UNIVERSITI TENAGA NASIONAL, M sia. Keywords: Mathematical Modeling; Finite Element Method; Temperature History

Transcription

1 Utilization o Numrical chniqus to Prdict h hrmal Bhavior o Wood Column Subjctd to Fir Part : Using Finit Elmnt Mthods to Dvlop Mathmatical Modl or Wood Column Mohamd ElShayb 1, a, Faisal.Fairag 2, Zolman Hari 3, Norhaida b Razak 1, Jacqulin Eng Ling Siang 1,b, Fazril Idris 1 1 Mchanical Eng. Dpartmnt, Collg o Eng. 3 Scinc & Maths Dpartmnt, UNIVERSII ENG NSIONL, M sia 2 Mathmatic Dpartmnt, KING FHD UNIVERSIY OF PEROLEUM ND MINERLS. a smohamd@unitn.du.my, b jacqulin@unitn.du.my Kywords: Mathmatical Modling; Finit Elmnt Mthod; mpratur History bstract. h mathmatical modl to prdict th tmpratur history or wood column is ndd in ordr to dtrmin its ir rsistanc whn xposd to ir. In this papr, an intllignt mthodology calld Finit Elmnt Mthod (FEM) o prorming analysis or th squar and circular wood columns by virtually or artiicially dvloping a tmpratur history mathmatical modl. Numrical simulation modl has bn dvlopd or th wood column by using two-dimnsional mathmatical modl. h two-dimnsional mathmatical modl was dvlopd by using Galrkin s Wightd Rsidual tchniqu. his modl ocuss on th rgional matrial o th wood column or dscribing its thrmal bhavior. Whn th tmpratur history in a column and rlvant matrials proprtis ar known, th strngth o th column can b calculatd at any tim during ir. hror, th dvlopmnt o th tmpratur history mathmatical modl is a must bor any urthr study to b carrid out or th wood columns. h low o convction will rsult in minimal incras in th rat o hat nrgy raching th column cor. h analysis shows that th tmpratur o th column incrass with rspct to th duration o xposur to ir. Calculation Procdur Division o cross sction into lmnts. nalysis is don or two typs o columns, namly squar and circular columns. Du to its symmtrical proprty, only on-quartr o th cross sction is considrd. hn, th cross sctional ara o th on-quartr o th cross sction is subdividd into a numbr o lmnts, i.. into many triangular lmnts. h triangular lmnt is shown in Fig.1 blow. Fig.2 and Fig. 3 shows th quartr mshing or circular and squar cross sction column. y k (x k, y k ) j (x j, y j ) i (x i, y i ) Fig. 1 riangular Elmnt x Fig. 2 Quartr mshing o squar column h hat lux rom th ir is assumd to b transrrd into th cross sction through ithr on or two o th boundary by convction and radiation mods. Onc th hat lux ntrs th boundary, ths hat luxs will b transrrd into th innr part o th cross sction through conduction mod.h valuation o th tmpratur in init lmnt mthod is basd on lmntwis considration.fig. shows th shmatic gnral two dimnsioanl hat conduction problm.

2 Insulatd WOOD Fir q rl q s q cv L q cvb q r B y Insulatd x Fig.3: Quartr mshing o circular crosssction Fig. : Schmatic Gnral two-dimnsional Hat Conduction Problm By using th unstady stat partial dirntial quation, kt + kt + h ( ) + ( ) + q + Qt = ct s x x y y εσ ρ t (Eq.1) whr: = tmpratur gradint in x-dirction, o C/m, x = tmpratur gradint in y-dirction, o C/m k = thrmal conductivity, W/m. o C, t = thicknss, m h = sum o th convctiv hat transr coicints on th on latral acs o z-axis, W/m 2. o C = ir tmpratur, o C, = body tmpratur, o C ε = sum o th on latral surac along z-axis surac missivitis on th two latral acs, dimnsionlss σ = Stan-Boltzmann constant, W/kg.K Q s = th sum o th hat luxs imposd on th on latral acs o z-axis, W Q = th hat gnration pr unit tim and volum, W/s ρ = dnsity, kg/m 3, c = spciic-hat, J/kg. o C, = tmpratur gradint with tim, o C/s t o suit th ormulation into init lmnt analysis, Galrkin s wightd rsidual mthod is applid with th rsidual { } = R kt + kt + h ( ) + εσ( ) + q + Qt ρct (Eq.2) r s x x τ h Galrkin s wightd rsidual mthod is applid to Eqn. (2) in a vry straightorward mannr. h intgration o th shap unction tims th rsidual will minimiz th rsidual to zro as ollow: [ N ] { R }dv = [ N ] kt + kt + h ( ) + εσ( ) + q + Qt ρct dxdy = s x x y y t (Eq. 3) ( ) whr :{ R }= th rsidual contributd by lmnt () to th inal systm o quations [ N ] = th transpos o th lmnt shap unction By applying th Grn s horm on th trm involving scond-ordr drivativs yilds: [ N ] { R } dv = [ ] [ N ] [ N ] { } N kt nxdc kt dxdy + x x x C q cvb q rb z q s q r L q cvl

3 [ N ] C kt nydc [ N ] [ N ] { } kt dxdy [ ] dxdy - [ N ] h [ N ]{} + [N] hdxdy + 3 [ ] [ ]{}dxdy [ N ] εσ [ N ]{} N [ N ] εσ dxdy + [ N ] q s dxdy + [ N ] Qt dxdy { } dxdy.. [ N] ρc[ N] = (Eq.) τ By taking into considration th boundary conditions, intgration is prormd upon o all th trms in Eq. to obtain composit lmnt stinss matrix[ K ], composit nodal orc vctor [ ] and capacitanc matrix [C] rom th Galrkin s solution.h inal orm or th lmnt wis or local approximation will b in th orm o: [C] { & } + [K] {} = {} whr {& } { } = t (Eq.5),matrix o unknown variabl, whr is tmpratur and t is th tim. [K] = [ K ] [ ] xx + K yy, lmnt stinss matrix (Eq.6) ssmblag o th lmnt matrics to th global matrics o th domain. h assmblag stp is ncssary in all init lmnt analysis. In this stp th rgion bing analyzd is put back togthr or assmbld rom th individual lmnts comprising it. h assmblag stp is basd on th principl o compatibility; i.. a particular nod on any givn lmnt must b idntical to thos associatd with th sam nod on ach lmnt that shars this nod. h global must b assmbld to orm th assmblag stinss matrix, assmblag capacitanc matrix, and assmblag nodal orc vctor. h basic ida bhind this stp is that th unknown paramtr unction must hav th sam valu at any givn nod rgardlss o th lmnt containing th nod. ssmbling th stinss matrix, hat capacitanc matrix, and th orc vctor o all lmnts o th domain, into. global matrics as ollows: [ K ] g {} + [ C] g = { } g (Eq.7) whr [ K ] g = [ K ] + [ K ] + [ K ] +... tc.. = rat o chang o tmpratur with = global stinss matrix, (n x n) { } = unknown tmpratur, (n x 1) rspct to tim, (n x 1) [ C] g = [ C] + [ C] + [ C] +... tc. { } g = { } + { } + { } +... tc. =global hat capacitanc matrix, (n x n) = global orc vctor, (n x 1) im Dpndnt Solution. h solution mthod usd in this problm is th Backward Dirnc Schm (BDS). h BDS mans th drivativ, i.. tim (t), is writtn in th backward dirction (with rspct to tim) and th tim stp is not zro, t. Rsults and Discussions. h compltd init lmnt quations or th abov tmpratur history mathmatical modls will b incorporatd into FORRN computr program. h computr program will comput th tmpratur history by using th Backward Dirnc Schm solution with th maximum loating-point prcision. hr ar our important parts in th computr program to achiv th objctiv: a. Dvlopmnt o mshing program. b. Dvlopmnt o tmpratur history program. c. Dvlopmnt o strngth dtrmination program. d. Dvlopmnt o post-procssing program. h gnral dimnsion o th column is givn in Fig. 5 and th mshing is givn in Fig. 2.

4 abl 1: Comparison column cntr tmpratur and nduranc tim at dirnt boundary conditions or squar column (Kruing wood, and applid load 138 kn) width Fig.5: Squar wood column dpth yp o boundary Fir rom On sid Fir rom wo sids Fir rom hr sids Fir rom all sids mpratur o column cntr atr 3 min. 2 o C 2 o C 25 o C 27 o C mpratur history o squar column. h numbrs o th slctd lmnts ar 5 and 25 (on th column surac), 125 and 16 (at th irst third o column quartr), 28, 291(at th scond third o th column quartr) and lmnt 36 (at th column cntr). abl 1 summarizs th comparison o th columns cntr tmpratur at dirnt ir boundary or 3 minut. Fig.6 and Fig. 7 show th tmpratur history or th slctd lmnt at dirnt ir boundary. his rsults shows that th wood has a vry low thrmal conductivity. From th tmpratur history graphs, at th ir boundary th tmpratur starts to incras rapidly at 11 o C. h rason is that som o th ir nrgy at th bginning o pyrolysis is consumd to vaporat moistur; latr on this ir nrgy will rais th surac column tmpratur. It is obsrvd rom Fig. 6 that th intrsction o th twotmpratur history proils or lmnt 25 on th column boundary and lmnt 291 insid th column. Elmnt 25 is closr to ir than lmnt 291, this is why th tmpratur o lmnt 25 starts to incras at th starting o pyrolysis. It is obsrvd that th rat o tmpratur incrass or lmnt 25 is starting to dcras atr 7 minuts, on contrary to lmnt 291. It is suggstd that th rason bhind this, is du to that lmnt 291 has mor sids subjctd to hat conduction than lmnt 25. lso lmnt 25 has on sid subjctd to atmosphric tmpratur, which will dissipat som o th hat spcially at highr tmpratur. 35 mratur dgrs C im (min) 15 El El 5 El 25 El 125 El16 El 28 El 291 El 36 Fig.6: mpratur variation o slctd lmnts on th squar column surroundd with ir rom on sid or Malaysian Kruing timbr

5 mpratur dgrs C im (min.) El 25Sq El 5 Sq EL16Sq El 125 Sq El 291Sq El 28 Sq El 36Sq Fig.6: mpratur variation o slctd lmnts on th squar column surroundd with ir rom all sids or Malaysian Kruing timbr mpratur history or circular cross sction column. h snsitivity analysis or this column will b don such that th ir boundary is surrounding th cross-sction. h column diamtr is 21 mm. h tmpratur historis or th slctd lmnts ar prsntd in Fig. 8. h quartr mshing o circular cross-sction column is shown in Fig. 2. h charrd ara or th quartr o th column is 592 mm mratur dgrs C im (m in.) El 55 El 88 El 128 El 173 Fig.8: mpratur variation o slctd lmnts on circular cross sction column surroundd with ir rom all sids (Malaysia Kruing timbr) Comparison o columns cntr tmpratur or dirnt column cross sctions surroundd with ir rom all sids. h tmpratur rsults or dirnt conigurations o th column cntr ar shown in Fig 9. h highst column cntr is squar column. h simulation rsults indicat that th bst cross sction is th circular cross sction. h last circumrnc is th circular column bcaus th ara xposur to ir is last. Conclusion two-dimnsional numrical simulation modl basd on Finit Elmnt Mthods had bn dvlopd or wood column. his mathmatical modl was latr convrtd into computr modls using FORRN programming languag or simulation purposs as shown in rsults, discussion

6 and prsntd in this rsarch papr as Part. h latr subsctions (Part B, [5], and C, [6]) o th rsarch paprs will discuss th analysis o th column s tmpratur and ir rsistanc mpratur dgrs C Squ El36 Rc El36 6SEl 136 8SEl 336 I Sc El235 Cir El im(min) Fig.9: mpratur variation or dirnt column coniguration at th column cntr (Malaysian Kruing timbr) Rrncs [1] M. Elshayb, Y.K. Bng, pplication o Finit Dirnc and Finit Elmnt Mthods or hrmal Problms, Univ. Malaysia Sabah (21) [2] F.N. Istas, Estimation o Strngth o Enginring Componnt Exposd to Fir, Mastr o Scinc in Mchanical Enginring, Univrsiti knologi Malaysia (1997) [3].H. Mian, Using Numrical chniqus or Prdicting th Fir Rsistanc o Stl Column during Fir, MSc. diss., Dpt. o Mch. Enginring, Univrsiti knologi Malaysia (2) [] H..E. Gouda, Prormanc o Structural Wood Column Exposd to Fir, MSc. Diss., Dpt o Mch. Enginring, Univrsiti knologi Malaysia (1999) [5] Mohamd ElShayb, Faisal Fairag, Zolman bin Hari, Norhaida bt b Razak,Jacqulin Eng Ling Siang, Utilization o Numrical chniqus to Prdict h hrmal Bhavior o Wood Column Subjctd to Fir Part B: nalysis o Column mpratur and Fir Rsistanc, Procding o FEOFS 25, Bali, Indonsia. [6] Mohamd ElShayb, Faisal Fairag, Zolman bin Hari, Fazril Idris,Jacqulin Eng Ling Siang, Norhaida bt b Razak, Utilization o Numrical chniqus to Prdict h hrmal Bhavior o Wood Column Subjctd to Fir Part C: Snsitivity nalysis, Procding o FEOFS 25, Bali, Indonsia.