A childhood question about heat

When I was a little kid, I asked some adults the following question.

If hot things cool, and cool things warm up, could something hot cool down and warm back up?

The people I asked didn’t understand my question and just laughed. I have no idea how old I was, but I wasn’t old enough to articulate what I was thinking.

Here’s what I had in mind. I knew that hot things like a cup of coffee grew cold. And I knew that cold things, say a glass of milk, get warm. Well, could the coffee get so cold that it becomes a cold thing and start to warm back up?

Could the coffee become as cold as the glass of milk? Common sense suggests that can’t happen. When we say coffee grows cold, we mean that it becomes relatively colder, closer to room temperature. And when we say the milk is getting warm, we also mean it is getting closer to room temperature. We’ve never left a hot cup of coffee on a table and come back later to find that it has cooled off so much that it is colder than room temperature. But could there be small fluctuations?

As the coffee and milk head toward room temperature, could they overshoot the target, just by a little bit? Say room temperature is 70 °F, the coffee starts out at 150 °F, and the milk starts out at 40 °F. We don’t expect the coffee to cool down to 40 °F or the milk to warm up to 150 °F. But could the coffee cool down to 69.5 °F and then go back up to 70 °F? Could the milk warm up to 70.5 °F and then cool back down to 70 °F?

I didn’t get a satisfactory answer to my childhood question until I was in college. Then I found out about Newton’s law of cooling. It says that the rate at which a warm body cools is proportional to the difference between its current temperature and the ambient temperature. This law can be written as a differential equation whose solution shows that the temperature of a warm body decreases exponentially to the ambient temperature. The temperature curve always slopes downward. It doesn’t wiggle even a little on its journey to room temperature. Cold bodies warm up the opposite way, exponentially approaching room temperature but never exceeding it.

In case this seems obvious, think about thermostats. They don’t work this way. Say the temperature in a room is 85 °F and you’d like it to be 72 °F, so you turn on the air conditioning. Will the temperature steadily lower to 72 °F? Not exactly. If you were to plot the temperature in the room over time and look at the graph from far enough away, it would look like it is steadily going down to the desired temperature. But if you look at the graph more closely, you’ll see wiggles. The AC may cool the room to a little below 72 °F, maybe to 70 °F. The AC would cut off and the temperature would rise to 72 °F. Unlike the cup of hot coffee, the AC will often overshoot its target, though not by too much. The temperature may feel constant, but it is not. It oscillates around the desired temperature.

RelatedConsulting in differential equations

10 thoughts on “A childhood question about heat

  1. It’s been a while since I took thermodynamics, but I think that the Second Law of Thermodynamics is relevant here as well. In particular, the heat formulation of the Second Law, which states that heat cannot flow from a cooler material to a warmer material without external work. Thus, heat will flow from hot coffee into the cup until they are both at the same temperature. However, once they are at the same temperature, energy transfer will cease. So while Newton’s law of cooling describes the rate of heat transfer, the Second Law explains why the transfer can only proceed in one direction.

  2. Coffee cooling and milk heating are phenomena of a two systems reaching thermal equilibrium. For example, if you had the hot coffee and the cold milk put next to each other with a metal plate separating them, conduction through the plate would make the milk warmer and the coffee cooler. But imagine you have 4 eq. of milk and 1 eq of coffee … then you will end up with the coffee cooling significantly more and the milk heating up much less. You can imagine attributing this to thermal energy dissipating in the larger cold sink (the milk) much more, hence drawing more energy from the coffee before equilibrium is achieved.

    The HVAC example is relevant in process design of PID (Proportional-integral-derivative) controllers. Yeah, AC systems overshoot because they measure at a point some distance away from the feed stream (the air vent). A fine tuned controller will not over/under shoot, however the amount of variables to account for (solar heat through window, doors, ice cold beer, cooking stove), will cause localized fluctuations that will cause the controller to react to the measured temperature.

    Let’s make an imaginary system with a set point at 70. The thermostat will see that the temperature is rising to 70 … so it shuts down the system in anticipation that by the time all the heating elements and vents shut down, the temperature will settle at 70. However, just being mis-programmed, it shut off too late and over heated (Underdampened system). This over heat causes the cooling to kick in … and the same programming error will cause an undershoot. Worst case, it misses it’s target even more than the first time. Then you have an oscillating system that will never reach it’s set point. (While make people feel like this happens, this is rarely true. More often than not, it will eventually find the set point, it will just take much longer than hoped). A similar case can happen if it’s too EARLY to react (overdampened system). While the oscillation will not occur, instead, it will take a significant amount of time to reach any set point.

    Sorry, I didn’t mean to end up in a lecture. I HAD the same question as a child, and I ended up studying chemical engineering in college and I learned to enjoy thermodynamics & process dynamics and control. It is very fascinating to study and very relevant (in my opinion) with many things in everyday life.

  3. The physics is primary, but I’m more familiar with differential equations. I know more about Newton’s law of cooling and the heat equation than about thermodynamics.

    I had a roommate in college who was a mechanical engineer. I remember his summary of thermodynamics: you can’t win, you can’t break even, and you can’t quit playing the game!

  4. A very good summary of thermodynamics indeed!

    Yeah, thankfully, a lot of the physics in this case can be reduced to differential equations. I say thankfully, because you can end up playing with the values in a closed loop system and get an idea of how an mal-dampened system will react. Unthankfully … finding a solution to create a perfectly dampened system can become nearly impossible if you move away from the most basic closed loop systems. That’s why most modern systems utilize a computational approaches that can try to change on the fly to adjust to unknown variables.

  5. That’s all very well, but a really useful contribution toward global enlightenment would be an explanation that would convince my (otherwise highly intelligent) wife that setting the thermostat to the lowest (or highest) possible value will not make the temperature reach the desired level of comfort any faster and will in fact, if left unattended, have other unexpected (by her) consequences in the temperature department.

  6. The thermodynamics and differential equation approaches reflect the aggregate behaviour of the system, as if temperature were a continuous quantity. From a statistical mechanics view, however, temperature is just the mean of the speeds of the molecules, which are exchanging energy by collisions which happen randomly. So, it is possible for a “colder” molecule to transfer energy to a “hotter” molecule, but with probability < .5, whereas the opposite transfer is more likely. So, it is statistically possible for the coffee to become colder than room temperature, and it certainly will by some infinitesisemal amount, but to do so by even .5 degree will likely never happen in the lifetime of the universe.

  7. Mike W.: To help your intelligent wife understand how dumb our thermometers are (unless you have a supercomputer controlling yours) … a fine tuned controller working alone with little disturbance will reach your set point the fastest and remain at the desired temp. An underdampened controller (one that oscillates significantly) will over/undershoot your desired set point if you over compensate even more, cause more oscillations, causing you to either freeze or melt because your HVAC system went all crazy. IF it’s OVERDAMPENED (if it has no oscillations and really takes forever to reach a set point) … your very intelligent wife is right.

    Walt: True, thermo & differentials approaches analyze the total system. I follow your statistical mechanics example that specific point particles can over/undershoot their desired thermal equilibrium because what is temperature but the RMS average of the particle speeds/vibrations. But you do not really measure temperature as a single point particle, but as a entire system. This phenomena is calculated in a Boltzmann Distribution. While the microstates will fluctuate within the system and that distribution will cool or heat depending on the heat transfer in or out of the system, it’s overall value, and temperature, will depend on thermal equilibrium. Can there be a temperature drift where hot and cold spots form? It’s possible, but that is dealing with fluid mechanics and heat transfer principles in order to ‘push’ the system away from an ideal thermal equilibrium. (I could be wrong, and please feel free to correct me, it’s been about two years since I’ve thought about these types of theories)

  8. whlie we are on the subject of childhood questions that have gone unanswered. Bear with me – this will no doubt seem like a very retarded question to those who really know what they are on about. I am working on the following basis, such is my (very limited) understanding of cosmology etc. Assumption 1 is that the universe is finite (i.e. it has an edge and is expanding, although at what rate is very much up for debate and also whether or not will reach a static point and/or collapse back or continue expanding). Assumption 2 is that we have some idea of the speed of expansion (ok velocity, but assuming away from the centre of the big bang). Assumption 3, light has mass (at its usual speed). Assumption 4, the speed of light is faster than the speed of the expansion of the universe.
    So why isnt the edge of the universe determined by where the most far out photon is? i.e. if light has mass, surely that point would be the edge of known space; the first photon would now be further away from the centre of the universe than where we would expect the edge to be. I am sure i must be missing something very critical and one or more of my assumptions are probably wrong, but I would like to know the answer to this.

  9. Ahmet, re your first point; are you saying that if I keep my thermostat controller at the right moisture level, it will never over or undershoot its set point. So I keep it just keep it damp enough, but not too damp.

  10. Note that evaporation can cool things down. This is how most air conditioners work (but not the solid state ones). Trees do something similar (and also do a variety of other interesting things), so in addition to the shade the provide they also provide a cooling effect.

    So, for example, if you let milk warm up at 100% humidity and then moved it to another room at 5% humidity, it should cool down (and might also get sour).

    Meanwhile, fire can warm things up (and so can biological activity — such as the souring process that can happen with milk).

    I’m not sure how I would recognize a child being interested in these sorts of scope issues, nor am I sure how I could convey them to a child.

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