Inversion of the Critical Back Pressure Relation in Isentropic Nozzle Flow

Abstract

We consider the one-dimensional isentropic flow equation that relates the nozzle expansion area ratio to the critical back pressure that must not be exceeded in avoidance of a subsonic throat condition. To eliminate guesswork and numerical root solving in deducing the critical back pressure, we apply asymptotic tools to invert this relation analytically. Our perturbation technique is based on the reciprocal of the nozzle area expansion ratio which, in most applications, does not exceed 0.3. A five-term approximation for the critical back pressure ratio with respect to the total pressure is readily obtained. By extending our series approximation to higher orders, we unravel a recursive formula that permits the efficient calculation of the pressure ratio to arbitrary level of precision. Favorable agreement with the numerical solution at several gas compression ratios is demonstrated as the relative error in a three-term approximation is found to be a mere 0.28 percent for a nozzle area ratio of 0.3 and γ = 1.4. The error slightly decreases as the gas compression ratio is increased. Using a newly derived formula for the exit pressure (Majdalani, J. and Abu-Irshaid, E. M., “General Solutions for Some Isentropic Equations in Variable Area Duct Flow,” AIAA Paper 2005-4382), we evaluate the ratio for optimal expansion. In concert with the present solution, we then present an explicit solution for the ratio of critical back pressure and the optimal exit pressure for a given nozzle expansion area ratio. All solutions are numerically verified.