* Wheat falls on harvest supplies, Black Sea exports

* Soybeans climb on fresh export sales

* Corn eases on better Midwest weather

CHICAGO, Aug 3 (Reuters) - Chicago soybean futures climbed on Thursday, energized by fresh export sales, although cooler, rainy forecasts capped gains.

Wheat fell to a three-week low, as the market assessed global supply prospects marked by large Russian exports and war-curtailed shipments from Ukraine.

Corn also touched a three-week low, pressured by crop-friendly weather.

The most-active soybean contract on the Chicago Board of Trade (CBOT) added 4-1/2 cents to $13.25-3/4 a bushel, by 10:41 a.m. (1641 GMT).

CBOT wheat fell 11-1/2 cents to $6.28-1/2 a bushel, after earier hitting its lowest since July 13 at $6.26-1/2.

CBOT corn dipped 5 cents to 4.95-1/2 a bushel.

Private sales of 134,000 metric tons of U.S. soybeans to China, reported by the U.S. Department of Agriculture Thursday morning renewed optimism around export demand.

"Beans have walked back from their highs considerably. I think we’re finding that demand and that’s keeping us above $13," said Dan Hussey, senior market strategist at Zaner Group.

The flash sale adds to recent export activity. The USDA reported soybean export sales the week ended July 27 at 2.721 million metric tons near the high end of trade expectations.

Wheat export sales of 421,300 metric tons were in line with expectations, while 456,400 metric tons of corn neared the low end of predictions.

August weather remains in focus, after commodity brokerage Stonex projected the average U.S. 2023 soybean yield at 50.5 bushels per acre (bpa), below the USDA's latest projection of 52.0 bpa. Such a drop would significantly tighten U.S. stocks.

"Everyone’s wondering how the dry June, July, across the belt have affected things," said Andrew Jackson, southeast cash grain trader for Action Ag. "Analysts are saying ‘we can still push trendline,’ but farmers are just really doubting that."

Stonex pegged U.S. corn yield at 177.0 bpa, just below the USDA's latest projection of 177.5 bpa but still a record high, if realized.

Wheat remains under pressure as U.S. harvest progresses and Black Sea routes remain open to Russian wheat, exemplified by a recent tender from Egypt's state grains buyer for wheat of Russian and Romanian origin. (Reporting by Gus Trompiz in Paris and Naveen Thukral in Singapore; Editing by Rashmi Aich, Sonia Cheema and Alexander Smith)