Rankine- Hugoniot Equa0ons and Detona0ons. Forman A. Williams UCSD

Transcription

1 Rankine- Hugoniot Equa0ons and Detona0ons Forman A. Williams UCSD

2 Steady, Planar, Exothermic Reac0on Fronts To ask what kinds of steady, planar reac0on fronts may exist, derive the overall conserva0on equa0ons for the flow illustrated in the diagram at the right. The resul0ng jump condi0ons are the Rankine- Hugoniot equa/ons. All gradients vanish at an infinite distance upstream (subscript 0) and downstream (subscript ). Mass conserva0on then becomes where m is the mass flow rate per unit area, and momentum conserva0on is. Flows for which these two equa0ons both apply are called Rayleigh- line flows when viewed in a plane of p and 1/ρ, auer elimina0ng v, resul0ng in a straight line having a nega0ve slope in that plane. This applies irrespec0ve of the forms of the energy and species conserva0on equa0ons.

3 The Hugoniot Equa0on The equa0on for energy conserva0on becomes with both thermal and chemical contribu0ons to the enthalpies. Species conserva0on provides the composi0on upstream, and the downstream composi0on should be determined by chemical equilibrium. Equa0ons of state also are needed. The Rayleigh- line equa0on can be wriven as which from mass conserva0on is an expression for the square of the product of velocity and density. Use of that to eliminate veloci0es from the above energy conserva0on, that is, expressing the kine0c- energy term through state variables, results in the Hugoniot equa/on which is a rela0onship among thermodynamic proper0es alone. Given the final specific volume 1/ρ this equa0on, along with the M equa0ons for chemical equilibrium and the L atom- conserva0on equa0ons (M + L = N) (where the ν s are the number of atoms of kind j in species i) serves to determine all downstream proper0es in terms of known upstream proper0es.

4 Calcula0on of the State Downstream The final state of the system can be determined as the intersec0on if the Rayleigh line with the Hugoniot curve as illustrated at the right. Given the mass flow rate per unit area m, the Rayleigh line can be drawn with its nega0ve slope, and the final state is the point where it intersects the Hugoniot curve. Given the qualita0ve shape of the Hugoniot shown in the figure, it can be inferred that, depending on the value of m, there may or may not exist an intersec0on point, marked solu0on.

5 The Simplest Model Explicit solu0ons can be obtained by introducing a simplified model of the system. Assume constant values for the average molecular weight of the mixture and the specific heat at constant pressure:. Then the equa0on of state for an ideal gas can be employed, along with the associated enthalpy expression, in which the heat released per unit mass of the mixture is expressing the Hugoniot equa0on as the factor involving γ having come from the ra0o of the specific heat to the gas constant per unit mass. Given the ini0al state and constant values of γ and q, this is the explicit Hugoniot fo the plane of 1/ρ and p.

6 Nondimensional Form of the Equa0ons The ini0al pressure and density can be employed to define a nondimensional pressure p and specific volume v as and. It then becomes convenient to define a nondimensional heat relase and square of mass flow rate as and The Hugoniot equa0on can then be solved for p to give The corresponding nondimensional expression for the Rayleigh line becomes Graphs of the Hugoniot formula for different values of α and γ can be helpful.

7 Hugoniot Diagrams γ = 1.4 γ = 1.2

8 Different Intersec0ons of the Rayleigh Line with the Hugoniot Curve The Rayleigh line intersects the Hugoniot curve only if the magnitude of its slope exceeds a cri0cal value or is less than a different cri0cal value. At these cri0cal values the curves are tangent at the intersec0ons. Such intersec0ons are called Chapman- Jouguet points. They are points B (upper, larger p) and E (lower, smaller p) shown. Two dis0nct branches are seen, the upper (Detona/on) branch and the lower (Deflagra/on) branch. Except at the C- J points, any given intersec0ng Rayleigh line intersects twice, resul0ng in the Strong and Weak shown solu0ons.

9 Chapman- Jouguet Points in the Simplified Model By differen0a0ng the Hugoniot formula and using the Rayleigh- line formula it can be shown that at the C- J points p = v/[(γ+1)v γ]. Subs0tu0on of this into the Rayleigh- line formula then gives μ = γp/v, which, in dimensional variables, is, the sound speed in the burnt gas. The final velocity therefore always is sonic at C- J points. Explicit solu0ons are (+ for upper, - for lower): These result in This leads to a formula for the propaga0on Mach number M 0 = v 0 /a 0, which is supersonic for C- J detona0ons and subsonic for C- J deflagra0ons.

10 C- J Results for the Simplified Model with γ = 1.3 The formulas show especially large pressure increases for detona0ons and all changes increasing with increasing heat release. The C- J results approach those of ordinary sound waves as the heat release approaches zero. Gas mixtures have frozen and equilibrium sound speeds, and it is the laver (lower) that is approached downstream in general.

11 Summary of Rankine- Hugoniot Results

12 A Simplified Model for the Structures of Deflagra0ons and Detona0ons A sequence of 10 assump0ons can simplify the conserva0on equa0ons to a point at which both deflagra0on and detona0on structures can be addressed in a phase plane of only 3 dimensions. The first 6 assump0ons are: The momentum mass, species, diffusion, and energy equa0ons then become, respec0vely:

13 The next 2 addi0onal assump0ons: lead to the formulas and These facilitate introduc0on of the normalized stagna0on thermal enthalpy. Energy conserva0on becomes: The denominator here and above is the heat release per unit mass of mixture, q. Another assump0on is: With P species 2, this leads to a single reac0on- progress variable which will appear directly and simplify the species conserva0on equa0ons.

14 The extreme simplifica0on then is that there are only two chemical species, 1 and 2, the reactant R and product P, so that the mass frac0on of one is determined by the mass frac0on of the other, and the normalized mass- frac0on variable of P can be introduced as. The normalized product mass and flux frac0ons thus are zero- one variables, like τ. In the two- component mixture, the diffusion equa0ons become simply and the Lewis number is It becomes convenient to introduce the nondimensionl distance variable. With this variable, the diffusion and equa0ons become simply and One species conserva0on equa0on remains. The final assump0on is: With this assump0on and the preceding equa0ons for Y and τ it becomes evident that Y = τ, since their differen0al equa0ons and boundary condi0ons are iden0cal. The diffusion equa0on therefore no longer needs to be considered. AVen0on then rests on only three first- order ordinary differen0al equa0ons, momentum, energy and species conserva0on. In terms of the nondimensional variable momentum conserva0on can be shown to become with the defini0on where

15 Given the ideal gas law, and selec0ng the Arrhenius reac0on- rate expression the equa0on for species conserva0on can be shown to be expressible as Temperature T in the rate is found from This equa0on, along with the previous momentum and energy equa0ons, and define the simplest possible descrip0ons for deflagra0on and detona0on structure. The zero ini0al condi0on for the reac0on- progress variable is not consistent with the fact that the Arrhenius rate does not vanish at τ = 0. This is the cold- boundary difficulty of combus0on; the Arrhenius rate is replaced by zero at low temperature.

16 A Simplified Solu0on for Planar Detona0on Structure The preceding equa0ons describe a simplified structure in a three- dimensional phase plane of ε, τ, and ϕ. To understand the character of the solu0on consider a slice of this three- dimensional space for constant ε. From the defini0ons, the viscous term in momentum conserva0on vanishes when which defines the parabolas representa0ve being shown at the leu. These are the Rayleigh lines in this plane. Four intersec0ons, a, b, c, d. are illustrated with a representa0ve final state, ε=τ=1. Important cri0cal or singular points are:

17 A Simplified Solu0on for Planar Detona0on Structure (cont.) The intersec0ons a and b occur for parabolas with the ini0al Mach number M 0 >1, and the intersec0ons c and d occur for parabolas with M 0 <1. The first two therefore correspond to detona0ons and the last two to deflagra0ons. Intersec0ons a and c correspond to the last of the preceding four equa0ons, and intersec0ons b and d to the third. The ini0al state, ϕ=1, also must lie along a parabola and thus is given by the first of the four previous equa0ons for detona0ons and by the second for deflagra0on. Then, point c is the hot boundary point for a weak deflagra0on, and point d for a strong deflagra0on, while point a is the hot boundary point for a strong detona0on, point b for weak detona0on. For C- J waves, the square root vanishes, and the parabola is tangent to the horizontal line at the final state, resul0ng in only one intersec0on. Characteris0cs of the solu0ons depend on expansions about the cri0cal points. Near the cold boundary, when the cold- boundary difficulty is removed, τ increases ini0ally with ε=0, and the hot boundary in that plane is a saddle point, at which the relevant slope is. The important plane to study, however, in inves0ga0on the behavior of the solu0ons in the vicinity of the cri0cal points is the plane of ϕ and τ. Expansions about the cri0cal points at τ=0 and at τ=1 in this plane are most revealing.

18 Illustra0on of Projec0ons of Solu0on Curves on the Plane of ϕ and τ

19 Illustra0on of Projec0ons of Solu0on Curves on the Plane of ϕ and τ (cont.) The cri0cal points at the lower leu and upper right are saddle points into which only two integral curves arrive, while those at the upper leu and lower right are nodes into which an infinite number of integral curves arrive. The cold boundary is on the leu for deflagra0ons and on the right for detona0ons. Curve b is a typical solu0on for a weak deflagra0on; with the present simplifica0ons, there is no solu0on for the structure of strong deflagra0ons (curves from lower leu to upper right in this diagram). Curve f would be a solu0on for a weak detona0on; like the weak deflagra0on, there is only one such curve. Since the integral curves in the third dimension approach the final cri0cal point only for par0cular reac0on- rate func0ons when the ini0al velocity is given, these structures involve eigenvalues, in that when reac0on- rate func0ons are specified, solu0ons exist only for par0cular propaga0on veloci0es. For strong detona0ons both boundary points are nodes in this plane, and so there are an infinite number of integral curves between them. Curves c, d, and e are examples. Solu0ons for structures of strong planar detona0ons therefore should exist for any reac0on- rate func0on. Curve c is typical; curve e would require the chemistry to be too fast, as will be seen later. Line a in this figure corresponds to a non- reac0ng shock wave.

20 Illustra0ve Integral Curves in Three the Dimensions, Progress Variable, Velocity, and Heat Release All curves require τ to exceed ε away from the boundaries and therefore lie above a slanted plane. The velocity typically increases con0nuously in the deflagra0on but decreases in the detona0on before finally increasing to a value less than the ini0al value; the ini0al decrease from ϕ = 1 occurs almost along the ϕ axis.

21 ZND Detona0on Structures Recall that while Detona0on propaga0on veloci0es are high, so that m is large, causing the reac0on- rate term F to be small in detona0on structure. This forces the non- dimensional distance ξ over which proper0es change by chemistry in detona0ons to be large, so that the last equa0on above becomes approximately which places the en0re solu0on prac0cally on the slanted plane in the previous figure and reduces the first equa0on above to one involving only τ and ϕ. In this approxima0on, then, the detona0on structure is described by a single ordinary differen0al equa0on that defines a trajectory in the previously illustrated plane of ϕ and τ.

22 ZND Detona0on Structures (cont.) Since the ini0al mixture is too cold for reac0ons to occur, detona0on structures begin with a shock wave, line a in the plane of ϕ and τ. The shock wave heats the mixture to a temperature at which the chemistry begins. Following the shock, in momentum conserva0on the deriva0ve, represen0ng viscous effects, is negligible because changes occur slowly in the ξ co- ordinate, so that vanishes. The curve b in the figure, which follows a therefore is very close to the Rayleigh line shown. This structure was iden0fied independently by Zel dovich, von Neumann, and Doring and so is called the ZND structure. Along this Rayleigh line,, and so the solu0on for the simplified model is simply Given the ZND approxima0on, the structure following the shock is readily found by forward integra0on of the set of ordinary differen0al equa0ons for mole frac0ons, obtained from the chemical kine0cs,.

23 Planar Detona0on Structure in a Pressure- Volume Diagram At the right is a pressure- volume diagram illustra0ng detona0on structures. The labeling of the curves is the same as in the previous figure for the plane of ϕ and τ. The ZND structure is a followed by b, but with b coincident with the Rayleigh line. The lead shock a is strong, so that kine0c theory rather than the Navier- Stokes equa0ons is needed to describe its structure. While curves c, d, and e are not encountered in real deona0ons, curve f may represent condensa0on shocks seen in wind tunnels.

24 Illustra0on of a Representa0ve Planar Detona0on Structure in a Physical Plane

25 Detona0on Propaga0on in Tubes Consider a planar detona0on treated as a discon0nuity propaga0ng from the closed end of a tube. The velocity as a func0on of distance in a co- ordinate system fixed with respect to the moving detona0on is shown at the right. Since the gas in the laboratory is at rest ahead of the detona0on and at the closed end, the velocity must increase with distance behind the front. That velocity profile corresponds to am expansion wave, Since the velocity behind the wave for a strong detona0on is subsonic with respect to the wave, the sonic expansion wave overtakes the strong detona0on and thereby weakens it. This weakening con0nues un0l the velocity behind the wave, with respect to the wave, becomes sonic, which occurs at the C- J point, subsequent to which the expansion ceases to reach thee detona0on. This is the fundamental reason that C- J detona0ons are found in experiments.

26 Equilibrium versus Frozen Sound Speeds The diagram shows a series of frozen Hugoniots drawn at different points along an equilibrium Hugoniot. The C- J velocity is seen to be smaller for the equilibrium Hugoniot. That is the correct one because the evolu0on following the Neumann spike of the lead shock would reach D before C. Further weakening thus must occur un0l the end state is B.

27 Effects of Tube Walls Again consider a reference frame fixed with respect to the detona0on. Since the side wall is fixed with respect to the gas ahead of the detona0on, the velocity profile in the boundary layer is as illustrated in the figure. In this steady- flow configura0on the flow rate in the boundary layer exceeds that in the free stream, so there must be ou{low from the free stream. This curves the detona0on, convex towards the fresh mixture, resul0ng in free- stream cross- sec0onal area expansion with increasing distance downstream. The heat release and area expansion compete, resul0ng in a smooth transi0on through M = 1, from M < 1 to M > 1, so that the C- J condi0on is applied where these two effects balance. This leads to a reduc0on in the detona0on propaga0on velocity v 0 with decreasing tube diameter d given in terms of the displacement thickness of the boundary layer by where the factor 2.1 is par0ally empirical.

28 Stability of Planar Detona0ons Sufficiently strong planar detona0ons are stable. That is, at high enough overdrive (final pressures high enough to make the detona0on very strong), every planar detona0on becomes stable. Shock- tube studies of chemical kine0cs behind shock waves in principle always involve such highly overdriven detona0ons if the overall chemistry is exothermic. All real detona0ons of prac0cal interest, however, are unstable when planar. There are two types of instabili0es, planar and non- planar. Non- planar instabili0es are dominant and lead to cellular detona/ons. The non- planar modes involve transverse vor0city fluctua0ons and exhibit instability even when the heat release is small compared with the thermal enthalpy of the fresh mixture and insensi0ve to temperature (J. J. Erpenbeck, Physics of Fluids 4 (1961) 481,5 (1962) 604, 9 (1966) 1293). The planar instability, on the other hand, is a consequence of the increase of the rate of heat release with increasing temperature, and it dominates the structures and dynamics of both planar instabili0es that lead to galloping detona/ons, as well as the ul0mate states of cellular detona0ons. This mechanism is oscillatory. Planar detona0on structures generally involve a strongly temperature- sensi0ve induc0on zone, in which radical concentra0ons build up with livle heat release, followed by a heat- release zone, of a size comparable with that of the induc0on zone, in which radical recombina0on releases heat. The oscillatory mechanism is explained most simply by assuming that the heat release occurs instantaneously, auer the shock- heated reactants have traversed an induc0on length l e behind the Neumann spike.

29 Instability Mechanism; Oscilla0on Frequency Let v be the gas velocity with respect to the shock at the Neumann condi0on and t e be the strongly temperature- dependent explosion 0me at the Neumann state. Then In terms of the specific- heat ra0o, in the strong- shock limit, at the Neumann state the ra0o of the velocity to the propaga0on velocity, which equals the ra0o of the ini0al density to the Neumann- state density, is If the propaga0on velocity in which n is the pressure exponent of the reac0on rate (overall order). The strong- shock rela0ons then give with E the overall ac0va0on energy and α the temperature exponent of the rate. Since the factor on the right is generally nega0ve, an increase in v 0, which increases T and p, decreases l e, which sends a compression wave back to the shock at velocity a v, (where a is the sound speed), further strengthening the shock. This drives and oscilla0on of period where the Mach number at the Neumann state is.

30 Cellular Detona0ons For more recent informa0on on stability and dynamics see: E. S.Oran and F. A. Williams, The Physics, Chemistry and Dynamics od Explosions, Philosophical Transac0ons of the Royal Society of London, Volume 370, Number 1960, Pages , 13 February, Cellular detona0ons involve triple- shock structures like that illustrated at the leu. The transmived shock (the Mach stem) is the strongest and so ini0ates the the heat release in detona0on, as illustrated. The pressure is highest at the triple point, which therefore makes marks on sooty walls.

31 Cellular Detona0ons (cont.) The trajectory of the triple point traces the illustrated diamond pavern of soot removal when a smoked foil is placed on wall of a detona0on tube. The illustrated transverse spacing s is the cell size which depends on the chemical reac0on rates, for example through l e. Not all detonable mixtures exhibit cells as regular as illustrated, and for near- limit mixtures in round tubes the markings are spiral (spinning detona/ons). It is found that, approximately, s/l e = 13, for typical specific- heat ra0os (1.2). Also, tube diameters must be about 13 s (13 cells) for a detona0on to propagate from the open end of the tube into the open. Thus, 13 is a magic number for detona0ons. Standing detona/ons can be established in diverging nozzles by Mach reflec0on. Also, rota/ng detona/ons in annuli and pulsed detona/ons in tubes have been inves0gated for propulsion purposes, showing promise but not yet prac0cal.

32 Detona0on Failure; Detonability Limits If a tube is too small or a detonable layer is too thin (as illustrated at the right), then a detona0on will fail to propagate. In the figure, the oblique shock causes flow deflec0on, producing expansion of the area A of the detonable layer. The expansion competes with the heat release in the reac0on zone and can quench the heat release. A similar effect in a tube arises from the increased flow into the boundary layer (men0oned earlier), but that expansion is smaller. Confinement therefore aids propaga0on. Propaga0on veloci0es are so large that confinement is iner0al; polyethylene boundaries can provide nearly as much confinement as steel in preven0ng failure. Failure is enhanced in weak (diluted or low- pressure) mixtures, leading to the mixtures being beyond detonability limits.

33 Criteria for Detona0on Failure A failure criterion is that the area- expansion effect exceeds the heat- release effect at Neumann condi0ons. In variable- area isentropic flow, and in constant- area flow with heat addi0on, Here T 0 is the stagna0on temperature. Failure occurs unless the decrease of M from the first is less than the increase from the second. For strong shocks, this criterion for propaga0on becomes approximately A rough approxima0on to this formula is da/dx < A/l e. In terms of the hydraulic diameter H of a tube, the effec0ve free- stream area increase becomes and when the frac0onal distance over which the displacement thickness increases is approximated as l e here, the limi0ng condi0on for failure For a laminar boundary layer this thickness will be about l e divided by the square root of the Reynolds number, giving tube diameters required for failure in the millimeter range for typical detonable mixtures.

34 Direct Ini0a0on of Detona0on Detona0ons are produced in two different ways. A common way is the transi0on from deflagra0on to detona0on (DDT), in which a deflagra0on, for example in a tube, generates pressure waves, from hot spots, typically at walls, that develop into detona0on. This is a complicated process that con0nues to be inves0gated. A sufficiently energe0c charge, however, can ini0ate a detona0on directly. In planar, k =1, cylindrical, k =2, or spherical, k =3 geometry the energy U k (k=3), energy per unit length U k (k =2), or energy per unit area U k (k =1) required for direct ini0a0on with rapid (effec0vely instantaneous) deposi0on can be es0mated as where C 1 = 1, C 2 = 2π, and C 3 = 4π, and the func0on Γ is of order unity. When the explosion 0me is known, this formula can readily be applied for an es0mate. Recent literature should be consulted for more up- to- date results. A future authorita0ve source will be: P.Clavin and G. Searby, Combus0on Waves and Fronts in Flows (Flames, Shocks, Detona0ons, Abla0on Fronts, and Explosions of Stars), Cambridge University Press, to appear, July, 2016.