Acta Astronautica 61 (2007) 866 – 872 www.elsevier.com/locate/actaastro The optimization of rough surface supersonic nozzles D. Vlassov a , J.V.C. Vargas a , ∗ , J.C. Ordonez b a Departamento de Engenharia Mecânica, Universidade Federal do Paraná, CP 19011, Curitiba, PR, 81531-990, Brazil b Department of Mechanical Engineering and Center for Advanced Power Systems, Florida State University, Tallahassee, FL 32310, USA Received 11 April 2005; accepted 4 January 2007 Available online 12 April 2007 Abstract In this work, a theoretical optimization study of compressible gases adiabatic flows was conducted for conical nozzles with internal rough surface. A mathematical model was developed for allowing the determination of thrust accounting for nozzle friction and divergence losses. In the first part of the study, it is shown that in rough surface nozzles there are critical values of the nozzle opening half angle, under which, additional increase in gas velocity is not obtained. It is also observed that the minimum nozzle opening half angle value increases as the flow velocity and the nozzle internal surface roughness increase. In the second part of the study, it is shown that there are optimal values of nozzle opening half angle for minimum hydraulic loss across the nozzle, for each combination of the hydraulic resistance caused by the internal surface roughness and the nozzle outlet cross sectional area. © 2007 Elsevier Ltd. All rights reserved. Keywords: Gas dynamics; Supersonic nozzles; Hydraulic resistance; Nozzle outlet cross sectional area 1. Introduction The continuous development of thermal engines that utilize supersonic nozzles to create thrust demand the reduction of nozzle losses to a minimum level. The high gas flow velocity at the divergent part of the nozzle and the roughness of the nozzle internal surface increase considerably friction losses. It is well known that a rocket engine develops max- imum thrust when the pressure at the Laval’s nozzle outlet is equal to the atmospheric pressure. As the altitude varies with the rocket trajectory, the atmo- spheric pressure varies and the engine loses part of its thrust. To compensate such thrust loss it is necessary to increase (or reduce) the length and the area at the ∗ Corresponding author. Tel.: +55 41 3361 3307; fax: +55 41 3361 3129. E-mail address: jvargas@demec.ufpr.br (J.V.C. Vargas). 0094-5765/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2007.01.068 nozzle outlet, according to the trajectory, i.e., accord- ing to the actual ambient conditions around the rocket position. Therefore, it is necessary to use nozzles that allow a dynamic variation of their geometry. One of the designs that allows the length and the area at the nozzle outlet to vary consists of two parts, a short and a longer one which is the continuation of the former. Such noz- zles are manufactured with a flexible material, which is obviously rough. The roughness of the nozzle inter- nal surface, determined by the structure of the mate- rial, increases the hydraulic loss even more with the gas expansion increase in the nozzle [1]. In conventional nozzles, in actual operating condi- tions, the increase in internal surface roughness can be caused by several reasons. In solid fuel rocket engines, the condensed fluid in the liquid phase sticks on the nozzle surface, and changes to the solid phase. The condensed phase particles accumulate on the surface, turning it similar to a sandpaper surface.