1905-10 was the production period for Barrel-hood according to Franklin Club's site. But in this case I think that 1910 is ruled out, as the same soucre states fine mesh hood altought this very same picture is captioned as a 1910 Model K [sic!] elsewhere:
http://momentcar.com/franklin/1910/franklin-model-k/
Again H. H. Franklin Club's guide to identification states that the barrel hood models are Series G, D and H.