CLOSED LOOP CONTROL OF GAS JET FLAMES DISTRIBUTION USING PROBABILITY DENSITY FUNCTION SHAPING TECHNIQUES

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1 CLOSED LOOP CONROL OF GAS JE FLAMES DISRIBUION USING PROBABILIY DENSIY FUNCION SHAPING ECHNIQUES X. B. Su ad H. Wag Cotrol Systems Cetre, Deartmet of Eletrial Egieerig ad Eletrois, UMIS, Mahester, UK Istitute of Automatio, Chiese Aademy of Siees, Beijig, P R Chia hog.wag@umist.a.uk Keywords: Stohasti system, robability desity futios, flames distributio otrol, stati otrol. Abstrat his aer resets a losed-loo otrol algorithm for the D Flame emerature Distributio (FD of gases usig the reetly develoed stohasti distributio otrol tehiques [11]. At first, the temerature distributio of the flame alog the ross setio is aroimated. Uder the assumtio that the flames are symmetrial with reset to the vertial ais, the shae otrol of FD is trasferred ito the otrol of the Flame Eergy Distributios (FED. A otrol algorithm is the desiged usig the outut robability desity futio otrol oet, where a equivalet stati model is established usig the B-slie futios. Simulatios have bee erformed where the mehaial models of the flames ad the FED otroller are embedded ito a losed-loo otrol struture. he results show that the FED a be made to effetively trak a give shae. 1 Itrodutio he effiiey of the boiler is mostly determied by the flame status of eah burer [1 3]. he fuel of the boiler a be either blast furae gas or ulverized oals. I the blast furae gas boiler, the flame at eah burer is i a form of gas turbulet jet flame whe the fuel jet veloity is higher tha a ertai threshold value, ad i the form of gas flamelet jet flame whe the veloity is lower. I ulverized oal boilers, the flame at eah burer is formed by two-hase turbulet reative flows. Aordig to Bilger [4], for turbulet reative flows, there are maily two modellig methods, amely the robability desity futio (PDF based modellig [5] ad the oditioal momet losure (CMC based modellig [6]. he former alulates the roerties of turbulet reative flow fields, where at eah oit i the flow field, a omlete statistial desritio of the state of the fluid is established for the veloity-omositio joit PDF. O the other had, the CMC method redits the oditioal averages ad higher momets of quatities suh as seies mass fratios ad ethaly. Based o these models, umerial simulatio a be erformed to rodue the FD of the flame, leadig to a timeosumig roess. For the gas flamelet jet flame, a stati model a be established [7] ad used i the losed-loo otrol system. For eamle, Has at el [8] desiged a losed-loo otrol system of the gas jet flame, where, oyge ad ombustible gas are used as the otrol iuts ad their ombustio flame is the outut. CCD ameras are used to ature multi-setral images that are roessed to rodue flame features (brightess ad height. Zhou et al [9] realized a losed-loo otrol of the boiler flame through the radiat eergy obtaied from the temerature of the flame. However, sie traditioal boilers otrol the iut fuels aordig to the oyge ratio i the flue gas, it is geerally diffiult to obtai a desired losed-loo erformae due to the eistee of a log time delay. As suh, serial otrol is emloyed [9] to solve the large iertia roblem by takig the radiat eergy as a itermediate variable. I the aalysis of flames, the FD is geerally used as a arameter to estimate the situatio of the boiler. I may ases, FDs a be oe-to-oe maed ito the Flame Eergy Distributios (FED that a be regarded as a PDF. his idiates that the reetly develoed stohasti distributio otrol theory (that aims at otrollig the shae of systems PDFs a be diretly alied to formulate a losed-loo otrol that otrols o-lie the shae of the FED. I reet years, the shae otrol of the PDFs of system variables has bee roosed [11 1], where the aim of the otrol iut desig is to guaratee the trakig of outut PDF to a give PDF. It has bee show that these grous of methods a be widely used i hemial roessig ad aermakig systems. As the flames eergy distributio a be regarded as a PDF, i this aer a PDF shaig method will be used to otrol the gas flamelet jet flame based o the stati model. I this losed-loo otrol system, the jettig rate of the fuel is used as the otrol iut ad the FED is the outut. It is assumed that istrumets suh as CCD ameras are available to ature the flame images that a be further roessed to etrat the FED iformatio, whih is take as the feedbak sigal for the losed-loo otrol. Flame model resetatio ad simulatio.1 Flame model resetatio I ratie, the gas flamelet jet flame is a 3-D flame just like a torh flame. Beause the flamelet jet flame is a aially

2 symmetrial flame, a -D temerature distributio model a be well used i most ases to rereset the flame temerature distributio as show i Figure 1, where d is the diameter of the fuel ijetio ozzle, u is the seed of the fuel ijetio at the ozzle, is the vertial oordiate, r is the horizotal oordiate, ad b( is the flame boudary. he two ars i Figure 1 are the isotherms of the jet flame. Figure 1: Flame temerature distributio demostratio. Boudary oditios ad oservatios equatios o obtai a stati model the flame is assumed to be a uomressible steady flow ad the olumed free jet flame. All the variables oered are defied as follows: r Horizotal oordiate Vertial oordiate u Vertial veloity v Horizotal veloity emerature emerature at,that is temerature i eviromet Yi Mass fratio for omoet i, where Yi = 1 ρ ρi Desity of the miture Seifi heat aaity of the fuel wi Reatio rate of the omoet i Qi Reatio heat of the omoet i γα,,d Movemet visosity oeffiiet, diffusio oeffiiet, heat diffusio oeffiiet of lamiar flow resetively γ, α,d Movemet visosity oeffiiet, diffusio oeffiiet, heat diffusio oeffiiet of turbulet flow resetively Usig these otatios, the followig otiuity, mometum, eergy ad mass balae equatios a be obtaied ( ru ( ru + = (1 u u ru u + rv = ( γ + γ ( r ( ( ( ru + rv ( wqr i i = ( α + α ( r + ρ Yi Yi Yi wi ru + rv = ( D + D ( r + r (4 ρ where idies i (= 1,, 3 stad for fuel (F, oidat (OX, ad the rodut of ombustio. hese equatios are suffiiet to reset the evolutio of the flame. o obtai a aalytial solutio, the followig boudary oditios are eeded: d Whe = ad r u = u - = YOX YOX, =, YF = YF, Whe r= ad u/ r= u = u m (- / r m = =,, / r = YOX YOX, = YOX, m, YF / r = YF = Y F, m he suffi reresets the value ifiite he suffi reresets the value at the ozzle he suffi m reresets the value at etral ais.3 Solutio of the equatios (1-(4 Beause the distributio of veloity, temerature ad mass fratio i ay ross setio is similar, a Gaussia tye futio i Figure a be used to desribe suh a distributio, leadig to r = e[ K( ] (5 m where K is a oeffiiet betwee 8. ad 9.. o simlify the model, the followig double biases distributio is used to aroimate the Gaussia distributio r = 1, r b( (6 m b( A satisfatory solutio a be obtaied from the oservatio equatios (1-(4 after suh a aroimatio. (3

3 Figure. emerature distributio i a ross setio before ad after aroimatio Ideed, the iitial oditios of these oservatio equatios whe = ad r d / are give as follows: u/ u = u ( / = YF / r (7 YF = Y = F,, / YOX YOX, = YP, ( Y Y, / O the other had, whe r = ad, it a be show that u/ (- / = (8, / ( YP Y, / Based o the temerature distributio aroimatio ad the iitial oditios, a solutio a be obtaied from the oservatio equatios. For this urose, defie ( BF = + YF (9 Q F Beause of the similarity of the FD i eah ross setio, F( r, is defied i the followig to rereset the ratio of the value at the oordiate (, r to the oe at the ozzle. ( + YF BF QF F(, r = = (1 B (, F + YF Q F As suh, the FD a be formulated to give: YQ F F YQ F F (, r = [( ] F(, r (11 For turbulet flows, the defiitio of F(, r i (1 is give by 3 1 r 3 1 F(, r = (1+ 8 [1 (1+ 8 ] (1 d 3 d d Whilst for the flamelet flows, F(, r i (1 is give by: 8 1 r 8 1 F(, r = (1 + [1 (1 + ] Re d 3 d Re d (13 ud Re = γ From (13 it a be see that the otrol iut u otrols the value of F(,r. Usig (11, it a be see that u diretly otrols the FD (i.e., (, r. As suh, model (1 (13 a be regarded as a stati model for the flame system. Fuel jettig Veloity u Flame emerature Distributio Model FD (,r Figure 3. Simulatio theory demostratio I the simulatio, the iut is the fuel jet veloity u ad the outut is the FD, amely (, r. he roess arameters are seleted as follows: Fuel CO = 78.6 J / Kg K Q f = 8.84 KJ / mo l d =.1m u = 1m/s -5 γ= = 5K K is the absolute thermometri sale C=73.15K l (14 Usig model (1-(13, differet FDs a be obtaied whe u hages. If differet temerature values are rereseted by differet olour, the a oloured temerature distributio image a be obtaied as show i Figure 4. I Figure 4, the height of the simulated flame is 8m, the width is 3m ad the highest temerature is K whe the CO jet veloity is u = 1m/s. Beause the Reyolds umber of CO is betwee 48 ad 5, the flamelet flame hages to a turbulet flame whe u 17.7m/s, leadig to the followig arameters used to simulate the turbulet flame FD as show i Figure 5. Fuel CO = 78.6 J / K g K Q f = 8.84 K J / m o d =.1m u = 3m /s =.18 = 5K (15.4 Simulatio of the FD model Based o the solutios (1-(13, a FD model is a be ostruted as show i Figure 3. Figure 4. FD of Flamelet flame Figure 5.FD of the turbulet flame

4 3 System modellig ad otrol he roess to be otrolled is the stati jettig flame model as show i (1-(13, where the outut is the flame temerature distributio, whih is a D distributio alog (, r diretios (see Figure 1. However, sie the flame is symmetri, aother distributio, amely the Flame Eergy Distributio, a be used as a feedbak for the losed-loo otrol. Physially, the FED is the sum of all the temerature values i eah horizotal ross setio ad is therefore alulated from + (, r dr γ ( u, ( k = + + (16 d (, r dr Suh a γ(, u (k, that orresods to the (, r as show i the left had side of Figure 6, is dislayed i the right had side of Figure 6. I this ase, the iut is the fuel jet veloity u( k [,] (where k stads for the urret samle time. As the FD a be retrieved from γ(, u (k usig the symmetrial ature of the flames, the aim of the otrol desig is to make γ(, u (k as lose as ossible to a give flame eergy distributio. Figure 6. FD ad its orresodig FED As γ(, u (k is a ratio of the eergy i eah ross setio to the total eergy as show i (16, it a also be regarded as a robability desity futio, where the stohasti distributio theory [11] a be diretly alied to otrol the shae of γ(, u (k. A losed-loo flame distributio otrol system a be established as show i Figure 7. slies FD model a be obtaied i the same way as desribed by Wag i ([11]. γ (, u ( k = C ( V ( u ( k + L ( (17 B ( B1( B 1( d, B ( d ( B B( B ( d, 1( 1 C ( = B ( d (18, B ( B 1( B 1( d B ( d where B i ( (i = 1,,, -1 are a set of re-seified B- slie basis futio. ( 1 1 Vu ( ( k [ wu (, w( u,, w ( u] = (19 1 k k 1 k L ( = ( B( d B( Where w i (uk (i = 1,,, -1 are the weights of the B- slie basis futio. he erformae futio should be J ( u( k = ( C( V( u( k + L( g( d ( where g( is a give target FED futio. Deote Σ= C ( C( d ( 1 ( 1 ( ( ( ( η = g L C d 1 ( 1 ( g ( L( d (1 γ = the the otrol iut a be alulated from V( u u( k = u( k 1 µ ( V ( u Σ η u = u( k 1 u ( where µ > is a re-seified ste legth. 4 Simulatio results ad solutios Forty d order basis futios are used to aroimate the flame eergy distributio urve. For oe iut a geerate oe FED urve, the give FED urve is determied by a give iut ug. Let µ =.1 i equatio (. Figure 8 shows the resose of the otrol iut as alulated from (, ad Figure 9 gives the resose of the erformae futio i (. he 3D lots i Figures 1-11 show how the losedloo otrol a be realized so as to otrol the distributio of FED towards its target distributio. + - Cotroller Feedbak Flame model Figure 7. Close-loo otrol system Whe u hages from to, the FED urves a be obtaied from the stati model ((1 (16. Usig the FD data geerated from the stati model, the followig equivalet B- Figure 8. he resose of the otrol iut sequee

5 Akowledgemet he work is fuded by the UK Leverhulme rust (F/38/D ad the Chiese NSF (61833, these are gratefully akowledged. Referees Figure 9. he resose of the erformae futio Figure 1. he losed loo resose of the Flame Eergy Distributio 5 Colusios I this aer a stati model of flames distributio systems is derived ad used for the losed-loo otrol desig. Uder the assumtio that the flame is symmetri with reset to the vertial ais, the flame eergy distributio futio is used to realise the otrol of the flames temerature distributio. As suh a distributio a ow be measured through video amera ad image roessig, the roosed otrol a be used to ostrut a effetive losed-loo otrol for the flames. A simulated eamle has bee give where desired results have bee obtaied. [1] Q. Fa. Power Plat Boiler Faility ad Oeratio, Chia Eletri Power Press, Beijig, Chia, (1. [] F. Hua. Boiler Combustio heory ad Aliatio, Shaghai Jiao og Uiversity Press, Shaghai, Chia, (1999. [3] C. Ha. Pulverized Coal Combustio, Siee Press, Beijig, Chia, (1. [4] R.W. Bilger., Future rogress i turbulet ombustio researh, Progress of Eergy Combustio Si. 6, , (. [5] S. B. Poe. PDF methods for turbulet reative flows, Prog. Eergy Combustio Si., 11, , (1985. [6] A.Y. Klimeko, R. W. Bilger. Coditioal Momet Closure for urbulet Combustio, Prog. Eergy Combustio Si., 5, , (1999. [7] W. Fu, Combustio heory, Higher Eduatio Press, Beijig, Chia, (1989. [8] J.. We, H. Burkhardt. Aliatio of Fuzzy Logi ad Neural Networks to the Cotrol of a Flame Proess, Pro. of the Seod Iteratioal Coferee o Itelliget Systems Egieerig,.35-4, (1994. [9] H. Zhou. Model Establishmet of Fuel Cotrolled Objetive i Utility Boilers Based o Sigal of Radiative Eergy from Furaes ad Simulatio o Its Cotrol, Proeedigs of the CSEE16, 4,.6-9, (1996. [1] J. Wag, J. Du, S. Wag. Study ad Aliatio of Eergy Balae Method ad O-lie hermal Effiiey, Eletri Power, 1,.38-41, (1995. [11] H. Wag. Bouded Dyami Stohasti Distributios: Modellig ad Cotrol, Sriger-Verlag Ltd, Lodo, UK, (. [1] H. Yue, H. Wag. Reet develomets i stohasti distributio otrol: a review, Measuremet ad Cotrol, 36,.9-15, (3. Figure 11 he estimated Flame Eergy Distributio Usig B- slie Model.