Summer 23 Lecture 5 Stochometry Dmensonless Parameters Stochometry Gven the reacton M dn or n terms of the partal masses dm ( )W N X dn j j M we have seen that there s a relaton between the change n the number of moles of the speces;.e., for any two speces and j ( j j dm j j )W j Snce the total of mass (unlke the total number of moles) n the system s unchanged by chemcal reacton, we also have ( dy )W ( j dy j j )W j Unversty of Illnos at Urbana- Champagn
Summer 23 Consder a global reacton descrbng the combuston of a sngle fuel for example, the combuston of a hydrocarbon fuel C m H n F C mh n + O O 2! CO 2 CO 2 + H 2 O H 2O wth the stochometrc coe cents F, O m + n/4, CO 2 m, H n/2 2O where F was taken equal to one, arbtrarly. whch may be wrtten as (prmes are unnecessary) F Fuel + O Oxdzer! Products then O/ F s the rato of the stochometrc coe cents, and dy F dy O OW O Integratng dy F dy O OW O between the ntal unburned state (subscrpt u) and a later state Y F Y O Y Ou OW O A fuel-ar mxture s referred to as a stochometrc mxture, f the fuel-to-oxygen rato s such that both reactants are entrely consumed;.e. when combuston to CO 2 and H 2 O s completed. Y Ou st O W O mass-weghted stochometrc rato Unversty of Illnos at Urbana- Champagn 2
Summer 23 In general, the ntal state may not be at stochometry. A measure of the departure from stochometry s gven by the equvalent rato /Y Ou /Y Ou /Y Ou st / O W O Y Ou < < stochometrc mxture < lean mxture (n fuel) >rchmxture(nfuel) for combuston n ar, the equvalence rato s often expressed as the fuel-to-ar rato F/Ar F/Ar st Adabatc Flame Temperature If a gven combustble mxture s made to approach chemcal equlbrum by means of an sobarc, adabatc process, then the temperature attaned by the system s the adabatc flame temperature T a. For an adabatc, sobarc process dh. Integratng from the unburned to the burned state h u h b Y u h Y b h Z Tu Y u h o + c pu dt T o Z Tb Y b h o + c pb dt T o Unversty of Illnos at Urbana- Champagn 3
Summer 23 Y u Y b h o Z Tb T o c pb dt Z Tu T o c pu dt where the specfc heats are those of the mxture, calculated wth the mass fractons of the unburned/burned gas, respectvely c pu Y u c p (T ), c pb Y b c p (T ) For a one-step global reacton, where we consder j to be the fuel, denoted by F, the equaton dy dy j ( )W ( j j )W j ( dy )W dy F F W F can be ntegrated to gve ( Y u Y b Y F b )W Usng ths last relaton n the conservaton of energy equaton: Y F b Y u Y b h o {z } ( k )W h o {z } Q Z Tb T o c pb dt Z Tu T o c pu dt Z Tb T o c pb dt Z Tu T o c pu dt Y F b If the reference temperature T o T u, for complete combuston of fuel (Y F b ) the adabatc flame temperature T a s calculated from Q Z Ta T u c pb dt Q Unversty of Illnos at Urbana- Champagn 4
Summer 23 It s convenent to rewrte ths relaton usng the the molar heat capactes C p. Z Ta T u c pb dt Q ) Z Ta T u C pb dt Q Example: CH 4 + 5 (.2 O 2 +.79 N 2 )! b CO 2 + b 2 H 2 O+b 3 N 2 + b 4 O 2 atom conservaton ) b b 2 2 b 3.85 b 4.5 Z Ta T u C pb dt Q Let T u 298 K Q h o CH 4 h o CO 2 2h o H 2O 9.755 kcal Z Ta use an teratve procedure T u [C p +2C p2 +.85C p3 +.5C p4 ] dt 9.755 kcal for T a 2 K, LHS 23.94 for T a 7 K, LHS 87.9 ) T a 732 K ths s not accurate because we dd not account for product dssocaton. Wth product dssocaton CH 4 + 5 (.2 O 2 +.79 N 2 )! CO 2,H 2 O, N 2,O 2, NO, H, OH, O, N, CO, O 3, NO +,etc.. and the determnaton of the fnal compostons requres, n addton to the atom conservaton equatons, chemcal equlbrum equatons. CO 2 H 2 O 732. CO 2 H 2 O CO 73.9 CO 2 H 2 O CO O 73.8 CO 2 H 2 O CO O N NO 727.3 CO 2 H 2 O CO O N NO, H, OH 725.7 The ncluson of other products such as O 3,CH 3, HO 2,CH 2 O, C 2 H 6, NO 2, HCO etc., dd not change the adabatc flame temperature. T a Unversty of Illnos at Urbana- Champagn 5
Summer 23 Smplfed expresson For complete combuston of fuel (Y F b ) the adabatc flame temperature T a s calculated from Z Ta T u c pb dt Q Assume c p s nearly constant, the adabatc temperature for a lean mxture, where the fuel s totally consumed (Y F b ) s T a T u + (Q/c p) For a rch mxture, the oxdzer s totally consumed (Y O b n an analogous way ), and we obtan T a T u + (Q/c p)y O u O W O The two expressons are dentcal when TABLE.2 Approxmate Flame Temperatures of Varous Stochometrc Mxtures, Intal Temperature 298 K Fuel Oxdzer Pressure (atm) Temperature (K) Acetylene Ar 26 a Acetylene Oxygen 34 b Carbon monoxde Ar 24 Carbon monoxde Oxygen 322 Heptane Ar 229 Heptane Oxygen 3 Hydrogen Ar 24 Hydrogen Oxygen 38 Methane Ar 22 Methane Ar 2 227 Methane Oxygen 33 Methane Oxygen 2 346 a Ths maxmum exsts at φ.3. b Ths maxmum exsts at φ.7. lean rch Glassman & Yetter, 28 Unversty of Illnos at Urbana- Champagn 6
Summer 23 Law, 26 Constant volume equlbrum temperature If a gven combustble mxture s made to approach chemcal equlbrum n an sochorc (constant volume), adabatc process, then the temperature attaned by the system T b s hgher than the adabatc flame temperature. e u e b If the reference temperature T o T u, for complete combuston of fuel (Y F b ) the adabatc flame temperature T a s calculated from Z Tb T o c pb dt Assumng constant specfc heats Q + R W (T b T u ) and T b s clearly larger than the adabatc flame temperature T a. T b T u + (Q/c v), T b T u + (Q/c v)y O u O W O for lean/rch mxtures Unversty of Illnos at Urbana- Champagn 7
Summer 23 Smplfed Model and Dmensonless Equatons In the followng the chemstry wll be represented by a global one-step (rreversble) reacton F Fuel + O Oxdzer! Products descrbng the combuston of a sngle fuel. The reacton wll be assumed of order n F,n O, wth respect to the fuel/oxdzer, and an overall order n n F + n O. The reacton rate wll be assumed to obey an Arrhenus law nf no YF YO! B e E/RT W F wth an overall actvaton energy E and a pre-exponental factor B (treated as constant). W O Unversty of Illnos at Urbana- Champagn 8
Summer 23 For smplcty, we wll treat µ,, D constants (although some of the theoretcal development could accommodate temperature-dependent transport wthout much d culty) so that Dv t + v p + µ 2 v + 3 ( v) + g DY D r 2 Y W!, F, O c p DT r 2 T Dp + p RT/W + Q nf no YF YO! B e E/RT W F W O Characterstc values: Non-dmensonal Equatons the fresh unburned state p,,t (satsfyng p RT /W ) for p,, T a characterstc speed v to be specfed the d uson length l D D th /v for dstances, where D th mxture thermal d usvty / c p s the the d uson tme l D /v for t Ths choce s clearly not unque and there may be other length, tme, and velocty scales that, for a gven problem, could be more relevant. We wll use the same varables for the dmensonless quanttes;.e. after substtutng ṽ v/v say, we remove the. Unversty of Illnos at Urbana- Champagn 9
Summer 23 + t Dv p + Pr 2 v + 3 ( v) + Fr 2 M DYF DYO DT v LeF r2 YF eg LeO r2 YO r2 T Dp + Pr M 2 +q! p T! D n YFnF YOnO e /T Moshe Matalon Dmensonless Parameters M p Pr v Mach number p / µcp Fr Prandtl number heat release parameter Q/ F WF c p T Froude number / cp D Le Note that Pr v ld /µ Re where Re s the Reynolds number based on the d uson length. The Reynolds number based on a hydrodynamc length L wll be large because typcally L ld. q v2 /ld g Lews number of speces E/RT actvaton energy parameter O WO F WF mass weghted stochometrc coe. numerator s the heat release per unt mass of fuel D (ld /v ) [( /WF ) n F ( /WO ) n O F B] flow tme reacton tme Damko hler number Note: the unts of F B, lke the unts of k s [(conc)n set F n wrtng the chemcal reacton equaton). Moshe Matalon Moshe Matalon Unversty of Illnos at Urbana- Champagn (tme)] (one usually
Summer 23 Low Mach Number Approxmaton The propagaton speed of ordnary deflagraton waves s n the range - cm/s, namely much smaller than the speed of sound (n ar a 34, cm/s). M momentum equaton ) rp p P (t)+ M 2 ˆp(x,t)+ wll all other varables expressed as v + M 2ˆv + Dv rˆp +Pr r 2 v + 3 r(r v) +Fr e g DT r 2 T T P (t) dp dt + q! acoustc dsturbances travel nfntely fast, and are fltered out. Unless P (t) s specfed, we are mssng an equaton, snce p has been replaced by two varables P and ˆp. An equaton n bounded problems can be obtaned as follows: @( T ) @t Z V @T @t + v rt r2 T + r ( vt ) r 2 T dp dt dp dt dv on the surface S, v n and for adabatc condtons @T/@n. @ + r v /T @t r (P v rt )+q! Z S dp dt + q! dp dt + q! Z (P v rt ) n ds + q V dp dt!dv q V Z V!dV Unversty of Illnos at Urbana- Champagn
Summer 23 The low Mach number equatons are (wth the hat n p removed), therefore Dv DY F DY O DT t + v rp +Pr r 2 v + 3 r(r v) +Fr e g Le F r2 Y F Le O r2 Y O r 2 T T P dp dt + q! and when the underlyng pressure does not change n tme, P. Otherwse, unless t s specfed, we need an equaton for P (t) (thepressurep has been replaced by two varables P and ˆp), whch can be obtaned by a global ntegraton across the entre volume (as dscussed below). Couplng Functons For unty Lews numbers the operator on the left hand sde of these three equatons s the same. DY F DY O DT Le F r2 Y F Le O r2 Y O r 2 T q! The combnatons H F T + qy F and H O T + qy O / (and hence Y F satsfy reacton-free equatons Y O / ) DH r 2 H leavng only one equaton wth the hghly nonlnear reacton rate term. Ths s a great smplfcaton, but as we shall see, small varatons of the Lews numbers from one produce nstabltes and nontrval consequences. Unversty of Illnos at Urbana- Champagn 2
Summer 23 The constant-densty approxmaton Thermo-D usve model One must abandon the equaton of state ( @v +(v r)v) @t rp t + v +Pr r 2 v + 3 r(r v) +Fr e g solve to determne v T ( @Y F @t ( @Y O @t + v ry F ) Le F r2 Y F! + v ry O ) Le O r2 Y O! ( @T @t + v T ) r2 T q! wth the gven v solve for T,Y F,Y O. Can be derved systematcally by assumng that the heat release q Unversty of Illnos at Urbana- Champagn 3