Chapter 2 Thermodynamcs of Combuston 2.1 Propertes of Mxtures The thermal propertes of a pure substance are descrbed by quanttes ncludng nternal energy, u, enthalpy, h, specfc heat, c p, etc. Combuston systems consst of many dfferent gases, so the thermodynamc propertes of a mxture result from a combnaton of the propertes of all of the ndvdual gas speces. The deal gas law s assumed for gaseous mxtures, allowng the deal gas relatons to be appled to each gas component. Startng wth a mxture of K dfferent gases, the total mass, m, of the system s m ¼ XK m ; (2.1) where m s the mass of speces. The total number of moles n the system, N, s ¼1 N ¼ XK ¼1 N ; (2.2) where N s the number of moles of speces n the system. Mass fracton, y, and mole fracton, x, descrbe the relatve amount of a gven speces. Ther defntons are gven by y m m and x N N ; (2.3) where ¼ 1,2,...,K. By defnton, X K ¼1 y ¼ 1 and X K ¼1 x ¼ 1: S. McAllster et al., Fundamentals of Combuston Processes, Mechancal Engneerng Seres, DOI 10.1007/978-1-4419-7943-8_2, # Sprnger Scence+Busness Meda, LLC 2011 15
16 2 Thermodynamcs of Combuston Wth M denotng the molecular mass of speces, the average molecular mass, M, of the mxture s determned by P M ¼ m N M N ¼ ¼ X N x M : (2.4) From Dalton s law of addtve pressures and Amagat s law of addtve volumes along wth the deal gas law, the mole fracton of a speces n a mxture can be found from the partal pressure of that speces as P P ¼ N N ¼ V V ¼ x ; (2.5) where P s the partal pressure of speces, P s the total pressure of the gaseous mxture, V the partal volume of speces, and V s the total volume of the mxture. The average ntrnsc propertes of a mxture can be classfed usng ether a molar base or a mass base. For nstance, the nternal energy per unt mass of a mxture, u, s determned by summng the nternal energy per unt mass for each speces weghted by the mass fracton of the speces. P u ¼ U m u m ¼ m ¼ X y u ; (2.6) where U s the total nternal energy of the mxture and u s the nternal energy per mass of speces. Smlarly, enthalpy per unt mass of mxture s h ¼ X y h and specfc heat at constant pressure per unt mass of mxture s c p ¼ X y c p; : A molar base property, often denoted wth a ^ over bar, s determned by the sum of the speces property per mole for each speces weghted by the speces mole fracton, such as nternal energy per mole of mxture ^u ¼ X x ^u ; enthalpy per mole of mxture ^h ¼ X x ^h ;
2.2 Combuston Stochometry 17 and entropy per mole of mxture ^s ¼ X x ^s : Assumng constant specfc heats durng a thermodynamc process, changes of energy, enthalpy, and entropy of an ndvdual speces per unt mass are descrbed as follows: Du ¼ c v; ðt 2 T 1 Þ (2.7) Dh ¼ c p; ðt 2 T 1 Þ (2.8) Ds ¼ c p; ln T 2 T 1 R ln P ;2 P ;1 (2.9) P,1 and P,2 denote the partal pressures of speces at state 1 and state 2, respectvely. R s the gas constant for speces (R ¼ ^R u =M ¼ unversal gas constant/molecular mass of speces ). The overall change of entropy for a combuston system s DS ¼ X m Ds : A summary of the thermodynamc propertes of mxtures s provded at the end of the chapter. 2.2 Combuston Stochometry For a gven combuston devce, say a pston engne, how much fuel and ar should be njected n order to completely burn both? Ths queston can be answered by balancng the combuston reacton equaton for a partcular fuel. A stochometrc mxture contans the exact amount of fuel and oxdzer such that after combuston s completed, all the fuel and oxdzer are consumed to form products. Ths deal mxture approxmately yelds the maxmum flame temperature, as all the energy released from combuston s used to heat the products. For example, the followng reacton equaton can be wrtten for balancng methane-ar combuston CH 4 þ? O 2 þ 79 21 N 2!?CO 2 þ?h 2 O þ?n 2 ; (2.10) where ar consstng of 21% O 2 and 79% N 2 s assumed. 1 The coeffcents assocated wth each speces n the above equaton are unknown. By balancng the atomc
18 2 Thermodynamcs of Combuston abundance on both the reactant and product sdes, one can fnd the coeffcent for each speces. For nstance, let s determne the coeffcent for CO 2 : on the reactant sde, we have 1 mol of C atoms; hence the product sde should also have 1 mol of C atoms. The coeffcent of CO 2 s therefore unty. Usng ths procedure we can determne all the coeffcents. These coeffcents are called the reacton stochometrc coeffcents. For stochometrc methane combuston wth ar, the balanced reacton equaton reads: CH 4 þ 2ðO 2 þ 3:76N 2 Þ!1CO 2 þ 2H 2 O þ 7:52N 2 : (2.11) Note that on the reactant sde there are 2 (1 + 3.76) or 9.52 mol of ar and ts molecular mass s 28.96 kg/kmol. In ths text, the reactons are balanced usng 1 mol of fuel. Ths s done here to smplfy the calculatons of the heat of reacton and flame temperature later n the chapter. Combuston stochometry for a general hydrocarbon fuel, C a H b O g, wth ar can be expressed as C a H b O g þ aþ b 4 g ðo 2 þ3:76n 2 Þ!aCO 2 þ b 2 2 H 2Oþ3:76 aþ b 4 g N 2 : (2.12) 2 The amount of ar requred for combustng a stochometrc mxture s called stochometrc or theoretcal ar. The above formula s for a sngle-component fuel and cannot be appled to a fuel consstng of multple components. There are two typcal approaches for systems wth multple fuels. Examples are gven here for a fuel mxture contanng 95% methane and 5% hydrogen. The frst method develops the stochometry of combuston usng the general prncple of atomc balance, makng sure that the total number of each type of atom (C, H, N, O) s the same n the products and the reactants. 0:95CH 4 þ 0:05H 2 þ 1:925ðO 2 þ 3:76N 2 Þ! 0:95CO 2 þ 1:95H 2 O þ 7:238N 2 : The other method of balancng a fuel mxture s to frst develop stochometry relatons for CH 4 and H 2 ndvdually: CH 4 þ 2ðO 2 þ 3:76N 2 Þ!CO 2 þ 2H 2 O þ 2 3:76N 2 H 2 þ 0:5ðO 2 þ 3:76N 2 Þ!H 2 O þ 0:5 3:76N 2 Then, multply the ndvdual stochometry equatons by the mole fractons of the fuel components and add them:
2.2 Combuston Stochometry 19 0:95 fch 4 þ 2ðO 2 þ 3:76N 2 Þ!CO 2 þ 2H 2 O þ 2 3:76N 2 g 0:05 fh 2 þ 0:5ðO 2 þ 3:76N 2 Þ!H 2 O þ 0:5 3:76N 2 g )0:95CH 4 þ 0:05H 2 þ 1:925ðO 2 þ 3:76N 2 Þ! 0:95CO 2 þ 1:95H 2 O þ 7:238N 2 2.2.1 Methods of Quantfyng Fuel and Ar Content of Combustble Mxtures In practce, fuels are often combusted wth an amount of ar dfferent from the stochometrc rato. If less ar than the stochometrc amount s used, the mxture s descrbed as fuel rch. If excess ar s used, the mxture s descrbed as fuel lean. For ths reason, t s convenent to quantfy the combustble mxture usng one of the followng commonly used methods: Fuel-Ar Rato (FAR): The fuel-ar rato, f, s gven by f ¼ m f m a ; (2.13) where m f and m a are the respectve masses of the fuel and the ar. For a stochometrc mxture, Eq. 2.13 becomes f s ¼ m f M f m ¼ a stochometrc ða þ b 4 g 2 Þ4:76 M ; (2.14) ar where M f and M ar (~28.84 kg/kmol) are the average masses per mole of fuel and ar, respectvely. The range of f s bounded by zero and 1. Most hydrocarbon fuels have a stochometrc fuel-ar rato, f s, n the range of 0.05 0.07. The ar-fuel rato (AFR) s also used to descrbe a combustble mxture and s smply the recprocal of FAR, as AFR ¼ 1/f. For nstance, the stochometrc AFR of gasolne s about 14.7. For most hydrocarbon fuels, 14 20 kg of ar s needed for complete combuston of 1 kg of fuel. Equvalence Rato: Normalzng the actual fuel-ar rato by the stochometrc fuelar rato gves the equvalence rato, f. f ¼ f f s ¼ m as m a ¼ N as N a ¼ N O2s N O2;a (2.15) The subscrpt s ndcates a value at the stochometrc condton. f <1 s a lean mxture, f ¼ 1sastochometrc mxture, and f >1sarch mxture. Smlar to f, the range of f s bounded by zero and 1 correspondng to the lmts of pure ar and fuel respectvely. Note that equvalence rato s a normalzed quantty that provdes the nformaton regardng the content of the combuston mxture. An alternatve
20 2 Thermodynamcs of Combuston varable based on AFR s frequently used by combuston engneers and s called lambda (l). Lambda s the rato of the actual ar-fuel rato to the stochometrc ar-fuel rato defned as l ¼ AFR AFR s ¼ 1=f 1=f s ¼ 1 f =f s ¼ 1 f (2.16) Lambda of stochometrc mxtures s 1.0. For rch mxtures, lambda s less than 1.0; for lean mxtures, lambda s greater than 1.0. Percent Excess Ar: The amount of ar n excess of the stochometrc amount s called excess ar. The percent excess ar, %EA, s defned as %EA ¼ 100 m a m as ¼ 100 m a 1 m as m as (2.17) For example, a mxture wth %EA ¼ 50 contans 150% of the theoretcal (stochometrc) amount of ar. Convertng between quantfcaton methods: Gven one of the three varables (f, f, and %EA), the other two can be deduced as summarzed n Table 2.1 wth ther graphc relatons. In general, the products of combuston nclude many dfferent Table 2.1 Relatons among the three varables for descrbng reactng mxtures f (fuel ar rato by mass) f (equvalence rato) %EA (% of excess ar) f ¼ f s f f ¼ 100 f f ¼ f %EA ¼ 100 1 f f s s f %EA þ 100 100 f ¼ %EA ¼ 100 1 f =f s %EA þ 100 f =f s fuel ar rato (mass) 0.5 10 f s = 0.05 0.4 8 f s = 0.05 0.3 6 0.2 4 0.1 2 0 0 2 4 6 8 10 0 0 0.1 0.2 0.3 0.4 0.5 Equvalence rato, φ Fuel ar rato (mass) Equvalence rato, φ % of excess ar 400 300 200 100 0 100 0.5 1.0 1.5 2.0 Equvalence rato, φ fuel ar rato (mass) 0.5 0.4 f s = 0.05 0.3 0.2 0.1 0 0 200 400 % of excess ar Equvalence rato, φ 10 8 6 4 2 0 100 0 100 200 300 400 % of excess ar % of excess ar 400 300 200 100 0 100 0 f s = 0.05 0.1 0.2 0.3 0.4 0.5 Fuel ar rato (mass)
2.2 Combuston Stochometry 21 speces n addton to the major speces (CO 2,H 2 O, N 2,O 2 ), and the balance of the stochometrc equaton requres the use of thermodynamc equlbrum relatons. However, assumng that the products contan major speces only (complete combuston) and excess ar, the global equaton for lean combuston fb1 s C a H b O g þ 1 f a þ b 4 g ðo 2 þ 3:76N 2 Þ! 2 aco 2 þ b 2 H 2O þ 3:76 a þ b f 4 g N 2 þ a þ b 2 4 g (2.18) 1 O 2 f 1 2 100 In terms of %EA, we replace f by and the result s %EA þ 100 C a H b O g þ %EA 100 þ 1 a þ b 4 g ðo 2 þ 3:76N 2 Þ! 2 aco 2 þ b 2 H 2O þ 3:76 %EA 100 þ 1 a þ b 4 g N 2 þ a þ b 2 4 g %EA 2 100 O 2 (2.19) The amount of excess ar can be deduced from measurements of exhaust gases. The rato of mole fractons between CO 2 and O 2 s x CO2 x O2 ¼ a a þ b 4 g %EA 2 100! %EA 100 ¼ a a þ b 4 g 2 xco2 x O2 or usng Table 2.1 100 f ¼ 100 þ %EA! f ¼ 1 a 1 þ a þ b 4 g 2 xco2 x O2 (2.20) For rch combuston (f>1), the products may contan CO, unburned fuels, and other speces formed by the degradaton of the fuel. Often addtonal nformaton on the products s needed for complete balance of the chemcal reacton. If the products are assumed to contan only unburned fuel and major combuston products, the correspondng global equaton can be wrtten as C a H b O g þ 1 f a þ b 4 g ðo 2 þ 3:76N 2 Þ! 2 a f CO 2 þ b 2f H 2O þ 3:76 a þ b f 4 g N 2 þ 1 1 (2.21) C a H b O g 2 f
22 2 Thermodynamcs of Combuston Example 2.1 Consderng a stochometrc mxture of sooctane and ar, determne: (a) the mole fracton of fuel (b) the fuel-ar rato (c) the mole fracton of H 2 O n the products (d) the temperature of products below whch H 2 O starts to condense nto lqud at 101.3 kpa Soluton: The frst step s wrtng and balancng the stochometrc reacton equaton. Usng Eq. 2.12, C 8 H 18 þ 8 þ 18 4 0 ðo 2 þ 3:76N 2 Þ!8CO 2 þ 9H 2 O þ 3:76 8 þ 18 4 0 N 2 C 8 H 18 þ 12:5ðO 2 þ 3:76N 2 Þ!8CO 2 þ 9H 2 O þ 3:76 12:5 N 2 From here: (a) x C8 H 18 ¼ N C8 H 18 N C8 H 18 þ N ar ¼ M f 1 1 þ 12:5 4:76 ¼ 0:0165 114 (b) f s ¼ ða þ b 4 g 2 Þ4:76 M ¼ ar 12:5 4:76 28:96 ¼ 0:066 N H2 O 9 (c) x H2 O ¼ ¼ N CO2 þ N H2 O þ N N2 8 þ 9 þ 3:76 12:5 ¼ 0:141 (d) The partal pressure of water s 101 kpa 0.141 ¼ 14.2 kpa. A saturaton table for steam gves the saturaton temperature at ths water pressure ff 53 C. Example 2.2 How many kg (lb) of ar are used to combust 55.5 L (~14.7 US gallons) of gasolne? Soluton: We wll use sooctane C 8 H 18 to represent gasolne. The stochometrc fuel-ar rato s M f f s ¼ ða þ b 4 g 2 Þ4:76 M ar 114 kg=kmol ¼ ð8 þ 18=4 0Þ4:76 28.84 kg/kmol ¼ 0:066 One gallon of gasolne weghs about 2.7 kg (6 lb). The total fuel thus weghs about 40 kg (88 lb). The requred ar weghs about 40/f s 610 kg 1,300 lb. Ths s a lot of weght f t must be carred. Hence, for transportaton applcatons, free ambent ar s preferred.
2.3 Heatng Values 23 Example 2.3 In a model can-combustor combuston chamber, n-heptane (C 7 H 16 ) s burned under an overall lean condton. Measurements of dry exhaust gve mole fractons of CO 2 and O 2 as x CO2 ¼ 0.084 and x O2 ¼ 0.088. Determne the %EA, equvalence rato f, and l. Soluton: To avod condensaton of water nsde the nstruments, measurements of exhaust gases are taken on a dry mxture that s obtaned by passng the exhaust gases through an ce bath so that most water s condensed. Further removal of water can be done wth desccants. The mole fractons measured under dry condtons wll be larger than at real condtons snce water s removed. However, ths wll not mpact the relaton deduced above, as both x CO2 and x O2 are ncreased by the same factor. %EA 100 ¼ a a þ b 4 g 2 ¼ 66:7 xco2 x O2 ¼ 7 ¼ 0:667! %EA ð7 þ 16=4 0Þð0:084=0:088Þ Next we use the relatons gven n Table 2.1 to convert %EA to f and l f ¼ 100 %EA þ 100 ¼ 100 66:7 þ 100 ¼ 0:6 l ¼ 1 f ¼ 1:67 2.3 Heatng Values Heatng values of a fuel (unts of kj/kg or MJ/kg) are tradtonally used to quantfy the maxmum amount of heat that can be generated by combuston wth ar at standard condtons (STP) (25 C and 101.3 kpa). The amount of heat release from combuston of the fuel wll depend on the phase of water n the products. If water s n the gas phase n the products, the value of total heat release s denoted as the lower heatng value (LHV). When the water vapor s condensed to lqud, addtonal energy (equal to the latent heat of vaporzaton) can be extracted and the total energy release s called the hgher heatng value (HHV). The value of the LHV can be calculated from the HHV by subtractng the amount of energy released durng the phase change of water from vapor to lqud as LHV ¼ HHV N H2O;PM H2O h fg N fuel M fuel (MJ/kg), (2.22)
24 2 Thermodynamcs of Combuston where N H2O,P s the number of moles of water n the products. Latent heat for water at STP s h fg ¼ 2.44 MJ/kg ¼ 43.92 MJ/kmol. In combuston lterature, the LHV s normally called the enthalpy or heat of combuston (Q C ) and s a postve quantty. 2.3.1 Determnaton of HHV for Combuston Processes at Constant Pressure A control volume analyss at constant pressure wth no work exchanged can be used to theoretcally determne the heatng values of a partcular fuel. Suppose reactants wth 1 kmol of fuel enter the nlet of a control volume at standard condtons and products leave at the ext. A maxmum amount of heat s extracted when the products are cooled to the nlet temperature and the water s condensed. Conservaton of energy for a constant pressure reactor, wth H P and H R denotng the respectve total enthalpes of products and reactants, gves Q rxn;p ¼ H R H p : (2.23) The negatve value of Q rxn,p ndcates heat transfer out of the system to the surroundngs. It follows from above that the heatng value of the fuel s the dfference n the enthalpes of the reactants and the products. However, n combuston systems, the evaluaton of the enthalpes s not straghtforward because the speces enterng the system are dfferent than those comng out due to chemcal reactons. Q rxn,p s often referred to as the enthalpy of reacton or heat of reacton, wth the subscrpt p ndcatng that the value was calculated at constant pressure. The enthalpy of reacton s related to the enthalpy of combuston by Q rxn,p ¼ Q C. 2.3.1.1 Enthalpy of Formaton In combuston processes, reactants are consumed to form products and energy s released. Ths energy comes from a rearrangement of chemcal bonds n the reactants to form the products. The standard enthalpy of formaton, D^h o, quantfes the chemcal bond energy of a chemcal speces at standard condtons. The enthalpy of formaton of a substance s the energy needed for the formaton of that substance from ts consttuent elements at STP condtons (25 C and 1 atm). The molar base enthalpy of formaton, D^h o, has unts of MJ/kmol, and the mass base enthalpy of formaton, D ^h o, has unts of MJ/kg. Elements n ther most stable forms, such as C (graphte),h 2,O 2, and N 2, have enthalpes of formaton of zero. Enthalpes of formaton of commonly encountered chemcal speces are tabulated n Table 2.2. A departure from standard condtons s accompaned by an enthalpy change. For thermodynamc systems wthout chemcal reactons, the change of enthalpy of an deal gas s descrbed by the sensble enthalpy,
2.3 Heatng Values 25 Table 2.2 Enthalpy of formaton of common combuston speces Speces D^h o (MJ/kmol) Speces D^h o (MJ/kmol) H 2 O (g) 241.83 H +217.99 CO 2 393.52 N +472.79 CO 110.53 NO +90.29 CH 4 74.87 NO 2 +33.10 C 3 H 8 104.71 O +249.19 C 7 H 16 (g) (n-heptane) 224.23 OH +39.46 C 8 H 18 (g) (sooctane) 259.25 C (g) +715.00 CH 3 OH (g) (methanol) 201.54 C 2 H 2 (acetylene) +226.73 CH 3 OH (l) (methanol) 238.43 C 2 H 4 (ethylene) +52.28 C 2 H 6 O (g) (ethanol) 235.12 C 2 H 6 (ethane) 84.68 C 2 H 6 O (l) (ethanol) 277.02 C 4 H 10 (n-butane) 126.15 Fg. 2.1 Constant-pressure flow reactor for determnng enthalpy of formaton Q = 393,522 kj (heat out) 1 kmol C 1 kmol O 2 C + O 2 CO 2 1 kmol CO 2 @ 25 C, 101.3 kpa CV @ 25 C, 101.3 kpa Z T ^h s ¼ T o ^c p ðtþdt; where the subscrpt refers to speces, T 0 denotes the standard temperature (25 C), and ^ ndcates that a quantty s per mole. Note that the sensble enthalpy of any speces s zero at standard condtons. The absolute or total enthalpy, ^h, s thus the sum of the sensble enthalpy and the enthalpy of formaton: 2 ^h ¼ D^h o þ ^h s (2.24) One way to determne the enthalpy of formatonofaspecesstouseaconstantpressure flow reactor. For nstance, the enthalpy of formaton of CO 2 s determned by reactng 1 kmol of C (graphte) wth 1 kmol of O 2 at 25 C at a constant pressure of 101.3 kpa. The product, 1 kmol of CO 2, flows out of ths reactor at 25 C as sketched n Fg. 2.1. An amount of heat produced n the reacton s transferred 2 When phase change s encountered, the total enthalpy needs to nclude the latent heat, ^h ¼ D^h o þ ^h s þ ^h latent.
26 2 Thermodynamcs of Combuston out of ths system, therefore the enthalpy formaton of CO 2 s negatve D ^h o CO2 ¼ 393.52 MJ/kmol. Ths means that CO 2 at 25 C contans less energy than ts consttuent elements C (graphte) and O 2, whch have zero enthalpy of formaton. The enthalpy of formaton s not negatve for all chemcal speces. For nstance, the enthalpy formaton of NO s D^h o NO ¼ +90.29 MJ/kmol, meanng that energy s needed to form NO from ts elements, O 2 and N 2. For most unstable or radcal speces, such as O, H, N, and CH 3, the enthalpy of formaton s postve. 2.3.1.2 Evaluaton of the Heat of Combuston from a Constant-Pressure Reactor Usng the conservaton of energy equaton (2.23), we can now evaluate the enthalpes of the reactants and products. Insertng the expresson for the total enthalpy, Q rxn:p ¼ H R H p ¼ X N ;R D^h o ;R þ ^h s;r X N ;P D^h o ;P þ ^h s;p " ¼ X N ;R D^h o ;R X # N ;P D ^h o ;P þ X N ;R ^h s;r X (2.25) N ;P ^h s;p ; where N s the number of moles of speces. The sensble enthalpes of common reactants and products can be found n Appendx 1. When the products are cooled to the same condtons as the reactants, the amount of heat transfer from the constant-pressure reactor to the surroundngs s defned as the heatng value. At STP the sensble enthalpy terms drop out for both reactants and products and the heat release s Q 0 rxn;p ¼ X N ; R D ^h o ;R X N ;P D^h o ;P (2.26) Usually excess ar s used n such a test to ensure complete combuston. The amount of excess ar used wll not affect Q 0 rxn;p at STP. Unless the reactant mxtures are heavly dluted, the water n the products at STP normally wll be lqud. 3 Assumng that water n the products s lqud, HHV s determned: HHV ¼ Q0 rxn; p N fuel M fuel : (2.27) The negatve sgn n front of Q 0 rxn;p ensures that HHV s postve.
2.3 Heatng Values 27 2.3.2 Determnaton of HHV for Combuston Processes from a Constant-Volume Reactor A constant-volume reactor s more convenent than the constant-pressure reactor to expermentally determne the HHV of a partcular fuel. For a closed system, conservaton of energy yelds Q rxn;v ¼ U R U p (2.28) Because of the combuston process, the same type of accountng must be used to nclude the change n chemcal bond energes. The nternal energy wll be evaluated by usng ts relaton to enthalpy. Note that relaton h ¼ u+pv s mass based and the correspondng molar base relaton s ^h ¼ ^u þ ^R u T. At STP (T ¼ T 0 ¼ 25 C), the total nternal energy of the reactants, U R, nsde the closed system s U R ¼ H R PV ¼ H R X N ;R ^R u T 0 ¼ X N ;R D ^h o ;R X N ;R ^R u T 0 (2.29) The total nternal energy of products s evaluated n a smlar manner: U P ¼ X N ;P D^h o ;P X N ;P ^R u T 0 (2.30) Usng the nternal energy relatons, we can re-express the heat release at constant volume n terms of enthalpes as Q 0 rxn;v ¼ U R U P ¼ X N ;R D^h o ;R X N ;R ^R u T 0 " # X N ;P D ^h o ;P X N ;P ^R u T 0 ¼ X N ;R D^h o ;R X N ;P D ^h o ;P þ X N ;P X N ;R!^R u T 0 (2.31) Therefore, HHV for combuston processes s calculated as Q 0 rxn;v P N ;P P N ;R ^R u T 0 HHV ¼ ; (2.32) N fuel M fuel
28 2 Thermodynamcs of Combuston where N fuel s the number of moles of fuel burned and M fuel s the molecular mass of the fuel. The negatve sgn n front of Q 0 rxn;v s to make sure that HHV s postve. For a general fuel, C a H b O g, the dfference between Q rxn,v and Q rxn,p s X N ;P X N ;R!^R u T 0 ¼ DN ^R u T 0 ¼ b 4 þ g 2 1 ^R u T 0 (2.33) and s usually small n comparson to HHV; therefore normally no dstncton s made between the heat of reacton at constant pressure or constant volume. 2.3.2.1 Expermental Determnaton of HHV: The Bomb Calormeter To expermentally measure the heatng value of a fuel, the fuel and ar are often enclosed n an explosve-proof steel contaner (see Fg. 2.2), whose volume does not change durng a reacton. The vessel s then submerged n water or another lqud that absorbs the heat of combuston. The heat capactance of the vessel plus the lqud s then measured usng the same technque as other calormeters. Such an nstrument s called a bomb calormeter. A constant-volume analyss of the bomb calormeter data s used to determne the heatng value of a partcular fuel. The fuel s burned wth suffcent oxdzer n a closed system. The closed system s cooled by heat transfer to the surroundngs such that the fnal temperature s the same as the ntal temperature. The standard condtons are set for evaluaton of heatng values. Conservaton of energy gves U P U R ¼ Q 0 rxn;v (2.34) Strrer Thermocouple Ignter Insulated contaner flled wth water Reacton chamber (bomb) Sample cup Fg. 2.2 Bomb calormeter
2.3 Heatng Values 29 Because the fnal water temperature s close to room temperature, the water n the combuston products s usually n lqud phase. Therefore the measurement leads to the HHV from a constant-volume combuston process as descrbed by Eq. 2.32: ( HHV ¼ Q 0 rxn;v X N ;P X ) N ;R!^R u T 0 = N fuel M fuel ; where N fuel s the number of moles of fuel burned and M fuel s the molecular weght of the fuel. The negatve sgn n front of Q 0 rxn;v ensures that HHV s postve. In a bomb calormeter, f the fnal temperature of the combuston products s hgher than the reactants by only a few degrees (<10 C), the error s neglgble. The amount of heat transfer s estmated by Q 0 rxn;v ¼ðm steel c p;steel þ m water c p;water ÞDT; (2.35) where DT s the temperature change of the water and the steel contaner. The bomb calormeter can also measure the enthalpy of formaton of a chemcal speces. For nstance, to determne enthalpy of formaton of H 2 O, we start out wth 1 mol of H 2 and 0.5 mol of O 2. These element speces have zero enthalpy of formaton; therefore X N ;R D^h 0 ;R ¼ 0: The only product s the speces of nterest, namely H 2 O. We therefore can wrte the enthalpy of formaton of H 2 O, D ^h 0 where D ^h 0 ;P ¼ Q0 rxn;v þ ;P,as P N ;P P N ;R ^R u T 0 N ;P ¼ Q0 rxn;v þ DN ^R u T 0 N ;P (2.36) DN ¼ X N ;P X N ;R : 2.3.3 Representatve HHV Values Lsted n Table 2.3 are hgher heatng values of some common and less common fuels. Example 2.4 A table of thermodynamc data gves the enthalpy of formaton for lqud water as D^h H 0 2 OðlÞ ¼ 285.8 kj/mol. A bomb calormeter burnng 1 mol of H 2 wth O 2 measures 282.0 kj of heat transfer out of the reacted mxture. Estmate the error of the enthalpy measurement.
30 2 Thermodynamcs of Combuston Soluton: We start out wth the combuston stochometry H 2 ðgþþ0:5o 2 ðgþ ¼ H 2 OðlqÞ; DN ¼ 1:5 ðchange n moles of gas n the mxtureþ Applyng the deal gas approxmaton to the energy balance wth Q 0 rxn;v ¼ 282.0 kj, D^h 0 H 2 OðlÞ ¼ Q0 rxn;v þ DN ^R u T 0 ; ¼ 282:0 kj/mol 1molþð 1:5 mol 8:314 J/mol K 298K 0:001 kj/jþ ¼ð 282:0 3:72ÞkJ ¼ 285:7 kj The error s (285.8 285.7)/285.8 ¼ 0.03%. In ths case, more heat s gven off f the reacton s carred out at constant pressure, snce the P-V work (1.5 ^R u T 0 ) due to the compresson of 1.5 mol of gases n the reactants would contrbute to D ^h 0 H 2 OðlÞ. However, ths dfference s only about 1 2% of the enthalpy of formaton. The enthalpy of formaton for gaseous H 2 O s obtaned by addng the latent heat to D ^h 0 H 2 OðlÞ : D ^h 0 H 2 OðgÞ ¼ D^h 0 H 2 OðlÞ þ ^h fg ¼ 241:88 kj/mol; Table 2.3 Heat values of varous fuels Heatng value Fuel MJ/kg BTU/lb kj/mol Hydrogen 141.8 61,100 286 Methane 55.5 23,900 890 Ethane 51.9 22,400 1,560 Propane 50.35 21,700 2,220 Butane 49.5 20,900 2,877 Gasolne 47.3 20,400 ~5,400 Paraffn 46 19,900 16,300 Desel 44.8 19,300 ~4,480 Coal 15 27 8,000 14,000 200 350 Wood 15 6,500 300 Peat 6 15 2,500 6,500 Methanol 22.7 9,800 726 Ethanol 29.7 12,800 1,368 Propanol 33.6 14,500 2,020 Acetylene 49.9 21,500 1,300 Benzene 41.8 18,000 3,270 Ammona 22.5 9,690 382 Hydrazne 19.4 8,370 622 Hexamne 30.0 12,900 4,200 Carbon 32.8 14,100 393.5
2.4 Adabatc Flame Temperature 31 where ^h fg ¼ 43:92 kj/mol: Example 2.5 The heat released by 1 mol of sugar n a bomb calormeter experment s 5,648 kj/mol. Calculate the enthalpy of combuston per mole of sugar. Soluton: The balanced chemcal reacton equaton s C 12 H 22 O 11 ðþþ12o s 2 ðgþ ¼ 12CO 2 ðgþþ11h 2 OðlqÞ Snce the total number of moles of gas s constant (12) n the products and reactants, DN ¼ 0. Therefore, work s zero and the enthalpy of combuston equals the heat transfer: 5,648 kj/mol. 2.4 Adabatc Flame Temperature One of the most mportant features of a combuston process s the hghest temperature of the combuston products that can be acheved. The temperature of the products wll be greatest when there are no heat losses to the surroundng envronment and all of the energy released from combuston s used to heat the products. In the next two sectons, the methodology used to calculate the maxmum temperature, or adabatc flame temperature,wllbepresented. 2.4.1 Constant-Pressure Combuston Processes An adabatc constant-pressure analyss s used here to calculate the adabatc flame temperature. Under ths dealzed condton, conservaton of energy s: where H P ðt P Þ¼H R ðt R Þ; (2.37) H P ðt P Þ¼ X N ;P ^h;p ¼ X N ;P ½D ^h o ;P þ ^h s;p ðt P ÞŠ and H R ðt R Þ¼ X N ;R ^h ;R ¼ X N ;R ½D ^h o ;R þ ^h s;r ðt R ÞŠ: Fgure 2.3 s a graphc explanaton of how the adabatc flame temperature s determned. At the ntal reactant temperature, the enthalpy of the product mxture
32 2 Thermodynamcs of Combuston H R (T R ) Ο H R (T) H R (T R ) = H p (T P ) x Enthalpy Energy Release H P (T) Ο H P (T R ) Reactant Temperature Temperature, T Adabatc Flame Temperature Fg. 2.3 Graphcal nterpretaton of adabatc flame temperature s lower than that of the reactant mxture. The energy released from combuston s used to heat up the products such that the condton H P ðt P Þ¼H R ðt R Þ s met. The task s fndng the product temperature gven the enthalpy of reactants. Three dfferent methods can be used to obtan T P : 1. Usng an average c p value, 2. An teratve enthalpy balance, 3. Fndng the equlbrum state usng computer software (such as Cantera). The frst two methods can be performed manually f complete combuston s consdered and provde only quck estmates. An equlbrum state solver takes nto account dssocaton of products at hgh temperature, makng t more accurate than the frst two methods. Method 1: Constant, average c p From conservaton of energy, H p ðt p Þ¼H R ðt R Þ, whch can be expressed as X N ;P ½D^h o ;P þ ^h s;p ðt P ÞŠ ¼ X N ;R ½D^h o ;R þ ^h s;r ðt R ÞŠ Rearrangng yelds ( X N ;P ^h s;p ðt P Þ¼ X N ;P D^h o ;P X ) N ;R D^h o ;R þ X N ;R ^h s;r ðt R Þ ¼ Q 0 rxn;p þ X N ;R ^hs;r ðt R Þ (2.38)
2.4 Adabatc Flame Temperature 33 wth Q 0 rxn;p ¼ X N ;R D ^h o ;R X N ;P D^h o ;P : (2.39) Note that water n the products s lkely n gas phase due to the hgh combuston temperature; therefore Q 0 rxn;p ¼ LHVN fuelm fuel ¼ LHVm f when the fuel s P completely consumed. The second term, N ;R ^h s;r ðt R Þ, n Eq. 2.38 represents the dfference of sensble enthalpy between T R and T 0 (25 C) for the reactant mxture. Wth the assumpton that the sensble enthalpy can be approxmated by hˆs,p(t P ) ĉ p (T P T 0 ) wth ĉ p constant, we have ðt P T 0 Þ X N ;P^c p ^c p ðt P T 0 Þ X N ;P ¼ Q 0 rxn;p þ X N ;R ^h s;r ðt R Þ (2.40) Rearrangng the equaton one fnds T P as Q 0 rxn;p þ P N ;R ^h s;r ðt R Þ T P ¼ T 0 þ P N ;P^c p T R þ Q0 rxn;p P N ;P^c p ¼ T R þ LHV N fuel M P fuel ; N ;P^c p (2.41) where the followng approxmaton has been appled 4 P P N ;R ^h s;r ðt R Þ N ;R^c p;r ðt R T 0 Þ P ¼ P T R T 0 N ;P^c p N ;P^c p When reactants enter the combustor at the standard condtons, the above equaton reduces to (as sensble enthalpes of reactants are zero at T 0 ) T P ¼ T 0 þ LHV N fuel M P fuel : (2.42) N ;P^c p 4 P N ;R^c p ;R and P N ;p^c p are assumed to be approxmately equal.
34 2 Thermodynamcs of Combuston The above procedure s general and can be appled to any mxture. Note that the specfc heat s a functon of temperature, so the accuracy of ths approach depends on the value selected for the specfc heat ĉ p. If the heatng value of a fuel s gven, a mass-based analyss for the same control volume can be conducted. The ntal mxture conssts of fuel and ar wth m f and m a, respectvely. By mass conservaton, the products have a total mass of m f +m a.the sensble enthalpy of the products s approxmated by H s,p ¼ (m a +m f ) c p;p (T P T 0 ), where c p;p s an average value of specfc heat evaluated at the average temperature of the reactants and products,.e., c p;p ¼ c p ð TÞ; where T ¼ðT p þ T R Þ=2. Smlarly, the sensble enthalpy of the reactants s estmated by H s,r ¼ (m a +m f ) c p;r (T R T 0 ), where c p;r s an average value of specfc heat evaluated at the average temperature of reactants and the standard temperature,.e., c p;r ¼ c p ð TÞ, where T ¼ðT R þ T 0 Þ=2. From conservaton of energy, H s,p equals the amount of heat released from combuston plus the sensble enthalpy of the reactants, H s,p ¼ Q 0 rxn;p þ H s;r ¼ m fb LHV þ H s,r,wherem fb s the amount of fuel burned. For fb1, m fb ¼ m f snce there s enough ar to consume all the fuel n a lean mxture. For rch combuston (f > 1), the lmtng factor s the amount of ar avalable, m a. Therefore, for f>1, the amount of fuel burned (wth ar, m a )sm fb ¼ m a f s, where f s s the stochometrc fuel/ar rato by mass. Then the adabatc flame temperature s calculated for a lean mxture as fb1 T P ff T 0 þ m f LHV þðm a þ m f Þc p;r ðt R T 0 Þ ðm a þ m f Þc p;p T R þ m f LHV ¼ T R þ m f =m a LHV ðm a þ m f Þc p;p ð1 þ m f =m a Þc p;p ¼ T R þ f LHV ¼ T R þ f f s LHV ð1 þ f Þc p;p ð1 þ f f s Þc p;p (2.43) where c p;r c p;p s used n dervng the second lne. Smlarly, for the rch mxtures one gets fr1 T p ¼ T R þ f s LHV f s LHV ¼ T R þ (2.44) ð1 þ f Þc p;p ð1 þ f f s Þc p;p Note that f s s very small for hydrocarbon fuels (e.g., f s ¼ 0.058 for methane). As such, the product (flame) temperature ncreases almost lnearly wth equvalence rato, f, for lean combuston as shown n Fg. 2.4. As expected, the flame temperature peaks at the stochometrc rato. In rch combuston, the flame temperature decreases wth f. Method 2: Iteratve enthalpy balance A more accurate approach s to fnd the flame temperature by teratvely assgnng theflametemperaturet p untl H p (T p ) H R (T R ). The enthalpy of reactants s assumed gven. The enthalpy of products can be expressed n the followng form
2.4 Adabatc Flame Temperature 35 Temperature (K) 2500 2000 1500 1000 Estmate wth constant c p Enthalpy balance Smulated flame Equlbrum 500 0 0.1 1 10 Equvalence Rato, φ Fg. 2.4 Comparson of flame temperatures wth dfferent approaches H P ðt P Þ¼ X N ;P ^h ;P ¼ X N ;P ½D ^h o ;P þ ^h s;p ðt P ÞŠ ¼ H R ðt R Þ¼ X N ;R ^h ;R Next, we rearrange the above equaton to fnd an expresson for the sensble enthalpy of the products as X N ;P D ^h o ;P þ X N ;P ^h s;p ðt P Þ¼ X N ;R D ^h o ;R þ X N ;R ^h s;r ðt R Þ X N ;P ^hs;p ðt P Þ¼ X N ;R D^h o ;R X N ;P D ^h o ;P þ X N ;R ^hs;r ðt R Þ (2.45) X N ;P ^h s;p ðt P Þ¼ Q 0 rxn;p þ X N ;P ^h s;r ðt R Þ: Wth an ntal guess of flame temperature, T p1, one evaluates H p (T p1 ) from tables such as those n Appendx 3. If H p (T p1 ) < H R (T R ), we guess a hgher flame temperature, T p2. One repeats ths process untl the two closest temperatures are found such that H p (T f1 ) < H R (T R ) < H p (T f2 ). The product temperature can be estmated by lnear nterpolaton. Ths method, although more accurate, stll assumes complete combuston to the major products. Method 3: Equlbrum State (Free software: Cantera; Commercal software: Chemkn) Dssocaton 5 of products at hgh temperature (T > 1,500 K at ambent pressure) can take a sgnfcant porton of energy from combuston and hence the product 5 Dssocaton s the separaton of larger molecules nto smaller molecules. For example, 2H 2 O 2H 2 +O 2.
36 2 Thermodynamcs of Combuston temperature s lower than that calculated wth only major components as products. The equlbrum state determnes the speces concentratons and temperature under certan constrants such as constant enthalpy, pressure, or temperature. The equlbrum flame temperature s expected to be lower than the temperatures estmated wth Method 1 or Method 2. In addton, the chemcal equlbrum state s often used n combuston engneerng as a reference pont for chemcal knetcs (the subject of Chap. 3) f nfnte tme s avalable for chemcal reactons. At ths deal state, forward and backward reacton rates of any chemcal reacton steps are balanced. By constranng certan varables such as constant pressure and enthalpy, the chemcal equlbrum state can be determned by mnmzng the Gbbs free energy, even wthout knowledge of the chemcal knetcs. Computer programs (such as STANJAN, Chemkn, Cantera) are preferred for ths task, as hand calculatons are tme consumng. 2.4.2 Comparson of Adabatc Flame Temperature Calculaton Methods The presented methods of estmatng adabatc flame temperature wll produce dfferent values from each other. Predcted adabatc flame temperatures of a methane/ar mxture at ambent pressure usng these methods are compared n Fg. 2.4 for a range of equvalence ratos. Also ncluded are the results from a flame calculaton usng a detaled, non-equlbrum flame model. On the lean sde, the results agree reasonably well among all methods, as the major products are CO 2, H 2 O, unburned O 2, and N 2. Vsble devatons arse near stochometrc condtons and become larger n rcher mxtures. One reason for the devaton s the assumptons made about product speces n the rch mxtures. For rch mxtures at the equlbrum state, CO s preferred over CO 2 due to the defcency n O 2. Because the converson of CO nto CO 2 releases a large amount of energy, the rch mxture equlbrum temperatures are lower than those from the flame calculaton, whch has a resdence tme of less than 1 s. Among the methods, the results from the detaled flame model calculatons are closest to realty, as real flames have fnte resdence tmes and generally do not reach equlbrum. Example 2.6. Estmate the adabatc flame temperature of a constant-pressure reactor burnng a stochometrc mxture of H 2 and ar at 101.3 kpa and 25 C at the nlet. Soluton: The combuston stochometry s H 2(g) þ 0.5 (O 2(g) +3.76 N 2(g) )! H 2 O (g) þ 1.88 N 2(g) Q 0 rxn;p ¼ X N ;R D^h o ;R X N ;P D^h o ;P ¼ D ^h o H2 þ 0:5D^h o O2 þ 1:88D^h o N2 1 D ^h o H2O ¼ 0 þ 0 þ 0 1 mol ð 241:88 kj/molþ ¼241:88 kj
2.4 Adabatc Flame Temperature 37 Method 1: Assumng a constant (average) ^c p at 1,500 K, ^c p; H 2 Oð1; 500 KÞ ¼0:0467 kj/mol K and ^c p;n2 ð1;500 KÞ ¼0:0350 kj/mol K: Q 0 rxn;p þ P N ;R ^h s;r ðt R Þ T p ¼ T 0 þ P N ;p^c p; ð241:88 þ 0ÞkJ=mol ¼ 300 þ ð0:047 þ 1:88 0:035Þ kj/mol K 2;148 K The average temperature of the products and reactants s now (2,148 K + 298 K)/ 2 ~ 1,223 K, ndcatng that the ntal assumpton of T ave ¼ 1,500 K was too hgh. Usng the new average temperature of 1,223 K to evaluate the specfc heats, the calculated flame temperature becomes T p ~ 2,253 K. The average temperature s now T ave ¼ 1,275 K. Ths new average temperature can be used to calculate the specfc heats and the process should be contnued untl the change n the average temperature s on the order of 20 K. By dong ths procedure, we obtan T P ~2,230K. Method 2: Iteratve enthalpy balance: X N ;p D ^h o ;p þ X H P ðt P Þ¼H R ðt R Þ N ;p ^h s;p ðt p Þ¼ X N ;R D^h o ;R þ X N ;R ^h s;r ðt R Þ N H2 OD ^h o H 2 O þ N H 2 O ^h s;h2 OðT P ÞþN N2 D ^h o N 2 þ N N2 ^h s;n2 ðt P Þ ¼ N H2 D ^h o H 2 þ N H2 ^h s;h2 ðt R ÞþN O2 D ^h o O 2 þ N O2 ^h s;o2 ðt R Þ þ N N2 D ^h o N 2 þ N N2 ^hs;n2 ðt R Þ 1 D ^h 0 H 2 O þ ^h s;h2 OðT P Þþ0 þ 1:88 ^h s;n2 ðt P Þ¼0 þ 0 þ 0 þ 0 þ 0 þ 0 D ^h 0 H 2 O þ ^h s;h2 OðT P Þþ1:88 ^h s;n2 ðt P Þ¼0: The frst step s to guess the product temperature. For ths case, let s pck T P ¼ 2,000 K. We now plug n the value for the heat of formaton of water and use thermodynamc property tables to evaluate the sensble enthalpy terms. T P (K) H P (T P ) (MJ) 2,000 K 241.83 + 72.69 + 1.88 56.14 ¼ 63.6 MJ 2,500 K 241.83 + 98.96 + 1.88 74.31 ¼ 3.1 MJ
38 2 Thermodynamcs of Combuston Our ntal guess of T P ¼ 2,000 K was too low. The process was repeated wth a hgher guess of T P ¼ 2,500 K whch resulted n a much smaller remander, mplyng that T P ~ 2,500 K. For more accuracy, we can use lnear extrapolaton (or nterpolaton f we bracketed the real value): T P 2; 500 2;500 2;000 ¼ 0 þ 3:1 3:1 þ 63:6 T P ¼ 2;526K Method 3: Cantera. Assume H 2,O 2, and H 2 O are the only speces n the system; equlbrum temperature s 2,425.1 K. The equlbrum mole fractons are lsted below Mole fractons Speces x reactant x product H 2 0.2958 0.0153 O 2 0.1479 0.0079 N 2 0.5563 0.6478 H 2 O 0 0.3286 Note that there s a small amount (~1.5%) of H 2 exstng n the products due to the dssocaton of H 2 O at hgh temperature. Results of the above three methods agree wth each other wthn 100 200 K whch s less than 12% of the flame temperature. If radcals, such as H, OH, and O, are also ncluded n the products, the equlbrum temperature drops to 2,384 K because addtonal dssocaton occurs. Ths 41 K dfference s about 1.7% of the flame temperature. Example 2.7 The space shuttle burns lqud hydrogen and oxygen n the man engne. To estmate the maxmum flame temperature, consder combuston of 1 mol of gaseous hydrogen wth 0.5 mol of gaseous O 2 at 101.3 kpa. Determne the adabatc flame temperatures usng the average c p method. Soluton: The combuston stochometry s H 2ðgÞ þ 0:5O 2ðgÞ! H 2 O ðgþ Q 0 rxn;p ¼ LHV of H 2 at constant pressure Q 0 rxn;p ¼ X N ;R D^h o ;R X N ;P D^h o ;P ¼ D^h o H2 þ 0:5D^h o O2 1D^h o H2O ¼ 0 þ 0 1 molð 241:88 kj/molþ ¼241:88 kj Guessng a fnal temperature of about 3,000 K, we use average specfc heats evaluated at 1,500 K
2.4 Adabatc Flame Temperature 39 Q 0 rxn;p þ P N ;R ^h s;r ðt R Þ T P ¼ T 0 þ P N ;P^c p 241:88 kj=mol ¼ 300 K þ 0:047 kj/mol K 5; 822 K Dscusson: Ths temperature s evdently much hgher than the NASA reported value of ~3,600 K. What s the man reason for such a BIG dscrepancy? The estmated temperature s well above 2,000 K and one expects a substantal dssocaton of H 2 O back to H 2 and O 2. That s, H 2 (g) þ 0.5 O 2 (g) H 2 O (g). Now we use Cantera or a commercal software program, such as Chemkn, to compute the equlbrum temperature wth only three speces H 2,O 2, and H 2 O. The predcted adabatc flame temperature drops to 3508.7 K. The mole fractons of these three before reacton and after combuston are lsted below. Speces Reactant Product H 2 0.6667 0.2915 O 2 0.3333 0.1457 H 2 O 0 0.5628 As seen n the table, the dssocaton s very sgnfcant; about 30% of the products s H 2. Let s fnd out how much fuel s not burned by consderng the followng stochometrc reacton: H 2 ðgþþ0:5o 2 ðgþ! X H 2 þ 0:5X O 2 þ ð1 XÞH 2 Og ð Þ The mole fracton of H 2 n the products s x H2 ¼ X X þ 0:5X þ 1 X ¼ X 0:5X þ 1 : Wth x H2 ¼ 0.2915, we get X ¼ 0.3412. If we assume 66% of fuel s burned, a new estmate based on ^c p at 1,500 K leads to 0:66 241:88 kj=mol T p ¼ 300 K þ 0:047 kj/mol K 3;700 K that s n much better agreement wth the equlbrum result. If we estmate ^c p at 1,800 K we get 0:66 241:88 kj=mole T p ¼ 300 K þ 0:04966 kj/mole K 3;514:7K:
40 2 Thermodynamcs of Combuston If we nclude addtonal speces, H, OH, and O n the products, the predcted equlbrum temperature drops to 3,076 K. The table below shows the mole fractons of each speces n ths case. Speces Reactant Product H 2 0.6667 0.1503 O 2 0.3333 0.0510 H 2 O 0 0.5809 OH 0 0.1077 O 0 0.0330 H 0 0.0771 Evdently, the radcals OH, H, and O take some energy to form; note that ther values for enthalpy of formaton are postve. Because the space shuttle engne operates at 18.94 MPa (2,747 ps, ~186 atm) at 100% power, the pressure needs to be taken nto consderaton as the combnaton of radcals occurs faster at hgher pressures. The predcted equlbrum temperature at 18.94 MPa s 3,832.4 K and the mole fractons are lsted below. Speces Reactant Product H 2 0.6667 0.1169 O 2 0.3333 0.0336 H 2 O 0 0.7051 OH 0 0.1005 O 0 0.0143 H 0 0.0296 The energy needed to vaporze lqud H 2 and O 2 and heat them from ther bolng temperatures to 25 C are estmated to be 8.84 kj/mol and 12.92 kj/mol (energy ¼ latent heat + sensble energy from bolng pont to STP). Wth H 2 þ 0.5O 2, the total energy requred s then 8.84 þ 0.5 12.92 or about 15.3 kj/mol. The temperature drop due to ths process s about ~15.3 kj/(0.049 kj/mol-k) ¼ 148 K. Wth ths, we estmate the space shuttle man engne temperature s 3,832 148 K or ~3,675 K. The followng nformaton s used for estmatng energy to vaporze H 2 and O 2 : (1) for H 2, latent heat of vaporzaton 445.7 kj/kg, bolng temperature ¼ 252.8 C, c p ~ 4.12 kj/kg-k; (2) for O 2, latent heat of vaporzaton 212.7 kj/kg, bolng temperature ¼ 183 C, c p ~ 0.26 kj/kg-k. 2.5 Chapter Summary The followng shows the relatons among dfferent thermodynamcs propertes expressed n terms of mass fractons and mole fractons.
2.5 Chapter Summary 41 Property Mass fracton, y Mole fracton x Speces denstyr (kg/m 3 ) ry r xm Mole fracton, x [ ] y =M P K y j=m j Mass fracton, y, x M Mxture molecular mass, M (kg/kmol) 1 Internal energy of mxture, u (kj/kg) Enthalpy of mxture, h (kj/kg) Entropy of mxture, s (kj/kg-k) Specfc heat at constant pressure c p (kj/kg-k) Specfc heat at constant volume c v (kj/kg-k) Internal energy of mxture, ^u (kj/kmol) P K y j=m j P K P K P K P K P K M PK P K x jm j P K x jm j P K x j M j P 1 y j u K j M x j ^u j P 1 y j h K j M x j ^h j P 1 s j h K j M x j ^s j P 1 y j c K pj M x j ^c pj P 1 y j c K vj M x j ^c vj y j u j P K x j ^u j Enthalpy of mxture, ^h (kj/kmol) Entropy of mxture, ^s (kj/kmol-k) Specfc heat at constant pressure ^c p (kj/kmol-k) Specfc heat at constant volume ^c v (kj/kmol-k) M PK M PK M PK M PK y j h j y j s j y j c pj y j c vj P K P K P K P K x j ^h j x j ^s j x j ^c pj x j ^c vj Defntons Enthalpy of combuston or heat of combuston: Ideal amount of energy that can be released by burnng a unt amount of fuel. Enthalpy of reacton or heat of reacton: Energy that must be suppled n the form of heat to keep a system at constant temperature and pressure durng a reacton.
42 2 Thermodynamcs of Combuston Enthalpy of formaton or heat of formaton: Heat of reacton per unt of product needed to form a speces by reacton from the elements at the most stable condtons. Combuston stochometry for a general hydrocarbon fuel, C a H b O g C a H b O g þ a þ b 4 g 2 ðo 2 þ 3:76N 2 Þ!aCO 2 þ b 2 H 2O þ 3:76 a þ b 4 g 2 N 2 Varables to quantfy combustble mxtures Fuel/ar rato by weght: f ¼ m f m a For stochometrc mxture: f s ¼ m f m as Equvalence rato: f ¼ f f s ¼ m as m a Normalzed ar/fuel rato l ¼ AFR AFR s ¼ 1=f Percent of excess ar 1=f s ¼ 1 f =f s ¼ 1 f %EA ¼ 100 ðm a m as Þ ¼ 100 m a 1 m as m as ¼ 100 1 f 1 Global equaton for lean combuston fb1 C a H b O g þ 1 f a þ b 4 g ðo 2 þ 3:76N 2 Þ 2! aco 2 þ b 2 H 2O þ 3:76 a þ b f 4 g N 2 þ a þ b 2 4 g 1 2 f 1 O 2 n terms of l C a H b O g þ l aþ b 4 g ðo 2 þ 3:76N 2 Þ 2! aco 2 þ b 2 H 2O þ 3:76 l a þ b 4 g 2 N 2 þðl 1Þ a þ b 4 g 2 O 2 Global equaton for rch combuston f>1wth the assumpton that products contan unburned fuel C a H b O g þ 1 f a þ b 4 g ðo 2 þ 3:76N 2 Þ 2! a f CO 2 þ b 2f H 2O þ 3:76 a þ b f 4 g N 2 þ 1 1 C a H b O g 2 f Enthalpy of formaton (heat of formaton) determned by bomb calormeter D ^h o ¼ Q0 rxn;v þ DN ^R u T 0 N ;P DN ¼ X N ;P X N ;R ¼ b 4 þ g 2 1
2.5 Chapter Summary 43 where Q 0 rxn;v s the heat released from a constant-volume reactor where the products and reactants are at STP. Heatng values at STP (T 0 ) from a constant-volume reactor P N ;R D^h o ;R P N ;P D ^h o ;P þ HHV ¼ N fuel M fuel P N ;p P N ;R ^R u T 0 ðmj=kgþ LHV ¼ HHV N H2O;PM H2O h fg ; h fg ¼ 2;440kJ=kg N fuel M fuel Heatng values at STP (T 0 ) determned from a constant-pressure reactor P N ;R D ^h o ;R P N ;P D ^h o ;P HHV ¼ N fuel M fuel Adabatc flame temperature for reactants at standard condtons Method 1: Estmate based on average ^c p values N fuel M fuel LHV þ P N ;R ^h s;r ðt R Þ T P ¼ T 0 þ P T P T R þ N fuelm fuel LHV P N ;P^c p N ;P^c p or f mxture s not stochometrc: mass-base analyss usng LHV and f Method 2: Enthalpy Balance f b 1 T P ¼ T R þ f LHV ð1 þ f Þc p ¼ T R þ f f s LHV ð1 þ f f s Þc p f > 1 T P ¼ T R þ f s LHV ¼ T R þ f s LHV ð1 þ f Þc p ð1 þ f f s Þc p H P ðt P Þ¼H R ðt R Þ H P ðt P Þ¼ X N ;P ^h;p ¼ X N ;P ½D ^h o ;P þ ^h s;p ðt P ÞŠ Tral and error of T P such that H P (T P ) matches H R (T R )
44 2 Thermodynamcs of Combuston Exercses 2.1 Consder an sentropc combuston system wth a total of K speces. Assumng constant specfc heats, show that the mxture temperature and pressure at two dfferent states are related to the respectve pressures as T 2 ¼ P ðg 1Þ=g 2 T 1 P 1 where g ¼ P K ¼1 P K ¼1 m c p; m c v; : 2.2 Measurements of exhaust gases from a methane-ar combuston system show 3% of oxygen by volume (dry base) n the exhaust. Assumng complete combuston, determne the excess percentage of ar, equvalence rato, and fuel/ar rato. 2.3 There has been a lot of nterest about replacng gasolne wth ethanol, but s ths really a good dea? We re gong to compare a blend of ethanol (70% ethanol and 30% gasolne by volume) to gasolne. Calculate the lower heatng value (LHV) of a 70% ethanol/30% sooctane mxture n terms of kj/mol of fuel. Assume complete combuston. How does ths compare to the tabulated value for gasolne (sooctane)? Assumng a 20% thermal effcency, f you need to get 100 kw of power from an engne, how much of each fuel (n mol/ s) do you need? If you have a stochometrc mxture of the ethanol/gasolne blend and ar n your 100 kw engne, how much CO 2 are you emttng n g/s? How does ths compare to the same engne runnng a stochometrc mxture of 100% gasolne and ar? 2.4 Gasolne s assumed to have a chemcal composton of C 8.26 H 15.5. (a) Determne the mole fractons of CO 2 and O 2 n the exhaust for an engne wth normalzed ar/fuel rato l ¼ 1.2 wth the assumpton of complete combuston. (b) The enthalpy of formaton of C 8.26 H 15.5 s 250 MJ/kmol. Determne the LHV of gasolne n terms of MJ/kg. The molecular mass of C 8.26 H 15.5 s 114.62 kg/kmol. (c) Usng an average c p for the products at 1,200 K, estmate the adabatc flame temperature at constant pressure of 1 atm for the lean (l ¼ 1.2) mxture. 2.5 A mxture of methane gas and ar at 25 C and 1 atm s burned n a water heater at 150% theoretcal ar. The mass flow rate of methane s 1.15 kg/h. The exhaust gas temperature was measured to be 500 C and approxmately
Exercses 45 Q Addtonal propane Q 3-way catalyst heater Staton 1 Staton 2 T 1 = 500K T 2 T 0 φ = 0.8 Fg. 2.5 Exercse 2.7 1 atm. The volumetrc flow rate of cold water (at 22 C) to the heater s 4 L/mn. (a) Draw a schematc of the water heater and name ts most mportant elements. (b) Usng Cantera, determne the amount of heat generated from burnng of 1 kg of methane. (c) Calculate the temperature of the hot water f the heat exchanger were to have an effcency of 1.0,.e., perfect heat transfer. 2.6 An acetylene-oxygen torch s used n ndustry for cuttng metals. (a) Estmate the maxmum flame temperature usng average specfc heat c p. (b) Measurements ndcate a maxmum flame temperature of about 3,300 K. Compare wth the result from (a) and dscuss the man reasons for the dscrepancy. 2.7 A space heater burns propane and ar wth ntake temperature at T 0 ¼ 25 C and pressure at 1 atm (see Fg. 2.5). The combustble mxture enters the heater at an equvalence rato f ¼ 0.8. The exhaust gases ext at temperature T 1 ¼ 500 K and contan CO 2,H 2 O, O 2, and N 2 only at staton 1. In order to use a 3-way catalyst for exhaust treatment, addtonal propane s njected nto the exhaust to consume all the remanng oxygen n the exhaust such that the gases enterng the catalyst contan only CO 2,H 2 O, and N 2 at staton 2. Assume that the entre system s at P ¼ 1 atm and complete combuston occurs n both the heater and n the exhaust secton. (a) The volumetrc flow rate of propane enterng the heater s 1 L/mn. Determne the njecton rate of propane nto the exhaust between staton 1 and staton 2 (see Fg. 2.5). Note that the propane at the njecton staton s at the same condtons as heater nlet,.e., T ¼ 25 C and P ¼ 1 atm. (b) Wth the assumpton of constant specfc heats for the gases, estmate the temperature at staton 2, T 2. The specfc heat can be approxmated by that of N 2 at 700 K as ^c p ¼ 30:68 kj=kmol K,
46 2 Thermodynamcs of Combuston Fuel: T fuel = 25 C P fuel = 1 atm Ar: T ar = 427 C P ar = 1 atm. Q loss Products Fg. 2.6 Exercse 2.9 2.8 Two grams of sold carbon, C(s), are combusted wth pure oxygen n a 500 cm 3 bomb calormeter ntally at 300 K. After the carbon s placed nsde the bomb, the chamber s evacuated and then flled wth gaseous oxygen from a pressurzed tank. (a) Determne the mnmum O 2 pressure nsde the bomb necessary to allow complete combuston of the sold carbon. (b) When the bomb s cooled back to ts ntal temperature of 300 K, determne the pressure nsde the bomb. 2.9 Consder the combuston chamber n a jet engne at crusng alttude. For smplcty, the combustor s operated at 1 atm of pressure and burns a stochometrc (f ¼ 1) mxture of n-heptane (C 7 H 16 ) and ar. The ntake condtons are as ndcated n Fg. 2.6. (a) Wrte the stochometrc chemcal reacton for the fuel wth ar. (b) If the mass flow rate of fuel s 1 kg/s, what s the mass flow rate of ar? (c) What s the rate of heat loss from the combuston chamber f 10% of the LHV (heat of combuston) of the fuel s lost to surroundngs? (d) What s the temperature of the products? (e) How does the temperature change f we burn fuel rch (f > 1)? How about fuel lean (f < 1)? (Hnt: Easest to show wth a plot) 2.10 An afterburner s a devce used by jet planes to ncrease thrust by njectng fuel after the man combustor. A schematc of ths system s shown n Fg. 2.7. In the man combustor, hexane s burned wth ar at an equvalence rato of f ¼ 0.75. The products of the man combustor are CO 2,H 2 O, O 2 and N 2, all of whch enter the afterburner. In the afterburner, addtonal hexane s njected such that the equvalence rato s f ¼ 1.25. In the afterburner the hexane reacts wth the excess O 2 from the man combustor to form CO, H 2 O, and CO 2 only. Combned wth the products of the man combustor, the gases extng the afterburner are CO, CO 2,H 2 O, O 2 and N 2. The entre system s
Exercses 47 ar T = 20 o C man combustor T 1 =? CO 2, H 2 O, O 2, N 2 Afterburner CO, CO2, H 2 O, O 2, N 2 T 2 =? hexane T = 20 o C hexane T = 20 o C Fg. 2.7 Exercse 2.10 nsulated, and the pressure everywhere s atmospherc. The nlet temperature of the hexane and ar s 20 C. Determne the temperature of the exhaust gases at each stage (Fg. 2.7). Note: An approxmate answer s suffcent and t can be assumed that the specfc heats for the gases are constant and approxmately equal to that of N 2 at 1,000 K.
http://www.sprnger.com/978-1-4419-7942-1