I want to build something similar to this but instead of plain copper tubing, have the cold water pump into a heat exchanger of sorts to make it more efficient. Any suggestions on where I can find something about the size of a box fan? Or maybe a radiator roughly that size? Thanks!

Go find a scrap semi radiator.........................

The contraption in your link would have very little efficiency.  You can calculate BTU/hr of heat removal, but it just will not work well at all.

Mastercool makes portable evaporative coolers on wheels that are sold by Home Depot, Lowes, and many others.  I use one in my garage and have a huge one on my house tied into my HVAC.  They work very well in low humidity environments and easily drop 20 degrees below ambient.  

Goto the auto parts store and buy a radiator I once work in a factory that cool the building with creek water. It was pumped though cast iron radiators and sprayed on the metal roof. It work so well that on an 80 or 90 degree day you would need a long sleave shirt or a light jacket on while in the building.

evaporative coolers are also called swamp coolers, just an FYI

not sure where in Iraq you are, but I know Al Asad has a hell of a DRMO and they always have some serious swamp coolers sitting around. bet you could score one of those and get it going

or... make friends with some HVAC troops in your local USAF CES or RED HORSE unit

Thanks for all the replies guys.

I realize the one shown in my link would be very inefficient, which is why I'm thinking about using a heat exchanger. That would be what an evaporative cooler would be right (heat exchanger with cold water + fan)? There's also another one on the same site where a guy found copper pipes with aluminum fins so he welded a bunch together. Basically built a radiator, so I'm thinking why not just use a radiator (I don't have welding equipment)?

Ironically, this isn't for use here in Iraq (there's enough portable AC's and huge swamp coolers to go around). This is actually for when I get home to Southern California. Our house is pretty old and has never had a working AC as far as I can remember. I've grown accustomed to sucking it up and using fans, but why live in unnecessary misery when the energy consumption of a DIY cooler is justifiable?

Swamp coolers generally blow air directly across the wetted surface/media - ie big ass sponge..  Adding a heat exchanger so that you have a separate closed loop will be incredibly inefficient since there is little temp difference between the working fluids.

Brian

Most places in SOCAL are too humid for a swamp cooler to work.  Save your pennies and get an AC system.