Mole Fraton Temerature (K) Transort Shool of Aerosae Engneerng Equatons for Multomonent Flows (ontnue) Jerry Setzman 0.2 2500 0.15 2000 0.1 0.05 0 CH4 H2O HCO x 1000 Temerature Methane Flame 0 0.1 0.2 0.3 Dstane (m) 1500 1000 500 0 TransortEqns2-1 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston Shool of Aerosae Engneerng Summary So far storage routon flux(out) t storage+onv ffusve flux DY Sees Conservaton W YV Momentum Dv Y f P Conservaton D Mass Conservaton 0 v 0 v Energy??? sees ffuson veloty stress tensor (mom. ffuson) TransortEqns2-2 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston 1
Shool of Aerosae Engneerng Conservaton of Energy Statement of 1 st Law for oen systems s basally balane of storage an transort of energy arre by flu mass nternal energy e knet energy v 2 /2 otental energy, e.g., gravty (oul nstea treat as work term) transort of energy not through mass heat transfer, Q work, W e o Total Energy 1 e v v g r 2 TransortEqns2-3 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston Shool of Aerosae Engneerng Mehanal Energy Conservaton General statement of total energy onservaton De o Q W work ue to t t normal/shear stresses De Dv Q v g v v P t Last term v P v P P : v Dv from g momentum Cons. of mehanal energy Dv 2 v v P v g v v 2 e.g., v t t TransortEqns2-4 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston 2
Shool of Aerosae Engneerng Thermal Energy Conservaton Can show by searatng flow work terms v from stress tensor an ombnng art wth nternal energy to form enthaly Dh D q : v qr Pressure term usually neglgble for low see flows t Term n ( ) usually alle vsous ssaton onverson of bulk KE to thermal energy (ranom KE) usually neglgble for low see flows TransortEqns2-5 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. ffusve energy flux work ue to vsous stresses D net nut of raatve energy v AE/ME 6766 Combuston Shool of Aerosae Engneerng KE an Vsous Dssaton Examle magntues v (m/s) T=v 2 /2 (K) v/x (s -1 ) (N/m 2 ) T=v/v 0 0 0 0 0 10 0.05 10000 0.2 0.0002 30 0.45 30000 0.6 0.0006 50 1.25 50000 1.0 0.001 100 5.00 100000 2.0 0.002 200 20.0 200000 4.0 0.004 =1 kj/kgk e.g., x=1 mm =210-5 Ns/m 2 =1 kg/m 3 So tyally neglgble for subson ombuston TransortEqns2-6 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston 3
Shool of Aerosae Engneerng Energy (Moleular) Dffuson Three maor terms (normal) thermal ffuson (temerature graent) thermal/hem. energy arre by mass ffuson Dufour effet (thermal flux ue to on. gra.) usually neglgble q T h RT D W D T V V note hange of symbol from revous symbol k (onfuse wth rate onstants ffusve mass flux vetor TransortEqns2-7 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston TransortEqns2-8 Shool of Aerosae Engneerng Smlfe Energy Equaton Fous on enthaly Neglet KE, PE raaton vsous ssaton an any work but flow work Dufour effet Dh D q : v qr Dh D T h Dh D T h Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston 4
TransortEqns2-9 Shool of Aerosae Engneerng Unty Lews # Aroxmaton Reue further usng exresson for enthaly of mxture (erf. gas) h T hy mx Into (ffusve) heat flux vetor q T h q h h Y Relae wth effetve Le (normal ffuson only) Le D D h Y 1 DMY h Y Le 1 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. q h =0 f all Le =1 all sees have same mass ffusvty, an no net flux ue to hem/ fferenes AE/ME 6766 Combuston DM D Shool of Aerosae Engneerng Unty Le: Energy Conservaton For all Le =1, general equaton for enthaly reues Neglgble: KE, PE, raaton, vs. ss., external work Dh D q Neglgble: Dufour, Soret, Le=1 q h Dh 1 D h Note: no soure term n total h hemal h thermal h (for no external q, no work but flow work) TransortEqns2-10 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston 5
Shool of Aerosae Engneerng Temerature Equaton (Perf. Gas) Dh Total ervatve of enthaly, by efn. h T hy ffusve mx flux Dh DT 1 D W h q mx q Into energy equaton DT D T T W h mx omressve thermal ffuson of work onuton sens. enthaly by mass neglete vsous ssaton, Dufour, raaton TransortEqns2-11 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. hemal soure T hemal energy onverson ( soure ) AE/ME 6766 Combuston h Shool of Aerosae Engneerng Comarson to Sees Conserv. Can smlfy further f varatons small omare to other terms an ~small ressure hanges DT 1 h T T W TransortEqns2-12 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. small f,.e., nearly same T for all sees; 0 DT T W h Comare to sees eq. wth only normal ffuson DY 1 Le Y W equatons same for Le =1 (f roerly normalze) AE/ME 6766 Combuston 6
Shool of Aerosae Engneerng Lews Number Varatons Arox. values for varous sees n CH 4 -ar flame TransortEqns2-13 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. Le Le CO 2 1.39 HCO 1.27 O 2 1.11 CO 1.10 H 2 O 0.83 OH 0.73 CH 4 0.97 O 0.70 H 2 0.30 H 0.18 ref: Smooke an Govanggl, Leture Notes n Physs 384,. 29, Srnger-Verlag Berln 1992. Le=1 assumton ~okay exet for lght sees (H, H 2 ) an ossbly heavest AE/ME 6766 Combuston Shool of Aerosae Engneerng Shvab-Zelovh Formulaton Orgnal formulaton showng equaton smlarty for Le=1 assumes steay, no boy fores, normal ffuson, negl. raaton an vsous ssaton, one D to esrbe all mass ffuson Energy equaton o vh h h W sens sens Sees equaton vy DY W Smlar for Le=1 (=D), f TransortEqns2-14 Coyrght 2004-2005 by Jerry M. Setzman. All rghts reserve. AE/ME 6766 Combuston 7