This thesis considered two passive safety measures: plug formation and heat pipes. During a reactor core meltdown, the molten corium material can access cooling pipe connections. There is a chance that the passive plugging of melt flow due to solidification can occur, provided there is an adequate heat sink. A numerical model was created to simulate corium flow through an empty vertical pipe. The numerical model was validated through experimental work using gallium and verified using a previously built analytical model. Heat pipes are passive, two-phase heat exchangers with excellent heat transfer capabilities. They can be used in passive reactor core cooling and spent fuel pool cooling. Heat pipes have different operating limits that impact their operating conditions and heat transfer capabilities. A numerical approach was used to determine the operational limits of a liquid metal heat pipe that can be used in nuclear applications.