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Analytical and empirical solutions of the oblique and normal shock waves with application to real ammonia

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Summary

The analytical solution for a shock wave in real ammonia (NH3) is obtained by the use of the shock wave governing equations which are valid in general and the best state equation of NH3 being available. The results obtained from the analytical solution have an excellent accuracy but they need a large amount of a computer time. The accurate results of the analytical solution are approximated by explicit empirical relations having mathematical forms similar to those of the ideal gas shock wave, but with new, numerically calculated parameters (coefficients and exponents). The obtained accuracy of the empirical results is very good in comparison with these of the analytical ones (relative deviation less than 0.1%). The whole procedure is applied in a region with practical significance, for the general case of the oblique shock wave, while the normal shock is obtained as a limiting case corresponding to specific geometrical conditions.

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Abbreviations

A i :

constant in NH3 model equation (i=1, 2, 3)

a i :

constant in NH3 model equation (i=1, 2, ..., 36)

a i 0 :

constant in NH3 model equation (i=1, 2, ..., 6)

c :

flow velocity, m/s

C i :

constant in NH3 model equation (i=1, 2)

c p :

constant pressure heat capacity, J/kg,K

c v :

constant volume heat capacity,J/kg,K

h :

specific enthalpy,J/kg

k :

isentropic exponent

M :

Mach number

M id :

ideal partial function in NH3 model equation

M r :

real partial function in NH3 model equation

p :

pressure, N/m2

R :

gas constant,J/kgK

r i :

constant in NH3 model equation (i=1, 2, ..., 31)

s :

specific entropy,J/kgK

T :

absolute temperature,K

t :

temperature, °C

T 0 :

constant in NH3 model equation,K

t i :

constant in NH3 model equation (i=1, 2, ..., 31)

u :

specific internal energy,J/kg

v :

specific volume, m3/kg

v 0 :

constant in NH3 model equation, m3/kg

x ij :

parameters in empirical shock wave relations of a real gas

x idij :

parameters in shock wave relations of an ideal gas

α:

sound velocity, m/s

δ:

flow deflection angle

ϱ:

density, kg/m3

σ:

angle between shock wave and upstream flow velocity

n :

denotes direction normal to the shock wave

t :

denotes direction tangential to the shock wave

0:

denotes the stagnation or total state

1:

denotes properties upstream from the shock

2:

denotes properties downstream from the shock

References

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  3. Kakatsios, X. K.: Numerical evaluation of steam normal shock waves. Acta Mech.110, 183–197 (1995).

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Author notes

  1. X. K. Kakatsios & G. E. Houzouris

    Present address: Mechanical Engineering Department-Thermal Section, National Technical University of Athens, Petission 42-10682, Athens, Greece

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    1. X. K. Kakatsios
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    2. G. E. Houzouris
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    Kakatsios, X.K., Houzouris, G.E. Analytical and empirical solutions of the oblique and normal shock waves with application to real ammonia. Acta Mechanica 120, 141–156 (1997). https://doi.org/10.1007/BF01174321

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    • DOI: https://doi.org/10.1007/BF01174321

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