# What is the limit on ISP for cooled physical nozzles — and how hard is it?

High performance chemical rockets, as well as hypothetical gas-core nuclear thermal rockets, can operate with chamber temperatures above the failure point of any available substance, because internal cooling and film cooling allow the walls of the chamber and nozzle to be cooler than the gas in the thrust chamber.

Thermal rocket ISP is fundamentally based on temperature -- the higher, the better.

Conceptual designs for fusion rockets generally use a magnetic nozzle -- but at what point (in ISP) do physical nozzles become impossible to use -- and how distinct is this point -- is it a specific ISP for a specific propellant, or heavily variant on tolerable mass, thrust, etc?

The point is fairly distinct, as improved materials and designs only yield diminishing returns with respect to temperature. It's a square law, with Isp being proportional to the square root of temperature

$$I_{sp} \propto \sqrt{T}$$

If some miracle breakthrough in material science resulted in materials standing twice as high temperatures, it would still merely improve the efficiency of thermal rockets by 40%, still short of ion propulsion.

As you suspected, this is also depended on the propellant, with molar mass ($$M$$) being the deciding factor. Lighter molecules move faster at equal temperatures

$$I_{sp} \propto \sqrt{\frac{T}{M}}$$

(proportionality gotcha: this also depends on the heat capacity ratio of the molecule)

This is the prime motivation for using $$H_2$$ in nuclear rockets, as it has the second lowest molar mass of any molecule. Above an Isp of 1,000s, $$H_2$$ starts disassociating into monoatomic $$H$$. Improvements in the operational temperature of nuclear rockets therefore scale somewhat better than temperature scaling alone would suggest, as they hydrogen starts disassociating. But beyond that point, no improvements in molar mass can be made.

Chemical rockets are not bottlenecked by nozzle temperature, but instead by the energy contents of the fuel.

Cooling itself comes with some scaling problems. In space, it can only be done in two ways:

1. Throwing mass overboard