5. COMBUSTIO AD THERMOCHEMISTRY Ch5 1
Overview Definition & mathematical determination of chemical equilibrium, Definition/determination of adiabatic flame temperature, Prediction of composition and temperature of combusted gases as a function of initial temperature, Prediction of amounts of fuel & oxidizer, Thermochemical changes during expansion process in nozzle. Performance Parameters: C F γ +1 2γ 2 2 γ 1 p γ 1 γ +1 1 e p 0 γ 1 γ + p e p a p 0 ε c * RT 0 γ γ +1 2 γ +1 γ 1 Performance depends on: T, MW, p 0, p e, p a, γ Ch5 2
Overview Important Concepts & Elements of Analysis Conversion of Chemical Energy to Heat Simple Treatment of Properties of Gases Balancing Chemical Reactions - Stoichiometry Adiabatic Flame Temperature Chemical Equilibrium and Gibbs Free Energy ozzle Expansion Effects Thermochemical Calculations Ch5 3
5.1 THERMODYAMICS OF GAS MIXTURES Ch5 4
Perfect Gas Perfect Gas Law relates pressure, temperature and density for a perfect gas/ mixture of gases : pv nrt mrt p v RT Universal Gas Constant: Gas Constant: R 8.314 R R M J mol K Calorically Perfect Gas: Internal Energy Enthalpy du c v dt u 2 u 1 c v (T 2 T 1 ) dh c p dt h 2 h 1 c p (T 2 T 1 ) Specific Heat Relationships: c p c v R γ c p c v Definition of Mole : A mole represents the amount of gas, which contains Avogadro s number of gas molecules: 6.02 10 23 molecules/mol. Ch5 5
Gibbs-Dalton Law Properties of a mixture is determined by the properties of constituents according to Gibbs Dalton Law: The pressure of a mixture of gases is equal to the sum of the pressure of each constituent when each occupies alone the volume of the mixture at the temperature of the mixture. The internal energy and the entropy of a mixture are equal, respectively, to the sums of the internal energies and the entropies of its constituents when each occupies alone the volume of the mixture at the temperature of the mixture. V Container T Container p Container Temperature Pressure Volume Energy Entropy Enthalpy T mix T 1 T 2 T p mix p 1 + p 2 + p 3 + p V mix m mix v mix m 1 v 1 m 2 v 2 m v E mix m mix e mix m 1 e 1 + m 2 e 2 + + m e i1 p i i1 m i e i Bar denotes Property with respect to Molar Quantity S mix m mix s mix m 1 s 1 + m 2 s 2 + + m s s mix S mix n mix H mix m mix h mix m 1 h 1 + m 2 h 2 + + m h h mix H mix n mix Ch5 6
Mixture of Gases Composition of a gas mixture is expressed by either the constituent mass fractions or mole fractions. Definition of Mass Fraction: y i m i m mix m i 1 m i y i 1 i1 Equivalent Molecular Weight: M mix equiv 1 ( y i M i ) i1 m mix n mix V Container T Container p Container Perfect Gas Law Pressure (Gibbs-Dalton Law) Enthalpy Entropy where species entropy is p i V m i R i T n i RT p i1 h mix p i i y i h i s mix (T,p) y i s i (T,p i ) i s i (T, p i ) s i (T, p ref ) R ln p i p ref Ch5 7
Mixture of Gases Definition of Mole Fraction: Equivalent Molecular Weight: x i n i n mix M mix equiv n i 1 x i M i i1 n i m mix n mix x i 1 i1 Perfect Gas Law Pressure (Gibbs-Dalton Law) Partial Pressure: Enthalpy Entropy where species entropy is p i V m i R i T n i RT p i1 p i x i p h mix p i i x i h i s mix (T,p) x i s i (T,p i ) i s i (T, p i ) s i (T, p ref ) Rln p i p ref V Container T Container p Container Ch5 8
Mixture of Gases Relationship between Mass and Mole Fractions: x i y i M mix M i Other Relationships for a Gas Mixture: Specific Heat: c p,mix i1 c p,i y i Ratio of Specific Heat: γ mix c p,mix c v,mix c p,mix c p,mix R mix Ch5 9
5.2 1 st LAW OF THERMODYAMICS Ch5 10
1 st LTD - Fixed Mass First law of thermodynamics embodies the fundamental principle of conservation of energy. Q and W are path functions and occur only at the system boundary. E is a state variable (property), E is path independent. Q System Boundary enclosing Fixed Mass m, E W Q W ΔE 1 2 Heat added to system in going from state 1 2 Work done by system on surrounding in going from state 1 2 Change in total system energy in going from state 1 2 Q W de dt q w de dt E m u + 1 2 v2 + g z Ch5 11
1 st LTD - Control Volume Conservation of energy for a steady-state, steady-flow system. Control Surface (CS) enclosing Control Volume (CV) ( ) m e + p v inlet ( ) outlet m e + p v dm CV dt 0 de CV dt 0 Assumptions: Q CV Q CV W CV m e outlet m e inlet + m p o v o p i v i Rate of heat transferred across the CS, from the surrounding to the CV. Rate of all work done by CV, including shaft work but excluding flow work. Q CV W CV m h o h i Rate of energy flowing out of CV. ( ) + 1 2 v o Control Volume is fixed relative to the coordinate system. Eliminates any work interactions associated with a moving boundary, Eliminates consideration of changes in kinetic and potential energies of CV itself. Properties of fluid at each point within CV, or on CS, do not vary with time. Fluid properties are uniform over inlet and outlet flow areas. There is only one inlet and one exit stream. W CV Rate of energy flowing into CV. 2 2 ( v i ) + g z o z i ( ) ( ) et rate of work associated with pressure forces where fluid crosses CS, flow work. Ch5 12
TD PROCESSES in CHEM. SYSTEMS Chemical systems (chemical reactions) are treated as either constant-volume or constant-pressure processes. Energy Equation (1st Law of TD) E U + E potential + E kinetic Q W shaft W flow Inside a rocket combustion chamber, fluid velocity (E kin ) is small and height changes of the fluid mass (E pot ) is negligible. Energy contained in the fluid is governed by the internal energy of the hot combustion gas. E U de du (δq δw shaft δw flow ) Work contribution in a rocket combustion chamber results from changes in specific volume of pressure. The fluid doesn t perform any mechanical work (W shaft 0). V 2 W p (ext ) dv δw flow p dv V 1 Constant Volume (Isochoric) Process: du Q Constant Pressure (Isobaric) Process: du Q p dv H U + pv dh Q Ch5 13
5.3 REACTAT AD PRODUCT MIXTURES Ch5 14
STOICHIOMETRY The stoichiometric quantity of oxidizer (substance A) is just that amount needed to completely burn a quantity of fuel (substance B): An oxidizer-fuel mixture is LEA, when there is more than a stoichiometric quantity of oxidizer in the mixture. An oxidizer-fuel mixture is RICH, when there is less than a stoichiometric quantity of oxidizer in the mixture. Stoichiometric Chemical Reaction: Examples: CH 4 + 2O 2 CO 2 + 2H 2 O One mole of methane and 2 moles of oxygen form one mole of carbon dioxide and 2 mole of water. H 2 + 1 2 O 2 H 2 O One mole of H 2 and a half mole of O 2 form one mole of H 2 O. Ch5 15
STOICHIOMETRY Stoichiometric Oxidizer-Fuel Ratio: O m oxidizer F m stoic fuel stoic A F stoic m air m fuel stoic n air n fuel M air M fuel 4.76 a 1 M air M fuel Equivalence Ratio Φ : ( ) stoic Φ O F O F F O ( F O) stoic where This ratio is a quantitative indicator whether a fuel-oxidizer mixture is Lean:! < 1 Rich:! > 1 Stoichiometric:! 1 O F n oxygen n fuel M oxygen M fuel Other Parameters: Percent Stoichiometric Oxidizer: Percent Excess Oxidizer: % stoichiometric oxidizer 100% Φ % excess oxidizer (1 Φ) Φ 100% Ch5 16
AIR (O2)/FUEL COMBUSTIO Stoichiometric Combustion of Air and Fuel (Hydrocarbon) C x H y + a O 2 + 3.76 2 x & y define the hydrocarbon fuel! ( ) x CO 2 + y 2 H 2 O + 3.76a 2 a x + y 4 Lean Combustion of Air and Fuel ( ) b CO 2 + c H 2 O + d O 2 + 3.76 a 2 C x H y + a O 2 + 3.76 2 Balancing Chemical Reaction: C : x b b x H : y 2c c 1 2 y O : 2a 2b + c + 2d a x + 1 4 y + d Rich Combustion of Air and Fuel ( ) b CO 2 + c H 2 O + d C x H y + 3.76 a 2 C x H y + a O 2 + 3.76 2 Balancing Chemical Reaction: C : x b + x d b x (1 d) H : y 2c + yd c 1 y (1 d) 2 O : 2a 2b + c a (x + 1 y) (1 d) 4 Ch5 17
Examples Example #1: A small, low-emission, stationary gas-turbine engine operates at full load (3,950 kw) at an equivalence ratio of 0.286 with an air flowrate of 15.9 kg/s. The equivalent composition of the fuel (natural gas) is C 1.16 H 4.32. Determine the fuel mass flow rate and the operating air-fuel ratio for the engine. Ch5 18
Examples Example #2: A natural-gas-fired industrial boiler operates with an oxygen concentration of 3 mole percent in the flue gases. Determine the operating air-fuel ratio and the equivalence ratio. Treat the natural gas as methane. Ch5 19