There are four factors to consider when speculating on the future of slavery in an independent Confederacy. The first two would seem to support the idea of an early end to slavery, but the other two would seem to argue against this.
Supporting the idea of an early (i.e. 1880s) end to slavery is the fact that industrial agricultural techniques and inventions were making slavery obsolete and uneconomical. Setting any notion of ethics or morality aside, slavery was simply not going to pay in the long run. We might hopefully imagine that Southern slaveowners would see that it was cheaper to hire low-wage labor for their fields, rather than deal with slaves (who must be fed, given medical attention, and taken care of when old).
Another factor that would support an early end to slavery would be the international condemnation the Confederacy would no doubt endure if they attempted to maintain the institution. IOTL, supporters of the Confederacy in Britain largely persuaded themselves that the South would abolish slavery pretty soon after gaining independence. Had they discovered that the South had no intention of doing so, there would have been a serious backlash. The people of the North, obviously, would have maintained a hostile attitude towards slavery in the South. Because new sources of cotton had been developed in Egypt and India during the war, I would expect to see organized boycotts of Confederate cotton in both Britain and the United States, which would have hit the Southern economy hard.
But there are two major obstacles to an early Confederate abolition of slavery. First, the Confederate Constitution made it almost impossible for the Confederacy to abolish slavery. Article I, Section 9, says specifically that "no bill of attainder, ex post facto laws, or law denying or impairing the right of property in negro slaves shall be passed". Article IV, Section 3, says that slavery will automatically exist in any new territory acquired by the Confederacy.
Another reason that an early emancipation scenario is unlikely is how the culture of the South had adapted to slavery between the Revolutionary era and secession. Around 1800, the general feeling was that slavery was a problem that needed to be solved or, at best, a necessary evil. Jefferson, Madison, and Washington had all been slave-owners but had all also acknowledged that slavery was an abomination and that the world would be better off without it.
By 1860, however, the pendulum had radically shifted, with society's leaders maintaining that slavery was a positive good. A few, like Edmund Ruffin, went so far as to say that poor whites should be enslaved, too. For two generations, Southerners had felt themselves pushed against the wall by rising abolitionist sentiment in the North and had therefore changed their views on slavery, transforming it in their minds from a necessary evil to a positive good. A few men, such as Robert E. Lee, still maintained the old-fashioned attitude that slavery was morally wrong but that nothing could be done about it, but most of the men who mattered in society saw it as something to be celebrated and maintained rather than something that should be gradually done away with.
All things considered, I would expect slavery to continue in the Confederacy until either significant social change takes place or some sort of convulsion from within or without forces the Confederacy towards emancipation at the point of a gun. Even then, I would expect to see an apartheid society.