{"id":1753,"date":"2023-05-29T06:03:57","date_gmt":"2023-05-29T03:03:57","guid":{"rendered":"https:\/\/fti.dp.ua\/conf\/?p=1753"},"modified":"2023-05-29T06:03:59","modified_gmt":"2023-05-29T03:03:59","slug":"05291-0602","status":"publish","type":"post","link":"https:\/\/fti.dp.ua\/conf\/2023\/05291-0602\/","title":{"rendered":"Statement of the problem of designing a liquid rocket engine dual bell nozzle of the maximum thrust using the direct method of the calculus of variations"},"content":{"rendered":"\n

Statement of the problem of designing a liquid rocket engine dual bell nozzle of the maximum thrust using the direct method of the calculus of variations<\/h1>\n\n\n\n
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Ivan Dubrovskyi<\/h5>\n\n\n\n

ORCID: https:\/\/orcid.org\/0000-0002-0707-0074<\/a><\/em><\/p>\n\n\n\n

Oles Honchar Dnipro National University, Dnipro<\/em><\/em><\/p>\n<\/div><\/div>\n\n\n\n

<\/div>\n\n\n\n
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Valerii Bucharskyi<\/h5>\n\n\n\n

ORCID: https:\/\/orcid.org\/0000-0002-8245-5652<\/a><\/em><\/p>\n\n\n\n

Oles Honchar Dnipro National University, Dnipro<\/em><\/em><\/em><\/p>\n<\/div><\/div>\n\n\n\n

<\/div>\n\n\n\n

Introduction<\/h3>\n\n\n\n

A rocket engine with a standard single-section profiled nozzle provides maximum thrust only at a certain value of atmospheric pressure [1]. To expand the range of atmospheric pressure values at which the optimum engine operation mode is achieved, a dual bell nozzle can be used [2]. A typical diagram of such a nozzle is shown in Fig. 1. Its main elements are three cross-sections a<\/strong>, b<\/strong>, c<\/strong> and two profiled section elements a-b<\/strong>, b-c<\/strong> bounded by them. The fig. 1 shows the general case of the contour of such a nozzle, for which a corner in the generatrix is located at points a<\/strong> and b<\/strong>. In addition, the section a<\/strong> coincides with the critical section of the engine chamber.<\/p>\n\n\n\n

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\"\"<\/figure><\/div>\n\n\n
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Figure 1 \u2013 Dual bell nozzle contour<\/p>\n\n\n\n

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Let\u2019s consider the principle of operation of such a nozzle. When operating at low altitude, with a high value of atmospheric pressure pe1<\/sub><\/em>, the thrust is generated only by the first profiled section a<\/strong>–b<\/strong>. With an increase in flight altitude and a corresponding drop in atmospheric pressure to pe2<\/sub> <\/em>(pe1 <\/sub><\/em>> pe2<\/sub><\/em>), the flow of combustion products expands, begins to flow around the second profiled section and create additional thrust. Thus, an engine with a dual bell nozzle will generate more average flight time thrust than an engine with a single nozzle designed to operate at one of the atmospheric pressures pe1 <\/sub><\/em>or pe2<\/sub><\/em>.<\/p>\n\n\n\n

Usually, the method of characteristics [3] is used for profiling a dual bell nozzle. However, it has the following disadvantages:<\/p>\n\n\n\n