AME 513 Principles of Combustion Lecture 8 ... - Paul D. Ronney

Rankine-Hugoniot relations" What premixed flame propagation speeds are possible in 1D? Assumptions Ideal gas, steady, 1D, constant area, constant C P , C v , γ ≡ C P /C v Governing equations Equations of state: P = !RT;h 2! h 1= C P(T 2!T 1) Mass conservation: !m / A = ! 1u 1= ! 2u 2 Navier-Stokes, 1D, no viscosity:!u !u!x + !P!x = 0, !u = !m A = const. " !u2 + P = const " P 1+! 1u 2 21=P 2+! 2u 2 Energy conservation, no work input/output:h 1+ u 2 1/ 2 + q = h 2+ u 2 2/ 2q = heat input per unit mass = fQ R if due to combustion Mass, momentum, energy conservation eqns. can becombined yielding Rankine-Hugoniot relations:! " P 2" 1%$ !1'! 1 " P 2%"$ !1' " %1$ +1' = q ; u ! u 2 1= (u = " 1!1! !1#P 1" 2 & 2 # P 1 &#" 2 & RT 1u 1u 1" 2AME 513 - Fall 2012 - Lecture 8 - Premixed flames I31D momentum balance - constant-area duct "Coefficient of friction (C f )!C f"Wall drag force1 2#u 2 $ (Wall area)d(mu)! Forces = !dt! Forces = PA " (P + dP)A " C f(1/ 2!u 2 )(Cdx)d(mu)! = ! !mu = !mu " !m(u + du)dtCombine: AdP+ !mdu + C f(1/ 2!u 2 )Cdx = 0If C f= 0 : AP+ !mu = const; !m = !uA # P + !u 2 = const# P 1+ ! 1u 1 2 = P 2+ ! 2u 22Velocity upressure Pdensity !duct area Amass flux = !uAVelocity u + dupressure P + dPdensity !duct area Amass flux = !uAThickness dx,circumference C,wall area CdxAME 513 - Fall 2012 - Lecture 8 - Premixed flames I4• 2