Why are subway stations platforms +15 F/+8 C hotter than ambient temperature in summer

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An S7 underground train, as used on the Circle Line, weighs 213.7 tonnes empty, and carries up to 1045 passengers at speeds of up to 100km/h. A train carrying 500 passengers of average mass 60kg decelerating from 80km/h (22m/s) to stationary must get rid of 60MJ of energy via its brakes, which will be dissipated as heat into the environment.

If the platform is 120m long, and is of semi-circular cross-section with radius 5m, that energy is dumped into at most 5,000m^3 of air. My faithful old Science Data Book says that average air has a density of 1.293 kg/m^3 and a specific heat capacity of 993 J/kgK, so each train's energy causes a temperature rise of around 9K (9C, ~16F) in that air.

There are lots of confounding variables; you asked about New York, for a start. S7 stock has regenerative brakes that are able to shunt about 20% of that energy back into the network, trains on short inner-city runs probably don't get up to 80km/h between stations, the passage of the trains pulls air in and out of the tubes, most platforms have ventilation systems, the brakes are unlikely to cool to ambient temperature in the short time it takes to let passengers on and off, and so on. But on the face of it, trains do provide a significant heat source to the platform environment.

Edit: because comments threads can disappear around these parts, particularly on closed questions, I wish to link to an excellent article on the subject, drawn to my attention by Patricia Shanahan, to whom much thanks. It provided a much more detailed analysis of the contributors to heat in the system, and helpfully confirms that braking is the dominant contributor. It also discusses in much more detail how heat is removed from the system, and concludes that the battle can only be won by decreasing dissipated brake heat.

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