Gas law cooler

I was thinking of a setup like this:

Basically a pump, it could be run with an engine but I show it here with a foot pump, connected through an insulative container and to an array of very thin thermally conductive tubing. there is an opening on the top of this tubing so water can fill it, but it is thin enough that when air is pumped into it the air replaces the water in the tubing without forming bubbles. It was hard to diagram but the tubing has to be arranged so when no air is coming out of the pump water can fill the tubing, so it would be more horizontal and on a slant. The volume of the tubing is the same as one cycle of the pump, let's say 1 Liter.
The operation is that the pump presses one time, forcing air to fill the tubing, which forces the water out, then as the pump is refilling with air from the outside water drains into the tubing and the air that was in the tubing exits from the exhaust tube. Then this repeats. This gradually cools the water.
The reason it cools the water is the pump has 1 liter of air and the tubing also holds 1 liter of air. But there is much more surface area over the tubing so the pressure of the air drops as it fills the tube. This is because pressure is a force per area and the force doesn't change and the area increases so the pressure drops.
by the ideal gas law:

 Since n and R on the right side are constant, and the volume remains constant, if the pressure drops the temperature also has to drop. I figure for the shape of the pump and tubing the pressure could drop 1/20th so it should be efficient in it's cooling.

**Edit**
I think maybe an even better design would be with a motorized pump that's fast enough to keep the tubing from filling with water, and have the exhaust pipe connected back to entry of the pump so the same air is cooled over and over, the pump would be built so it could radiate heat well to the outside air when it pressurizes the air coming into it.