# Which theory does this reentry heating rate equation come from?

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

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

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

• I've formatted your equation using MathJax, can you double check it? Thanks! – uhoh Apr 13 '20 at 4:15
• 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. – Paul Apr 13 '20 at 5:40
• @Paul that's a great explanation, but can you give a reference source (even a textbook)? – Carl Witthoft Apr 13 '20 at 13:34
• 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. – Paul Apr 13 '20 at 22:15