Inversion of the Fundamental Thermodynamic Equations for Isentropic Nozzle Flow Analysis

Abstract

The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann’s theorem of classical analysis. Bosley’s technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable.