Learjet 45 Captain reported that a CAB DUCT OVHT warning resulted in smoke in the cockpit and a diversion to the nearest airport.

Narrative

This medevac flight was from ZZZZ to ZZZ1 and was being operated at FL270. At that flight level; in cruise; and with the pressurization system in manual mode; we were able to maintain a cabin altitude less than 1000 ft as was necessary for the patient we were repatriating. In cruise; a loud bang could be heard coming from the cabin. Seconds later; a CAB DUCT OVHT CAS (Crew Alerting System) message was displayed on DU2 (Display Unit) followed by smoke starting to fill up behind the cockpit. The FO and I wore our oxygen masks and I called for the QRH to run the checklist related to the CAS message. The 'Cabin/Cockpit Fire; Smoke; or Fumes' checklist would have us run through a process of elimination to identify the source of the smoke; but the source was clearly related to the bleed air system given the CAS message. The CAB DUCT OVHT checklist had us pulling both L and R bleed circuit breakers. Seconds later; the smoke in the cabin started dissipating so I opted to not dump or depressurize the cabin to 13700 ft as the Learjet 45XR is designed to do. After pulling the two breakers; we were no longer able to maintain a near-sea-level cabin altitude for the patient. I opted to divert to ZZZ for its proximity; available services and runway length. We requested priority handling and diverted to ZZZ. Immediately North/North East of our position was a line of cumulonimbus clouds. No weather deviation was necessary and we proceeded to ZZZ via direct. We landed on the active runway for the prevailing winds which was runway XX and shot the ILS for that runway.We believe the regulator valve on the ACM (Air Cycle Machine) failed in the open position which would have caused two things to happen. For one; the excess bleed air would overspin the ACM turbine which in turn would spit out oil. Secondly; the rush of hot air would have heated up the duct. We suspect that the smoke accumulated in the cabin was likely the ACM turbine oil coming into contact with the overheated duct.