The square root formula is derived by mathematical methods after making a series of assumptions. Its assumptions are as follows:
　　1The casting is a flat plate casting with an infinite length and width. With this assumption, the original heat transfer problem in three directions can be simplified to a unidirectional heat transfer problem, that is, only the heat transfer perpendicular to the mold wall is considered, and the heat transfer at the end face of the casting is not taken into account.

　　2In the derivation process, it is assumed that the mold wall and the casting always maintain close contact without any gap. This can greatly simplify the analysis of the heat transfer process. Otherwise, the heat transfer problem of the intermediate gap layer also needs to be considered. Shortly after the actual casting is poured, when the solidified layer of the shell begins to contract linearly, a gap appears between it and the mold, and the heat transfer rate slows down. Assuming that the metal and the mold are always in close contact, this problem can be simplified to a semi-infinite heat transfer process for solution, thus simplifying the problem.

　　3During the derivation process, it is assumed that the temperature at the interface between the metal and the mold remains constant. In sand casting, the temperature at this interface does not change much.

　　4During the entire solidification process, the physical constants of the mold and the metal, such as heat capacity and thermal conductivity coefficient, are all constants.

　　5It is applicable to pure metals and eutectic composition alloys.

　　Since the square root law is derived based on the above assumptions, it can only be applied within a certain range. Experiments show that for carbon steel castings of flat type (where the contact surface between the mold and the casting is flat) and with a low carbon content, the actual solidification time is basically the same as or very close to the solidification time obtained by the square root law. So, the square root law reflects some fundamental laws of the solidification process, but it has limitations and is an approximate calculation method.

　　The square root law is the fundamental method for calculating the solidification time, and other calculation methods have all been developed on this basis. When calculating, various conditions are taken into account, such as overheating of the alloy, gaps between the casting and the mold, etc. The aim is to take all factors into account to make the calculation more accurate.

　　To find the solidification time of carbon steel plates, just take the wall thickness of1/2The time obtained by substituting into the square root law is the solidification time of the casting.

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