 a α and b β 5.8 Le dvde e nerval [a,b] no wo subnervals [a,c] and [c,b]. Le us defne wo IVP s as follows IVP# defned for a c `` f,, ` 5.83 a α and b γ 5.84 IVP# defned for c b

22 `` f,, ` 5.85 a α and b γ 5.86 Te soluon of e BVP s composed of e soluons of wo IVPs n s way a c c b 5.87 To ensure connuy of bo and a c we requre en a c c ' c ' c 5.88 Tese are wo nonlnear algebrac equaons for wc e unnowns are γ and γ. Tey can be solved by ewon meod. Ts procedure can be eended by dvdng e orgnal nerval [a,b] many subnervals. Cec references [56] for some deals. 5.5 ISL ROUTIES Some ISL rounes o solve OE-BVP are as follows: Roune BVPF BVPS Feaures Solve a sysem of dfferenal equaons w boundary condons a wo pons, based on fne dfference meod. Solve a sysem of dfferenal equaons w boundary condons a wo pons, usng a mulple-soong meod.

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