You really should know more about thermodynamics and enthalpy to see if your idea is even possible. There is a finite number of Joules of energy in one AA battery. You need to know also how much mass you will be trying to cool to a certain temperature and from what initial temperature. If this mass will not be going through a change in phase state then the total energy required to be removed will be less. As Tcmtech highlighted, peltier devices will likely be your best bet.

The devilish details in this problem are just a few critical items:

• Thermally isolating what you wish to chill from the rest of the world but recognize that heat (power, joules per second) will still be coming in from the outside to heat what you want chilled.
• Identify all of the mass that will be chilled, not just the desired object.
• Removing the waste heat from your heat pump (peltier) without letting it to easily return back to the cooled item.
• Avoid any phase change in matter as possible. Condensing moisture out of the air and freezing water to ice will take a large amount of total energy.

JB -

You could find some really great ideas for your project if you do some research on the millions, if not billions, of $$$ spent on this technology for several industries - Space (NASA), Fire and Rescue, Military, Medicine, and Sports and Recreation.

Big Players on the NASA Constellation extravehicular environmental systems (all new temperature regulating undergarments for space suits) included Dow, Gore, 3M, DuPont, Oceaneering, JPL, Boeing, and many more. Genius people with essentially unlimited funding. They have a system that works pretty good if you have a few million for the suit, and the vacuum of space to dump the heat into, and the weightless environment to help lug around the 215 pound device, plus the insulated suit.

What you want to do with your cooling pack is physically possible. Financially? Practically? Safely? Maybe not so much...

As others have said, a few of your problems are going to be the need for a huge power supply if you want this to work for more than a few seconds at best, how to get rid of the heat generated by the power supply as it creates the power you need for the cooling cycle, along with the heat your device sucks out of the gel/pack, and the overall mass/bulk of the system.

Please continue to work on it. Imagine the rush of figuring this out and being the guy who can say - "I just looked at it from different angles and kept going till I found something that worked."

Meanwhile, ice is pretty safe and cheap.