Simplification of the Rankine-Hugoniot equations

The temperatures Tq and may be eliminated from equation (21) by using equation (18). It can be seen that the ratio c liR /W) will enter into the result. Since the specific heat at constant volume is given by c = Cp — R°/W for an ideal-gas mixture, it follows that Cpl R°lW) = Cp/(Cp — c ) = y/(y — 1), where y s Cp/c is the specific-heat ratio for the final mixture. Solving equation (18) for and Tq and substituting these results into equation (21) therefore yields 

Equations (9) and (22) show that the Hugoniot equation may be written in the form 

From equation (23) an explicit algebraic expression for as a function of along the Hugoniot curve may be obtained. Equations (8) and (23) therefore determine the complete solution for the simplified system. 

The nature of the results will become more transparent if dimensionless variables are introduced. A dimensionless final pressure is the pressure ratio 

A dimensionless final specific volume is the density ratio 

7/(7 — 1), where 7 = c /c is the specific-heat ratio for the final mixture. Solving equation (18) for and Tq and substituting these results into equation (21) therefore yields 

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