617 Appendix H Equilibrium Primer Consider a general reaction: or equivalently (H.1) Here, j is an index from 1 to m reactants, r j refers to the number of moles of the j th reactant, R j is the j th reactant, the double-headed arrow (↔) means that both the forward and reverse reactions occur (typically at different rates), k is an index from 1 to n products, p k is the number of moles of the k th product, and P k is the k th product. For Reaction H.1, we may define an equilibrium constant with the follow- ing relation: (H.2) The brackets denote the volume concentrations of the enclosed species. If we wish to express K eq in terms of mole fraction rather than concentration for gases, then, using the ideal gas law, we obtain (H.3) rr pp 11 22 11 22 RR PP++↔++$$ rp jj j m kk k n R == ∑∑ ↔ 11 P K pp rr k = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ PP RR P 12 12 12 12 $ $ pp k n j r j m k j = = ∏ ∏ ⎡ ⎣ ⎤ ⎦ 1 1 R K P RT yy yy ss rr S s S s r = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ++−−− 12 12 1 1 2 2 1 1 $$ $ R RR S R 2 2 1 1 r sr s k n r j m P RT y y kk k k k j $ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ∑ = = ∏ ∏ © 2006 by Taylor & Francis Group, LLC 618 Modeling of Combustion Systems: A Practical Approach Please note that P and R (italicized) are the pressure and universal gas constant, while P and R (nonitalicized) refer to products and reactants, respectively. For constant-pressure systems such as combustion, we may also incorpo- rate the pressure dependence as part of the equilibrium constant, as : (H.4) In either case, the equilibrium constant is defined as products over reactants. K (or K y ) is a function of temperature according to an Arrhenius relation: (H.5) Thus, K (or K y ) is only constant if the reaction temperature is constant. Since temperature is in the exponential, it usually overwhelms the temperature effects shown in Equation H.3. In particular, note that K y is neither necessarily dimensionless nor independent of pressure, except in the case that the num- ber of moles remains invariant. KK RT P yy pr kk = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ∑ K y y y p k n r j m k k k j = = = ∏ ∏ P R 1 1 KAe b T = − © 2006 by Taylor & Francis Group, LLC . 2006 by Taylor & Francis Group, LLC 6 18 Modeling of Combustion Systems: A Practical Approach Please note that P and R (italicized) are the pressure and universal gas constant, while P and R. reactant, the double-headed arrow (↔) means that both the forward and reverse reactions occur (typically at different rates), k is an index from 1 to n products, p k is the number of moles of. (nonitalicized) refer to products and reactants, respectively. For constant-pressure systems such as combustion, we may also incorpo- rate the pressure dependence as part of the equilibrium constant,