16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 6: Heat Conduction: Thermal Stresses

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1 16.512, okt Proulon Prof. Manul Martnz-Sanhz Ltur 6: Hat Conduton: Thrmal Str Efft of Sold or Lqud Partl n Nozzl Flow An u n hhly alumnzd old rokt motor. 3 2Al + O 2 Al 2 O 2 3 m C, b C In modrn formulaton, wth 20% Al by ma, th Al2O3 ma fraton of th haut an b 35-40%. Th matral do not and, o thr mut b a lo n t vloty, hn n I. m Aum ma flow (a) m (old), non-onvrtn. Th momntum quaton m du + m du + Ad 0 Call th (ma of old)/(volum) (not th dnty of th old, thory) u du + u du + d 0 Dfn a ma flu funton m m + u u + u m u du + du + d 0 udu d udu Th nry quaton mlarly, ( )( ) ( ) 1 dt + u du + dt + u du , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 1 of 10

2 Subttut hr udu from abov: d dt u du + dt + u du 0 ( ) d ( ) dt + dt u u du 1 + wth no artl (0), th v d P T T dt P P 0 T0 1 Wth artl, w nd to know th htory of th vloty l u u and of th tmratur l T T. Th a dffult roblm, rqurn dtald modln of th moton and hatn/ooln of th artl. But w an look at th trm a aly. (a) Vry Small Partl ood ontat. For ub-mro artl (not a bad rrntaton of ralty), w an ay that u u u, T T T. Thn d + dt Not that th man f hat ( and ar r unt ma) ( ) ( ) 1 + v ( )( v) ( ) v v + and alo 1 So that d dt d d dt T dt P P T and dfnn an fftv by th uual, v P P 1 T T , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 2 of 10

3 Th quaton of moton now ( + ) udu + d 0 Or udu + d 0 P P T 1 T ( ) P udu + d 0 T From th two bod quaton w that vrythn from hr an rod a f th a wr ml, but wth molular ma M M 1 (or ( ) ), and wth ( ) +. For aml, P u 2 T 1 P 1 T,P n hambr t. For ntvty analy t may b of ntrt to lnarz th for albra tdou, but on t, 1. Th u 1 1+ u 2 0 ( η ) ln( η ) η , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 3 of 10

4 P wth, η 0 P and, of our, 1 P u 2 0 T 1 1 P 1 W from th that f < 1 ( <, whh ommon), thn u < u (and v-vra). For a numral aml, look at Problm 2 (attahd) 0 (b) Vry Lar Partl Hard to quantfy, but robably for damtr > 100 µ m or o, th artl hav too muh nrta (and thrmal nrta) to follow th a alraton and ooln. W thn hav du du ; T T ( at hambr) T du 0; dt 0 or turnn to th d dt d quaton, t now look a f thr wr no artl: (.., artl jut do not artat n th dynam or n th thrmal 1 P T balan). So, w tll hav. Th do not man zro P0 T0 rforman fft, thouh. W do not t th full a t vloty 1 P u 2 T 1 P but th artulat do not ontrbut to thrut, bau thy t at u u : mu + mu I ( ) I m + m , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 4 of 10

5 Th atually mor lo than n th mall artl a (about tw a muh, dndn on ). From th aml, th a rou lo n old rokt. Crtron for Sl π πµ ( ) τ µ µ du du m 6 u u u u dt 9 dt 9 du u u dt τ all u u u u du d d du + dt dt τ dt τ dt Say τ du and dt t τ a ar ontant a τ + C ( 0) 0 C aτ 2 ε 1 1 ε ε ε 2 aτ t τ and u a t u t τ τ t t τ t τ 1 t 2 τ... 1 t τ , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 5 of 10

6 So, mall l for t τ L u τ L 2 u 9 µ 2 9 µ L 2 u Say 5 µ 3 10 K/m/ L 0.3m K/m u m/ m 3µ m So, 3µ m no l 3µ m full la , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 6 of 10

7 Problm Problm 2 A notd n la, th fft of arryn a ma fraton of fn old artl n th andn a n a rokt nozzl an b aountd for by un an avra f hat rato ( 1 ) ( 1 ) + + v and an avra molular ma M M 1 For Al O th hh tmratur f hat 1260 J/K/K. 2 3 Condr a old rokt wth ( ) , P P 0.01, M 18 / mol. For a 20% alumnum loadn n th rollant, of th ordr of 37%. Calulat th mathd f mul of th rokt and omar to what t would b for th am T 3300 K, but wth no artl , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 7 of 10

8 Problm 2 Sf hat of lan a 1.25 r J/K /K r 1 M v J / K / K r Th f hat of th old (or lqud) Al2O J / K / K. Th avra f hat rato thn 2309 ( ) + ( ) r ( 1 ) v + ( ) And th avra molular ma (M ) M 18 M /mol 0.37 P Th t d for 0.01 P and T 3300 K thn 1 Q P u 2 T m / 1 M P 2521 NOTE: Altrnatvly, and ar to do, you an u P u 2C T P 1 wth ( ) C J/K/K (o M not rally ndd) , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 8 of 10

9 A a hk, M J/K/K a t hould. Undr rur-mathd ondton, thr no t rur ontrbuton to thrut or, and hn I 2613 I Wthout artulat but wth th am P,P and T, w would obtan 1 2 Q P u T 3199 m/ 0 1 M P and I Thr thrfor a lo of % % n I It ntrtn to tt th auray of th lnar aromaton vn n la for mall : ( η ) ln( η ) u P η ; η 0 1 u P 1 u W fnd η , and thn u (16.3% lo) (not too dffrnt, dt lar ) , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 9 of 10

10 NOTE: Altrnatvly, and ar to do, you an u P u 2C T P 1 wth ( ) C J/K/K (So M not rally ndd) A a hk, M J/K/K a t hould , okt Proulon Ltur 6 Prof. Manul Martnz-Sanhz Pa 10 of 10

Equil. Properties of Reacting Gas Mixtures. So far have looked at Statistical Mechanics results for a single (pure) perfect gas

Equil. Properties of Reacting Gas Mixtures. So far have looked at Statistical Mechanics results for a single (pure) perfect gas Shool of roa Engnrng Equl. Prort of Ratng Ga Mxtur So far hav lookd at Stattal Mhan rult for a ngl (ur) rft ga hown how to gt ga rort (,, h, v,,, ) from artton funton () For nonratng rft ga mxtur, gt mxtur