METHODOLOGY

Thermal Analysis

Aerodynamic heating analysis for predicting surface temperatures during high-speed flight through the atmosphere.

01

Aerodynamic Heating

When a rocket travels through the atmosphere at high speed, air molecules impacting the surface transfer kinetic energy as heat. The heating rate increases dramatically with velocity, scaling approximately with the cube of velocity.

Heating mechanism:

  1. Air compresses at the vehicle surface (stagnation regions)
  2. Viscous friction in the boundary layer generates heat
  3. Surface temperature rises toward the recovery temperature
  4. Radiation and conduction provide cooling pathways
02

Recovery Temperature

The recovery temperature is the equilibrium temperature a surface reaches in high-speed flow:

Recovery Temperature
Tr=T∞×(1+r⋅γ−12⋅M2)T_r = T_\infty \times \left(1 + r \cdot \frac{\gamma - 1}{2} \cdot M^2\right)
TrT_rRecovery temperature (K)T∞T_\inftyFreestream static temperature (K)rrRecovery factorγ\gammaRatio of specific heats (1.4 for air)MMMach number

Recovery Factor

r=0.89r = 0.89 for turbulent flow, r=0.85r = 0.85 for laminar flow
03

Stagnation Temperature

At stagnation points (e.g., nose cone tip), the recovery factor is 1.0, giving the maximum possible aerodynamic temperature:

Stagnation Temperature
T0=T∞×(1+γ−12⋅M2)T_0 = T_\infty \times \left(1 + \frac{\gamma - 1}{2} \cdot M^2\right)

MACH 1

+20%

above ambient

MACH 2

+80%

above ambient

MACH 3

+180%

above ambient

04

Heating Rate

The convective heat flux to the surface depends on the temperature difference and convective heat transfer coefficient:

Heat Flux
q=h×(Tr−Twall)q = h \times (T_r - T_{wall})
qqHeat flux (W/m²)hhHeat transfer coefficient (W/m²·K)

Simplified Heating Correlation

For engineering estimates, heating rate scales with:
q∝ρ0.5×V3q \propto \rho^{0.5} \times V^3

This explains why maximum heating occurs at max-Q (maximum dynamic pressure), not at maximum velocity—density decreases with altitude faster than velocity increases.

05

When Peak Heating Occurs

All surface locations experience peak heating at approximately the same moment: max-Q. For typical HPR flights, this occurs 2-5 seconds after launch.

ParameterLaunchMax-QBurnoutApogee
Velocity0ModerateMaximum0
DensityMaximumHighLowerVery Low
Dynamic Pressure0MaximumLower0
Heating Rate0MaximumModerate0
06

Heating by Component

Nose Cone

Nose Cone

Experiences the highest temperatures due to flow stagnation at the tip. Full stagnation temperature applies at the tip; recovery temperature along the surface.

qstag∝ρ0.5×V3Rnq_{stag} \propto \frac{\rho^{0.5} \times V^3}{\sqrt{R_n}}

Blunter noses (larger RnR_n) have lower peak heating but more total heat

Body Tube

Body Tube

Experiences primarily skin friction heating. Temperature is slightly higher near the nose cone junction, cooling along length as boundary layer thickens.

qbody=0.5×ρ×V3×Cfq_{body} = 0.5 \times \rho \times V^3 \times C_f

Fins

Fins

Experience multiple heating sources: leading edge quasi-stagnation, flat surface skin friction, and interference heating at the root junction (10-20% higher than freestream).

Leading Edge

90%

stagnation + 10% recovery

Flat Surface

Recovery

temp + 15% interference

07

Fin Sweep Angle Effects

Fin sweep reduces leading edge heating by reducing the velocity component normal to the edge. This correction factor is derived from NASA TP-2914.

Sweep Correction
Heating Factor=1−0.18×sin⁡2(Λ)\text{Heating Factor} = 1 - 0.18 \times \sin^2(\Lambda)
Λ\LambdaSweep angle = arctan(sweep length / span)
Sweep AngleHeating FactorReduction
0°1.0000%
30°0.9554.5%
45°0.9109.0%
60°0.86513.5%
08

Leading Edge Radius Effect

Based on Fay-Riddell theory, stagnation point heating scales inversely with the square root of the leading edge radius. Sharper leading edges experience higher peak heating.

Fay-Riddell
qstag∝1Rq_{stag} \propto \frac{1}{\sqrt{R}}
Cross-SectionLE Heating FactorNotes
Square2.0Sharp blunt edge, highest heating
Double Wedge2.0Sharp wedge angle
Biconvex1.4Curved compression surface
Rounded1.0Finite radius reduces peak
Airfoil1.0Smooth NACA-style curvature
09

Radiation Cooling

At elevated temperatures, surfaces radiate heat. Radiation becomes significant above ~400 K and provides a natural temperature ceiling.

Stefan-Boltzmann
qrad=ε⋅σ⋅(Twall4−Tambient4)q_{rad} = \varepsilon \cdot \sigma \cdot (T_{wall}^4 - T_{ambient}^4)
ε\varepsilonSurface emissivity (0.8-0.95)σ\sigmaStefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²·K⁴)
10

Material Thermal Limits

Different materials have different maximum service temperatures. Peak surface temperature should stay 20% below material limits for safety.

MaterialMax Temp (°C)Notes
PLA50-60Softens easily
ABS90-100Better than PLA
Cardboard150Chars above this
Birch Plywood150Fire risk above
Fiberglass (G10)180Epoxy limited
Carbon Fiber200Epoxy limited
Phenolic250Ablative properties
Aluminum300+Strength degrades
11

Example Calculations

Case 1: Typical HPR at Max-Q

  • Velocity: 300 m/s (Mach 0.88)
  • Altitude: 1000 m (T = 282 K)
Tr=282×(1+0.89×0.2×0.77)=321 K (48°C)T_r = 282 \times (1 + 0.89 \times 0.2 \times 0.77) = 321 \text{ K } (48°C)

Minimal thermal concerns at subsonic speeds.

Case 2: High-Performance Rocket

  • Velocity: 600 m/s (Mach 1.8)
  • Altitude: 5000 m (T = 256 K)
Tr=256×(1+0.89×0.2×3.24)=403 K (130°C)T_r = 256 \times (1 + 0.89 \times 0.2 \times 3.24) = 403 \text{ K } (130°C)

PLA and cardboard would fail. G10 fiberglass or phenolic recommended.

Case 3: Supersonic Research Rocket

  • Velocity: 900 m/s (Mach 2.7)
  • Altitude: 10000 m (T = 223 K)
Tr=223×(1+0.89×0.2×7.29)=512 K (239°C)T_r = 223 \times (1 + 0.89 \times 0.2 \times 7.29) = 512 \text{ K } (239°C)

Metal or phenolic nose cone required. Fin leading edges are critical.

12

Limitations

VALID ASSUMPTIONS

  • Quasi-steady heating (short flight duration)
  • Zero angle of attack (stable flight)
  • Turbulent boundary layer (Re > 500,000)
  • Continuum flow (below ~80 km altitude)

KNOWN LIMITATIONS

  • No transient heat soak-through
  • Simplified geometry (component-average)
  • No ablation modeling
  • Simplified shock treatment

SOURCES

References

Academic Papers

NASA & Technical Reports

Textbooks

  • Hypersonic and High-Temperature Gas Dynamics — Anderson, J.D. (2006)

    Comprehensive textbook on aerothermodynamics (2nd Edition, AIAA)

  • Hypersonic Aerothermodynamics — Bertin, J.J. (1994)

    AIAA Education Series - Theory and applications of high-speed heating

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