Which is hell: punishment, or revenge? Let me first explain my understanding of these words (punishment &amp; revenge):

Punishment is typically when you do something to one person to make things better for society, or for that person.  It is either something unpleasant, so that the person will not want to do the bad thing, because they know the punishment is there, or they do the crime, but because of the punishment (death penalty / chopping off someone’s arm / being locked away) they cannot do the crime again or will think twice before doing it again.

Punishment, in theory, serves the majority of people.  It is a good thing.  It has positive benefits to most people (negative to one person – the crime committer – temporarily to solve a social problem). 

Revenge is completely different.  That is when you cannot control your anger and hurt someone because they hurt you.  No positive benefit, other than making the angry person feel better.  It is, as I understand it, all about emotions.

Romans 12:19: “It is mine to avenge, I will repay, says the Lord”.  Hell does not seem like punishment, it seems like revenge.  This makes no sense, given the "good" and "just" descriptions of God.

So, based on the above, what is God, who is good and just, intending by sending people to hell:  to punish, or to take revenge?  Please include a logical explanation with your answer.

EDIT

I've been asked to define what I understand as "hell", and the doctrinal context that I want this answered in, so here's my attempt to do that:

I understand hell to be a place of eternal suffering that God created, where He sends people who do not believe in Jesus.

This is what I understand to be the typical protestant view of hell, as described in the bible.  If you have a different view, then you may not be the best person to answer this question, because I really want an answer from those who believe this point of view.

I would also like all answers to include some kind of reference, if their source is not simply logic.
Neither. Both if these options are built around a human centered view of justice.

In reality, God isn't working under the same constraints as we would be and he has other purposes in mind. It still comes down to justice, but not quite the same kind of justice that would be metered out on a court on earth.

Here on earth, our purposes in "punishment" are usually either corrective or protective. Some punishments are given to try to convince the wrongdoer that "crime doesn't pay". Others guard against future harm, sometimes at the expense of giving the offender any more chances.

Also in human terms, "revenge" is in some way trying to regain something that was lost whether possessions or honor, or at least restore a balance between victim and perpetrator but reversing the crime and setting the original perpetrator back.

God's judgement against sin is not quite either of these things. The nature of His person is not such that he has lost anything that would need to be restored and he is certainly not trying to hurt somebody else the way he was hurt just to get even.

On the other hand all of our ideas in this are in some way just echos of something more real and true; our ideas of "punishment" are but a memory destroyed by our own sin of what God's idea of punishment is.

Hell is a just punishment for the crime humanity has committed. In eternally damning some humans to there, God is doing the only just thing. He is also giving men just what he asked for. In his pride man set himself up as god apart from God, and in sin he continues to demand his independence from God. Being permanently cut off from Him is just the logical extension of that.

At the same time there is an element of vengeance in it. God is extracting from the rebels everything they tried to steel from Him. He is demonstrating for all eternity that HE is Lord, there is no other, and that he will not tolerate evil. Never will it be said of Him that he let the wicked prosper unjustly. While human revenge is unjust, God eradicating evil is not.