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Essay10 - ME 5446 Es say 10 COMBUSTION IN A CONSIiLVT-PRE...

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Unformatted text preview: ME 5446 Es say 10 COMBUSTION IN A CONSIiLVT-PRE SSL'RE REACIOR A constant-pressure reactor is pictured at the right. The reaction occurs in a space bounded above by a freely mos-'ing piston. The ambient air and the weight of the piston, taken together. exert a Free”- * a. moving constant pressure on the reaction zone. There is a spark plug situated % Wm” . . . . . N at the top of the combustion chamber that is used to initiate the E at." Reaction . . N. Zone chemical reaction. *2 s § I I I I . \\\\“Q The reaction to be investigated is nA+£iB—>cC+o'D [1) ts-‘hereA, B, C, and D are chemical compounds or elements, and a, b, c. and r3 are stoichiometric coefficients. Prior to the initiation of the reaction, the reaction space contains (rig), and (rig), moles of :f. aiidB. respectively. [n addition, the initial. pre-comloustion s-‘olume is If , and the initial temperature is I, . The mass in of the contents of the reaction zone is constant. Heat :2 is exchanged between the reaction zone and the space outside of the reactor. The energy transfers are gos-‘enied the first 1mm of thennodynamics. If the piston motion is very slow, the rate at which work is done by the gaseous mixture on the piston is ,mfici‘l’fdt) . With this, the first law can be written as d—E=Q-Tf’=Q-ntdl’i’a'o [2) (1 From the definition of the enthalpy, it = h— pt’, and ii = L'lilm . With these, Eq. (3] becomes an [33. arr—d! pair—m ' ri'r so that, n'i'r _ E _ [4} E lit:- Ihe enthalpy I? of the mixture is the sum of the enthalpies of the component compounds, so that . 1 l — _ r'?=;Zm_.h_ +5214}; [3) and after differentiation there results _ _ i' fi=iTnfl+iTam is) 0’: mf' ' art in?l ' (it The next step is to eliminate the MI in favor of El _ which gives, after noting that HI = PE, dial.- _ V nip—JI- _ JV — _ _ "— ? n'r r1": ' air [ 2' When Eqs. (6] and are substituted into the first law, Eq. [4). there follows all—r —. (fir—1; _ n'V O = T I".- —' T I’ ' — -— 3 i f p. ff: .K d! a d? E) Furthermore, since the enthalpy is a unique function of temperature, it is advantageous to write (a? GE. a'r arr —=—— =? _-— 9 n'r d1" dr P" (fir i 2' With this. Eq. (8] becomes I [WET __ .— firllli—j (fl? _—_ T — —V r' . ' —"_— ' .— I'l’ ‘9‘? r +pl‘?.-'+V;”£ fir! Igr? Enrpr— a." R (In) [t is appropriate to take stock of the unknowns that appear in E4]. (JO). In this regard, it may be noted that and ER..- are functions of temperature. Therefore, the unlmowns in Eq. [ID] are: Ht], Wt), and EU] . Since r' = A. B, C, and B: there are six tutlmowris To find additional equations to match the number of mimosa-us, use can be made of the reaction rate equations and of the ideal gas law. For the latter. if the molar form is used: [\J PER? {11) Since the pressure is constant. it follows that. ___ _ _ __"£"L _ :0 — (9.4 + .03 + pa + pa )— E" I {12) and. after differentiation. ME. : 0'53 : art—ac + dfis‘tzt’ p: 1 df {m L d: dr (1’! (it ,t' LEI-x T1 at, ' It only remains to write the reaction rate equations for A. B. C. and D. They are (fat—9.4 , — a — w? {a}?! — a'— .fi' (if =-m = -ak(f}(p_tl [pal =-rIAe (to) {pg} {14) (f— _.. _ . ,_ - fi = am. “Matt‘th t a 5] dr (1'— _ . fiwe “Vomit—73f (161 (it L2” = dAe‘Ecmwoit {m There are six governing equations. Eqs. [10). (13). and [Isl-1?). to match the six unknowns. These equations are to be solved numerically subjected to six initial conditions. which are V=V[O). I=T{0]. fiat—alto} (LS) The nutnerical solution proceeds as follows: Step 1: ‘With the known initial conditions. the slopes art—)3 [alt . ($153 / df. dfic fdt . and (fit—JD [alt can be determined from Eqs. {14-1 T"). Step '2: The output of step 1 is then used in conjunction with Eq. (13) to find dT/a't . Step 3: Then. (fl/fa?! is obtained from Eq. (10). Step 4: A time step At is chosen such that t1 = O + At . Step 5: Calculate the values off. V. ,5}. . at II from T61] = + Mir/(£050 ~ V0?) = _ (dV/df)..<.- 5.103] = )t +(dEB/derfi‘ {1-9) Step 6: Repeat steps 1 through 3 by using the values of T._ V, and . Step 7': Continue this marching procedure. ...
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Essay10 - ME 5446 Es say 10 COMBUSTION IN A CONSIiLVT-PRE...

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