Re:Cooling a cup of coffee with help of a spoon
by gus_massa on Thursday February 17, 2011 @06:48PM
(#2233449)
I think that the explanation is wrong (or at least it is a oversimplification.)
The main problem is that the explanation ignores completely the air. Static air is a good isolator, therefore much of the heat must be transported by thermal convection or forced convection (blowing). If the air doesn't move, it wouldn't be useful to make each part of the coffee visit the surface.
The second problem is that the it says that most of the heat is lost by evaporation of the water molecules (latent evaporation heat). I think that this is true when the temperature is over 195 F/90 C. But when the water is cooler I think that thermal conduction is more important, and it is not related to the "high-kinetic-energy outlier water molecules". And also some heat is lost by the floor and walls of the cup, but I think that most of the heat is lost by the surface. [More details in "experiment" bellow.]
The third problem is more an oversimplification. It is incorrect imagine that some of the molecules of the water are "high-kinetic-energy outlier". Let imagine that we mentally "paint" the fast molecules at initial time. Very soon these molecules will bounce with other molecules and change their velocity, so a few instants later the fast molecules will be other molecules. To estimate the time that we have to wait until the fast molecules bounce we can use the "mean free time". This time is useful in a gas where the molecules are far apart, and for air in usual conditions it is ~5E-10seg. Obviously water is not a gas, but the molecules are more close together, so this the time we should wait is much smaller. But the time that a drop of water needs to go from the bottom to the top of the cup is
~.1 seg (rough estimation). So the molecules have time to bounce and bounce. Moreover, while the drop is traveling though the surface the molecules have time to bounce and reach thermal equilibrium after some of then evaporate. So I think that a macroscopic model that only consider the temperature should work.
[experiment]
A few yeas ago, I worked at a school and we need an experiment for a physic olympiad. We measure the cooling of a plastic cup of water in different conditions. The idea was that the temperature follow a law like:
Temp = Constant * exp(-time/tau) + Temp_external
Taking logarithms
log(Temp-Temp_external) = - time/tau + Konstant
So it is possible to make a graphic of the log(Temp-Temp_external) and from the slope get the value of tau. In an ideal case, tau is a constant (or almost constant) that depends on the materials and the sizes of the experiment.
Really tau is not constant. When the temperature if below 195 F/90 C the value of tau is almost constant and the graphic of the log is a nice line.
But when the water is hotter tau is not constant, the bigger the temperature, the bigger is tau (IIRC perhaps like twice the cold value). So the graphic of the log is curved and the water cooled faster than expected.
When the temperature if above 195 F/90 C something strange was happening. Looking at the cups we see that above that temperature there was a lot of steam over the cups and bellow that temperature the seam almost disappear. Looking at the graphics of the vapor pressure of water we saw that above 195 F/90 C the vapor pressure is bigger than the value bellow 195 F/90 C. So we expect to have a lot of evaporation when the water is hot and few evaporation when it is cold.
So the conclusion was that when the water is cold most of the heat is exchanged by conduction and convection, but when the water is hotter the evaporation is and additional important part of the cooling effects.
Note 1: We did some additional experiments that are consistent with this explanation, but perhaps there is something we forgot to check.
Note 2: The change between the "hot" and "cold" water above and bellow 195 F/90 C was not sharp. While the water was cooling the steam get thiner softly, tau get lower softly and the vapor pressure gets null softly. So the number 195 F/90 C is only an approximation. (And it was a few years ago.)
Note 3: Heat flow is difficult to modelate in paper, so better measure it experimentally. "In theory, practice is the same as theory. In practice, it differs."
The main problem is that the explanation ignores completely the air. Static air is a good isolator, therefore much of the heat must be transported by thermal convection or forced convection (blowing). If the air doesn't move, it wouldn't be useful to make each part of the coffee visit the surface.
The second problem is that the it says that most of the heat is lost by evaporation of the water molecules (latent evaporation heat). I think that this is true when the temperature is over 195 F/90 C. But when the water is cooler I think that thermal conduction is more important, and it is not related to the "high-kinetic-energy outlier water molecules". And also some heat is lost by the floor and walls of the cup, but I think that most of the heat is lost by the surface. [More details in "experiment" bellow.]
The third problem is more an oversimplification. It is incorrect imagine that some of the molecules of the water are "high-kinetic-energy outlier". Let imagine that we mentally "paint" the fast molecules at initial time. Very soon these molecules will bounce with other molecules and change their velocity, so a few instants later the fast molecules will be other molecules. To estimate the time that we have to wait until the fast molecules bounce we can use the "mean free time". This time is useful in a gas where the molecules are far apart, and for air in usual conditions it is ~5E-10seg. Obviously water is not a gas, but the molecules are more close together, so this the time we should wait is much smaller. But the time that a drop of water needs to go from the bottom to the top of the cup is ~.1 seg (rough estimation). So the molecules have time to bounce and bounce. Moreover, while the drop is traveling though the surface the molecules have time to bounce and reach thermal equilibrium after some of then evaporate. So I think that a macroscopic model that only consider the temperature should work.
[experiment]
A few yeas ago, I worked at a school and we need an experiment for a physic olympiad. We measure the cooling of a plastic cup of water in different conditions. The idea was that the temperature follow a law like:
Taking logarithms So it is possible to make a graphic of the log(Temp-Temp_external) and from the slope get the value of tau. In an ideal case, tau is a constant (or almost constant) that depends on the materials and the sizes of the experiment.Really tau is not constant. When the temperature if below 195 F/90 C the value of tau is almost constant and the graphic of the log is a nice line.
But when the water is hotter tau is not constant, the bigger the temperature, the bigger is tau (IIRC perhaps like twice the cold value). So the graphic of the log is curved and the water cooled faster than expected.
When the temperature if above 195 F/90 C something strange was happening. Looking at the cups we see that above that temperature there was a lot of steam over the cups and bellow that temperature the seam almost disappear. Looking at the graphics of the vapor pressure of water we saw that above 195 F/90 C the vapor pressure is bigger than the value bellow 195 F/90 C. So we expect to have a lot of evaporation when the water is hot and few evaporation when it is cold.
So the conclusion was that when the water is cold most of the heat is exchanged by conduction and convection, but when the water is hotter the evaporation is and additional important part of the cooling effects.
Note 1: We did some additional experiments that are consistent with this explanation, but perhaps there is something we forgot to check.
Note 2: The change between the "hot" and "cold" water above and bellow 195 F/90 C was not sharp. While the water was cooling the steam get thiner softly, tau get lower softly and the vapor pressure gets null softly. So the number 195 F/90 C is only an approximation. (And it was a few years ago.)
Note 3: Heat flow is difficult to modelate in paper, so better measure it experimentally. "In theory, practice is the same as theory. In practice, it differs."