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

Question

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?

Answer 1

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.

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

1. Throwing mass overboard
2. Radiating heat

Option 1) is far more efficient than 2), but the only way of doing this in a way that does not destroy performance is to use the propellant itself for coolant.

But propellant flow is limited. When the engine simply operates hotter, the cooling requirements increase without the amount of coolant available increasing. The only way of compensation for this is to note that the surface area of the nozzle increases with the square of the size of the engine, while thrust and mass flow increases with the cube. That's the opposite problem of expander cycle engines, which have a maximum possible thrust. Regeneratively cooled engines have a minimum thrust, growing proportionally to the cube of the temperature.

At some point, the massive size of the minimum engine starts being bulkier than you can afford.

Option 2) on the other hand starts of far worse, but scales less badly than option 1). Space is an efficient insulator, so to get rid of large amounts of excess heat, large radiators are needed. This sets limits for thrust, as the mass of the radiators is simply gets too large. A spacecraft bottlenecked by radiators has low acceleration.

However, increasing coolant requirements only scale linearly with radiators. Twice as much heat, twice as many radiators. Improvements in material science even bring some very favourable efficiency gains, as radiation is proportional to the fourth power of the temperature.

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Thermal rockets have been demonstrated up to about a 1,000s. Even with some slight boost from hydrogen disassociation, 2,000s seems completely unachievable, as it would require materials with almost quadruple capabilities. Even 1,500s is very optimistic.