Jan 20 (Reuters) - Shares of Baker Hughes rose sharply on Thursday after the company reported an adjusted quarterly profit and topped analysts" earnings expectations as higher crude oil and natural gas prices fuel demand for its equipment and services. Oil prices surged more than 50% last year amid a global economic recovery from the COVID-19 pandemic and as OPEC+ cut supplies. Higher crude prices have spurred more drilling activity, with the U.S. rig count rising 68% year-over-year to 586 at the end of the fourth quarter, according to Baker Hughes data. "We believe the broader macro recovery should translate into rising energy demand for 2022 and relatively tight supplies for oil and natural gas," said Baker"s Chief Executive Officer Lorenzo Simonelli. In North America, Simonelli anticipates private energy companies to continue driving growth, while publicly traded firms remain disciplined. Shares of Baker Hughes closed up 1.64% at 26.72. They had climbed around 5% earlier in the day. Wall Street analysts said the report was positive, praising Baker"s $645 million in free cash flow, which topped forecasts, and higher than expected EBITDA. Investment firm Tudor, Pickering, Holt & Co called the earnings "a very nice way to rebound," pointing to revenue and operating income across three of Baker"s four reporting segments that beat expectations. The company reported some $7.7 billion of orders in 2021 in its Turbomachinery and Process Solutions unit, driven largely by those related to liquefied natural gas (LNG). "We believe that the step-up in LNG order activity provides a solid indication that a new LNG cycle is beginning to take shape," Simonelli told investors during a conference call. He anticipates 2022 orders to be in line with 2021. Simonelli said that disruptions from the pandemic continued to impact operations, and warned that supply chain and inflationary pressures that drove higher costs and delivery issues in its oilfield services and digital solutions business units would likely continue in the first half of 2022. Adjusted net income for the fourth quarter was $224 million, or 25 cents per share, missing analysts" estimates by 3 cents, according to data from Refinitiv IBES. The same quarter last year, Baker reported a loss of $50 million, or 7 cents per share. Revenue for the quarter was $5.52 billion, which beat Wall Street forecasts of $5.49 billion. For the full year, the company reported a loss of $219 million, versus a loss of $9.94 billion in 2020.