# How cold does this ice have to be to freeze this water bottle solid?

We are at sea level in a room that is 21 celsius.

We have 1 liter of sterile water with a temperature of 21 celsius in a normal plastic bottle.

We have a 20 liter bucket of ice cubes, consisting of sterile frozen water. Each cube is one cubic centimeter.

We will place the bottle into the centre of the mass of ice cubes.

The water in the bottle needs to be frozen completely solid. What is the highest temperature the ice can have upon contact for this to happen?

I would also like to see a formula for calculating this.

If we accept a large error margin of 5-10 degrees, and make assumptions regarding the shapes, materials, etc, is it possible on pen and paper?

• This sounds like a homework problem. I assume the water in the bottle is room temperature at the start. In simplest, but in accurate terms, take the energy to change the state of water and the current temp of the water and divide by 20 and there's your answer, but in practice, that wouldn't work because much of the cold in the ice would cool the air. – userLTK May 22 '16 at 11:08
• @userLTK Hi. Yes, I realize this must indeed sound like a homework problem. I can assure you, though, I asked it out of pure curiosity. I'm not engaged in any type of formal education, and I did not pull this question out of anyone's homework. (I don't know if me assuring you of this really matters at all.) – Fiksdal May 22 '16 at 11:13
• Are you interested in a practical answer, which means, probably 2/3rds of the cooling would likely go into the air (actually heat moves, cold doesn't move, but you know what I mean), and practical melt time or are you interested in a 100% heat exchange answer. The 2nd one is far easier and far less accurate. – userLTK May 22 '16 at 11:56
• @userLTK I'm interested in what would actually be the case in the real world (roughly). – Fiksdal May 22 '16 at 11:59

## 1 Answer

For the hypothetical case of a thermally perfectly insulated system, I'm sure you can work out yourself from the specific heat and the enthalpy of fusion for water. Given that the enthalpy of fusion (330 kJ/kg) and the specific heat of ice (2 kJ/kg-K) have a ratio of 165 K, and you need the entire ice bucket to stay below the melting point, and your 20:1 ratio of ice/water, I can give a quick estimate that it will definitely not freeze down for an ice temperature above -165/20=-8.2 K. (I neglect the heat to cool down the water from 21 °C to 0 °C).

For the practical case of heat entering from the outside world into the bucket, it gets much more complicated. What you need is the heat-transfer coefficient from the ice cubes into the bottle and the heat-transfer coefficient from the environment into the bucket. Once you know those, you can write a system of coupled differential equations (temperature of the ice cubes, temperature of the bottle, fraction frozen in the bottle) and you can find out how quickly the ice cubes reach the melting point versus how quickly the bottle cools down. You have not provided enough data to estimate either of the two heat transfer coefficients. In particular, the heat inflow from the environment will depend on the presence of air circulation and on the humidity - condensation of ambient humidity will result in much more heat transfer than in the case of dry air at the same temperature.

The heat transfer coefficient from the bucket to the bottle will be on the order of $h=\lambda/D$, where $\lambda$ is the thermal conductivity of ice and $D$ is the diameter of the bottle. (The effective thermal conductivity of ice cubes with air gaps is another difficult thing to estimate.)

Heat transfer from ambient to the bucket is much more difficult to estimate. For free convection (air circulation only due to temperature differences) combined with thermal radiation, $h=10 \mathrm{~Wm^{-2}K^{-1}}$ is a typical value. For heat transfer with forced convection and/or condensation, you'll have to consult a book on heat transfer. Even then, your question would probably require computer simulations for a reasonable answer. It's easier to set up an experiment. My gut feeling is that you won't get the bottle to freeze down even if you start with freezer-temperature ice (-18 °C, 0 °F) because I know from experience that freezing a 1L bottle of water in the freezer will take at least 12 hours and the temperature in your un-insulated bucket will rise to above -8 °C much faster than that.

Update If you want to get some feeling for how to calculate it, you could proceed as follows.

1. Find an estimate for the heat transfer coefficient due to the combined effects of condensation, convection and thermal radiation. These three processes are additive; the convection and radiation will be around 10 W/(m2K). As for condensation, you might estimate it by putting a bottle of frozen water (-18 °C) out and measure how quickly it accumulates ice/water at the surface. Divide by area, multiply by enthalpy of vaporization.

2. Estimate the thermal conductivity $\lambda$ of the ice-cube/air mixture. This will depend on how they are stacked. Let's say, half of the conductivity of solid ice.

3. Calculate the thermal diffusivity, $\alpha=\lambda/(\rho C)\approx$6e-07 m^2/s. Estimate how long it will take for heat entering from the outside to travel to the middle (where your bottle is), using $R=\sqrt{2\alpha t}$, where $R$ is the radius of your bucket.

4. Estimate how long it takes for the water to freeze as a function of temperature at the bottle wall (assume that the ice is cold enough not to melt). For this, assume an effective heat transfer coefficient $h=Nu\lambda/D$, where $Nu=4$ is a typical Nusselt number for a cylindrical bottle of diameter $D$. ($D$ and $\lambda$ here are different from those above for the entire bucket)

5. Now you have to combine the data from the steps above. For a given initial temperature, is the bucket time scale (step 3) fast compared to the bottle time scale (step 4)? Then assume that the ice bucket changes temperature as a whole. From there on, it is relatively straightforward Is it slow? Then you can neglect the heat transfer at the outside of the bucket; you have to solve the heat transfer equation only near the bottle. Even if you assume cylindrical symmetry (temperature only dependent on radial coordinate), this will probably require a numerical analysis program to solve.

• That's a lot of insight. I agree that it almost certainly will not happen with -18 degrees. I would love to run experiments, but unfortunately I don't have access to cooler ice than that, nor such a cool room. I live in the tropics. – Fiksdal May 22 '16 at 13:43
• So why are you interested in the anwser? If you want to freeze water without access to a freezer, then put the ice cubes in an insulated container and add salt. – Han-Kwang Nienhuys May 22 '16 at 13:53
• I am not interested in this answer for practical purposes. I have a freezer. I'm interested in this question for theoretical purposes. I'm curious as to how cold the ice would need to be. For the purposes of this question we can assume that we can use the air circulation in a normal room with no fans, and the current humidity in Venice, Italy. – Fiksdal May 22 '16 at 14:57
• For me, interesting theoretical problems are the ones where the processes are dominated by a single mechanism. Not the case here; the answer will depend on a myriad of practical details such the size of air gaps between the ice cubes, environmental temperature and humidity, shape of the bucket, shape of the bottle, whether or not there is a lid on the bucket, air flow. An engineering company with proper simulation software would probably spend two weeks on setting up and running a simulation model. And it's not an interesting engineering challenge either. – Han-Kwang Nienhuys May 22 '16 at 16:05
• I didn't realize it was this hard to find this out. It seems I will probably not get any good answer. Yet @userLTK thought this sounded like a homework problem. Anyway. I think you're saying the only way to get an answer to this is to actually run experiments or use an advanced simulation software? – Fiksdal May 22 '16 at 16:24