Suggested solutios TEP470 Heat ad combustio techology 5 May 06 by Ivar S Ertesvåg Revised 3 May 06 ) Itroduce the Reyolds decompositio: ui ui u i (mea ad fluctuatio) ito the give equatio Average the equatio Apply the rules for averagig (obtaiable from the defiitio of averagig): a a a b a b ; a ' 0; ; a b a b ; a b a b t t here a ad b are variables (also products of variables) p ( u i u i) ( uiu j uiuj u iu j u iuj ) ij fi t x x x j i j Note: i priciple, the desity ρ ca be decomposed This is ot required here Desity ca be assumed costat, or (more precise) ot to correlate ith the velocity compoets The body-force acceleratio f i is usually costat (gravity) The terms icludig the average of oe fluctuatig compoet ill be zero, eg u u u u 0 i j i j The other terms have to be ept It is customary (but, strictly, ot required) to move the to-fluctuatio term to the RHS: p ( ui ) ( uiu j ) ij u iu j fi t x x x j i j Note : The pressure ca be decomposed ito mea ad fluctuatio I this equatio this is ot ecessary, sice the pressure appears i the gradiet as a term o its o The average pressure appears he this term is treated accordig to the procedure The average behaviour is maitaied, hile much of the iformatio of the istataeous variatio of the velocity ad pressure (ad desity) is lost Some effects of fluctuatios o the mea flo are retaied i the Reyolds stresses The Reyolds stresses origiates from the covective term ad expresses mea mometum trasfer due to turbulet motios (fluctuatig velocities) See from the mea flo, this is a diffusive effect (ie stresses he the matter is mometum) The effect is lie ehacig the stresses of the flo ) The Reyolds stresses are i geeral expressed from u u i j u l u iu j t ij ij, here t C x j xi 3 x l 3 is the turbulece viscosity, is the turbulece eergy ; is the dissipatio rate of turbulece eergy t (si term of the equatio); C is a modellig costat; ij is = for i j, =0 for i j The Reyolds fluxes of species mass ca be expressed from Y t Y Y u j Dt, x x j Y j here Y is the mass fractio of species ; Schmidt umber (usually set as costat) Dt is the turbulece mass diffusivity; Y is the turbulece
3) u Defiitios: u ; / u u / xu ; x ; here u is the streamise mea velocity of a boudary layer flo; x is the coordiate ormal to the all; is the shear stress at the all; is the desity of the fluid; is the iematic viscosity of the fluid; u is the shear velocity Near a all, gradiets parallel to the all are much smaller tha those ormal to the all, ad velocities approaches zero, so that the mea mometum equatio simplifies to 0 d uu dx, or uu cost The costat has to equal the all shear stress, du Very close to the all, viscous stress domiates; thus ( uu ) ad dx dx The expressio ca be rearraged to du dx u or u ux d d, assumig that the u viscosity is costat This ca be ritte as du dx, ad itegrated: u x C Sice u( x 0) 0, the itegratio costat C is zero, ad e have obtaied the first part of the give expressio: u x du For somehat larger values of x, experimets sho that dx u, here is a costat x du u dx d( xu / ) dx Rearragig this gives d or du u u x ( xu / ) x Itegratig this gives u l x C, hich is the secod part of the give expressio ad C are empirical costats The practical use is as a all-fuctio boudary coditio for velocity i CFD Avoid very fie resolutio ear a all 4) A coserved scalar is a property ith zero source (or si) term Examples: The elemetal mass fractios; the ethalpy if all source terms ca be eglected (radiatio, pressure effects, body-force effects, viscous dissipatio etc); combiatios of mass fractios that give zero source: Yfu Yox r ; Yfu Ypr r ad r Yox Ypr, he the reactio balace is assumed as r g fu + r g ox ( r) g pr I priciple, the mixture fractio (belo) is also a coserved scalar
mix Mixture fractio, defiitio: ; here is a coserved scalar (other tha the mixture fractio itself), ad are its values at to differet iflos, ad mix its value i the mixture The mixture fractio of a stoichiometric mixture of propae ad air (% O, 79 % N, molar): Alterative I: Use the coserved scalar Yfu Y ox r ; ilet is : Yfu, ; Ilet is air (oxidizer): Yox, r r ; stoichiometric mixture meas Y Y r ad thus 0 fu ox mix 079 079 m 5MO 0 MN 5 3 ox 0 8 The stoichiometric amout of air (mass based): r 56 m M 44 0 r Combied: st 0060 r r 56 Alterative II: Use the oledge that the mixture fractio is the ratio of the mass from ilet to the total mass (+air) Thus, for stoichiometric coditios: st ( r), ith further calculatios as i Alterative I 5) d[oh] f[o][ho] r[oh] f3[co][oh] r3[co ][H] f4[h][o ] r4[oh][o] dt I Reactio forard, the reactats are stable species All the other reactios (icludig Reactio reverse) ivolve a radical as reactat, hich is very reactive ad these reactios are liely to be fast Thus Reactio forard ca be suspected to be sloer 6) Borghi diagram, premixed fu C3H8 u' u Thic amiar u ' log( ) u ReT Wriled cost Da Da K ' log( ) / u ' : turbulece velocity ( ) ' : turbulece legth scale u : lamiar flame speed : lamiar flame thicess Da c : Damöhler umber Da : Damöhler umber K c c u : chemical timescale ' u' : large turbulece timescale : Kolmogorov timescale Re ( ) u' '/ : turbulece Reyolds umber T 3
7) Mai parts of EDC ad purposes: - The cascade model: Provide a li betee the fie structures (small scales), here reactios occur, ad the large scales, hich are represeted by a turbulece model It express characteristic quatities of the fie structures i terms of quatities obtaied by the turbulece model - The reactor model: Provide the reactio rates for fie structures ad the mea reactio rates EDC provides the source term for the averaged species mass balaces For CFD (here restricted to solutios of averaged basic equatios, aa RANS), the folloig is eeded i additio: - Turbulece model (provide the Reyolds stresses ad a turbulece time scale; if relevat, a turbulece viscosity) - A model for turbulece diffusio of scalars(the Reyolds fluxes of scalars) - Radiatio model (if radiatio heat trasfer is importat) - Costitutive las (Fourier s la, Fic s la, Neto s viscous stress relatio) ad thermal properties (coductivity, diffusivity, viscosity) - Relatios betee ethalpy (ad/or specific heat) ad temperature - A chemical mechaism (ca be simple or complex) 8) The expressio is based o the reactor model of EDC Put up the mass balace for a species: M M Y M Y R o FS i out Assume that the reactor is steady state, Mi Mout M Itroduce m M M FS, ad rearrage to R o m Y Y M FS / M tot is the ratio of mass i fie structures to the total mass Assume that all reactios tae place i the fie structures ; sice 4 R is reactios per volume MFS Mtot (=mass/desity) ad time, this meas R R R This is the case if all the fie structure reacts Sice this may ot be the case, the reactios ca tae place i a certai fractio of the fie structures, hece R R With the expressio for R itroduced, this gives R o m Y Y Usig the symbol m m obtaied: R ( Y ) Y m ad the mass-fractios relatio give i the problem, the requested relatio is 9) The overall (global) reactio balace ca be ritte as: CH (O 376N ) CO H O+( )O 376N NOx 4 NOx
The amout of NOx is very small, thus the assumptio NOx ca simplify some calculatios The amout of substace of flue gas icludig H O ( et ) per mol of (eglectig NOx): et ( ) 75 95 O The (actual) molar fractio of O (et): ( ) 03 et 95 (0395) 03 79 et This is the actual air excess ratio, hich also gives 756 dry Excludig the H O ( dry ): ( ) 75 95 9579 556 dry et HO Alteratively: 756 556 The cotet (molar fractio or volume fractio) of NOx at dry coditios NOx NOx et 6 756 6 00 60 6 ppm 556 dry et dry The regulatios ( referece ) are specified for O,ref et,ref ref ( 05 95) 05 ref 376 This is the referece air excess ratio, hich gives et,ref ( ref ) 95 ref 05 95376 3680 The corrected or referece molar fractio of NOx becomes NOx NOx et 6 756 6 X NOx 00 500 50 ppm dry 3680 et,ref et et,ref The emissio idex: m M M 30 g 6 g NO NO NO NO et NO 6 3 EINO 00 756 03 0 mfu M et M Mass of NO per loer heatig value: mno mno 3 g g 6 g EINO 030 06 0 H m h h g 50 MJ MJ HV HV 0) Exhaust gas recirculatio - Reduce oxyge cotet (mole fractio) - Icrease specific heatig capacity cp of flue gas loer temperature (less thermal NOx) - May reduce cocetratio (less Feimore/prompt NOx) - May reduce temperature fluctuatios, ie pea temperatures 5