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I'm after the math, which I don't understand yet and can't find elsewhere.

The variables are:

  • Two identical cups, A and B
  • The same volume of fluid is contained in each cup, test fluid is water.
  • Both cups are in the same temperature controlled environment of room temperature being 23°C
  • Cup A is heated to 80°C and B to 50°C, without affecting the environment temperature.

What will be the difference in the curve of heat loss for each cup's?

Will cup A lose heat faster or the same as cup B?

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Heat transfer can occur by conduction, by convection, and by radiation.

If you consider conduction through the bulk of the cup, the rate of heat transfer is directly proportional to the temperature difference across the material of the cup. As your experiment holds all variables equal except the temperature difference, cup A will lose heat at a faster rate than cup B, according to this version of Newton's Law of Cooling:

$$∆Q/∆t = −kA(∆T /L)$$

where $(∆Q/∆t)$ is the rate of heat conduction ($\mathrm{kJ/s}$), $∆T$ is temperature difference across the material, $L$ is thickness of the layer ($\mathrm{m}$), $A$ is area of the material ($\mathrm{m}^2$), and $k$ is thermal conductivity of the material per unit thickness ($\mathrm{kJ/m/s/°C}$).

The heat loss curve for cup A should be steeper than the curve for cup B. As you can see, the greater temperature difference ($∆T$) results in a greater rate of heat transfer from the cup to the ambient space.

Convection also plays a part in heat transfer. Convection is the movement of fluid inside the cup that circulates heat throughout the cup. The rate of heat transfer through convection is also directly proportional, to the temperature difference between the cup's surface and the fluid within the cup.

Another consideration is evaporative cooling at the liquid/air interface. The hotter the liquid, the faster it will evaporate and the more heat energy will dissipate per unit time. This is because hotter water molecules jiggle with greater energy, and more readily break the inter-molecular bonds that create surface tension. However, the rate of evaporation will increase only until the ambient air is saturated. The rate of evaporative cooling generally is dependent on more variables than conduction or convection, but it is directly proportional to temperature difference.

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A will lose heat faster. Conduction and convection scale as the temperature difference, which is almost twice as large for A as for B. The warmer water in A will also evaporate faster, removing more heat as it does.

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  • $\begingroup$ And the above is the best short answer you are likely to get. $\endgroup$ – David White Oct 1 '15 at 23:24
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A hot cup of coffee left in a cool room will cool down because the room is colder than the coffee. A person standing outside on a cold frosty day will lose heat and so cool down. On a hot sunny day you will warm up if you stand outside because you are receiving more heat energy from the Sun than you are emitting to the surroundings Leaving the door of a fridge open will allow the contents to warm up because they are colder than the room.

From this schoolphysics page on cooling.

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  • $\begingroup$ If this is a quotation, you ought to state where you have copied it from. Also, how does this answer the question? $\endgroup$ – sammy gerbil Jan 28 '17 at 21:32
  • $\begingroup$ Copy-paste job here is from this schoolphysics page $\endgroup$ – Kyle Kanos Jan 28 '17 at 21:45

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