As we have learned, kelim used for cooking must be kashered by means of hagala. However, there are different temperature levels in cooking, and the operative principle is: ke-bole’o kakh polto.
A kli rishon on the flame: The highest level of intensity of cooking, and as it is used for ḥametz, so it is kashered: in boiling water, in a kli rishon on a flame. Even if the water in the kli rishon on the fire was not boiling when it was used for ḥametz, it must boil during kashering, because it is impossible to estimate the temperature at which the kli absorbed ḥametz, so to dispel uncertainty, it must undergo hagala in boiling water.
A kli rishon removed from the flame: A kli that was heated over a flame and then removed, or if the flame beneath it was then extinguished, still can cook a bit. Therefore, kelim that became forbidden in this manner can be kashered in a kli rishon that is no longer on a flame if the water in it remains very hot.
Liquid poured (“irui”) from a kli rishon: This has the capacity to cook the surface layer (“kedei klipa”) of a food item. Kelim that became forbidden through irui can be kashered by pouring boiling water over them from a kli rishon.
A kli sheni: If a dish that was cooked on a fire was poured into another kli, the second kli is called a kli sheni. The poskim disagree about whether it is necessary to kasher kelim that were placed in a kli sheni. In practice, the ruling is that they must be kashered by means of hagala in a kli sheni (SA 451:5).
Some are even stringent regarding a kli shlishi and beyond (i.e., kelim at least twice removed from the kli in which the food was cooked), maintaining that as long as the food remains at the temperature of yad soledet bo, a kli into which it was inserted must be kashered. Even though most poskim are lenient and maintain that it is sufficient to rinse them, due to the gravity of the ḥametz prohibition, le-khatḥila the custom is to kasher them by means of hagala (SAH 451:34).
A kli that is used at a temperature that never reaches yad soledet bo, that is, it remains under 45oC, even a kli rishon, does not require hagala, and it is sufficient to rinse it.
Le-khatḥila, many have the custom to kasher all kelim used with hot ḥametz foods in a kli rishon on the fire, even if they were used only for irui or as a kli sheni, as we are concerned that they were once used as a kli rishon, but that this was forgotten. (See Rema 451:6; Kaf Ha-ḥayim 451:107.) However, when the level of forbidden use is certain, it is not necessary to kasher the kli using a higher level of hagala.