Heat transfer between shell and rigid body through the thin heat-conducting layer taking into account mechanical contact

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1 Advanced Cmpuainal Meds in Hea Tansfe X 8 Hea ansfe beween sell and igid bdy ug e in ea-cnducing laye aking in accun mecanical cnac V. V. Zzulya Cen de Invesigación Cienífica de Yucaán, Méida, Yucaán, Meic Absac A pblem f ea cnducing and unilaeal cnac f a sell wi a igid bdy ug e ea-cnducing laye is fmulaed. An appac cnsiss in cnsideing a cange f laye ickness in e pcess f e sell defmain. F mdelling emelasic sae f e sell classical Kicff-Lve's mdel is epled. F mdelling ea cnduciviy f e sell epansin in a plynmial Legende seies in ems f e ickness is used. Cnac cndiins a ake in accun pssibiliies f unilaeal mecanical cnac and cange f ea ansfe cndiins beween sell and igid bdy ae fmulaed. Numeical eamples f e unilaeal emelasic cnac f e cylindical sells and igid bdy ug e ea-cnducing laye ae cnsideed. Influence f pysical and gemeical paamees f e sell and ea cnducing laye is invesigaed. Keywds: ea-cnduciviy, cylindical sell, ea-cnducing laye, mecanical cnac. Inducin Cnac ineacin is e ms cmmn way ansfe lad fm ne bdy ane. In e case f cnacing bdies aving diffeen empeaues beween em ea-cnac ineacins ake place. Teefe, n nly e cndiin f e mecanical cnac, bu als cndiins f e emal cnac ave be cnsideed. Usually, pefec emal cnac is assumed, i.e. i is assumed a e empeaue and e emal flu f e cnacing bdies in e cnac aea ae e same []. In numeus publicains [, 4 5] i was swn a in many cases ese cnac cndiins ae n accepable because ey can n ake in WIT Tansacins n Engineeing Sciences, Vl 6, 8 WIT Pess ISSN 74-5 (n-line) di:.495/ht88

2 8 Advanced Cmpuainal Meds in Hea Tansfe X accun pysical pcesses elaed defmain and ea ecange. In ese publicains e pblem f emelasic cnac f plaes and sells ave been cnsideed ug a ea-cnducin laye wi cnsideing cange f e laye ickness duing e plaes and sells defmain. Numeical eamples pesened ee sw a in many impan cases f science and engineeing e esul bained using pefec emelasic cnac cndiins and e cndiins wi cnsideing cange f e laye ickness in e pcess f defmain ae vey diffeen. In sme cases e diffeence is n nly quaniaive bu als qualiaive. Teefe, i is vey impan cnside cnac cndiins wic elae defmains and ea ansfe in e pblems wee in-walled sucues may ave cnac ug e ea-cnducing laye in e inensive empeaue field. Suc kinds f pblems ake place in many impan sucues, equipmen, and devices in cemical, aispace, nuclea indusies, ec. Te develped appac ave been applied e plaes and sells emelasic cnac pblems in [4 7, 5], e laminaed cmpsie maeials wi e pssibiliy f delaminain and emelasic cnac in empeaue field in [8, 9], and e pencil-in nuclea fuel ds mdeling in []. In is pape sme new esuls elaed unilaeal emelasic cnac f e aisymmeical cylindical sell ug e ea-cnducing laye ae fmulaed. Te cnneced equains f emelasiciy and ea cnduciviy ae ceaed. Tese equains ake in accun cange f e cndiins f ea ecange beween e sell and e igid bdy duing is defmain and pssibiliy f clse unilaeal mecanical cnac. Numeical eamples f e ea cnduciviy f e cylindical sells ug e ea-cnducing laye ae cnsideed. Te emmecanical effecs caused by cnac ineacin and ei influence n e emmecanics paamees ae invesigaed. Saemen f e pblem Le us cnside an aisymmeical cylindical sell wi paamees: is a + adius, is a ickness, l is a leng, and Ω ae middle, eenal and inenal sufaces f e sell. We cnside w siuains illusaed in e fig.: a) a igid punc is placed inside f e sell wi gap ( ) b) a cylindical sell placed inside f e cylindical le in e igid bdy wi gap ( ). Te ea is ansfeed fm e bdy suface ψ e sell and invesely ug e ea cnducing laye wic is n esised sell defmain. Te sell can be subjeced eenal mecanical lad p() and empeaue T (). Cnsequenly, e sell is defmed and can cme in mecanical cnac wi e igid bdy. As a ± esul is esablised as unknwn befe clse cnac aea Ωe = Ω ψ and fces f cnac ineacin g (). Tus we ave a siuain wi defmain influence n ea ecange and empeaue influence n defmain. In suc fmulain we ave cnneced e emelasic and ea cnducing pblem. In is case, equains f emelasiciy and ea cnduciviy cann WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

3 Advanced Cmpuainal Meds in Hea Tansfe X 8 be slved sepaaely. Te pssibiliy f unilaeal mecanical cnac ave be als aken in accun. Figue.. Equains f emelasiciy f e sell We will use ee a classical Kicff-Lve's sell ey. Accding is ey aial ε and cicula ε θ defmains f e sell middle suface ae defined by e equains du + dv du d w π ( z + w) π ( z) w w ε = = + z, ε = θ = () d d d π ( z) z and cespnding sesses by = E du d w w σ + z + + T ν α ( ν ), ν = E du d w w σ + z + + T θ ν ν α ( ν ), () ν wee u and w ae displacemens in aial and cicula diecins, E and ν ae mdulus f elasiciy and Pisn ai, T = T + T z is empeaue f e sell. Diffeenial equains f em-elasiciy ave e fm 4 d w 4 d T + 4β w = ( p g) a 4 T + a, () d D d 4 ( ν ) α ( ν ) α ( + ν ) wee β = a =, a = 4. Equains f ea cnduciviy f e sell Fllwing [4] we epesen empeaue f e sell in e fm z = T T, T = T + Tdz, T = Tzdz (4) WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

4 84 Advanced Cmpuainal Meds in Hea Tansfe X Ten diffeenial equains f ea cnduciviy ave e fm d T + + ( Q Q ) Q = d d T + + ( Q + Q ) Q + Q = (5) d Hee T i and Q i ae cefficiens f e plynmial epansin f e empeaue and ea flu. Tey ae elaed by equains = ( ) + Q T T, ( ) + 5 Q = T + T, Q = 5Q T, Q + Q = Q + Q Q + Q = Q (6) + + wee T, T, Q and especively. ( ). Bunday and cnac cndiins +, Q ae ei values n e sufaces Ω + and F finie leng sell we cnside e fllwing mecanical bunday cndiins. d w d w dw =, = - fee end; w L p =, = - fied end; d d d L p L p L p d w w L p =, = - simply supped end. (7) d L p F infinie leng sell we cnside e fllwing cndiins a infiniy dw d w d w w,,, f (8) d d d Bunday cndiins f e equains f ea cnduciviy ae L T L = T - if empeaue is pescibed; Ω dt L λ = q - if ea flu is pescibed; (9) d L Ω T α L + ( T T ) = - f cnvecive ea ansfe λ Ω L wee λ is a cefficien f ea cnduciviy, α is a cefficien ea ecange ug a suface. Mecanical and emal cnac cndiins ae pesened in e fm [4 7] w < g =, Ω + \ Ωe, w = g >, Ωe + λ( w) T + λ* T T = T k, T k = () λ( w) + λ* I is impan menin a emal cnac cndiins include sell deflecin w nnlinealy and eefe equain () elaes e equains f sell emelasiciy () and ea cnduciviy (5). As a esul we ave Ω WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

5 Advanced Cmpuainal Meds in Hea Tansfe X 85 cnneced nnlinea pblems f emelasiciy and ea cnduciviy wi unilaeal cnac cndiins..4 Tansfmain e inegal equain In [4, 6, 7, 5] i was swn a e diffeenial equains f e sell emelasiciy () and ea cnduciviy (5) can be ansfmed in e inegal equains f Hammesein's ype G (, y) F ( y) dy = T G, y F T y dy = wee F, ( ) ( ) l l W (, y) [ p( y) g( y) ] β F ( y) dy = w, () D l =.5ε ( T + Tk ) + ( Tk T ), F.5 ( Tk T ) + ( Tk + T )- T F = β ( F + ε T )- β T = ε, Te kenels in ese inegal equains ae fundamenal sluins f cespnding diffeenial peas. Tey ave e fm G (, y) = ep( ε y )/ ε, G (, y) = ep( ε y )/ ε () W (, y) = ep( β y )[ cs( β y ) + sin( β y )] 8β D wee 5 ( ν ) ατ ( ν ) α ε =, ε =, β =, β = + τ E, D = ν.5 Algim f e pblem sluin ( ) Algims f e pblem s sluin cnsiss in an ieaive pcess f e nnlinea inegal equains f Hammesein's ype sluin and in e case if unilaeal cnac aking place an addiinal ieaive algim is used. Te algim as been elabaed in [4, 6]. In e pblems unde cnsideain e algim is cnvegen and cnvegence is fas enug..6 Sess calculain Sesses in e aisymmeical cylindical sell ae calculaed by fmulas (), wic f cnvenience may be pesened in e fm E d w z σ = z ( + ν ) α T, (4) ν d σ θ = Ew α T E E + ν d w ν z ( + ν ) α T d z WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

6 86 Advanced Cmpuainal Meds in Hea Tansfe X Deiving fm () epessin f e secnd deivaive f e displacemens and subsiuing i in (4) we bain e fllwing inegal epesenains f sess = l d G( ξ, ) σ ( ) b T ( ξ ) dξ + (5) d wee b b l + [ ] d G( ξ, ) G( ξ, ) T ( ξ ) dξ bt ( ) b p( ξ ) q( ξ ) dξ d σ θ l [ ( ) + T ( z] Ew( ) ( ) = νσ ( ) + α E T ) 5 Eα z =, b ( + ν ) 6 z α b =, 6 ( +ν ) Eα z = τ, b 4 ( ) α τ 4 E z =, ( ν ) Ez ν α E z β =, z β =, β =. 8 Mizes sesses ae calculaed by e equains σ i = ( σ ) + ( σ )( σ ) + ( σ ), (6) σ ( q) ( ) ( ) q i = σ + σ σ + σ + Using equains (5) and (6) sesses in e sell can be easy calculaed. Invesigain f e em-mecanical sae f e sell We will cnside ee em-mecanical sae f e sell wi cnsideing influence f e sell defmain n e ea ansfe beween i and a igid bdy. Calculain ave been dne f e daa: maeial 5 5 ppeies: E =.5 MPa, ν =.5.5 α τ = C, λ = V m C and gemeical paamees =. 5m, =. m, l b =, Eample. We cnside an aisymmenical cylindical sell f infinie leng placed in e igid siup wi a mgeneus iniial gap as is swn in fig.. Tempeaue n e siup suface is n mgeneus and equal + T ( ) = Tm + Tb sin π / lb, Tb = 6 C Tm = C. (7) On e sell suface ac mgeneus lad p( ) = MPa, empeaue n e inenal suface f sell is T = C, iniial gap is equal =. 5 and λ = λ *. In fig. ae pesened: Mizes sesses n eenal σ + and inenal σ sufaces f e sell, fce f cnac ineacin g, nmalized bending W = w / and empeaue n cnac suface T k. Te dased lines cespnd a sluin f pefec emal cnac wiu cuning influence f e sell defmain n e ea ecange and e slid lines cespnd e sluin pesened ee. WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

7 Advanced Cmpuainal Meds in Hea Tansfe X 87 Figue. Figue. + Mizes sesses n eenal σ and inenal σ sufaces f e sell, fce f cnac ineacin g, f e same daa and iniial gap =. ae pesened in fig 4 and 5 eenal and inenal punc especively. Figue 4. Figue 5. Analysis f ese daa sws a n mgeneus empeaue disibuin cause significan sell defmains and clse mecanical cnac wi a igid siup. As a esul significan sess ccus in e sell. Calculains assuming pefec emal cnac lead significan inaccuacy, wic is n nly quaniaive bu als qualiaive. Eample. Hee we cnside an aisymmenical cylindical sell f infinie leng placed in e igid siup wi n mgeneus iniial gap, as swn in e Fig.6. Te gap is given by e funcin * ( ) = + sin π / l, =. 5, b = /, l b =. b b Figue 6. Figue 7. WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

8 88 Advanced Cmpuainal Meds in Hea Tansfe X In fig.7 ae pesened: Mizes sesses n eenal σ + and inenal σ sufaces f e sell, nmalized bending W = w / and empeaue n cnac suface T k. Te dased lines cespnd sluin f pefec emal cnac wiu cuning influence f e sell defmain n e ea ecange and e slid lines cespnd e sluin pesened ee. Analysis f ese daa sws, a n mgeneus iniial gap cause significan sell defmains. As a esul, in e sell, n significan mgeneus sess and empeaue disibuin ccus. Calculains assuming pefec emal cnac lead significan inaccuacy. Sme values f emmecanical sae diffe. Eample. We als invesigae influence f diffeen paamees n sell defmain and ea ecange. In fig. 8 ae pesened defmains and empeaue n e cnac suface f fied-end sell in mgeneus + empeaue field T = 5 C, T = C, f λ = 4λ*, =. 5. Figue 8. Figue 9. I is ineesing pin u a in e case wen T > T + accuning influence f e sell defmain n e ea ecange deceases is defmain, sess and empeaue disibuins. In fig. 9 is pesened dependence f e sell deflecin n is adius. Calculains ave been dne f e fllwing daa: λ = 5λ* =. 5, cuve f e eenal punc and Tb = 6 C, cuve f e inenal punc and Tb = C. I is impan menin a f eenal punc wen / > 5 dependence becmes nnlinea and e sell is aaced e punc, slid cuve. All e abve calculains ave been dne f e case wen n e sufaces Ω and ψ is pescibed empeaue. In fig. and fig. dependences w = f ( λ / λ* ) and w = f ( / ) f e eenal punc ae pesened. Cuves cespnd e case wen n e sufaces ψ empeaue and n e suface Ω ea flu ae se. Cuves cespnd e case wen n e sufaces Ω empeaue and ψ n e suface ea flu ae se. In b cases empeaue is disibued in accdance wi equain (7), ea ansfe ae is equal 4 4 q = [ V / m K] and q =.4 [ V / m K] f e fis and secnd cases WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

9 Advanced Cmpuainal Meds in Hea Tansfe X 89 especively. In e fis case sell deflecin is minimal f λ λ and / * = inceases wen / decease. In e secnd case sell deflecin is maimal f λ / λ* = and des n depend muc n /. In b cases sell deflecin des n cange muc f λ λ. / * > Figue. Figue. 4 Cnclusins Te esuls pesened ee sw a in many siuains in-walled sucues subjeced ig empeaue pefec emal cnac cndiins ae n accepable because ey cann ake in accun pysical pcesses elaed defmain and ea ecange. Suc kinds f pblems may ccu in diffeen field f science and engineeing, f eample in nuclea, aespace, cemical indusies, ec. In suc siuains e appac develped ee and in u pevius publicains ave be used. Refeences [] Bley B.A. and Weine J.H. Tey f emal sess, Wiley, New Yk, 96. [] Kan B. Ya., Zzulya V.V. Cnneced pblem n cnac plae wi igid bdy ug e ea-cnducing laye, Dclady Akademii Nauk Uk. SSR, 4, pp., 988. (in Russian) [] Psigac Ya. S, Sves P. N. Yu. M., Temelasiciy f in sells, Kiev, Naukva dumka, 978. (in Russian) [4] Zzulya V.V. Cnac cylindical sell wi a igid bdy ug e eacnducing laye, Dclady Akademii Nauk Uk. SSR, 989,, pp (in Russian) [5] Zzulya V.V. Te cmbined pblem f em- elasic cnac beween w plaes ug a ea cnducing laye, Junal Applied Maemaics and Mecanics, 5(5), pp.7 77, 989. [6] Zzulya V.V. Bending f a plae in empeaue field unde esicins, Izvesiya vuzv. Engineeing,, pp. 4 7, 99. (in Russian) WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess

10 9 Advanced Cmpuainal Meds in Hea Tansfe X [7] Zzulya V.V. Cnac cylindical sell wi a igid bdy ug e eacnducing laye in ansiinal empeaue field, Mecanics f Slids, Vl., pp.6 65, 99. [8] Zzulya V. V. Nnpefec cnac f laminaed sells wi cnsideing debnding beween laminas in empeaue field, Teeical and Applied Mecanics, Vl. 4, pp.9 97, 6. [9] Zzulya V.V. Laminaed sells wi debnding beween laminas in empeaue field, Inenainal Applied Mecanics, Vl. 4(7), pp. 5 4, 6. [] Zzulya V.V. Maemaical Mdelling f Pencil-Tin Nuclea Fuel Rds, in Pc. SMIRT9 Cnf. Sucual Mecanics in Reac Tecnlgy, A. Gupa Ed., Tn, 7, pp. C4 C. [] Zzulya V.V. Cnac f a sell and igid bdy ug e ea-cnducing laye empeaue field, Inenainal Junal f Maemaical Mdels and Meds in Applied Sciences, (), pp.8 45, 7. [] Zzulya V.V. Cnac f e in-walled sucues and igid bdy ug e ea-cnducing laye, in Pc. Teeical and Epeimenal aspecs f ea and mass ansfe, Acapulc, Meic, pp. 45 5, 8. [] Zzulya V.V., Aguila M., Tem-elasic cnac and ea ansfe beween plaes ug e ea-cnducing laye, in Hea ansfe, B. Sunden and C.A. Bebbia Eds. Cmpuainal Mecanics Publicains, Suampn, UK and Bsn, USA, pp.,. [4] Zzulya V.V., Bdenk Yu. N. Temelasic cndiin f cylindical sell, wic ineacin wi a igid bdy ug e ea-cnducing laye, Izvesiay vuzv. Engineeing, 8, pp. 47 5, 99. (in Russian) [5] Zzulya V.V., Bdenk Yu. N. Cnnecing pblem n cnac f cylindical sells wi a igid bdy in empeaue ug e eacnducing laye, Dclady Akademii Nauk Uk. SSR, 4, pp.5 4, 99. (in Russian). WIT Tansacins n Engineeing Sciences, Vl 6, ISSN 74-5 (n-line) 8 WIT Pess