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Advanced Transport Phenomena Module 4 Lecture 13 Momentum Transport: Shock Waves Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Momentum Transport: Shock Waves STEADY 1D COMPRESSIBLE FLUID FLOW Steady, frictionless flow of a nonreacting gas mixture in a constant-area duct with heat addition: Conservation equations may be written as: G const (mass), du dp u d d dh0 u q ''' d (momentum), (energy ), Gdu dp Gc p dT0 (q ''' Ad ) Gdq A STEADY 1D COMPRESSIBLE FLUID FLOW Mass & momentum conservation equations yield: p Gu const p 0 Since u G / Gv, p G p0 2 This locus on the p-v (or corresponding T-s) plane Rayleigh line (locus) STEADY 1D COMPRESSIBLE FLUID FLOW Local stagnation (or total) temperature u2 T0 T 2c p For a constant cp-gas mixture between any two duct sections 1 & 2, change in T0 is governed by heat addition per unit mass: q 12 c p (T0,2 T0,1 ) Since Tds c p dT dp (Gibbs ), Entropy change & Ma at each point along Rayleigh line may be calculated. STEADY 1D COMPRESSIBLE FLUID FLOW Steady one-dimensional flow of a perfect gas (with g1 .3) in a constant area duct, frictionless flow with heat addition STEADY 1D COMPRESSIBLE FLUID FLOW Steady compressible flow of a nonreacting gas mixture in a constant-area duct with friction but without heat addition: Conservation equations may be written as: G=constant (mass), du dp P G w d d A ho =constant (energy) 2 (Fanno Locus) 1G 1 2 C p T0 T ) Gv ) 2 2 STEADY 1D COMPRESSIBLE FLUID FLOW Steady one-dimensional flow of a perfect gas (with g1 .3) in a constant area duct, adiabatic flow with friction SHOCK WAVES Discontinuity separating two adjacent continua e.g., mixture of perfect gases, same EOS valid on both sides of discontinuity How do we apply conservation laws on field variables? Assume locally planar discontinuity, fixed in space, fed by a gas stream with known velocity normal to it, known thermodynamic state properties SHOCK WAVES Control volume and station nomenclature for applying conservation principles across a gas dynamic discontinuity separating two regions of flow in which diffusion processes can be neglected SHOCK WAVES Consider a macroscopic control volume, shrunk down to a “pillbox” of unit area straddling the figure Field variables “jump” across discontinuity “jump operator” ) 2 ) 1 SHOCK WAVES Conservation equations across a discontinuity without chemical reaction: (total mass ), u 0 ui 0 ( species mass), uu p (momentum), uh0 0 (energy ) (entropy ) us 0 SHOCK WAVES Mass flux G u , G 0 (total mass), i 0 ( special mass), G u p (normal momentum), (energy ), h0 0 (entropy ), s 0 SHOCK WAVES In general, G, wi, h0 are continuous (no jump) across discontinuity; u, p, T, v (≡ 1/), s jump. Compatible with conservation principles, relevant EOS Combining total mass & normal momentum relations: p u 1 u 2 SHOCK WAVES When discontinuity becomes a sufficiently weak compression wave, positive entropy jump is negligible; hence 1/2 p u 1 u2 a sconst For a perfect gas, then: 1/2 p a s g RT M 1/2 SHOCK WAVES For a discontinuity of arbitrary strength, final state must lie on intersection of Fanno and Rayleigh loci passing through initial state on T-s diagram, corresponding to common mass flux G Rayleigh line links all states with same p + Gu (irrespective of heat addition) Fanno locus links all states with same stagnation enthalpy irrespective of viscous dissipation SHOCK WAVES Fanno and Rayleigh loci for the same mass flux G, displayed on the T-s plane. The normal shock transition goes from the supersonic intersection to the subsonic intersection SHOCK WAVES Rankine – Hugoniot interrelation: 1 h p .2 1 ) . 2 Defines a locus (“shock adiabat”) on p-v plane along which final state must lie For a perfect gas, this relation is given by g 1 2 g 1 1 p2 , p 1 g 1 2 g 1 . 1 1 exp s R/M p0, 1 p0, 2 1/ g 1) 2g g 1 2 .Ma 1 g 1 g 1 2 g 1 . 2 g 1) Ma 1 g 1 g /(g 1) SHOCK WAVES For strong compression waves, upstream flow is supersonic, downstream subsonic Rules out possibility of rarefaction Ma 1 1 shocks SHOCK WAVES Rankine-Hugoniot “shock adiabat” on the p-v plane SHOCK WAVES In terms of upstream (normal) Mach number: p2 2g 1 Ma 21 1 p1 g 1 T 2 T1 and ) 2 2 g Ma g 1) g 1 2 1 2 1 Ma 1 , 2 2 2 g 1 Ma ) 1 2 g 1) 1 g 1) Ma 21 2 Ma 21 As Ma1 ∞, p2/p1 and T2/T1 also ∞; however, 2 / 1 approaches the finite limit (g1)/g1) SHOCK WAVES Normal shock property ratio as a function of upstream (normal) Mach number Ma ( for g =1.3 ) DETONATION / DEFLAGRATION WAVES Abrupt transitions accompanied by chemical reactions i 0 h must include chemical contributions Reaction may be seen as adding heat q per unit mass to a perfect gas mixture of constant specific heat g e.g., many fuel-lean/ air mixtures DETONATION / DEFLAGRATION WAVES Generalized R-H conditions then become: G 0, G u p , c p T0 q, q s T2 DETONATION / DEFLAGRATION WAVES Detonation adiabat is above shock adiabat by an amount depending on heat release q Detonations propagate with an end-state at or near Chapman-Jouguet point CJ (figure on next slide) Singular point at which combustion products have minimum possible entropy, and normal velocity of products is exactly sonic (Ma 2 = 1) DETONATION / DEFLAGRATION WAVES Rankine -Hugoniot “detonation adiabats” on the p-v plane DETONATION / DEFLAGRATION WAVES Imposing Ma2 = 1 yields: Ma 1 1 H ) H 1/2 1/2 {+ sign upstream Mach number for a CJ-detonation (compression wave) - sign upstream Mach number for a CJ-deflagration (subsonic combustion wave)} where g 2 1 Mq H . 2g RT 1 MULTIDIMENSIONAL INVISCID STEADY FLOW Even neglecting diffusion & non-equilibrium chemical reaction, equations governing local conservation of mass, momentum & energy for steady flow of a perfect gas remain PDE’s In field variables v , ,T and p Must be solved subject to conditions “at infinity” & vn along body surface DETONATION / DEFLAGRATION WAVES Simpler procedure: Reduce PDEs to one (higher order) PDE involving only one unknown– velocity potential, v x ) ,where: v = gradv All inviscid compressible flows admit such a potential, with constant gradient far from the body v / n 0 everywhere along body surface MULTIDIMENSIONAL INVISCID STEADY FLOW Scalar function v x ) must satisfy non-linear 2nd order PDE: 2 1 2 a div(gradv ) gradv .grad gradv ) 2 where a 2 g RT0 M g 1 2 grad v ) 2 when a2 ∞(e.g., incompressible liquid): div grad v ) 0 (Laplace’s Equation) MULTIDIMENSIONAL INVISCID STEADY FLOW Hence, many simple inviscid flows can be constructed using “potential theory” & analytical methods (rather than numerical) Nature of solution depends on whether local flows are supersonic or subsonic In case of upstream supersonic conditions, shock waves can appear within flow field– piecewise continuous Energy can then be dissipated even in inviscid fluids Interplay of diffusion & convection determines structure of discontinuities