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In a hybrid LOX rocket, only the flow of LOX may be controlled. The amount of paraffin melted by heat is determined by the very complex heat transfer from gas to solid paraffin. If the mixture of propellants is too fuel rich or too oxygen rich, the efficiency will suffer. The temperature of the paraffin before launch may influence the efficiency. A start in a very hot climate may be impossible.

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You just found the primary shortcoming of hybrid engines: fuel flow cannot be controlled on the fly. The engine can be shut down and restarted unlike common SRB, but throttlability is lousy and lossy. And lots and lots of promising fuels were abandoned due to bad melting or evaporation temperatures.

You can prepare the optimal fuel a'priori by choosing the parafin composition - longer hydrocarbons will have higher melting point. Other additives may reduce impact of parafin melting; composite the parafin with fibrous material (cellulose/paper, cotton, polyethylene fiber) and despite parafin melting, the shape will be retained until the fiber burns out. Or just tune your combustion parameters (oxidizer supply, chamber pressure) to such that parafin surface ablates before the heat from the surface gets to transfer deeper.

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  • $\begingroup$ (you may also try cooling the parafin with the LOX. Say, the LOX must flow on the outside of the combustion/fuel chamber before being injected; parafin is cooled, LOX is heated/evaporated providing more energy. Of course as you'll have a lot of trouble controlling the temperature, this may introduce combustion stability problems as mix of gaseous and liquid oxygen is injected... well, no easy solution here, it's rocket science!) $\endgroup$ – SF. Jan 19 '17 at 16:47
  • $\begingroup$ But in solid parafin heat transport is very low, it is a bad thermal conductor. $\endgroup$ – Uwe Jan 19 '17 at 17:21

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