Lecture 7 etonation Waves p strong detonation weak detonation weak deflagration strong deflagration / 0 v =/ University of Illinois at Urbana- Champaign
eflagrations produce heat Thermal di usivity th 0 5 m 2 /s cold fresh mixture Chemical time t c 0 3 0 5 s hot products l f S L Speed S L p th /t c S L 0 m/s Wave thickness l f p th t c l f 0 cm p /, T /T 0 6 propagation by di usion in the laboratory frame p, u = ( )S p T, Y L S L i u 0 =0 0, 0 T 0, burned gas cold, fresh gas l f Y i0 in the frame attached to the wave p, u = S L u 0 = S L, 0 T, Y i T 0, burned gas cold, fresh gas Y i0 fluid particle velocity u wave = u lab V wave V wave = S L University of Illinois at Urbana- Champaign 2
etonations rapid, violent, spectacular detonated products l R fresh mixture shock pressures up to 500,000 atm temperatures up to 5,500 K power density 20 0 9 Watt/cm 2 shock followed by a fast flame propagation by shock compression in the laboratory frame p, u, 0 T, Y i u 0 =0 T 0, cold, fresh gas Y i0 in the frame attached to the wave p,, ( u) u 0 0 = T, Y i T 0, cold, fresh gas Y i0 fluid particle velocity u wave = u lab V wave V wave = University of Illinois at Urbana- Champaign 3
Governing Equations - Euler equations + r u =0 t u t = rp h t = p t Y i t =! i i usion e ects, all important in flames, are negligible in detonations because of their extremely high propagation velocity. Here u is the gas particle velocity, and v =/ will be used to denote the specific volume (volume per unit mass). These equations are supplemented with an equation of state, which for ideal gases is p = RT, and a caloric equation of state h = P N i= Y ih o i + R T c T o p dt where c p = P N i= Y ic pi is the mixture specific heat. Assuming equal specific heats, with c p independent of temperature, the enthalpy of the mixture is h = NX Y i h o i + c p (T T o ) i= The chemical reaction is assumed to be a one-step irreversible reaction R! P, and the reaction rate is described in terms of a single progress variable (for example, the mass fraction of the products P) such that = 0 corresponds to the unreacted material and = to completed reaction. The reaction rate is assumed of the form t =!! = k( ) e E/RT R h = Y R h o R + Y P h o P + c p (T T o )=( )h o R + h o P + ( ) T + const. R = (h o R h o P )+ T + const. = Q + pv + const. where use has been made of c p = R Caloric equation of state Equation of state h = pv = pv Q cp T Speed of sound c = p pv University of Illinois at Urbana- Champaign 4
laboratory frame wave-attached frame u u 0 =0 ( u) u0 = cold, fresh gas cold, fresh gas following flow u 0 = The following flow is determined by the rear boundary condition, and usually a rarefaction. u u 0 =0 ( u) For a steadily propagating detonation the shock velocity is constant and the flow in the reaction zone steady in a frame attached to the shock (the following rarefaction is necessarily unsteady). State ahead of the shock, v 0 (= 0 ), u 0, 0(= 0) State within the reaction zone p, v(= ), u, Steady, one dimensional conservation laws d ( u) =0 dx u du dx = dp dx u dh dx = u dp dx ) d ( u) =0 dx d dx p + u2 =0 d dx h + 2 u2 =0 in a frame attached to the wave 0 = ( u) + 0 2 = p + ( u) 2 h 0 + 2 2 = h + 2 ( u)2 these relations connect the state at any point within the reaction zone to the state ahead of the shock ) p + p = m 2 (v v 0 ), m = 0 v = + v 0 + 2 +2 + + Q v 0 Rayleigh line and Hugoniot University of Illinois at Urbana- Champaign 5
S The Hugoniot is parametrized by, with = 0 corresponding to an inert shock and = to a completely reacted state. p W = v v 0 =0 The ZN structure Zel dovich, von Neueman, öring shock followed by a fast flame reaction zone fire T induction lead shock p Ahead of the wave, the gas is quiescent and there is insignificant reaction. Passage through the lead shock the gas is compressed, the pressure increases tremendously and its temperature rises thousands of degrees. The ensuing chemical reaction goes to completion very rapidly in a relatively thin reaction zone (or fire) behind the shock. University of Illinois at Urbana- Champaign 6
The two extreme Hugoniot curves correspond to = 0 and =. For a given shock velocity, all states within the reaction zone must lie on the corresponding Rayleigh line p = 2 0 2 (v v 0 ) Since the Hugoniot reaction must also be satisfied, the portion of the Rayleigh line relevant to the ZN structure is that bounded by the two extreme Hugoniot curves (i.e., the solid portion NS). Starting with an initial state (,v 0 ), the state of a gas particle jumps to the point N along the shock-hugoniot (i.e., corresponding to = 0) upon passage through the lead shock. As the particle reacts, increases and the state of the particle slides down along the Rayleigh line towards the end point S crossing Hugoniot curves of increasing (i.e., corresponding to partial reaction Hugoniot curves). At the end of the reaction zone, the particle reaches the = Hugoniot at the final state S. The flow at the point S is subsonic. The lowest possible Rayleigh line is the one tangent to the complete-reaction Hugoniot (i.e., corresponding to = ). The final state in this case is the Chapman-Jouguet () state. The corresponding detonation speed,,isthe minimum speed consistent with the conservation laws. The flow at the point is sonic. N S N - von Neumann point S - strong detonation W - weak detonation - Chapman-Jouguet detonation p W = v v 0 = =0 University of Illinois at Urbana- Champaign 7
N S N - von Neumann point S - strong detonation W - weak detonation - Chapman-Jouguet detonation p N W = v v 0 = =0 The ZN structure is not possible for weak detonations. The ZN structure does not restrict the propagation speed for strong detonations. Therefore, wave speeds depend on the experimental configuration, or on the rear BCs. Strong detonations are therefore overdriven detonations, namely forced to run at velocity > by being pushed from behind by a piston, say. The question remains on how to determine. The detonation, is an unsupported detonation, namely one that is not pushed from behind, and travels at a speed determined by the conservation laws. University of Illinois at Urbana- Champaign 8
The rear BC can be thought to be a hypothetical piston following the wave. The question is how to determine for a given piston velocity u p (in a laboratoryfixed frame). We denote by u the particle speed at the end of the reaction zone and u corresponding value for the detonation. the u p >u u p reaction zone The detonation is overdriven. The detonation speed is chosen such that u p = u (), and the following flow is uniform. steady following flow u = u p u As u p is reduced towards u, the same qualitative picture remains, with the Neuman state N on the shock-hugoniot dropping lower and the final state S approaching the point. x As u p is reduced towards u, the final state S approaches the point and =. What happens when u p is reduced further? u p <u A further reduction in u p below u leaves the detonation speed and the reaction zone unchanged, since the final state has reached its lowest value on the fully-reacted Hugoniot. u p rarefaction reaction zone u The detonation wave (including the reaction zone) is now unsupported and continue to propagate at speed, una ected by the following flow. constant state x The following flow, however, must now be reduced to match the BC. And, unlike the previously uniform state, it is replaced by a (time-dependent) rarefaction wave, which could be followed by a constant state as necessary. The smaller u p, the larger the amplitude of the rarefaction. University of Illinois at Urbana- Champaign 9
The state of the gas immediately behind the shock is easily obtained from where m = 0. p + p = m 2 (v v 0 ) 2 v = +2 + v 0 + + + Q v 0 Using the first relation to eliminate v/v 0, one gets a quadratic equation for p, with two solutions ( p = + ( + 0 2 p0 2 2 ) /2 ) ± 2 Q + + + 2 0 2 The inert shock solution (denoted by subscript s) is found by setting = 0. One of the two solutions is the undisturbed state p =,v = v 0. The other is the state of an inert shock: p s = + 2 + (M 2 0 ) 2(M0 2 ) v s = v 0 ( + )M0 2 v 0 u s = s where M 0 = /c 0 and c 2 0 = / 0 The spatial distribution behind the shock is determined from t =!! = k( ) e E/RT which in a frame attached to the shock is given by ( u) d dx = k( ) e R/RT which can be integrated to give Z x = 0 [ u( )] k( ) ee/rt d The end state is found when =. A natural length scale is the half-reaction length scale, obtained by setting =/2, namely Z /2 [ u( )] `/2 = k( ) ee/rt d 0 University of Illinois at Urbana- Champaign 0
Summer 203 Reaction zone structure of an unsupported detonation, with all variables plotted as a function of the distance from the lead shock, scaled with `/2. A prominent feature is the appearance of an induction zone, where there is only a small amount of reaction, followed by a rapid reaction zone that is well-separated from the shock.! istance from the shock Moshe Matalon u T p istance from the shock istance from the shock Moshe Matalon Moshe Matalon University of Illinois at Urbana- Champaign Ficke& & avis, 979
Curved detonations reacted products shock unreacted mixture n For weak curvature, in a frame attached to the shock @ ( (u )) + appleu =0 @n (u ) @u @n + @p @n =0 @h @n @p @n =0 (u ) @ @n =! These equations are quasi-steady (and therefore independent of initial data) and quasi-planar (requiring only knowledge of the state of immediately behind the shock). A solution exists only if = (apple); i.e., and apple satisfy an eigenvalue relation that depends on the kinetics. etonation velocity vs curvature Bdzil & Stewart (2007) 9 n(mm/µsec) 8 High velocity branch 7 (Ignition) 6 ( n) 5 (Extinction) 4 ( n) 0 Low velocity branch x 3 c0 κ κ2 2 0 0.5.0.5 κ(mm) - For weak curvature apple, and for a rate law of the form! = k( following relations are obtained ) the apple for 0 < < appleln apple apple for = University of Illinois at Urbana- Champaign 2
The planar structure is highly unstable and is prone to result in transient threedimensional structures. Strehlow, 968 Cellular structure of a hydrogen-oxygen mixture University of Illinois at Urbana- Champaign 3