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Thermodynamics, Flame Temperature and Equilibrium

Thermodynamics, Flame Temperature and Equilibrium
Thermodynamics, Flame Temperature and Equilibrium CEFRC Combustion Summer School 2014 Prof. Dr.Ing. Heinz Pitsch Copyright ©2014 by Heinz Pitsch. This material is not to be sold, reproduced or distributed without prior written permission of the owner, Heinz Pitsch. Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Thermodynamic quantities • Thermodynamics, flame • Flame temperature at complete conversion temperature, and equilibrium • Chemical equilibrium • Governing equations • Laminar premixed flames: Kinematics and Burning Velocity • Laminar premixed flames: Flame structure • Laminar diffusion flames 2 Thermodynamic Quantities First law of thermodynamics balance between different forms of energy • Change of specific internal energy: du specific work due to volumetric changes: dw = pdv , v=1/r specific heat transfer from the surroundings: dq • Related quantities specific enthalpy (general definition): specific enthalpy for an ideal gas: • Energy balance: 3 Multicomponent system • Specific internal energy and specific enthalpy of mixtures • Relation between internal energy and enthalpy of single species 4 Multicomponent system • Ideal gas  u and h only function of temperature • If c is specific heat capacity at constant pressure and h is reference pi i,ref enthalpy at reference temperature T , temperature dependence of partial ref specific enthalpy is given by • Reference temperature may be arbitrarily chosen, most frequently used: T = 0 K or T = 298.15 K ref ref 5 Multicomponent system • Partial molar enthalpy H is i and its temperature dependence is where the molar heat capacity at constant pressure is • In a multicomponent system, the specific heat capacity at constant pressure of the mixture is 6 Determination of Caloric Properties • Molar reference enthalpies of chemical species at reference temperature are listed in tables • Reference enthalpies of H , O , N and solid carbon C were chosen as zero, 2 2 2 s because they represent the chemical elements • Reference enthalpies of combustion products such as CO and H O are 2 2 typically negative 7 Determination of Caloric Properties • Temperature dependence of molar enthalpy, molar entropy, and molar heat capacities may be calculated from polynomials • Constants a for each species i are listed in tables j 8 Determination of Caloric Properties NASA Polynomials for two temperature ranges and standard pressure p = 1 atm 9 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Thermodynamic quantities • Thermodynamics, flame • Flame temperature at complete conversion temperature, and equilibrium • Chemical equilibrium • Governing equations • Laminar premixed flames: Kinematics and Burning Velocity • Laminar premixed flames: Flame structure • Laminar diffusion flames 10 Flame Temperature at Complete Conversion • First law of thermodynamics for an adiabatic system at constant pressure (dq = 0, dp = 0) with only reversible work (dw = pdv) • From first law with follows • Integrated from the unburnt (u), to burnt (b) gives or 11 Flame Temperature at Complete Conversion • With and follows • Specific heats to be calculated with the mass fractions of the burnt and unburnt gases 12 Flame Temperature at Complete Conversion • For a onestep global reaction, the left hand side of may be calculated by integrating which gives / h i,ref and finally 13 Flame Temperature at Complete Conversion • Definition: Heat of combustion • Heat of combustion changes very little with temperature • Often set to: • Simplification: T = T and assume c approximately constant u ref p,b • For combustion in air, nitrogen is dominant in calculating c p,b • Value of c somewhat larger for CO , somewhat smaller for O , while that pi 2 2 for H O is twice as large 2 • Approximation for specific heat of burnt gas for lean and stoichiometric mixtures c = 1.40 kJ/kg/K p 14 Flame Temperature at Complete Conversion • Assuming c constant and Q = Q , the flame temperature at complete conversion p ref for a lean mixture (Y = 0) is calculated from F,b  Coupling function between fuel mass fraction and temperature • With n = n' follows F F 15 Flame Temperature at Complete Conversion • For a rich mixture should be replaced by • One obtains similarly for complete consumption of the oxygen (Y = 0) O ,b 2 16 Flame Temperature at Complete Conversion • Equations and may be expressed in terms of the mixture fraction • Introducing and and specifying the temperature of the unburnt mixture by where • T is the temperature of the oxidizer stream and T that of the fuel stream 2 1 • c assumed to be constant p 17 Flame Temperature at Complete Conversion • Equations and then take the form • The maximum temperature appears at Z = Z : st 18 Flame Temperature at Complete Conversion BurkeSchumann Solution: Infinitely fast, irreversible chemistry 19 Flame Temperature at Complete Conversion • The table shows for combustion of pure fuels (Y = 1) in air (Y = 0.232) F,1 O ,2 2 with T = 300 K and c = 1.4 kJ/kg/K u,st p stoichiometric mixture fraction stoichiometric flame temperatures for some hydrocarbonair mixtures 20 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Thermodynamic quantities • Thermodynamics, flame • Flame temperature at complete conversion temperature, and equilibrium • Chemical equilibrium • Governing equations • Laminar premixed flames: Kinematics and Burning Velocity • Laminar premixed flames: Flame structure • Laminar diffusion flames 21 Chemical Equilibrium • Assumption of complete combustion is approximation, because it disregards the possibility of dissociation of combustion products • More general formulation is assumption of chemical equilibrium Complete combustion then represents limit of infinitely large equilibrium constant (see below) • Chemical equilibrium and complete combustion are valid in the limit of infinitely fast reaction rates only, which is often invalid in combustion systems Importance of kinetics 22 Chemical Equilibrium • Chemical equilibrium assumption Only good for hydrogen diffusion flames For hydrocarbon diffusion flames • Overpredicts formation of intermediates such as CO and H 2 for rich conditions by large amounts • Equilibrium assumption represents an exact thermodynamic limit 23 Chemical potential and the law of mass action • Partial molar entropy S of chemical species in a mixture of ideal gases depends i on partial pressure where p = 1 atm and 0 depends only on temperature • Values for the reference entropy S are listed in tables i,ref 24 Chemical potential and the law of mass action • Partial molar entropy may be used to define chemical potential where is chemical potential at 1 atm th • Condition for chemical equilibrium for l reaction is given by 25 Chemical potential and the law of mass action • Using in leads to • Defining the equilibrium constant K by pl one obtains the law of mass action 26 Chemical potential and the law of mass action • Equilibrium constants for three reactions 27 Equilibrium Constants • Calculation of equilibrium constants K (T) from the chemical pk potentials with:  Enthalpies of formation  Entropies of formation  Heat capacities • Approximation  Neglect temperature dependence of heat capacities 28 Approximation for Equilibrium Constants • Equilibrium constants: • With it follows for constant c p,i • Approximation: 29 Approximation for equilibrium constants • With follows 30 Properties for gases at T = 298,15 K ref M h s i i,m,ref i,m,ref  A,i B,i kg/kmol kJ/mol kJ/mol K 1 H 1,008 217,986 114,470 1,2261 1,9977 2 HNO 31,016 99,579 220,438 1,0110 4,3160 3 OH 17,008 39,463 183,367 3,3965 2,9596 4 HO 33,008 20,920 227,358 ,1510 4,3160 2 5 H 2,016 0,000 130,423 2,4889 2,8856 2 6 H O 18,016 241,826 188,493 1,6437 3,8228 2 7 H O 34,016 136,105 233,178 8,4782 5,7218 2 2 8 N 14,008 472,645 153,054 5,8661 1,9977 9 NO 30,008 90,290 210,442 5,3476 3,1569 10 NO 46,008 33,095 239,785 1,1988 4,7106 2 11 N 28,016 0,000 191,300 3,6670 3,0582 2 12 N O 44,016 82,048 219,777 5,3523 4,9819 2 31 Properties for gases at T = 298,15 K ref M h s i i,m,ref i,m,ref   B,i A,i kg/kmol kJ/mol kJ/mol K 13 O 16,000 249,194 160,728 6,85561 1,9977 14 O 32,000 0,000 204,848 4,1730 3,2309 2 15 O 48,000 142,674 238,216 3,3620 5,0313 3 16 NH 15,016 331,372 180,949 3,0865 2,9596 17 NH 16,024 168,615 188,522 1,9835 3,8721 2 18 NH 17,032 46,191 192,137 8,2828 4,8833 3 19 N H 30,032 212,965 218,362 8,9795 5,4752 2 2 20 N H 31,040 153,971 228,513 17,5062 6,9796 2 3 21 N H 32,048 95,186 236,651 25,3185 8,3608 2 4 22 C 12,011 715,003 157,853 6,4461 1,9977 23 CH 13,019 594,128 182,723 2,4421 3,,0829 24 HCN 27,027 130,540 201,631 5,3642 4,6367 32 Properties for gases at T = 298,15 K ref M h s i i,m,ref i,m,ref  A,i B,i kg/kmol kJ/mol kJ/mol K 25 HCNO 43,027 116,733 238,048 10,1563 6,0671 26 HCO 29,019 12,133 224,421 ,2313 4,2667 27 CH 14,027 385,220 180,882 5,6013 4,2667 2 28 CH O 30,027 115,896 218,496 8,5350 5,4012 2 29 CH 15,035 145,686 193,899 10,7155 5,3026 3 30 CH OH 31,035 58,576 227,426 15,3630 6,6590 2 31 CH 16,043 74,873 185,987 17,6257 6,1658 4 32 CH OH 32,043 200,581 240,212 18,7088 7,3989 3 33 CO 28,011 110,529 197,343 4,0573 3,1075 34 CO 44,011 393,522 213,317 5,2380 4,8586 2 35 CN 26,019 456,056 202,334 4,6673 3,1075 36 C 24,022 832,616 198,978 1,9146 3,5268 2 33 Properties for gases at T = 298,15 K ref M h s i i,m,ref i,m,ref  A,i B,i kg/kmol kJ/mol kJ/mol K 37 C H 25,030 476,976 207,238 4,6242 4,6367 2 38 C H 26,038 226,731 200,849 15,3457 6,1658 2 2 39 C H 27,046 279,910 227,861 17,0316 6,9056 2 3 40 CH CO 43,046 25,104 259,165 24,2225 8,5334 3 41 C H 28,054 52,283 219.,468 26,1999 8,1141 2 4 42 CH COH 44,054 165,979 264.061 30,7962 9,6679 3 43 C H 29,062 110,299 228,183 32,6833 9,2980 2 5 44 C H 30,070 84,667 228,781 40,4718 10,4571 2 6 45 C H 44,097 103,847 269,529 63,8077 14,7978 3 8 46 C H 50,060 465,679 250,437 34,0792 10,0379 4 2 47 C H 51,068 455,847 273,424 36,6848 10,8271 4 3 48 C H 56,108 16,903 295,298 72,9970 16,7215 4 8 34 Properties for gases at T = 298,15 K ref M h s i i,m,ref i,m,ref  A,i B,i kg/kmol kJ/mol kJ/mol K 49 C H 58,124 134,516 304,850 86,8641 19,0399 4 10 50 C H 70,135 35,941 325,281 96,9383 20,9882 5 10 51 C H 72,151 160,247 332,858 110,2702 23,3312 5 12 52 C H 84,152 59,622 350,087 123,2381 25,5016 6 12 53 C H 86,178 185,560 380,497 137,3228 28,2638 6 14 54 C H 98,189 72,132 389,217 147,4583 29,6956 7 14 55 C H 100,205 197,652 404,773 162,6188 32,6045 7 16 56 C H 112,216 135,821 418,705 173,7077 34,5776 8 16 57 C H 114,232 223,676 430,826 191,8158 37,6111 8 18 58 C H O 44,054 51,003 243,044 34,3705 2 4 59 HNO 63,016 134,306 266,425 19,5553 3 60 He 4,003 0,000 125,800 35 Example 1: Equilibrium Calculation of the NOair system • Calculation of the equilibrium concentration ppm of NO in air  Temperatures up to 1500 K  p = p = 1 atm 0  Global reaction:   iB iA N 3,6670 3,0582 2 O 4,1730 3,2309 2 NO 5,3476 3,1569 36 Example 1: Equilibrium Calculation of the NOair system   iB iA N 3,6670 3,0582 2 O 4,1730 3,2309 2 NO 5,3476 3,1569 37 Example 1: Equilibrium Calculation of the NOair system • Law of mass action: • Assumption: (air) unchanged 38 Result: Equilibrium Calculation of the NOair system Result: T K X ppv NO . 16 . 10 300 3,52 10 3,52 10 . 8 . 2 600 2,55 10 2,55 10 . 5 1000 3,57 10 35,7 . 3 1500 1,22 10 1220 6 6 1 ppv = 10 = X 10 parts per million (volume fraction) i 6 6 1 ppm = 10 = Y 10 parts per million (mass fraction) i 39 Result: Equilibrium Calculation of the NOair system • Mole fraction of NO in equilibrium: • Equilibrium values for T = 2000 K and T = 400 K differ by 10 orders of magnitude Cooling due to expansion and heat losses in • High temperatures during exhaust system combustion lead to high Catalytic NOconcentration reduction Combustion • NO is retained to a large extent if gas is cooled down rapidly 40 Example 2: Equilibrium Calculation of the H air system 2 • Using the law of mass action one obtains for the reaction 2 H + O = 2 H O 2 2 2 the relation between partial pressures where was approximated using and the values for from the JanafTable 41 Example 2: Equilibrium Calculation of the H air system 2 • Introducing the definition the partial pressures are written with as where the mean molecular weight is 42 Example 2: Equilibrium Calculation of the H air system 2 • The element mass fractions of the unburnt mixture are • These are equal to those in the equilibrium gas where while Z remains unchanged N 43 Example 2: Equilibrium Calculation of the H air system 2 • These equations lead to the following nonlinear equation for G H O,b 2 44 Example 2: Equilibrium Calculation of the H air system 2 • Equation has one root between G = 0 and the maximum values H O,b 2 G = Z /2W and G = Z /W H O,b H H H2O,b O O 2 which correspond to complete combustion for lean and rich conditions in the limit • The solution, which is a function of the temperature, may be found by successively bracketing the solution within this range • The temperature is then calculated by employing a Newton iteration on leading to 45 Example 2: Equilibrium Calculation of the H air system 2 • The iteration converges readily following where i is the iteration index • The solution is plotted here for a hydrogenair flame as a function of the mixture fraction for T = 300 K u 46 Result: Equilibrium Calculation of the H air system 2 • Equilibrium mass fractions of H , O and H O 2 2 2 for p = 1 bar and p = 10 bar and different temperatures • T↑  Y ↓ H2O • p↑  Y ↑ H2O 47 Conclusion: Pressure and temperature dependency of the equilibrium constant • Temperature dependence  Exothermic reactions: Dh 0  dK /dT 0 m,ref p  Equilibrium is shifted towards educts with increasing temperature • Pressure dependence  Less dissociation at higher pressure  Le Chatelier‘s Principle Equilibrium tries to counteract the imposed changes in temperature and pressure 48 Summary Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Thermodynamic quantities • Thermodynamics, flame • Flame temperature at complete conversion temperature, and equilibrium • Chemical equilibrium • Governing equations • Laminar premixed flames: Kinematics and Burning Velocity • Laminar premixed flames: Flame structure • Laminar diffusion flames 49
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