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AME 436 Energy and Propulsion Lecture 3 Chemical thermodynamics concluded: Equilibrium thermochemistry

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2 AME Spring Lecture 3 - Chemical Thermodynamics 2 Outline Why do we need to invoke chemical equilibrium? Degrees Of Reaction Freedom (DORFs) Conservation of atoms Second Law of Thermodynamics for reactive systems Equilibrium constants Application of chemical equilibrium to hydrocarbon-air combustion Application of chemical equilibrium to compression/expansion

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3 AME Spring Lecture 3 - Chemical Thermodynamics 2 Why do we need chemical equilibrium? (From lecture 2) What if we assume more products, e.g. 1CH 4 + 2(O N 2 ) ? CO 2 + ? H 2 O + ? N 2 + ? CO In this case how do we know the amount of CO vs. CO 2 ? And if if we assume only 3 products, how do we know that the “preferred” products are CO 2, H 2 O and N 2 rather than (for example) CO, H 2 O 2 and N 2 O? Need chemical equilibrium to decide - use 2nd Law to determine the worst possible end state of the mixture

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4 AME Spring Lecture 3 - Chemical Thermodynamics 2 Degrees of reaction freedom (DoRFs) If we have a reacting “soup” of CO, O 2, CO 2, H 2 O, H 2 and OH, can we specify changes in the amount of each molecule independently? No, we must conserve each type of atom (3 in this case) Conservation of C atoms: n CO + n CO2 = constant O atoms: n CO + 2n CO2 + 2n O2 + n H2O + n OH = constant H atoms: 2n H2O + 2n H2 + n OH = constant 3 equations, 6 unknown n i ’s 3 degrees of reaction freedom (DoRFs) # of DoRFs = # of different molecules (n) - # of different elements Each DoRF will result in the requirement for one equilibrium constraint

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5 AME Spring Lecture 3 - Chemical Thermodynamics 2 Conservation of atoms Typically we apply conservation of atoms by requiring that the ratios of atoms are constant, i.e. the same in the reactants as the products C atoms: n CO + n CO2 = constant O atoms: n CO + 2n CO2 + 2n O2 + n H2O + n OH = constant H atoms: 2n H2O + 2n H2 + n OH = constant Specifying n O /n H also would be redundant, so the number of atom ratio constraints = # of atoms - 1 What are these “constants?” Depends on initial mixture, e.g. for stoichiometric CH 4 in O 2, n C /n O = 1/4, n C /n H = 1/4

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6 AME Spring Lecture 3 - Chemical Thermodynamics 2 2nd law of thermo for reacting systems Constraints for reacting system First law: dE = Q - W = Q - PdV Second law: dS ≥ Q/T Combine: TdS - dU - PdV ≥ 0 for any allowable change in the state of the system For a system at fixed T and P (e.g. material in a piston- cylinder with fixed weight on top of piston, all placed in isothermal bath: d(TS-U-PV) ≥ 0, or per unit mass d(Ts-u-Pv) ≥ 0 Define “Gibbs function” g h - Ts = u + Pv - Ts Thus for system at fixed T and P: d(-g) ≥ 0 or dg ≤ 0 Thus at equilibrium, dg = 0 (g is minimum) Similarly, for system at fixed U and V (e.g. insulated chamber of fixed volume): ds ≥ 0, at equilibrium ds = 0 (s is maximum)

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7 AME Spring Lecture 3 - Chemical Thermodynamics 2 2nd law of thermo for reacting systems For a mixture of ideal gases with 1 DoRF that reacts according to A A + B B C C + D D, e.g. 1 CO 2 1 CO +.5 O 2 A = CO 2, A = 1, B = nothing, B = 0, C = CO, C = 1, D = O 2, D = 0.5 with dg = 0, “it can be shown”

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8 AME Spring Lecture 3 - Chemical Thermodynamics 2 Equilibrium constants Examples of tabulated data on K - (double-click table to open Excel spreadsheet with all data for CO, O, CO 2, C, O 2, H, OH, H 2 O, H 2, N 2, NO at 200K K) Note K = 1 at all T for elements in their standard state (e.g. O 2 )

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9 AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium - example For a mixture of CO, O 2 and CO 2 (and nothing else, note 1 DoRF for this case) at 10 atm and 2500K with C:O = 1:3, what are the mole fractions of CO, O 2 and CO 2 ? 1 CO 2 1 CO +.5 O 2 3 equations, 3 unknowns: X CO2 = , X O2 = , X CO = At 10 atm, 1000K: X CO2 =.6667, X O2 = , X CO = 2.19 x At 1 atm, 2500K: X CO2 = , X O2 = , X CO = With N 2 addition, C:O:N = 1:3:6, 10 atm, 2500K: X CO2 = , X O2 = , X CO = , X N2 = (X CO2 /X CO = 29.3 vs without N 2 dilution) (note still 1 DoRF in this case) Note high T, low P and dilution favor dissociation

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10 AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium How do I know to write the equilibrium as 1 CO 2 1 CO +.5 O 2 and not (for example) 2 CO + 1 O 2 2 CO 2 For the first form which is the appropriate expression of equilibrium for the 2nd form - so the answer is, it doesn’t matter as long as you’re consistent

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11 AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium - hydrocarbons Reactants: C x H y + rO 2 + sN 2 (not necessarily stoichiometric) Assumed products: CO 2, CO, O 2, O, H 2 O, OH, H, H 2, N 2, NO How many equations? 10 species, 4 elements 6 DoRFs 6 equil. constraints 4 types of atoms 3 atom ratio constraints Conservation of energy (h reactants = h products ) (constant pressure reaction) or u reactants = u products (constant volume reaction) Pressure = constant or (for const. vol.) = 12 equations How many unknowns? 10 species 10 mole fractions (X i ) Temperature Pressure = 12 equations

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12 Equilibrium constraints - not a unique set, but for any set Each species appear in at least one constraint Each constraint must have exactly 1 DoRF Note that the initial fuel molecule does not necessarily appear in the set of products! If the fuel is a large molecule, e.g. C 8 H 18, its entropy is so low compared to other species that the probability of finding it in the equilibrium products is negligible! Example set (not unique) AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium - hydrocarbons

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13 AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium - hydrocarbons Atom ratios Sum of all mole fractions = 1 Conservation of energy (constant P shown)

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14 AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium - hydrocarbons This set of 12 simultaneous nonlinear algebraic equations looks hopeless, but computer programs (using slightly different methods more amenable to automation) (e.g. GASEQ) existGASEQ Typical result, for stoichiometric CH 4 -air, 1 atm, constant P

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15 AME Spring Lecture 3 - Chemical Thermodynamics 2 Chemical equlibrium - hydrocarbons Most of products are CO 2, H 2 O and N 2 but some dissociation (into the other 7 species) occurs Product is much lower than reactants - affects estimation of compression / expansion processes using Pv relations Bad things like NO and CO appear in relatively high concentrations in products, but will recombine to some extent during expansion By the time the expansion is completed, according to equilibrium calculations, practically all of the NO and CO should disappear, but in reality they don’t - as T and P decrease during expansion, reaction rates decrease, thus at some point the reaction becomes “frozen”, leaving NO and CO “stuck” at concentrations MUCH higher than equilibrium

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16 AME Spring Lecture 3 - Chemical Thermodynamics 2 Adiabatic flame temp. - hydrocarbons

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17 AME Spring Lecture 3 - Chemical Thermodynamics 2 Adiabatic flame temp - hydrocarbons Adiabatic flame temperature (T ad ) peaks slightly rich of stoichiometric - since O 2 is highly diluted with N 2, burning slightly rich ensures all of O 2 is consumed without adding a lot of extra unburnable molecules T ad peaks at ≈ 2200K for CH 4, slightly higher for C 3 H 8, iso-octane (C 8 H 18 ) practically indistinguishable from C 3 H 8 H 2 has far heating value per unit fuel mass, but only slightly higher per unit total mass (due to “heavy” air), so T ad not that much higher Also - massive dissociation as T increases above ≈ 2400K, keeps peak temperature down near stoichiometric Also - since stoichiometric is already 29.6% H 2 in air (vs. 9.52% for CH 4, 4.03% for C 3 H 8 ), so going rich does not add as many excess fuel molecules CH 4 - O 2 MUCH higher - no N 2 to soak up thermal energy without contributing enthalpy release Constant volume - same trends but higher T ad

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18 AME Spring Lecture 3 - Chemical Thermodynamics 2 Compression / expansion As previously advertised, compression / expansion are assumed to occur at constant entropy (not constant h or u) Three levels of approximation Frozen composition (no change in X i ’s), corresponds to infinitely slow reaction Equilibrium composition (X i ’s change to new equilibrium), corresponds to infinitely fast reaction (since once we get to equilibrium, no further change in composition can occur) Reacting composition (finite reaction rate, not infinitely fast or slow) - more like reality but MUCH more difficult to analyze since rate equations for 100s or 1000s of reactions are involved Which gives best performance? Equilibrium – you’re getting everything the gas has to offer; recombination (e.g. H + OH H 2 O) gives extra enthalpy release, more push on piston or more exhaust kinetic energy Frozen - no recombination, no extra heat release Reacting - somewhere between, most realistic

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19 AME Spring Lecture 3 - Chemical Thermodynamics 2 Entropy of an ideal gas mixture Depends on P AND T (unlike h and u, which depend ONLY on T) (1) (2) (3) (1) Start with gases A, B and C at temperature T and P o = 1 atm (2) Raise/lower each gas to pressure P ≠ 1 atm (same P for all) A P = P ref B P = P ref C P = P ref A P ≠ P ref B P ≠ P ref C P ≠ P ref A, B, C P ≠ P ref

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20 (3) Now remove dividers, V A V A + V B + V C (V = volume) Can also combine X i and P/P ref terms using partial pressures P i ; for an ideal gas mixture X i = P i /P, where (as above) P without subscripts is the total pressure = P i AME Spring Lecture 3 - Chemical Thermodynamics 2 Entropy of an ideal gas mixture

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21 AME Spring Lecture 3 - Chemical Thermodynamics 2 Compression / expansion Most useful form in terms of s = S/m, since mass (m) is constant = entropy of species i at reference pressure (1 atm) and temperature T (not necessarily 298K) (see my tables) ln(P/P ref ) - entropy associated with pressure different from 1 atm - P > P ref leads to decrease in s X i ln(X i ) - entropy associated with mixing ( X i < 1 means more than 1 specie is present - always leads to increase in entropy since - X i ln(X i ) > 0) Denominator is just the average molecular weight Units of s are J/kgK Use s reactants = s products constant for compression / expansion instead of h reactants = h products or u reactants = u products All other relations (atom ratio constraints, X i = 1, equilibrium constraints) still apply for compression & expansion processes

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22 AME Spring Lecture 3 - Chemical Thermodynamics 2 Compression / expansion Example - expansion of CO 2 -O 2 -CO mixture from 10 atm, 2500K to 1 atm in steady-flow control volume (e.g. nozzle) or control mass (e.g piston/cylinder) Initial state (mixture from lecture 2 where h was calculated): X CO = , X O2 = , X CO2 = , T = 2500, P = 10 atm h = kJ/kg, u = kJ/kg

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23 AME Spring Lecture 3 - Chemical Thermodynamics 2 Compression / expansion Expand at constant entropy to 1 atm, frozen composition: T = 1738K, X CO = , X O2 = , X CO2 = ,, h = kJ/kg, u = kJ/kg, s = 7381J/kgK Work done (control volume, steady flow) = h before - h after = kJ/kg Work done (control mass) = u before - u after = +762 kJ/kg Expand at constant entropy to 1 atm, equilibrium composition: T = 1794K, X CO = , X O2 = , X CO2 = (significant recombination) h = kJ/kg, u = kJ/kg, s = 7382 J/kgK Work done (control volume, steady flow) = kJ/kg (1.6% higher) Work done (control mass) = 787 kJ/kg (3.3% higher) Moral: let your molecules recombine!

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24 AME Spring Lecture 3 - Chemical Thermodynamics 2 Summary - Lecture 3 In order to understand what happens to a chemically reacting mixture if we wait a very long time, we need to apply 1st Law of Thermodynamics (conservation of energy) - but this doesn’t tell us what the allowable direction of the reaction is; A B or B A are equally valid according to the 1st Law 2nd Law of Thermodynamics (increasing entropy) - invokes restrictions on the direction of allowable processes (if A B is allowed then B A isn’t) Equilibrium occurs when the worst possible end state is reached; once this point is reached, no further chemical reaction is possible unless something is changed, e.g. T, P, V, etc. The statements of the 2nd law are ds ≥ 0 (constant u & v, e.g. a rigid, insulated box) dg = d(h - Ts) ≤ 0 (constant T and P, e.g. isothermal piston/cylinder)

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25 AME Spring Lecture 3 - Chemical Thermodynamics 2 Summary - Lecture 3 The application of the 2nd law leads to complicated looking expressions called equilibrium constraints that involve the concentrations of each type of molecule present in the mixture These equilibrium constraints are coupled with conservation of energy, conservation of each type of atom, and the ideal gas law (or other equations of state, not discussed in this class) to obtain a complete set of equations that determine the end state of the system, e.g. the combustion products

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