Will coffee in two different size styrofoam cups in the same room conditions cool at the same rate? One cup is 8 oz. and the other would be 12 oz.
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2 Answers
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Let's assume the same form factors of the cups and that the styrofoam is perfectly insulting from the surrounding air, which is cooler than the temperature of the coffee. In this case, heat exchange will happen only at the upper surface of the liquid. Whether that is by evaporative cooling, conduction, or convection, heat loss only happens at this surface. Setting up the heat diffusion equation and its boundary conditions reveals that, all else equal, the cup with the greater ratio of "heat loss surface" to volume will cool faster. For identical shaped cups the top surface scales a height^2 but the volume goes as height^3. The cup with greater volume has greater height, and thus a smaller surface to volume ratio and thus will cool more slowly.
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If we may assume that the cups have the same form factor, or that the cups are identical and that the 12 oz just fills it up farther; in either case the 12 oz. will stay warmer longer.
Agreed with Georg: that the dominant heat loss mechanism in this case will be evaporative cooling
