The paper by Wang et al. at Scitech 2018 uses the following constraint on the rocket heating rate (equation (3) in the paper):

$$k_Q \sqrt{\rho} V^{3.15} \le \dot{Q}_{max}$$

(screen shot)

I am wondering what is the original source for this heating rate model? Where does this equation originate from?

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    $\begingroup$ I've formatted your equation using MathJax, can you double check it? Thanks! $\endgroup$ – uhoh Apr 13 '20 at 4:15
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    $\begingroup$ Heating rate is proportional roughly to the cube of velocity. It’s an order of magnitude estimation based on the fact that heating comes from a combination of stagnation enthalpy (proportional to $v^2$ at high velocities) and convective heat transfer (proportional to $v^1$). I think the 3.15 power is a fudge factor for other uncertainties. $\endgroup$ – Paul Apr 13 '20 at 5:40
  • $\begingroup$ @Paul that's a great explanation, but can you give a reference source (even a textbook)? $\endgroup$ – Carl Witthoft Apr 13 '20 at 13:34
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    $\begingroup$ It’s a standard estimate for high speed air flow. Most text books on compressible supersonic/hypersonic flow usually mention this as a back of the envelope estimate of heating. One standard textbook that briefly mentions this estimate is J D Anderson’s Intro to Aerodynamics. $\endgroup$ – Paul Apr 13 '20 at 22:15

A Detra-Kemp-Riddell model for stagnation heating.

Detra, R. W.; Kemp, N. H.; and Riddell, F. R.: Addendum to "Heat Transfer to Satellite Vehicles Re-entering the Atmosphere." Jet Propulsion, vol. 27, no. 12, Dec. 1957, pp. 1256-1257. published online

Equation 32 (page 20) and Ref 6 in NASA TM X-2058 A General Transient Heat-Transfer Computer Program for Thermally Thick Walls L. Bernard Garrett and Joan I. Pitts, 1970


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