The Cooling Coffee Cup Paradox · AI Generator #GEN-3BB1FFBC
Physics · thermodynamics
Problem
A barista notices something peculiar about their thermal coffee cups. When a hot coffee at Th=85°C is poured into a well-insulated cup and left in a room at Tc=20°C, the coffee cools according to Newton's law of cooling with time constant τ=1200s. However, when the same coffee is poured into an identical cup that contains a small amount of room-temperature milk (negligible thermal mass), the initial temperature drops to T0=80°C, but then the cooling time constant becomes τ′=900s. The barista hypothesizes that the milk changes the surface tension and contact angle of the coffee, effectively increasing the exposed surface area for heat transfer by a factor of k. Assuming the heat transfer coefficient remains constant and that Newton's law of cooling applies in both cases, find the factor k by which the surface area increases when milk is added.