The shroud allowed the gun to fire at 10 rounds per second, or 600 rpm. But would it be sufficient to cope with 1200 rpm like on the MG42? Maybe, as the higher rate of fire would have pulled more air through the shroud. Using non-Cordite .303 ammunition might have been additionally helpful for cooling as Cordite burns very hot.
The shroud likely increased manufacturing cost both because of complexity and through use of precious aluminium for its internal cooling fins. We take aluminium for granted nowadays, but it's actually very energy-intensive to extract. In terms of lack of simplicity, compare with the simplicity of the Rheinmetall MG42 -- a stamped metal wonder of simplification and reduction of operating costs compared to the MG38, or the Bren.
But when the Lewis Gun came out its comparator was the heavy, water-cooled Maxim MG with similar rate of fire. The Maxim took three men to operate and move about, whereas the Lewis could both be operated and carried by one man (though in practice, I believe, usually by two men).
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