Both these processes are used in concentration of bauxite.
They use different reagents.
Which is more commonly used (wikipedia says Bayer's) and why?
When is one preferred over the other?
Which is more efficient and why?
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From this source
Baeyer's process: This method is mainly applied when ferric oxide is present as chief impurity.
Serpeck's process: This process is used when silica is present in considerable amount of bauxide ore.
I couldn't find any information on the efficiency , but I suspect both are efficient in their own way. We cannot compare these two processes as they are used for two different impurities.
During extraction of Al from Bauxite containing more than 10% of silica as impurity, Baeyer's process will be uneconomical due to generation of insoluble sodium aluminium silicate. In such cases Serpeck's process can be preferred in which Silicon can be separated as volatile substance at higher temperature.
Baeyer's process is used to leach Red Bauxite, which has Fe2O3 as the chief impurity. Serpeck's process, on the other hand, is used to leach White Bauxite, which has Silica as main impurity. Both are efficient for their respective ores.
But Baeyer's process is more frequently used, as Bauxite, more often than not, is also a source of Gallium, which can be extracted from Baeyer's process.
I guess this is why Baeyer's process is more frequently used.
Source of Information : Wikipedia