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# Hugoniot curve

For steady, one-dimensional flow of a combustible gas that burns to completion, equations relating initial and final conditions are called the Rankine-Hugoniot relations and they provide jump conditions across the front, from upstream (subscript 0) to downstream (subscript \infty ) . These equations are: Figure 1. Schematic locus of burnt-gas states for combustion
waves (LinanA:1993).

m_{0} \equiv \rho _{0} u_{0} =\rho _{\infty } u_{\infty }

P_{0} \equiv \rho _{0} u_{0}^{2} +p_{0} =\rho _{\infty } u_{\infty }^{2} +p_{\infty }

The sequence of final states obeying

p_{\infty } +m_{0}^{2} /\rho _{\infty } =P_{0} \equiv p_{0} +m_{0}^{2} /\rho _{0}

obtained by substituting first two equations, is the Rayleigh line, a straight line in the plane of final pressure p_{\infty } , and specific volume, 1/\rho _{\infty } , such as Fig.1.

Use of energy conservation equation together with second jump condition provides a relationship among thermodynamic properties, the Hugoniot curve, which can be written as:

\left(\frac{\gamma }{\gamma -1} \right)\left(\frac{p_{\infty } }{\rho _{\infty } } -\frac{p_{0} }{\rho _{0} } \right)-\frac{1}{2} \left(\frac{1}{\rho _{\infty } } +\frac{1}{\rho _{0} } \right)\left(p_{\infty } -p_{0} \right)=h_{0}

where h_{0} is the total amount of chemical heat release per unit mass of the mixture,

The Hugoniot curve is shown schematically in Fig. 1 for a representative combustion system. The final state is determined by the intersection of the Rayleigh line with the Hugoniot curve.

The Hugoniot curve has two branches, an upper branch of large \rho _{\infty } and p_{\infty } , called the detonation branch, and a lower branch of small \rho _{\infty } and p_{\infty } , called the deflagration branch. There is a minimum propagation velocity for detonations, corresponding to tangency at the upper Chapman-Jouget point. The Rankine-Hugoniot equations can be solved by e.g., STANJAN code, examples of results from such computations for hydrogen-oxygen and hydrogen-air mixtures are given in Table 1.

Table 1. Calculated detonation properties for hydrogen mixtures with oxygen and air initially at 298 K and 1 atm (ReynoldsWC:1986).

 Reactants Fuel (1 mole) H2 H2 O2 (moles) 0.5 0.5 N2 (moles) 0 1.88 Detonation Products (mole fraction) H2O 0.5304 0.2943 O2 0.0486 0.0078 OH 0.1370 0.0183 O 0.0386 0.0021 H 0.0811 0.0060 NO 0 0.0078 H2 0.1641 0.0317 N2 0 0.6319 Detonation parameters UD [m/s] -- CJ detonation velocity 2842 1971 MaD -- detonation Mach number 5.28 4.84 Tp [K] -- temperature of products 3683 2949 pp/pr -- detonation pressure ratio 18.85 15.62 \gammap -- products specific heat ratio 1.129 1.163 ap [m/s] -- products sound speedd 1546 1092 Mp products molecular mass 14.5 23.9

#### Overdriven detonation

Under certain circumstances, it is possible for the detonation wave to move faster than the unique steady-state velocity given by CJ theory. This usually occurs because another event causes the detonation products to move faster than the velocity they would have in a CJ wave. As a result, the pressure associated with the overdriven detonation front can be significantly higher.

The extent of pressure increases that can occur can be seen in Table 2 for hydrogen-air and hydrogen-oxygen detonations.

Table 2. Theoretical pressure (bar) for overdriven detonation in stoichiometric hydrogen mixtures (TeodorczykA:1992).

 M/MCJ 1 1.05 1.1 1.2 1.3 1.4 H2 – air 16.1 22.75 27.01 35.14 43.52 52.31 H2 – O2 19.44 27.83 32.94 42.76 52.81

Linan A. and Williams F.A. (1993) Fundamental Aspects of Combustion. Oxford University Press.(BibTeX)
Reynolds W.C. (1986) The element potential method for chemical equilibrium analysis: implementation in the interactive program STANJAN. Technical report, Mechanical Engineering Department, Stanford University.(BibTeX)
Teodorczyk A. (1992) Calculation of thermodynamic parameters of combustion products behind overdriven detonation wave. Biuletyn Informacyjny ITC Politechniki Warszawskiej, 76:21-45.(BibTeX)