The applications of the non-linear equations systems algorithms for the heat transfer processes

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1 MHEMICL MEHODS, COMPUIONL ECHNIQUES, INELLIGEN SYSEMS he applatos o the o-lear equatos systems algorthms or the heat traser proesses CRISIN PRSCIOIU, CRISIN MRINOIU Cotrol ad Computer Departmet Iormats Departmet Petroleum Gas Uversty o Ploest Ploest, 39 Bd. Buurest, ROMNI patrasouu@upg-ploest.ro marou_@yahoo.om bstrat: he paper presets the author s researhes the heat traser mathematal models ad the mplemetato o the umerally algorthms or solvg the o-lear equatos systems. he artle has three parts. I the rst part s preseted the mathematally model o the heat exhager. he seod part s dedated to study o the umerally algorthms or solvg the o-lear equatos systems. he authors have studed three algorthms: the Newto-Raphso algorthm based o aalytally expressos o the Jaobea matrx, the Newto-Raphso algorthm based o umerally values o the Jaobea matrx ad the Broyde algorthm. he last part otas a aalyss o the perormaes o theses algorthms rapport o alulus eort ad alulus preso rtera. Key-Words: o-lear equatos system, Newto-Raphso algorthm, Broyde algorthm, mathematal modelg, heat exhager, umerally aalyze Itroduto he mathematal modellg o the hemal proesses represets a mportat problem or the proess desg ad or the hemal proess operatg. lass o the hemal proesses s represeted by the heat traser proesses. he authors have studed the heaters proess ad they have modeled the radato seto o the tubular heater []. he researhes have permtted or the authors to buldg the statally haratersts o the tubular heater there ad have otrbuted to developed the optmal ombusto otrol struture []. I the last years, the authors have studed the modelg o the heat exhager, or desg the otrol systems whh otaed heat exhagers [3]. O mportat problem o the heat exhagers modelg s the mathematal algorthm used to solve the model. Beause the mathematal model o these proesses s a equatos system, the authors have studed the algorthms used to solve the model. he struture ad the mathematal model o the heat traser proess most mportat lass o the heat traser proesses s deed by the heat exhagers. he authors have studed, modeled ad smulated the shell ad tube heat exhager havg luxes outer low. I gure s preseted a seto o the shell ad tube heat exhager [3]. Fg. he shell ad tube heat exhager seto he heat exhager s haraterzed by our let ad two outlet varables, gure. he let varables are: h, Q hot the put hot temperature ad the hot low rate, l, Q old the put old temperature ad the low rate o old lud. he outlet varables are: h - the outlet temperature o the hot lud, the outlet temperature o the old lud. ISSN: ISBN:

2 MHEMICL MEHODS, COMPUIONL ECHNIQUES, INELLIGEN SYSEMS Fg. he shell ad tube heat exhager struture Mathematal model o heat exhager ossts to heat balae equato assoated wth the hot low ad the old low ad the Delaware model o the heat exhager [4]. he ompat orm o the mathematal model o the exhager s a o-lear equato system wth two equatos ad two varables h,, h. he o lear utos o the equatos system are the ext orm: Q = Q ; hot p,hot h h old p,old h h h = Qhot p,hot h h k. 3 h l 3 he algorthms used to solvg the o-lear equato systems Let the o-lear equato system x,x, Kx x,x, Kx x,x, Kx respetvely X, 4 F, 5 where the X vetor represets the o-lear equato system varables [ x,x, K,x ] X =. 6 he vetor F otas the utos deed to o-lear equato system 4 [,, K, ] F =. 7 he equato system soluto may be approxmated usg the suessve alulatos, started at tal estmato [ ] X = x,x, K,x. 8 he soluto o the o-lear system s determated usg the Newto method [5]. hs method s based o the aylor lear approxmato ormula = X + 0 x. 9 X X = 0 X I wll geeralzed the aylor lear approxmato o the all utos o the system 4 wll be obtaed 0 X + 0 x, =, K, X. 0 x X X = = Usg the matreal ompoets deed to relatos 6 ad 7, the relato 0 wll have a ew orm F 0 X = F X he X 0 + J X X. J represets the Jaobea matrx assoated to o-lear equato system 4 L L J X =. L L L L L he applatos o the Newto method are the Newto Raphso algorthm ad the Broyde algorthm. ISSN: ISBN:

3 MHEMICL MEHODS, COMPUIONL ECHNIQUES, INELLIGEN SYSEMS 3. he Newto - Raphso algorthm he Newto-Raphso algorthm ossts to ew approxmato o the soluto o the o-lear equato system, approxmato deed by relato + X = X + X. 3 he orreto vetor X s obtaed usg the F X =, respetvely equato, where 0 J k X X = F X. 4 I the Jaobea matrx J X s a o-sgular matrx, the soluto o the lear system 4 has the orm h =Q hot p,hot k X =J X F X. 5 he umeral values o the soluto 5 are obtaed by usg the Gauss algorthm. he overgee odtos o the array 0 { K X, X,, X,K} o the system soluto estmatos are preseted [5]. he stop rtera o the Newto-Raphso algorthm s ormulated by the relatos X ε, =, K,. 6 he Jaobea matrx assoated to o-lear equatos system has the ollowg ompoets: ; 7 =Q old p,old ; 8 h =Q hot p,hot k ed l h h l h h h h h + ; 9 =k ed l h h l h h h h h +. 0 he most mportat dsadvatage o the Newto- Raphso method ossts to alulatg eort to evaluate the elemets o the Jaobea matrx J X ad the utos o the vetor F X [7]. 3. he umeral evaluato o the Jaobea matrx he authors have studed the estmato o the partally dervates o the utos,, K, o the olear system 4. For the urret pot X have bee deed the small varatos h. 000 x, =, K,. For evaluato o the Jaobea matrx the pot X, the authors have used the geeral relato or the evaluato o the elemet o the Jaobea matrx x, K, x + h, K, x X =, k h =, K,; =, K,. 3.3 he Broyde algorthm he Broyde algorthm redues the Jaobea elemets alulus eort usg the approxmato o the Newto-Raphso Jaobea matrx: ISSN: ISBN:

4 MHEMICL MEHODS, COMPUIONL ECHNIQUES, INELLIGEN SYSEMS F X = + X, 3 X where relato 0 0 = J X ad F s deed by k F = F X F X 4 ad X = X X. he matrx most to satsy the propertes: a o veryg the equato J X X = F X ; b o mmzg the deree m. heses two propertes o the matrx have demostrated [6]. he stop rtera o the Broyde algorthm are smlarly to the stop rtera o the Newto-Raphso algorthm. he steps o the Broyde algorthm are desrbed below [7]: Step. Varables talzato k, X 0 0 = J, X 0 0 F = F. Step. Italzato usg the Newto-Raphso algorthm X F = X. Step 3. Whle the stop rtero has alse value does: k = k+ ; F = F X ; F X = + X X ; + k X = X F. Step 4. Stop he maor dsadvatage o the Broyde algorthm s represeted by the evaluato o the verse matrx or eah terato k. better soluto o ths problem has bee deed by Sherma-Morrso [7]. he orgal orm o the Sherma-Morrso ormula s uv + uv =, 5 + v u where s a osgular matrx ; v ad u are dmesoal vetors wth the property v u. he authors have osdered the ollowg expressos or the u ad v varables: F X u= ; 6 X =. 7 v X Usg the ew varables u ad v, the relato 3 wll have the orm = + u v. 8 Usg 5 to 8 ormula s obtaed a ew orm o the verse o the matrx k = + u v = = uv v + u. 9 Wth ths result, the thrd step o the Broyde algorthm s trasormed a ew orm: Step 3. Whle the stop rtero has alse value does: k = k+ ; F = F X ; F X u= ; X v = X ; = uv v + u k ; + k X = X F. ll the modatos have otrbuted to rease the perormaes o the Broyde algorthm. ISSN: ISBN:

5 MHEMICL MEHODS, COMPUIONL ECHNIQUES, INELLIGEN SYSEMS 4 he results o the smulato o the heat traser proess he authors have elaborated the umeral programs dested to smulato o the shell ad tube heat exhager. he programs have a modular struture whh otaed: a the sub-algorthms or readg the put data le the geometrally heat exhager data, the propertes ad the operatg parameters o the heat ad old lud; b the sub-algorthm or solvg the o-lear equato system; the ma module the put data readg ad the heat traser algorthm. he authors have elaborated three umeral programs, deretated by the sub-algorthm used to solve the o-lear equato systems. Frst program uses the Newto-Raphso algorthm wth the aalytally expressos o the Jaobea matrx. he seod program mplemeted the Newto- Raphso algorthm wth the umerally expressos o the Jaobea matrx. he last program s based o the Broyde algorthm. ll the sub-algorthms used to solvg the o-lear equato system are made by the authors [8]. he shell ad tube heat exhager used by the authors s desrbed [3, 9]. For the struture showed gure, the heat exhager s haraterzed by the ext put data: Q = kg/h; h Q = kg/h; h = 80 C; 3 C. For all the algorthms the stop rtera parameter 6 has the value ε 0 5 =, =, kj/h, ad the 0 tal soluto has the ompoets: x = 60 C, 0 x = 30 C. For the Newto Raphso algorthm whh uses the umerally expressos or evaluato o the Jeaobea matrx, the values o the small varato 4 are h = 0 4 x C. he results obtaed wth the three umeral programs are preseted table. able. Comparatve results to solver the o-lear equato system o the heat exhager lgorthm he Newto-Raphso algorthm based o aalytally expressos o the Jaobea matrx he Newto-Raphso algorthm based o umerally values o the Jaobea matrx he Broyde algorthm Iterato 9 6 he varable o the system x 0 [ C] [kj/h] E E E E E E+03 he soluto o the o-lear equatos system preseted [9] s x = 40 C ad x = 8 C. For the tal soluto assumed by the authors ad or the stop rtero preseted allow, the soluto o the o-lear equato system s obtaed varous teratos depedg o the algorthm. he program based o the Newto-Raphso algorthm determes the most exatly soluto the bgger terato umber. For the system, the Newto- Raphso algorthm based o umerally values o the Jaobea matrx s the most qukly. ll the obtaed solutos are oetrated aroud the pot h = 39 C ad = 8 C. he luee o the stop rtera parameter to the preso o the soluto o the o-lear equato system s preseted table. he olusos o umeral aalyss are ollowg: a he derease o the stop rtera value wll rease the alulus preso; wll derease the values o the utos ad ad wll rease the umber o terato. ISSN: ISBN:

6 MHEMICL MEHODS, COMPUIONL ECHNIQUES, INELLIGEN SYSEMS b lthough the umeral values o the utos ad derease at oe by the stop rtera value, the modato o the soluto o the o-lear equatos system s very small, that the supplemetary alulus eort s ot usted. able. he luee o the stop rtera value o the soluto o the o-lear equatos system solved by the Newto-Raphso algorthm Stop rtera value [kj/h] Equato o the system x 0 [ C] [kj/h] Iterato umber E E E E E E E E Beause the errors betwee the values o the solutos obtaed by usg the three algorthms ad the orgal soluto are very small, the authors osder that the mathematal model o the heat exhager, the algorthms or the solvg the olear equatos systems ad the umeral programs are valdated. 4 Coluso he artle presets the author s researhes o the applatos o the umeral algorthm or heat proesses. For the heat exhager, the mathematal model s represeted by a o-lear equatos system. For solvg the mathematal model, the authors have studed three algorthms: the Newto- Raphso algorthm based o aalytally expressos o the Jaobea matrx, the Newto- Raphso algorthm based o umerally values o the Jaobea matrx ad the Broyde algorthm. he authors have aalyzed the perormaes o theses algorthms rapport o alulus eort ad alulus preso rtera. Reerees: [] Pătrăşou C., Marou V., dvaed Cotrol System or the tubular heaters o the vauum ad rude ut - I. Mathemat Modellg o the Combusto ad Heat raser, Revsta de hme r.4, 997 Romaa. [] Pătrăşou C., Marou V., Modellg ad Optmal Cotrol o a Idustral Furae - DYCOPS-5, 5 th IFC Symposum o Dyams ad Cotrol o Proess Systems, Coru, Greee, Jue 8-0, 998. [3] Patrasou C., he Steady-State Modelgad Smulatoo a Heat Exheger, Petroleum-Gas Uversty o Ploest Bullet, ehal Seres, Vol LXI, No [4] Serth, De R. W., Proess heat traser: prples ad applatos, ISBN , Elsever Ltd, p.89-95, 007. [5] Demdovh B. P., Maro I.,., Computatoal mathemats, Mr Publshers, Mosow, 98. [6] 58.pd [7] emethodmod.html [8] Patrasou C., Numeral methods appled hemal egeerg PSCL applatos, MatrxRom, Buurest, p.7-4, 005 Romaa. [9] Dobresu D., Heat traser proesees ad spe deves, Edtura Ddata s Pedagga, Buurest, p , 983 Romaa. ISSN: ISBN: