Cooling an open bowl holding tens of thousands of people, in a climate where the outside air can exceed 40 degrees Celsius, sounds impossible. It is not, but only if you stop thinking about "making cold air" and start thinking about moving thermal energy.
The refrigeration cycle
A cooling system uses a working fluid, the refrigerant, driven around a loop in four stages.
A compressor raises the refrigerant's pressure, which raises its temperature above that of the outside air. In the condenser, the hot refrigerant transfers thermal energy to the outside and condenses to a liquid. An expansion valve drops the pressure sharply, and the liquid's temperature falls well below room temperature. In the evaporator, the cold refrigerant absorbs thermal energy from indoor air and evaporates, and the cycle repeats.
Energy is conserved throughout. The heat removed from inside, plus the work done by the compressor, is exactly the heat dumped outside. Every cooled space warms its surroundings by more than it cools itself.
Efficiency you can calculate
The useful measure is the coefficient of performance: heat removed divided by electrical work input. A good chiller has a COP between 3 and 6, meaning one joule of electricity moves three to six joules of thermal energy. That is possible because the machine is transporting energy, not generating it.
COP falls as the temperature difference between inside and outside grows — which is why the same unit performs worse on the hottest afternoon, exactly when it is needed most.
Cooling people, not the sky
The stadium designs used in Qatar apply a simple principle: cool the occupied zone only. Chilled air is delivered at low velocity through nozzles under the seats and at pitch level, forming a cool layer roughly two metres deep. Because cool air is denser, it tends to stay low. Above that layer, the air is left warm.
The rest of the strategy is about reducing the load in the first place: shading the bowl with a roof so seats and pitch never absorb direct sun, choosing reflective materials, shaping the structure to block hot wind, and pre-cooling the space overnight when outside air is cooler and electricity demand is lower. Chilled water is produced centrally at a district cooling plant, and some of it is made at night and stored as ice.
The lesson generalises well beyond stadiums: the cheapest joule to remove is the one you never let in.