^ This issue was the core reason for the Chernobyl disaster.
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The biggest problem at Chernobyl was the design of the reactor, which resulted in a positive void coefficient. The positive void coefficient produced a runaway positive feedback loop during the strange experiment they were running. In short, boiling in the reactor produced more power, which resulted in more boiling, and even more power. Things got out of control very quickly and the reactor tore itself apart with a large steam explosion from the quick power spike.
US plants do not have this kind of design problem, and would never be licensed to operate in the US. BTW, the plant in question in Ukraine is a VVER, which is more closely similar to the western pressurized water reactor (PWR). The Chernobyl-style RBMKs are no longer operated in Ukraine, but Russia does still have them in operation.
All of the discussion in other posts about emergency power ignores things like steam-driven emergency feedwater pumps. Those safety systems are very common in the US, and something like them exist in PWRs as well as BWRs. Steam is produced in either the steam generators or reactor. The steam goes to drive a small turbine connected to a pump. The pump then delivers cooling water back to the steam generators or reactor. No outside power is needed drive the pump. There might be a small amount of electricity needed to run the control systems for the pump and regulation valves, but that is a small draw.
- Pete