Commercial Fixed Wing
Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Time Pressure; Troubleshooting
Aircraft Equipment Problem Critical; Inflight Event / Encounter Fuel Issue
Flight Crew Landed As Precaution; Flight Crew Inflight Shutdown; Flight Crew Diverted
Aircraft
I first noticed something awry as the center pumps were nearing empty. The mains should have been perfectly balanced; yet they weren't. When the center tank was empty; I did a quick crossfeed to balance the 3 to 400 lbs. imbalance.Soon after; we crossed the first fix on our fuel/position estimates printout. We were 2 minutes early; 400 over burn. Usually; our flight plans are more accurate than that. We had leveled off at our first cruise altitude; but already we were ready to climb to our final cruise; so we requested and received clearance to climb. The mains indicated another imbalance occurring; approximately 500 lbs. We leveled off again; but my concern over the situation was growing slowly - I balanced the mains; and then watched the results carefully. After going back to four pumps on; crossfeed closed; the left main dropped 500 lbs in approximately 3 minutes. At this; I was on high alert. I pulled out the QRH; and began going through the checklist. I didn't want to take action too quickly; so I stopped short of taking any irreversible action for a few minutes.I called the FAs (Flight Attendant); and requested they look for signs of fuel leak. They didn't observe any misting behind the engine; but they did see liquid around the engine pylon; and at the trailing edge of the flaps. This was inconclusive; since we'd deiced fully - type 1 and 4 - before takeoff.The initial drop to 500 lbs imbalance tapered off; but continued; so over the next ten minutes or so; the imbalance reached 900 lbs. We were approaching our next fix on the plan; and we were now 1900 lbs. over expected fuel burn. At this point; I decided it was time to commit to the checklist; which required shutting down the leaking engine; and divert. An outstation was convenient at that point in time; so we made that the plan. I shut the engine down per the QRH; and walked the panel - Dispatch; FAs; and passengers.Communication was difficult as we descended to our drift down altitude. We were in mountainous terrain; and had trouble reaching the Local Approach Controller; as well as raising Operations on company frequency. I didn't have the chance to try DTMF microphone before [we] were able to get back in contact with ATC. [At] this point; I was too busy to get back to Dispatch over the radio; when the essentials of re-dispatch had already been accomplished.We landed slightly overweight; approximately 46;000 lbs; with a flaps 15 max of 44;200. The landing was smooth; with a minimal sink rate. I think the whole situation proceeded very smoothly. It's a very busy checklist; so I did my best to take my time and [to] get everything done properly. By the time we landed; I felt we were ready and well-prepared; rather than rushed or stressed.
Air carrier Captain reported a fuel imbalance event during cruise; which resulted in an engine shutdown and was followed by a successful diversion.