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Once your ship temperature gets to 150%, you start taking an increasing amount of hull damage until your ship explodes.
Reducing Heat Generation
There doesn't appear to be a method of preventing heat from building up while running silent, although toggling the power state of the various ship modules in your ship and the amount of throttle provided to the engines will affect how quickly the heat builds up.
For example, running through the overheat process with various systems on I found:
Powered: Engines, Life Support, Power Distribution, Sensor Suite, Engine throttle at 55%: 3 minutes 4 seconds
Powered: Engines, Life Support, Power Distribution, Sensor Suite, Engine throttle at 141%: 1 minute 54 seconds
Powered: Engines, Life Support, Power Distribution, Sensor Suite, Engine throttle at 159%: 1 minute 46 seconds
Powered: Engines, Life Support only, Engine throttle at 159%: 2 minutes 13 seconds
All tests were performed by starting from 0% total heat generated, toggling on all of the systems I was using for the test and running silent mode at the same time to provide the most accurate numbers. All tests used the default power distribution.
To confirm whether the amount of power sent to the engine through power distribution mattered I did one more run with identical parameters to above but with all power diverted to the weapons (which were disabled on the modules screen) and the shields (which were also disabled on the modules screen). This capped the maximum throttle percentage at 99%;
Powered: Engines, Life Support, Power Distribution, Sensor Suite, Engine throttle at 99%: 1 minute 46 seconds
I found this odd, but then considered that perhaps (a little illogically), the heat generated was tied to the actual throttle percentage rather than what is displayed on the screen. Since with no power going to the engines, 99% on screen is in fact 100%, while with two blocks of power going to the engine 159% on screen was in fact 100%, that perhaps the best solution to getting the best efficiency out of running silent would be to divert as much power as possible to the engines. Since shields are disabled while running silent, I diverted 4 blocks of power to the engines and 2 blocks of power to the weapons (which were in fact disabled on the modules screen), and then attempted to get the throttle to 99% on screen while running with the same configuration as before:
Powered: Engines, Life Support, Power Distribution, Sensor Suite, Engine throttle at 108%: 2 minutes 38 seconds
Managing Heat Generation
It is possible to purchase a "heat sink launcher", which requires heat sinks as a form of ammunition but allows you to eject heat sinks to reduce your current heat level. Additionally, it is possible to use the heat sink launcher while in silent running mode.
Conclusion
I concluded from the above testing that the best way to get maximum time out of running silent without having to stop to cool down would be:
Full power to the engines in power distribution
Low throttle % - something like 50-75
Turn off shields
This should provide you about three minutes (ish) of time in running silent without having to stop to cooldown.
Another thing I did notice during testing, the engines generate a lot of heat - so much in fact, that even with all of the other systems off and the ship at a stand still, if the heat percentage is above 0% when you active silent running, the heat percentage will climb by itself, regardless of whether the ship is moving or not.
