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If the holder is a Poison-type, it will recover 1/16th of its maximum HP rounded down at the end of each turn. All other types will take damage equal to 1/8th of their maximum HP if holding this item.
Black Sludge can be summarized as an exclusive alternative to Leftovers, with nasty side-effects for the non-designated holder. Like Leftovers, they should be placed on (Poison-type, naturally) Pokemon that require either extra bulk or use it to supplement their natural walling abilities. Leftovers and Black Sludge are generally used to restore lost HP, and often allows for extended longevity. Black Sludge may often be used on offensive Pokemon so that passive damage does not add up so quickly, allowing a sense of bulk on sweepers over Pokemon with items such as Life Orb. More defensive Poison-types can also utilize the item effectively, gaining HP back with ease.
The main advantage of Black Sludge opposed to Leftovers does not apply to the standard metagame: on teams with Poison-types, these Pokemon can forfeit Leftovers to free up a slot under Item Clause. The second advantage is that it is a "weapon" of sorts against Trick Pokemon, discouraging them from Tricking, or used to damage the Trickers themselves. The HP regained from Black Sludge can be used to set up Substitutes or snatch an opportunity to boost one's stats; while there are few Poison-types which commonly use set-up moves in OU, in the lower tiers, there are several options available. Where Life Orb would kill a sweeper too quickly, Black Sludge can be used instead to give a sense of bulk. It also negates sandstorm and hail, which wear at frailer teams quickly.
There are several disadvantages to using Black Sludge as well. The extra HP will often be useless on a sweeper, and therefore the power lost through item choice can be devastating for a team, as it may leave certain Pokemon unable to KO the opponents when specifically needed. While EVs can be saved, the EVs not used defensively due to the effect of Black Sludge may leave a Pokemon in danger to threats if sandstorm or hail is in play. Lastly, the key reason to using Leftovers instead of Black Sludge is that if Black Sludge is Tricked onto an opponent, that opponent then can Trick it onto a non-Poison-type member of your own team.
In short, there are few reasons to use Black Sludge over Leftovers, but under Item Clause it can be a handy alternative. It is also interesting to note that there are two Poison-types with the ability Sticky Hold, which renders attempts to steal the item futile (Swalot and Muk).
When using either item, it is advisable to make your HP divisible by 16, +1. This will allow for a "magic Leftovers number" (a number divisible by 16), while the +1 allows essentially for a "free" hitpoint due to the floor function in the damage formula.