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High-Altitude Rocket Nozzle Design - Rao's Method

  • Hector Ruiz
  • Jun 18
  • 2 min read

Explore the design process, equations, results and MATLAB code by clicking the link below. Join EngineerX in advancing rocket engineering innovation!



Analysis Overview


In this MATLAB project, I designed a bell-shaped rocket nozzle optimized for a target thrust of 10,000 N at a design altitude of 10,000 meters, using Rao’s method to minimize flow separation and maximize efficiency. Below is the methodology, key results, and resources. Full details, including the MATLAB code and methodology,

are included in the pdf.


The Design Challenge


The goal is to create an efficient, compact rocket nozzle for high-altitude propulsion systems, balancing thrust, specific impulse, and geometry. Starting with a target thrust of 10,000 N, I computed the nozzle’s geometry, expansion ratio, and performance across altitudes (0–80,000 meters). I used an 85% length ratio (relative to a 15-degree conical nozzle). Rao’s method optimizes the bell contour to reduce divergence losses, ensuring robust performance for rocket applications.


Process Overview


The MATLAB script produces the nozzle geometry and produces a figure for visualization. The following is the analysis process overview.

  1. Define Input Parameters: Set chamber pressure (5 MPa), target thrust (10,000 N), gas properties (R = 340 J/kg·K, γ = 1.25), chamber temperature (3,000 K), and design altitude (10,000 m).

  2. Interpolate Rao’s Angles: Use empirical tables to find initial (θ_N) and exit (θ_E) divergence angles based on expansion ratio (3.5–100) and length ratio (85%).

  3. Atmospheric Conditions: Interpolate ambient pressure (26,500 Pa at 10,000 m) and gravity (9.776 m/s²) for accurate performance.

  4. Compute Flow: Solve for exit Mach number (M_e ≈ 3.85) and expansion ratio (ε ≈ 17.1) using isentropic relations.

  5. Nozzle Geometry: Calculate throat (D_t = 39.15 mm) and exit (D_e = 161.80 mm) diameters, with nozzle length (252.44 mm).

  6. Divergence Efficiency: Compute λ = 0.994 to account for bell nozzle efficiency over conical designs.

  7. Performance Analysis: Evaluate thrust, specific impulse (I_sp), and exhaust velocity (V_e) across altitudes.

  8. Contour Design: Construct the Rao nozzle contour using circular arcs and a quadratic Bézier curve, visualized with control points in the output figure.

  9. Visualize: Plot the nozzle profile (entrant, exit, bell sections) with annotations for throat, exit, and length.


Key Calculation Results

  • Throat Diameter: 39.15 mm

  • Exit Diameter: 161.80 mm

  • Expansion Ratio: 17.10

  • Nozzle Length: 252.44 mm

  • Divergence Efficiency (λ): 0.994

  • Thrust: 10,000 N

  • Specific Impulse: 261 s at 10,000 m (181 s at sea level, 295 s at 80,000 m)


The MATLAB output performance table shows thrust and Isp increasing with altitude due to reduced backpressure, with the nozzle contour optimized for minimal divergence losses. Adjusting parameters like length ratio or chamber pressure can further tailor the design for specific rocket missions.


Takeaway


My MATLAB tool empowers aerospace engineers to design efficient, high-performance rocket nozzles for high-altitude propulsion. By sharing this methodology and code, EngineerX aims to accelerate innovation in rocket engineering, driving the next generation of space exploration.

 
 
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