What are fire bricks which are used in Down's Cell? What is its role?

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Hint: A fire brick, also known as a firebrick or refractory, is a ceramic block used to line furnaces, kilns, fireboxes, and fireplaces. The primary purpose of a refractory brick is to endure high temperatures, but it also has a poor thermal conductivity to improve energy efficiency. Dense firebricks are typically employed in applications that are subjected to intense mechanical, chemical, or thermal stresses, such as the inside of a wood-fired kiln or furnace, which is subjected to abrasion from wood, fluxing from ash or slag, and high temperatures.

Complete answer:
The Downs' process is an electrochemical technique for commercially producing metallic magnesium that involves electrolyzing molten \[MgC{{l}_{2}}\] in a specific device known as the Downs cell.
A carbon anode and an iron cathode are used in the Downs cell. Magnesium chloride has been heated to a liquid condition as the electrolyte. Although solid magnesium chloride is a poor conductor of electricity, when it is molten, the magnesium and chloride ions get mobilised and become charge carriers, allowing electric current to be conducted.
Anode \[2C{{l}^{-}}\to \text{ }C{{l}_{2}}\left( g \right)\text{ }+\text{ }2{{e}^{-}}\]
Cathode \[M{{g}^{2+}}+\text{ }2{{e}^{-}}\to \text{ }Mg\text{ }\left( l \right)\]
Overall \[M{{g}^{2+}}+\text{ }2C{{l}^{-}}\to \text{ }Mg\text{ }\left( l \right)\text{ }+\text{ }C{{l}_{2}}\left( g \right)\]
A ceramic substance called firebrick is used to line furnaces and kilns.
Firebrick is heat resistant and has a poor thermal conductivity.
A Downs cell works at 600 degrees Celsius. It is made out of a steel shell with a firebrick lining.
The firebrick has three purposes:
It shields the steel shell from the molten \[NaCl\,(or)\,CaC{{l}_{2}}\]mixture's corrosive effects.
It works as a thermal insulator, trapping heat inside the cell and increasing efficiency.
It functions as a self-sealing membrane. The electrolyte freezes in the gap before it reaches the steel shell because the firebrick is thick enough.

Calcium chloride, barium and strontium chlorides, and, in certain cases, magnesium fluoride are added to the electrolyte to lower the temperature necessary to maintain the electrolyte liquid. Sodium chloride melts at 801 degrees Celsius (1074 degrees Fahrenheit), while a salt combination comprising 33.2 percent \[MgC{{l}_{2}}\] and 66.8% \[CaC{{l}_{2}}\] may be maintained liquid at temperatures as low as 600 degrees Celsius. When using pure magnesium chloride, a metallic magnesium emulsion forms in the molten \[MgC{{l}_{2}}\], making separation difficult. As a result, a combination of \[MgC{{l}_{2}}\] (42 percent) and \[CaC{{l}_{2}}\] (58 percent) is one possibility.