Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

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 withheat addition: • Conservation equations may be written as:

STEADY 1D COMPRESSIBLE FLUID FLOW Mass & momentum conservation equations yield: Since , This locus on the p-v (or corresponding T-s) plane  Rayleigh line (locus)

STEADY 1D COMPRESSIBLE FLUID FLOW Local stagnation (or total) temperature For a constant cp-gas mixture between any two duct sections 1 & 2, change in T0 is governed by heat addition per unit mass: Since 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 g=1 .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 withoutheat addition: • Conservation equations may be written as: (Fanno Locus)

STEADY 1D COMPRESSIBLE FLUID FLOW Steady one-dimensional flow of a perfect gas (with g=1 .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”

SHOCK WAVES • Conservation equations across a discontinuity without chemical reaction:

SHOCK WAVES • Mass flux ,

SHOCK WAVES • In general, G, wi, h0 are continuous (no jump) across discontinuity; u, p, T, v (≡ 1/r), s jump. • Compatible with conservation principles, relevant EOS • Combining total mass & normal momentum relations:

SHOCK WAVES • When discontinuity becomes a sufficiently weak compression wave, positive entropy jump is negligible; hence For a perfect gas, then:

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: • Defines a locus (“shock adiabat”) on p-v plane along which final state must lie • For a perfect gas, this relation is given by

SHOCK WAVES • For strong compression waves, upstream flow is supersonic, downstream subsonic • Rules out possibility of rarefaction shocks

SHOCK WAVES Rankine-Hugoniot “shock adiabat” on the p-v plane

SHOCK WAVES • In terms of upstream (normal) Mach number: and As Ma1 ∞, p2/p1and T2/T1also  ∞; however, r 2 /r 1approaches the finite limit (g+1)/(g-1)

SHOCK WAVES Normal shock property ratio as a function of upstream (normal) Mach number Ma ( for =1.3 ) g

DETONATION / DEFLAGRATION WAVES • Abrupt transitions accompanied by chemical reactions • 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:

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: {+ sign  upstream Mach number for a CJ-detonation (compression wave) - sign  upstream Mach number for a CJ-deflagration (subsonic combustion wave)} where

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 • 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, ,where: • All inviscid compressible flows admit such a potential, with constant gradient far from the body • everywhere along body surface

MULTIDIMENSIONAL INVISCID STEADY FLOW Scalar function must satisfy non-linear 2nd order PDE: where when a2  ∞(e.g., incompressible liquid): (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

Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras