I suppose you can generate some secondary alcohol from the primary alcohol with sulfuric acid. Presumably, protonation of the alcohol, which when heated can eliminate water and to rearrange from a primary carbocation to a more stable secondary carbocation. If this is captured by the water, it would give the secondary alcohol. However, I think it is a little unfair to expect that one can efficiently do so as loss of water would be a faster reaction from the secondary alcohol than the primary alcohol. Also, for the loss of water to occur, conditions for removal also are not good for trapping of a carbocation with water.