MINNEAPOLIS — United Airlines plans to demonstrate new fuel-saving methods on two flights between the U.S. and Europe tomorrow.

The airline says it expects to save about 940 gallons of fuel, about 3 percent of the fuel it would usually burn, on the flights. The testing includes a trip from Frankfurt to Chicago on a United 777, and a return trip on the same plane. Both are regular flights with paying passengers.

Commercial flights usually stay at a precise altitude. But this flight will drift up and down as much as 3,000 feet. That way the pilots won't need to burn extra fuel maintaining a precise altitude.

It also lets them choose the best altitude depending on wind and other conditions, said Joe Burns, a United captain and managing director of technology and flight tests.

He said those small adjustments wouldn't make much difference on a short domestic flight, but they add up on an eight- or 10-hour overseas flight.

"A thousand feet can make a big difference in our fuel burn," he said.

The flight is also being planned with a flight-planning system that computes the best route, altitude and speed depending on the wind and the plane's weight, which changes during the flight as it burns fuel. Other changes include only filling the drinking water tanks high enough for the planned passenger load and flight time, instead of automatically filling them to the top.

"It's just a matter of putting it all together and seeing what we could do if we could really optimize all our flights," Burns said.

The flight has clearance from the Federal Aviation Administration as well as air traffic authorities in Canada and Europe. One issue that keeps airlines from using some of those fuel-saving practices all the time is that they can make it harder for air traffic controllers to keep the required minimum distances between planes.

United and its regional partners burned 564 million gallons of jet fuel last year, costing almost $1.19 billion.

Several airlines have been testing ways to cut their fuel bill.

A year ago, American Airlines tested fuel-saving technology on a Paris-to-Miami flight. It took a direct route guided by global-positioning technology instead of staying on the aerial highways normally used by jetliners. That plane climbed continuously out of Paris instead of at intervals of 2,000 feet or more at a time. It also descended gradually into Miami instead of the usual stair-step decline.

Alaska Airlines also tested a gradual descent into Seattle in August, along with other fuel-saving measures.

United used synthetic fuel on a test flight in April.