I saw the following image while searching for this answer in Sung Hwan Kim's thesis Germanium-Source Tunnel Field Effect Transistors for Ultra-Low Power Digital Logic It plots power density in Watts per square centimeter. Data points are for microprocessors, but it also includes indicators for 'Hot Plate", "Nuclear Reactor", "Rocket Nozzle", and "Sun's Surface".

The value plotted for the Rocket Nozzle seems to be 1,000 W/cm^2. Is that a sort-of canonical number with many engines used for launch purposes being somewhat similar, or is that an extreme example?

enter image description here

The Sun

The solar constant is the total electromagnetic radiation power per unit area at 1AU (about 150 million km) and is about 1361 W/m^2. Scale that by $1/r^2$ $to the radius of the Sun (about 0.696 million km) and that's 6300 W/m^2 which agrees nicely with the plot.

A Rocket Nozzle

I'll work through one example as a proposed way to estimate this.

Merlin engine with the smaller nozzle for first-stage atmospheric operation.

From this answer and an image in this answer I'll call the exit diameter $D$ of 90 centimeters and so radius $R$ of 45 centimeters.

From Wikipedia's Merlin (rocket engine family) I'll use the sea level $I_{SP}$ of 282 seconds and thrust (force) $F$ of 845 kN to get the total mass flow rate.

$$\dot(m) = \frac{F}{v=gI_{SP}} \approx 305 \ \text{kg/s}$$

Start by assuming correct stoichiometry as an approximation CH2 + 1.5O2 → CO2 + H2OI get that 23% of the mass flow rate comes from the CH2 or kerosene.

$$\dot{m_K} \approx 69 \ \text{kg/s}$$

The energy density of kerosene $U$ is about 43 MJ/kg. Add in a fudge factor of 0.8 for incomplete burning, and I get:

$$I = \frac{P=\dot{m}U}{A=\pi R^2} \approx \ 466,000 \ \text{W/cm^2}$$

or 466 times larger than the number in the plot. This means I'm dramatically mis-interpreting something about the plot or I've mad a mistake in my math.

So the exhaust is certainly cooled by the expansion and not all of the chemical energy released in combustion is still present as heat at the exit of the exhaust, hopefully most of it has been converted to directed kinetic energy.

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    $\begingroup$ This paper mentions about the SSME " the heat flux in the throat region is around 160 MW/m^2". sciencedirect.com/science/article/pii/S1000936117301024 $\endgroup$
    – Paul S
    Feb 7 '19 at 6:23
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    $\begingroup$ The paper has a typo. Using the figure from the mfg rocket.com/space-shuttle-main-engine and converting horsepower to watts (ugh...), I get ~160GW/m^2 for the throat energy flux of the SSME. $\endgroup$
    – Paul S
    Feb 7 '19 at 8:01
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    $\begingroup$ I think you've misunderstood what the authors wanted to compare with this graph. I think the "Rocket nozzle" means the heat conduction (like a CPU conducts heat into the heatsink which is a big problem - keyword dark silicon) into the nozzle wall not the heat convection through the nozzle. Typical values given are usually 10 - 100 MW/m^2 which is equal to 1 -10 kW/cm^2. This paper measures heat flux of 1-10 MW/m^2 (page 335, notice the *10^6 on the y axis) for a subscale chamber. $\endgroup$
    – Christoph
    Feb 7 '19 at 11:06
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    $\begingroup$ I was thinking along the same lines as @Christoph this morning! This paper servidor.demec.ufpr.br/CFD/bibliografia/propulsao/… gives about 16 KW/cm^2 for the SSME nozzle coolant circuit, and about half that for a test chamber. Still too big but closer. $\endgroup$ Feb 7 '19 at 14:45
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    $\begingroup$ @OrganicMarble If we take an average over the whole inner surface we get to a few kW/cm² only. And the SSME is a bit larger than the 'typical' engine, so about 1 kW might be about right. $\endgroup$
    – asdfex
    Feb 7 '19 at 22:34

Using Wikipedia as the source, I repeated your calculations for the Space Shuttle Main Engine (SSME) and compared it to the manufacturer's specification and got good agreement. So your method appears valid. The chart in Sung Hwan Kim's thesis must be referring to some old rocket technology such as the V-2.

For SSME with a sea level $I_{SP}$ of 366 seconds and thrust $F$ of 1859kN

$$\dot m = \frac{F}{v=gI_{SP}} \approx 518 \ \text{kg/s}$$

And a stoichiometry of 1/9 part by weight of Hydrogen (16 for O 1x2 for H) I get

$$\dot{m_K} \approx 57.6 \ \text{kg/s}$$

The energy density $U$ of H2 is 142 MJ/kg and the nozzle exit diameter is 230cm. Calculating the energy flux:

$$I = \frac{P=\dot{m}U}{A=\pi R^2} \approx \ 197,000 \ \text{W/cm^2}$$

Now, comparing to the manufacturer's specification of 12 million horsepower (careful, don't run this engine out to Mars!): I convert 12 million HP to Watts and get 8,950 Megawatts. Dividing this by the area of the nozzle:

$$I = \frac{P}{A=\pi R^2} \approx \ 215,000 \ \text{W/cm^2}$$

Though if you note the manufacturer of the SSME says this is at 109% power, not liftoff power as the Wikipedia article indicates. Dividing by 1.09 yields the same answer.

Too tired to calculate for the V-2, but here's the link to the required info

I also add that the comparison to the CPU to engine nozzle exit is not fair. The equivalent of the nozzle exit is the external area of the CPU heat sink. A more fair comparison would be to the throat of the combustion chamber. With an expansion ratio of 77.5 on the SSME, you get to multiply by 77.5 to get the energy flux at the throat, or $15.3 \ \text{MW/cm^2}$.

Those rocket designers are dealing with energy fluxes that exceed that of the surface of the Sun by three orders of magnitude!

  • $\begingroup$ Sweet! Thank you very much for digging into this. $\endgroup$
    – uhoh
    Feb 7 '19 at 9:10
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    $\begingroup$ no problem. Been playing with Kerbal and had been wondering about energy densities myself. The ISP of 500,000 and 3MN engine is kinda ridicules on the density side. As usual it's material science that's the limit on most physical things we wish to do. $\endgroup$
    – Paul S
    Feb 7 '19 at 9:12
  • $\begingroup$ It's not a energy density (J/cm³), it's a power flux (W/cm²). To your last sentence: be careful in comparisons with the Sun. The Suns surface is static and radiating the heat. The rocket engine is not - there's (literally) a ton of material shot out every two seconds carrying this energy. The kinetic energy alone is 3.7 GW or 45 kW/cm² at the nozzle. The comparison isn't too impressive if you compare a square meter of surface to a tube of a imaginary 4 km length and one square meter diameter. $\endgroup$
    – asdfex
    Feb 7 '19 at 20:02
  • $\begingroup$ Funny I googled for energy density last night, and suddenly it's changed in the morning. I'll do a second round of search and replace. As far as the Sun, it's ejecting material called the Solar Wind as well as photons. IR not an expert in that at all, so feel free to suggest a true value. As far as a tube vs surface, well, that's volume vs surface. The original comparison was energy for surface area, so we should stick with that, since it directly impacts the materials engineering that's behind this discussion. $\endgroup$
    – Paul S
    Feb 7 '19 at 21:48
  • $\begingroup$ Sure, the Sun surface is not fully static - but the flux is really low. For materials engineering the flux in the nozzle doesn't play a role - only temperature and heat transfer to he walls. And temperature is lower in the rocket exhaust compared to the Sun. $\endgroup$
    – asdfex
    Feb 7 '19 at 22:11

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