A mechanical engineer could work in the building services industry. They mainly cover HVAC systems, which is a broad but extremely vital part of any building. HVAC, that is, heating, ventilation and air conditioning, but also extends to refrigeration.
Large buildings and especially a collection of large buildings, such as a university, need more than your typical wall mounted AC unit to cool down the entire building during summer. One major method to achieve large scale cooling is through what we call chillers. Chillers generate chilled water, which combined with "air handling units", provide cool air through "supply air grilles" that you see on the ceilings of many large buildings. Alternatively, parts of large buildings can use a "direct expansion" system to provide cooling. That is, air to directly passed over refrigerant coils to provide cool air. Your wall units and ducted systems at home will almost certainly use a DX system. For larger scaled buildings, DX can be used in conjunction with VRVs (technically VRF but Daikin was the inventor) to provide independent cooling to several rooms.
But before all that can be decided, a mechanical engineer will need to know how much energy is contained within the building. Not all buildings are built and designed the same. It might be north facing, which yields a much higher energy level than a south facing building. The engineer will need to know how that affects AC design, and what system is appropriate for that area. We 'size' the AC by determining how much energy is within that area and over what time period.
This is literally a light touch on what the building services industry is like. A mechanical engineer could also be part of building commissioning, fire safety, fire engineering, and working with the controls, i.e. the building management system (BMS). You'll have to do your own research if interested, since there is far too much to cover in a single post.
